Heat Mass Transfer DOI 10.1007/s00231-017-2103-7

ORIGINAL

Adiabatic partition effect on natural convection heat transfer inside a square cavity: experimental and numerical studies S. Mahmoudinezhad 1 & A. Rezania 1 & T. Yousefi 2 & M. S. Shadloo 3 & L. A. Rosendahl 1

Received: 21 February 2017 / Accepted: 8 July 2017 # Springer-Verlag GmbH Germany 2017

Abstract A steady state and two-dimensional laminar free convection heat transfer in a partitioned cavity with horizontal adiabatic and isothermal side walls is investigated using both experimental and numerical approaches. The experiments and numerical simulations are carried out using a Mach-Zehnder interferometer and a finite volume code, respectively. A horizontal and adiabatic partition, with angle of θ is adjusted such that it separates the cavity into two identical parts. Effects of this angel as well as Rayleigh number on the heat transfer from the sideheated walls are investigated in this study. The results are performed for the various Rayleigh numbers over the cavity side length, and partition angles ranging from 1.5 × 105 to 4.5 × 105, and 0° to 90°, respectively. The experimental verification of natural convective flow physics has been done by using FLUENT software. For a given adiabatic partition angle, the results show that the average Nusselt number and consequently the heat transfer enhance as the Rayleigh number increases. However, for a given Rayleigh number the maximum and the minimum heat transfer occurs at θ = 45°and θ = 90°, respectively. Two responsible mechanisms for this behavior, namely blockage ratio and partition orientation, are identified. These effects are ex-

* A. Rezania [email protected]

1

Department of Energy Technology, Aalborg University, Aalborg, Denmark

2

Department of Mechanical and Industrial Engineering, Ryerson University, Toronto, Canada

3

CORIA-UMR 6614, CNRS-University & INSA of Rouen, Normandie University, 76800 St. Etienne du Rouvray, France

plained by numerical velocity vectors and experimental temperatures contours. Based on the experimental data, a new correlation that fairly represents the average Nusselt number of the heated walls as functions of Rayleigh number and the angel of θ for the aforementioned ranges of data is proposed. Nomenclature C Gladstone–Dale coefficient (m3/kg) Polynomial coefficients Ci g Gravitational acceleration (m/s2) H Cavity side length (m) Local heat transfer coefficient (W / m2. K) hy have Average heat transfer coefficient (W / m2. K) k Thermal conductivity of air (W/m K) L Partition length (m) Nuave Average Nusselt number Modified Nusselt number Num Local Nusselt number on the heated wall Nuy P Pressure (Pa) R Gas constant (J/kg K) Ra Rayleigh number based on the cavity side length T Temperature (K) W Partition width (m) x Horizontal direction (perpendicular to the hot surface) y Vertical direction (parallel to the hot surface) Greek symbols ε Fringe shift λ Laser wave length (m) θ Angel of the partition (Degree) Subscripts f Film condition ref Reference condition (ambient condition)

Heat Mass Transfer

∞ SH SC

Ambient condition Hot surface Cold surface

1 Introduction The study of natural convection heat transfer is important because of its usage in many engineering applications. Natural convection is a cheap cooling technology, and is independent of electromechanical equipment. Therefore, it is widely used in many industries such as solar collectors, fire research, electronic cooling, aeronautics, chemical apparatus and solar energy systems. In most of these applications natural convection occurs inside a single or a set of cavities. Since the size of cavity directly affects the heat transfer from walls, changing the cavity size, for instance by placing an interior partition, gives a promising approach to control effectiveness of the heat transfer. Natural and forced convection heat transfer was studied for different geometries [1, 2]. Natural convection inside a cavity has been extensively studied using both numerically and experimentally [3–5]. Many of the studies in heat transfer in a cavity consider natural convection in enclosed square or rectangular cavities [6–19]. The influences of the attendance of a hot rectangular cylinder, at different horizontal and diagonal eccentric locations in a cold square enclosure, on fluid flow and heat transfer were investigated by Dash and Lee [6]. Numerical studies were presented for Rayleigh numbers in the range 103–106. Local and averaged Nusselt number variations as functions of Rayleigh number and eccentricity were presented and also a comprehensive analysis of isotherms, streamlines was done. Nardini et al. [7] carried out an experimental and numerical investigation of natural convection in a square cavity with partially active thermal side walls and an additional active source located in the center of the bottom wall for Rayleigh numbers changing between 104 and 105. The study was carried out experimentally through holographic interferometry and numerically using CFD software. Interferograms, streamlines, isotherms and velocity maps were examined for three different positions of the hot source. The results clearly show that different positions of the hot source have an effect on heat transfer and also the numerical tests presented good agreement with the holographic interferograms obtained. Optimal shape of cavity with variation of hot walls geometry was obtained using Particle swarm optimization algorithm (PSOA) by azimifar and payan [8]. Continuity, momentum and energy equations were discretized by finite volume method and solved by SIMPLER algorithm. The results were found for range of Rayleigh numbers (104 to 105) and different boundary conditions. Results indicated that PSOA is able to obtain size and location of changes in

