General expressions for the calculation of air flow and heat transfer rates in tall ventilation cavities

Building and Environment 46 (2011) 2069e2080

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General expressions for the calculation of air flow and heat transfer rates in tall ventilation cavities Guohui Gan* Department of Architecture and Built Environment, University of Nottingham, University Park, Nottingham NG7 2RD, UK

a r t i c l e i n f o

a b s t r a c t

Article history: Received 8 February 2011 Received in revised form 11 April 2011 Accepted 11 April 2011

Solar heated tall ventilation cavities including solar chimneys are used to enhance natural ventilation of buildings. A validated CFD model was used to predict the buoyancy-driven air flow and heat transfer rates in vertical ventilation cavities with various combinations of heat distribution on two vertical walls ranging from symmetrical to fully asymmetrical heating. The natural ventilation rate and heat transfer rate have been found to vary with the total heat input, heat distribution on the cavity walls, cavity width and height and inlet opening position. General expressions for these variables have been obtained and presented in non-dimensional terms, Nusselt number, Reynolds number, Rayleigh number and aspect ratio (H/b), as Nu ¼ f(Ra, H/b) and Nu ¼ f(Ra, Re) or Re ¼ f(Ra, Nu), for natural ventilation design. Ó 2011 Elsevier Ltd. All rights reserved.

Keywords: Tall cavity Buoyancy Natural ventilation Flow rate Heat transfer coefficient Heat distribution

1. Introduction Design of a natural ventilation system requires determination of air flow and heat transfer rates. This can be achieved using analytical or numerical methods in addition to physical measurement. Analytical methods are generally based on an assumption of uniform air temperature in a space or zone. Such methods may be adequate for estimating buoyancy-driven ventilation of a building with simple vent openings but would not be of sufficient accuracy for design of tall ventilation structures such as solar chimneys, Trombe walls and double facades where the air temperature and velocity vary considerably. Heat transfer through tall cavities consisting of two parallel plates/walls has been extensively studied. Most studies were concerned with symmetrical flow where both walls were heated at the same rate or fully asymmetrical flow where only one wall was heated while the other wall was insulated [1e4]. The flow in conventional solar chimneys [5e8] and Trombe walls [9] can be considered to be nearly fully asymmetrical as the exterior skin of these structures is made of glass to allow solar heat transmission with negligible absorption and storage of heat compared with the interior storage wall which acts as the main source of buoyancy.

* Corresponding author. Tel.: þ44 115 9514876. E-mail address: [email protected]. 0360-1323/$ e see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.buildenv.2011.04.014

However, when photovoltaic devices that transform most of the absorbed solar radiation into heat are used as a part or whole of the exterior skin of such a tall cavity structure [10], both the exterior and interior skins can behave as heated walls with different heat transfer rates. The flow patterns and heat and air flow rates through a cavity with two differentially heated walls differ from those with one heated wall or two walls heated at the same rate. Rodrigues et al. [11] numerically investigated natural convection in an asymmetrically heated channel with a fixed heat flux ratio of 1/5 between the cold wall and hot wall to model the flow in a solar collector and found that the flow rate increased with channel width and total heat flux. Nguyen et al. [12] measured the wall temperature along an asymmetrically heated 0.8 m high channel for a range of aspect ratios and percentages of heat distribution. They also used the heat balance model to calculate the flow rate which increased with channel size. Burek and Habeb [13] also measured the heat transfer and air flow in a 1 m high channel for different channel widths and heat fluxes. The mass flow rate through the channel was found to increase with channel width and heat input. The natural convection in such short channels could be laminar or involve mixed flow. Miyamoto et al. [14] experimentally studied turbulent natural convection heat transfer through 5 m tall parallel plates and the induced flow rate through the tall cavity with different widths and heat fluxes. The results were presented in dimensionless terms. Olsson [15] noticed from a review of literature on buoyancy-induced flow in symmetrically heated vertical

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G. Gan / Building and Environment 46 (2011) 2069e2080

Nomenclature

Ra

a b c1 c2 g H H/b hc k m n Nu

Rab Rac

Nuc Q q q1 q2 qr qw

exponent in Equation (10a) cavity width (m) constant in Equation (9) constant in Equation (10) gravitational acceleration (m/s2) cavity height (m) aspect ratio (dimensionless) convective heat transfer coefficient (W/m2K) thermal conductivity (W/mK) exponent in Equation (9) exponent in Equation (10) Nusselt number (¼ hc b/k), see Equation (3) (dimensionless) Nusselt number in Equation (13) (¼ qw/(tw-ta) b/k) (dimensionless) volumetric flow rate per unit length (m3/s-m) total heat flux (¼ q1 þ q2) (W/m2) heat flux on the opposite wall (W/m2) heat flux on the inlet wall (W/m2) heat distribution ratio (¼ q2/(q1 þ q2)) (dimensionless) convective component of heat flux on one heated wall (W/m2)

channels that formulae (based on analytical solutions and measurements) for the calculation of heat transfer for various geometries had been published extensively while formulae for flow rate calculation had not received the same attention. No such formulae are available for the calculation of heat transfer or flow rate in asymmetrically heated cavities with different proportions of heat distribution. The author of the present paper has numerically investigated both air flow and heat transfer rates in vertical cavities of various sizes and with different heat fluxes and heat distribution ratios [16,17]. The material properties of the cavity walls for radiation heat transfer were not taken into account in the investigations because uniform heat fluxes were prescribed on the wall surfaces. However, a wide range of total heat fluxes on the vertical walls from 100 W/m2 to 1000 W/m2 and a full range of proportions of heat distribution between two vertical walls from fully asymmetrical heating to symmetrical heating encompassed most practical wall surface properties in terms of thermal radiation. For example, the cover of a solar collector or solar chimney may have different degrees of solar heat absorption ranging from a transparent cover with little heat capacity to a cover composed of opaque photovoltaic cells (photovoltaic-thermal collector) which could absorb more heat than does the absorber behind under bright sunshine. The balance between the solar heat absorption by the cover and the absorber, radiation heat transfer between the two surfaces, the convection and radiation heat losses from the cover to the ambient and heat removal from the collector gives rise to two heat fluxes on the opposing surfaces of the cover and absorber which can be represented by the sum and proportion of heat fluxes on two wall surfaces of a ventilation cavity. For a simple analysis, an estimated heat absorption by the collector can be used for calculating the air flow and heat transfer rates because the numerical results were based on the total heat input rather than the convective component of heat input as used in [14]. These investigations have been conducted using computational domains larger than the size of cavity geometries for simulation of buoyancy-driven flow in tall ventilation cavities with a vertical or horizontal inlet opening. Based on the results of the investigations,

