Heat Transfer Correlations for Gas Turbine Cooling LITH-IKP-EX--05/2313--SE. Jenny Sundberg

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1 Heat Transfer Correlations for Gas Turbine Cooling LITH-IKP-EX--05/2313--SE Jenny Sundberg

2 Acknowledgements I would like to thank the members of the group working with aerodynamics and cooling of blades and vanes, GRCTA at Siemens in Finspång for always taking time to answer my questions. Also, special thanks to my supervisors at Siemens Mats Annerfeldt and Yuri Mamon and to my supervisors at LiTH Dan Loyd and Johan Svensson for your support. In the Siemens office in Lincoln, I would like to thank John Maltson, Anthony Davis and Andrew Down for helping me with the Multipass program code. I would also like to thank Mats Kinell for helping me with the Q3D program code and Xiufang Gao and Daniel Lörstad for your help with correlations

3 Abstract A first part of a Heat Transfer Handbook about correlations for internal cooling of gas turbine vanes and blades has been created. The work is based on the cooling of vanes and blades 1 and 2 on different Siemens Gas Turbines. The cooling methods increase the heat transfer in the cooling channels by increasing the heat transfer coefficient and/or increasing the heat transfer surface area. The penalty paid for the increased heat transfer is higher pressure losses. Three cooling methods, called rib turbulated cooling, matrix cooling and impingement cooling were investigated. Rib turbulated cooling and impingement cooling are typically used in the leading edge or mid region of the airfoil and matrix cooling is mostly applied in the trailing edge region. Literature studies for each cooling method, covering both open literature and internal reports, were carried out in order to find correlations developed from tests. The correlations were compared and analyzed with focus on suitability for use in turbine conditions. The analysis resulted in recommendations about what correlations to use for each cooling method. For rib turbulated cooling in square or rectangular ducts, four correlations developed by Han and his co-workers [3.5], [3.8], [3.9] and [3.6] are recommended, each valid for different channel and rib geometries. For U-shaped channels, correlations of Nagoga [3.4] are recommended. Matrix cooling is relatively unknown in west, but has been used for many years in the former Soviet Union. Therefore available information in open literature is limited. Only one source of correlations was found. The correlations were developed by Nagoga [4.2] and are valid for closed matrixes. Siemens Gas Turbines are cooled with open matrixes, why further work with developing correlations is needed. For impingement cooling on a flat target plate, a correlation of Florschuetz et al. [5.7] is recommended for inline impingement arrays. For staggered arrays, both the correlations of Florschuetz et al. [5.7] and Höglund [5.8] are suitable. The correlations for impingement on curved target plate gave very different results. The correlation of Nagoga is recommended, but it is also advised to consult the other correlations when calculating heat transfer for a specific case. Another part of the work has been to investigate the codes of two heat transfer programs named Q3D and Multipass, used in the Siemens offices in Finspång and Lincoln, respectively. Certain changes in the code are recommended.

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5 Nomenclature The original parameters of the references have been used for most cases, to facilitate comparison to references and to adjust to commonly used parameters for each cooling method. Therefore, one parameter sometimes has multiple meanings. For those cases, the cooling method it is valid for is written after the explanation of the parameter. General a = channel cross-section area [m 2 ] a A = channel flow cross-section area at matrix side bound [m 2 ] A = area [m 2 ] A f = open area ratio, i.e. ratio of jet hole area to opposing heat transfer surface area [-] A i = index of alloy heat resistance, see equation (4.11), 6 Ai = for ally of interest here [Pa] A r = coefficient in Q3D for Nu calculation on ribbed side wall [-] A row = area of all holes in a spanwise impingement row [m 2 ] A s = coefficient in Q3D for Nu calculation on smooth side wall [-] b = width of a 2D slot with equivalent area of a single row [m] of impingement C = friction enhancement factor [-] C d = discharge coefficient [-] C d,s = discharge coefficient for a thin orifice [-] C p = specific heat at constant pressure [J/(KgK)] C v = specific heat at constant volyme [J/(KgK)] cbs = vector in function Sfun in the Q3D program code [-] cfs = vector in function Sfun in the Q3D program code [-] d = hydraulic diameter of channel (matrix cooling) [m] d = hole diameter (impingement cooling) [m] D = effect of centrifugal force on flow in matrix side bound [-] D h = hydraulic diameter in ribbed channel [m] D p = diameter of curved target plate [m] e = rib height [-] e = rib height [m] + e = roughness Reynolds number [-] = average friction factor for flow in duct with ribs on 2 f r 2 opposite sides [-] F = total area increase for SR- and SSR- schemes [-] f = friction factor [-] mass exchange effect of opposite sub channels through F C = the inter-ribs windows, F C =z [-] effect of flow contraction-diffusion in matrix side F CD = bound [-] F ρ = heat transfer area of ribs themselves [-] F s = heat transfer surface area of a corresponding smooth duct [-]

6 F 1 = total leading edge air outlet area [m 2 ] F 4 = total trailing edge air outlet area [m 2 ] G c = channel crossflow mass velocity based on channel cross section area [kg/(m 2 s)] G f = summary of cooling air mass flow [kg/(m 2 s)] G g = mass flow after the compressor [kg/(m 2 s)] G j = jet mass velocity based on jet hole area [kg/(m 2 s)] G j = average jet mass velocity [kg/(m 2 s)] G ( e + ) = heat transfer roughness function, non-dimensional temperature at rib tip [-] h = rib height [m] H = channel height [m] I c = cross flow interference paramterer [-] k = width of rib top [m] K = heat transfer enhancement factor [-] K A = heat transfer enhancement due to increased heat transfer surface area [-] K = heat transfer enhancement factor, including effect of increased heat transfer surface area [-] l = length of matrix channel (matrix cooling) [m] l = length of control section (rib turbulated cooling) [m] l = lenght of target surface arc on curved target plate (impingement cooling) [m] L = length of matrix (matrix cooling) [m] L = channel length (rib turbulated cooling) [m] L = length of impingement hole (impingement cooling) [m] L c = characteristic length according to Table 2.1 [m] L es = lenght of equivalent slot [m] LE = Leading Edge m = distance between ribs, m = t-k (matrix cooling) [m] m = distance from stagnation line (impingement cooling) [m] M * = cross flow-to-jet flow mass velocity ratio, m & c m& j [-] n = index of alloy heat resistance in equation (4.11), n = 6060 for alloy of interest here [-] n c = number of streamwise rows in an impingement array [-] N p = number of holes in a spanwise impinement row [-] = Reynolds number exponent in Q3D Nu calculation for n r ribbed side wall [-] = Reynolds number exponent in Q3D Nu calculation for n s smooth side wall [-] Nu = Nusselt number [-] Nu s = average Nu over the full length of a smooth duct [-] Nu = averaged Nu over an impingement array [-] Nu 1 = Nu for the first row, without cross flow [-] p = pressure [N/m 2 ] p 0 = plenum pressure (impingement cooling) [N/m 2 ] p 2 = channel pressure (impingement cooling) [N/m 2 ] P = rib pitch [m] p = pressure loss [N/m 2 ]

7 Pr = Prandtl number [-] PS = Pressure Side Q = heat transfer [W] R = radius of U-shaped channel (rib turbulated cooling) [m] R = radius of curved target plate (impingement cooling) [m] r = specific gas constant, r = [kj/(kgk)] R.A.M = Reduced Area Method Re = Reynolds number [-] R ( e + ) = heat transfer roughness function, non-dimensional velocity at rib tip [-] s = width of rib base [m] S = perimeter of blade or vane cross-section (rib turbulated cooling) [m] S = area of blade cross section with intensifiers of the cooling method (S<S 0 ) (matrix cooling) [m 2 ] S = with of equivalent slot according to Nagoga s definition (impingement cooling) S * = area of through flow section in matrix [m 2 ] S 0 = area of shell cross section of initial smooth channel without intensifiers [m 2 ] SS = Suction Side SSR = Segmented Rib cooling scheme SR = Surrounding Rib cooling scheme S T = rib top area [m 2 ] St = Stanton number [-] t = rib pitch [m] T = temperature [K] T diff = temperature difference between the core flow and in the flow near the wall [K] T = cooling air temperature in the sub channels [K] f T G = stagnation temperature [K] T w = metal temperature [K] T = temperature difference [K] T temperature change in the blade wall in control w = section [K] T initial temperature in the blade wall in a smooth w,0 = channel [K] TE = Trailing Edge u = mean axial velocity of fluid [m/s] U + ( e + e ) = R ( e + ) [-] v = velocity [m/s] v = mean velocity [m/s] w = cooling air velocity in channel (matrix cooling) [m/s] w = rib width (rib turbulated cooling) [m] w,x = maximum value of w [m/s] W = channel width (rib turbulated cooling) [m] W/H = aspect ratio [-] W = matrix width (matrix cooling) [m]

8 W r = ribbed wall width at a cross-section [m] W s = smooth wall width at a cross-section [m] x = local distance measured from channel inlet [m] x = for U-shaped channel: distance from duct inlet to the control section (rib turbulated cooling) [m] x = distance from stagnation line (impingement cooling) [m] x = x/d [-] x n = streamwise hole spacing [m] xbs = vector in function Sfun [-] xfs = vector in function Sfun [-] y n = spanwise hole spacing [m] z = number of channels in matrix (matrix cooling) [-] z = jet plate-to-impingment plate spacing (impingement cooling) [m] z a = streamwise location from impingement line [m] Z = 2W/(W+H) [-] Greek Symbols α = heat transfer coefficient [W/(m 2 K)] α = rib angle (rib turbulated cooling) [rad] β = angle of longitudinal rib [rad] δ = distance between matrix top and bottom [m] ε = surface roughness = matrix end clearance, see Figure 4.2 [m] φ = rib flank angle [-] Φ = rotation parameter [-] ϕ = angle of flow rotation [rad] ϕ T = ability to carry load of section S T [-] η = rib efficiency [-] κ = relative depth of matrix channel, channel width-toheight ratio [-] κ = specific heat ratio, κ = C p Cv [-] λ = thermal conductivity of fluid [W/(mK)] µ = dynamic viscosity of fluid [kg/(ms)] ν = kinematic viscosity of fluid [m 2 /s] Π = portion of the duct perimeter covered by the rib system, Π = F Fs [-] Θ = cooling effectiveness, relative cooling depth [-] Θ = ( T G Tw ) ( TG T f ) ρ = density [kg/m 3 ] σ = effect of tension stresses of cooling method [-] τε = blade life with cooling method used [cycles] τ = effect on life on turbine blade or vane [-] ξ hydraulic resistance in the spatial turns at matrix [-] m = side bound Ψ = streamfunction [-] Ψ = hydrodynamic energy effectiveness [-] Ψ = correction factor, comparisons of cooling methods [-]

9 Subscripts air = cooling air AV = average b = bulk b = rib base (matrix cooling) rib = total rib heat transfer surface c = corrected c = cross flow (impingement cooling) conv = convective d = jet hole diameter e = reduced area method f = film (fluid) j = jet flow l = channel length n = normal distance new = value calculated with one of the correlations from the literature study original = value used in Q3D today r = ribbed side in channel with 1, 2, 3 or 4 ribbed walls rc = circular channel r4 = channel with four ribbed sides s = smooth channel s4 = channel with four smooth sides ST = line along concave matrix side wall w = wall x = local value x = local distance in channel from matrix side wall (matrix cooling) x- = axial component 0 = smooth duct 0 = jet discharge condition (impingement cooling) = value near the wall ϕ = tangential component

10 Table of Contents 1 INTRODUCTION BACKGROUND PROBLEM DESCRIPTION OBJECTIVE LIMITATIONS METHOD Method Criticism DISPOSITION AND READING INSTRUCTIONS THEORY FLOW AND HEAT TRANSFER BASICS Heat Transfer Friction Flow GAS TURBINE VANES AND BLADES REFERENCES THEORY RIB TURBULATED COOLING THEORY CORRELATIONS FROM ARTICLES Results Analysis Conclusions CORRELATIONS IN Q3D Results Analysis Conclusions CORRELATIONS IN MULTIPASS Results Analysis Conclusions REFERENCES RIB TURBULATED COOLING MATRIX COOLING THEORY What is Matrix Cooling? Fin Effect of Ribs Summary of Literature Study CORRELATIONS FROM LITERATURE Results Analysis Conclusions RIB EFFECTIVENESS IN Q3D Results Analysis Conclusions REFERENCES MATRIX COOLING IMPINGEMENT COOLING THEORY What is Impingement Cooling? Summary of Literature Survey FLAT TARGET PLATE Results Analysis Conclusions...12

11 5.3 CURVED TARGET PLATE Results Analysis Conclusions REFERENCES IMPINGEMENT COOLING CONCLUSIONS RIB TURBULATED COOLING MATRIX COOLING IMPINGEMENT COOLING FUTURE WORK...12 APPENDIX 1 - RIB TURBULATED COOLING...12 APPENDIX A CORRELATIONS...12 APPENDIX B CORRELATIONS FOR U-SHAPED CHANNELS...12 APPENDIX C RESULTS FOR CASE A...12 APPENDIX D EFFECT OF RIB ANGLE...12 APPENDIX E EFFECT OF PITCH-TO-RIB HEIGHT...12 APPENDIX F EFFECT OF ASPECT RATIO...12 APPENDIX G EFFECT OF RIB HEIGHT...12 APPENDIX H Q3D CORRELATIONS...12 APPENDIX I MULTIPASS RESULTS...12 APPENDIX J INTERESTING ARTICLES...12 APPENDIX 2 - MATRIX COOLING...12 APPENDIX A ZINC TEST...12 APPENDIX B CORRELATIONS AND RANGES OF [4.2]...12 APPENDIX C CORRELATIONS AND RANGES OF [4.5]...12 APPENDIX D RESULTS FOR A TYPICAL TURBINE BLADE...12 APPENDIX E INTERESTING ARTICLES...12 APPENDIX 3 IMPINGEMENT COOLING...12 APPENDIX A CORRELATIONS AND RANGES...12 APPENDIX B FLAT TARGET PLATE, CASE B...12 APPENDIX C FLAT TARGET PLATE, COMPONENTS...12 APPENDIX D FLAT TARGET PLATE, RANDOM TESTS...12 APPENDIX E FLAT TARGET PLATE, EFFECT OF PARAMETERS...12 APPENDIX F ANALYSIS OF CORRELATIONS...12 APPENDIX G CURVED TARGET PLATE, COMPONENTS...12 APPENDIX H CURVED TARGET PLATE, RANDOM TESTS...12 APPENDIX I CURVED TARGET PLATE, EFFECT OF PARAMETERS...12 APPENDIX J INTERESTING ARTICLES...12

12 Table of Figures Figure 2.1 Boundary Layer [2.3]. 6 Figure 2.2 Velocity Profiles in Pipe Flow [2.1]. 7 Figure 2.3 Schematics of a Gas Turbine [[2.6]. 8 Figure 2.4 Schematic Drawing of SGT -800 [2.7]. 8 Figure 2.5 Turbine Blade (Right), and Turbine Vane (Left ) [2.8]. 9 Figure 2.6 Typical Cooling Concept [2.9] 9 Figure 3.1 A Ribbed Channel. 11 Figure 3.2 Flow around Ribs [3.1]. 11 Figure 3.3 Comparison of K and C. Ribbed Duct is Number 3 [3.3]. 12 Figure 3.4 Geometry of SR-Scheme, Front View (Left) and Top View (Right). 12 Figure 3.5 Geometry of SSR-Scheme, Front View (Left) and Top View (Right). 12 Figure 3.6 Secondary Flow induced by Angled Ribs[3.1]. 12 Figure 3.7 Flow Patterns for Different Rib Spacings [3.13]. 12 Figure 3.8 Triangular Ducts Investigated by Metzger & Vedula [3.1]. 12 Figure 3.9 Triangular Ducts Investigated by Zhang et al [3.1]. 12 Figure 3.10 Ranges for Correlations and Channels in SGT -700 and SGT Figure 3.11 Nu on Ribbed Side Wall for Case A. 12 Figure 3.12 Friction Factor on Ribbed Side Wall for Case A. 12 Figure 3.13 Nu on Smooth Side Wall in Ribbed Duct for Case A. 12 Figure 3.14 K on Smooth Side Wall in Ribbed Duct for Case A. 12 Figure 3.15 Results for SR- and SSR-Scheme Calculations. 12 Figure 3.16 Cross-Section 4, Vane 2, SGT Figure 3.17 Temperature in Vane 2, SGT Figure 3.18 Temperature Difference in Vane 2, SGT Figure 3.19 Q3D Interface for Ribbed Channels. 12 Figure 3.21 Original Interface (Left) and Corrected Interface (Right). 12 Figure 4.1 Example of a Matrix Geometry [4.1] 12 Figure 4.2 Open Matrix. 12 Figure 4.3 Geometry of Longitudinal Rib. 12 Figure 4.4 Geometry in Matrix Side Bounds [4.2]. 12 Figure 4.5 Line ST on Matrix Shell inner Concave Surface. 12 Figure 4.6 Comparsion of Cooling Depht [4.2]. 12 Figure 4.7 Three Matrixes Investigated by Jurchenko and Malkov [4.3]. 12 Figure 4.8 Nusselt Number Enhancement for the Tests with End Clearances [4.3]. 12 Figure 4.9 Height and Chord of an Airfoil. 12 Figure 4.10 Alfa as a Function of Re d in Initial Section, Re d = Figure 4.11 Alfa as a Function of x/d in Inital Section, Re d = Figure 4.12 Alfa as a Function of x/d for Intial and Basic Sections, Re= Figure 4.13 Rib effectiveness interface in Q3D. 12 Figure 4.14 Q3D Rib Geometry. 12 Figure 5.1 Impingement Cooling Setup. 12 Figure 5.2 Curved Target Plate[5.1]. 12 Figure 5.3 Impingement Cooling in Turbine Vane [5.2]. 12 Figure 5.4 Impinging Jet (Left, [5.3]) and Free Jet (Rigth) [5.4]. 12 Figure 5.5 Re and G c /G j as a Function of Row Number for x n /d=y n /d =3, z/d=2.75 and Re d =29900 and [5.14]. 12 Figure 5.6 Alfa for Case C. 12 Figure 5.7 Q3D Comparison for Case C. 12

13 Introduction 1 Introduction In this opening chapter, the background, problem description and objective of the work is presented, together with an explanation of the methods used. The limitations of the work and the disposition of the report are also offered. 1.1 Background The efficiency and power output of a gas turbine increases with higher turbine inlet gas temperature. Modern gas turbine vanes and blades are exposed to gas with temperatures which far exceeds the melting point of the component material. Thus, the blades and vanes have to be cooled in order to lower the temperature. When cooling the component it is important to know the correct boundary conditions, to avoid creating too large temperature gradients. Large temperature gradients cause thermal stresses and significantly decrease the component life. Both internal and external cooling is used in turbine blades and vanes. The cooling air is extracted from the compressor. The cooling affects the gas turbine in two ways. First, less mass flow is available for combustion in the combustion chamber. Second, the trailing edge thickness has to be increased, which creates a larger wake behind the trailing edge which affects the aerodynamics negatively. The extraction of air decreases the efficiency of the turbine, since less air is available for power generation. Maximum cooling with minimum cooling air is therefore desired. This work concerns internal cooling of turbine vanes and blades 1 and 2 on Siemens Gas Turbines 700 and 800, shortly called SGT -700 and SGT These components are situated next to the combustor chamber, and are therefore exposed to the highest temperatures. The report is primarily addressed to gas turbine cooling engineers. Blades and vanes are cooled by internal channels, through which the cooling air flows in different schemes and configurations. The cooling air decreases the channel wall temperature by convective cooling. A number of cooling methods are applied to different part of the vane or blade. To make the cooling systems more efficient and spend a minimum of air, the cooling systems nowadays usually include features that increase the heat transfer coefficient and/or increasing the heat transfer surface area. The heat transfer coefficient is increased by enhancement of the flow turbulence and by breaking the flow boundary layer. The penalty paid for the increased heat transfer is higher pressure loss. Many methods exist in theory, but only a handful are widely used in practice. Three cooling methods are investigated here. They are called rib turbulated cooling, matrix cooling and impingement cooling. The flow of these cooling methods is complicated. Heat transfer coefficients and pressure losses can be calculated by Computational Fluid Dynamics programs, or CFD, but modeling is very time consuming. Since it is often necessary to investigate many different cases, it is convenient to use semi empirical correlations based on experiments. 1

