To validate the mathematical model, the store element of the ICS is analysed under two different working conditions. In the first, experimental data from other authors is available??. In the second, numerical solutions from more detailed numerical data (i. e. Direct Numerical Simulation, DNS) have been obtained in order to compare them with the solutions of the model here presented.

Case A. Constant heat flux problem.

The experimental test defined in [8] is reproduced numerically. The store element has a length of L = 154 cm and a width of cm. Heat losses through the sides of the

store are set to Ut = 0 and Ut = 0.375. Ambient and initial temperature are set at T0=7)=13

0C. At the beginning of the test, the water is at rest with a uniform temperature T*. The test consist of applying a constant uniform net solar irradiance of Gt-Gtrc/= 603.2 W/rri2 on the absorber. In the experiment, the absorber is modelled by means of electrical resistances. Experimental data is provided for the distribution of temperatures in the ж direction halfway the width of the store (i. e. y=4.05 cm) after 225 min of heating process.

To perform the numerical simulations, the second level of refinement (n = 10) is consid­ered. The time-step has been fixed to 2 seconds.

In Figure 2 numerical results are compared with experimental data. A general agreement is observed.

Case B. 450-Inclined differentially heated cavity problem.

A differentially heated cavity problem with water (Pr = 7.0), an inclination of 45° with respect to horizontal and an aspect ratio of L/W=A=10 is modelled both with the model here presented on five different levels of refinement, n=5, 10, 20, 40 and 80 and with a detailed numerical simulation (DNS). To do so, boundary conditions at the front and back plates have been changed from those described in the definition of the problem: the absorber and back plate surfaces are set at a uniform temperature (Dirichlet boundary condition) of To()S=80 0C and ТЬаск=20 0C, resulting into a Rayleigh number of, according

to the standard definition of Да for this kind of problems. Other sides are assumed adiabatic. Initially the fluid is considered at rest at 20 0C. And the transient evolution of the heat transfer and fluid flow is evaluated until the steady-state (or a steady-oscillating-state) is reached.

The goal of this study is to show the capacity of the numerical model here presented to simulate the water store in extreme working conditions.

Numerical solutions are compared to detailed numerical data from a Direct Numerical Simulation, where no assumptions of laminar flow are made. This DNS results are to be considered as the highest possible level of modelling, and are here taken as the reference solution (“real” solution) to see how far are the solutions of the numerical laminar model from reality. DNS results were obtained from a code developed in the Centre Tecnologic de Transferencia de Calor. In this code, the spatial discretization is carried out using a staggered mesh [4], and a spectro-consistent second-order scheme [10]. In this way the global kinetic-energy balance is exactly satisfied even for coarse meshes. For the tempo­ral discretization, a central difference scheme is used for the time derivative term and a fully explicit second-order Adams-Bashforth scheme for convection, diffusion and body force terms. An implicit first-order Euler scheme is employed for the pressure gradient term and the mass conservation equation. For more details see [9]. The computational domain is

discretized with 400 * 160 CVs. The mesh is also intensified at the walls along y-direction with a concentration factor of 1.0.

In figure 3, time averaged Nusselt and temperature profiles of the DNS are compared to instantaneous data obtained for three levels of refinement {n = 10, 20, 40) when the problem has reached a quasi-steady-state solution ( non-dimensional time 10^4). Good agreement is observed when the level of refinement is higher than.

This agreement is also observed in the mean Nusselt numbers (see table 3). The same agreement has been observed when comparing the solutions of the velocity fields.

Figure 3: Validation. Case B. Inclined differentially heated cavity problem. Numerical results for different levels of refinement (n=10, 20, 40) vs DNS data. (a) Nusselt at the cold wall. (b) Non-dimensional temperature profile aty=W/2.

Conclusions

Numerical methods on heat transfer and fluid flow have been consolidated during the last decades as an indispensable tool for the resolution of thermal and mechanical problems, being nowadays an essential complement to the experimental studies. These methods solve the multidimensional equations governing the fluid flow and heat transfer by converting them into a set of algebraic equations which are to be solved by means of some kind solver. In order to assess that the numerical solution is an appropriate solution of the mathematical
model, a large number of equations may be required (large number of CV), resulting in very large and time consuming calculations. However, and due to the huge recent improvement in the computation capacities and on the improvement of the numerical methods, nowadays is already possible to apply this kind of methods to solve problems of engineering interest.

This work has described a numerical model that solves the two-dimensional Navier — Stokes equations together with the energy equation in order to predict the heat transfer and fluid flow in a rectangular water store of a integrated collector storage system. With this model is possible to obtain a prediction of the behaviour of the store exposed to real outdoor weather data during 24 hours, only just with a few hours of computation (about 12 hours) on a standard personal computer.

Results have been presented in order to show that the numerical solutions obtained by the model are an appropriate approximation of the mathematical model used (verification), and in order to demonstrate that the numerical solutions fit with results obtained from exper­iments and from highest level simulation techniques (validation).

Acknowledgments

This work has been funded in part by The European Commission under the “Energy, Environment and Sustainable Development” Programme, Framework Programm V, 1998­2002, project contract number CRAFT-1999-70604.

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