hot walls. They found that these variations can increase heat transfer from cold wall between 13% and 40% for different boundary conditions. The effect of a small heating source, located in the side walls of a square cavity, was investigated by Nardini et al. [9]. Numerical and experimental analyses were performed to investigate natural convection heat transfer in a square cavity heated by hot strips in the side walls. The influence of placing the hot strips at two different positions was assessed. The temperature distribution and the Nusselt numbers at different Rayleigh numbers were experimentally measured using both real-time and double-exposure holographic interferometry. The isothermal patterns obtained through the holographic interferometry were compared with the temperature and velocity fields from a numerical study performed using the finitevolume code Fluent. A cavity of big dimensions was tested by Saury et al. [10]. Temperature levels imposed on active walls associated with the cavity dimensions lead to very large Rayleigh numbers reaching 1.2 × 1011. Velocity fields and Nusselt number distributions on the two active walls were presented. Radiation was considered in this study. Differences between measured and computed data were relatively small. Finally, NusseltRayleigh type correlations were proposed for all cases. Varol et al. [11], treated numerically and experimentally the case of a square section cavity. The active hot and cold walls were horizontal and located on the bottom and top respectively. The vertical walls were adiabatic and one of them was equipped with a fin of variable dimensions and locations, inclined with respect to the horizontal. The Nusselt number was studied, leading to proposing correlations which include, in addition to the Nusselt and Rayleigh numbers, the dimensions, the fin position on the wall and its angle of inclination relative to the vertical, changing between 30o and 120o. The Rayleigh number also varies between 8 × 105 and 4 × 106. Corvaro et al. [12] presented a numerical study complemented with experimental measurements using the Particle Image Velocimetry (PIV) technique to determine the dynamic fields in inclined cavities with this interesting non-invasive technique. Rezaei et al. [13] used an adaptive neuro-fuzzy inference system (ANFIS) to predict the free convection in a partitioned cavity consisting of an adiabatic partition. The training data for optimizing the ANFIS structure was obtained experimentally. They showed that ANFIS can precisely predict the experimental results for different cavity partition angle. Famouri and Hooman [14] studied entropy generation for natural convection in a partitioned cavity, with adiabatic horizontal and isothermally cooled vertical walls. They found that the fluid friction term has nearly no contribution to entropy production. They also showed that the irreversible heat transfer increases monotonically with an increment in the Nusselt number and dimensionless temperature differences.

Heat Mass Transfer

Oztop et al. [15] conducted a numerical study of natural convection for vertically and horizontally heated thin plate located inside an enclosure. They observed the heat transfer is enhanced about 20% when plate was located vertically. A similar problem was studied by Nag et al. [16]. They analyzed an adiabatic enclosure separated by a perfect conducting partition and found that the heat transfer at the cold wall increases irrespective of the position or length of the conducting partition. Tasnim and Collins [17] studied the natural convection in a square cavity with a thin baffle on the hot wall using finite volume method (FVM). They observed that the fin has a blocking effect on the fluid flow that depends on its position, length, and the Rayleigh number. In addition, a number of recirculating regions were formed above and under the baffle. Shi and Khodadadi [18] studied the effect of an almost perfect conducting partition on a hot surface. They identified two competing mechanisms that were responsible for flow and thermal modifications. One is due to the blockage effect of the fin, whereas the other is due to extra heating of the fluid that is accommodated by the fin. They showed that for high Rayleigh numbers the flow field is enhanced regardless of the fin’s length and position. Bilgen [19] solved the natural convection problem in cavities with a thin fin on a hot wall. He found that Nusselt number is an increasing function of Rayleigh number, and a decreasing function of the fin length and the relative conductivity ratio. This study is essentially the continuation of a previous work (Rezaei et al. [13]). In this framework, the aim of this paper is to investigate the effects of an adiabatic partition with the angle of θ on laminar free convection and fluid flow behavior inside a square cavity. This is done using both experimental and numerical approaches. The experiments are Fig. 1 Schematic of Mach–Zehnder interferometry experimental setup