14 Introduction To evaluate how effective these cooling methods are, the heat transfer and friction of the cooling methods are compared to a corresponding smooth channel. The augmentation of heat transfer and friction are expressed by the heat transfer enhancement factor and the friction enhancement factor. These factors are used in heat transfer programs for flow and temperature calculations of gas turbine components. 1.2 Problem Description The enhancement factors are calculated from correlations based on tests or approximated from experiments. A large number of correlations exist in open literature, and it is not obvious which ones are most suitable for different flow schemes and geometries. There is a need to gather information about correlations and to evaluate them, in order to find out which correlations to use for different cases. 1.3 Objective The objective of this work is to create a first part of a Heat Transfer Handbook for internal cooling of gas turbine vanes and blades. This part of the Heat Transfer Handbook will contain recommendations for how to calculate the heat transfer- and friction enhancement factors for rib turbulated cooling, matrix cooling and impingement cooling. A sub objective is to investigate the code of two heat transfer programs used at Siemens, in order to examine how well the code agrees with open literature correlations. 1.4 Limitations The effect of rotation is not considered. Only articles with correlations based on tests are considered, so for example articles about CFD analysis are not in the scope of this work. Concerning ribbed channels, correlations for ribs with 30 to 90 angles to the air flow velocity vector are investigated. Only continuous ribs with inline pattern are considered. Investigated ducts have rectangular, U-shaped or square cross-section. The limitations of matrix cooling have been that there are limited sources of information and that the work has a time limit. For impingement cooling, the average heat transfer of impingement of multiple jets on a concave or flat surface is investigated. The target plate is solid without roughness elements, film cooling holes or other irregularities. The jets have circular crosssection and impinge perpendicular to the jet target. Both single rows of jets and staggered or inline array patterns are investigated. 1.5 Method One cooling method was investigated at the time, first rib turbulated cooling, second matrix cooling and third impingement cooling. Literature studies were performed for each area, which covered both open literature and in-house reports. The information from the literature studies were summarized for each cooling method. Much information was also gathered by personal communication with cooling engineers in the Turbine Aerodynamic and Cooling department at Siemens, GRCTA. Articles and 2

15 Introduction reports containing correlations were studied with focus on what type of experiment had been performed, correlation ranges and the applicability to turbine conditions. For each cooling method, the correlations were compared and analyzed. The agreement between the correlations was also investigated, since if independent investigators have come to the same conclusion the correlations are more likely to be accurate. Conclusions about the correlations were drawn based on information in the sources of the correlations and from analyzing the results of the comparisons. Parallel to the investigation of correlations in literature, the codes of two in-house programs used at Siemens were investigated for each cooling method. In Finspång, a heat transfer program called Q3D is used for heat transfer calculations. In the Siemens office in Lincoln, a heat transfer program called Multipass is used. Q3D was studied for all three cooling methods. Multipass cannot be used for heat transfer for matrix and impingement cooling, and thus was only studied for rib turbulated cooling. The code was checked against the correlations it was based upon Method Criticism The recommendations of correlations are partly based on the results of a limited number of flow cases and geometries that are relevant for turbine conditions. It is possible that the correlations behave somewhat different for other test cases. It has not been doable to test all cases possible for all correlations. Parts of the Q3D code consist of numerical approximations of correlations, why it is hard to analyze. This can cause misinterpretations. The Multipass program itself was not available for use, but the subroutines were investigated separately from the program. That can cause mistakes in the code interpretations. However, the results were checked and accepted by the Lincoln office. 1.6 Disposition and Reading Instructions All information about each cooling method is gathered in one place, to facilitate use of the report for cooling design. Chapter 1 gives an introduction to the work and presents background, problem description, objective, limitations and the disposition of the report. Chapter 2 shortly presents basic flow and heat transfer theory and also the geometry and cooling fundamentals of turbine blades and vanes. The next three chapters summarize the work on rib turbulated cooling, matrix cooling and impingement cooling, in that order. Each chapter contains basic theory about the cooling method, a summary of the literature study, results, analysis and discussion and finally conclusions. Both the work about correlations and the program codes are presented in each cooling method chapter. References for the cooling methods are also included in each separate chapter. Next, Chapter 6 summarizes the conclusions drawn for all cooling methods. Chapter 7 presents future work. 3

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17 Theory 2 Theory This chapter presents basic flow and heat transfer relationships. A brief introduction to gas turbines and blade cooling is also offered. 2.1 Flow and Heat Transfer Basics Definitions of flow, friction and heat transfer parameters and terms used are shortly presented Heat Transfer Much of the heat transfer in blades and vanes internal cooling systems takes place by convective cooling, where heat is transferred from the hot wall to the cooling air. The heat transfer coefficient - α is defined by equation (2.1) [2.1]. ( T T ) Qconv = α A w air (2.1) To generalize heat transfer correlations, it is common to use non-dimensional parameters. The heat transfer coefficient is often made non-dimensional by the Nusselt number, defined in equation (2.2). α L Nu = c (2.2) λ The characteristic length L c varies for different geometries and some examples are given in Table 2.1 [2.2]. Table 2.1 Examples of Characteristic Lengths. Flow Case L c = flat plate, local value x flat plate with length L, the hole plate L cylindrical pipe d non-cylindrical pipe D h Another parameter that describes convective heat transfer is the Stanton number, see equation (2.3). St = Nu (2.3) Re Pr Friction The fanning friction factor is defined from the pressure loss according to equation (2.4). 2 f L ρ v p = (2.4) 4 Dh 2 The Darcy friction factor is defined according to equation (2.5). p = f D L D h ρ v 2 2 (2.5) 5

18 Theory The relationship between the fanning and the Darcy friction factor is displayed in equation (2.6). f f = D (2.6) Flow Flow can be turbulent, laminar or in the transitional region between laminar and turbulent. Laminar flow occurs for non-disturbed flow with relatively low velocities and is characterised by even velocities and orderly motions. The opposite, turbulent flow, occurs at higher velocities and is characterised by velocity fluctuations and disordered motions. The Reynolds number describes the flow regime, see equation (2.7). For example, flow in a pipe is laminar for Re < 2300 and turbulent for Re > 4000 approximately. The Reynolds number is defined as the ratio of the inertia forces to the viscous forces in the fluid. The inertia forces depend on the flow kinetic energy and are a function of the fluid density and the square of flow velocity. The viscous forces depend on the fluid viscosity and the flow velocity. The characteristic length is the same as for the Nusselt number, see Table 2.1. ρ v Lc Re = (2.7) µ Flow in a pipe or over a flat plate can be divided into two regions. First the boundary layer region, where the friction from the surface below affects the velocity profile. Second, the inviscid flow region, where the friction effect is negligible, see Figure 2.1. Figure 2.1 Boundary Layer [2.3]. In a pipe, where flow enters with uniform velocity, the boundary layer will grow thicker and thicker further from the pipe entrance and thus, the inviscid region will decrease. After a certain length from the inlet, the hydrodynamic entry length, the velocity profile is uniform and the flow is said to be hydrodynamically developed, see Figure

19 Theory Velocity boundary layer Velocity profile x Hydrodynamic entry region Hydrodynamically developed region Figure 2.2 Velocity Profiles in Pipe Flow [2.1]. If also the temperature profile is constant, the flow is called fully developed. If neither cooling nor heating occurs, hydrodynamically developed flow is equivalent of fully developed flow. [2.1] The Prandtl number describes the thickness of the boundary layer, see equation (2.8). ν Pr = (2.8) λ ( ρ ) C p The definition of mass flow is important for correlations. Mass flow is defined in equation (2.9). m& = ρ A v (2.9) A discharge coefficient is often used for mass flow definitions in correlations. It is defined in equation (2.10) [2.4]. C m& real d = (2.10) m& ideal Mass velocity G is defined in equation (2.11) [2.5]. m G = & (2.11) A 7

20 Theory 2.2 Gas Turbine Vanes and Blades Gas turbines consist of three main parts; the compressor, the combustor and the turbine see numbers (1), (2) respectively (3) in Figure 2.3. Figure 2.3 Schematics of a Gas Turbine [2.6]. The compressor increases the pressure of the inlet air before the air enters the combustion chamber. In the combustor, air is mixed with fuel and the gas mixture combusts at high temperatures. The temperature increase makes the gas expand. In the turbine, the gas first reaches vane 1, in which the expanding gas is directed towards blade 1. In blade 1, the gas stream is deflected, which causes a torque on the shaft. The torque brings the shaft to rotate, which is the useful movement of the engine. The rotational movement can then be used for various purposes, such as working oil and gas pumps or, as in Figure 2.3 operating a generator (4) that produces electricity which is transformed to high tension by a transformer (5). A picture of SGT -800 is seen in Figure 2.4. Figure 2.4 Schematic Drawing of SGT -800 [2.7]. Figure 2.6 shows the nomenclature for a blade or vane. The front part is called the leading edge and the back part the trailing edge. Low pressure and high pressure sides are called suction side and pressure side, respectively. 8

21 Theory Platforms Figure 2.5 Turbine Blade (Right), and Turbine Vane (Left ) [2.8]. Figure 2.6 describes a typical cooling concept of a turbine blade. Figure 2.6 Typical Cooling Concept [2.9] Impingement cooling is often applied at the leading edge. In the mid chord region, rib turbulated cooling is common and in the trailing edge, pin fin cooling or matrix cooling is used. Platforms on vanes are often cooled by impingement. Exampes of a blade and a vane are displayed in Figure

22 Theory 2.3 References Theory [2.1] Çengel, Y. A. and Turner, R. H. (2001). Fundamentals of Thermal-Fluid Sciences, McGraw-Hill Companies, New York (ISBN ) [2.2] Storck, K., Karlsson, M., Andersson, I., Loyd, D. Formelsamling i termo- och fluiddynamik,(2001), Institutionen för Konstruktion och Produktion, Linköpings Tekniska Högskola [2.3] Appelqvist, B. and Loyd, D. Grundläggande teknisk strömningslära, (1979), Institutionen för Konstruktion och Produktion, Linköpings Tekniska Högskola [2.4 ] Höglund, H. Experimental Investigation of Impingement Cooling Under a Staggered Array of Circular Jets, Thesis Work at the Department of Energy Tehnology Royal Institute of Technologi, KTH (1999) [2.5] Holman, J. P. (2002). Heat Transfer Ninth Edition, McGraw-Hill Companies, New York (ISBN ) [2.6] Vattenfall (2005). ( ) [2.7] Siemens Industial Turbomachinery, Finspång [2.8] Vontobel (2005). ( ) [2.9] Han, J.C., Dutta, S., & Ekkad S.V. (2000). Gas Turbine Heat Transfer and Cooling Technology, Taylor & Francis, New York (ISBN X) 10

23 Rib Turbulated Cooling 3 Rib Turbulated Cooling The concepts of rib turbulated cooling are described, together with a summary of the literature study. After that, the results, analysis and conclusions of investigated correlations and program codes are presented. 3.1 Theory In turbine vanes and blades, ribs are mostly used in the internal cooling channels in the middle of the component. The ribs are situated on opposite walls, almost always towards the pressure side and suction side, see Figure 3.1 [3.1]. Sometimes only one side has ribs, because the internal cooling has to match the external load, which can be different on pressure and suction side. Ribs W H Figure 3.1 A Ribbed Channel. The ribs cause separation from the flow at the rib tops, and reattachment to the flow between the ribs. This disturbs the boundary layer, which leads to increased heat transfer, see Figure 3.2. Separation and reattachment increases the turbulence of the flow, which mixes the fluid elements near the wall with the cooler ones in the middle of the flow.[3.1] A new thin boundary layer is started at the reattachment point after every rib. P α Figure 3.2 Flow around Ribs [3.1]. The increase of heat transfer depends mainly on the aspect ratio of the duct, the flow Reynolds number and the rib configuration [3.2]. Figure 3.2 shows the setup of a ribbed duct, and the important rib parameters such as rib height - e, pitch, and rib angle. Ribs that are orthogonal to the flow direction are called transverse ribs. Due to the complex flow that the ribs create, the flow has to be described by empirical correlations from experiments, instead of by analytical solutions [3.1]. To compare the results from different experiments and with turbine blade channels, certain dimensionless ratios of the geometry is of extra interest. Important 11

24 Rib Turbulated Cooling dimensionless parameters for ribbed ducts are the channel aspect ratio, the pitch-to-rib height ratio and the rib height-to-hydraulic diameter ratio. The definition of rib angleα is also important, see Figure 3.2. The disadvantage of ribbed duct is that the pressure drop is increased by the ribs. However, since the ribs are relatively small the pressure drop is often acceptable. The largest pressure drop usually occurs in the channel bends. A comparison of the heat transfer to the pressure drop was made for different cooling methods [3.3]. Curve number 3 in Figure 3.3 represents a duct with ribs on two opposing walls. It is clear that the friction grows faster than the heat transfer. K Figure 3.3 Comparison of K and C. Ribbed Duct is Number 3 [3.3]. The Nusselt number in a smooth duct is highest near the inlet, due to the turbulent developing flow. Further from the inlet the Nusselt number decreases. However, the smooth wall Nusselt number in a ribbed duct is about 20% to 60% higher than that in a smooth duct. In a ribbed duct, the Nusselt number on both the ribbed and smooth walls are also fluctuating, due to the separation from and reattachment between the ribs. The ribbed walls have higher fluctuations and higher Nusselt number than the smooth walls. [3.1] The leading edge channels are often U-shaped. Rib turbulated cooling of U-shaped ducts were investigated by Nagoga, [3.3]. He tested two different cooling methods, called the SR-scheme and the SSR-scheme. Both the SR- and SSR-scheme consist of a periodic array of ribs that are spaced at a pitch P either perpendicularly or at angle α to the flow. The ribs are attached to the concave side of the duct. SR stands for Surrounding Ribs, and the geometry can be seen in Figure 3.4. [3.4] C Figure 3.4 Geometry of SR-Scheme, Front View (Left) and Top View (Right). The SSR-scheme consists of segmented ribs, which are placed in a semi-circular longitudinal duct. Figure 3.5 describes the geometry of the SSR-scheme. [3.4] 12

25 Rib Turbulated Cooling Figure 3.5 Geometry of SSR-Scheme, Front View (Left) and Top View (Right). 13

26 Rib Turbulated Cooling Summary of Literature Study The heat transfer enhancement decreases with increased Re [3.5] [3.6]. The reason is that flow reattaches faster for higher Re [3.5]. For Re = , the flow reattached approximately 2 rib heights downstream, compared to 6 rib heights for Re = [3.5]. [3.7] found that for a rib angleα of 45, Nu is increased but f remains the same as for α =90. [3.8] and [3.9] reported that angled ribs create a Figure 3.6 Secondary Flow induced by Angled Ribs[3.1]. secondary flow according to Figure 3.6. For α = 60 to 30, the secondary flow makes Nu decrease along the rib axes, from left to right in Figure 3.6 Secondary Flow induced by Angled Ribs[3.1]. [3.9]. The ribs cause Nu to have a periodic distribution, due to the continuous separation and reattachment of the flow. For a square duct with transverse ribs, Nu decreases from the inlet along the channel in the streamwise direction, until x/d h > 3 where the periodic Nu reaches a constant value. For a square duct with angled ribs, Nu also decreases after the inlet, but for x/d h > 3 Nu increases again, due to the secondary flow induced by the rib angle. For ducts with larger W/H than 1, this effect is gradually decreased. [3.9] Han et al. [3.8] investigated the same phenomenon as described above but in channels with aspect ratios less than 1. They found that for α = 90 or 30, local Nu is periodically distributed between the ribs after x/d h > 3, i.e. Nu neither increase nor decrease. For α = 60 or 45 [3.8] found that the periodic Nu increases along the channel due to the secondary flow described above. Han and Park [3.9] also found that for a duct with square cross-section, highest Nu and f were obtained for α =60. For a duct with W/H = 4, the highest Nu and f were obtained for α = 90. The best heat transfer performance, i.e. high Nu and low f, were obtained for α = 30 and 45 [3.10]. Ribs with 60 and 45 angle to the flow had 25 to 30 % better heat transfer performance than transverse ribs [3.8]. Concerning aspect ratio, [3.5] investigated ducts with aspect ratios from ¼ to 4 and found that ducts with small aspect ratios had better heat transfer performance than large aspect ratio ducts. [3.8] investigated ducts with aspect ratios less than 1, and also found that channels with smaller aspect ratios perform better. The effect of rib height was investigated by [3.7]. It was concluded that increased e/d h led to increased friction factor. An experimental study took place at Siemens in Finspång in 2002, which aimed to investigate the effect of rib height [3.11]. The outcome was that the influence of rib height on the Nu heat transfer decreases with an 14

27 Rib Turbulated Cooling increased Re. An increase of rib height from 5 mm to 7 mm had little effect on the heat transfer coefficient, but increased the friction factor by 2 [3.11]. The effect of rib pitch was discussed in many articles, and many investigators came to the same conclusions. For P/e no less than 10, Nu increases with decreased P/e [3.7],[3.5],[3.12]. Maximum heat transfer is obtained for P/e =10 [3.7], [3.10]. For smaller rib spacing, the reattachment between ribs cannot occur, see Figure 3.7 [3.13]. Figure 3.7 Flow Patterns for Different Rib Spacings[3.13]. [3.6] investigated the effect of number of ribbed walls. They found that Nu and f increased with the number of ribbed walls. For example, K increased from 2.04 for 1 ribbed wall to 2.63 for 4 ribbed walls when Re = They also found that a duct with 4 ribbed wall and W/H = ½, obtained 29% higher Nu than a duct with 1 ribbed wall and Re = A duct with 4 ribbed walls and W/H = 2 obtained 4% higher Nu than a duct with 2 ribbed walls when Re= [3.6] In 1987, Metzger and Vedula studied heat transfer performance in ribbed channels with a triangular cross-section. Different configurations of angled ribs were used, which created secondary flow according to Figure 3.8. [3.1] Figure 3.8 Triangular Ducts Investigated by Metzger & Vedula [3.1]. Metzger and Vedula found that the downstream angled ribs transported the main flow towards the ribbed sides, and the upstream angled ribs carried the main flow towards the smooth side wall. This resulted in higher heat transfer coefficient on the ribbed 15

28 Rib Turbulated Cooling sides for the downstream angled ribs, and higher heat transfer coefficient on the smooth side wall for the upstream angled ribs. [3.1] Experiments for the spiral ribs showed that for 60 angle, all three walls reached nearly the same Nusselt number for fully developed flow. They also found that the ribbed side wall for spiral ribbed had lower Nusselt number than the ribbed side wall of the downstream and upstream angled ribs. [3.1] Zhang et al. investigated channels with triangular cross-section in Experiments were made on six geometries, see Figure 3.9. Figure 3.9 Triangular Ducts Investigated by Zhang et al [3.1]. They found that Nu for the three-wall partial ribs were approximately 10 % higher than for three-wall full ribs. For transverse ribs K=2-2.3 and C = and for 45 angled ribs K = and C = In the 55 corner, K = for transverse ribs and K = for 45 angled ribs. In the 35 corner, heat transfer enhancement was low for all configurations, both transverse and for 45 angled ribs. [3.1] 16

29 Rib Turbulated Cooling 3.2 Correlations from Articles The results, analysis and conclusions drawn of the investigation of correlations from articles are presented below Results concerning Correlations from Articles Investigated Correlations Following correlations from published articles were examined. Webb, Eckert &Goldstein, 1970 [3.13] Han, 1988 [3.5] Chandra, Niland & Han, 1997 [3.6] Han, Glicksman & Rohsenow, 1978 [3.7] Han & Park, 1988 [3.9] Han, Ou, Park & Lei, 1989 [3.8] Han, 1984 [3.12] Han, Park and Lei, 1985[3.10] Nagoga, 2001 [3.4] The correlations are summarized in Appendix A. The correlation ranges are showed in Figure 3.10, together with ranges of interest for turbine cooling. The investigated reports from Siemens did not contain correlations for ribbed ducts. More articles about rib turbulated cooling are listed in Appendix J. The correlations are valid for different ranges of Reynolds number - Re, rib angle, aspect ratio, pitch-to-rib height ratio and rib height-to-hydraulic diameter ratio. Ranges for the correlations derived from articles are shown in Figure 3.10, together with ranges that are of interest for typical gas turbine blades and vanes in the first and second stages. The ranges of Nagoga s correlations for U-shaped ducts are displayed in Appendix B. 17

30 Rib Turbulated Cooling Reynolds Number Re*10-3 Aspect Ratio W/H Pitch-to-Rib Height Ratio P/e Rib Height-to-Hydraulic Diameter Ratio Webb et al Han et al Han 1984 Han 1985 Han & Park 1987 Han 1988 Han et al Chandra et al 1997 Turbine components e/d h Rib Angle Rib Angle Figure 3.10 Ranges for Correlations and Channels in SGT -700 and SGT

31 Rib Turbulated Cooling Comparison of Correlations The correlations were compared and analyzed by using Matlab. The effect of rib angle, aspect ratio, pitch-to-rib height ratio, rib height-to-hydraulic diameter ratio and Reynolds number was investigated. To be able to compare as many correlations as possible in their valid ranges, a test case with rib- and duct geometry according to Table 3.1 was chosen. This case is referred to as Case A. All correlations except for Webb et. al s 1970, Chandra et. al s 1997 and Han et. al s 1978 are valid for Case A. The latter are therefore analyzed separately. Table 3.1 Case A, Test Case for Ribbed Ducts. P [mm] 3.15 e [mm] D h [mm] 5 W [mm] 5 H [mm] 5 Number of ribbed walls 2 Re [-] α [ ] 90 W/H [-] 1 P/e [-] 10 e/d h [-] Figure 3.11 and Figure 3.12 display the results for the Nusselt number and friction factor for Case A. Nu for a smooth channel is calculated with the Dittus-Boelter equation, see equation (3.1). The Q3D correlations will be handled in chapter but they are also presented here to avoid displaying almost the same figures twice Han 1988 Han & Park 1988 Han et. al 1989 Nusselt Number for a Ribbed Duct Nusselt Number Han 1984,OBS! Nu average Han et al Q3D Smooth Channel Reynolds Number x 10 4 Figure 3.11 Nu on Ribbed Side Wall for Case A. 19

