Numerical Determination of Effects of Wall Temperatures on Nusselt Number and Convective Heat Transfer Coefficient in Real-Size Rooms

Abstract

A modeled room was numerically heated from a wall and cooled from the opposite wall in order to create a real-room simulation. The cooled wall simulated heat loss of the room, and the heated wall simulated the heat source of enclosure. The effects of heated and cooled wall temperatures on convective heat transfer coefficient (CHTC) and Nusselt number in the enclosure were investigated numerically for two- (2D) and three-dimensional (3D) modeling states. Different hot wall and cold wall temperatures were applied in order to obtain correlations that contained characteristic length in Rayleigh numbers. Results were compared with the results of previously reported correlations that have been suggested for vertical room surfaces in enclosures. In addition, new correlations for Nusselt number and average CHTC for enclosures for isothermal boundary conditions within indicated Rayleigh number ranges were derived through solutions. Average deviations of new correlations obtained for CHTC and Nusselt number from the numerical data were found 0.73% and 1.76% for 2D study, 3.01%and 0.49%for 3D study. It was observed that the difference between the 2D and 3D solutions in terms of CHTC and Nusselt number was approximately 10%.

1. Introduction

Because of their importance in industry and living places, natural convection and temperature and velocity distribution problems in enclosures have been widely studied not only numerically but also experimentally in appropriate real-size building geometries. In the era in which we have been encountering many energy crises, natural convection problems in real-size rooms also has great importance since it remarkably affects energy consumption in buildings. Heating and cooling systems with vertical walls, floor, and ceiling (radiant panels) have not been extensively utilized for many years. Nevertheless, especially heating technologies with low water temperature have begun to be reassessed due to the austere energy policies pursued by governments throughout the world. As well as the progression of renewable energy techniques and low temperature heating systems, improving computational methods and descending solution periods for calculation of natural convection problems in real-size building rooms provide us to use computational fluid dynamics (CFD) programs in order to obtain convective heat transfer coefficients (CHTCs) using proper numerical solution preferences. Although conduction and radiation heat transfer simulation models in room size enclosures have been thoroughly described by numerous researchers, in convection there are still some uncertainties. Difficulties in accurate convection modeling especially analytical and numerical methods arise from complexity of the enclosure geometry, solution of fluid dynamics problems (governing equations), and differences at each air flow pattern in each heating choice.

ASHRAE HVAC Systems and Equipment Handbook—Fundamentals Handbook [1] lists many CHTC values and correlations. However, these equations were derived with the assumption that CHTC in an enclosure is equal to free edge isolated plates. Nonetheless, air flows at surrounding walls, even if they are not heated or cooled, affect the flows on adjacent walls. Also, the air flow on all surfaces affects the overall flow pattern in the enclosure. Thus, correlations for free plates cannot be accurately used for natural convection problems in enclosures [2]. It can also be correlations derived through 2D enclosure modeling that cannot be utilized due to adjacent wall effects that are neglected in analytical expressions or numerical procedures.

Furthermore, experimental and computational studies have revealed that convection CHTCs of real-room surfaces are affected by the depth or height of the room, boundary conditions of the surface, smoothness/roughness of surfaces, and whether forced convection is present in the room.

In their study, Beausoleil-Morrison [3] showed the effect of CHTC correlations over building heating load predictions. They conducted experimental studies in a well-insulated test buildings that had radiant heating systems. Utilizing measurements from these test buildings and many different CHTC correlations, Beausoleil-Morrison found an 8% difference among the different simulations. He also found that building load amounts were more sensitive to CHTC correlations and the control set point of the room than to building fabric thermal properties or air infiltration. Peeters et al. [2] published an extensive review on experimentally derived CHTCs in enclosures and over free plates. They classified these correlations according to heating conditions, flow intensity, and also reference temperature preference in the enclosure. They questioned robustness of the correlations derived by numerous researchers and carried out new experiments in order to validate correlations. They concluded that large discrepancies were present among existing correlations in the literature. The differences were attributed to values of predicted coefficients, chosen reference air temperatures, and formats of the correlations. They also indicated that determination of appropriate choice of characteristic length was crucial. Nonetheless, it was asserted that choice of these parameters in building simulation programs was limited due to the common single node approach.