carried out using Mach–Zehnder interferometer while the numerical cases are simulated by an FVM based code. The study is performed for seven different partition angles from θ = 0°to θ = 90° and for five different Rayleigh numbers from1.5 × 105 to 4.5 × 105. The main achievement of the present study is proposing a new correlation that properly illustrates the average Nusselt number of the heated wall as functions of Rayleigh number and the angel of θ for the abovementioned ranges of data.

2 Experimental approach 2.1 Interferometer The experimental study was carried out using Mach–Zehnder interferometry (MZI) technique. The interferometer consists of a light source, a micro lens, a pinhole, two doublets, three mirrors and two beam splitters. Figure 1 shows the interferometer setup. Beam splitters BS1 and BS2, along with plane mirrors M1 and M2 constituted the basic MZI. The laser beam gets expanded after passing through spatial filter and the Doublet1. The expanded beam is split into two equal beams by BS1. One beam passes through the test section and the other through the undisturbed field. These two beams again recombine at BS2. If the four optical plates, M1, M2, BS1, and BS2 are parallel, then infinite fringe interferograms will be formed. Further information about MZI can be found in [20–22]. The light source is a 10 mW Helium–Neon laser with a 632.8 nm wavelength. All the interferograms were digitized with a BARTCAM-320P^ 1/2″ CCD camera with3.2 M pixels. To acquire the interferograms the camera was connected to a PC.

Heat Mass Transfer

2.2 Experiment test section Details of the partitioned cavity used in the experiments are shown schematically in Fig. 2. The length of each isothermal wall was chosen as 100 mm, which causes the induced flow to be two-dimensional. Also, the wooden end caps with thermal conductivity of 0.05 W/m K [23] were installed on each aluminum plate bases to minimize the end effects. By passing electricity through each heater that placed back of each aluminum plate and considering relatively thick-walled aluminum plate, we could achieve constant surface temperature. By measuring temperature at three different locations, the uniformity of temperature in each plate surface can be validated experimentally. The differences in temperature readings for each aluminum plate surface were about 0 . 1°C. The local surface temperatures of each heated aluminum plate were recorded via three type-K thermocouples, embedded vertically in the aluminum plate wall (see Fig. 2). Two other thermocouples of the same type were used to measure the ambient and the reference temperatures for data reduction. All the temperatures were monitored continuously in a PC by a selector switch and a BTESTO 177 T4^ four channel data logger. The laboratory pressure was recorded during all the experiments. The maximum uncertainties of temperature and pressure measurement for the present test condition were ±0 . 1°C and ±100 Pa, respectively. In all the experiments, the heater voltage and current is recorded. In order to ascertain the accuracy of the measurements, the energy balance calculation for many

cases was done by calculating free convection heat transfer from the fringe patterns of the Mach–Zehnder interferometer and by measuring electrical power input to the heaters. Comparison between the heat transfer coefficients obtained by two methods shows an excellent agreement. The partition was held using two holding rods; each with a 1.2 mm diameter, which was attached to the two ends of partition. The rods were connected to a special stand that could be adjusted to provide partition parallelism with the laser beam by independent movement of each one in the vertical and horizontal direction. Two windows were used on both side of the cavity to prevent external air from entering the cavity. In order to eliminate the effect of any other air disturbances on the experimental test section, the entire interferometer table was housed within a top open transparent plastic enclosure with dimensions of 3 × 1.5 × 1.5 m3. 2.3 Data reduction The aim of the data reduction procedure was to determine the local and average Nusselt numbers. For each case three interferograms with ten seconds interval were captured for assurance of experiment repeatability. In order to determine the local and average Nusselt number on the hot wall, a MATLAB code has been developed. The temperature of the interference fringes as well as their distance from the surface of the aluminum plate in horizontal direction is calculated by the method explained by Eckert and Goldstein [21] and Hauf

