A New Forced Convection Heat Transfer Correlation for 2D Enclosures

Abstract

This work presents a new parametric correlation for 2D enclosures with forced convection obtained from CFD simulation. The convective heat transfer coefficient of walls for enclosures depends on the geometry of the enclosure and the inlet and outlet openings, the velocity and the air to wall temperature difference. However, current correlations not dependent on the above parameters, especially the position of the inlet and outlet, or the temperature difference between the walls. In this work a new correlation of the average Nusselt number for each wall of the enclosure has been developed as a function of geometrical, hydrodynamic and thermal variables. These correlations have been obtained running a set of CFD simulations of a 3 m high sample enclosure with an inlet and outlet located at opposite walls. The varying parameters were: a) the aspect-ratio of the enclosure (L/H = 0.5 to 2), b) the size of the inlet and outlet (0.05 m to 2 m), c) the inlet and outlet relative height (0 m to 3 m high), and d) the Reynolds number (Rein = 103 to 105). Furthermore, a parametric analysis has been performed changing the temperature boundary conditions at the internal wall and founds a novel correlation function that relates different temperatures at each wall. A specifically developed numerical model based on the SIMPLER algorithm is used for the solution of the Navier–Stokes equations. The realisable turbulence k-ε model, and an enhanced wall-function treatment have been used. The heat transfer rate results obtained are expressed through dimensionless correlation-equations. All developed correlations have been compared with CFD simulations test cases obtaining a R2 = 0.98. This new correlation function could be used in building energy models to enhance accuracy of HVAC demands calculation and estimate the thermal load.

1. Introduction

The mechanisms for this energy transport are conduction, radiation and convection. While the study of conduction and radiation processes are founded on well-established analytical and numerical models, the treatment of convection is currently much less rigorous and precise (J. [1, 2]).

To characterise convection phenomena, fluid dynamics problems must be solved through computational fluid dynamics (CFD). The geometric complexity and variety of possible air-flow patterns and the integration of fluid dynamics problems with thermal simulation programs for buildings greatly complicate the treatment of these problems.

Heat-transfer coefficients inside of a building quantify the heat transfer rate between a wall and the surrounding air. Currently, thermal simulation programs assume that this coefficient has a constant value or can be calculated by assuming correlations for flat plates or using empirical correlations [3]. These hypotheses produce underestimations or overestimations of the heat-transfer coefficient and directly influence the air-conditioning demands for the building (J. [4]).

Heat-transfer coefficients are difficult to calculate because of the thermal and hydrodynamic boundary layer produced between the wall and fluid. Such calculations can be achieved through CFD tools and the use of dense meshes in close proximity to the walls. These procedures can capture temperature and velocity gradients, which are used to obtain the heat flow of the walls and heat-transfer coefficient. Because of the high computational complexity of these problems, heat-transfer coefficients for an entire building are calculated according to interconnected individual spaces.

However, the reference air temperature is a key factor in the calculation of heat-transfer coefficients according to Newton’s law of cooling. According to the study conducted by Saena [5], significant discrepancies are found among reported results, with many authors calculating heat-transfer coefficients using the air temperature in the immediate proximity of the wall, air temperature 10 mm away from the wall, and fluid temperature at the inlet; however, the majority use the average temperature of the room.

Thus far, studies have not addressed the general problem of a two-dimensional (2D) room under forced convection in which all of the geometric and hydrodynamic parameters of the problem are varied. Studies have obtained correlations that depend on some but not all of the geometric parameters. For example, the work by Al-Sanea [6] analyses the fixed positions and sizes of the inlet and outlet openings and the influence of the aspect ratio of the room and Reynolds number on the heat-transfer ratio at the roof. In the case of ASHRAE [3], the room is treated as a flat plate, and the flow pattern within the enclosure, which is determined by its geometry and the positions and sizes of its openings, is ignored.

Another interesting study of 2D models was conducted by Saeidi [7], who produced a 2D model with a fixed position of the inlet and variable position of the outlet, the size of both openings and the Reynolds number. That study, however, is centred in the laminar flow regime, and the working fluid is water; in addition, the authors presented results for the local Nusselt number at the walls.

One of the first studies on ventilated inner enclosures was conducted by [8], who focused on 2D models of k-ε turbulence. The results of that work have been experimentally verified.

In the works by Novoselac et al. [9, 10, 11], the effects of discrete heat flows in the heat transfer by convection in enclosures with displacement ventilation systems were studied. Regarding the correlation proposed by the aforementioned authors regarding the floor geometry, a correction factor is used in the correlation (Tw-Tin)/(Tw-Ta) to calculate the heat-transfer coefficient in forced convection, and it is a function of the temperatures of the floor and air entering the room.

In addition, a correlation study conducted by Beausoleil-Morrison [1, 12] characterised flow regimes that are commonly found in real buildings, and the most appropriate correlations were selected from among all of the available correlations. Their goal was to create an adaptive algorithm for modelling heat transfer by convection in programs for thermal simulations in buildings. The most adequate correlation was proposed by Churchill and Usagi [13] in mixed convection using an exponent n = 3 and employing the natural convection correlations of Alamdari and Hammond [14] and corrections for forced convection by Fisher [15].