Awbi and Hatton [4] studied natural convection in two different enclosures. The enclosures' dimensions were 2.78 by 2.30 by 2.78 m and 1.05 by 1.01 by 1.05 m. One wall in each of the enclosure was used as a “heat sink” through an air conditioner placed in a small room next to the large enclosure. Opposite and adjacent walls to the “heat sink” wall have been heated with impregnated flexible sheets that had a 200 W/m² output. Thermal sensors were located inside on both sides of the surfaces. The aim of placing sensors on the outer surface of the enclosure was to calculate heat loss from heated surfaces. Reference air temperature for wall heating system was determined as 100 mm from heated surface and defined as “undisturbed air temperature” or in other words the temperature outside the thermal boundary layer. Thermal radiation was calculated through measuring emissivity of the surfaces and was extracted from total heat flux. Because they also heated the walls partially, characteristic length was determined as equal to hydraulic diameter. The results demonstrated that CHTCs of the heated wall were lower for the small enclosure that had a volume of approximately 1 m³ than for the larger enclosure. Nonetheless, in order to assess whether the difference was because of the heating plate sizes or enclosure sizes, more experiments were carried out with small plates placed on surfaces. From these experiments, it was determined that there was a close agreement between CHTCs calculated with whole wall heated and CHTCs calculated with small plates heated. Consequently, the authors asserted that rather than the heated area on a wall, the size of the enclosure significantly affects CHTC. They also compared their results with the correlations in the extant literature and found that their data fell in the middle of the curves. The correlations that they suggested for walls are presented in Table 1.

Table 1:

Correlations derived for CHTC and Nusselt number in enclosures.

Correlation	Conditions	Reference temperature

Awbi and Hatton [4]
$h = \frac{1.823}{D^{0,121}} {(Δ T)}^{0,293}$ Nu = 0.289 (Gr)^0,293	Heated wall (2.78 × 2.30 × 2.78 m)	Local air temperature (100 mm from wall)
Khalifa and Marshall [5]h = 2.3 (ΔT)^0,24	For a room with a heated wall and a room heated by a radiator under a window (2.35 × 2.95 × 2.08 m)	Average room temperature
h = 2.10 (ΔT)^0,23	For a room heated by a fan heater (2.35 × 2.95 × 2.08 m)	Average room temperature
ASHRAE [1]
Nu = 0.117 (Gr)^0,117	Turbulent flow 10⁸ < Gr < 10¹² (for free plate)	Average room temperature
Li et al. [6]
h = 3.08 (ΔT)^0,25	Normal conditions, occupied room	Average room temperature
Min et al. [7]
$h = 1.646 \frac{{(Δ T)}^{0,32}}{H^{0,05}}$	Heated floor or heated ceiling	Average room temperature

Awbi [8] presented the results of a CFD study on natural CHTCs of a heated wall, a heated floor, and a heated ceiling. Two turbulence models were used: standard k-∊ model using wall functions and low Reynolds k-∊ model. The results were compared with experimental results.

Fohanno and Polidori [9] aimed to develop a theoretical model of convective heat transfer between an isolated vertical plate and natural convective flow. They assumed a constant heat flux on surface, and the model they produce allowed average and local CHTCs to be calculated in laminar and turbulent regime. The correlation they derived for average CHTC was produced through local CHTC results. They indicated that there was a good agreement between Alamdari and Hammond's correlations. The applicability of the new correlations to real-size buildings was considered, and despite the three-dimensionality of real-size rooms, a 10% difference was found between data taken from experimental and mathematical work.