A

C

C

A

B

B

Section A-A

Thermocouple locations Section B-B

Electrical heaters

Partition (wood fiber) Aluminum plates

Wood fibers End-Cap (wood fiber)

Holding rods locations

Fig. 2 Details of the partitioned cavity used in the current experiments

Section C-C

Heat Mass Transfer

and Grigull [20]. The local air temperature gradient at the aluminum plate surface is obtained from following equation:

and




dT

dT

dε

¼ : dx
x¼0 dε
x¼0 dx
x¼0

Nuave ¼ ð1Þ

x¼0

Here R = 287 J/kg, K is the gas constant, P is the ambient pressure, λis the wavelength of Helium-Neon laser beam, C is the Gladstone–Dale constant, and Trefis the temperature of reference fringe. In the Fig. 3 the reference (infinity) temperature and the fringe shift value are displayed. The local heat transfer coefficient is determined as follows:
dT

1 hy ¼ −k SH
dx x¼0 T SH − T ∞

ð3Þ

Then the local Nusselt number is obtained from: hy y kf

ð4Þ

Where Tf is the film temperature and T f ¼ T SH 2þT ∞ , kSH and kf are the thermal conductivity of air evaluated at the surface temperature of TSH and the film temperatureof Tf, respectively. The average heat transfer coefficient and Nusselt number are calculated as follow: have ¼

ð6Þ

2.4 Uncertainty analysis

where ε is the fringe shift value, x is the perpendicular distance to the heated plate surface and T is the temperature magnitude. Using the ideal gas and Lorenz–Lorenz equations is calculated as follow [20]:   L P
3C dT

λ R ¼   ð2Þ 2 : dε
x¼0 3 L P C − 2ε 2 λ RT ref

Nuy ¼

have L kf

1 L ∫ hy dy L 0

ð5Þ

ε=0

Thermocouple (reference temperature)

ε=1 ε=0 ε=0.5

Fig. 3 Reference temperature and fringe shift values for angle of θ = 30° and Rayleigh number of Ra = 4.5 × 105

Uncertainty analysis has been carried out using the method proposed in [22, 24]. The standard uncertainties in the gas constant, the thermal conductivity values, the fringe shift and the Helium-Neon laser wavelength have been neglected. The uncertainties of the parameters T∞, Tw and P∞ can be estimated from the measuring devices precision and the uncertainty of the cavity sided measuring devices. The uncertainty of the parameter Δx, the difference in horizontal distance from the plate surface, is related to the precision of reading the digitized interferograms. The error sources are collected in the Table 1. Using these parameters, the uncertainty in the measurement of local Nusselt number has been evaluated to be 3.2 ± 0.9%.

3 Numerical approach 3.1 Description of the model A finite volume method (FVM) code is used for the numerical simulations presented in the current work. The problem is considered to be two-dimensional. The computational domain is a square box with the side length of H. A partition with the length of L, and width of W is inserted in the middle of computational box such that it divided the domain into two identical parts (in opposite sense). For each numerical experiment, the partition has a prescribe angle of θ with the horizontal line (the clock wise angle with x−direction). Figure 4 illustrates a schematic of the problem. Geometry and mesh creation has carried out with Gambit software. Triangle unstructured meshes are used for the domain. In the near of isothermal surfaces due to thermal gradients are very closer, boundary layer mesh is used. Figure 5 shows this Triangle unstructured meshes in the cavity. The results are then calculated for laminar steady-state solution. The no-slip boundary condition is adjusted for all walls. The partition and both horizontal wall are assumed to be

Table 1

The sources of errors in laboratory experiments

Error sources

Variables

Bias errors

Temperature Pressure Cavity Dimensions Fringe Distances

T P L,H Δx

0.1 °C 100 Pa 0.02 mm 0.03 mm

Heat Mass Transfer

equations solved by finite volume based CFD programs to calculate the flow pattern and associated scalar fields.  ∂ ∫CV ρϕdV þ ∫A n:ðρϕuÞdA ¼ ∫A n:ðΓgrad ϕÞdA þ ∫CV S ϕ dV ð7Þ ∂t