Recently, attempts have been made to couple CFD and building energy simulation (BES) techniques to obtain more precise results. Zhai and Chen [16] recommended an iterative coupling method that transfers the temperatures of the inner room surface from BES to CFD and returns the convection, heat-transfer coefficient, and air temperature in the room from CFD to BES.

J.M. Salmeron [17] studied the efficiency of cross ventilation at night as a function of the air flow rate, flow pattern and distribution of thermal mass and considered the positions of the inlet and outlet openings as well as their influence on the heat-transfer coefficients of the walls through the use of CFD techniques; however, the sensitivity was not analysed for all of the geometric variables of the problem in this study.

Sudhir [18] a numerical simulation is carried out to analyse the effect of turbulent intensity on the flow behaviour of flow past two dimensional bluff bodies. Triangular prism, diamond and trapezoidal shaped bodies with the same hydraulic diameter D, a dimensionless length scale are taken into consideration as bluff bodies. The study reveals that transition SST Model can be efficiently used to cover both laminar and turbulent flow regimes to estimate the heat transfer. However, this study does not show attention to the size of the inlets and outlets made in this study for rectangular enclosures.

All of the published research works thus far have been strongly centred on turbulence models in the type of code used to solve the Eqs. (Q. [19]) or to reproduce the experimental results [8]. However, only limited studies have considered all of the variables required to calculate heat-transfer coefficients inside of an enclosure and obtained correlations that can be easily implemented in applications for thermal simulations. The latter is the goal of this work, where the precision of the results is a function of the accuracy of the correlations.

Our aim is to obtain a series of correlations that calculate the convective heat-transfer coefficients at each wall of a 2D model under forced convection, with inlet and outlet openings in opposite walls. The most general solution of this problem assumes that the correlations depend on the airflow velocity upon entry, dimensions of the enclosure and positions and sizes of the inlet and outlet openings.

First, we studied cases in which all of the walls are at the same temperature to obtain a correlation that is independent of the temperatures of the walls or air. Then, we studied cases where the walls are at different temperatures to introduce a correction factor into the correlation that is dependent on the temperatures of the walls and air entering the enclosure.

2. Material and methods

2.1 Formulation

An enclosure with 2D geometry under forced convection have been studied. This model corresponds to enclosures where the inlet and outlet openings are as large as the third dimension. In these cases, the 3D effects of flow are negligible compared with the effects that occur in the 2D plane [5]. In a 2D enclosure under forced convection with only one inlet and one outlet, three distinct typologies occur: typology 1 includes an inlet and outlet that are located on opposite walls; typology 2 includes an inlet and outlet that are located on adjacent walls; and typology 3, includes an inlet and outlet that are located on the same wall. The present work focuses on typology 1, which is the most common in construction, although the developed methodology is valid for typologies 2 and 3 as well. Figure 1 shows the geometric variables of the case being studied, including the dimensions of the enclosure, the positions and dimensions of the inlet and outlet openings and the temperature of the entry and walls.

Figure 1.

Definition of the variables for enclosures with inlet and outlet openings on opposite walls.

For the formulation of the problem working with dimensionless numbers, the most important being the following:

$$\mathit{Gr}=\frac{g·\beta ·\left(T_s-T_{\infty }\right)L_c^3}{{\nu }^2}$$E1

$$\mathit{Pr}=\frac{\nu ·\rho ·\mathit{Cp}}{k}$$E2

$${\mathit{Ra}}_L=\mathit{Gr}·\mathit{Pr}$$E3

$${\mathit{Re}}_{\mathit{\text{in}}}=\frac{V_{\mathit{\text{in}}}·W_{\mathit{\text{in}}}}{\nu }$$E4

$${\mathit{Nu}}_i=\frac{h_i·W_{\mathit{\text{in}}}}{k}$$E5

$$\mathit{Ri}=\frac{\mathit{Gr}}{{\mathit{Re}}^2}$$E6

$$y^{+}=y·\frac{u_{\tau }}{\nu }$$E7

The Richardson number indicates the relative importance of natural convection with respect to forced convection in mixed convection processes. This number determines the processes that are more important for convection: For low Richardson numbers (Ri < <1), the Reynolds number is larger than the Grashof number, and forced convection is predominant. In the opposite case (Ri > > 1), the effects of forced convection are negligible compared with that of natural convection. However, if the Richardson number lies somewhere between these two limits, both effects are important, and convection is considered mixed. The present work studies cases of predominant forced convection and does not consider the effects of natural convection.

2.2 CFD governing equations

To calculate the heat-transfer coefficients, the velocity, temperature and pressure fields of the enclosure must be determined. Thus, we employed the Navier–Stokes equations, which describe the fluid motion for a given set of boundary conditions. These equations along with the turbulence model and energy equation are solved at each node of the mesh.