To calculate CHTCs for all surfaces in an enclosure, Khalifa and Marshall [5] utilized an experimental enclosure that had dimensions similar to those of a real-size room. Sixty-five aluminium thermistors were used to measure air and surface temperatures in the enclosure. Inner and outer surfaces of the enclosure were coated with aluminium. Radiant heat exchange was not counted in the CHTC calculation process. Also, a sensitive uncertainty analysis was employed. In this analysis, temperature measurements, conductivity of the materials, and the lack of inclusion of long wave radiation in the low emissivity chamber were considered in order to take account of all possible error sources [2]. They derived many general correlations, including the two that are presented in Table 1.

Khalifa and Khudheyer [10] conducted an experimental investigation on the effects of 14 different configurations of partitions on natural convection heat transfer in enclosures. Like other studies, the experiment considered vertical hot and cold walls, while the other walls were insulated. Investigation was conducted for Rayleigh numbers between 6 · 10⁷ and 1.5 · 10⁸ with the aspect ratio of 0.5. Correlations for the test configurations were derived. Khalifa [11] presented an extensive review of studies regarding isolated vertical and horizontal surfaces. Comparisons between correlations for heat transfer coefficients were conducted, and the discrepancies were determined to be up to a factor of 2 for isolated vertical surfaces, up to a factor of 4 for isolated horizontal surfaces facing upward, and up to a factor of 4 for isolated horizontal surfaces facing downward.

Khalifa [12] presented a wide review of two- and three-dimensional natural convection problems, focusing primarily on heat transfer in buildings. He determined the discrepancies between correlations to be up to a factor of 5 for vertical surfaces, a factor of 4 for horizontal surfaces facing upward, and up to a factor of 8 for horizontal surfaces facing downward.

In their study, which can be considered as the first experimental investigation for CHTCs in enclosures, Min et al. [7] studied within the range of Rayleigh number 10⁹ to 10¹¹ and with enclosure dimensions 3.60 by 7.35 by 2.40 m, 3.60 by 7.35 by 3.60 m, and 3.60 by 3.60 by 2.40 m. These correlations were derived for nonventilated conditions. Surfaces which were not heated were kept at constant temperature. Temperatures of the surfaces and heat fluxes applied to the enclosure were measured. Also, radiation effects were recorded. While the temperatures of the surfaces that were not heated varied between 4.4°C and 21.1°C, temperatures of floor surfaces varied between 24°C and 43.3°C and temperatures of ceiling surfaces varied between 32.2°C–65.6°C [2].

Li et al. [6] investigated natural convection in an occupied office room with normal working conditions up to a temperature difference of 1.5°C.

Karadağ [13] numerically investigated the relationship between radiation and convection heat transfer coefficients at ceilings when the floor surface was isolated. To achieve this goal, first Karadağ neglected radiative heat transfer at surfaces (∊_w = ∊_c = 0) for different room sizes (3 by 3 by 3 m, 4 by 3 by 4 m, and 6 by 3 by 4 m) and thermal boundary conditions (T_c = 0–25°C, T_w = 28–36°C). Then Karadağ determined radiative heat transfer for different surface emissivities (∊_w = ∊_c = 0.7–0.8 and 0.9). Numerical data were compared with results in the literature. The ratios of radiative heat transfer to convective heat transfer coefficients were calculated, and it was observed that ratios varied from 0.7 to 2.3.

Karadağ et al. [14] numerically analyzed changes in Nusselt number with ceiling and floor temperatures and room dimensions. While wall temperatures were kept at constant, ceiling temperature ranged from 10 to 25°C for different room dimensions. It was observed that when the temperature between the ceiling and air was raised, the Nusselt number over the floor also increased. Correlations in the literature that did not take into account the ceiling and floor temperatures deviated up to 35% from the results of this study. As a result, it was indicated that a new correlation for Nusselt number over the floor that encompasses the effect of thermal conditions and all room dimensions must be explored.

Although many other laminar regime natural problems in an enclosure have been solved in the literature, the turbulence natural convection problem in an enclosure of a similar size as a real room that was heated from one wall and cooled from the opposite wall has not been thoroughly researched with numerical methods, especially due to the length of solution periods. The main purpose of this study is to numerically investigate the effect of hot and cold wall temperatures of the room and characteristic length on average Nusselt number and CHTC for an enclosure modeled two- and three- dimensionally. Then, the results were compared with the correlations in the literature that are presented in Table 1. The enclosure's right and left walls were heated and cooled with constant wall temperature, while the other walls of the enclosure were kept adiabatic with the heat flux input “0 W/m²”.