32 Rib Turbulated Cooling Nu = Re Pr (3.1) As can be seen in Figure 3.11, Han 1984 results in a lower Nusselt number than the other correlations. The reason for this is that this correlation concerns the average Nusselt number for the ribbed and smooth walls in a duct with two ribbed walls. The other correlations are for the Nusselt number on the ribbed side wall, which is of more interest for turbine blade design Friction factor for a Ribbed Duct Friction Factor Han 1988 Han & Park 1988 Han et. al 1989 Han 1984, OBS! Nu average Han et al Q3D Smooth duct Reynolds Number x 10 4 Figure 3.12 Friction Factor on Ribbed Side Wall for Case A. The results concerning the friction factor are displayed in Figure All correlations except Q3D result in approximately the same value for the friction factor. All correlations of Han et al. calculate an average friction factor, which is based on a weighted average between a totally smooth duct and a totally ribbed duct, see Appendix A. Figure 3.13 and Figure 3.14 presents Nu and K for a smooth wall in a duct with two ribbed walls. 20

33 Rib Turbulated Cooling Nusselt Number on Smooth Wall in a Ribbed Duct Han 1988 Han & Park 1988 Han et al Q3D Smooth Channel 200 Nusselt Number Reynolds Number x 10 4 Figure 3.13 Nu on Smooth Side Wall in Ribbed Duct for Case A. Nusselt Number Enhancement K on Smooth Wall in a Ribbed Duct 2 Han 1988 Heat Transfer Enhancement Factor Han & Park 1988 Han et al Q3D Reynolds Number x 10 4 Figure 3.14 K on Smooth Side Wall in Ribbed Duct for Case A. 21

34 Rib Turbulated Cooling Figure 3.14 shows that there is quite a large difference on the enhancement factor between the correlations. Results for K, C and eff of a ribbed side wall are presented in Appendix E. The effect of α, P/e, e/d h and W/H on Nusselt number enhancement, K, friction factor enhancement, C and efficiency, eff for Case A has been studied. The results are presented in Appendices C to G. U-shaped Channels The friction factors and enhancement factors in U-shaped ducts cooled by SR- and SSR-schemes are displayed in Figure 3.15 for the geometry according in Table 3.2. Table 3.2 Geometry for SR- and SSR- Calculations. e D h R P α T f T w Nu Nusselt Number SR SSR K Nusselt Number Enhancement Factor K SR SSR Re x 10 4 Friction Factor Re x 10 4 Friction Enhancement Factor C 3.5 f SR SSR C SR SSR Re x 10 4 Figure 3.15 Results for SR- and SSR-Scheme Calculations Re x 10 4 Temperature Analysis A temperature analysis was made in order to investigate the effect on surface temperature for a change of K. Vane 2 on SGT -700 was chosen since the duct s geometries are easily approximated to fit the correlations. A cross-section in the middle of the vane, named section 4 in Q3D, was used, see Figure

35 Rib Turbulated Cooling Figure 3.16 Cross-Section 4, Vane 2, SGT The K values in channels 1, 2 and 3 were altered, see Figure The temperatures are calculated for the 2D-section, and are shown as the temperatures of the external surface. Temperature is calculated at points along the perimeter. The distance along the external perimeter, S, starts with 0 at the suction side trailing edge and is defined as S max at the pressure side trailing edge. The points along the perimeter are made dimensionless by dividing S by S max. For channel 1, only the Nagoga correlation was applicable, for channel 2, only Han & Park 1988 was valid, and for channel 3 only Han et al was valid. Table 2 display the geometries and both original and new K values. The dimensions of the channel are taken from the CAD program Catia. Channels 2 and 3 were approximated to have a rectangular cross-section. The Reynolds number and enhancement factors were taken from a Q3D model of the vane. The formula used for the calculations of the original K values is not known. Index s indicates value for smooth wall and index r value for ribbed wall. Table 3.3 Parameters for Channels in Vane 2, SGT Channel 1 Channel 2 Channel 3 Correlation Nagoga [3.4] Han & Park [3.9] Han et al. [3.8] W H e P D h α R W/H P/e e/dh Re K r,orig K r,new K s,orig K s,new K r,new /K r,orig K s,new / K s,orig

36 Rib Turbulated Cooling The Nagoga correlations does not consider the smooth side wall, therefore the original value was kept for channel 1 for the smooth side wall. Figure 3.17 and Figure 3.18 display the results of the temperature calculations. SGT 700 Vane 2 Section Temperature [K] Original SGT700 New SGT ,2 0,4 0,6 0,8 1 1,2 Position S/S_max Figure 3.17 Temperature in Vane 2, SGT Temperature Difference, New-Original Temp. Difference [K] Temp. differences 0 0,2 0,4 0,6 0, Position, S/Smax Figure 3.18 Temperature Difference in Vane 2, SGT

37 Rib Turbulated Cooling Analysis and Discussion of Correlations from Articles Correlations from eight articles have been investigated, in order to find which ones are most reliable to use in turbine blade and vane cooling channels with ribs. Figure 3.10 gives an overview of the different ranges of the correlations together with the ranges of interest for turbine design. Unfortunately, no correlation covers the total area of interest. Therefore different correlations could be of use for different cooling channels. The friction factors in the correlations are the average friction factors, weighed between f of a fully smooth duct and that of a fully ribbed duct. The friction factor on a ribbed side wall is therefore higher than the values displayed in Figure Webb et al. s correlation from 1970 covers a lot of interesting ranges. However, the experiments it is based on were performed on fully ribbed circular pipes, instead of rectangular or square ducts with only two ribbed sides. This makes the correlation unsuitable for turbine blade design. The correlations of Han 1988 cover a wide range of parameters. J.C. Han has made a large amount of research in this area, and when it comes to correlations of this kind, it is difficult to find articles that he has not been involved in. His results from 1987, 1988 and 1989 are similar but valid for different duct- and rib geometries. The main difference is that Han 1988 only is valid for transverse ribs and the other two for both transverse and angled ribs, see Figure 4. The correlations from 1987 are valid for aspect ratios between 1 and 4, and those from 1989 are valid for smaller aspect ratios, between ¼ and 1. Experiments from these articles are comparable to each other and all aim to gather information to use for turbine blade cooling. Figure 3.11 shows that the Han 1988 correlations give a slightly lower value of Nu compared with Han & Park 1988 and Han et al Since it is safer to choose a too low Nu than a too high, it would be wiser to use the correlation from 1988 for transverse ribs. Figure 11 in Appendix D shows that for W/H = 1, where both the correlations from 1987 and 1989 are valid, the later gives a higher value of the Nusselt number. Therefore it would be safer to use the 1987 correlation for an aspect ratio of one. His earlier work, in 1984 and 1985, also aims for turbine blade design. However, the correlations from 1984 only concerns average values for the Stanton number and friction factor, instead of the actual St and f of a ribbed or smooth wall which are used in heat transfer programs. It is therefore not suitable for turbine blade design. The correlation from 1985 concerns angled ribs, but is valid for a smaller range for e/d h and W/H, than the correlations from 1987 and The values of St and f of Han 1985 differs from those of Han et al and 1987, Figure 3.11 and Figure The difference between the correlations is that in Han 1985, calculations are made + with the average roughness Reynolds number, e and the average roughness + function, R ( e ) and in the other Han correlations calculations are based on the + roughness Reynolds number e and roughness function R(e + ). Definitions of these 25

38 Rib Turbulated Cooling parameters is found in Appendix A. Since the most recent and the majority of articles use the later way of calculations, that method it likely to be the most reliable. The article of Han, Glicksman and Rohsenow from 1978 covers a relatively small range of Reynolds number and is valid for very large aspect ratios, see Figure Therefore the correlations are often not applicable for turbine vane and blade design. The experiments were performed using parallel plate geometry instead of a rectangular or square duct, which also contributes to the unsuitability of using these correlations for the turbine components of interest. However, it can be of interest for other areas, for example combustor cooling. Chandra, Niland and Han performed experiments to investigate the effect of different number of ribbed walls in The correlations are only valid for a small range of W/H and e/d h and may therefore not be suitable to use for duct with two ribbed walls, where there exist more applicable correlations. However, the correlations can be useful for ducts with one, three or four ribbed walls. Effect of Rib Angle The correlations from articles all display similar trends concerning rib angle, according to Figure 1 to Figure 3 in Appendix D. The highest Nusselt number appears for α = 60 for all correlations except for Han et al. 1989, which calculates the highest Nusselt number for α = 75. However, the experiments from 1989 were only made for α = 30, 45, 60 and 90, therefore it is difficult to say how correct the value is for α = 75. The highest Nusselt number is unfortunately followed by the highest friction factor, as seen in Figure 5, Appendix D. Therefore the efficiency, Figure 3 in Appendix D, has a minimum for α around 60 and 75. Even thought Figure 3 in Appendix D show a minimum of eff for α around 60 and 75, rib angles of 60 are often used in turbine vanes and blades. This is because the pressure loss due to the ribs in cooling ducts in are relatively small compared to other losses, for example that caused by the channel bends. Therefore rib angle is often chosen with respect to the highest Nusselt number instead of efficiency. Effect of Pitch-to-Rib Height Ratio Appendix E show the results concerning P/e. These correlations are not valid for P/e < 10, but for the valid range it is clear that the highest heat transfer coefficient occur for P/e = 10. The correlation of Han et al is valid for a larger span of P/e. Figure 4, Appendix E indicates that P/e = 10 is the optimal pitch-to-rib height ratio. However, this correlation is not valid for Case A, so the result should be handled with caution. The Han correlations agree well with the theory in chapter 3.2. Effect of Aspect Ratio The effect of aspect ratio is displayed in Appendix F. The Han correlations from 1987, 1988 and 1989 give similar results, whereas that from 1985 deviate from the trend. The graphs confirm the theory that smaller aspect ratios results in higher efficiency than larger aspect ratios. An increased aspect ratio leads to increased K and C, according to Figures 1 and 2, Appendix F. The increase of C is larger than the increase of K. Therefore, the highest efficiency is received for small aspect ratios. 26

39 Rib Turbulated Cooling Effect of e/dh Figure 14 to Figure 16 in Appendix G display the effect upon C, K and eff for various e/dh. Han 1988 and 1985 returns a lower value than Han et al and Han 1984 returns lower values than the other correlations, which is the same pattern as for previous results. The effect on K is relatively low compared with the other dimensionless parameters investigated. The friction factor is more sensitive to rib height according to Figure 2, Appendix G. It is logical that increased rib height leads to increased pressure drop. This agrees well with the theory about the effect of e/dh. The fact that the friction factor increases while the Nusselt number is rather unaffected by increased rib height, makes the highest efficiency for small rib height-to-hydraulic diameter ratios, as seen in Figure 16, Appendix G. Temperature Analysis As seen in Figure 3.17 and Figure 3.18, the changes of K values affect the surface temperature on the vane. Since the values of K new where all lower than the original values, it should result in higher surface temperatures. Since K original is higher than K new, the model was not based on calculations from Q3D, since the Q3D enhancement factors result in lower values than all Han correlations except that from 1984, see Figure The model of Vane 2, SGT -700 was built in Moscow several year ago, and it is not known what correlation the enhancement factor is bases on. The surface temperature became higher with K new implemented, except at the trailing edge. At the trailing edge, the temperature in the original model is approximately 12 C higher than in the modified model. The explanation is that since the K values are lowered in channels 1 to 3, less heat is transferred to the cooling air which results in colder cooling air in the trailing edge. Colder trailing edge cooling air results in colder surface temperature in this region, which is displayed in Figure 3.17 andfigure The largest temperature difference appears next to the leading edge at the pressure side. This is natural since the K values are lowered most in channel 1 and 2. The new revised model s surface temperature is 20 C higher than the original model. Since the enhancement factor K has such impact on the surface temperature, and the difference of the original and new value of K for this example is as much as 18.3 %, it is recommended to look into the routine of choosing Nusselt number enhancement factors. 27

40 Rib Turbulated Cooling Conclusions concerning Correlations from Articles In conclusion, following recommendations are made for what correlations to use. Han 1988 [3.5] for transverse ribs in ducts with two opposing ribbed walls Han et al [3.8] for angled ribs in ducts with two opposing ribbed walls with 1 4 W H < 1. Han & Park 1988 [3.9] for angled ribs in ducts with two opposing ribbed walls with 1 W H 4. Chandra et al [3.6] for duct with one, three or four ribbed walls. Nagoga 2001 [3.4] for U-shaped ducts with SR- or SSR-schemes. 28

41 Rib Turbulated Cooling 3.3 Correlations in Q3D The results from an investigation of the Q3D program code are presented and analyzed below Results concerning Q3D The correlation in Q3D is an in-house correlation based on Han test data. Q3D calculates the Nusselt number and the friction factor. The input variables are duct height, duct width, rib height and the hydraulic diameter. The Nusselt number is calculated according to equation (3.2). The Q3D correlations are used to choose A and n. Nu n = A Re (3.2) The Q3D interface for ribbed channels is showed in Figure Figure 3.19 Q3D Interface for Ribbed Channels. The path to the Channel box is H_Hydro Hydraulic Net HN Branches- FE Grid Channels As seen in Figure 3.19, both the Nusselt number for a ribbed duct, Nu r and for a smooth duct, Nu s are calculated. The exact values and expressions used in Q3D were extracted from the code and are displayed in Appendix H. Reduced Area Method The area and hydraulic diameter used in Q3D to calculate the friction factor and pressure drop are calculated by the so called Reduced Area Method - R.A.M. This means that the area and hydraulic diameter are calculated by the reduced height H e, instead of the total height H according to equation (3.3) and (3.5) and Figure

42 Rib Turbulated Cooling H e = H 2 e (3.3) D e W H e = 2 (3.4) W + H e A = W (3.5) H e D W H = 2 h W + (3.6) H A = W H (3.7) The alternative to the Reduced Area Method is the Total Area Method, when H and D h are used. The Total Area Method is used in most of the open literature on the subject. To compare Q3D with the other correlations, the code was recalculated to the Total Area Method. e H Figure 3.20 Cross-Section of Channel with 2 Ribbed Walls. Whether Nu is calculated with the Reduced or Total Area Method depends on in what form the hydraulic diameter input value is entered. Q3D Compared to the Han Correlations Nu and f calculations of Q3D compared to the different Han correlations are displayed in Figure 3.11 tofigure 3.14 and in Appendices A to E. The Q3D Nusselt number is lower than those of the Han correlations. The Q3D friction factor is considerably higher than those of the other correlations, see Figure The high friction factor is probably due to a bug in the code. The code contains calculations to adjust f according to the number of ribbed walls, but the calculations are immediately overwritten. The overwritten equations use three parameters Lch, LRibs and m_lbr. Lch represents the channel perimeter and LRibs the part of the perimeter with ribs. The meaning of m_lbr is unknown. These parameters should be input parameters, but they can t be entered into the program. A bug was found in the interface. Three boxes were excluded from the interface concerning number of ribbed walls. The original and the corrected interface are showed in Figure

43 Rib Turbulated Cooling LRib m_lbr Lch Figure 3.21 Original Interface (Left) and Corrected Interface (Right). This is not the only reason for f being so high, because f is already too high before this stage of calculation. When reasonalble values are entered in the three boxes for Lch, LRib and m_lbr, they are immediately overwritten by some default value and are not used in the calculations Analysis and Discussion of the Q3D Correlations Figure 3.11 and Figure 3.12 show that the Nusselt number is lower and f is much higher in Q3D than the Han correlations. The code is very different from the equations used in the Han correlations, and is probably curve-fitted models of the original equations and data. Whether the differences between the original correlations and the code are deliberate corrections or mistakes is difficult to tell. One reason of the high friction factor could be that the friction factor from Q3D is the one for a ribbed side wall instead of an average between a fully ribbed and fully smooth duct. The low value for the Nusselt number could be chosen due to safety reasons, since it is better to underestimate the value of Nu than vice versa. To localize the fault concerning both f and Nu is difficult, since the code is based on numerical approximations which are hard to connect to a physical duct. However, the three hidden input parameters is part of the solution. They need to be investigated further. The Q3D code also differs from the original correlations since P/e in Q3D affects neither the value of f or Nu for Q3D, see Figure 39 Figure 40 in Appendix E. The reason is that the code is based on a constant value of P/e = 10. Figure 14 in Appendix 31

44 Rib Turbulated Cooling G shows that for Q3D, Nu is unaffected of e/d h, which is not the case for the Han correlations. It is also notable that Q3D s correlation has a different Reynolds dependency than the other correlations. Again, the reason for this is difficult to detect without more knowledge about the basis of the correlations. Q3D is based on the Reduced Area Method. For the Nusselt number, the only difference from the Total Area Method is that the reduced hydraulic diameter, D e is expected as input instead of D h, see equations (3) and (5). For the friction factor, the difference from the Total Area Method is that H e is used instead of H, see equation (2). Therefore it would be easy to change the code to use the Total Area Method, which is used in most open literature Conclusions Concerning Q3D Following conclusions were drawn from the investigation of the Q3D code. Nu in Q3D is lower than Nu from the Han correlations it is based on. The friction factor in Q3D is considerably higher than the Han correlations, probably due to a bug that needs further investigation. Three input parameters concerning number of ribbed walls are missing in the interface. The parameters are part of the solution to the high friction factor, but needs further investigations. It would be easy to change the code from the Reduced Area Method to the Total Area Method, which is recommended. 32

45 Rib Turbulated Cooling 3.4 Correlations in Multipass Correlations used in the Multipass program have been investigated. The results, analysis and conclusions of the investigation are presented below Results concerning Multipass Four subroutines calculate the friction factor and Stanton number, each based on different correlations. The subroutines have been investigated and compared to the original correlations. The results are presented graphically in Appendix I. The comparisons concern the Nusselt number and friction factor for a ribbed wall in a duct with two ribbed walls. The Nusselt number is obtained from equation (2.3). Subroutine Ribbed 1 The equations in Ribbed 1 are based on the work of Han, Glicksman and Rohsenow [3.7]. In Figure and Figure 18 in Appendix I, the Nusselt number and friction factor from Han et al. s correlation is displayed together with that from Ribbed 1. The geometry used for comparison is quite different from the other test geometries, since this correlation has anogher validity range, see Figure Subroutine Ribbed 2 Calculations in Ribbed 2 are based on Han and Park [3.9]. A bug in the code was discovered. Figure 19 and Figure 20 compare the original correlation with the Nu and f used in ribbed 2, both with the bug corrected and not corrected. Subroutine Ribbed 3 The code is based on correlations from work by Han, Ou, Park and Lei [3.8]. A bug was detected for the calculation of the friction factor. Figure 21 and Figure 22 display the Nusselt number and friction factor concerning Ribbed 3, both with and without the bug corrected. Subroutine Ribbed 4 This subroutine is based on a mixture of Ribbed 2 and Ribbed 3, i.e. it is a mixture of correlations from Han & Park 1988 and Han et al A bug in the Stanton number due to a mix up of friction factors was detected and corrected. The results are displayed in Figure 23 and Figure Analysis and Discussion of the Multipass Correlations Subroutine Ribbed 1 The Stanton number agrees with the original correlation of [3.7]. The friction factor in Ribbed 1 is first calculated according to the work of [3.7], but then manipulated numerically. The numerical manipulations results in a friction factor that differs slightly from the original correlation, see Figure 18. In [3.7], the fanning friction factor, f is derived. In Ribbed 1 and Ribbed 3, a subroutine called FF is used for calculations of the friction factor, which calculates the Darcy friction factor, f D. For better comparisons, the friction factors from Multipass are recalculated to the fanning friction factors here. 33

46 Rib Turbulated Cooling Subroutine Ribbed 2 The equations in the code are almost identical with the Han and Park s correlations. There is one difference, where Ribbed2 uses the friction factor of a ribbed wall in a duct with two ribbed walls, f r2 instead of duct with four ribbed walls, f r4 in for calculations of e +. This is probably a bug in the code and is easily corrected. Subroutine Ribbed 3 The Stanton number is calculated almost exactly as the original correlation, whereas f is numerically manipulated, which results in a much higher value than that of Han et al., see Figure 22. Except for the numerical solution of f, there are also a few differences between the code s calculations and that of Han et al. s correlation. It is probably a bug and is easily corrected. Figure 21 and Figure 22 display the Nusselt number and friction factor concerning Ribbed 3, both with and without the bug corrected. Subroutine Ribbed 4 One difference between the code and the correlation exists, similar with the one in Ribbed 2. In one equation, Ribbed 4 uses the friction factor for a ribbed wall in duct with 2 ribbed walls, f r2 instead of the friction factor for a fully ribbed duct, f r4. This effects the Stanton number but not the friction factor. Figure 23 and Figure 24 show the results Conclusions concerning Multipass In Ribbed 1, the Stanton number is the same as in [3.7]. The friction factor is numerically manipulated and is not equal to the original correlation. In Ribbed 2, the Stanton number differs from [3.9], due to a mix up of friction factors, which are used in the Stanton number calculation. The friction factor is equal to [3.9]. In Ribbed 3, the Stanton number differs from the [3.8], due to a small difference from the base correlation. The friction factor is numerically manipulated and differs from the [3.8]. In Ribbed 4, the Stanton number also differs from the Han correlation, due to a mix up of friction factors that are used in the St calculation. The friction factor is equal to the original correlation. 34