The turbulence model employed here is the realisable k-ε model. This model differs from the standard k-ε model through a new formulation of turbulent viscosity and transport equation for ε. The equations for the 3D model are provided in tensor notation, where $x_i$ represents the variables X, Y and Z and $u_i$ represents the corresponding velocity components.

The continuity equation is as follows:

$$\frac{\partial \rho }{\partial t}+\frac{\partial \left(\rho u_i\right)}{\partial x_i}=0$$E8

The equation for conservation of momentum is as follows:

$$\frac{\partial \left(\rho u_i\right)}{\partial t}+\frac{\partial \left(\rho u_i{u}_j\right)}{\partial x_j}=-\frac{\partial p}{\partial x_i}+\frac{\partial }{\partial x_j}\left[\mu \left(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i}-\frac{2}{3}{\delta }_{\mathit{ij}}\frac{\partial u_k}{\partial x_k}\right)\right]$$E9

The energy equation is as follows:

$$\frac{\partial \left(\mathit{\rho T}\right)}{\partial t}+\frac{\partial \left(\rho u_{i}T\right)}{\partial x_i}=\frac{\partial }{\partial x_i}\left(\left(\frac{\mu }{\mathit{Pr}}+\frac{{\mu }_t}{{\mathit{Pr}}_T}\right)\frac{\partial T}{\partial x_i}\right)$$E10

The shear viscosity equation is as follows:

$${\mu }_t=\rho C_{\mu }{k}^2/\epsilon$$E11

The variable $C_{\mu }$ is calculated as follows:

$$C_{\mu }=\frac{1}{4.04+A_s\frac{k{U}^{\ast }}{\epsilon }}$$E12

where

$$U^{\ast }\equiv \sqrt{S_{\mathit{ij}}{S}_{\mathit{ij}}+{\stackrel{\sim }{\mathrm{\Omega }}}_{\mathit{ij}}{\stackrel{\sim }{\mathrm{\Omega }}}_{\mathit{ij}}};{\stackrel{\sim }{\mathrm{\Omega }}}_{\mathit{ij}}={\mathrm{\Omega }}_{\mathit{ij}}-2{\epsilon }_{\mathit{ijk}}{\omega }_k;A_s=\sqrt{6}\mathit{\text{cosϕ}};S_{\mathit{ij}}=\frac{1}{2}\left(\frac{\partial u_j}{\partial x_i}+\frac{\partial u_i}{\partial x_j}\right)$$E13

The “k” transport equation in the turbulence model is as follows:

$$\frac{\partial \left(\mathit{\rho k}\right)}{\partial t}+\frac{\partial \left(\mathit{\rho k}{u}_j\right)}{\partial x_j}=\frac{\partial }{\partial x_j}\left[\left(\mu +\frac{{\mu }_T}{{\sigma }_k}\right)\frac{\partial k}{\partial x_j}\right]+G_k+G_b-\mathit{\rho \epsilon }+S_k$$E14

where G_k represents the production of turbulent kinetic energy, which is common to all k-$\epsilon$ turbulence models and is given by

$$G_k=-\overline{u{´}_{i}u{´}_j}\frac{\partial u_j}{\partial x_i}$$E15

The term G_b represents the generation of turbulent kinetic energy because of buoyant forces when the system is under a gravitational field, and it is calculated as follows:

$$G_b=\beta g_i\frac{{\mu }_T}{{\mathit{Pr}}_t}\frac{\partial T}{\partial x_i}$$E16

where Pr_t = 0.72 is the Prandtl number for energy and β is the thermal expansion coefficient, which is calculated as follows:

$$\beta =-\frac{1}{\rho }\left(\frac{\partial \rho }{\partial x_i}\right)$$E17

The transport equation for $\epsilon$ from the turbulence model is as follows:

$$\frac{\partial \left(\epsilon \right)}{\partial t}+\frac{\partial \left(\mathit{\rho \epsilon }{u}_j\right)}{\partial x_j}=\frac{\partial }{\partial x_j}\left[\left(\mu +\frac{{\mu }_T}{{\sigma }_{\epsilon }}\right)\frac{\partial \epsilon }{\partial x_j}\right]+{\mathit{\rho C}}_1\mathit{S\epsilon }-{\mathit{\rho C}}_2\frac{{\epsilon }^2}{k+\sqrt{\mathit{\nu \epsilon }}}+C_{1\epsilon }\frac{\epsilon }{k}{C}_{3\epsilon }{G}_b+S_{\epsilon }$$E18

The source terms $S_{\epsilon }$ and $S_k$ can be defined for each case and are optional.

