Available online at www.sciencedirect.com ScienceDirect Procedia Engineering 157 (2016 ) 44 49 IX International Conference on Computational Heat and Mass Transfer, ICCHMT2016 SIMULATION OF TEMPERATURE DISTRIBUTION AND HEAT TRANSFER COEFFICIENT IN INTERNALLY RIBBED TUBES Grądziel Sławomir a *, Maewski Karol a a Cracow University of Technology, Al. Jana Pawla II 37, 31-864 Cracow, Poland Abstract This paper puts forward modelling thermal and flow phenomena in internally rifled tubes. The proposed model is a distributed parameter model based on solving balance equations describing the principles of the mass, momentum and energy conservation. The model enables an analysis of transient-state processes. The aim of the calculations is, among others, to find the distribution of the fluid enthalpy, mass flow and pressure in internally rifled tubes and to determine the heat transfer coefficient. The analysis concerns tubes arranged vertically and operating at supercritical steam parameters. The numerical calculation results will be compared to values obtained from CFD modelling. 2016 The Authors. Published by by Elsevier Elsevier Ltd. Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of ICCHMT2016. Peer-review under responsibility of the organizing committee of ICCHMT2016 Keywords: supercritical boiler; internally rifled tubes; mass, momentum and energy conservation equations; thermal load 1. Introduction Both internally rifled and smooth tubes are now in common use in many industrial devices and processes. They are widely used in refrigerating engineering (small-diameter tubes in particular) and in power boilers. Owing to such solutions, the heat exchanger size can be reduced and the permissible tube wall temperature is not exceeded in flows threatened with a boiling crisis, i.e. in cases where departure from nucleate boiling (DNB) occurs. The application of rifled tubes with internal spiral ribs in power boilers makes it possible to avoid many costly failures arising from the * Corresponding author. Tel.: +048126283553; fax: +048126283560. E-mail address: gradziel@mech.pk.edu.pl 1877-7058 2016 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of ICCHMT2016 doi:10.1016/.proeng.2016.08.336
Grądziel Sławomir and Maewski Karol / Procedia Engineering 157 ( 2016 ) 44 49 45 material overheating. The issues related to lengthening the life of the boiler components, including thick-walled elements, and to improving their efficiency are presented in [1]. The use of internally rifled tubes with spiral ribs in the boiler evaporator involves a change in thermal and flow processes. The fluid moves spirally inside the tubes, which helps to intensify the heat transfer, and if a boiling crisis occurs makes it possible to maintain the water film on the tube surface. This is due to the centrifugal force arising in the fluid, which throws heavier droplets onto the wall. The process enables steady collection of a large heat flux even for a high content of the gaseous phase in the fluid and, as a result, makes it possible to keep a safe temperature of the tube wall. In the case of supercritical boilers, rifled tubes are placed in zones with the highest local thermal load. Such a solution is adopted in circulating fluidized-bed (CFB) boilers, where the wing walls are made of internally rifled tubes with spiral ribs, and where the spiral tube arrangement of the furnace chamber waterwalls is not applied due to enhanced erosion caused by the circulating material. This paper presents the calculation results of thermal and flow phenomena occurring in internally rifled tubes for a medium with supercritical parameters. The results were obtained by means of a numerical analysis and CFD modelling. It also presents the mass, momentum and energy conservation equations with distributed parameters. The equations were used in an in-house program written in the Fortran language and intended for numerical calculations [2]. The CFD modelling was carried out using the Ansys Fluent software package [3]. Nomenclature A surface area, m 2 c specific heat, J/(kgK) d diameter, m g gravitational acceleration, m/s 2 G mass flux, kg/(m 2 s) h enthalpy, J/kg k thermal conductivity, W/(mK) L length, m m mass flow, kg/s p pressure, Pa q heat flux, W/m 2 r radius, m t temperature, C Greek letters α heat transfer coefficient, W/(m 2 K) Θ temperature, C ρ density, kg/m 3 τ time, s φ tube inclination angle, deg Criterial numbers Nu Pr Re Nusselt Prandtl Reynolds