47 Rib Turbulated Cooling 3.5 References Rib Turbulated Cooling [3.1] Han, J.C., Dutta, S., & Ekkad S.V. (2000). Gas Turbine Heat Transfer and Cooling Technology, Taylor & Francis, New York (ISBN X) [3.2] von Karman Institute for Fluid Dynamics, Heat Transfer and Cooling in Gas Turbine, Lecture Series , May [3.3] Nagoga, G. Intensification of the Heat Transfer in the Cooling Ducts of the Gas Turbine Blade, Finspång HTC Database, Part I, Russia (2000) [3.4] Nagoga, G. Intensification of Heat Transfer in the Cooling Ducts of Gas Turbine Blade, Finspång HTC Database, Part IV, Moscow, 2001 [3.5] Han, J. C. Heat Transfer and Friction Characteristics in Rectangular Channels With Rib Turbulators, Journal of Heat Transfer, 110, (1988), [3.6] Chandra, P. R., Niland, M. E. and Han, J. C. Turbulent Flow Heat Transfer and Friction in a Rectangular Channel With Varying Numbers of Ribbed Walls, Journal of Turbomachinery, 119, (1997), [3.7] Han, J. C., Glicksman, L. R. and Rohsenhow, W. M. An Investigation of Heat Transfer and Friction for Rib-Roughened Surfaces, International Journal of Heat Mass Transfer, 21, (1978), [3.8] Han, J. C., Ou, S., Park, J. S. and Lei, C. K. Augmented Heat Tranfer in Rectangular Channels of Narrow Aspect Ratios with Rib Turbulators, International Journal of Heat Mass Transfer, 32, (1989), [3.9] Han, J. C. and Park, J. S. Developing Heat Transfer in Rectangular Channels with Rib Turbulators, International Journal of Heat Mass Transfer, 31, (1988), [3.10] Han, J. C., Park, J. C. and Lei, C. K. Heat Transfer Enhancement in Channels With Turbulence Promoters, Journal of Engineering for Gas Turbines and Power, 107, (1985), [3.11] Larsson, T.(2002) Investigation of the Effect of Different Rib Configurations on Heat Transfer in the GT10C Blade 1 Leading Edge, Siemens Finspång Technical Report [3.12] Han, J. C. Heat Transfer and Friction in Channels With Two Opposite Rib- Roughened Walls, Journal of Heat Transfer, 106, (1984), [3.13] Webb, R. L. Heat Transfer and Friction in Tubes with Repeated-Rib Roughness, International Journal of Heat Mass Transfer, 14, (1971),

48

49 Matrix Cooling 4 Matrix Cooling The basic concept of matrix cooling is presented, togheterher with a summary of the literature study. Next, the results, analysis and conclusions concerning matrix cooling are accounted for. 4.1 Theory The flow and geometry of matrix cooling is described. After that, a summary of the literarure study is presented What is Matrix Cooling? A matrix consists of two layers of opposite angled longitudinal ribs. The ribs create a system of channels, in which the cooling air flows and continuously changes direction as it changes channel on its way to the matrix exit, see Figure 4.1. l Figure 4.1 Example of a Matrix Geometry [4.1] The heat transfer coefficient is increased due to inlet effects, where a new thin boundary layer is developed at the entrance of each channel. When the flow passes from one channel to another, a swirl is created and the turbulence of the flow is increased. Heat transfer is also increased due to increased heat transfer surface area from the longitudinal ribs. Another positive effect is that the longitudinal ribs enhance the strength of the component. The angle β of the channels has a large influence on the matrix heat transfer enhancement.[4.2] The matrix in Figure 4.1 is a so called closed matrix, where the channels reach the side wall. In a closed matrix, the flow that reaches the end of a channel flows through the bend and into the channel on the opposite side without mixing with the flow from the other channels. An alternate design is called open matrix, where there is a clearance between the channels and the side wall, see 1 and 2 in Figure 4.2.[4.3] In an open matrix, the flow that reaches the end of a channel mixes with the flow in the clearances 1 or 2. The air that flows in to new channels starting at the clearances, is taken from the flow in the clearance channel. 37

50 Matrix Cooling Matrix cooling is often used in the trailing edge of turbine vanes and blades. In Siemens Gas Turbines produced in Finspång, only open matrixes are used. Usually the flow in open matrixes is axial and that in closed matrixes is radial. 1 2 W Figure 4.2 Open Matrix. This cooling method is relatively unknown in west, but has been used for many years in the former Soviet Union. Therefore available information in open literature is limited Fin Effect of Ribs Fins, or in this case ribs, increase heat transfer due to increased heat transfer surface area. However, the ribs are not of uniform temperature, but are cooler at the tip than close to the hot wall. The effective temperature difference is therefore lower compared to an ideal rib where the whole rib is at wall temperature. This effect is described by the rib efficiency - η, which is defined as η = Ideal Actual Heat Transfer of the Rib Heat Transfer from Rib if Entire Rib was at Wall Temperature The equations for rib efficiency depend on whether the rib top is isolated or convective, see equations (4.1) to (4.4) and Figure 4.3. [4.4] Isolated Rip Top: ( M h) η = tanh (4.1) M h where 2 α M = (4.2) λ s Convective Rip Top: ( M hc ) η = tanh (4.3) M hc where k h c = h + (4.4) 2 38

51 Matrix Cooling k h s Figure 4.3 Geometry of Longitudinal Rib. The fin increases the heat transfer due to increased heat transfer surface area. The fin effectiveness -ε is defined according to equation (4.5). Heat Transfer with Rib Arib ε = = η Heat Transfer if there was No Rib A b (4.5) The area A b is the rib base area, based on length s. A rib is the heat transfer surface of the rib Summary of Literature Study A summary of the investigated literature is presented below. Bunker, 2004 [4.1] Tests were made to study the pressure losses and local and average heat transfer enhancements in matrix cooling channels. Two methods were used. First, acrylic models were tested and the effect of increased heat transfer surface area was not taken into account, since the rib material was insulating. Heat transfer on the matrix shell, corresponding to suction and pressure sides, was investigated. Liquid crystal technique was used for temperature measurements. Second, metal models were tested and the effect of increased heat transfer area was included. Temperature was measured with an infrared camera. The heat transfer in the channels was compared to that in a smooth duct, calculated with Dittus-Boelter, equation (3.1). Following conclusions were drawn from the results of the tests. Average K on duct shell was equal to 2.5. After turns, local K could reach a value of 3. Narrow sub channels provide higher overall heat transfer enhancement, with K 3, than wider sub channels which have less turn effects and K 3. The effect of increased heat transfer surface area is of great importance. It is appropriate to treat the ribs as simple fins, and each rib surface has approximately the same heat transfer coefficient as the shell of the matrix channel. 39

52 Matrix Cooling Nagoga, 2000 [4.2] This is the only work found that contains heat transfer and friction correlations for matrixes. The correlations and their ranges are summarized in Appendix B. To describe heat transfer and friction enhancement factors, Nagoga compared the matrixes investigated to a matrix with straight channels, i.e. for which β = 0. He did not think it was reliable to compare the matrixes to the flow in a long smooth duct. Nagoga found that a matrix increases the heat transfer compared to a geometry with straight channels, i.e. with β = 0. Heat Transfer on Base Shell The base shell corresponds to the suction and pressure sides on a turbine blade. The matrix can be divided into two parts, the initial and basic section. The initial section consists of the channels that begin at the inlet of the matrix and end at the side bound. The basic section consists of the channels beginning at a side bound and ending either at a side bound or at the outlet of the matrix. Channels in the initial section have the same heat transfer and flow behavior as a smooth duct. Heat transfer in the basic section was higher than that in a straight duct. The local enhancement factor varied from 1.28 to 2. For increased Reynolds number and x/d, the enhancement factor decreased. Both the local Nusselt number and the Nusselt number averaged on the length of the channel depend on β, and increase with increased β up to β = 45. The average Nusselt number decreases with increased channel length. The local Nusselt number reaches a maximum right after the turn and then reduces along the channel. Local Nu has a value similar to that in a smooth duct when x d 30. Nagoga concluded that the rib pitch, relative channel depth, form of channel crosssection and the type of side bound, concave or flat for example, did not affect local and average heat transfer in the scope of interest for turbine blades and vanes. Experiments showed that the effect of the relative depth of the channels must not be connected with the effect of cross flow interaction with the co-planar sub channels on opposing sides. Heat Transfer on Side Bounds In order to understand the heat transfer on the internal surface on the semi-cylindrical side bounds, which corresponds to the leading edge on a turbine blade, a number of new parameters were defined, see Table 4.1. Table 4.1. Definition of Ddimensionless Paramteres. D Describes the effect of the centrifugal force on flow, defined as D = R ( 1+ cot 2 ( β )) d. F Describes contraction-diffusion of the flow at the turn A in Figure 4.4. CD FCD = a A a, see Figure 4.4 for a A and a. F C Describes the mass exchange of opposite sub channels through the interribs windows. F C = 2 S cos β t = z = number of matrix channels 40

53 Matrix Cooling Figure 4.4 Geometry in Matrix Side Bounds [4.2]. The investigation concluded that the heat transfer on the side bounds were higher than for a duct with straight sub channels. The heat transfer in this area depends on the parameters D, F CD and F C described above, which in turn depend on angle β. The local and average Nusselt numbers along a line ST on the inner concave surface of the side bound shell according to Figure 4.5 were measured and correlations developed, see equation (109) in Appendix B. Figure 4.5 Line ST on Matrix Shell inner Concave Surface. No explicit correlation of the local Nusselt number on the side bound is reported. The local Nusselt number is also determined by parameters D, F CD and F C, and the rate of dependence is the same as for the average Nusselt number. Hydraulic Losses The pressure losses and hydraulic friction in the matrix were studied. The hydraulic friction in the initial section did not differ from that in a smooth, straight duct with β = 0. In the basic section, the hydraulic friction exceeded that in a duct with straight sub channels. The enhancement of hydraulic friction was always higher than the heat transfer enhancements. Equations (94) and (104) in Appendix B display friction correlations in the initial and basic section, respectively. An energy efficency was defined according to equation (113), Appendix B. It was concluded that the energy efficiency of heat transfer intensification for a matrix depends only on the value 2 β. Maximum energy efficiency was achieved for 2 β =

54 Matrix Cooling Study of the Mechanism of Heat Transfer and Friction Flow visualization showed that in the initial section, the flow was axial, i.e. the full velocity vector w, coincided with the axis of the sub channels. The heat transfer was identical to that in a straight smooth duct, and no increase of heat transfer or friction occurs. In the basic section, the flow was twisted due to the turn and overflows from one channel to another at points A in Figure 4.4, why heat transfer and friction exceeded that of a smooth straight duct. Three parameters describe the twisted flow, the angle of flow rotation - ϕ, the rotation parameter - Φ and the maximum value of the axial velocity component - w,x. The angle ϕ is defined in equation (4.6) and w, x is defined in equation (4.7). The relationship between Φ and ϕ is displayed in equation (4.8) according to [4.2]. wϕ ϕ = arctan w x w (4.6) w, x w, x = w (4.7) AV tanϕ = 1.18 Φ (4.8) 0.76 The rotation parameter Φ describes the ratio of the angular momentum flow to the axial momentum flow. Experimental dependencies were developed for Φ and w, x, see equations (117) to (122) in Appendix B. Index indicates values near the wall. The angle of flow rotation - ϕ, and the rotation parameter - Φ both have a maximum close to the side bound and reduces downstream the channel. The increase of heat transfer and hydraulic friction depend on the flow rotation at the inlet of the channels. The flow rotation is caused by the contraction and diffusion of flow in the spatial turn at point A in Figure 4.4. The rotation intensity, described by ϕ, Φ and w, x, depend on β and the distance x/d from the turn to the control section. The local Nusselt number in the basic section channels turned out to be the same as for a round straight channel with inlet rotation. Therefore, the same correlations for the local Nusselt number can be used for these two cases. Rotation at the inlet of a channel causes the cooling air velocity to rise in the areas close to the walls and the mass centrifugal forces and secondary vortexes to form. Equations (123) and (124) in Appendix B display the correlation for the local Nusselt number in the basic section. The hydraulic friction in the basic section differed from that in a smooth straight channel with inlet rotation. Therefore, the same friction correlations could not be used for the two cases. The difference could be explained by the open U-type form of the cross-section of the matrix s sub channels determining the interaction of the flow in the opposite coplanar channel. Equations (125) and (126), Appendix B, display the correlation for the friction factor in the basic section sub channels. However, these correlations turned out only to be valid for Re > For Re < 10 5, the hydraulic 42

55 Matrix Cooling friction factor in sub channels in the basic section exceeded the friction factor calculated by equation (125) by 30 % to 60 %. Effectiveness of the Matrix Three indexes are important for evaluating the effectiveness of a cooling method concerning turbine blades and vanes: the hydrodynamic energy effectiveness - Ψ the relative depth of cooling or cooling effectiveness - Θ the effect on life of turbine blade or vane -τ Hydrodynamic energy effectiveness The matrix energy effectiveness - Ψ is defined in equation (113), Appendix B. It is only depending on angle β. A number of comparisons of the hydrodynamic energy effectiveness of matrix to other cooling methods were made. Main emphasis was put on rib turbulated cooling and pin fin cooling. Pin fin cooling consists of cylindrical pins in a flat channel. Enhancement factors K and C for a matrix are based on comparisons between a matrix with angled sub channels and an analogous geometry but with straight sub channels, i.e β = 0. The other cooling methods calculate K and C through comparisons between the cooling method and a single smooth duct. Since not the same bases are used, a correction factor Ψ was calculated and included in the comparisons. Also, the friction enhancement factor has to be corrected for an accurate comparison to other methods. The results showed that the matrix has lower heat transfer enhancement than the methods mentioned above. Cooling effectiveness The cooling effectiveness is defined in equation (4.9). T Θ = T G G T T w f (4.9) An increased cooling depth without increase of the relative cooling air mass flow is desired. Comparisons of cooling effectiveness depth with following other convective cooling methods were made. 1. a single ribbed duct 2. several connected ribbed ducts with film cooling holes 3. closed matrix 4. impingement The results showed that the matrix method had the largest relative cooling depth. This is shown in Figure 4.6, where the cooling depth as a function of relative mass flow for different cooling methods is shown. The numbers in Figure 4.6 correspond to the list above. 43

56 Matrix Cooling Figure 4.6 Comparsion of Cooling Depht [4.2]. The relative mass flow is defined in (4.10). G f g = (4.10) G g Effect on Life of Turbine Blade or Vane Since the blade is exposed to tension stresses -σ due to centrifugal forces, the hydrodynamic energy effectiveness and relative cooling depth is not sufficient to express the effectiveness of a convective cooling method. The effect of life of the blade or vane -τ must also be investigated. Equation (4.11) definesτ. n Tw Tw,0 Tw,0 τ 1 A i τ = = σ (4.11) τ 0 σ 0 The effect of tension stresses for the cooling method is calculated according to equation (4.12). σ 1 σ = = σ 0 S S T + (4.12) ϕ T S0 S0 The matrix method was compared to following convective cooling methods concerning blade life. pin fin turbulators rectangular ducts with two opposite ribbed walls dimple methods (spherical hollows) Following results were obtained. Blade or vane life increased 44

57 Matrix Cooling 3,7 to 4 times for pin turbulator method times for rib turbulated cooling 20 times and more for the whirlwind method 42 times for the matrix method For matrixes, maximum τ was received for β = 30. However, the report does not cover the effect of material and temperature on the increased blade life due to matrix cooling. Experiments were made to compare the matrix method to the impingement cooling method, with cross cylindrical pin turbulator in the leading edge. It was found that the matrix method increased the operation duration of the blade in 8 to 9 times until reaching the same radial strain for the screen insertion method. Conclusions of Nagoga [4.2] 1. Heat transfer and flow studies of the matrix base shell and side bounds were studied and correlations for heat transfer and friction were developed. 2. The intensification of heat transfer and friction in matrix channels are caused by the flow rotation induced by the turn of the flow at the side bound of the matrix. At the side bound, the cooling air overflows from one channel to the opposite in a spatial turn, see Figure 4.4. The intensity of friction, heat transfer and rotation is maximal right after the spatial turn, and then decreases with increased distance from the side bound. All three factors depend on angle β, with maximum for β = 45. The friction, heat transfer and rotation in a sub channel are not affected by the coplanar crossing flow of opposite sub channel. 3. For a scope valid for turbine blade, with β = 28.6 to and Re d = 5000 to , as an example, the heat transfer enhancement K = and friction factor enhancement C = for a closed matrix, which is better than most known cooling methods. 4. Following results were obtained for the heat transfer on the concave surface of the matrix side bounds, corresponding to the leading edge on a turbine blade. Heat transfer is increased for o reduced relative width F C of the matrix o increased curvature D o increased contraction F CD of the turn near the side bound. Heat transfer changes along perimeter of the concave surface and is maximal right after the middle point of the turn, in the flow direction. Maximum heat transfer is always received for β = 45. Heat transfer is higher than that in a smooth straight channel and higher than in serpentine-shaped channels. Heat transfer is less than that of impingement on a concave surface 45

58 Matrix Cooling 5. The hydraulic resistance in the spatial turns at the side bound - ξ m is independent of Re d, increases at the reduction of number of sub channels and is not higher than It is established that the area of the rib tops can carry load and take static and cyclical tension loads in full measure, if the angle between the force vector and ribs does not exceed 30. A matrix can handle a larger number of thermal cycles until appearance of cracks in the shells than for example a blade with pin fins in the trailing edge. This is difficult to prove theoretically, but experience at Siemens in Finspång has shown that cracks have not been a problem for matrix cooled areas. 7. Comparisons of cooling effectiveness were made between matrices and a number of other cooling methods used in turbine blades and vanes. The results showed that matrices had better performance concerning cooling effectiveness than any other method tested. 8. The effect on the blade and vane life was investigated. Numerical and practical tests showed that the matrix increased the blade life in high pressure turbines in 40 times, which was 3 to 4 times more than the ability of for example the methods pin fins and ribbed ducts. 46

59 Matrix Cooling Jurchenko & Malkov, 1995 [4.3] Three matrixes were investigated to gather information about heat transfer and hydraulic resistance. The geometry of the models is displayed in Figure 4.7. Figure 4.7 Three Matrixes Investigated by Jurchenko and Malkov [4.3]. Matrix geometries are described in Table 4.2 to 47

60 Matrix Cooling Table 4.4. There was no direct channel connection between the inlet and outlet of the matrixes. Table 4.2 Rib heights of the Three Models. model M1 model M2 model M3 h 1 [mm] h 2 [mm] Table 4.3 Geometry for test matrix. W L [mm] H/L 2 β [ ] t [mm] δ [mm] [mm] The experiments with the three models in Table 4.2 were carried out in two stages, first with clearances 1 and 2 between the matrix and the side wall, and second, tests with the same models but without clearances between the matrix and the side wall, see Figure 4.7. The values of the through flow section areas and the end clearances are displayed in 48

61 Matrix Cooling Table 4.4. In the cases without end clearances, i.e. 1= 2 =0, the through flow section area S * is equal to 32 mm 2. 49

62 Matrix Cooling Table 4.4 End Clearances and Through Flow Section Areas. model M1 model M2 model M3 1 [mm] [mm] S * [mm 2 ] The hydraulic characteristic for models with end clearances was investigated. As expected, the maximum mass flow was obtained in model M1. The mass flow of the other models were 20 % to 30 % lower than for M1, due to the smaller through flow area. Next, the hydraulic characteristic for models without end clearances was studied. Also here, model M1 had the highest mass flow. The mass flow for M1 was approximately 2 times lower than for the model with end clearance. Corresponding values for M2 and M3 were approximately 2.6 to 2.8. Zinc tests were performed to investigate the heat transfer. For information about zinc tests, see Appendix A. The highest zinc crust thickness, i.e. the highest heat flow, was obtained in the middle of the profile on model M1. That was expected since this model had the maximum mass flow. The test showed that the heat transfer was lower at the inlet of the matrix, due to that the flow in the initial channels has not yet been exposed to turning movements at the matrix side bounds. The heat flux also reduces from the inlet to the outlet of the matrix because the cooling air is heated. The enhancement of heat transfer, here labeled K was expressed through the enhancement of the Nusselt number and the effect of increased heat transfer area K A, see equations (4.13) to (4.15). K = Nu Nu 0 K A (4.13) where ( T w T ) Nu 0 = Re Pr f (4.14) α d Nu = (4.15) λ f Figure 4.8 shows the results of the heat transfer enhancement factor for the three models and a fourth model with similar angle β as the other models but with aspect ratio W/H = The rib tip contact plane displacement turned out not to affect the heat flow significantly. 50