The coefficient $C_{3\epsilon }$ is calculated as follows:

$$C_{3\epsilon }=\mathit{tanh}\left|\frac{v}{u}\right|$$E19

The constants used in the realisable k-$\epsilon$ model are as follows:

$${\sigma }_k=1.0,{\sigma }_{\epsilon }=1.3,C_{1\epsilon }=1.44,C_{2\epsilon }=1.92,C_2=0.43,C_2=1.9$$E20

The heat-transfer coefficient between the surface of the wall and fluid in motion at different temperatures is provided by Newton’s law of cooling and is dependent on the total heat of the wall, the transfer area and the temperature difference between the surface and unperturbed fluid:

$${\overline{h}}_i=\frac{Q_i}{A_i\left(T_{\mathit{si}}-T_{\infty }\right)}$$E21

The total heat flow that is transferred between the fluid and the wall is calculated from the temperature gradient produced inside of the thermal boundary layer of the fluid through an integration of Fourier’s law at each wall. Because the heat flow depends on the temperature gradient, it is solved using CFD. Thus, the heat flow at a wall is obtained by integrating the temperature gradient along the wall, and it is affected by the fluid’s conductivity, as shown in Eqs. (22) and (23):

$$Q_{\mathit{CFD}}{\left|{}_X={\int }_0^{L}k\frac{\partial T}{\partial y}\mathit{dx}\right|}_{y=0}$$E22

$$Q_{\mathit{CFD}}{\left|{}_Y={\int }_0^{H}k\frac{\partial T}{\partial x}\mathit{dy}\right|}_{x=0}$$E23

Therefore, at the fluid–solid interface, the heat transfer by conduction equals the heat transfer by convection, and the average heat-transfer coefficients are as follows:

$${\overline{h}}_{\mathit{ix}}=\frac{{\left.{\int }_0^{L}k\frac{\partial T}{\partial y}\mathit{dx}\right|}_{y=0}}{A_i\left(T_{\mathit{si}}-T_{\infty }\right)}$$E24

$${\overline{h}}_{\mathit{iy}}=\frac{{\left.{\int }_0^{H}k\frac{\partial T}{\partial x}\mathit{dy}\right|}_{x=0}}{A_i\left(T_{\mathit{si}}-T_{\infty }\right)}$$E25

Eqs. (24) and (25) show that the temperature difference between the wall (T_si) and free fluid (T_∞) is proportional to the temperature gradient of the thermal boundary layer. Thus, increases of the temperature transferred between the wall and fluid correspond to increases of the gradient within the boundary layer, which maintains a constant ratio and indicates that the heat-transfer coefficient is temperature independent. This finding is valid for flat plates where the fluid is not perturbed by other walls and for enclosures with walls whose temperatures are all equal. However, this finding is not valid for cases in which the walls have different temperatures.

2.3 Computational model

The computational model presented here considers a steady flow, 2D geometry and incompressible Newtonian fluid. All of the fluid properties remain constant, and it behaves as an ideal gas. All of the properties are evaluated at the fluid’s average temperature within the enclosure. The phenomenon under study is forced convection; therefore, the velocities are large enough for all buoyancy effects to be negligible and for gravity to be ignored. The CFD results are obtained by solving the Navier–Stokes equations and energy equation through the finite volume method using the commercial software package Fluent V14 [20]. The SIMPLE (Semi-Implicit Method for Pressure Linked Equations) numerical algorithm developed by Patankar and Spalding [21] is also used. The equations for mass, momentum and energy are solved iteratively for the corresponding boundary conditions using numerical methods until convergence is reached for the variables of interest (velocity, temperature, pressure and heat flow at the walls).

One of the most important aspects to consider when using CFD tools is the construction of the mesh for the computational environment. The mesh utilised in the simulation is built from rectangular elements, and it enlarges from the walls toward the centre of the enclosure in a uniform fashion. This type of mesh produces a better convergence because there is a higher density of elements within the thermal boundary layer where the heat-transfer coefficient is calculated. To ensure the correct solution within the boundary layer, nodes must be placed within the viscous sub-layer. Thus, the first element must be located at a maximum distance of 1 mm from the wall, and the growth rate toward the centre of the enclosure must be between 10% and 15% [22].

The parameter that controls the correct solution of the viscous sub-layer is y+. This dimensionless parameter depends on the turbulence model [Eq. (26)]. Thus, for k-ε turbulence with “enhanced wall treatment,” the parameter y + must have a value of approximately one. Figure 2 shows an example of an enclosure represented by a mesh with an aspect ratio of one.

Figure 2.

Uniform-growth mesh for a 2D enclosure under forced convection.

$$y^{+}=y·\frac{u_{\tau }}{\nu }$$E26

2.4 Validation of the CFD methodology

To validate the CFD methodology employed in 2D enclosures under forced convection, the following problem with a known solution is used as a reference: the case of a flat plate. By solving this problem using the CFD method and comparing the results with those obtained by correlating the flat plate in both the laminar and turbulent regimes, the CFD methodology is validated. The problem to be solved is that of a horizontal plate of length L with an incident air flow at velocity v_in in the parallel direction. The circulating air is at temperature T_∞, whereas the flat plate is at temperature T_s. The computational domain is large enough so that it does not influence the solution. It is necessary to ensure that the number of nodes and type of mesh do not depend on the solution. The correct solution depends on the parameter y+, which must have a value of approximately 1. Therefore, the number of mesh elements and elements located within the limiting thermal layer for each case are adjusted to obtain y + ≈1.