RaH Raq Re S4 t1 t2 ta tw Ui V xi y

a b G4 n 4

Rayleigh number (¼ gb ½ (q1 þ q2) b4/(nak) b/H), see Equation (5) (dimensionless) Rayleigh number with b as the main variable Rayleigh number in Equation (13) (¼ gb qw b4/(nak) b/H) (dimensionless) Rayleigh number with H as the main variable Rayleigh number with q as the main variable Reynolds number (¼ Q/n), see Equation (4) (dimensionless) source term for flow variable 4 in Equation (1) temperature of opposite wall surface (K) temperature of inlet wall surface (K) ambient temperature (K) temperature of one heated wall (K) xi component of mean velocity, see Equation (1) (m/s) mean cavity velocity (m/s) Cartesian coordinates (m) vertical distance from cavity inlet (m) diffusion coefficient (m2/s) thermal expansion coefficient (1/K) diffusion coefficient for flow variable 4 in Equation (1) kinematic viscosity (m2/s) flow variable in Equation (1)

this paper presents general expressions for calculating buoyancydriven air flow and heat transfer rates in tall ventilation cavities for a wide range of Rayleigh numbers at any heat distribution ratio. The expressions can be used for design of tall cavity structures for natural ventilation of buildings. 2. Methodology Commercial computational fluid dynamics software FLUENT version 6.3 [18] was used for simulation of air flow and heat transfer through tall ventilation cavities. Air flow through a tall ventilation cavity would likely involve both laminar and turbulent flow and the RNG k-e turbulence model [19] was employed for modelling of both strongly turbulent and less turbulent flow. For an incompressible steady-state flow, the time-averaged air flow and heat transfer equations can be written in the following form


 vðrUi 4Þ v v4 G4  ¼ S4 vxi vxi vxi

(1)

where 4 represents the mean velocity component Ui in xi direction, turbulent parameters and mean enthalpy, G4 is the diffusion coefficient and S4 is the source term for variable 4. Details of the model equations are described in [20]. To solve the equations for air flow through a tall cavity, the computational domain was extended from the cavity around the four sides by up to 10 times the cavity width [16]. The grid size of a computational domain, e.g., for a 3 m tall and 0.6 m wide cavity with a vertical inlet was 468,784 with dense cells distributed inside the cavity and near the cavity walls. Two types of boundary were used for prediction of cavity flow e no-slip wall for solid wall surfaces and uniform pressure inlet/outlet for the extended domain boundary. The thermal boundary condition for a wall surface was prescribed with a heat flux ¼ (0 for insulated or unheated wall). Near a solid wall, use was made of the enhanced wall treatment [18] to resolve the viscosity-affected inner region using a one-equation model and the fully turbulent region using the k-e turbulence model. A blending function was used to smoothly blend the two regions. Accurate

G. Gan / Building and Environment 46 (2011) 2069e2080

computation of turbulent heat transfer near a solid wall can be achieved using a low-Reynolds number model [21] instead of wall functions but this requires a very fine mesh near the wall followed by a dense mesh as a transition to the rest of the domain to avoid solution instability. The enhanced wall treatment is similar to the low-Reynolds number model in terms of near-wall direct modelling, but is flexible with the mesh distribution requirement, e.g., a very fine mesh for the wall boundary inside the cavity and a relatively coarse mesh for the wall boundary outside the cavity in the extended domain, without incurring solution instability. The air flow model had been validated for natural convection in three tall ventilation cavities with different inlet and outlet opening positions e i) vertical inlet and outlet openings [16], ii) horizontal inlet opening and vertical outlet opening [20] and iii) horizontal inlet and outlet openings [20] using larger computational domains than the cavities. The validated model was used to conduct a series of parametric analysis for two-dimensional vertical cavities of different heights ranging from 1 m to 6 m and widths from 0.05 m to 0.6 m. Each cavity had one vertical opening at the top as an air outlet and another opening as an air inlet either vertically at the bottom [16] or horizontally through a wall near the bottom [17]. The variation of cavity width and height gave aspect ratios (height/width) between 5 and 60. Fig. 1 illustrates the configurations of two types of vertical cavity e one with a vertical inlet and another with a horizontal inlet. For the inlet positioned horizontally through either of the walls, the inlet height was equal to the cavity width and the height of the heated sections of the vertical walls was the same. The horizontal bottom wall and part of the wall opposite to and with the same height as the inlet opening were insulated. The (right) wall with the inlet is referred to as the inlet wall (with heat flux q2 in W/m2) and the other vertical wall (heated section) as the opposite wall (with heat flux q1). The ratio of the heat flux from the inlet wall to the total heat flux from both walls is the heat distribution ratio (qr),

qr ¼

q2  100% q1 þ q2

(2)

The parametric analysis was undertaken with total heat fluxes from both walls varying between 100 W/m2 and 1000 W/m2 and for the entire range of heat distribution ratios from 0% (heat from one/left wall only), to 50% (heat equally distributed on two walls) and to 100% for heat from another/right wall only. It should be mentioned that at a much higher heat flux imposed on one wall than the other, radiation heat transfer would play an important role in redistributing heat between the two opposing wall surfaces. The simulation and analysis involved convection heat transfer only but the likely effect of radiation heat transfer can be accounted for by different proportions of heat distribution (i.e., heat distribution ratio) for different emissivities of wall surfaces and cavity Outlet

Heated wall q2 b

Heated wall q 1

H

Heated wall q 1

Heated wall q 2

configurations. In other words, e.g., results for a heat distribution ratio of 67% can be interpolated for an application where radiation heat transfer from one heated wall to the other insulated wall is at the same rate as that by convection from the heated wall to air (i.e. q1 ¼ ½ q2). Further simulation involving radiation heat transfer in tall cavities indicates that the proportion of heat transfer depends on the cavity aspect ratio and the surface emissivity but the effect of inlet position is insignificant unless the cavities are very wide. This is because radiation heat transfer principally takes place between the opposing vertical wall surfaces and the proportion of radiation heat transfer from vertical walls to the horizontal surfaces or openings is small and because the small amount of heat transferred to the bottom opening of a cavity with a vertical inlet would be similar to that transferred to a part of a cavity with a horizontal inlet below the heated wall section including the vertical opening, even though calculation of the view factor for the latter would be much more complex. To put it simply, the radiation heat loss through the horizontal opening is similar to that through the vertical opening of the same size. Take 3 m tall cavities for example, if the emissivity of walls (say 0.9) is close to the effective emissivity of the openings (¼ 1 for a black body surrounding), the calculated difference in the radiation heat transfer between the cavity with a horizontal inlet and that with a vertical inlet is generally within 2% and the maximum difference is less than 5% for aspect ratios greater than 6. The maximum difference for the widest cavities with an aspect ratio of 5 is about 10%. It can be inferred that calculation of average radiation heat transfer between two opposing wall surfaces of a tall cavity can generally be simplified as that for two parallel plates regardless of the position of openings. However, radiation heat transfer can lead to non-uniform convective and radiative wall heat distribution near the openings but using a uniform heat flux on each wall surface of a tall cavity would not result in significant errors in the prediction of overall heat transfer and air flow. For example, for a 3 m tall and 0.3 m wide cavity with a vertical inlet, the difference in the predicted flow rate is less than 1% between simulation one with combined convection and radiation heat transfer and simulation two with convection heat transfer only but with uniform wall heat fluxes which are obtained from simulation one. Fig. 2 shows the predicted air flow patterns near the inlet and outlet of 3 m tall and 0.6 m wide cavities with a large heat flux of 1000 W/m2 fixed on the right wall. The velocity vectors were plotted for only one in twelve cells in either of the two dimensions. The air flow was very asymmetrical even in the symmetrical cavity with vertical inlet and outlet as a result of heat flow from the right wall only. It can be seen that air flowing into the vertical inlet formed a small recirculation zone near the unheated (left) wall. For the cavity with horizontal inlet, air flowed along the opposite wall after turning from horizontal at the inlet opening to vertical direction. A large recirculation zone was formed near the bottom of the unheated wall and a smaller zone above the inlet opening. For both cavities, non-uniform reverse flow away from the heated wall occurred over a large area near the top. Such flow patterns and resulting ventilation and heat transfer rates could not be obtained using analytical solutions or numerical methods without employing a large extended computational domain [16].