Fig. 4 Numerical model geometry of the partitioned cavity

thermally adiabatic (zero flux condition). Two left and right vertical walls are kept respectively at the constant temperatures of TSHand TSC. The ambient fluid is air at initial constant temperature of T∞. The gravity g is acting downward in y−direction. The effects of various partition angles,θ, and Rayleigh numbers, Ra, on the heat transfer from the vertical hot wall of the air filled cavity are then investigated. A commercial code Fluent, (Fluent software) has used for solving the problem. In Fluent after checking the grid the solver is determined. Two dimensional and steady state models are used. Energy equation is activated and in the properties of air, Boussinesq approximation is used for the density. Under relaxation factors for pressure, density, body force and momentum are 0.3, 1, 0.9 and 0.6, respectively. Underrelaxation factors are included to stabilize the iterative process. Also convergence can be accelerated by using appropriate under-relaxation factors. The type of discretization for pressure is PRESTO [25] and for momentum and energy is Second Order Upwind [26]. Pressure-Velocity coupling type was chosen SIMPLE [27] and the magnitude of residual for convergence was assumed 10−6. 3.2 Governing equations Equation (7) represents the general conservation equation in integral form. This is the actual form of the conservation

Fig. 5 Triangle unstructured meshes in the cavity with prism layer mesh near walls

In this equation ϕ is the property (mass, momentum, energy) per unit mass, u is the flow velocity of the fluid, n is the outward-pointing unit-normal vector, and S represents the sources and sinks in the flow, taking the sinks as positive. The first term in the left side of this equation represents the rate of increase of ϕ and the second term shows the net rate of decrease of ϕ due to convection across boundaries. On the right side, the first term illustrates the net rate of increase of ϕ due to diffusion across boundaries and the last term is the net rate of creation of ϕ. Governing equations for the above-mentioned problem are the Navier-Stocks equations, which in their dimensional form and with considering Boussinesq approximation can be written as follow: 0

0

∂u ∂v þ ¼0 ∂x0 ∂y0 " 0 #  0 0 0 0 μ ∂2 u ∂2 u 1 ∂p 0 ∂u 0 ∂u u 0 þv 0 ¼ þ − ρ ∂x0 2 ∂y0 2 ρ ∂x0 ∂x ∂y " 0 #  0 0 0 0  0  μ ∂2 v ∂2 v 1 ∂p 0 ∂v 0 ∂v u 0 þv 0 ¼ þ −T − þ gβ T ∞ 0 ρ ∂x0 2 ∂y0 2 ρ ∂y ∂x ∂y 0 0 0 0

K ∂2 T ∂2 T 0 ∂T 0 ∂T þ u 0 þv 0 ¼ ρC p ∂x0 2 ∂x ∂y ∂y0 2 ð8  1Þ

Where u′ and v′ are respectively the velocity components in x and y′ directions. T′denotes the temperature,p′ pressure, μ viscosity, ρ density,βthermal expansion coefficient, Kthermal conductivity coefficient of the fluid, and Cpis the specific heat capacity at constant pressure in their dimensional forms. ′

Heat Mass Transfer

Now if we use (x, y) = (x′, y′)/H, (u, v) = (u′, v′)H/α, 0

0

−T ∞ p ¼ ρup 2 , and T ¼ T ΔT for scaling the position, velocity,

Table 2 Independency of the numerical simulations from mesh resolution for Ra = 4.0 × 105 and θ = 15°

∞

Number of cells

pressure and temperature fields, respectively, the set of Eq. (8-1) will be reduced to their dimensionless forms as: ∂u ∂v þ ¼0 ∂x ∂y 2

∂u ∂u ∂ u ∂2 u ∂p − ¼ Pr þ u þv ∂x ∂y ∂x2 ∂y2 ∂x 2

∂v ∂v ∂ v ∂2 v ∂p − þ Ra:Pr:T þ u þ v ¼ Pr ∂x ∂y ∂x2 ∂y2 ∂y 2

∂T ∂T ∂ T ∂2 T þv ¼ þ u ∂x ∂y ∂x2 ∂y2

Average Nusselt number

Relative error (%)

Coarse (Mesh-1)

11,706

11.544

0.164

Intermediate (Mesh-2) Fine (Mesh-3)