To generalise the studied cases, the Reynolds number is set between 3 x·10³ and 7 x·10⁶ along the length of the plate in the direction of the air velocity. Thus, both the laminar and turbulent regimes are explored. The studied cases are presented in Table 1. To validate the methodology for the case of a flat plate, the CFD results are represented in terms of the average Nusselt number along with the results obtained through the flat plate correlations by Pohlhausen [23] and Reynolds analogy. As shown in Figure 3, the match between the CFD results and correlations is good and presents a relative error of less than 3%. Thus, the methodology employed for the mesh, convergence criterion and turbulence model are correct and can be used in the solution of 2D enclosures under forced convection.

Case	L(m)	V(m/s)	Re	Regime	Mesh	Δ(1^stnode)	yplus	Nu
1	1	0.5	3.31E+04	Laminar	300Vx100H	0.001	—	56.1
2	1	1	6.63E+04	Laminar	300Vx100H	0.001	—	78.3
3	1	2	1.33E+05	Laminar	300Vx100H	0.001	—	109.7
4	1	3	1.99E+05	Laminar	300Vx100H	0.001	—	133.8
5	1	4	2.65E+05	Laminar	300Vx100H	0.001	—	154.2
6	2	0.1	1.33E+04	Laminar	300Vx200H	0.001	—	36.5
7	2	0.5	6.63E+04	Laminar	300Vx200H	0.001	—	79.0
8	2	1	1.33E+05	Laminar	300Vx200H	0.001	—	110.5
9	2	1.5	1.99E+05	Laminar	300Vx200H	0.001	—	134.6
10	2	2	2.65E+05	Laminar	300Vx200H	0.001	—	154.8
11	2	3	3.98E+05	Laminar	300Vx200H	0.0001	—	188.8
12	3	3.5	6.96E+05	Turbulent	300Vx300H	0.0001	0.63	880.1
13	3	4	7.95E+05	Turbulent	300Vx300H	0.0001	0.71	981.5
14	3	4.5	8.95E+05	Turbulent	300Vx300H	0.0001	0.79	1082.5
15	3	5	9.94E+05	Turbulent	300Vx300H	0.0001	0.87	1181.8
16	3	6	1.19E+06	Turbulent	300Vx300H	0.0001	1.03	1375.9
17	6	7	2.78E+06	Turbulent	300Vx600H	0.0001	1.11	2722.1
18	6	10	3.98E+06	Turbulent	300Vx600H	0.0001	1.53	3661.3
19	6	15	5.96E+06	Turbulent	300Vx600H	0.0001	1.21	5161.7
20	6	17	6.76E+06	Turbulent	300Vx600H	0.0001	1.48	5754.2

Table 1.

Validation cases for a flat plate and forced convection.

Figure 3.

Validation of the CFD methodology for a flat plate and forced convection.

2.5 Case studies

2.5.1 Enclosures with walls at the same temperature

In 2D enclosures under forced convection where all of the walls are at the same temperature, which is generally different from the air temperature at entry, the variables can be varied continuously. For the present work, only three values are considered for each variable. The typical enclosure height employed for construction is 3 m; therefore, this variable is kept fixed and used to rescale all other variables and make them dimensionless. Table 1 shows the values used for each dimensionless variable. To solve the problem, a factorial experiment is conducted with six factors and four responses [24]. The variables are labelled as (X_j), and the variable of interest is labelled as a response Y_i. The relationship between factor and response is provided by the function Y_i = f(X_i).

For the 2D case being studied, there are four responses: the average heat-transfer coefficients at each wall (Nu₁, Nu₂, Nu₃ and Nu₄) and six factors or independent variables (V_in/H, L/H, W_in/H, W_out/H, H_in/H and H_out/H). Table 2 shows the numerical value of each variable. The goal is to determine the function f that best fits the results from the simulation, and it is therefore necessary to solve the full range of cases to obtain the most general correlation possible.

Vars. Adim.	Min.	Med.	Max.
Win/H	0.017	0.067	0.4
Wout/H	0.017	0.067	0.4
Hin/H	0.025-0.083-0.2166	0.5	0.975-0.917-0.783
Hout/H	0.025-0.083-0.2166	0.5	0.975-0.917-0.783
L/H	0.5	1	2
Rein(v = 0.5)	1657	6627	39764
Rein(v = 1)	3314	13255	79529
Rein(v = 1.5)	4971	19882	119293

Table 2.

Range of application of the variables (dimensionless).

To obtain enough data and the best fit to the results, all of the variables must be combined with their three possible values so that any possible combination within the variable applicability range is contemplated. Thus, the six variables can take three values each, which produces a total of 3⁶ = 729 cases. However, because the mechanism being studied is forced convection, gravity does not influence the motion of the fluid, which means that the fluid motion is independent of the spatial orientation of the enclosure. Therefore, we can infer that among all possible cases, certain cases will be symmetric with respect to the positions of the inlet and outlet (H_in and H_out). Figure 4 shows all of the possible cases when only the positions of the inlet and outlet are varied.