63 Matrix Cooling Figure 4.8 Nusselt Number Enhancement for the Tests with End Clearances [4.3]. For Reynolds number larger than 6000, the difference of K between the models does not exceed 7-10 %. For Re < 6000, model M2 had the highest heat transfer enhancement. For the tests with no end clearance, the highest values of heat transfer were observed for model M1. The results can probably not be used for matrixes with 2 β < 120 and with direct channels between inlet and outlet of the matrix. It was found that for the same available pressure drop, heat transfer was higher for the matrixes with end clearances, i.e. open matrixes, than without. However, heat transfer intensification was found to be connected to an increase of cooling air mass flow. For the matrixes without end clearances, heat transfer increased for constant mass flow, but with higher hydraulic resistance compared to matrixes with end clearances. This concludes that for a case where maximum heat transfer is wanted for specified pressure loss, matrixes without channels that reach from matrix inlet to matrix outlet without reaching an end wall, must have end clearances. A positive side effect of matrixes with end clearances is increased blade strength. [4.3] 51

64 Matrix Cooling Filipov and Bregman, 2005 [4.6] Recommendations about matrix cooling are presented. Matrix should only be used for airfoils with height-to-chord ratio 2, see Figure 4.9. chord height Figure 4.9 Height and Chord of an Airfoil. For larger height-to-chord ratios, the inlet-outlet-area ratio of the matrix will be too small. This ratio should not be less than 2. Advantages of matrix cooling are increased blade strength effective blocking of internal cavities high air velocity, even for cases with low cavity blocking Draw backs with matrix cooling are lower average heat transfer enhancement factors, K , than pin fins (K 2) and ribbed ducts (K 2-2.5) fin effectiveness low near trailing edge outlet due to small slot thickness 52

65 Matrix Cooling 4.2 Correlations from Literature Correlations for matrices were investigated and compared to correlations for smooth channels. The results, analysis and conclusions of the investigation are presented in this chapter Results concerning Correlations from Literature Nagoga [4.2] is the only literature that presents matrix correlations, except for the shorter summary of [4.2] presented by Shukin and Nagoga [4.5]. However, several correlations for smooth ducts with inlet effect are available and can be used for comparisons. More articles about matrix cooling are listed in Appendix E. According to [4.2], the matrix can be divided into two sections. First, the initial section, which are the channels beginning at the matrix inlet and ending at the side wall. The flow in these channels is the same as in a smooth channel [4.2]. Second, the basic section, which are the channels beginning at the side wall and ending at a side wall or at the matrix outlet. The heat transfer enhancement factors on a blade in a Siemens Gas Turbine that is cooled by two matrixes are today based on the results of crystal and zinc tests. The crystal test resulted in knowledge of gas temperature and the temperature profile on the blade. To measure the gas temperature, thermal crystals are glued at the end of small sticks situated on the blade. For measurements of the blade temperature profile, small holes are EDMed and the thermo crystals are glued in the holes. The thermal crystals measurements result in knowledge of the highest temperature that the crystal has been exposed to during the tests. It turned out that the blade temperature profile calculated in Q3D differed from the measured temperatures. Among other things, the difference depended on an error in the K value in Q3D. After the crystal test, a zink test was performed, which resulted in that more correct enhancement factors were obtained. For information about zink tests, see Appendix A. The enhancement factors in the channels of the large matrix were calculated to 1.3. The correlations from [4.2] were applied to two test channels in a typical cooling system. Since these correlations were developed for a closed matrix, the results of the test example should be handled with care. In the correlations, the relevant input parameters are the channel length, the local coordinate, the Reynolds number, the thermal conductivity of air and the channel hydraulic diameter. The matrix in the example contains channels that reaches from the matrix inlet direct to the matrix outlet. They are here called the direct section. Nagoga [4.2] did not develop correlations for the direct section, but the flow is assumed to be similar to that in a smooth duct. Input for each test channel is displayed in Table

66 Matrix Cooling Table 4.5. Input parameter for Test Channels. Initial Section Basic Section l [mm] x [mm] 1 1 Re d [-] λ [W/m/K)] d [mm] T w [K] T f [K] The correlations of [4.2] and the results of the calculations in the example are presented below. Initial Section Equations (90) to (131) in Appendix B describe correlations from [4.2] for the initial section. The correlations from [4.2] were compared to other correlations for smooth ducts, see below. The correlation named Q3D, equation (4.25), is used in Q3D as a default value for Nu in a smooth duct. Kays and London [4.7]: Nu =.02 Re 0.8 Pr 0.3 T 0. f T (4.16) ( ) w Larsson [4.7]: 0.80 Nu = Re (4.17) Cohen et al. [4.8]: Nu = Re Pr ( T f Tw ) ( x d ) (4.18) ISBN [4.7]: x T f Nu = Re Pr ( T ) + f Tw 1 (4.19) d Tw Dittus-Boelter [4.4]: Nu = Re Pr (4.20) Sieder and Tate [4.4]: Nu =.027 Re 0.8 Pr 1 3 µ µ 0. (4.21) ( ) 14 0 d w f Gnieliski [4.4]: Nu = Re 100 Pr (4.22) ( ) 4 Nusselt [4.4]: Nu = Re 0.8 Pr 1 3 d l 0. d (4.23) ( ) 055 Petukhov [4.9]: Re Pr ( f 8) Nu = (4.24) f 8 Pr 1 ( ) ( ) 54

67 Matrix Cooling where f = ln Re ( ( ) 1.64) 2 QD3: 0.8 Nu = Re (4.25) Figure 4.10 and Figure 4.11 display the results of the comparison. For a better comparison of different Nu definitions, the heat transfer coefficient α is displayed. Equation (91) is labeled Nagoga Nu l and equation (92) Nagoga Nu d since they are both correlations for average Nu but based on different Nu definitions Nagoga, local Nagoga, Nu l Nagoga, Nu d Alfa in Initial Section Alfa Larsson 2002 Kays & London 1983 Cohen et. al 1996 ISBN Dittus-Boelter Gnielinski Nusselt Sieder and Tate Petukhov Q3D Re x 10 4 d Figure 4.10 Alfa as a Function of Re d in Initial Section, Re d = Figure 4.10 display the heat transfer coefficients from each correlation in the following order with highest first; ISBN , Nusselt, Cohen et al., Sieder and Tate, Nagoga local i.e. equation (90), then Larsson and Dittus-Boelter close togheter, Gnielinski, Q3D, Nagoga Nu d, Petukhov, Kays and London and last Nagoga Nu l. Figure 4.10 is plotted for x =1mm, where the inlet effects are significant. Equation (90) is calculated for K = 1, which is the case for the initial section [4.2]. 55

68 Matrix Cooling Alfa Alfa in Initial Section Nagoga, local Nagoga, Nu l Nagoga, Nu d Larsson 2002 Kays & London 1983 Cohen et. al 1996 ISBN Dittus-Boelter Gnielinski Nusselt Sieder and Tate Petukhov Q3D x/d Figure 4.11 Alfa as a Function of x/d in Inital Section, Re d = Basic Section Equations (97) to (108) in Appendix B display the correlations developed by Nagoga [4.2] for the basic section in a matrix. Shukin and Nagoga [4.5] summarized the work [4.2]. The correlations presented in [4.5] are rewritten but are basically the same as in [4.2], see Appendix C. The results of heat transfer and friction calculations are presented in Appendix D. The heat transfer coefficients for the initial and basic sections are compared in Figure Correlations for a smooth duct are also displayed for comparison. For clarity, not all correlations for a smooth duct earlier discussed are included. 56

69 Matrix Cooling Alfa Alfa in Basic and Intial Sections, for the Intial Test Geometry Nagoga,inital local Nagoga,initial av. Nu l Nagoga, initial av. Nu d Cohen et. al 1996 ISBN Dittus-Boelter Q3D Nagoga average, basic Nagoga local, basic x/d Figure 4.12 Alfa as a Function of x/d for Intial and Basic Sections, Re= Equation (91) is labeled Nagoga Nu l and equation (92) Nagoga Nu d since they are both correlations for average Nu but based on different Nu definitions. 57

70 Matrix Cooling Analysis of Correlations from Articles Initial Section Figure 4.10 and Figure 4.11 show that the correlations give very different results. In Figure 4.10 it is seen that equation (4.19) results in a much higher Nu than the other correlations. The explanation is that it considers inlet effect and since Figure 4.10 is taken at x = 1 mm the inlet effects significant. Figure 4.11 display the dependency of x/d, and it shows that equation (4.19) is only much higher than the other correlations near the channel inlet. Below equation (4.19) in Figure 4.10 comes the correlations of Nusselt and Cohen et al., which also display high heat transfer coefficients. The reason is that also these correlations consider the inlet effects. The Nusselt number correlation is only valid at the entrance region. It calculates an average Nu for not fully developed flow, why it results in higher values than the correlations for fully developed flow. The Cohen et al. correlation considers inlet effects and is dependent of x/d. It is seen in Figure 4.11 that it displays lower Nu for larger x/d, i.e. where there are no inlet effects. Below Nusselt and Cohen et al. are the correlations of Sieder and Tate, Larsson and Nagogas correlation for local Nu, equation (90) gathered in Figure It is seen in Figure 4.10 that equation (90) is not similar to the Larsson and Sieder and Tate for larger x/d, but is significantly lower. Sieder and Tate and Larson display relatively high heat transfer coefficients. The reason that Larsson is high is because the correlation was developed for not fully developed flow. The Sieder and Tate correlation consider the effect of the temperature difference between fluid near the hot wall and the cooler fluid in the core flow. Other correlations that also consider this effect are Kays and London, Cohen et al. and equation (4.19), which also display high values with exception of Kays and London. The other correlations are not valid for high temperature differences between fluid and wall. Dittus-Boelter, Petukov and Gnielinski do not condsider inlet effects or temperature difference effects, which explain the relatively low values. Nagoga Nu d, Petukov and Nagoga Nu l also display relatively low values. Nagoga Nu d and Nagoga Nu l consider temperature difference effects but not inlet effects. The Dittus-Boelter equation is probably developed for small temperature differences, whereas the Q3D and Nagoga correlations are based on zinc tests and thus probably valid for higher temperature differences. Basic Section Figure 4.12 display the heat transfer coefficient of the basic section compared to the initial section and other smooth duct correlations as a function of x/d for Re = It is clear that for all correlations that consider inlet effects, in Nagoga s case the correlations for local values, the trends are similar. At the channel entrance, new thin boundary layers are created which causes turbulent flow and high heat transfer coefficients. The heat transfer then decreases along the channel length, as x/d increases. When comparing the local Nagoga correlations for the initial and basic sections, the basic section has approximately 2.4 times higher heat transfer coefficient at the channel inlet. Closer to the channel end, for x/d > 15, the basic section has approximately 1.5 times higher heat transfer coefficient than the smooth section. In a comparison of the average heat transfer coefficients of Nagoga, it is seen that the 58

71 Matrix Cooling basic has approximately 2.5 times higher heat transfer coefficient than the smooth section. More plots of correlations of the basic section are displayed in Appendix D. Correlation of [4.2] and [4.5] differ only due to different round ups. Nagogas s correlations result in slightly lower values, why it is better to use, see Figure 26 to Figure 29. The local heat transfer enhancement factor for the test channel in Table 4.5 is approximately 2.52 and the average is approximately 1.87 according to Figure 28 for Nagoga s correlation. The average value of 1.87 is higher than K = 1.3, which is used today. However, the value 1.3 was chosen conservatively to avoid an overestimation of K. Open matrices have lower heat transfer than closed matrixes [4.3]. According to Figure 31 that display Nagoga s correlation, K goes from 2.6 at the inlet to 1.6 at the end of the channel. The inlet effects are thus significant. Since Nagoga s correlations were developed for closed matrixes, they are expected to result in higher values than what are used for open matrices. The friction is increased by a factor of 3.4 according to [4.2] or 3.5 according to [4.5], see Figure 27. According to Figure 29, the friction factor in the basic section test channel is approximately and in a corresponding smooth channel f would be approximately

72 Matrix Cooling Conclusions concerning Correlations from Articles Matrix cooling is a cooling method that is relatively unknown in west. The only correlations found in the literature study are those form [4.2], which are valid for closed matrixes. Flow in initial matrix channels can be approximated to flow in smooth ducts. Several correlations for flow in smooth ducts are available. Since the channels are relatively short, correlations that consider inlet effects are most suitable. In Siemens Gas Turbines, open matrix cooling is used. The enhancement factors used today are estimated based on tests, since there are no correlations available for open matrix cooling. Further work is needed to develop reliable correlations for open matrices. 60

73 Matrix Cooling 4.3 Rib Effectiveness in Q3D Since a matrix consists of longitudinal ribs, it was of interest to investigate box Rib Effectiveness in Q3D. It is reached by following pathway. H_Hydro Toolbar, Utilities Rib Effectiveness The interface with input and output parameters are displayed in Figure Figure 4.13 Rib effectiveness interface in Q3D. The Q3D code was studied in order to evaluate the rib effectiveness calculations. Comparisons were made with correlations from open literature Results concerning Rib Effectiveness in Q3D From the input parameters, a rib geometry according to Figure 4.3 is defined. The channel area and the channel perimeter are calculated by a numerical method, which is difficult to relate to a physical geometry. As a comment in the code, an easier calculation of channel area and perimeter is noted. This may be interpreted as a hint of what area and perimeter the other code refers to. The shape factor -K f is defined in the code is displayed in equation (4.26). L1 K f = (4.26) t K f is a measurement of the heat transfer surface enhancement, where L 1 is the length of the dotted line in Figure

74 Matrix Cooling L 1 h t s Figure 4.14 Q3D Rib Geometry. The rib effectiveness -ε Q3D is defined according to equations (4.27) and (4.28) in Q3D. ( mu h) ε Q D = tanh 3 (4.27) mu h 2 α mu = (4.28) λ s Analysis of Rib Effectiveness in Q3D Equation (4.27) and (4.28) correspond to the definition of rib efficiency for ribs with isolated tops, see equations (2.6) and (2.9). In Figure 4.13, the term Rib Effectiveness has been mixed up with Rib Efficiency. Q3D assumes that the rib tops are isolated. In a matrix, approximately half the rib top area is in contact with opposing rib top and can be approximated to be isolated. The other half is exposed to convective cooling. Therfore, it would be better to implement at formula where X percentage of the efficiency is calculated for convective rib tops cooling and the rest, 1-X, for isolated rib tops according to equation (4.29). ( M h ) 1 X tanh( M h) X tanh c η = + (4.29) 100 M h 100 M h c Conclusions concerning Q3D Rib Effectiveness The implementation in Q3D calculated the rib efficiency for isolated rib tops. Since approximately half of the rib tops in a matrix is exposed to convective cooling, it would be better to implement a formula according to equation (4.29), which considers both the isolated and convective rib tops. The phrase Rib Effectiveness in the Q3D interface should be replaced by Rib Efficiency. 62

75 Matrix Cooling 4.4 References Matrix Cooling [4.1] Bunker, R. S. Latticework (Vortex) Cooling Effectiveness Part 1: Stationary Channel Experiments, Proceedings of ASME Turbo Expo 2004, Power for Land, Sea and Air, June 14-17, 2004, Vienna, Austria [4.2] Nagoga, G. Intensification of the Heat Transfer in the Cooling Ducts of the Gas Turbine Blade, Finspång HTC Database, Folder No 1, Alstom,Russia, (2000) [4.3] Jurchenko, V. and Malkov, V. Technical Report on Results of the Work: Experimental Investigation of Hydraulic Resistance and Heat Exchange in Vortex Matrices, Formed by Parallel Ribs and Channels with Assymetric Location of Ribs Tips Contact:,(1995) [4.4] Holman, J. P. (2002). Heat Transfer Ninth Edition, McGraw-Hill Companies, New York (ISBN ) [4.5] Shukin, S. and Nagoga, G. Comparative Efficiency Analysis of the Known Schemes of Heat Transfer Intensification in the Cooling Channels of Gas Turbine Blades, The Program of Further Development and Proposal on Rational Cooling Scheme of Modern Gas Turbine Blades, Alsom, (2001) [4.6] Filipov, V. and Bregman, V. SGT -800 Gas Turbine Criteria of the Matrix Cooling Application, Ref.: /TR045 Rev. 1, Moscow 2005/09/30 [4.7] Larsson, T. Investigation of the Effect of Different Rib Configurations on Heat Tranfer in the GT10C Blade 1 Leading Edge, RTT10_10/02 (2002) [4.8] Cohen, H., Rogers, G. F. C. and Saravanamuttoo (1987). Gas Turbine Theory, 3 rd Edition, Longman Scientific & Technical, Harlow (ISBN X) [4.9] Çengel, Y. A. and Turner, R. H. (2001). Fundamentals of Thermal-Fluid Sciences, McGraw-Hill Companies, New York (ISBN ) 63

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77 Impingement Cooling 5 Impingement Cooling The concepts of impingement cooling and a summary of the literature study are presented below. The chapter also contains the results, analysis and conclusions of the investigation of correlations and the Q3D code. 5.1 Theory Impingement cooling is described below, and a summary of the literature study is presented What is Impingement Cooling? Figure 5.1 gives an example of impingement cooling. High pressure air flows through holes in a perforated plate. When the jets formed by the holes hit the surface, the surface is cooled. Inline pattern Staggered pattern Figure 5.1 Impingement Cooling Setup. Cross Flow The cooled surface is called target plate and the perforated plate is called jet plate. Different impingement cooling systems exist, for example with variations in the number of jets, target plate configurations, jet hole configuration and jet angle of attack. Figure 5.1 display an inline array of jets impinging on a flat surface. Figure 5.2 gives example of impingement on a curved target plate, which for example is the case for cooling of the leading edge on a vane or blade. Figure 5.2 Curved Target Plate[5.1]. 65

78 Impingement Cooling Where the jets hit the target plate, the flow is highly turbulent and the boundary layer very thin, which leads to a very high heat transfer coefficient. Impingement is mainly used where there is need for very high heat transfer coefficients. Best effect is achieved for geometries with plenty of mass flow available but stricter limitations of allowable pressure drop. The impingement holes often situated on inserts which are not load-bearing, see Figure 5.3. It is also used on thicker parts of components such as the leading edge and mid chord regions. Platforms are also often cooled by impingement. Figure 5.3 Impingement Cooling in Turbine Vane [5.2]. The impingement geometry is described by the hole diameter, the streamwise and spanwise jet-to-jet distances and the distance between jet orifice and target plate, according to Figure 5.1. For a curved target plate, such as the leading edge, the target plate curvature also has to be considered. For heat transfer calculations, these parameters are often made dimensionless by division of the hole diameter in order to generalize the correlations. The aerodynamics of a single impinging jet is described in Figure 5.4. The flow is divided into three regions, the free jet region, the stagnation region and the wall jet region. Figure 5.4 Impinging Jet (Left, [5.3]) and Free Jet (Rigth) [5.4]. Before striking the target plate, the impingement jet acts as a free jet. The flow of a free jet is divided into the potential core zone, the developing zone and the fully developed zone, see Figure 5.4. When the free jet leaves the impingement hole, the 66

79 Impingement Cooling outer part of the jet is mixed with the surrounding air and causes it to mix with the jet. This increases the jet diameter and the turbulence of the flow. The part of the jet that is not affected by the surrounding air is called the potential core. The velocity in the potential core is constant and equal to the exit velocity of the jet. The influence of the surrounding air increases as the distance from the impingement hole increases, and finally the potential core is non-existent.[5.5] In the developing zone, the axial velocity profile changes toward the fully developed profile which is achieved in the fully developed zone. The free jet region is followed by the stagnation region, which is located where the jet impinges on the target plate. It is surrounded by developing boundary layers. After impingement, the spent jet causes a highly turbulent flow which increases the heat transfer. In theory, no heat transfer can occur in the stagnation point as the velocity is zero, but in reality the stagnation point is very unstable and moves all the time, so the heat transfer coefficient in the stagnation region is very high due to the thin laminar boundary layer [5.6]. The heat transfer coefficient decreases with increased distance from the stagnation point. However, when the wall jet changes from laminar to turbulent flow, a second heat transfer maxima can occur for certain Reynolds number and z/d distances. In the wall jet region, the flow is parallel to the impingement plate, and its velocity profile is described in Figure 5.4. The wall jet increases in thickness due to a build up of the boundary layer.[5.2] The impingement cooling system has two chambers, the jet creating or plenum chamber and the target chamber. The jet creating chamber has higher pressure than the target chamber. In turbine cooling, multiple jets are often used, either as a row of jets or as arrays of jets. This complicates the flow structure. Nearby jets affect each other, and the upstream jets creates a cross flow, which flow perpendicular to the jet flow along the target plate, see Figure 5.1. [5.6] 67

