HEAT TRANSFER BY CONVECTION AND CONDUCTION FROM THE FLUID MOVING AT SOLID WALLS Associate Professor Ph.D. Amado George STEFAN, Lt.Eng., doctoral student Constantin NISTOR MILITARY TECHNICAL ACADEMY Abstract. The paper shows the convective heat transfer in flat plate with similitude Nusselt criterion. The report Pr Pr 025, f / p takes account of changes in thermal properties of the boundary layer thickness. Thermal conductivity is the only possible way to transmit heat in solids mass. If fluid transfer occurs only when it is at rest or when macroscopic flow is laminar them, but in the last Heat Transfertransmite boundary layer of heat and fluid flow, whether laminar or turbulent character's current. Size a / c [m 2 /s] is called the temperature diffusivity or temperature conductivity. The size of the report depends on the speed with which transmits a temperature variation in substance considered. Heat transfer is modeled shoulders from a central jet of warm air from outside coaxial jet of cold air. Keywords: heat transfer, thermodynamics. 1. INTRODUCTION The study of thermal phenomena using similarity is very important. The first to use this method for convection, otherwise known process beforehand and successfully applied in hydrodynamics was Nusselt. În this way he introduced order into the multitude of existing experimental results at that time in this area. Until then, each experimental result was only valid for the case investigated by the experimenter, without any method by which this result can be generalized to all similar phenomena. Nusselt applied in this case the most rigorous way to determine the similarity criteria, based on the phenomenon of differential equations. 2. CONVECTIVE HEAT TRANSFER. NUSSELT CRITERION. FLAT PLATE In Figure 1 is the flat plate boundary layer scheme. With the help of experimental corrections, where rolling motion similarity relation to the distance X x/l, the flat plate is: 0,25 Pr 0,5 0,5 0,33 f Nu f 0.33X Re f Prf, (1) Pr p where the subscript "f" refers to the fluid in motion, and "p" to the wall. Report (Pr f /Pr p ) 0,25 take account of changes in the thermal properties of the boundary layer thickness. Number Pr gas depends very little on temperature. With very good accuracy can be considered (Pr f /Pr p ) 0,25 = 1. In equation (1) is adopted: 5 Re Re 5 10, 0,6 Pr 1,5. cr Fig. 1. Laminar and turbulent boundary layer on flat plate: δ d dynamic boundary layer thickness; δ T thermal boundarylayer thickness; T f temperature of the fluid; T p temperature of the wall. Using relation (1) is calculated following numerical values Nu f (Re f,x,pr): Nu Re f (Re f ) Nu f (Re f ) Nu f (Re f ) f X = 0.3, Pr = 0.6 X = 0.5, Pr = 0.6 X = 0.7, Pr = 0.6 1 10 5 48.291 62.343 73.765 2 10 5 68.294 88.167 104.32 3 10 5 83.642 107.982 127.765 4 10 5 96.582 124.686 147..531 5 10 5 96.582 139.404 164.945 TERMOTEHNICA 2/2013 91
Amado George STEFAN, Constantin NISTOR Fig. 2. Variation of the numerical values of the number Nu f (Re f,x,pr). Blasius's solution for laminar boundary layer: x x 0.5 1/3 Nux 0.332 Rex Pr, Pr 0.6 (2) for: Pr 0,05; Pe = Re Pr > 100 x x 0.5 Nu x 0.565Pex (3) For any number Pr is valid correlation Churchill Ozoe: Nu x 0,5 1/3 x 2/3 1/4 x x 0,3387 Re Pr 1 0,0468 / Pr Pex 100. (4) For flat plate turbulent boundary layer 4/5 1/3 f x f Nu 0.0296 Re Pr, 0,6 < Pr < 60. (5) 3. CONVECTIVE HEAT TRANSFER IN CIRCULAR CYLINDERS Figure 3 shows the flow in a cylinder and the variation of convection coefficient depending on boundary layer: In the case of steady movement 08 043 Prf,, Nu f 0. 