38,758 64,524

11.562 11.563

0.008 0.000

Very fine (Mesh-4)

91,942

11.563

--

ð8  2Þ

In the following sections the effect of the inclined adiabatic partition and Rayleigh number on heat transfer from the heated wall are investigated in details. To do that, the numerical experiment is initially validated with the laboratory data. Then the fluid behaviors calculated by simulations are utilized to explain and justify the thermal behavior along the heated cavity wall. Both numerical and laboratory experiments are performed for Rayleigh numbers, based on the cavity side wall length H, ranging from Ra = 1.5 × 105 to Ra = 4.5 × 105 and for various inclined partition angles ranging from θ = 0° to θ = 90°. More than overall 40 test cases are solved using each method (more than 80 test cases in total). Only the insightful resulted are plotted and shown in the following sections.

−N uMesh−4 calculated from: Relative error ð%Þ ¼ N uMesh−i  100, N uMesh−4 th where i shows the i row of the same table. Having converged mesh, the accuracy of the numerical data are validated against the experimental results by comparing the isotherms as well as the average and local Nusselt numbers. Isotherms computed using FVM is plotted against its equivalent experiment in Fig. 6. This test case corresponds to the cavity with inclined partition at the angle of θ = 30° andRa = 4.5 × 105. As can be seen in this figure, the numerical temperature contours qualitatively resemble those obtained by the experiment especially near the heated wall where the maximum heat transfer appears. In the Table 3 thermal boundary conditions for different Rayleigh numbers obtained from the experiments and applied in the simulations is presented for θ = 30°. In many references streamline [28, 29] or vorticity contours [30, 31] were used to describe rotational flow but in the present investigation velocity vectors have been used. For this case, streamlines and vorticity contours are also presented in Fig. 6. To quantify the accuracy of the results, the local Nusselt number along the vertical heated wall is further shown in Fig. 7 for the same test case. It is seen that the numerical simulation predicts both the local Nusselt number value and its pattern very well, hereby, confirming the precision of FVM code. Additionally, the average Nusselt number for the cavity with the partition angle of θ = 30° at different Rayleigh numbers is examined and illustrated in Fig. 8. It can be observed that the numerical values are in close agreement with those of experiments having the maximum deviation of less than 3.5%.

4.1 Numerical convergence and validation

4.2 Local Nusselt number

Independency of solution from the number of meshes used in the numerical simulation is examined for the test case with Ra = 4.5 × 105 and θ = 15°, as is shown in Table 2. It is seen that both intermediate (Mesh-2) and fine (Mesh-3) mesh resolutions are providing very accurate results in comparison with the finest mesh (Mesh-4). Therefore, having a good balance between the computational cost and numerical accuracy, the intermediate grid resolution (Mesh-2) is used for the numerical simulations presented in the rest of this work. In Table 2 the relative error is

Figure 9 represents the variation of the experimental local Nusselt number along the heated wall with respect to vertical distance from the bottom of cavity for the case with horizontal partition θ = 0°. As can be seen, the experimental local Nusselt number has a maximum value at the bottom of the heated wall, where the boundary layer starts to develop. The maximum local Nusselt number for θ = 0° is around 23 but this number for θ = 15° (see Fig. 10) is around 20 and it shows that in θ = 0° there is a higher heat

Where Ra ¼ gβHαϑΔT , Pr ¼ αϑ, and ΔT = TSH − T∞. To investigate the given problem precisely and to justify the experimental results a numerical code was developed using FVM. As shown in Fig. 5, triangle unstructured meshes are used inside of the cavity. Due to the importance of the boundary layer and as consequence temperature gradients near walls, a prism layer mesh is used in those regions. 3

4 Results and discussion

Heat Mass Transfer Fig. 6 a Isotherms obtaining from numerical simulation, (b) their corresponding experimental fringes for the inclined partition, (c) Streamlines and (d) vorticity contours obtaining from numerical simulation with the angle of θ = 30° and Rayleigh number of Ra = 4.5 × 105

at farther vertical distance from the cavity bottom that is approximately aty = 31 mm. To describe these behaviors infinite fringe interferograms for the Rayleigh number of Ra = 4.0 × 105and for the partition angle of θ = 0°and θ = 15°, are depicted in Figs. 11 and 12, 35 30

