Turbulence Modeling of Air Flow in the Heat Accumulator Layer

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1 PIERS ONLINE, VOL. 2, NO. 6, Turbulence Modeling of Air Flow in the Heat Accumulator Layer I. Behunek and P. Fiala Department of Theoretical and Experimental Electrical Engineering Faculty of Electrical Engineering and Communication Brno University of Technology, Kolejni 2906/4, Brno, Czech Republic Abstract The article deals with one layer of a heat accumulator which is suitable for solar systems. Here is presented numerical coupled model with turbulent flow thermal field depends on time. There is a numerical model of the air turbulence, heat transfer, conduction and also phase change of CaCl 2.6H 2 O which is used to increase the density of stored energy. The numerical solution was done by the help of finite element method (FEM in ANSYS software. DOI: /PIERS INTRODUCTION Phase changes of materials are a perspective way of thermal energy storage. The application of PCM offers a lot of advantage. We can reach higher density of stored energy. Table 1 shows the calculation for classical materials and PCM. The initial temperature 20 C and final temperature 50 C at the end of heating are supposed. Next advantage is a possibility to store heat at low temperature. We don t need to have such good thermal insulation and solar collectors work with better efficiency so the demand for area of collectors decreases. Table 1: The comparison of classical materials and PCM. material density of storage energy [kwh m 3 water 34.5 gravel 23.0 paraffine wax 62.4 CaCl 2.6H 2 O Na 2 CO 3.10H 2 O Na 2 HPO 4.12H 2 O MATHEMATICAL AND NUMERICAL MODEL OF TURBULENCE The model of air velocity distribution is derived for incompressible fluid as for a stabile state of flow stands from the energy conservation law. We suppose a turbulent flow div v = 0 (1 div v = 0 (2 rot v = 2ω (3 where ω is the angular velocity of fluid. If we use the Stokes theorem, the Helmoholtz theorem for moving particle and continuity equation, we can formulate from the balance of forces the Navier- Stokes equation for the fluid element t + v gradv = A 1 grad p + v v (4

2 PIERS ONLINE, VOL. 2, NO. 6, where A is an external acceleration and v kinematic viscosity. In the Equation (4 we can substitute pressure losses grad p = (K x v x v + fdh v x v + C x µv x u x (K y v y v + fdh v y v + C y µv y u y (K z v z v + fdh v z v + C z µv z u z (5 where K are suppressed pressure losses, f resistance coefficient, D h hydraulic diameter, C air permeability of system, µ dynamic viscosity and u unit vector of the Cartesian coordinate system. The resistance coefficient is obtained from the Boussinesq theorem f = are b (6 where a, b are coefficient from [3. The model of the velocity field is formulated from the condition of steady-state stability which is expressed fd + tdγ = 0 (7 where f are specific forces in area and t are pressures, tensions and shear stresses on the interface of area Γ. By means of the transformation into local coordinates we obtain a differential form for the static equilibrium f + div 2 T v = 0 (8 where T v is a tensor of the internal tension [ Xx X y X z T v = Y x Y y Y z (9 Z x Z y Z z where X, Y, Z are stress components which act on elements of area. It is possible to add a form of specific force from (1 (3 to the condition of static equilibrium. The form of specific force is obtained by means of an external acceleration A, by condition of pressure losses and shear stresses τ ( t + v gradv N s A F 1 + div 2 T v = 0 (10 where F 1 are discrete forces. The model which covers forces, viscosity, and pressure losses is ( t + v gradv N s A F 1 + grad p v v = 0 (11 We can prepare a discretization of Equation (20 by means of the approximation of velocity v and acceleration a v = a = N ϕ v vk W k (x, y, z, (x, y, z, k=1 N ϕ a vk W k (x, y, z, (x, y, z, (12 k=1 where v v, a v are immediate node values, W is a base function, N ϕ is a number of mesh nodes. If we apply the approximation (10 and Galerkin principle in (11 we get the semidiscrete solution ( vi W j [ W i t + N v N v N v N s W i v vi gradv vi + W i grad p i W i A vi W i F li d [ W j W i divv grad v vi d Γ [ W i W i X i dγ = 0, j = 1, 2,..., N V (13