021Re f Prf Pr p 4. THERMAL CONDUCTIVITY, 025,. (6) The heat transfer phenomena are functions of time. On the other hand, the heat transfer is irreversible phenomenon, which occurs because the temperature difference can never be infinitely small, such as thermodynamic reversibility require. Fig. 3. Circular cylinder turbulent boundary layer and the variation qualitative factor. Intrinsic properties of substances are included in the coefficient of thermal conductivity. The value of depends on the substance and its mode of presentation and phase. Coefficient has the dimension Wm K in S.I. Thermal conductivity is the only possible way to transmit heat in solids mass. In the case of fluids, it occurs only when they are in their rest macroscopic or when the flow is laminar, and in the latter case only in the transverse direction to the direction of flow. All heat is transmitted by conduction in the boundary. Coefficient of thermal conductivity. For gases the coefficient of thermal conductivity is a function of temperature and pressure. We can write: f t, p. (7) Temperature dependence is written by the approximate formula: 0 (1 t ), (8) where λ 0 depends on the material and temperature. This conductivity is a linear function of the specific heat of the gas at constant volume and viscosity of the gas flow. Pressure dependence of the thermal conductivity is very similar to the pressure dependence of the specific heat. The amount λ 0 of the various gases of the technical at various temperatures is shown in Table 1. Metals and alloys. The metals are the best of all heat conductive materials, in particular in the pure state. In the case of high specific conductivity of metals (Ag, Cu, Au, Al etc.), even lower than the conductivity of the traces of impurities. 92 TERMOTEHNICA 2/2013
HEAT TRANSFER BY CONVECTION AND CONDUCTION FROM THE FLUID MOVING AT SOLID WALLS Table 1 The values of 10 3 λ 0 for gas, in [W/m K] Gas Temperature, C 50 0 100 150 200 250 300 400 500 H 2 He NH 3 (H 2 O)C O N 2 O 2 Aer CO 2 CH 4 146 13,6 17,8 19,7 19,7 19,7 9,8 15,5 267 141 17,4 21,7 23,6 23,6 23,6 14,1 24 215 167 24,8 23,3 30,6 30,6 30,6 19,5 250 29,0 26,6 33,8 33,8 33,8 21,1 272 25,0 31,8 31,8 31,8 26,7 254 33,2 39,9 39,9 39,9 316 36,6 42,8 42,8 42,8 357 43,2 48,4 48,4 48,4 397 49,8 53,8 53,8 53,8 Pure gold Pure nickel Technical nickel Bronze (86 Cu, 7 Zn, Sn6) Pure iron Technical metals Iron - Ni (5% Ni) The values of λ 0 for gas, in [W/m K] 310 93 69,7 60,4 73,2 46,4 58 34,8 Iron - Ni (20% Ni) Iron - Ni (35% Ni) Iron - Ni (50% Ni) Iron - Ni (80% Ni) Monel metal (29 Cu, 67 Ni, two Fe) Bronze (95%) Dural (94.5% Al, 4 Cu, 0.5 mg) Table 2 18,6 11 14,5 32,5 22 82,4 159 5. THE GENERAL EQUATION OF CONDUCTIVITY a Using the Laplace operator 2 and noting, we can write: c t 2 q a t (8) c this is the general differential equation of thermal conductivity Fourier. Size a [m 2 /s] is called the diffusivity of c temperature and conductivity on temperature. The size depends on the speed with which this report is transmitted through the substance temperature variation considered. If q = 0 there is considered that the body heat, Fourier's equation is: t 2 a t (10) Equation (10) includes the case of stationary, and if it is considered that this phenomenon occurs in one direction x, we can write: 2 d t 0 2 dx (11) in which by integration gives: Q t x t1 A (12) where t 1 is the highest temperature of the substance. From equation (12) it can be seen that the temperature of a flat wall varies linearly with x (Fig. 4). The higher value, the difference between t 1 and t is less than the same x. Fig. 4. Variation of the thickness of a flat wall temperature, for different values of λ. 6. THERMAL CONDUCTIVITY THROUGH THE TUBE WALLS 1) Simple walled tubes. Of particular interest for special presents, in addition to conduction through plane wall, conduction through the tube wall or simple compounds. Most of the time, the technique is considered stationary phenomena, that in steady state operation, without even periodic variations in temperature. If the tube has a constant cross-section length L, then by a cylindrical layer of thickness dr in which TERMOTEHNICA 2/2013 93