2. Numerical Method and Mathematical Formulation

Because of long solution periods of governing equations in large geometries, utilization of computational fluid dynamics (CFD) programs has not been practical until a few years ago. However, today's improving computer technology allows us to utilize CFD programs in this field. Despite the frequent mesh density in the boundary layers of room model, CFD programs provide results in reasonable periods, especially in 2D solutions but also in 3D solutions. In this study, we examined numerical methods for calculating average Nusselt number and CHTCs over the heated wall of an enclosure that had similar dimensions to a room of a building. 2D and 3D natural convection problems in enclosure with the dimensions of 4.00 by 2.85 by 4.00 were considered. In order to acquire similar wall temperatures with wall heating systems, the heated sidewall was heated within the range of 20°C to 35°C, while cooled sidewall was cooled to the temperatures between 5°C–15°C. The other walls of the enclosure were kept adiabatic.

The most significant aspect of solving a heat transfer problem in a real-size room by means of a CFD program is to model and properly mesh of the enclosure. For 2D and 3D modeling the enclosure, GAMBIT 2.4.6, a modeling and meshing program, was chosen. In simple geometries as in this study's model, quad/hex meshes provide more qualified solutions with fewer cells than a tri/tet mesh. Thus, all the surfaces of the enclosure were chosen as plane quad/hex meshes in this study. To decrease the meshing effects on the results in each modeling type (two- and three- dimensional) edges of both 2D and 3D modeled rooms have been meshed with different “interval counts.” In order to observe whether the mesh effect has been widely diminished, Nusselt numbers were taken from FLUENT for each modeling condition. Since, the difference between the last two solutions is so small, the interval count before the last one (120 × 100) was chosen as appropriate number. In the 2D model, distance between heated and cooled walls (characteristic length, L = 4,0 m) was divided into 120 intervals. These intervals were preferred to be frequent near walls due to the rapid temperature differences in surface boundary layers; nonetheless, the intervals were determined less frequent towards the center of the model (Figures 1 and 2) due to the fact that temperature fluctuation from outside of the boundary layer to center of the room is approximately zero, as can be seen in Figure 3. To achieve this, “first length”—the ratio of midpoint interval size at the edge to the first interval size at the corner of edge—was determined as 0.001. Also, in 3D solution, various interval counts on three axes were employed and appropriate interval counts were found (45 × 40 × 20). Air was chosen as the fluid existing in the enclosure.

Figure 1:

Meshing detail of two-dimensional room (L × H = 4,0 m × 2,85 m).

Figure 2:

Meshing detail of three-dimensional room (L × H × L = 4,0 m × 2,85 m × 4,0 m).

Figure 3:

Temperature variation along the center of enclosure, from midpoint of hot wall to cold wall (L × H × L = 4,0 × 2.85 × 4,0 m).

FLUENT 6.3 software, one of the most common used codes, was utilized to solve governing equations (energy, momentum, continuity, and turbulence). The program's solution technique is focused on control volume theory turning governing equations into algebraic equations in order to solve them. The control volume technique works by integrating the governing equations for each control volume and then generating discretization of the equations, which conserve each quantity based on control volume [15]. In the model, the key dimension used is characteristic length (L = 4.0 m)—which is the difference between heated and cooled walls, that is opposite walls. According to characteristic length, the Rayleigh numbers for the system were calculated, as shown in (1) for one example.