Experimental Data Numerical

25 20

Nuy

transfer which the reason will be presented in the next part based on the velocity vectors for θ = 0° and θ = 15°. Increasing the distance from the bottom, the local Nusselt number decreases linearly to a minimum value. It, then, followed by a sudden increment at the middle of the heated wall (approximately at y = 25mm). The difference between the values of Nuyfor these two points (i.e. local minima and maxima) is decreasing by a decrement in Rayleigh number. After this point, once more, the local Nusselt number decreases. However the slope of the decrement in this region is gentler in comparison with that in the first region. The same pattern is also observed for the cavity with partition angle of θ = 15°. However, this time the maximum experimental local Nusselt number appears Table 3 Temperatures of the hot and cold side of cavity, ambient temperature, reference temperature, film temperature and ambient pressure for different Rayleigh numbers and θ = 30° °

°

°

°

24,8 25,5 25,8 26,1 26,4

30 32,4 40,1 49,1 51,4

15 10

°

TSC( C) TSH( C) T∞( C) Tref( C) Tf( C) P∞(Pa) Ra = 1.5040 × 105 Ra = 2.0014 × 105 Ra = 3.0028 × 105 Ra = 4.0017 × 105 Ra = 4.5078 × 105

27,3 29 32,3 37 39,6

43,6 51,9 70,1 94,6 112

34.2 38.7 47.95 60.35 69.2

87,400 87,500 87,400 87,300 87,300

5 0

0

10

20

30

40

50

Y (mm) Fig. 7 Comparison between experimental and numerical values of the local Nusselt numbers for θ = 30° and Rayleigh number Ra = 4.5 × 105

Heat Mass Transfer 26

16

24 Experimental Data Numerical

14

22

Ra = Ra = Ra = Ra = Ra =

20 18 16

Nuy

Nuave

12

10

5

1.5× 10 5 2.0× 10 3.0× 10 5 4.0× 10 5 4.5× 10 5

14 12 10 8 6

8

4 6

2 0

100000

200000

300000

400000

500000

Ra Fig. 8 Comparison between experimental and numerical values of the average Nusselt numbers for different Rayleigh numbers at the partition angle of θ = 30°

respectively. It can be seen that in the region close to the bottom of the heated wall, isotherms are very close to each other. This means higher temperature gradient and as a consequence higher heat transfer. Moving upward, the distance among these fringes increases continuously. This causes a continuous decrement in the local Nusselt number down to minima at a distance close to the middle of cavity (see also Nuy in Fig. 9). This position corresponds to the point of maximum distance between fringes, as shown in the Fig. 11. Then, the partition affects the development of the thermal boundary

26 24 22

Ra = Ra = Ra = Ra = Ra =

20 18

Nuy

16

5

1.5× 10 2.0× 10 5 3.0× 10 5 4.0× 10 5 4.5× 10 5

14

0

0

10

20

30

40

50

Y (mm) Fig. 10 The experimental local Nusselt number variation along the vertical heated wall for different Rayleigh numbers at the partition angle of θ = 15°

layer and forces fringes to get closer into the heated wall. This is the point where a sudden increment in the local Nusselt number happens. Crossing the partition, the isotherms are continuously expanded until the top of the heated wall. With increasingθ, it is seen that fringes in both sides of the heated wall (before and after the partition) get less influences from the plate. Therefore, the effect of the partition on an unexpected raise in the local Nusselt number is reduced. Based on this results it can be concluded that since the blockage effect a sudden rise in local Nusselt number would be happened at any angle less than 15 degrees. Furthermore, for both cases, the isotherms before the plates diverge faster than those after it. Therefore, the slope at which Nuy decreases is sharper in this region. Figure 13 indicates the variation of the experimental local Nusselt number of the cavity with an adiabatic partition at the angle of θ = 45°. The maximum local Nusselt number is higher than 26 and it’s a higher value in comparison with θ = 0° and θ = 15°. Unlike previous cases, it is seen that the local Nusselt

12 10 8 6 4 2 0

0

10

20

30

40

50

Y (mm) Fig. 9 The experimental local Nusselt number variation along the vertical heated wall for different Rayleigh numbers at the partition angle ofθ = 0°

Fig. 11 Effect of the partition on the thermal boundary layer and isotherms for Ra = 4.0 × 105 and θ = 0°