H

Outlet

2071

b

Inlet

Inlet

b

3. Dimensionless numbers for heat transfer rate and air flow rate

Insulated walls Vertical inlet

Horizontal inlet

Fig. 1. Schematic of two types of cavity in terms of inlet position.

Etheridge [22] pointed out a number of benefits of using nondimensional graphs generated from sophisticated numerical models for natural ventilation design. To derive general expressions based on dimension analysis for heat transfer and air flow in ventilation cavities, dimensionless numbers are introduced including

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G. Gan / Building and Environment 46 (2011) 2069e2080

Fig. 2. Predicted air flow patterns near the inlet and outlet of 3 m tall and 0.6 m wide cavities with 100% heat distribution ratio.

Nusselt number for the heat transfer rate, Reynolds number for the air flow rate and Rayleigh number associated with the cavity size and boundary conditions. The heat transfer rate or coefficient in a tall cavity is represented by the Nusselt number, Nu, defined as

q1 þ q2 b hc b 2  Nu ¼ ¼
t1 þ t2 k  ta k 2

(3)

where hc is the convective heat transfer coefficient (W/m2K), b is the cavity width (m), k is the thermal conductivity of air (W/mK), t1 and t2 are the temperatures of two heated wall surfaces (K), ta is the temperature of ambient/inlet air flowing into the cavity (K). The thermal properties of air such as the conductivity are based on the average temperatures of cavity wall surfaces and inlet air, i.e., [½ (t1 þ t2) þ ta]/2. The air flow rate through a tall cavity is represented by the Reynolds number, Re, and for two-dimensional flow, it is defined as the ratio of the flow rate to the kinematic viscosity of air:

Re ¼

Vb Q ¼ v v

(4)

where V is the mean velocity of air through the cavity (m/s), Q is the flow rate for a unit length of two-dimensional cavity (m3/s-m) and v is the kinematic viscosity of air (m2/s). The Reynolds number can also be based on the hydraulic diameter of a cavity, useful for determining the type of forced convection flow. For a two-dimensional cavity flow, the hydraulic diameter is twice the cavity width and Re would also be twice the value given by Equation (4).

The predicted air flow rate and heat transfer rate for the vertical cavities have been shown to vary with cavity size, total heat input and heat distribution ratio [16,17]. The effects of cavity size and heat input can be combined into a single parameter represented by the Rayleigh number, Ra, as follows:

Ra ¼

gb

q1 þ q2 4 b b 2 nak H

(5)

where g is the gravitational acceleration (m/s2), b and a are the thermal expansion coefficient (1/K) and diffusion coefficient (m2/s) of air, respectively, H is the height of cavity (heated walls) (m) and H/b is the aspect ratio of cavity. The variations of cavity size and heat input gave rise to a very wide range of Rayleigh numbers from 1.99  104 to 3.07  1010. Figs. 3 and 4 show examples of variations in air flow and heat transfer rates with heat distribution ratio for cavities with an aspect ratio of 10 and a total heat flux of 1000 W/m2. The variations in air flow and heat transfer rates with heat distribution ratio in a cavity with vertical inlet are symmetrical along the point of 50% heat distribution ratio and so data are presented only for heat distribution ratios from 0 to 50%. It is seen from Fig. 3 that for any cavity size the air flow rate increased from the minimum when heat flowed from one wall only (qr ¼ 0) to the maximum when heat was equally distributed on both walls (qr ¼ 50%) of a cavity with a vertical inlet or when the heat distribution ratio was between 60% and 65% for a cavity with a horizontal inlet. In contrast, Fig. 4 shows that for a cavity with vertical inlet the heat transfer rate decreased with the increase in the heat distribution ratio from the maximum when heat flowed from one wall only to the minimum when heat was equally distributed on both walls whereas for a cavity with

G. Gan / Building and Environment 46 (2011) 2069e2080

30000

Height/Width 1/0.1

a

2/0.2

20000

Re

3/0.3

Nu

a

2073

200

Height/Width 1/0.1

150

2/0.2

100

3/0.3 4/0.4

4/0.4

10000 0

0

10

20

30

40

5/0.5

50

5/0.5

6/0.6

0

6/0.6

0

50

10

Cavity with vertical inlet

30

40

50

Cavity with vertical inlet

30000

Height/Width 1/0.1

20000

2/0.2

b

10000

5/0.5 6/0.6

0

Nu

4/0.4

250

Height/Width 1/0.1

200

3/0.3

Re

b

20

2/0.2

150

3/0.3

100

4/0.4

50

5/0.5 6/0.6

0 0

20

40

60

80

100

0

Cavity with horizontal inlet Fig. 3. Effect of heat distribution ratio on the flow rate in cavities with an aspect ratio of 10 and a total heat flux of 1000 W/m2.

horizontal inlet it decreased with increasing heat distribution ratio. The effect of heat distribution on the variations in both air flow rate and heat transfer rate increased with cavity size. 4. Correlations for heat transfer rate and air flow rate The Nusselt number can be correlated with the Rayleigh number and Reynolds number using either data obtained for all individual heat distribution ratios or values averaged over the entire range of heat distribution ratios. The correlations for two types of cavity in terms of inlet position have been found to have the same forms of power-law relationship between the Rayleigh number, Reynolds number and Nusselt number. 4.1. Correlations based on average values To illustrate how the dimensionless numbers can be correlated, values of Nu, Ra and Re for the entire range of heat distribution ratios were averaged to obtain one set of data for each cavity width, cavity height and total heat flux. That is, the variations of the dimensionless numbers with heat distribution ratio were not explicitly considered. Log linear relations were found to exist between the averaged Nu, Re and Ra values as shown in Fig. 5 for the two types of cavity. It is seen that the predicted heat transfer rate for cavities with a horizontal inlet (Nu_h) was slightly higher than that with a vertical inlet (Nu_v) except for very narrow cavities with an aspect ratio H/b  30 which had a slightly higher heat transfer rate whereas the air flow rate with a horizontal inlet (Re_h) was lower than that with a vertical inlet (Re_v) for different cavity widths, heights or total heat fluxes. The higher averaged heat transfer rate for a wide cavity with a horizontal inlet than that with a vertical inlet can also be seen from Fig. 4. The heat transfer rates