3 PIERS ONLINE, VOL. 2, NO. 6, where X are the known conditions on the interface of area. We substitute pressure losses in (13 for the function (5 and we obtain the model of air flow W j W i d dv ( vi dwi + W j v vi dx u x dw i dz u z dv vi ( dwi + W j dx u x dw i dz u z dp i W j da i W j df li Γ ( dwj dx + dw j dy On the interface there are conditions + dw j dz v i ( dwi dx u x + dw i dy u y + dw i dz u z dv vi W j dγx i = 0, i, j = 1, 2,..., N v. (14 on the border Γ vr1 where n is a normal vector to direction of air flow n (v = 0 (15 n (p = 0 (16 on the border Γ vr2 where Γ vr1 Γ vr2 is the interface between the solid body and the liquid. We can write [ { } C f dv vi +[K sx K F x {v vi }+[K cx p=[k gx {A i}+[f bx {F li }+[F sx {X i } i, j =1, 2,..., N v. (17 Matrixes C f, Ksx, KF x, Kcx, Kgx, F bx, KF x can rewrite a form for an element of mesh [ { } Ce f dv vi + [ Ke sx K F x e {vvi }+[Ke cx p i =[Ke gx, F sx are related to coefficients of equations (14. We {A i }+[Fe bx {F li }+[Fe sx {X i } e=1, 2,..., N e. (18 For the solution we used the standard k-ε model, the standard k-ω model and the SST (Shear Stress Transport Model. The standard k-ε gives exact results and use two equations for turbulent kinetic energy and its dissipation. Model k-ω solves the equations for turbulent kinetic energy and its specific dissipation rate. This model gives better results in the nearness of wall but worse in the distance from wall. The SST model combines and switch between k-ε and k-ω model in order to get the best result (see [5, 6, NUMERICAL MODEL OF HEAT ACCUMULATOR LAYER There is geometric model of one layer of accumulator in the Fig. 1. It consists from 26 PVC pipes in the square configuration. Inside of pipes there are 9.36 liters of modified CaCl 2.6H 2 O. The air flows through the layer and transfers heat into pipes. Progress of numerical solution had two parts. First we solved turbulence model and got heat transfer film coefficients. These results were the input of solution for second part when thermal model was calculated. Time dependence of temperature distribution in the layer is final result. Figure 1: Geometric model of layer with mesh of elements.

PIERS ONLINE, VOL. 2, NO. 6, 2006 665 Initial and boundary conditions inlet temperature of the air is 50 C, inlet velocity of the air is 0.4 m s 1, outlet pressure is 101.

4 PIERS ONLINE, VOL. 2, NO. 6, Initial and boundary conditions inlet temperature of the air is 50 C, inlet velocity of the air is 0.4 m s 1, outlet pressure is kpa + 10 Pa, initial temperature of the air, PVC and CaCl 2.6H 2 O is 20 C There are distributions of velocity in Fig. 2 and next results for distribution of the turbulent kinetic energy, dissipation, temperature, and pressure in following pictures. Figure 2: Velocity distribution of the air. Figure 3: Distribution of kinetic energy, dissipation. Figure 4: Distribution of temperature and pressure. 4. CONCLUSION We presented a numerical model of one layer of heat accumulator. This heat accumulator exploits advantages and suppresses disadvantages of water and gravel accumulator. The numerical model

5 PIERS ONLINE, VOL. 2, NO. 6, was solved by means of FEM in ANSYS software. We solved coupled problem of air flow turbulence, heat transfer, conduction, convection, and also phase change. There are also problems with solution stability during iteration in ANSYS FLOTRAN. Finite element method (FLOTRAN is not suitable for turbulence modeling. The application of finite volume method (e.g., ANSYS CFX, FLUENT for more complicated tasks is necessary. The paper was prepared within the framework of the research plan No. MSM of the Ministry of Education, Youth and Sports of the Czech Republic. REFERENCES 1. Ansys User s Manual, Svanson Analysys System, Inc., Huston, USA, Specht, B., Modellierung von beheizten laminaren und turbulenten Ströungen in Kanälen beliebigen Querschnitts, Technische Universität Carolo-Wilhelmina, Braunschweig, Germany, Piszachich, W. S., Nonlinear models of flow, diffusion and turbulence, Teubner Verlagsgessellschaft, Leipzig, Germany, Wilcox, D. C., Turbulence Modeling for CFD, DCW Industries, Inc., La Canada, California, USA, Zhang, Y., Finite elemente zur Berechnung instationärer Strömungen mit bewegten Wandungen, Universität Stuttgart, Stuttgart, Germany, 1996.

Differential relations for fluid flow

Differential relations for fluid flow In this approach, we apply basic conservation laws to an infinitesimally small control volume. The differential approach provides point by point details of a flow