Figure 4.

Study cases for the positions of the inlet and outlet openings for typology 1.

Figure 4 shows that there are nine possible typologies as a function of the positions of the inlet and outlet openings. It is only necessary to study five out of these nine because the remaining typologies are symmetric cases because of forced convection. In these symmetric cases, the heat-transfer coefficients at the inlet and outlet (walls one and three, respectively) are equal; the coefficient of wall two is equal to that of wall four in its symmetric case, and wall four is equal to that of wall two in its symmetric case. Therefore, symmetry considerations reduce the number of cases to be studied from nine to five, thus affecting the values used for the variables H_in and H_out. The corresponding number of cases is reduced from 729 down to 405 (3·x 3·x 3·x 3·x 5). To avoid convergence problems and inconsistent results, all of the cases where the inlet opening is much larger than the outlet opening (W_in> > W_out) are eliminated. Thus, out of the nine possible combinations of W_in and W_out, three are eliminated. This reduced the number of cases from 405 to 270 (3·x 3·x 6·x 5).

Through this analysis, we have reduced the number of study cases as well as the simulation time and work involved in elaborating the meshes. Therefore, the parametric study is feasible, with 270 cases to be solved, and statistical methods are not required to reduce the range of study cases.

2.5.2 Enclosures with walls at different temperatures

In the cases where the wall temperatures are all different, the heat-transfer coefficient depends on the temperature distribution of the enclosure walls regardless of the occurrence of forced convection. Table 3 shows a comparison of the results for walls at equal temperatures with those for walls at different temperatures to demonstrate this influence.

Case	Tin	T1	T2	T3	T4	Ta	T∞	h1	h2	h3	h4
1	20	30	30	30	20	23.28	23.3	2.08	2.41	2.92	5.10
2	20	40	40	40	30	26.56	26.6	2.08	2.41	2.92	5.10
3	20	50	50	50	20	46.25	46.2	2.08	2.41	2.92	5.10
4	40	30	30	30	20	36.72	36.7	2.08	2.41	2.92	5.10
5	30	20	20	50	20	45.62	27.0	0.59	0.76	16.39	1.46
6	30	40	50	20	10	28.74	31.7	1.14	2.10	4.17	2.30
7	20	50	30	50	40	38.83	28.5	4.64	0.23	5.63	9.36
8	50	10	10	20	10	33.27	39.0	2.55	2.96	4.13	7.51
9	40	40	10	10	10	48.65	33.6	2.54	1.57	1.86	2.49
10	50	20	50	10	30	43.87	40.1	2.23	4.13	2.61	4.21

Table 3.

Casos estudiados en 2D con convección forzada y temperaturas de las Paredes distintas.

Table 3 shows that the heat-transfer coefficients obtained for different wall temperatures do not match those for walls at equal temperatures because the fluid temperature considered in Newton’s law of cooling is the average enclosure temperature (T_a), which is different from that of unperturbed fluid (T_∞). In cases where the wall temperatures are equal, both temperatures (T_a and T_∞) are consistent, and the heat-transfer coefficients are equivalent for all cases. However, when the walls temperatures are not equivalent, the mean temperature of enclosure T_a is not consistent with the unperturbed fluid temperature T_∞, which leads to different heat-transfer coefficients.

In these cases, a correction factor for temperature must be obtained that considers this phenomenon. This factor (F_i) relates the heat-transfer coefficient in the case of different wall temperatures (h^Ti≠) to the corresponding value for equal wall temperatures (h^corr) as follows:

$$h_i^{T_i\ne }=h_i^{\mathit{\text{corr}}}·F_i$$E27

Therefore, for a given $T_{\infty i}$ for cases solved using CFD, a correcting factor for the temperature jump $\left({\mathit{Ts}}_i-T_a\right)$ must be obtained so that the heat-transfer coefficient and the average air temperature $T_a$ can be calculated from this correcting factor. Newton’s law of cooling and the heat-transfer coefficient $h_i^{\mathit{\text{corr}}}$ for the case of walls with equal temperatures are applied, and the correcting factor is provided by Eq. (30):

$$Q_{\mathit{\text{wall}}\_i}^{\mathit{CFD}}=A_i{h}_i^{\mathit{\text{corr}}}\left({\mathit{Ts}}_i-T_{\infty i}\right)$$E28

$$Q_{\mathit{\text{wall}}\_i}^{\mathit{CFD}}=A_i{h}_i^{\mathit{\text{corr}}}\left({\mathit{Ts}}_i-T_a\right)·F_i$$E29

$$F_i=\frac{\left({\mathit{Ts}}_i-T_{\infty i}\right)}{\left({\mathit{Ts}}_i-T_a\right)}$$E30

Eq. (30) shows that the correction factor F_i depends on T_∞i, which is calculated using CFD through Newton’s cooling law as well as through the heat flow and heat-transfer coefficient obtained from the following correlation:

$$T_{\infty i}=\frac{Q_{\mathit{\text{wall}}\_i}^{\mathit{CFD}}}{A_i{h}_i^{\mathit{\text{corr}}}}$$E31

However, the heat flow is only known for CFD cases; therefore, a correction factor must be calculated that does not depend on T_∞ but depends on a known variable, such as T_in. This temperature was employed by Novoselac [10, 11] to obtain the correction factor in his experiments. Figure 5 shows the temperature difference using T_in and T_∞i, which produces an excellent match; thus, the variable T_in can be used in the correction factor.