20

40

60

80

100

Cavity with horizontal inlet Fig. 4. Effect of heat distribution ratio on the heat transfer coefficient in cavities with an aspect ratio of 10 and a total heat flux of 1000 W/m2.

for a cavity with a horizontal inlet were higher for a wide range of heat distribution ratios while the minimum values were generally not less than those for a cavity with a vertical inlet. This resulted from a more turbulent flow along the vertical walls after air moved from the horizontal inlet, even though at a lower mean velocity, than air flow from the vertical inlet along the vertical cavity. 4.1.1. Vertical inlet Correlations can be obtained for any cavity size and heat flux. The best approximations between two of the three dimensionless groups for 3 m tall cavities with a vertical inlet and with a total heat flux on both walls of 100 W/m2 (Fig. 5a) are as follows:

Nu ¼ 1:7 Ra0:18 b Re ¼ 542 Ra0:124 b



 R2 ¼ 1



Nu ¼ 0:000198 Re1:44

(6a) 

R2 ¼ 0:997 

 R2 ¼ 0:997

(6b) (6c)

where Rab represents the Rayleigh number that varies with cavity width b but for a fixed cavity height and under a constant total heat flux. From Equations (5) and (6a), one obtains that Nu is proportional to b5  0.18 ¼ 0.9 and so hc is proportional to b0.1 according to the definition of hc ¼ Nu k/b. Therefore, in terms of trend of variation, the effect of cavity width on the convective heat transfer coefficient would differ from that on the Nusselt number, with Nu increasing while hc decreasing with the increase in cavity width. The heat transfer coefficient or heat transfer rate discussed in this paper is based on the Nusselt number.

2074

Re_h

Re_v

Nu_h

Nu_v

100

Nu

100000

Re

a

G. Gan / Building and Environment 46 (2011) 2069e2080

10000

1000

10

1.0E+051.0E+061.0E+071.0E+081.0E+091.0E+10

Rab

Re_h

Re_v

Nu_h

Nu_v

100

Nu

100000

Re

b

10000

1000



Nu ¼ 0:319 Ra0:268 H

10

1.0E+08

where Raq is the Rayleigh number that varies with total heat flux q ¼ (q1 þ q2) and associated physical properties of air for a given cavity (fixed cavity width and height). The increasing effect of the Rayleigh number on the Nusselt number resulted from decreasing relative change in wall temperature (average for two walls) and thus increasing change in the Nusselt number with the increase in the total heat flux. For example, at a total heat flux of 100 W/m2, the average wall temperature increased by 0.8 K at an average temperature rise of 11.6  C from the inlet air when the heat distribution ratio varied from 0 to 50%; the relative increase was 0.8/11.6 ¼ 7%. At a total heat flux of 1000 W/m2, the corresponding wall temperature increase was 1.3 K but at a much higher average temperature rise of 67  C, and so the relative increase was only 1.3/67 ¼ 2%. In addition, the range of variation in Reynolds number was larger for higher total heat fluxes and consequently the rate of change in the Nusselt number with Reynolds number was smaller. The effect of Reynolds number on the Nusselt number was however similar to that for forced convection heat transfer in turbulent flow over a flat plate, i.e., Nu f Re0.8. The predicted Nusselt number and Reynolds number were also found to vary with cavity height for the same aspect ratio (Fig. 5c) but the effect of cavity height was very close to that of total heat flux. For an aspect ratio of 10 and a total heat flux of 100 W/m2, the corresponding correlations are as follows:

1.0E+09 Re ¼ 10:9 Ra0:332 H

Raq



R2 ¼ 1

Nu ¼ 0:0461 Re0:808 H

Re_h

Re_v

Nu_h

Nu_v

100

Nu

100000

Re

c

10000

1000

10

1.0E+06 1.0E+07 1.0E+08 1.0E+09 1.0E+10

RaH

Fig. 5. Effect of parameters in Ra on Re and Nu based on averaged values for cavities with a vertical inlet and compared with those with a horizontal inlet.

When the total heat flux on both walls varied for each cavity width, the effect of the Rayleigh number was found to be more significant but the effect of the Reynolds number on the Nusselt number became less (Fig. 5b) and a different set of correlations can be obtained, e.g., for a 3 m tall and 0.3 m wide cavity:

Nu ¼ 0:311 Ra0:27 q

Re ¼ 10:2

Ra0:336 q

Nu ¼ 0:0484 Re0:802

  R2 ¼ 1 

 R ¼ 1 2



 R2 ¼ 1

(7a)

R2 ¼ 1





(8a)



(8b)

R2 ¼ 1



(8c)

where RaH is the Rayleigh number varying with cavity height H for a fixed aspect ratio and under a constant total heat flux. Similar correlations can be obtained for other cavity sizes and heat fluxes. However, the resulting constants and exponents in the powerlaw correlations may change with these parameters. For example, for 6 m tall cavities with a total heat flux of 100 W/m2, the constant and exponent become, respectively, 1.9 and 0.187 for Equation 6a, 719 and 0.14 for Equation 6b and 0.0003 and 0.133 for Equation 6c. Hence, such simple correlations are applicable only to a specific combination of cavity size and heat flux and derivation of general expressions requires consideration of all these parameters simultaneously. To take account of all the effects of total heat flux and cavity size on the Rayleigh number, the relationship between Nu and Ra and in combination with Re can be represented by the following expressions:

"
 #m H 3=2 Nu ¼ c1 Ra b

(9)

and

 n Nu ¼ c2 Ra1=3 Re

(10)

where c1 and c2 are constants and m and n are exponents to be determined from regression analysis. For Re and Nu values averaged over the entire range of heat distribution ratios as shown in Fig. 6, the correlations between Nu, Re and Ra are:

0:265  Nu ¼ 0:135 RaðH=bÞ3=2

(7b)

and

(7c)

 0:397 Nu ¼ 0:135 Ra1=3 Re





 R2 ¼ 0:999

 R2 ¼ 1

(11)

(12)