Full Rayleigh numbers table can be seen in Tables 2 and 3. Because all the Rayleigh numbers calculated were greater than 10⁹, a turbulence model was applied in the flow pattern. The first order upwind scheme was utilized to discretize governing equations. The under relaxation factors for density, momentum, turbulence kinetic energy, turbulence dissipation rate, turbulent viscosity, and energy (1.0, 2.0, 0.8, 0.8, 1, and 0.9) were preferred to converge the solution. Simulations were performed on a laptop with an Intel Core i5 processor and the following specifications −2430 M CPU 2.40 GHz, 6 GB Ram, Windows 7 Home Basic 64 Bit SP1 operating system. The required solution period for each appropriate (meshing effect minimized) grid model was about 12 hours:

Ra = \frac{g β L^{3} (T_{H} - T_{C})}{ν α} = \frac{9.81 \times 0.0034 \times 4^{3} (30 - 10)}{15.26 \times 1 0^{- 5} \times 2.15 \times 1 0^{- 5}} = 1.3 \times 1 0^{11} > 1 0^{9} .

(1)

The governing equations for turbulence 3D flow can be written as follows.

Table 2:

Change of Nusselt numbers and CHTCs at various wall temperatures (2D solution results, L × H × L = 4.00 * 2.85 * 4.00 m).

T_H (°C)	T_c = 5°C			T_c = 10°C			T_c = 15°C
T_H (°C)	Ra	Nu	h (W/m²K)	Ra	Nu	h (W/m²K)	Ra	Nu	h (W/m²K)

20	1.1E + 11	721.52	4.53	7.03E + 10	614.2	3.88	3.38E + 10	472.16	3.01
25	1.41E + 11	783.07	4.96	1.01E + 11	695.23	4.43	6.51E + 10	590.85	3.8
30	1.69E + 11	832.38	5.31	1.3E + 11	754.31	4.85	9.39E + 10	669.71	4.34
35	1.95E + 11	870.14	5.59	1.57E + 11	801.39	5.2	1.21E + 11	727.26	4.75

Table 3:

Change of Nusselt numbers and CHTCs at various wall temperatures (3D solution results, L × H × L = 4.00 * 2.85 * 4.00 m).

T_H (°C)	T_c = 5°C			T_c = 10°C			T_c = 15°C
T_H (°C)	Ra	Nu	h (W/m²K)	Ra	Nu	h (W/m²K)	Ra	Nu	h (W/m²K)

20	1.1E + 11	647.11	4.06	7.03E + 10	550.66	3.48	3.38E + 10	423.68	2.7
25	1.41E + 11	703.08	4.44	1.01E + 11	623.9	3.98	6.51E + 10	530.78	3.41
30	1.69E + 11	747	4.76	1.3E + 11	677.82	4.35	9.39E + 10	601.35	3.89
35	1.95E + 11	781.9	5.02	1.57E + 11	720.13	4.66	1.21E + 11	653.27	4.26

Continuity equation:

\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = 0 .

(2)

Momentum equations:

\begin{matrix} ρ (u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} + w \frac{\partial u}{\partial z}) = - \frac{\partial p}{\partial x} + μ (\frac{\partial^{2} u}{\partial x^{2}} + \frac{\partial^{2} u}{\partial y^{2}} + \frac{\partial^{2} u}{\partial z^{2}}) \\ ρ (u \frac{\partial v}{\partial x} + v \frac{\partial v}{\partial y} + w \frac{\partial v}{\partial z}) = - \frac{\partial p}{\partial y} + μ (\frac{\partial^{2} v}{\partial x^{2}} + \frac{\partial^{2} v}{\partial y^{2}} + \frac{\partial^{2} v}{\partial z^{2}}) \\ ρ (u \frac{\partial w}{\partial x} + v \frac{\partial w}{\partial y} + w \frac{\partial w}{\partial z}) = - \frac{\partial p}{\partial z} + ρ g + μ (\frac{\partial^{2} w}{\partial x^{2}} + \frac{\partial^{2} w}{\partial y^{2}} + \frac{\partial^{2} w}{\partial z^{2}}) . \end{matrix}

(3)

Energy equation:

ρ (u \frac{\partial T}{\partial x} + v \frac{\partial T}{\partial y} + w \frac{\partial T}{\partial z}) = α (\frac{\partial^{2} T}{\partial x^{2}} + \frac{\partial^{2} T}{\partial y^{2}} + \frac{\partial^{2} T}{\partial z^{2}}) .