80 Impingement Cooling Summary of Literature Survey Flat Target Plate Reynolds number -Re: The discharge coefficienct C d is vertically independent of Re d [5.7]. A weak dependence of Re exist depending of whether the hole has sharp edges or not [5.8]. Reynolds number based on the jet hole diameter gives a sufficiently good heat transfer correlation [5.9], and is what is used by most investigators. All articles investigated showed that the heat transfer increases with Re. According to Höglund [5.8], this trend is stronger for higher Re. Most investigators found that the heat transfer dependence of Re can be approximated by a power law [5.10], [5.11] and[5.7] to mention a few, but Kercher and Tabakoff, [5.12] disagree. They claim that heat transfer for a square inline array cannot be correlated by power functions equations of dimensionless parameters, and that the Reynolds number exponent is strongly dependent of x n /d. The heat transfer behavior changes distinctly at Reynolds number near 3000 [5.12]. Crossflow ratio-g c /G j : The cross flow decreases the impact from jets on the target plate surface and should be minimized to achieve maximum heat transfer [5.8], [5.10], [5.12], [5.13]. The Nusselt number of the last impingement row in an array is not significantly affected by the cross flow according to [5.13] and [5.7]. [5.13] investigated arrays with 6 to 12 rows and [5.7] studied arrays with 10 rows. For an inline, dense array of jets, Baileys and Bunker [5.14] found that G c /G j increased linerarly up to approximately row 6, see Figure 5.5. Figure 5.5 Re and G c /G j as a Function of Row Number for x n /d=y n /d =3, z/d=2.75 and Re d =29900 and [5.14]. Between row 6 and 8, G c /G j peaked due to a flow adjustment from impingement dominating flow to channel like flow, and thereby much increased convective cooling. A probable reason for the flow adjustment is the inline hole pattern. At the cross flow peak, Re decreases. After row 8, G c /G j increases again and the Reynolds number increases and peaks at the last row. This resulted in the highest Nu at the exit row and a Nu minimum in the middle rows. The tendency was most obvious for higher Reynolds number. A similar trend is reported by [5.8], for a staggered array. He found that for low G c /G j, heat transfer monotonically decreases for increased crossflow. Still, if the crossflow velocity is high, heat transfer will increase due to convection and Nu will rise at the end of the target plate, creating a heat transfer minima in the midsection of the array. 68

81 Impingement Cooling Cross flow changes the impingement location of the jets downstream, because it deflects the jets, and thus increase Nu downstream of the impingement location, [5.8],[5.10] and decreases the heat transfer upstream [5.10]. Florshuetz et al. [5.7] found that for small x n /d and y n /d and large z/d, Nu/Nu 1 decreases for increased G c /G j. However, for larger hole spacing and smaller z/d, Nu/Nu 1 increases slowly due to convective cooling effects of the crossflow, which does not interfere much with the jets for this kind of geometry. [5.7] also concluded that jets at the final rows are not much affected by the crossflow. Three complexities of the crossflow phenomena are identified in [5.7]. First that G c /G j used in calculations is average over the channel, instead of being lowest at the first row and then increasing downstream. Second, the correlations are valid for different x n /d and y n /d, with aspect ratios from to Geometries with small aspect ratios are not affected by z/d but geometries with large aspect ratios are much affected by z/d. Third, some jets impinge the target surface before they are developed, and some after. [5.13] investigated the effect of minimum, intermediate and maximum crossflow. They found that for a given massflow, the more densely placed holes are more sensitive to the effect of decreased heat transfer with increased crossflow. The tendency is more obvious for larger z/d, since it is easier for the cross flow to deflect a long and narrow jet. Jet-to-Impingement Target Spacing - z/d [5.7] found that heat transfer was not much dependent on z/d, except for the highest hole density tested, x n /d = 5 and y n /d = 4, where the average heat transfer decreased with increased z/d for a given cross flow ratio. The trend of z/d was stronger for higher x n /d and y n /d. The z/d range was 1 to 3 for the tests. The same trend was reported by [5.9], which tested z/d from 2 to 8. [5.15] found that the average Nu increases very slightly with increased z/d up to 4, where it reaches a maximum. For z/d > 4,the average Nu decreases with increased z/d. [5.11] found by analytical calculations that the optimal jet-to-target plate spacing is [5.12] found that without crossflow, heat transfer increases with increased z/d but with crossflow, heat transfer decreases with increased z/d. [5.8] reported that for small z/d, the average Nu is not much affected. The local Nu decreases at smaller z/d, but a second peak arises which makes the average Nu approximately the same. For large z/d, heat transfer decreases as z/d increases according to [5.8]. Jet-to-Jet Spacing x n /d and y n /d: [5.9] reported that very large jet-to-jet spacings, i.e. small open area -A f, results in a small average heat transfer. Up to A f = heat transfer increases rapidly with A f. For A f > 0.012, heat transfer still increases, but not as fast. Chance concludes that A f should be at least to The investigations of [5.13], [5.15], [5.12], [5.16] and [5.8] also concluded that heat transfer increases with smaller jet-to-jet spacings. [5.11] reports a calculated optimum open area of [5.8] reported that closely spaced jets results in that more of the cross flow channel is blocked by jets, which intensifies the interaction between jets and cross flow. Hole Pattern: [5.7] found that for large hole spacing, there is not much difference in heat transfer whether the array has inline or staggered hole pattern. For a decreased hole spacing and increased z/d, the staggered array showed decreased heat transfer coefficient 69

82 Impingement Cooling compared to the inline array. The reason was differences in the spanwise distribution of cross flow between the inline and staggered array. For the inline array, the cross flow becomes channeled between the rows of jets, which reduces the negative cross flow effect. [5.9] investigated the effect of hole pattern, and found that the difference in heat transfer between staggered and inline arrays was almost insignificant. Curved Target Plate On the stagnation line, Nu decreases with increased z/d, with stronger trend for smaller y n /d. For the average Nu, the same trend was observed, but with less z/d dependency. Nu x /Nu x=0 depends mostly on x n /d and showed only a weak dependency of z/d, especially for z/d >>1. x indicates the distance from the stagnation line. [5.1] For z/d = 2-8 correlations for average Nu should not display a dependency of z/d. A second heat transfer maxima was observed at 45 angular distance from the stagnation line. This maximum is strongly dependent on Re and occurs for relatively small z/d. At the stagnation point, the heat transfer and flow for a curved target plate is similar to that for a flat target plate. [5.17] Tests showed that z/d = 1 resulted in a heat transfer no less than 95 % of the maximum heat transfer obtained for other z/d tested. It was also found that a line of circular holes resulted in better heat transfer performance than slot jets. Smaller value of l/b led to increased average heat transfer. [5.18] Tests were done to investigate the effect of an elongated leading edge. For a semicircular leading edge, maximum heat transfer was obtained for z/d = 1 for circular jets. For a given mass flow, the average heat transfer increased monotonically as l/b increased. It was also found that slot jets and circular jets behaved similarly. [5.19] An investigation showed that the average Nu is proportional to Re 0.75 and inversely proportional to z/d with exponent The curvature of the target plate effects the heat transfer for the hole curvature surface, i.e. ± π 2.[5.20] 70

83 Impingement Cooling 5.2 Flat Target Plate The results, analysis and conclusions of the investiation of impingement on a flat target plate are presented in this chapter Results concerning Flat Target Plate Following correlations were investigated. Florschuetz, Truman and Metzger 1981 [5.7] Metzger and Korstad 1972 [5.10] Bailey and Bunker 2002 [5.14] Martin 1977 [5.11] Florschuetz and Isoda 1982 [5.21] Obot and Trabold 1987 [5.13] Behbahani and Goldstein 1983 [5.15] Höglund 1999 [5.8] Woldersdorf [5.23] Chance 1974 [5.9] Goldstein and Seol 1990 [5.16] The correlations and their ranges are summarized in Appendix A. The parameter Nu represents the average Nusselt number over an impingement row. More articles about impingement cooling are listed in Appendix J. The article of Woldersdorf has not been found, because it is an intern ABB report. All articles contained correlations for the Nusselt number, except for that of Florschuetz and Isoda that contained correlations for the friction factor and discharge coefficient. Höglund also developed correlations for the friction factor and discharge coefficient. A test case called Case B, that covers most ranges of the correlations, was used to compare the correlations for a flat target plate, see Table 5.1. Inline and staggered array correlations are compared separately. Nusselt number, discharge coefficient and friction factor were calculated for the fifth row in an array. The cross flow ratio could be estimated to 0.2 for this row [5.8]. Table 5.1 Case B, Geometry used for Flat Target Plate Comparisons. z/d [-] 2 y n /d [-] 4.5 x n /d [-] 5 N p 5 n c 10 T w [K] 1073 T j [K] 700 T c [K] T j C d [-] 0.79 G c /G j [-] 0.2 L [mm] 0.4 Results for Case B are presented for inline arrays and staggered arrays in Appendix B. 71

84 Impingement Cooling Calculations were also performed on the airfoil of a typical vane 1 cooling scheme, a single row upstream and two rows with inline pattern downstream, both on the pressure side. Results from the calculations are presented in Appendix C. Three tests with random geometry were also performed in order to investigate the correlations behavior further, see Appendix D. In Q3D, the correlation of Florschuetz et al [5.7] is used for heat transfer calculations on flat target plates. The Nusselt number correlation is identical with that of [5.7] and the mass flow calculation is also based on the same article. The massflow in Q3D is calculated according to equation (5.1). G j & j = GSR Arow (5.1) G j m The ratio G j G j is calculated according to equation (5.2) from Florschuetz et al. G G j j where x = x β = β N p cosh( β x xn ) = sinh( β N ) n ( i 0.5) Cd 2 ( π 4) ( y d ) ( x d ) n n c (5.2) and i = 1, 2, 3,, N p (5.3) The factor GSR is calculated according to equation (5.4), where G j in the numerator is calculated according to equation (5.5) and G G in the denominator according to equation (5.2). Cd G j GSR = (5.4) G G j j j j 2 2 p0 p2 G j = (5.5) R T This means that GSR = C d G, so that equation (5.1) actually is represented by equation (5.6). j row d j m& = G A C (5.6) For detailed information about the flow model, see [5.7]. The flow model described above is valid for incompressible flow. The streamwise pressure gradient is assumed to be due to the acceleration caused by the impining jets, and the wall shear is neglected. The pressure gradient can then be described according to equation (5.7) according to [5.7]. G c dg dp = 2 c (5.7) ρ 72

85 Impingement Cooling The impingement cooling flow in turbine components is compressible. Therefore, the incompressible model described above was compared to calculations for compressible flow, according to equations (5.8) and (5.9). The same expression for the pressure loss was used, i.e. equation (5.10). Ψ = 2 κ p κ 1 p κ p p 2 0 κ + 1 κ (5.8) p0 m& = A row Cd Ψ (5.9) r T The massflow for a geometry and flow case according to Table 5.2 was investigated. Table 5.2 Test Case for Q3D Mass Flow Comparsion. y n /d z/d T N p n c p 0 p 2 C d * * It turned out that the two calculatation came to somewhat different results, see Table 5.3. Table 5.3 Results of Mass Flow Comparison. Row m&, Q3D (Compressible) [g/s] m&, Incompressible Flow [g/s] Figure Figure 35 in Appendix B describe the C d obtained by using Höglund s and Florschuetz and Isoda s correlations for Case B. They give quite different results. An explanation could be that C d depends on a variety of parameters, such as how sharp the hole edge is, cross flow, pressure difference and manufacturing method. Effect of Different Parameters Appendix E contains the results of variations of parameters x n /d, y n /d, z/d, G c /G j and T c. The base geometry is called Case D, see Table

86 Impingement Cooling Table 5.4 Case D, Base Geometry for Effect of Parameters. z/d 2 x n /d 5 y n /d 4 N p 15 T w 1073 T j 700 T c 700 G c /G j 0.7 Hole Pattern Inline Analysis of Flat Target Plate The agreement between correlations, considered parameters and the validity of the ranges are analyzed. Appendix F contains a summary of the considered parameters and the validity of ranges. Inline Arrays Agreement between Correlations Figure 32 in Appendix B presents three correlations by Florschuetz, Truman and Metzger from A number of random tests were performed which are presented in Appendix D. The lowest curve is the base correlation which is used throughout this report. The correlation of the middle curved is a simplified version of the base correlation. The dotted curve is for the first line in an array with no crossflow present, subsequently it results in a higher Nu. Since the simplified correlation results in a higher Nu, it is safer to use the base correlation. Results from comparison of correlations are presented in Appendix B, C and D. The relative positions of the inline correlations are not the same for different cases, which complicates the choice of correlation. Still, some trends exist. Obot and Trabold and Goldstein and Seol often show the highest Nu. The latter is developed without concern of cross flow. The deviating behavior of Obot and Trabold could be explained by that it is slightly out of range for x n /d and y n /d. Metzger and Korstad always result in a relatively low Nu. The behavior of Wolfersdorf correlation is very inconsistent and, since the article is not available, difficult to analyze. Florschuetz et al., Martin, Chance and Bailey and Bunker often result in relatively similar curves, but the relative positions vary with the geometry. The conclusion is that the inline correlations of Florshuetz et al., Martin, Chance and Bailey and Bunker are those most similar to each other. For design of inline arrays, these correlations should be investigated and analyzed for the flow case and geometry of interest. Parameters Considered The considered parameters for each correlation are presented in Appendix F. Parameter T diff corresponds to the effect of the temperature difference between the cross flow and the impinging jets. All correlations consider the distance between target and jet plate -z/d and none considers the parameter T diff. The correlations of Martin and of Chance were developed for square arrays and therefore only consider one of y n /d and x n /d. Neither Wolfersdorf nor Bailey and Bunker consider the effect of y n /d. Obot and Trabold and Florschuets et al. are the ones that considers most parameters. 74

87 Impingement Cooling Validity Range The parameter ranges of the investigated correlations together with ranges relevant for gas turbines are displayed in Appendix A. The ranges for gas turbines are taken from impingement systems on vane 1 on SGT 600 and SGT -700 and blade 1 and vane 1 and 2 on SGT Appendix F contains a summary of whether the correlation ranges matches those relevant for turbine blades and vanes. Partly indicates that the correlation ranges match for approximately 50 % or more. Hardly indicates that the correlation ranges match for less than 50 %. The ranges of Chance and Obot and Trabold are not very suitable for turbine conditions. The Bailey and Bunker and the Florschuetz correlation matches the ranges of interest well, whereas the Martin and Wolfersdorf ranges matches the turbine ranges mediocrly. Staggered Arrays Agreement between Correlations Figure Figure 34 in Appendix B and Figure 38 in Appendix C present Nu for staggered arrays. The random tests in Appenix B also display results for a staggered array. The relative positions of the correlations vary from case to case. Trends observed are that Metzger and Korstad gives the lowest value for all cases tested. Goldstein and Seol often results in a relatively high value, but not always. This correlation was developed for Nu average over the hole array area, and thus deviates from the other correlations which are average over an impingement row. Martin s correlation agrees well with that of Behbahani and Goldstein. The results for Florschuetz, Chance, Wolfersdorf, Behbahani and Goldstein and Martin vary from case to case. Therefore it is difficult to make a recommendation of the use of a specific correlation based on agreement between them. Parameters Considered Höglund s correlation is the only one that considers T diff. Florschuet s correlation is the one that considers the most parameters. Martin s correlation is for square arrays, as mentioned earlier. Bebhahani and Goldstein and Wolfersdorf consider the same amount of parameters. Validity Range Florschuetz s and Höglund s correlations match the turbine ranges the best. Martin, Wolfersdorf and Bebhahani and Goldstein s ranges don t coincide with the turbines ranges very well, see Appendix F. Effect of hole spacing - x n /d and y n /d Chapter states that larger hole spacing decreases the average heat transfer. Figures Figure 49 and Figure 50 in Appendix E supports that theory. Figure Figure 49 describes the effect of x n /d. All correlations consider the effect of x n /d and display the same basic tendency, with decreased Nu for increased x n /d. Obot and Trabold deviates from the other with a significantly stronger dependence of x n /d. It should be noted that this correlation is slightly out of its validity range for open area ratio. The second strongest dependency is displayed by Baileys and Bunker. The flattest curve is that of Höglund. Behbahani and Goldstein coincides almost exactly with Martin. Woldersdorf, Chance, Höglund and Florshuetz have almost the same inclinations, but Woldersdorf display significantly lower Nu values. Largest agreement between 75

88 Impingement Cooling correlations is displayed by Höglund, Chance and Florschuetz. Therefore, the influence of x n /d is probably best described by these correlations. For effect of y n /d, Obot and Trabold deviates from the general trend again, with much stronger dependence. Goldstein and Seol, Florschuets et al., Metzger and Korstad and Chance display a weak dependence of y n /d. Behbahani and Goldstein, Martin and Wolfersdorf is independent of y n /d. In Höglunds correlation, y n /d is connected to x n /d so that yn = 2 3 xn for all tested plates. This leads to the conclusion that the spanwise jet-to-jet spacing is not of significant importance for the average Nusselt number. Effect of jet target-to-impingement target spacing - z/d Cross flow mass velocity G c was implemented as inversely proportional to z/d to consider the effect of increased cross-section of the crossflow channel. Kercher and Tabakoff [5.12] reported that for no crossflow, heat transfer increased with increased z/d but with crossflow heat transfer decreased with increased z/d. A similar trend is displayed in Figure 47 by Florshuetz et al.. Additional to the Nusselt number calculated from equation (45), Florschuetz et al. developed equation (47) for a row without crossflow. The correlation for no crossflow increases with increased z/d and the equation that considers crossflow decreases with increased crossflow. Tests were also done with lower G c /G j ratios and for those cases, the base equation presented similar behavior as that with no crossflow. The effect of increased Nu for increased z/d could be based on less jet-to-jet interference with larger z/d. A decreased Nu for larger z/d is probably due to decreased impact of the impinging jets. Höglund, Goldstein and Seol and Florshuetz et al. display a similar dependency of z/d. Florshuetz et al reported the highest dependency for x n /d = 5 and y n /d = 4, which is the case tested here. For denser spaced holes, the dependence should be less [5.7]. Behbhani and Goldstein differs form the other correlations in that Nu increases slightly with increased z/d up to z/d = 4. Baileys and Bunker, Martin, Obot and Trabold and Goldstein and Seol display a relatively high Nu dependency of z/d. Chance display a medium dependency of z/d. The z/d dependency without cross flow was also investigated, see Figure 48. The same pattern as above was displayed. Investigators have come to quite different conclusions concering the effect of jet plate-to-target plate spacing. The study of correlations led to the conclusion that for small z/d, the effect of jet plate-to-target plate spacing is low. For larger z/d, the heat transfer decrease with increased z/d, as seen in Figure 47 in Appendix E. Effect of aspect ratio - x n /y n Figure Figure 46 in Appendix E display the effect of x n /y n. The only correlation that tested different aspect ratios is that of Florschuetz et al, which show that increased aspect ratio leads to decreased heat transfer. The tendency is stronger for smaller aspect ratios. Chance did not report the range of aspect ratios used, why that correlation is not compared. Effect of Cross Flow Ratio G c /G j Martin, Behbahani and Goldstein and Goldstein and Seol don t considerate the effect of cross flow. Obot and Trabold investigated three different cross flow sizes, which is shown in the correlation by a parameter n obtained from Figure 31 in Appendix A. The effect is not seen by varying G c /G j. Metzger and Korstad consider cross flow, but 76

89 Impingement Cooling shows almost no effect on Nu. Höglund, Florschuetz et al., Chance and Baileys and Bunker show a relatively similar dependence of G c /G j, with strongest dependency for the two latter. Wolfersdorf deviates from the former mentioned group for G c /G j > 0.4 and is probably not valid for large cross flow since that results in negative Nu. For effect of G c /G j, Höglund and Florschuetz et al. are primarily recommended to use. 77

90 Impingement Cooling Conclusions concerning Flat Target Plate The investigation of impingement on flat and curvet target plates are summarized in Appendix F. After considering the agreement between correlations, considered parameters, validity of range and effect of different parameters, following correlations are recommended for impingement on flat target plates. Staggered Array: Höglund [5.8] or Florschuetz et al. [5.7] Inline Array: Florshuetz et al. Q3D : Q3D calculates Nu and m& according to [5.7] It is not obvious which one of Höglund or Florschuetz correlation that is most reliable. The advantage of Höglund equation is that the experimental study was performed at Siemens in Finspång and aimed for components of Siemens gas turbines. Since the tests were performed in Finspång, it is possible to look directly at the test data to locate the nearest measured value, instead of only using a curve fit of the measured values. The advantage of Florschuetz s correlation is that Florschuetz and his co-workers are experienced researchers that have performed a large number of investigations in this area. Höglund s report is a diploma work, and he has not presented any other material on the subject. For heat transfer calculations on turbine component, it is advised to compare the correlations and decide from case to case which one is more reliable, depending on range and geometry. 78

91 Impingement Cooling 5.3 Curved Target Plate The results, analysis and conclusions of the investigation of impingement on curved target plates are reported below Results concerning Curved Target Plate Following correlations were investigated. Chupp, Helmst, McFadden and Brown 1969 [5.1] Hrycak 1980 [5.17] Metzger, Yamashita and Jenkins 1969 [5.18] Nagoga 2000 [5.20] Kopelev 1988 [5.22] Metzger, Baltzer and Jenkins 1972 [5.19] The correlations and ranges are displayed in Appendix A. More articles about impingement cooling are listed in Appendix J. The correlations are developed for a single row of impingement. The literature about Kopelev s correlation is only available in Russian, why the correlation has not been genuinely analyzed. The range of Kopelev s correlation has not been found. No correlations for impingement arrays for curved target surfaces were found. A geometry called Case C was used for testing correlations for impingement on curved target plates, see Table 5.5. Table 5.5 Case C, Geomtery for Comparison of Curved Target Plate Correlations. D p /d [-] z/d [-] y n /d [-] T w [K] T j [K] TE outlet area [mm 2 ] N p [-] The correlations were tested with Reynolds number based on the width b of a so called equivalent slot. The equivalent slot is an imagined slot with the same area as the total area of all holes. The length L es and width b of the equivalent slot are defined in equation (5.10) and (5.11). L es = N d (5.10) p π d b = 4 2 N L p es (5.11) 79