46 Grądziel Sławomir and Maewski Karol / Procedia Engineering 157 ( 2016 ) 44 49 2. Mathematical model Equations describing the principles of the mass, momentum and energy conservation, respectively, will be solved to determine the distribution of the mass flow (mass flux) and the medium enthalpy (temperature) along the tube. The base balance equations are expressed in [4]. After appropriate reductions and transformations, mass, momentum and energy equations are written so that space derivatives should be obtained on the left side of the equations, whereas time derivatives occurring on the right side are replaced with backward difference quotients. This system of ordinary differential equations is solved using the Runge-Kutta method. After the changes described above are introduced, the energy balance equation takes the following form: dh A h h 4 t w dz m d (1) The fluid density is found as a function of enthalpy and pressure: f h, p (2) Solving the mass and momentum conservation equations, the following is obtained, respectively: dm dz A 2 2 2 m d 1 m m dp p gsin 2 dz A A dz (3) (4) The fluid temperature history is found as a function of enthalpy and pressure: t f h, p (5) Subscript in Equations (1-5) denotes the number of analysed cross-sections and varies from =1,2 M. All thermophysical properties of the fluid and of the wall, as well as the heat transfer coefficient, are determined on-line. The time- and space-dependent distribution of the tube wall temperature is also calculated using the transient thermal conductivity equation and assuming uniform heating of the tube with a heat flux with density q (Fig. 1) [5]: c 1 rk r r r w w w (6) under the following boundary conditions: k w r r ro q (7)
Grądziel Sławomir and Maewski Karol / Procedia Engineering 157 ( 2016 ) 44 49 47 rrin rr kw t t in r (8) Fig. 1. Analysed control volume of the tube. For the smooth tubes is many equations, that allow to calculate heat transfer coefficient. For presented values the Kitoh equation can be used [6]. In the case of rifled tubes, one of the ways to determine the convective heat transfer coefficient is to use the Chilton-Colburn factor. The quantity is determined based on the Reynolds number and the channel geometrical dimensions (Fig. 2a). Generally, the Chilton-Colburn parameter is found from the equation presented in [7]: St Pr 2 3 (9) Using the Stanton number definition [8], the convective heat transfer coefficient can be determined: p 1 3 h c GPr (10) In their studies, Webb et al. [9] determined the Chilton-Colburn factor as: e 0. 00933Re N di 0. 323 0. 181 0. 285 0. 505 (11) Zdaniuk and his team referred in their studies to equation (15) and proposed the following relation to define the factor [10]: e 0. 029 Re N di 0. 0877 0. 347 0. 253 0. 362 (12) 3. Numerical and CFD calculations In this section an internally rifled tube is analysed numerically. The numerical calculations was performed using the Fortran PowerStation 4.0 program and the commercial Ansys Fluent software package. The verification consisted of a comparison between results obtained by means of an in-house code (NUM) and those produced by the CFD program. To the calculations the initial temperature of water flowing through the Φ50.8 x 7.95 mm tube is t=313.4 C. Also the wall of a tube with length L = 60 m has the same initial temperature for time τ = 0. From the next time step on, an input function appears on the tube outer surface in the form of heating with heat flux. The numerical
48 Grądziel Sławomir and Maewski Karol / Procedia Engineering 157 ( 2016 ) 44 49 computations were performed using the following time and space steps: Δτ = 0.5 s and Δz = 0.5 m. The specific geometry dimensions of rifled tube shows the Table 1. The characteristic dimensions of rifled tubes and cross section analyzed tube are presented on the Fig. 2. Table 1. Dimensions of the used rifled tube. Dimension outer diameter, d o inner diameter (without ribs), d i minimum diameter, d min wall thickness, g rib height, h pitch, p rib at the base, a rib average width, b Value 50.8 mm 34.9 mm 32.9 mm 7.95 mm 1 mm 30 mm 5 mm angle between rib arms, α 45 rib helical angle, β 30 number of ribs, N 6 4.5 mm Fig. 2. Rifled