(4)

Residual values were determined as 10⁻⁶ for energy and 10⁻³ for momentum and continuity equations. The physical properties of air (thermal conductivity, specific heat, density, and viscosity) in the room were written into the program as the average temperature value of hot and cold walls from Incropera and DeWitt's physical properties of air tables [15].

FLUENT has two different solver types, “Pressure Based” and “Density Based.” In this study, we chose “Density Based,” and the Boussinesq approach was applied. Among many viscous models presented by the program, as suggested by Karadağ [13], “k-∊” was chosen, and the model of “k-∊” was indicated as “standard” which was utilized by Awbi as well. Studies in the literature suggest that the k-∊ standard model is the most appropriate for natural convection. It is the most widely used engineering turbulence model for industrial applications, as it is robust and reasonably accurate and contains submodels for compressibility, buoyancy and combustion, and so forth. Also, it is suitable for initial iterations, initial screening of alternative designs, and parametric studies, like this study: Fluent User's Guide, Introductory FLUENT v6.3 notes [16]. Turbulence kinetic energy, k, and dissipation rate, ∊, have been calculated through the transport equations (5) and (6), respectively,

\frac{\partial}{\partial x} (ρ k u) = \frac{\partial}{\partial y} ([(μ + \frac{μ_{t}}{σ_{k}}) \frac{\partial k}{\partial y}]) + G_{k} - ρ ∊

(5)

\frac{\partial}{\partial x} (ρ ∊ u) = \frac{\partial}{\partial y} ([(μ + \frac{μ_{t}}{σ_{∊}}) + \frac{\partial ∊}{\partial y}]) + C_{∊ 1} \frac{∊}{k} G_{k} - ρ C_{∊ 2} \frac{∊^{2}}{k},

(6)

where

\begin{matrix} μ_{t} = ρ c_{p} \frac{k^{2}}{∊} \\ G_{k} = - ρ \overset{—}{υ^{'} u} \frac{\partial υ}{\partial x}, \end{matrix}

(7)

where C_1∊ and C_2∊ are constants. σ_k and σ_∊ are the turbulent Pr numbers for k and ∊, respectively. Also, k-∊ model constants, C_μ, C_∊1, C_∊2, σ_k, and σ_∊ are equal to 0.09, 1.44, 1.92, 1.0, and 1.3, respectively (Fluent User's Guide, Introductory FLUENT v6.3 notes [16], Yılmaz and Öztop [17]).

Because the object of study was to find average CHTC and because the program gives CHTC with surface radiation heat transfer coefficient and these two could not be separated from each other, no radiation model was chosen from the “models” section of the program. In FLUENT, thermal boundary conditions can be defined with five different methods: constant heat flux, constant temperature, convection, radiation, and mixed. In this study, the wall surfaces that were not heated and cooled were assumed adiabatic and defined as “0 W/m²” to reach adiabatic boundary conditions in the program. In addition, thermal boundary conditions of heated and cooled walls that stood opposite were defined at a constant temperature. As suggested by Karadağ et al. [14], the surface Nusselt numbers of the heated walls were examined as references to decide whether the meshing effect on results was diminished. In Figures 4 –7 the CHTC and Nu number results that were obtained at different interval counts were presented and as stated above the optimum interval counts for 2D and 3D models were determined (120 × 100) and (45 × 40 × 20), in other words, “Grid Number 3,” because the difference between the last two grid number preferences is approximately 1%. To reduce the solution period and since the difference is about 1%, “Grid Number 4” was not chosen as the appropriate model.

Figure 4:

Grid independency of 2D results in terms of CHTC.

Figure 5:

Grid independency of 2D results in terms of Nu.

Figure 6:

Grid independency of 2D results in terms of CHTC.

Figure 7:

Grid independency of 2D results in terms of Nu.

