The limits of isothermal model validity in the case of natural convection cooling of heating blocks in a horizontal channel

Numerical simulations are conducted to study natural convection in a horizontal channel provided with heating blocks mounted on its lower wall. The blocks are submitted to a uniform heat generation and the channel upper surface is maintained at a cold constant temperature. The main objective of this study is to examine the validity of the model with isothermal blocks for the system under consideration. The controlling parameters of the present problem are the thermal conductivity ratio of the solid block and the fluid (0.1 ≤ k = ks ka ≤ 200) and the Rayleigh number (10 4 ≤ Ra ≤ 10). The validity of the isothermal model is examined for various Ra by using different criteria based on local and mean heat transfer characteristics.


Introduction
The study of convective heat transfer in systems provided with heating blocks has been motivated by its importance in many industrial problems such as cooling of heat generating components in the electrical, nuclear or chemical industries. In this kind of system, an important number of investigations have been performed experimentally [1][2][3] and numerically [4][5][6]. The literature review shows that the case when the blocks are heating with constant temperature (isothermal model) was considered in several studies including forced [3], natural [4] and mixed [6] convection. However, in various practical systems the blocks are submitted to a volumetric heat generation (case of volumetric heat dissipation in the electronic devices or that generated by chemical or nuclear reactions). It is well known that when the thermal conductivity of a solid block is large enough the temperature inside the latter becomes uniform (which leads to an isothermal block). Hence, the main objective of this study consists to determine the limits of isothermal model validity. Then, the calculations are performed by using two different models. A model with blocks generating a uniform volumetric power and another with blocks maintained at a constant temperature (deduced from the first model as an average surface temperature of the blocks).

Configuration and Modeling
The configuration studied is a horizontal channel of height ′ and having an upper surface maintained at a constant cold temperature. The channel is assumed sufficiently long and provided with an important number of rectangular heating blocks (of height ℎ′), periodically mounted on its lower adiabatic surface (Fig. 1). The present problem is studied using two models. In the first model (denoted M1) the blocks are submitted to a uniform volumetric heat generation, ′. The average temperature of the blocks surface, , calculated using M1 is then used as imposed temperature to the blocks in the second model (denoted M2). The flow is assumed two-dimensional and laminar and the coolant fluid (air) is assumed incompressible and obeying the Boussinesq approximation. Periodic conditions are applied to the system on the fictive boundaries 1 and 2 .

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The dimensionless equations in the fluid, written in stream function-vorticity formulation are: In the solid blocks, the equation of energy used for M1 is: The governing equations (1) to (5) were discretized by a finite difference method. The resolution of the corresponding algebraic system of the governing equations (1), (4) and (5), is done using the Alternating Directions Implicit method (ADI). The resolution of the Poisson equation (2) Fig. 2b and this, in the whole range of * considered in this study. For relatively high values of , two types of solutions (symmetrical and dissymmetrical) were obtained with M1. Each of these solutions leads to a specific temperature for the block and hereby two different solutions (symmetrical and dissymmetrical) for the M2. Then four solutions (two symmetrical and two dissymmetrical) are induced by M2 for a given set of governing parameters. Two of these solutions are found not to verify the energy balance between the volume and the surface of the block. Therefore, the latter are omitted in the following discussions.

Heat transfer
The threshold values of * above which the model M2 reproduces the quantity of heat Q corresponding to M1 with less than 5% difference will be denoted * . The evolution of * with the Rayleigh number for both symmetrical and dissymmetrical solutions is presented in Fig. 3. The denotations symmetrical and dissymmetrical are relative to the flow structures obtained with M1 at small values of * since the flow structures may change with * for some range of . In this figure, * is set equal to 0.1 when its value is lower than 0.1 since our study is limited to conductivities ratio * ≥ 0.1. Note that both symmetrical and dissymmetrical solutions are coexisting in the range 4 × 10 5 ≤ ≤ 10 7 . It can be seen from this figure that * corresponding to symmetrical solutions increases or decreases with depending on the range of the latter while it always increases with for the dissymmetrical solutions. The intersection between the curves of * indicates that, depending on the range of , the symmetrical solution hastens or delays (with respect to the dissymmetrical solution) the convergence of Q induced by the isothermal model towards that induced by M1. To determine the conductivity ratio above which the isothermal model allows the reproduction of the local profile of heat flux, a criterion based on relative local differences is defined as follows: In this expression, 1 and 2 denote the local heat fluxes corresponding to M1 and M2, respectively, and indicates the nodes on the block surface (as the summation is performed along the block surface). The value of the conductivity above which the condition (8) is satisfied is denoted * . A similar criterion is defined for local Nusselt (by replacing Q by in (8)) and the value of conductivity above which becomes lower than 5% is denoted * . Another conductivity ratio, * , characterizing the temperature uniformity inside the block is determined by a direct measure of the maximum relative temperature difference within the block, defined by the ratio . Similarly, * is defined as the value of * above which the ratio − ≤ 5%. The variations of * , * , * and * with are presented in Fig. 3. It can be seen that these quantities depend strongly on the solution type in the range of , where the multiplicity of solutions is obtained, and, generally, larger conductivities ratios correspond to . For > 10 5 , the different thresholds are such that * < * < * < * . For a relatively small range of ( < 7 × 10 4 ), it is found that * < * < * < * but * is slightly lower than * . The curves of * and * present sharp peaks at = 7.3 × 10 4 (for these peaks * = 79 and * = 81). In the vicinity of these peaks, the values of * , * are clearly much larger than those of * and * . In fact, for  Also the curves of Fig. 3

Conclusions
Natural convection induced in a horizontal channel provided with heated blocks is studied numerically. The validity of an isothermal model where the blocks are maintained at the average surface temperature deduced from the model with volumetric heat generation (model 1) is examined. For the cases where multiple steady solutions are obtained, it is found that some solutions induced by the isothermal model do not reproduce the results of the model 1. If one excludes the latter solutions, the * beyond which the validity of the isothermal model is confirmed depends on the Rayleigh number, the type of solution and the criterion used for the comparison. The criterion based on the total quantity of heat released by the blocks shows that the isothermal model is valid for some range of relatively large even for a * as low as 0.1. The conductivity ratio beyond which the isothermal model is valid may increase or decrease with depending on the range of the latter parameter and the type of solution.