Figure 5.

Relationship between T_in and T_∞i for each wall.

We conclude that the correction factor can be written as a function of T_in, T_a and T_si in the following equation:

$$F_i=\frac{{\mathit{Ts}}_i-T_{\infty i}}{{\mathit{Ts}}_i-T_a}=f\left(\frac{{\mathit{Ts}}_i-T_{\mathit{\text{in}}}}{{\mathit{Ts}}_i-T_a}\right)$$E32

This correcting factor depends on the flux and positions of the inlet and outlet openings. Thus, we have demonstrated how to calculate this factor and must determine a correlation expression for the factor for each typology of the enclosure. For that purpose, 100 cases are simulated using CFD, with 20 cases for each of the five enclosure typologies where the temperatures of the wall and inlet are varied randomly. Thus, a correlation of the correction factor can be obtained for each typology and for each wall of the enclosure.

3. Result and discussion

All of the cases defined in the previous section are solved using CFD by calculating the heat flow at every wall and the average temperature of the air in the enclosure. The solutions to all of the cases are considered to have converged when the convergence criteria adopted in the CFD simulations are met. In our case, the criteria are two-fold: the residues of the equations must be under 10⁻⁸; however, the monitored variable (heat flow at the wall) must converge to a constant value within at least 1000 iterations. With these constraints in the simulation software, most of the cases converge within 5000 iterations.

3.1 Proposed correlations for enclosures with walls at equal temperatures

To obtain a correlation that calculates the average heat-transfer coefficients of walls of a 2D enclosure based on geometric and hydrodynamic parameters, a mathematical treatment of the data is necessary to determine the function that best fits the simulation results. Additionally, the correlation must have the same form as the flat plate under forced convection. That is, it must depend on the following: the Reynolds number at the inlet to the power n; the Prandtl number for air (0.72); and a constant C that depends on all of the geometric parameters.

$${\mathit{Nu}}_i=C{·\mathit{Re}}_{\mathit{\text{in}}}^n\mathit{Pr}$$E33

$$C=f\left(\frac{L}{H},\frac{W_{\mathit{\text{in}}}}{H},\frac{W_{\mathit{\text{out}}}}{H},\frac{H_{\mathit{\text{in}}}}{H},\frac{H_{\mathit{\text{out}}}}{H}\right)$$E34

Once the simulation results are obtained, mathematical optimisation techniques are applied to obtain the correlation coefficients that best fit the results from the CFD simulations. The form of the correlation that best fits the CFD results is Eq. (35), which is obtained in dimensionless form through the Nusselt number and Eq. (36). The correlation coefficients for the walls are shown in Table 4.

2D Correlation coefficients forced convection
Wall		Wall 1	Wall 2	Wall 3	Wall 4
Coefficients	a	0.019512	0.041169	0.066311	0.009220
	b	−0.006984	−0.016679	−0.027515	−0.004926
	c	−0.005944	−0.033410	0.005247	−0.002107
	d	0.003044	−0.024028	0.005527	0.006855
	e	−0.005913	−0.002876	−0.003623	−0.001066
	f	0.016337	0.041022	0.020702	0.003956
	n	0.770041	0.782311	0.771561	0.868345

Table 4.

Correlation coefficients for 2D enclosures under forced convection.

$${\mathit{Nu}}_i=\left[a\frac{W_{\mathit{\text{in}}}}{H}+b\frac{W_{\mathit{\text{out}}}}{H}+c\frac{H_{\mathit{\text{in}}}}{H}+d\frac{H_{\mathit{\text{out}}}}{H}+e\frac{L}{H}+f\right]{·\mathit{Re}}_{\mathit{\text{in}}}^n$$E35

$$h_i=\frac{{\mathit{Nu}}_{i}k}{W_{\mathit{\text{in}}}}$$E36

Figures 6 and 7 show a comparison between the results from correlations and those obtained from the simulations. A mean error of approximately 15% was observed; however, the correlation describes the phenomenon with sufficient precision despite the complexity of the cases being studied.

Figure 6.

Comparison of Nu1 and Nu2 obtained through correlations and simulations.

Figure 7.

Comparison of Nu3 and Nu4 obtained through correlations and simulations.