Average Nusselt number and CHTC in the enclosure are calculated through the program's “area weighted average” speciality function and β, α, and γ are calculated at film temperature, namely, the average temperature of hot and cold wall temperatures. If we calculate the Rayleigh number for the 4,0 by 2,85 by 4,0 m room that can be seen in Figures 1 and 2 when hot and cold wall temperatures are 30°C and 10°C, respectively, it can be seen that Rayleigh number is above 10⁹, and consequently the flow pattern present in the room is turbulent. All calculated Rayleigh numbers at various wall temperatures are presented in Tables 2 and 3. Also, Nusselt numbers and CHTCs at corresponding Rayleigh numbers have been presented in these tables.

3. Results and Discussion

Computational solutions were conducted according to the depictions which were mentioned previously in Section 2. In these solutions, different hot and cold wall temperatures were applied in order to acquire various points on the Nusselt-Rayleigh number diagrams. Figure 8 illustrates the change of hot wall surface average Nusselt number with variation of cold wall temperatures for one constant room dimension (4.0 by 2.85 by 4.0 m), for both (2D) and (3D). The approximate difference between these results is 10% as also can be seen from Tables 2 and 3. It is obvious from the figures that when the cold wall temperature increases, the temperature difference between the hot wall and cold wall decreases, and thus the Nusselt number over the hot wall also decreases. Therefore, it can be stated that the temperature of cold wall in a room affects the Nusselt number and CHTC in the enclosure.

Figure 8:

Change of Nusselt number at different cold wall temperatures (T_c = 5–15°C, T_H = 20–35°C, L × H × L = 4.0 × 2.85 × 4.0 m).

In Figures 8 –10, results of this numerical study, implemented through FLUENT, for appropriate hot and cold wall temperatures similar to real wall temperatures in buildings (hot and cold wall temperatures for real-room surfaces) were compared with the results of correlations presented in the literature. For all the Rayleigh numbers that were presented in Tables 2 and 3, corresponding Nusselt numbers were obtained through the program and are illustrated in Figure 10. It can be deduced that in addition to the Rayleigh number, the Nusselt number depends on hot and cold wall temperatures. According to all these obtained numbers and Figure 10 we have derived two new Nusselt number correlations (8) for an enclosure similar with a real-size room of a building within a Rayleigh number range between 3.38 · 10⁹ and 1.95 · 10¹¹, while the temperature difference variant of Rayleigh number has been defined as the difference between hot and cold wall temperatures:

\begin{matrix} {Nu}_{L} = 0.1 {Ra}_{L}^{0,349} (2 D solution) \\ {Nu}_{L} = 0.09 {Ra}_{L}^{0,349} (3 D solution) . \end{matrix}

(8)

Figure 9:

Change of CHTC at various temperature difference values between hot and cold walls (T_c = 5–15°C, T_H = 20–35°C, L × H × L = 4.0 × 2.85 × 4.0 m).

Figure 10:

Variation of Nusselt number with Rayleigh number in the enclosure (L × H × L = 4,0 × 2,85 × 4, 0 m).

As the deviations of these new correlations from the Nusselt numbers obtained via 2D and 3D numerical study are 1.76% and 0.49%, deviations of 2D and 3D results from Awbi and Hatton's [4] correlation are 19.34% and 13.43%, respectively. Although 2D solutions have a much more sophisticated meshing infrastructure, 3D solution has a closer agreement with Awbi and Hatton's [4] experimental work. This situation can be explained with “adjacent wall effects.” Similar with vertical plate correlations presented and utilized for many years in the literature, 2D solutions also have a disadvantage. In 2D solutions, although the effect of x and y axis can be assessed, third side effect, in other words, “depth of the enclosure,” cannot be counted in the solution progression. Studies published in the literature show that all surrounding walls in the enclosure affect CHTCs, since they have significant effect on flow pattern. Hence, the difference between Awbi and Hatton's [4] experimental work and the 3D solutions of this study has better agreement than the 2D solution results as can be seen in Figure 9, although two-dimensional has more detailed meshing frequency especially near walls. Differences may also be interpreted with measurement precision quality of compared experimental study.