92 Impingement Cooling Figure 5.6 display the result for a comparison of the heat transfer coefficient for Case C. 2.5 x Chupp et al Hrycak Q3D Metzger et al Nagoga Kopelev 1988 Metzger et al Alfa as a Function of Re b Alfa Re b x 10 4 Figure 5.6 Alfa for Case C. The leading edge of vane 2 on SGT -800 is cooled by three impingement rows. Heat transfer calculations were performed for the middle row and the results are presented in Appendix G. The heat transfer coefficient used in the in-house cooling design program 3Dhtcp is approximately 2200 W/m 2 /K, which can be compared to that in Figure 53 of 5280 W/m 2 /K. The results are displayed in Appendix G. In Q3D, Nu for impingement on curved target plates are calculated by use of Chupp et al.. An investigation of the Q3D code showed that for a curved target plate, the code agreed completely with the base correlation, as seen in Figure

93 Impingement Cooling Q3D Comparison for Curved Target Plate Chupp et al Q3D Nu Re d Figure 5.7 Q3D Comparison for Case C. Since the correlations varied much, three random tests were also performed to investigate the tendencies, see Appendix H. The effect of different parameters was investigated and the results are displayed in Appendix I. 81

94 Impingement Cooling Analysis of Curved Target Plate Agreement between Correlations Since the Nusselt number is based on different characteristic lengths, the correlations are best compared by the heat transfer coefficientα. Results for the tested geometries in Appendix G and H show that the relative positions of the correlations remains approximately fixed. The values of α and the dependency of Re differs significantly between the correlations. The correlations of the two articles of Metzger et al [5.18] and [5.19] give very similar results and the highest α values. The correlations of Hrycak and Kopelev also result in similar values of the heat transfer coefficient, which are lower than that of Metzger et al s. Lowest α is obtained by the correlations of Chupp et al. and Nagoga, which results in very similar values. Since many correlations result in differentα, it is hard to choose a correlation. However, both Nagoga and Chupp et al. came to similar results. So did the two investigations by Metzger et al., but those correlations results in much higher α than Chupp et al. s and Nagoga s. It is better to estimate a too low value than a too high. Therefore, Nagoga s or Chupp et al s correlations are of interest. However, Chupp et al. s correlation is not valid for the range that is important for turbine cooling. Therefore, the Nagoga correlation is probably better to use. Parameters Considered All correlations consider D p /d and y n /d. Only Nagoga consider T diff. Only Chupp et al. and Nagoga consider the effect of z/d. Nagoga s correlations is the one that consider the most parameters, which promotes the use of Nagoga over the other in this aspect. Validity Range The range of Nagoga s correlation matches that of turbine components the best. Effect of Different Parameters Appendix I display the effect of different parameters. All tested correlations show approximately the same dependence of the hole spacing. Metzger, Baltzer and Jenkin s correlation deviates a little from the the other however. The strange behavior of the correlation is probably due to the form it is presented in, see Appendix A. Only Nagoga and Chupp et al. display a dependency of z/d within valid z/d range. Nu decreases slightly with increased z/d for both Nagoga and Chupp et al, with a little stronger dependency for Chupp et al.. Valid range for Kopelev s correlation is not known, but it displays the same trend as Nagoga s. Hrycak, Chupp et al. and Nagoga display approximately the same dependency of target plate curvature within valid ranges. Nu decreases with increased D p /d for all three correlations, except for small curvatures where Nu increases with increased D p according to the correlation of Chupp et al.. 82

95 Impingement Cooling Conclusions concerning Curved Target Plate The investigated correlations give very different results. However, taking into account ranges, parameters considered and the agreement between correlations, the Nagoga correlation [5.20] is recommended to use. If possible, it is also recommended to consult the other correlations when calculating Nusselt number for a specific case. The Q3D code agrees with the correlation it is based on, i.e. that of Chupp et al. [5.1]. 83

96 Impingement Cooling 5.4 References Impingement Cooling Literature [5.1] Chupp, R. E., Helmst, H. E., McFadden, P. W. and Brown, T. R. Evaluation of Internal Heat-Transfer Coefficients for Impingement-Cooled Turbine Airfoils, Journal of Aircraft, 6, (1969), [5.2] Nashar, Impingment Heat Transfer and Flow Characteristic in Gas Turbine Blade Cooling, ABB Technical Summary (1992) [5.3] Viskanta, R. Heat Transfer to Impinging Isothermal Gas and Flame Jets, Experimental Thermal and Fluid Science, 6, (1993), [5.4] Han, J.C., Dutta, S., & Ekkad S.V. (2000). Gas Turbine Heat Transfer and Cooling Technology, Taylor & Francis, New York (ISBN X) [5.5] Strand, T. Strömning och värmeöverföring i luftstrålar anströmmande vinkelrätt mot en väggyta, Chalmers Tekniska Högskola, Institutionen för Tillämpad Termodynamik och Strömningslära, (1972) [5.6] Han, J.C., Dutta, S., & Ekkad S.V. (2000). Gas Turbine Heat Transfer and Cooling Technology, Taylor & Francis, New York (ISBN X) [5.7] Florschuetz, L.W., Truman C. R. and Metzger D.E. Streamwise Flow and Heat Transfer Distributions for Jet Array Impingement with Crossflow, Journal of Heat Transfer, 103, (1981), [5.8] Höglund, H. Experimental Investigation of Impingement Cooling Under a Staggered Array of Circular Jets, Thesis Work at the Department of Energy Tehnology Royal Institute of Technologi, KTH (1999) [5.9] Chance, J. L. Experimental Investigation of air Impingement Heat Transfer Under an Array of Round Jets, Tappi, 57, (1974), [5.10] Metzger, D.E. and Korstad, R. J. Effects of Crossflow on Impingement Heat Transfer, Journal of Engineering for Power, January (1972), [5.11] Martin, H. Heat and Mass Transfer between Impinging Gas Jets and Solid Surfaces, Advances in Heat Transfer, Academic Press, 13, 1-60 (1977) [5.12] Kercher, D. M. and Tabakoff, W. Heat Transfer by a Square Array of Round Air Jets Impinging Perpendicular to a Flat Surface Including the Effect of Spent Air, ASME Journal of Engineering for Power, 73-82, (1970) [5.13] Obot, N.T. and Trabold, T.A. Impingement Heat Transfer within Arrays of Circular Jets: Part 1-Effects of Minimum, Intermediate, and Complete Crossflow for Small and Large Spacings, Transactions of the ASME, 109, (1987), [5.14] Bailey, J. C. and Bunker, R. S. Local Heat Transfer and Flow Distributions for Impinging Jet Arrays of Dense and Sparse Extent, Proc. of ASME TURBO EXPO, Amsterdam, June , Research and Development Center General Electric Company, Niskayuna, NY, (2002) [5.15] Behbahani, A.I, and Goldstein, R.J. Local Heat Transfer to Staggered Arrays of Impinging Circular Air Jets, Journal of Engineering for Power, 105, (1983), [5.16] Goldstein, R.J. and Seol, W. S. Heat Transfer to a Row of Impinging Circular Air Jets Including the Effect of Entrainment, Int. J. Heat Mass Transfer, 34, (1991), [5.17] Hrycak, P. Heat Transfer from a Row of Impinging Jets to Concave Cylindrical Surfaces, International Journal of Heat Mass Transfer, 24, , (1981) 84

97 Impingement Cooling [5.18] Metzger, D.E., Yamashita, T. and Jenkins, C. W. Impingement Cooling of Concave Surfaces With Lines of Circular Air Jets, Journal of Engineering for Power, 91, , (1969) [5.19] Metzger, D. E., Baltzer, R. T. and Jenkins, C. W. Impingement Cooling Performance in Gas Turbine Airfoils Including Effects of Leading Edge Sharpness, Journal of Engineering for Power, July, , (1972) [5.20] Nagoga, G. Intensification of Heat Tranfer in the Cooling Channels of Gas Turbine Blades, Part III, Moscow, (2001) [5.21] Florschuetz, L. W. and Isoda, Y. Flow Distributions and Discharge Coefficient Effects for Jet Array Impingement with Initial Crossflow, ASME Journal of Engine Power, 105, , (1983) [5.22] Kopelev (1988), ISBN Personal Communication [5.23] Xiufang Gao, department GRCCC at Siemens, Finspång, October

98

99 Conclusions 6 Conclusions Following conclusions were drawn for the three investigated cooling methods. 6.1 Rib Turbulated Cooling Recommended Correlations: Han 1988 [3.5] for transverse ribs in ducts with two opposing ribbed walls Han et al [3.8] for angled ribs in ducts with two opposing ribbed walls with 1 4 W H < 1. Han & Park 1988 [3.9] for angled ribs in ducts with two opposing ribbed walls with 1 W H 4. Chandra et al [3.6] for duct with one, three or four ribbed walls. Nagoga 2000 [3.4] for U-shaped ducts with SR- or SSR-schemes. Conclusions concerning Q3D: Nu in Q3D is lower than Nu from the Han correlations it is based on. Friction in Q3D is considerably higher than the Han correlations due to a bug that needs further investigation. Three input parameters concerning number of ribbed walls are missing in the interface. The parameters are part of the solution to the high friction factor, but needs further investigations. It would be easy to change the code from the Reduced Area Method to the Total Area Method, which is recommended. Conclusions concerning Multipass: Four sub routines, Ribbed 1 to Ribbed 4 are used. Ribbed 1 is based on [3.7]. St the same as in [3.7] but f is numerically manipulated and differs from [3.7]. Ribbed 2 is based on [3.5]. St is the same as in [3.5] but f is numerically manipulated and differs from [3.5]. Ribbed 3 is based on [3.8]. St differs from [3.8] due to a small bug and f is numerically manipulated and also differs form [3.8]. Ribbed 4 is based on a mixture of [3.5] and [3.8]. St differs from the original correlations due to a small bug in the code but f agrees with the original correlations. 87

100 Conclusions 6.2 Matrix Cooling Conclusions about Correlations: Literature about matrix cooling is limited. Only [4.2] contained matrix correlations. The correlations are valid for closed matrixes. Flow in initial or direct matrix channels can be approximated to flow in smooth ducts with inlet effects. Further work is needed to develop correlations for matrix cooling. Conclusions about Q3D: Q3D calculates rib efficiency for ribs with isolated tops. The code should be changed according to equation (4.29), which considers both the isolated and convective rib tops. The phrase Rib Effectiveness in the Q3D interface should be replaced by Rib Efficiency. 6.3 Impingement Cooling Flat Target Surface: Use Höglund [5.8] or Florschuetz et al. [5.7] for correlations for staggered impingement arrays. Compare the correlations and decide from case to case which one is more reliable, depending on range and geometry. Use Florschuetz et al s [5.7] correlation for inline impingement arrays. Q3D calculates Nu and m& according to [5.7] Curved Target Surface: The investigated correlations give very different results. Use the Nagoga correlation, but it is also recommended to consult the other correlations when calculating Nusselt number for a specific case. Q3D is based on the correlations of Chupp et al. [5.1]. The code agrees with the base correlation. 88

101 Future Work 7 Future Work The heat transfer handbook will also contain correlations and information about other cooling methods used for internal cooling of blades and vanes. Examples of such cooling methods are listed below. Pin Fin Cooling Dimple Cooling Cyclone Cooling Pin fin cooling consists of single or arrays of pins that increases heat transfer due to increased turbulence and increased heat transfer surface area. Dimple cooling consists of small circular cavities in the surface, dimples, which disturb the boundary layer and thus increases heat transfer. Cyclone cooling cools the leading edge under through cross supply of cooling air. For more information about these and other cooling methods, see [4.2]. Another part of the further work is to investigate the Q3D code more concerning the reported problems that are not fully solved. The next step of this work is also to implement the recommended correlations into the heat transfer program Q3D. 89

102

103 Appendix 1 - Rib Turbulated Cooling Appendix 1 - Rib Turbulated Cooling 91

104 Appendix 1 - Rib Turbulated Cooling Appendix A Correlations Valid ranges for the correlations are displayed in Figure Webb, Eckert and Goldstein 1970 [3.13] The correlations below are valid for ribbed tubes with circular cross-section. The estimated errors of the correlations were approximately 12 % for the friction factor and 15 % for the Stanton number. 2 f rc = 2 (1) D + 2.5ln R( e ) 2e P R ( e ) = 0.95 (2) e f rc St 2 rc = (3) f rc [ G( e ) Pr R( e )] 2 ( ) 4.50( ) G e = e (4) e f rc e = Re (5) D 2 Han 1988 [3.5] The correlations below are valid for rectangular ducts with ribs on two opposing walls. The deviations from measured data were ± 6 % for R(e + ), ± 8 % for G(e + ) and ± 10 % for G ( e + ). 2 f r 4 = e ( ) 2.5ln 2.5 (6) R e Z Dh f H W f r 4 + f s (7) W W + H = f s4 = 0.046Re (8) P R ( e ) = (9) e 92

105 Appendix 1 - Rib Turbulated Cooling 93 H W W Z + = 2 (10) ( ) + = ) ( ) ( r r r f e R e G f St (11) 0.28 ) 3.7( ) ( + + = e e G (12) ( ) 4 s4 r r s r r f f W W f f + = (13) Re = + r h f D e e (14) Chandra, Niland and Han 1997 [3.6] The correlations below are valid for rectangular ducts with 1, 2, 3 or 4 ribbed walls. The deviations from measured data were ± 4 % for R(e + ) and ± 4 % for G(e + ). + + = s r s r r s r f W W f W W f (15) ln ) ( 2 = + Z D e e R f h r (16) 3.41 ) ( = + e R (17) H W W Z + = 2 (18) Re = s f (19) ln ) ( 2 4 = + Z D e e G f St h r r (20) ) 1.427( ) ( + + = W W e e G r (21)

106 Appendix 1 - Rib Turbulated Cooling e f r 4 e = Re (22) D h 2 Han, Glicksman & Rohsenow 1978 [3.7] 2 f = e ( ) 2.5ln 2.75 R e D h (23) P e R( e ) = 4.9 α 10 where n = if P/e < 10 n = 0.53 ( α 90) if P/e 10 n (24) f St r = (25) + + [ H ( e ) R( e )] 2 f ( e 35) + 10 H ( e ) = (26) ( α 45) j 0.28 where j = 0.5 for α < 45 j = for α 45 e e = D Re 2 + f h 0.5 (27) Han & Park 1988 [3.9] The deviations from measured data were ± 6 % for R(e + ), ± 8 % for G(e + ) and ± 10 % for G ( e + ). f r 4 = R( e + 2 2e ) 2.5ln Dh Z (28) m W α α R ( e ) = ( P e 10) (29) H where m = 0 if α = 90 m= 0.35 if α < 90 if W/H > 2 let W/H = 2 94

107 Appendix 1 - Rib Turbulated Cooling H W f r 2 = f r 4 + f s4 W (30) W + H 0.2 f s4 = 0.046Re (31) St r 2 = 2 r [ G( e ) R( e )] f f r (32) ( W H ) ( e ) ( 90) m ( P e ) n + G( e ) = 2.24 α 10 (33) where m = 0.35 & n = 0.1 for square ducts m = 0 & n = 0 for rectangular ducts Smooth Side Wall: + + G ( e ) = 1.2 G( e ) (34) St = 2 r [ G ( e ) R( e )] f f r (35) St ( St ) W = St + (36) H s St r 2 e e = D Re 2 + f h 0.5 (37) Han, Ou, Park & Lei 1989 [3.8] The deviations from measured data were ± 6 % for R(e + ) and ± 8 % for G(e + ). 2 f r 4 = e ( ) 2.5ln 2.5 (38) R e Z Dh R( e + α α ) = W H m (39) where m = -0.5 if 60 α 90 m = 0.5 ( α 60) 2 if 30 <α < 60 m = 0 if α 30 95

108 Appendix 1 - Rib Turbulated Cooling H W f r 2 = f r 4 + f s4 W (40) W + H 0.2 f s4 = 0.046Re (41) St r 2 = 2 r [ G( e ) R( e )] f f r (42) + ( e ) n + G( e ) = C (43) where for ½ H for ¼ H < W 1 n = 0.35 C = 2.24 if α =90 C =1.80 if 60 α < 90 W ½ n = 0.35 ( W H ) C = ( ) 2.24 W H if α = C = ( ) 1.80 W H if 30 α < e f r 4 e = Re (44) D h 2 Han 1984 [3.12] The estimated errors of the correlations were approximately ± 10 % for the friction factor and ± 10 % for the Stanton number. f r 4 = R( e + 2 2e 2H ) 2.5ln ln + Dh H W 2 (45) ( P ) R( e ) = 0.95 e (46) W f s4 + H f r 4 f r 2 = (47) W + H W St s4 + H Str 4 Str 2 = (48) H + W St s = (49) Re 0.2 Pr 0.6 ( 2 R D ) 0. 2 av h 96

109 Appendix 1 - Rib Turbulated Cooling f r 2 2 Str 4 = + (50) 1+ f 2 r 4 + ( G( e ) R( e )) ( e ) Pr + G ( e ) = 4.5 (51) e f r 4 e = Re (52) D h 2 Han, Park & Lei 1985 [3.10] The deviations from measured data were ± 5 % for R(e + ) and ± 10 % for H(e + ). f r 2 = R( e + 2 2e ) 2.5ln 2.5 Dh 2 (53) 2 + α α R ( e ) = ) n ( P e 10) ( e ) (54) where n = 0 if α 45 n = 0.17 if α < e f r 2 e = Re (55) D h ( 90) ( ) H ( e ) = 3.74 α e (56) f r 2 St r 2 = + + f r 2 (57) 2 [ H ( e ) R( e )]

110 Appendix 1 - Rib Turbulated Cooling Appendix B Correlations for U-shaped Channels The correlations are valid for Re d P e 13, 0 α 90 and e = 0.1. D h SR-scheme Nu = C Nu for x 20 (58) r Nu s D h Nu = Nu, for x D h 20 (59) r, x CNu s x C Nu = ( 1+ K α ) 0.2 P α where K α is found in Table 1 (60) e Nu, Pr for x 20 (61) s x = Re x Tw D h Nu for x 20 (62) s = Re Pr Tw D h F ( 1 e R) 2 1 = 1+ η 0.94 < η 1 (63) e R Table 1 Values of K α. α K (SR) α K (SSR) α K α (SSR) f r = C f (64) f s 0.25 f s = K l Re or f (65) 0.25 s = Rel ( l D ) K l = 2.65 h (66) 0.2 P C f = ( α ) (67) e SSR-scheme Following correlations are valid for 1.67 P e 13, 30 α 90, 0.5 e R 1. 5 and Re Nu r = C Nu Nus for x D h 20, for Nu s, see equation (61) (68) Nu r = C Nu Nu s for D h 20 x (69) 98

111 Appendix 1 - Rib Turbulated Cooling Nu r, x = C Nu Nus, x for x D h 20,for Nu s, x, see equation (62) (70) C Nu P e = ( 1+ Kα α ) where K α is found in Table 1 (71) e R 0.2 x Nu s = Re Pr Tw (72) Dh f r = f s + ( ξ f ) r Π ( 1) C ξ s (73) C f = 1+ (74) Π 2 arccos 1 Π = 1 π + 8 ( e R) for e R in current study (75) π 1+ 2 e Π = 1 π + 8 R for e R in current study (76) f s = ξ (77) s f r = C f (78) f s ξ r = Cξ ξ s (79) 0.25 ξ s = K l Re for x D h 20 (80) ξ for x 20 (81) 0.25 s = 0.84 Re x D h 0.25 x K = 2.65 l (82) Dh ( + K α ) P e Cξ = 1 α where e R K α is found in Table 1 ( 83) 99

112 Appendix 1 - Rib Turbulated Cooling Appendix C Results for Case A Heat Transfer Enhancement Factor Nusselt Number Enhancement K for a Ribbed Duct Han 1988 Han & Park 1988 Han et. al 1989 Han 1984, OBS! Nu average Han et al Q3D Reynolds number x 10 4 Figure 1 K for Ribbed Side Wall for Case A. 18 Friction Factor Enhancement C for a Ribbed Duct 16 Friction Enhancement Factor Han 1988 Han & Park 1988 Han et. al 1989 Han 1984,OBS! Nu average, Nu average Han et al Q3D Reynolds number x 10 4 Figure 2 C for Ribbed Side Wall for Case A. 100

113 Appendix 1 - Rib Turbulated Cooling Efficiency, eff=k/c for a Ribbed Duct Han 1988 Han & Park 1988 Han et. al 1989 Han 1984,OBS! Nu average Han et al Q3D eff=k/c Reynolds Number x 10 4 Figure 3 eff for Ribbed Side Wall for Case A. 101