G. Gan / Building and Environment 46 (2011) 2069e2080

4.1.2. Validation of predicted dimensionless numbers The predicted dimensionless numbers for buoyancy-driven natural ventilation through vertical cavities were compared with experimental measurements from literature. Fig. 6 shows a comparison of the predicted heat transfer and air flow rates with the measurements by Miyamoto et al. [14] for an asymmetrically heated 5 m tall cavity with three widths of 0.04 m, 0.1 m and 0.2 m (aspect ratios of 125, 50 and 25, respectively) at a total heat flux of 104 W/m2 for which the flow rate and wall temperatures were available for calculating Ra, Re and Nu directly. It can be seen that the predicted values for the Nusselt number and Reynolds number agreed well with the measured results for the common range of Rayleigh numbers (for cavity widths of 0.1 m and 0.2 m). Miyamoto et al. also obtained the following relationship between the local Nusselt number and the Rayleigh number in the turbulent heat transfer region:


Nucy ¼ 0:4

0:484

1=2

Rac H=y

(13)

where y is the distance from the inlet of the cavity, Nucy is the local Nusselt number based on the convective component of the heat flux on the heated wall (qw) and the temperature difference between the heated wall and inlet air (Nuc ¼ qw/(twta) b/k), and Rac is the Rayleigh number also based on the convective component of the heat flux on the heated wall ¼ (gb qw b4/(nak) b/H). For comparison with the prediction in terms of average Nusselt number, the relationship is applied to the whole cavity as an approximation. Then, the average Nusselt number can be obtained through integration of Equation (13) over the wall height as

Nuc ¼ 0:775 Ra0:242 c

a

1000

100

10

10 1 1.0E+06

1000

Nu = 0.135(Ra(H/b)3/2 )0.265 R² = 0.999

Nu

100

The predicted Nusselt number in relation to the Rayleigh number is compared with Equation (14) in Fig. 7a. Again, good agreement was achieved between the predicted and measured results in the common range of Rayleigh numbers. The predicted data were scattered (with a coefficient of determination R2 ¼ 0.96) due to the large ranges of variations in the total heat flux, cavity height and width investigated, which required the forms of Equations (9) and (10) to obtain better correlations. By comparison, the true influence of the cavity height could not be taken into account in the measurements and the height in the Rayleigh number or aspect ratio was included purely for dimension analysis of the measured results. Besides, it was observed that the variation of the predicted Nusselt number with Rayleigh number was less than Equation (14). The exponent for the Rayleigh number in the form of Equation (14) would be about 0.2 for the predicted values which was consistent with turbulent buoyant heat transfer [12]. It should be stressed that the Nusselt number and Rayleigh number in Equation (14) were defined with the heat flux and temperature for the heated wall rather than the average of two walls. The measured results for Equation (14) could not be accurately corrected for the parameters used in the comparison for three reasons: i) both the Nusselt number and Rayleigh number depended on the heat flux which could partly negate the effect, ii) the Nusselt number varied with wall temperatures which were not given for all but three tests shown in Fig. 6, and iii) Equation (14) was approximately derived for heat transfer in the cavity involving both laminar and turbulent regions. Correction for the heat flux would increase the Nusselt number (Nu > Nuc) as the convection heat flux (in Rac) for the asymmetrically heated cavity was larger than one half of the total heat flux (in Ra) but the effect of correcting for the wall temperatures would depend on the increase in the temperature of the insulated wall above inlet air e a small temperature increase would lead to Nu > Nuc and vice versa.

Nu

a

(14)

2075

Predicted Miyamoto, et al. 1.0E+08

1.0E+10

1 1.0E+04

1.0E+12

Predicted Eqn (14) 1.0E+06

1.0E+08

1.0E+10

1.0E+12

Ra

Ra(H/b)3/2

Nu = f(Ra)

Nu = f(Ra, H/b)

b

1000

10000

Nu

100

Nu = 0.135(Ra1/3Re)0.397 R² = 1.000

10 1 1.0E+04

100000

Re

b

Predicted Miyamoto, et al. 1.0E+05

1.0E+06

1.0E+07

1.0E+08

Ra1/3 Re

1000

Predicted Eqn (16) for H/b = 50 Eqn (16) for H/b = 25 Eqn (17) for H/b = 50 Eqn (17) for H/b = 10

100 10 1.0E+04

1.0E+06

1.0E+08

1.0E+10

1.0E+12

Ra Re = f(Ra)

Nu = f(Ra, Re) Fig. 6. Effect of Ra and Re on Nu based on averaged values for cavities with a vertical inlet.

Fig. 7. Comparison between predicted and measured Re and Nu for cavities with a vertical inlet.

2076

G. Gan / Building and Environment 46 (2011) 2069e2080

In addition, Miyamoto et al. [14] presented graphically the air flow rate in relation to the Rayleigh number, indicating the following log-law relationship:

0

1

B C Re CflogðRac Þ  logB @ n bA Rac aH The graphical results can be approximately represented by


Rez1:4Ra0:3 c



aH nb

(15)

Assuming that radiation heat transfer from the heated wall to the insulated wall accounted for 20% of the total heat flux (15%e 20% according to [14]), then


Rez1:6Ra0:3



aH nb

(16)

The predicted values of Reynolds number are seen from Fig. 7b within the range of measured results for the 5 m tall cavity of 0.1 m and 0.2 m wide. Another set of comparison was with the measurements of heat transfer and flow rate by Burek and Habeb [13] for an asymmetrically heated 1 m high cavity with widths from 0.02 m to 0.11 m and heat fluxes from 200 W/m2 to 1000 W/m2. The measured air flow rate was related to the cavity size and heat flux in the following form (after transformation of the Rayleigh number based on the cavity height to that based on the cavity width): 0:572

Re ¼ 0:00172Ra


2:148 H b

(17)

This is also shown in Fig. 7(b) for the cavity of 0.02 m and 0.1 m wide as two curves. The predicted values of Reynolds number fell within the two curves in the measurement range. Results from [13] were more sensitive to the Rayleigh number than those from [14] perhaps because buoyancy-driven air flow in such a short cavity was less turbulent or laminar only. In fact, the estimated Rayleigh number (<10000) suggests that the measurements for the 0.02 m wide cavity (H/b ¼ 50) were in the laminar flow regime, the heat transfer for which would generally be more sensitive to the Rayleigh number (with an exponent of around 0.5) than that for turbulent flow (exponent for Ra z 0.2). The curve for H/b ¼ 50 in Fig. 7(b) was however extrapolated to the maximum Rayleigh number for the measurements and likewise the curve for H/b ¼ 10 was extrapolated to the minimum. The above comparisons give the confidence in the predicted dimensionless numbers. 4.1.3. Horizontal inlet Average values of Nu, Re and Ra can similarly be obtained for the entire range of heat distribution ratios to determine the effects of the cavity size and heat flux on the heat transfer and air flow rates. For example, for 3 m tall cavities with the same total heat flux on both walls of 100 W/m2, the effect of cavity width is represented by the following best approximations between Ra, Re and Nu in Fig. 5a:

Nu ¼ 1:46 Ra0:191 b Re ¼ 385 Ra0:136 b Nu ¼ 0:000354 Re1:4



 R2 ¼ 0:997

  R2 ¼ 0:997 

R2 ¼ 1

(18a) (18b)



(18c)

Similar to the cavities with a vertical inlet, when the cavity height was doubled, the constant and exponent become, respectively, 1.24 and 0.21 for Equation 18a, 544 and 0.147 for Equation 18b and 0.000137 and 0.145 for Equation 18c. The constant for Equation 18c was much smaller for taller cavities (with a similar exponent) and so the heat transfer rate in relation with the Reynolds number decreased with the height of cavities with a horizontal inlet. This was also true for cavities with a vertical inlet as indicated by Equation 10 but it was not apparent from Equation 6c for the two heights analysed due to different magnitudes of the exponent. The effect of total heat flux on the relationships between Nu, Re and Ra for a 3 m tall and 0.3 m wide cavity:



Nu ¼ 0:264 Ra0:283 q Re ¼ 8:46 Ra0:34 q

R2 ¼ 1



(19a)

  R2 ¼ 1 

Nu ¼ 0:0446 Re0:834

R2 ¼ 1

(19b) 

(19c)

And the effect of cavity height on the relationships between Nu, Re and Ra for an aspect ratio of 10 and a total heat flux of 100 W/m2:



Nu ¼ 0:325 Ra0:283 H Re ¼ 9:18 Ra0:335 H



R2 ¼ 1

R2 ¼ 1

Nu ¼ 0:0535 Re0:814





(20a)



(20b)

R2 ¼ 1



(20c)

Fig. 8 shows all the Nu, Re and Ra values averaged over the entire range of heat distribution ratios for any cavity size or total heat flux for cavities with a horizontal inlet compared with those for cavities with a vertical inlet (dashed lines according to Equations (11) and (12)). Data in the figure confirm that the heat transfer coefficient in cavities with a horizontal inlet is generally higher than that with a vertical inlet. The following general correlations between averaged Nu, Re and Ra values are obtained for cavities with a horizontal inlet:

0:282  Nu ¼ 0:098 RaðH=bÞ3=2



 R2 ¼ 0:997

(21)

and

 0:417 Nu ¼ 0:111 Ra1=3 Re



 R2 ¼ 0:998

(22)

4.2. Effect of heat distribution ratio The above correlations would be useful for estimating the heat transfer and air flow rates for buoyancy-driven cavity flow, if the variation with heat distribution ratio could be ignored as in circumstances where heat flows from two walls at a similar magnitude. When the heat transfer and air flow rates vary significantly with heat distribution ratio, individual data of Ra, Re and Nu for each heat distribution ratio, total heat flux, cavity width and height should be used as shown in Figs. 9 and 10. Each of the individual data point represents the Nusselt number, Reynolds number and Rayleigh number for one combination of cavity size, total heat flux and heat distribution ratio.

G. Gan / Building and Environment 46 (2011) 2069e2080

a

a

1000 Horizontal inlet Nu = 0.098(Ra(H/b)3/2 )0.282 R² = 0.997

Nu

10

10 1 1.0E+06

1.0E+08 1.0E+10 Ra (H/b)3/2

1 1.0E+04

1.0E+12

b

1000

1000

Horizontal inlet

Nu

10 1 1.0E+04

1.0E+06 Ra1/3

1.0E+07

1.0E+08

Nu = 0.139(Ra1/3 Re)0.394 R² = 0.998

100

Vertical inlet

10 1 1.0E+04

1.0E+05

1.0E+06 Ra1/3 Re

Nu

100

Nu = 0.111(Ra1/3 Re)0.417 R² = 0.998

1.0E+05

All data

Nu = f(Ra, H/b)

b

Nu = 0.124(Ra1/3 Re)0.405 R² = 0.985

100

Vertical inlet

Nu

100

1000

2077

1.0E+07

1.0E+05

1.0E+08

1.0E+06 Ra1/3 Re

1.0E+07

1.0E+08

Re Fig. 10. Effect of Ra and Re on Nu based on individual values for cavities with a vertical inlet.

Nu = f(Ra, Re) Fig. 8. Effect of Ra and Re on Nu based on averaged values for cavities with a horizontal inlet.

4.2.1. Vertical inlet The correlations between Nu, Re and Ra in Figs. 9 and 10 are:

would be most likely to occur) are ignored (Fig. 10b), a better correlation than Equation (24) can be obtained as follows:

0:267  Nu ¼ 0:132 RaðH=bÞ3=2

 0:394 Nu ¼ 0:139 Ra1=3 Re

  R2 ¼ 0:998

(23)

and

 0:405 Nu ¼ 0:124 Ra1=3 Re





R2 ¼ 0:985

(24)

The correlation coefficient for Equation (24) is however not as high as that for Equation (22). It is observed from Fig. 10a that some of the data for the correlation between Nu and Ra in combination with Re are scattered. This is an indication of the influence of heat distribution ratio and a major source for the influence is reverse flow near the outlet of wide cavities. If the data points for heat distribution ratios of less than or equal to 5% (at which reverse flow

1000

Nu = 0.132(Ra(H/b)3/2)0.267 R² = 0.998





R2 ¼ 0:998

(24a)

Equations (23) and (24) or (24a) appear to be independent of the heat distribution ratio. Even though the correlation coefficient for the equations is quite high, they could not reveal the effect of the heat distribution ratio for a small proportion of data points. To take full account of the effect of heat distribution ratio, data of Nu, Re and Ra for different cavity sizes and total heat fluxes are correlated for each heat distribution ratio. A typical set of data and correlations is shown in Fig. 11 for a heat distribution ratio of 30%. General correlations can then be obtained for all the data points with the regression coefficients (and constants) for Equations (9) and (10) being related to the heat distribution ratio. Fig. 12 shows the dependence of the regression coefficients on the heat distribution ratio. The coefficients are constant for wide ranges of heat distribution ratios, i.e.,

for 10%  qr  90%

(25a)

100

m ¼ 0:265

for 10%  qr  90%

(25b)

10

c2 ¼ 0:143

for 20%  qr  80%

(25c)

Nu

c1 ¼ 0:137

1 1.0E+06

n ¼ 0:391

1.0E+08 1.0E+10 Ra (H/b)3/2

1.0E+12

Fig. 9. Effect of Ra on Nu based on individual values for cavities with a vertical inlet.

for 20%  qr  80%

(25d)

The coefficients are similar to those in Equations (23) and (24a) but for smaller ranges of heat distribution ratios. For heat distribution ratios outside these ranges, the coefficients can be related to the heat distribution ratio as follows:

2078

G. Gan / Building and Environment 46 (2011) 2069e2080

a

a

1000 Nu = 0.137(Ra(H/b)3/2 )0.265 R² = 0.999

1000

Nu = 0.151(Ra2/5Re)0.364 R² = 0.999

100

Nu

Nu

100

10

10 1 1.0E+06

1.0E+08

1.0E+10

1 1.0E+04 1.0E+05 1.0E+06 1.0E+07 1.0E+08 1.0E+09 Ra2/5 Re

1.0E+12

Ra (H/b)3/2

qr

Nu = f(Ra, H/b)

b

1000 Nu =

b 1000

0.143(Ra1/3 Re)0.391

Nu = 0.183(Ra1/2 Re)0.318 R² = 0.999

100 Nu

R² = 0.999 100 Nu

10

10 1 1.0E+04

1 1.0E+05 1.0E+06 1.0E+07 1.0E+08 1.0E+09 1.0E+10 Ra1/2 Re

1.0E+05

1.0E+06

1.0E+07

1.0E+08

Ra1/3 Re

qr Fig. 13. Effect of Ra and Re on Nu for highly asymmetrically heated cavities with a vertical inlet.

Nu = f(Ra, Re) Fig. 11. Effect of Ra and Re on Nu for cavities with a vertical inlet and 30% heat distribution ratio.

c1 ¼ 0:119 þ 0:00354 qr  0:000187 q2r

for qr < 10%

(25e)

m ¼ 0:273  0:00157 qr þ 0:000845 q2r

for qr < 10%

(25f)

c2 ¼ 0:0745 þ 0:00789 qr  0:000235 q2r n ¼ 0:448  0:0069 qr þ 0:000218 q2r

for qr < 20%

for qr < 20%

(25g)

For heat distribution ratios equal to or smaller than 5% or greater than 95%, the correlation coefficients given as Equations (25g) and (25h) in the form of Equation (10) were not very high (R2 < 0.999). Better correlations can be obtained for Ra with a higher exponent. For example: For qr ¼ 5% (Fig. 13a),

(25h)

 0:364 Nu ¼ 0:151 Ra2=5 Re

Constant or Exponent

Because of the symmetrical cavity configuration, the above correlations can also be applied to the higher end of qr by replacing qr with (1qr). For example, for qr > 90%, c1 ¼ 0.119 þ 0.00354 (1qr)0.000187 (1qr2).

0.5



 R2 ¼ 0:999

(26a)



 R2 ¼ 0:999

(26b)

For qr ¼ 2% (Fig. 13b),

 0:318 Nu ¼ 0:183 Ra1=2 Re

Table 1 Coefficients and constants for correlations of heat transfer and air flow rate in cavities with a vertical inlet (after [20]).

0.4

Coefficient qr (%)

0.3

Correlation

0.2


3=2 m c1 H Nu ¼ c1 Ra b m

0e5

0.1

c1

m

c2

n

 n Nu ¼ c2 Raa Re

0

0

10

20

30

40

50

Fig. 12. Effect of heat distribution ratio on the correlation coefficient/constant for Nu for cavities with a vertical inlet.

c2 a n

For qr > 50%, replace qr with (1qr).

5e10 10e20 20e50

0.119 þ 0.00354 qr 0.000187 qr2 0.137 0.273e0.00157 qr þ 0.0000845 qr2 0.265 0.198e0.00893 qr 0.106 þ 0.00182 qr 0.591e0.0393 qr 1/3 0.288 þ 0.0152 qr 0.418 e0.0013 qr

0.143

0.391

G. Gan / Building and Environment 46 (2011) 2069e2080

1000

The above coefficients for Equations (9) and (10a) are summarised in Table 1. Equation (9) can be used to calculate the heat transfer rate/ coefficient for given cavity size, heat input and distribution on cavity walls. The air flow rate (represented by Re) can then be obtained from Equation (10a) in combination with Equation (9) as

Nu = 0.102(Ra(H/b)3/2)0.280 R² = 0.998

Nu

100 10

Re ¼

1 1.0E+06

1.0E+08 1.0E+10 Ra (H/b)3/2

1.0E+12

Fig. 14. Effect of Ra on Nu based on individual values for cavities with a horizontal inlet.

Hence, a better expression for qr ¼ 0%e5% is represented in the following form:

Nu ¼ c2 ðRaa ReÞ

n

2079

(10a)

where c2, a and n are given by

ðNu=c2 Þ1=n Raa

(10b)

However, because of the dependence of the thermal properties of air on the wall surface temperatures, more accurate values can be obtained using Equation (3) to update these parameters. 4.2.2. Horizontal inlet Similar correlations between Nu, Re and Ra can be obtained for cavities with a horizontal inlet below a heated vertical wall. The relationships between Nu, Re and Ra for cavities with a horizontal inlet (Figs. 14 and 15) can be represented by:

0:28  Nu ¼ 0:102 RaðH=bÞ3=2

c2 ¼ 0:198  0:00893 qr

(27a)

and

a ¼ 0:591  0:00393 qr

(27b)

 0:421 Nu ¼ 0:107 Ra1=3 Re

n ¼ 0:288 þ 0:0152 qr

(27c)

Equations (25g) and (25h) for 5% < qr < 20% can then be simplified as:

c2 ¼ 0:106  0:00182 qr

(25g)

n ¼ 0:418  0:0013 qr

(25h)

a

1000

Nu

10

1.0E+06 1.0E+07 Ra1/3 Re

1.0E+08

Nu = 0.120(Ra1/3 Re)0.410 R² = 0.998

Nu

100 10 1 1.0E+04

1.0E+05

1.0E+06 1.0E+07 Ra1/3 Re



R2 ¼ 0:987

(29)



 R2 ¼ 0:998

(29a)

for 5%  qr  60%

and

Constant or Exponent

1000

 0:41 Nu ¼ 0:12 Ra1=3 Re

c1 ¼ 0:0761 þ 0:000359 qr

All data

b

(28)

Fig. 15a indicates that data for the correlation between Nu and Ra in combination with Re are again scattered resulting mainly from reverse flow near the outlet of wide cavities. When the data points for heat distribution ratios qr < 5% and qr > 70% are excluded (Fig. 16b), a better correlation can be obtained as follows:

c1 ¼ 0:0961

1.0E+05



 R2 ¼ 0:998

Using the same method as before, the relationships between Ra, Re and Nu can be obtained for each set of heat distribution ratio and the effect of heat distribution ratio can then be determined for correlations (9) and (10). Fig. 16 shows that the regression coefficients and constants for correlations (9) and (10) are in general dependent on the heat distribution ratio. The coefficients and exponents are constant or vary slightly but linearly with heat distribution ratio as follows:

Nu = 0.107(Ra1/3 Re)0.421 R² = 0.987

100

1 1.0E+04



1.0E+08

Fig. 15. Effect of Ra and Re on Nu based on individual values for cavities with a horizontal inlet.

for 60%  qr  95%

(30a)

0.5 0.4 0.3 0.2 0.1 c1

0 0

20

m

40

c2

60

n

80

100

Fig. 16. Effect of heat distribution ratio on the correlation coefficient/constant for Nu for cavities with a horizontal inlet.