Two novel correlations were derived and presented in the figure, written as (9). It can be discerned that CHTC in an enclosure varies with temperature difference between room centre and heated or cooled wall. Figure 9 shows that discrepancies of 3D results of this study and the results of Khalifa and Marshall [5], Li et al. [6], Min et al. [7], and Awbi and Hatton [4] are 6.38%, 21.66%, 25.05%, and 27.7%, respectively. Differences between this study and the aforementioned studies can be interpreted as follows. In Min et al. [7] study no heating equipment on the vertical walls was utilized. Rather, the study used “heated floor on heated ceiling” and the difference can be explained due to this preference. Li et al. [6] worked in an “occupied room under normal conditions,” which could have caused the differences. Although Awbi and Hatton's [4] comprehensive study is the most similar to the present work, the researchers of the study preferred “constant heat flux” (200 W/m²) on vertical walls as boundary conditions. It is thought that this choice could bring about the discrepancy between the two studies. Results have not been compared with correlations derived for free plates, because, as previously stated, it was not appropriate to compare free plate results with enclosure results due to adjacent wall effects:

\begin{matrix} h = 2.21 Δ T^{0,343} (2 D solution) \\ h = 1.98 Δ T^{0,343} (3 D solution) . \end{matrix}

(9)

In addition, in Figures 11 and 12, 2D and 3D temperature and velocity flow patterns for the studied enclosure can be seen, respectively. Also, according to 2D and 3D models, maximum velocity in the room was found 0.27 m/s and 0.32 m/s, respectively. Both values are about 0.25 m/s, that is, the comfort velocity value in the rooms. Furthermore, velocity at overwhelming part of the enclosure is approximately zero.

Figure 11:

CFD predictions of temperature (K) and velocity magnitude (m/s) contours of 2D model's results, respectively (T_H = 30°C, T_C = 10°C; L × H × L = 4.0 × 2.85 × 4.0 m).

Figure 12:

CFD predictions of temperature (K) and velocity magnitude (m/s) contours of 3D model's results, respectively (T_H = 30°C, T_C = 10°C; L × H × L = 4.0 × 2.85 × 4.0 m).

4. Conclusions

Numerical case studies have been implemented to acquire Nusselt number and CHTC points on Rayleigh-Nusselt and CHTC-ΔT diagrams within the Rayleigh number range of 3.38 · 10⁹ to 1.95 · 10¹¹. To attain different points on diagrams and to observe the convective heat transfer behavior in the enclosure, different heated and cooled wall temperatures were applied. Since, all the Rayleigh numbers are above 10⁹, the study involves the area of turbulent convective heat transfer. This study has shown that many correlations are produced by room surfaces, particularly for heated and cooled vertical wall surfaces. Also, two new correlations were derived for both average CHTC and Nusselt number in an enclosure which has the dimensions of 4.00 by 2.85 by 4.00 and heated from one vertical wall and cooled from the opposite wall in order to consist a real-room situation.

The results obtained with this study lie within the range of data obtained from other correlations. The differences between correlation results are thought to be caused by reference temperature determination, whether the study considers 3D or 2D and heat losses and gains that could not be counted by some experimental studies.

Energy consumption and thermal comfort in buildings have great impact on people's productivity, and these factors must be combined at some point. Therefore, parameters for calculating heating and cooling loads in buildings must be well investigated. Correct usage of CHTC does significantly affect losses from wall surfaces to the atmosphere. Also, CHTCs are used with the conduction loss of windows while calculating their total heat transfer coefficient. The effect of correct usage has a significant impact on total heat loss amounts. Also, correct usage has a great impact on thermal comfort in living spaces.

The study carries weight with the determination of the heating load in buildings as well. In this respect, it is thought that this study's results will also be useful for the thermal comfort and energy efficiency in buildings that use wall heating radiant systems and provide an appropriate direction to engineers who calculate heating load via package programs.

Footnotes

Nomenclature

Acknowledgment

The authors gratefully acknowledge the financial support from the Scientific Research Projects Administration Unit of Yildiz Technical University (YTU-BAPK/27-06-01-03, 2007).

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