Amado George STEFAN, Constantin NISTOR there is loss of temperature dt, the heat is transmitted dt W 2rL. (13) dr When transmitting stationary heat and if the coefficient of thermal conductivity of the wall may be admitted to the temperature constant, then equation (13) is a differential equation ternperatura loss depending on the radius of the tube. By separation of variables and the temperature variation is obtained by integration along the radius of the tube (inside wall): W t ln rc, 2L Integration constant is determined by the initial and final temperatures conditions: W re ti te ln (14) 2L ri the index e relates to the outside of the tube and i in the interior. Resistance to conduction, in the case of single wall pipe, we can write: R c re ln ri. (15) 2 L 2) Multi-walled tubes. In the case of a tube wall composed of several layers of different materials, the total resistance is the sum of the partial resistances: r1 r2 re ln ln ln n1 ri r1 rn Rct R ci... i1 12L 22L n12l (16) and the heat flow is: ti te W. (17) Rct In all the above calculations it was assumed that t i > t e. But if t i < t e, it changes the meaning of heat transmission, since the flow calculations is always positive. The temperature of the layer n and n + 1 is determined by the relationship: W 1 r1 1 r2 tn t i ln ln... 2L 1 ri 2 r1.(18) 1 r ln n n rn 1 7. CASE STUDY Numerical modeling heat transfer in a central jet hot air from external coaxial jet of cold air. The numerical determination are made with Fluent software. Consider the case of a central air jet flow rate of 60 kg/s and a temperature of 1053 K flowing through a pipe with an internal diameter of 880 mm and a length of 1000 mm, pipe wall thickness is 2 mm. The first duct is coaxial with a second pipe internal diameter of 984 mm. Between the two ducts is the second stream of air, at 6 kg/s flow rate and a temperature of 300 K (Fig. 5). Fig. 5. The fluid-solid domain and boundary conditions. For the mesh it use rectangles cell, with 10 mm lengtht along the axis of revolution and 0.2 mm in radial direction for the wall, and for the fluid domain a variable dimension in radial direction, from 0.1 mm in the vicinity of the wall with a increase in geometrical progression with the ratio of 1.2 (Figure 6). Fig. 6. Detail of the mesh in the walls area. Turbulence model chosen is kε the standard wall position. At the inlet sections the turbulent flow intensity is considered to be 10%. The air is 94 TERMOTEHNICA 2/2013
HEAT TRANSFER BY CONVECTION AND CONDUCTION FROM THE FLUID MOVING AT SOLID WALLS considered compressible, with variable specific heat with temperature according to the law: C T 1161. 482 2. 368819 T p 2 5 3 0. 01485511T 5. 034909 10 T 8 4 9. 928569 10 T 10 5 14 6 1. 111097 10 T 6. 540196 10 T 17 7 1. 57358810 T 9. CONCLUSIONS Figures 7 and 8 present laws of variation for velocity and temperature at the exit sections for the hot and cold stream. Figure 9 shows the temperature field in the vicinity of the wall between the two flows in the output. It is noted heat dissipation of the stream of cold air downstream of the input section with the decrease in temperature in the central stream of hot air. In figure 10 is presented the variation of convection coefficient with axial distance (along the wall). Fig. 8. Velocity and temperature profile at the cold air outlet section. Fig. 9. Distribution of temperature (K) in the output (detail of the partition). Fig.7 Velocity and temperature profile at the hot air outlet section. Fig. 10. Variation of convection coefficient. TERMOTEHNICA 2/2013 95
Amado George STEFAN, Constantin NISTOR BIBLIOGRAPHY [1] Dobrovicescu, A., and others, Fundamentals of Technical Thermodynamics. Elements of technical thermodynamics, vol. I, Politehnica Publishing House, 2009. [2] Leonăchescu, N., Thermodynamics, Didactic and Pedagogic Publishing House, Bucharest, 1985. [3] Marinescu, M., Chisacof, A., Raducabu, P., Motoroga, A., O. Basics of Technical Thermodynamics. Heat and Mass Transfer. Fundamental Processes, vol. II, Politehnica Publishing House, 2009. [4] Nistor, C., Heat transfer by conduction. Conduction analysis of 1D, 2D and 3D scientific research report, Military Technical Acacemia, Bucharest, 2010. [5] Nistor, C., Heat transfer by convection. Research report, Military Technical Acacemia, Bucharest, 2011. [6] Olariu, V., Brătianu, C., Numerical finite element modeling, Technical Publishing House, Bucharest, 1986. [7] Ştefan A., G., Termohidrodinamice phenomena analysis by finite element method, Ed Mirton, 2003 96 TERMOTEHNICA 2/2013