**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 = 10<sup>3</sup> to 10<sup>5</sup> ). 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.

**Keywords:** Correlation heat transfer coefficient, Forced convection, CFD simulation, Square cavity, Rectangular enclosures

### **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 heattransfer 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 heattransfer 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. [9–11], 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

#### *A New Forced Convection Heat Transfer Correlation for 2D Enclosures DOI: http://dx.doi.org/10.5772/intechopen.99375*

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 heattransfer 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.