tube characteristic dimensions (a) and analyzed tube cross-section (b). During the CFD calculations of the flow in internally ribbed tubes a 3D model was prepared. All domain was divided into fifteen 4-meter long sections. Each section is divided into approximately 1.1 million elements, which gives about 16.2 million finite elements. The boundary conditions applied in the calculations are identical to those adopted previously. The steam-water properties were approximated using a spline curve for the average pressure value assumed for the calculations. The temperature values obtained using the numerical program and from the CFD calculations are listed in Table 2. The results of the heat transfer coefficient calculations based on relation (10) for selected cross-sections are shown in Fig. 3. Table 2 Comparison of the medium temperature calculation results for selected cross-sections of the rifled tube (steady state) L [m] t [ C] t [ C] L [m] CFD NUM CFD NUM 5 329.7 335.2 35 406.7 403.8 10 352.1 355.4 40 413.9 409.1 15 369.9 372.3 45 424.2 416.6 20 383.1 385.1 50 438.1 427.2 25 392.2 393.7 55 455.7 441.8 30 401.3 399.2 60 466.1 460.9 a) b)
Grądziel Sławomir and Maewski Karol / Procedia Engineering 157 ( 2016 ) 44 49 49 Fig. 3. Calculated heat transfer coefficient in selected cross-sections of the internally rifled tube: (a) based on the Chilton-Colburn -factor from correlation (11); (b) based on the Chilton-Colburn -factor from correlation (12). 4. Conclusions An analysis of the presented results indicates good convergence between the numerical program and CFD modelling. The divergences in the fluid temperature distribution may result from the application of a different model of the heat transfer coefficient determination and the different methods of determining thermophysical properties of water. The differences in the Fig. 3 are connected to the mathematical models developed for rifled tubes with smaller diameters, another geometry and greater number of ribs. The accuracy of the calculations is also the effect of the bigger number of sections resulting from the division, which is confirmed by the high demand for computational power. Considering the above, it can be observed that during the modelling of thermal and flow phenomena occurring in both smooth and internally rifled tubes, the selection of appropriate relations describing the heat transfer and the water thermophysical properties for the parameters under consideration is essential. References [1] Duda, P, Rząsa, D, 2015, A method for optimum heating and cooling boiler components of a complex shape, Journal of Thermal Science 24 (4) pp. 364-369. [2] Fortran PowerStation 4.0, 1995. Microsoft Developer Studio, Microsoft Corporation. [3] Ansys Fluent 13.1, Ansys INC. [4] Grądziel S., 2012 Modelowanie zawisk przepływowo-cieplnych zachodzących w parowniku kotła z naturalną cyrkulacą [Modelling thermal and flow phenomena occurring in the natural circulation boiler evaporator], Wydawnictwo Politechniki Krakowskie, series: Mechanika, Monografia 406, Kraków. [5] Zima W., Grądziel S., 2013, Simulation of transient processes in heating surfaces of power boilers, LAMBERT Academic Publishing. [6] Kitoh K., Koshizuka S., Oka Yo., 1999, Refinement of transient criteria and safety analysis for a high temperature reactor cooled by supercritical water, In: Proceedings of the 7th International Conference on Nuclear Engineering (ICONE-7), Tokyo, Japan, 19-23 April, Paper No. 7234. [7] Zdaniuk G., Chamra L., Mago P., 2008, Experimental determination of heat transfer and friction factor in helically-finned tubes, Experimental Thermal and Fluid Science 32, 761-775. [8] Hewit G. F., 1992, Handbook of heat exchanger design, Begell House, INC., New York. [9] R.L. Webb, R. Narayanamurthy, P. Thors, 2000, Heat transfer and friction characteristics of internal helical-rib roughness, Transactions of the ASME: Journal of Heat Transfer 122, pp. 134 142. [10] Zdaniuk G., Chamra L., Mago P., 2008, Experimental determination of heat transfer and friction factor in helically-finned tubes, Experimental Thermal and Fluid Science 32, pp. 761-775. [11] Maewski K., 2013, Concept of a measurement and test station for determining linear pressure drop and heat transfer coefficient of internally ribbed tubes, Journal of Power Technologies, 93 (5), 340-346.