2080

G. Gan / Building and Environment 46 (2011) 2069e2080

Table 2


3=2 m H for cavities with a horizontal inlet. Constant and exponent in Nu ¼ c1 Ra b Constant/Exponent

qr (%) 0e5

5e60

c1 m

0.0897 þ 0.00277 qr e 0.000233 qr2 0.29 e 0.000579 qr

0.0961 0.0761 þ 0.000359 qr 0.287 e 0.000116 qr

60e95

95e100 3.63 þ 0.0776 qr þ 0.000402 qr2 1.92 e 0.0341 qr þ 0.000178 qr2

Table 3  n Constant and exponents in Nu ¼ c2 Raa Re for cavities with a horizontal inlet. Constant/Exponent

qr (%) 0e5

5e20

20e90

90e95

95e100

c2 a n

0.206 e 0.0103 qr 0.667 e 0.0333 qr 0.265 þ 0.0161 qr

0.0712 qr0.132 1/3 0.446 e 0.00116 qr

0.0972 þ 0.000357 qr

0.0223 e 0.00104 qr

0.427  0.000275 qr

0.331 þ 0.00082 qr

0.0183 qr  1.61 0.0536 qr  4.75 2.87 e 0.026 qr

m ¼ 0:287  0:000116 qr c2 ¼ 0:00972 þ 0:000357 qr n ¼ 0:427  0:000275 qr

for 5%  qr  95% for 20%  qr  90% for 20%  qr  90%

(30b) (30c) (30d)

Outside these ranges, a second or higher order correlation with heat distribution ratio would be required, or the exponent for Ra in Equation (10) would vary with heat distribution ratio, in the form of Equation (10a). Tables 2 and 3 list the coefficients and constants in Equations (9) and (10a) for cavities with a horizontal inlet. Again, these expressions will allow the calculation of the heat transfer rate/coefficient and air flow rate in a cavity with horizontal inlet for given cavity size, heat input and distribution on the cavity walls. 5. Conclusions The air flow and heat transfer rates generated by buoyancy in tall cavities such as solar chimneys vary with cavity size, total heat input and heat distribution ratio on the two walls. The air flow and heat transfer rates are also affected by the position of inlet opening. For a cavity with horizontal inlet through a vertical wall as in the case of a solar chimney or double façade for room ventilation, the flow rate would be less than that for a cavity with vertical inlet at the bottom. The heat transfer rate for cavities with a horizontal inlet is generally higher that that with a vertical inlet. General expressions have been derived that correlate between the Nusselt number, Reynolds number and Rayleigh number. These can be used for the calculation of the air flow rate and heat transfer rate in tall ventilation cavities for given cavity width and height, heat flux and heat distribution ratio, and for design of such natural ventilation systems. The heat transfer rate and air flow rate are given by the following expressions:

" Nu ¼ c1


3=2 #m H Ra b

and

Re ¼

ðNu=c2 Þ1=n Raa

References [1] Aung W, Fletcher LS, Sernas V. Developing laminar free convection between vertical flat plates with asymmetric heating. International Journal of Heat and Mass Transfer 1972;15:2293e308. [2] Baskaya S, Aktas MK, Onur N. Numerical simulation of the effects of plate separation and inclination on heat transfer in buoyancy driven open channels. Heat and Mass Transfer 1999;35:273e80. [3] Badr HM, Habib MA, Anwar S, Ben-Mansour R, Said SAM. Turbulent natural convection in vertical parallel-plate channels. Heat and Mass Transfer 2006; 43:73e84. [4] Fedorov AG, Viskanta R. Turbulent natural convection heat transfer in an asymmetrically heated, vertical parallel-plate channel. International Journal of Heat and Mass Transfer 1997;40(16):3849e60. [5] Afonso C, Oliveira A. Solar chimneys: simulation and experiment. Energy and Buildings 2000;32(1):71e9. [6] Barozzi GS, Imbabi MSE, Nobile E, Sousa ACM. Physical and numerical modelling of a solar chimney-based ventilation system for buildings. Building and Environment 1992;27(4):433e45. [7] Bansal NK, Mathur R, Bhandari MS. Study of solar chimney assisted wind tower system for natural ventilation in buildings. Building and Environment 1994;29(4):495e500. [8] Gan G. Simulation of buoyancy-induced flow in open cavities for natural ventilation. Energy and Buildings 2006;38(5):410e20. [9] Gan G. A parametric study of Trombe walls for passive cooling of buildings. Energy and Buildings 1998;27(1):37e43. [10] Pedersen PV., Rasmussen A. Low energy ventilation system using PV modules. www.ecobuilding.dk. [11] Rodrigues AM, Canha da Piedade A, Lahellec A, Grandpeix JY. Modelling natural convection in a heated vertical channel for room ventilation. Building and Environment 2000;35:455e69. [12] Nguyen PH, Yerro P, Bernard JL, Sellitto P, Dupont M. Wall temperature experimental investigation in a thermally driven channel. Renewable Energy 2000;19:443e56. [13] Burek SAM, Habeb A. Air flow and thermal efficiency characteristics in solar chimneys and Trombe walls. Energy and Buildings 2007;39:128e35. [14] Miyamoto M, Katoh Y, Kurima J, Sasaki H. Turbulent free convection heat transfer from vertical parallel plates. In: Tien CL, Carey VP, Ferrell JK, editors. Proceedings of the 8th International heat transfer Conference, San Franscisco, CA, USA; 1986, 4: 1593e1598. [15] Olsson CO. Prediction of Nusselt number and flow rate of buoyancy driven flow between vertical parallel plates. Journal of Heat Transfer 2004;126: 97e104. [16] Gan G. Impact of computational domain on the prediction of buoyancy-driven ventilation cooling. Building and Environment 2010;45(5):1173e83. [17] Gan G. Simulation of buoyancy-driven natural ventilation of buildings Impact of computational domain. Energy and Buildings 2010;42(5): 1290e300. [18] FLUENT user’s guide. New Hampshire, USA: Fluent Inc; 2006. [19] Yakhot V, Orzsag SA. Renormalization group analysis of turbulence: I. basic theory. Journal of Scientific Computing 1986;1:1e51. [20] Gan G. Prediction of heat transfer and air flow in solar heated ventilation cavities. In: Murphy AD, editor. Computational fluid dynamics: theory, analysis and applications. New York: Nova Science Publishers; 2011. [21] Launder BE. On the computation of convective heat transfer in complex turbulent flows. Journal of Heat Transfer 1988;110:1112e28. [22] Etheridge DW. Nondimensional methods for natural ventilation design. Building and Environment 2002;37:1057e72.