3.2 Proposed correlations for enclosures with walls at different temperatures

In cases where the walls are at different temperatures, the heat-transfer coefficient is obtained from the corresponding coefficient through correlations for walls at the same temperature with the correcting factor. This factor depends on the flow and the positions of the inlet and outlet. Figure 8 shows a correlation of the correction factors for each wall of an enclosure of typology 2, where a variety of cases have been simulated, and the temperatures of the walls and air at the entry are varied. As shown in Figure 8, the fit and function that relates the correcting factor obtained through $T_{\infty i}$ to that obtained through $T_{\mathit{\text{in}}}$ are good. To appreciate the quality of the fit, Figure 9 shows the heat-transfer coefficient predicted using the correcting factor from CFD for each of the walls. The precision is high, which demonstrates that this factor must be employed to precisely calculate the heat-transfer coefficients when the walls have different temperatures. We proceed in the same fashion for the remaining typologies. Table 5 shows the coefficients of the correlation for the correction factor for each wall and for the five typologies. Eq. (37) provides the heat-transfer coefficient when the wall temperatures are different.

Figure 8.

Correlation between the correcting factor with T_∞i and T_in. Enclosure of typology 2.

Figure 9.

Heat-transfer coefficient calculated using the correcting factor compared with the heat-transfer coefficient calculated using CFD.

Table 5.

Correcting factors for all typologies.

$$h_i^{T_i\ne }=h_i^{\mathit{\text{corr}}}·\left(a_i\left(\frac{{\mathit{Ts}}_i-T_{\mathit{\text{in}}}}{{\mathit{Ts}}_i-T_a}\right)+b_i\right)$$E37

3.3 Effect of the ACH and W_in on the heat-transfer coefficients

To analyse the effect of the number of air changes per hour on the heat-transfer coefficients for each of the typologies, we employ the correlation obtained for cases where the walls are at the same temperature. Within the application range of the variables, Figure 10 shows the maximum values of the heat-transfer coefficients for each wall and typology compared with the number of air changes per hour. The maximum values are reached at wall two and its symmetric pair wall four; however, wall one shows the least sensitivity to variations in the number of changes. In cases where the enclosure is used for storing energy during the day and releasing it at night through ventilation, Figure 10 shows that typology 1 provides the highest heat transfer rates through cross ventilation. More specifically, the floor and roof (in the symmetric case) are the recommended enclosures for energy storage because they have higher heat-transfer coefficients and larger solid-to-fluid heat transfers compared with the rest of the enclosures. This is a good example that can be used as a design tool.

Figure 10.

Effect of the number of air changes per hour and typology on the heat-transfer coefficients at every wall.

To analyse the influence of the variable W_in on the heat-transfer coefficients as a function of the typology, Figure 11 shows the maximum values of the heat-transfer coefficients reached for each wall. The maximum values are obtained for typology 1, and in general, the coefficient decreases as the opening size increases because increases in opening size lead to decreases in the speed of the air that interacts with the walls and thus reduces the heat exchange between the wall and air.

Figure 11.

Effect of the size of the inlet opening and typology on the heat-transfer coefficients at every wall.

4. Conclusions

Heat-transfer coefficients are important parameters in the thermal modelling of a building and significantly influence the required air conditioning. The complexity of calculating these coefficients requires the use of CFD techniques that are not available to all users, which justifies the calculation of correlations that depend on the geometric parameters of a 2D enclosure under forced convection. We found that the CFD results are similar to those obtained using correlations for the flat plate model. The latter has the advantage of being velocity-independent in the proximity of the walls and is only dependent on the air speed at the inlet, which is known.

Another aspect highlighted in this study is the calculation of a correction factor that can be used when the wall temperatures are different. In addition, we obtained a correlation of the correction factor for each topology, which correct for the heat-transfer coefficients when all of the walls are at the same temperature.

Thus, our results indicate that the variables with the highest influence on the heat-transfer coefficients are the number of air changes in an enclosure followed by the size of the inlet.

Acknowledgments

This work is linked to two research projects, the first, and most socialising, is the project of National Plan “The air.es” entitled “Numerical modeling of combined heat inside buildings aeraulics oriented design eco-efficient “(MTM2012-36124-C02-00). This project is developed in collaboration with the research group FQM120 “Mathematical Modelling and Simulation of Environmental Systems”, University of Sevilla.

Nomenclature

H

enclosure height (m)

L

enclosure length (m)

Win

inlet opening dimension (m)

Wout

outlet opening dimension (m)

Hin

height inlet(m)

Hout

height outlet(m)

Tin

air inlet temperatura (°C)

Ti

wall temperature i (1, 2, 3, y 4)

Gr

Grashof number

Nu

Average Nusselt number

Pr

Prandtl number

Re

Reynolds number

Ri

Richardson number

Ra

Rayleigh number

Pr

Prandtl number

k

Thermal conductivity of air (W/mK)

ρ

density (kg/m3)

Cp

specific heat (J/kg·K)

ν

kinematic viscosity (m2/s)

β

coefficient of thermal expansion (1/K).

y+

y plus

μ_τ

friction velocity