114 Appendix 1 - Rib Turbulated Cooling Appendix D Effect of Rib Angle Effect of rib angle, α 3.5 Effect of alfa on K for a Ribbed Duct. Re = K Han & Park 1988 Han et. al 1989 Han et al Q3D Han alfa Figure 4 K as a Function of alfa, Case A Effect of alfa on C for a Ribbed Duct. Re = C Han & Park 1988 Han et. al 1989 Han et al Q3D alfa Figure 5 C as a Function of alfa, Case A. 102

115 Appendix 1 - Rib Turbulated Cooling Effect of alfa on Efficiency, eff=k/c for a Ribbed Duct. Re = Han & Park 1988 Han et. al 1989 Han et al Q3D 0.6 C alfa Figure 6 C as a Function of alfa, Case A. 103

116 Appendix 1 - Rib Turbulated Cooling Appendix E Effect of Pitch-to-Rib Height Effect of P/e Effect of P/e on K for a Ribbed Duct. Re = Han 1988 Han & Park 1988 Han et. al Han 1984, OBS! Nu average Han et al Q3D 2.2 K P/e Figure 7 Effect of Rib Pitch-to-Height ratio on K for Ribbed Side Wall. 16 Effect of P/e on C for a Ribbed Duct. Re = C Han 1988 Han & Park 1988 Han et. al 1989 Han 1984, OBS! Nu average Han et al Q3D P/e Figure 8 Effect of Rib Pitch-to-Height ratio on C for Ribbed Side Wall. 104

117 Appendix 1 - Rib Turbulated Cooling 0.5 Effect of P/e on Efficiency, eff=k/c for a Ribbed Duct. Re = eff=k/c Han 1988 Han & Park 1988 Han et. al 1989 Han 1984, OBS! Nu average Han et al Q3D P/e Figure 9 Effect of Rib Pitch-to-Height Ratio on eff for Ribbed Side Wall. 2.2 Effect of P/e on K for a Ribbed Duct. Re = K P/e Figure 10 Effect of Rib Pitch-to-Height Ratio for Han et al W/H=12, e/d h =

118 Appendix 1 - Rib Turbulated Cooling Appendix F Effect of Aspect Ratio Effect of aspect ratio Effect of W/H on K for a Ribbed Duct. Re = K Han 1988 Han & Park 1988 Han et. al 1989 Han 1984, OBS! Nu average Han et al Q3D W/H Figure 11 K as a Function of W/H, Case A Effect of W/H on C for a Ribbed Duct. Re = Han 1988 Han & Park 1988 Han et. al 1989 Han 1984, OBS! Nu average Han et al Q3D C W/H Figure 12 C as a Function of W/H, Case A. 106

119 Appendix 1 - Rib Turbulated Cooling Effect of W/H on Efficiency, eff=k/c for a Ribbed Duct. Re = Han 1988 Han & Park 1988 Han et. al 1989 Han 1984, OBS! Nu average Han et al Q3D eff=k/c W/H Figure 13 eff as a Function of W/H, Case A. 107

120 Appendix 1 - Rib Turbulated Cooling Appendix G Effect of Rib Height Effect of e/dh 2.8 Effect of e/dh on K for a Ribbed Duct. Re = K Han 1988 Han & Park 1988 Han et. al 1989 Han 1984, OBS! Nu average Han et al Q3D e/dh Figure 14 K as a Function of e/d h, Case A. 16 Effect of e/dh on C for a Ribbed Duct. Re = C Han 1988 Han & Park 1988 Han et. al 1989 Han 1984, OBS! Nu average Han et al Q3D e/dh Figure 15 C as a Function of e/d h, Case A. 108

121 Effect of e/dh on Efficiency, eff=k/c for a Ribbed Duct. Re = Han 1988 Han & Park 1988 Han et. al 1989 Han 1984, OBS! Nu average Han et al Q3D eff=k/c e/dh Figure 16 eff as a Function of e/d h, Case A. 109

122 Appendix H Q3D Correlations The value of A s depends on both rib angle and channel aspect ratio. Therefore it is not possible to explain the calculation of A s without use of a function called Sfun from the code. Appendix F contains the the equations from Q3D, function Sfun and the vectors xbs, cbs, xfs and cfs that are used for calculation of A s. The equations are presented in the reduced area form. A s is calculated by function Sfun, see below. The vectors xbs, cbs, xfs and cfs that are used for calculation of A s are displayed in Table1. The Q3D equations below are presented in the reduced area form. Rib angle of 90 A = n A n r s r s = 0.77 W f = ( 5, W H, xbs, cbs) Sfun = 100 = Rib angle of 75 4 Ar = W H n = A n r s s = Sfun = W f = Rib angle of 60 4 Ar = W H n = A n r s s = Sfun = W f = ( e Dh ) ( H 2 e) W + ( H 2 e) ( 4, W H, xfs, cfs) ( ( W H )) ( ( e Dh )) ( H 2 e) W + ( H 2 e) ( 4, W H, xfs, cfs) ( ( W H )) ( ( e Dh )) ( H 2 e) W + ( H 2 e) (84) (85) (86) 110

123 Rib angle of 45 4 Ar = W H n = A n r s s = Sfun = W f = Rib angle of 30 4 Ar = W H n = A n r s s = Sfun = W f = Smooth duct A = nr = 0.8 A = n s r s = 0.8 f = ( 4, W H, xfs, cfs) ( ( W H )) ( ( e Dh )) ( H 2 e) W + ( H 2 e) ( 4, W H, xfs, cfs) ( ( W H )) ( ( e Dh )) ( H 2 e) W + ( H 2 e) (87) (88) (89) 111

124 Sfun takes 4 variables, N, X, Y and C and returns a number. N and X are numbers and Y and C are vectors. The function Sfun in Visual Basic code: Function Sfun(N, X, Y, C) As Double Dim a As Double Dim mysfun As Double mysfun = C(N + 1) + C(N + 2) * X For i = 1 To N If X > Y(i) Then a = X - Y(i) mysfun = mysfun + C(i) * a * a * a Else Exit For End If Next i Sfun = mysfun End Function Table1 Vectors used to Calculate A s. xbs cbs xfs cfs

125 Appendix I Multipass Results Ribbed Nusselt Number for Ribbed 1 Ribbed 1,4 ribbed walls Han et al Ribbed 1,2 ribbed walls Nusselt Number Reynolds Number x 10 4 Figure 17 Nu Calculated for W=120 mm, H=10 mm, P=8mm, e=1.5 mm and alfa=90 deg Friction Factor for Ribbed 1 Ribbed 1 Han et al Friction Factor Reynolds Number x 10 4 Figure 18 f Calculated for W=120 mm, H=10 mm, P=8mm, e=1.5 mm and alfa=90 deg. 113

126 Ribbed Ribbed 2 Han & Park 1988 Ribbed 2, corrected Nusselt Number for Ribbed 2 Nusselt Number Reynolds Number x 10 4 Figure 19 Nu Calculated for W=5 mm, H=5 mm, P=3.15mm, e=0.315 mm and alfa=90 deg Friction Factor for Ribbed 2 Ribbed 2 Han & Park Friction Factor Reynolds Number x 10 4 Figure 20 f Calculated for W=5 mm, H=5 mm, P=3.15mm, e=0.315 mm and alfa=90 deg. 114

127 Ribbed Ribbed 3 Ribbed 3, corrected Han et al Nusselt Number for Ribbed Nusselt Number Reynolds Number x 10 4 Figure 21 Nu Calculated for W=5 mm, H=5 mm, P=3.15mm, e=0.315 mm and alfa=90 deg Friction Factor for Ribbed Friction Factor Ribbed 3 Ribbed 3, corrected Han et al Reynolds Number x 10 4 Figure 22 f Calculated for W=5 mm, H=5 mm, P=3.15mm, e=0.315 mm and alfa=90 deg. 115

128 Ribbed Ribbed 4 Ribbed 4, corrected Han & Park 1988 Han et al Nusselt Number for Ribbed 4 Nusselt Number Reynolds Number x 10 4 Figure 23 Nu Calculated for W=5 mm, H=5 mm, P=3.15mm, e=0.315 mm and alfa=90 deg. Friction Factor for Ribbed Ribbed 4 Han & Park 1988 Han et al Friction Factor Reynolds Number x 10 4 Figure 24 f Calculated for W=5 mm, H=5 mm, P=3.15mm, e=0.315 mm and alfa=90 deg. 116

129 Appendix J Interesting Articles Further information about rib turbulated cooling can be found in the articles below. Author Title Literature Hong & Hsieh Heat Transfer and Friction Factor Measurements in Ducts with Staggered and Inline Ribs Journal of Heat Tranfer, Vol. 115, February 1993 Kiml, Mochizuki & Murata Hwang Chandra, Fontenot & Han Taslim & Spring Acharaya, Eliades & Nikitopoulos Han, Zhang & Lee Korotky & Taslim Taslim & Wadsworth Effects of Rib Arrangements on Heat Transfer and Flow Behavior in a Rectangular Rib-Roughened Passage: Application to Cooling of Gas Turbine Blade Trailing Edge Heat Transfer-Friction Characteristic Comparison in Rectangular Ducts with Slit and Solid Ribs Mounted on One Wall Effect of Rib Profiles on Turbulent Channel Flow Heat Transfer Effects of Turbulator Profile and Spacing on Heat Transfer and Friction in a Channel Heat Transfer Enhancements in Rotating Two-Pass Coolant Channel With Profiled Ribs: Part 1- Average Results Augmented Heat Transfer in Square Channels with Parallell, Crossed, and V-shaped Angled Ribs Rib Heat Transfer Coefficient Measurements in a Rib- Roughened Square Passage An Experimental Investigation of the Rib Surface-Average Heat Transfer Coefficient in a Rib- Roughened Square Passage Journal of Heat Transfer, Vol. 123, August 2001 Journal of Heat Transfer, Vol. 120, August 1998 Thermophysics, Vol. 12, No. 1: Technical Notes, 1997 Journal of Thermophysics and Heat Transfer, Vol. 8, No. 3, July-Sept Journal of Turbomachinery, Vol. 123, Januray 2001 Journal of Heat Transfer, Vol. 113, August 1991 Journal of Turbomachinery, Vol. 120, April 1998 Journal of Turbomachinery, Vol. 119, April

130

131 Appendix 2 - Matrix Cooling Appendix 2 - Matrix Cooling 119

132 Appendix 2 - Matrix Cooling Appendix A Zinc Test Figure 25 describes the basics of a zink test. Figure 25 Zink Test Facility [4.3]. Following 6 steps describe how the tests were carried out. 1. The test object is connected to air supply, and is provided with pressure taps for static pressure measurements. A container with pure zinc is heated in a furnace to a temperature slightly higher than zinc crystallization temperature. 2. The test object is lowered into the container, which still is in the furnace, and stays there until temperature equalization between object wall and melt is reached. The test is monitored during this time to prevent metal crust formation on the object walls at the initial time of its submersion into the melt. 3. The container with the objects is taken out of the furnace. Cooling of the melt and object down to the crystallization temperature of zinc takes place. 4. When the crystallization point is reached for the melt and the test object, the test object is exposed to air blowing through its cooling channels which cools 120

133 Appendix 2 - Matrix Cooling the object walls intensively. A metal crust is formed on the surface of the test object. 5. After a certain time, the blowing of cooling air is stopped. The test object is taken out of the container and cooled down to room temperature. The container is replaced in the furnace. 6. The metal crust of the object is removed and marks along the object contour are made. Cut along the marks are made. Each test is performed three to five times, to eliminate errors. Crust thickness measurements are made. The heat flow determination error does not exceed 3 % to 8 %. [4.3] The overall heat transfer is evaluated from the mass of the zinc crust. Measuring the mass is more accurate than measuring the thickness. 121

134 Appendix 2 - Matrix Cooling Appendix B Correlations and Ranges of [4.2] Heat Transfer on Base Shell The ranges for which Nagoga s correlations are valid are displayed in Table 2. Table 2 Ranges for heat transfer correlations. Re d T w /T f [K] β [ ] 0-70 d [mm] L/W κ W/d 6-34 P [mm] l/d Initial Section T w Nu x = Re x Pr (90) T f T w Nu l = Rel Pr (91) T f Nu d T w = Re d Pr K (92) T f ( Re 10 ) K l 0.36 = d d (93) f 0.2 = 0.43 Re x (94) x Re x = Re d d (95) l Re l = Re d d (96) Basic Section T 0.55 n 0.4 w Nu x = A n Re x Pr (97) T f 0.55 T 2 A = sin 2β (99) n 0.4 w Nu l = A Rel Pr (98) T f ( ( )) 2 4 β n = (100) π 2 2 Κ = β sin 2β (101) ( ( ) ) ( ) ( ) m x Re x 122

135 [( ) 1] 2 Appendix 2 - Matrix Cooling m = β (102) K l 2 m [ sin ( 2 )] Re = β (103)* k 1 l n 1 x f = 0.43 C1 Re d (104) d 3 C = sin (2 ) (105) 1 β 2 4 β k = (106) π 3 = sin (2β ) Re (107)* Cl [ ] m l 2 4 β m = (108) π * In [4.2], Re x is used, which is probably a misprint, since the average enhancement factor cannot be calculated with the local Re. Side Bounds 0.55 Nu T 0.11 Ψ1 = F C (110) Ψ 0.4 = F (111) w 0.21 ST = Re d Pr Ψ1 Ψ2 D (109) T f CD Local hydraulic losses for flow turn near the matrix wall -ξ m ξ (112) m = FCD FC Index of Energy Efficiency 3 Cl ( sin(2β )) Ψ = = K sin(2β ) l ( ) 2 (113) Mechanisms of Heat Transfer and Friction wϕ ϕ = arctan w x w (114) w, x w, x = w (115) AV 0.76 tanϕ = 1.18 Φ (116) β = 30 Φ ( 0. x) = Φ in exp 04, (117) Φ, in = 0.3 (118) w x = x (119), 9 β = 45 Φ ( 0. x) = Φ in exp 04, (120) 123

136 Appendix 2 - Matrix Cooling Φ, in = 0.43 (121) w x = x (122), 2 Basic Section ( + 0. Φ ) w Nu x, β = Nu x, β = , x (123) , = Re Pr Tw Nu x β = x (124) T f λ β = λ β = 0 ( Φ ) w, x (125) 0.2 λ = 0 = 0.43 Re x (126) β 124

137 Appendix 2 - Matrix Cooling Appendix C Correlations and Ranges of [4.5] Correlations for the basic section as presented in Shukin and Nagoga [4.5] are displayed below. 2 [ sin ( 2 )] 0.55 n 0.4 w Nu x = n β Re x Pr (127) T f T n [ sin ( 2 )] Re Pr Tw Nu = + l β l (128) T f 2 4 β n = (129) π 2 n 0. 8 Κ x = ( n 0.8) [ sin ( 2β )] Re x (130) 2 n 0. 8 [ sin ( 2 )] Re K = β (131) l 3 n 0. 8 [ sin (2 )] Re l C = β (132) l 3 n 1 [ sin (2 )] Re l f = 0.43 β (133) l 125

138 Appendix 2 - Matrix Cooling Appendix D Results for a Typical Turbine Blade Alfa as a Function of Re d in Basic Section near Channel Inlet, x/d = 0.83 Nagoga average Nagoga local Shukin & Nagoga local Shukin & Nagoga Average 8000 Alfa Re x 10 4 d Figure 26 Alfa in Example for Typical Turbine Blade. Friction Enhancement Factor C l as a Function of Re d, Basic Section 4 C l Nagoga C l Shukin & Nagoga C l Re x 10 4 d Figure 27 C l in Example for Typical Turbine Blade, x/d =

139 Appendix 2 - Matrix Cooling Heat Transfer Enhancement K as a Function of Re d in Basic Section Nagoga local Nagoga average Shukin & Nagoga local Shukin & Nagoga average 2.2 K Re x 10 4 d Figure 28 K in Example for Typical Turbine Blade, x/d = Friction Factor as a Function of Re d for x/d= f Nagoga Basic f Shukin & Nagoga Basic 0.14 f Nagoga Inital f Re x 10 4 d Figure 29 f in Example for Typical Turbine Blade. 127

140 Appendix 2 - Matrix Cooling Alfa as a Function of x/d in Basic Section, Re=15000 Nagoga average Nagoga local Alfa x/d Figure 30 Alfa in Example for Typical Turbine Blade. 2.3 K as a Function of x/d in Basic Section, Re=15000 Nusselt Number Enhancement Factor K K average K local x/d Figure 31 K as a Function of x/d in Example for Typical Turbine Blade. 128

141 Appendix 2 - Matrix Cooling Appendix E Interesting Articles Further information about matrix cooling can be found in the articles below. Author Title Literature Gorelov, Y. G. Ways of Further Enhancement the Convective Cooling of Vortex-Matrix Rotor Blades in High-Temperature Turbines of Gas Turbines Engines Thermal Engineering, Vol. 51, No 11, 2004 Acharya, Zhou, Lagrone, Mahmood & Bunker Latticework (Vortex) Cooling Effectiveness Part 2: Rotating Channel Experiments Proceedings of ASME Turbo Expo 2004, Power for Land, Sea and Air, June 14-17, Vienna, Austria 129

142

143 Appendix 3 Impingement Cooling Appendix 3 Impingement Cooling 131

144 Appendix 3 Impingement Cooling Appendix A Correlations and Ranges Correlations and ranges for impingement on flat respectively curved target plates are presented below. The parameter Nu represents the average Nusselt number over an impingement row. Flat Target Plate The average heat transfer over an impingement row in an array is calculated unless other information is given. Florschuetz, Truman & Metzger 1981 [5.7] m n 1 3 Nu = A Re j { 1 B [ ( z d ) ( Gc G j )] } Pr (134) where n x n y n A, m, B and m = C x d y d z d ( ) ( ) ( ) z n n Table 3 Parameters for equation (). Inline Patterns Staggered Patterns C n x n y n z C n x n y n z A m B n The standard error of deviations in the equation above is estimated to 5.6 % for the inline pattern and 6.1 % for the staggered pattern. A simpler, alternative correlation where m and n are independent of geometrical parameters is described below. It has essentially the same confidence level as the former equation. nx ny nz Nu Nu ( ) ( ) ( ) ( ) n 1 = 1 C xn d yn d z d Gc G j (135) ( x d ) ( y d ) ( z d ) Re Nu 1 = n n j Pr (136) Table 4 Parameters for the Equations Above. C n x n y n z n Inline Staggered Metzger & Korstad 1972 [5.10] The correlation is valid for a single row of impingement St = M Re d (137) Bailey & Bunker 2002 [5.14] Tests were done with square inline arrays. Limiting case for a sparse array and dense array are 4 3 and number of jet rows, respectively. The first number refers to the axial direction and the second to the lateral direction. Nu = ( x d ) + ( z d ) ( ( z d )) ( x d ) Re d + (138) [ x d 28 z d ] Re d [ 4 10 ] ( G G ) ( ) ( ) c j 132

145 Appendix 3 Impingement Cooling Martin 1977 [5.11] The correlation is valid for square arrays Nu = Pr K a Re (139) a ( z d 6) a z d K = 1 + (140) 0.6 a π d 4 a = y n 2 for a square inline array (141) π a = 2 3 d y n 2 for a square staggered array (142) Florschuetz & Isoda 1982 [5.21] The correlations are valid for inline rectangular arrays. Table 5. Correlations for Cd x, y, z d d d Equation Range of G c /G j (5,4,1) C d = to 0.63 C = ξ ( G G ) > 0.63 d c j (5,4,2) C d = to 0.83 C = ξ ( G G ) > 0.83 d c (5,4,3) C d = to 0.90 C = ξ ( G G ) > 0.90 d c (5,8,1) C d = to 0.54 C ( G G ) to 1.5 C d = c j d ( G G ) c j j j = ξ > 1.5 (10,4,1) C d = to 0.54 C ( G G ) to 1.8 ( G G ) d = c j ( G G ) Cd = ξ c j > ξ ( Gc G j ) = (143) c j f = 24 Re c for Re c < 2000 (144) f = for 2000 < Re c < (145) Rec 133

146 Appendix 3 Impingement Cooling f = for Re c > (146) Rec Obot & Trabold 1987 [5.13] The correlations are valid for inline arrays. The flow scheme is defined in the left picture in Figure 31. Individual differences of less than 10 % were obtained for 85 % of the data, from 80 data points. Figure 31Cross Flow Definitions (Left) and Values of Exponent n (Right) [5.13]. n 0.8 z x 0 Re Af (147) Nu = A d 1.6 ξ ( Gc G j ) = G G (148) ( ) 602 c j Table 6. Parameters for Obot and Trabold. Flow Scheme A 0 n x Minimum see Figure Intermediate see Figure Maximum see Figure Behbahani & Goldstein 1983 [5.15] The correlation is valid for a staggered, square impingement array. Nusselt number averaged over the holes array is calculated. n 0.78 xn Nu = a Re j d Table 7. Parameters for Behbahani and Goldstein. z/d a n (149) 134

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