Heat Mass Transfer 26 24 22 20 18

Nuy

16 14 12 10

Fig. 12 Effect of the partition on the thermal boundary layer and isotherms for Ra = 4.0 × 105 and θ = 15°

8 6

number decreases continuously without having any local extreme. This can be justified by the fact that the partition is not anymore modifying the thermal boundary layer and as consequence isotherms; therefore, it is not causing any changes in Nuy curves. It is noted that the local Nusselt number decreases with a decrement in Rayleigh number, regardless of the partition angle. The latter is consistent with the results of Shi and Khodadadi [18] and Bilgen [19]. Figure 14 shows the variation of the experimental local Nusselt number along the heated wall for different partition angles at the Rayleigh number of Ra = 3.0 × 105. As justified before, it is found that the difference between the local Nusselt numbers is large in the first half of the heated wall and it reduces in its second half of the domain. Noted that, except the cavity with the vertical partition, which always has minimum Nuy among the reported case, the local Nusselt number for different cavity angles cross over each other several times. This makes their direct comparisons rather difficult.

4 2 0

0

10

20

30

40

50

Y (mm) Fig. 14 The experimental local Nusselt number variation along the heated wall for at different partition angles ofθ at the Rayleigh number of Ra = 3.0 × 105

Therefore, the averaged Nusselt number for different cases is investigated in the next section. 4.3 Average Nusselt number Figure 15 represents the variation of the experimental average Nusselt number of the heated wall as a function of the partition angle for different Rayleigh numbers. It is observed that for each partition angle, similar to Nuy, the average Nusselt number increases with an increment in the Rayleigh number. Also at each Rayleigh number the maximum Nuaveoccurs at θ = 45°

28

18

26 24

Ra = Ra = Ra = Ra = Ra =

22 20 18

14 12

14

Nuave

Nuy

16

16

5

1.5× 10 2.0× 10 5 3.0× 10 5 4.0× 10 5 4.5× 10 5

12 10 8

10 8 6

Ra = Ra = Ra = Ra = Ra =

6

4

4 2 0

2 0

10

20

30

40

50

Y (mm) Fig. 13 The experimental local Nusselt number variation along the heated wall for different Rayleigh numbers at the partition angle of θ = 45°

0

0

10

20

30

40

1.5× 105 2.0× 105 3.0× 105 4.0× 105 4.5× 105

50

60

70

80

90

Fig. 15 Variation of the experimental average Nusselt number versus partition angle for different Rayleigh numbers

Heat Mass Transfer 16 14 12

Nuave

10 8 6 4 2 0

0

100000

200000

300000

400000

500000

600000

Ra Fig. 16 Variation of the average experimental Nusselt number versus Rayleigh number for different partition angles

and the minimum average Nusselt number occurs at θ = 90°. Furthermore, there is a local minimum for average Nusselt number at θ = 15°. As a matter of fact, there is an unexpected local maximum in the average Nusselt number for θ = 0°. As it will be explained later, the reason is the strong symmetric circulating flow in each part of the cavity for θ = 0° which cause to have higher Nusselt number rather than θ = 15°. Figure 16 shows the variation of the experimental average Nusselt number for the heated wall as a function of Rayleigh number for different partition angles. It is further seen that, regardless of the partition angle, the average Nusselt number decreases with a decrement in Rayleigh number. This is consistent with the finding for the local Nusselt number in previous section. Additionally, confirming the results reported in Fig. 15, it is seen that the maximum and minimum values of the average Nusselt number at each Rayleigh number are related to the cavities with the partition angle of θ = 45°and θ = 90°, respectively. The latter observation contradicts the results reported by Oztop et al. [15] and Nag et al. [16] for cavities

with isothermal partitions where the wall heat transfer increased for a vertical partition. This variation in Nuavecan be described with utilizing velocity vectors and boundary layer behavior, which obtained from the numerical simulation. As shown in Fig. 17, whenθ = 0°, the partition divides the cavity into two equal rectangles. Although the partition hinders the development of the boundary layer, it helps the flow to produce a strong symmetric circulating flow in each part. This causes the relatively high average Nusselt number for this case. Velocity vectors which can be seen in Fig. 17a shows that there is two almost similar flow patterns in both up and down sides of the partition with high value (in red in the figure) of velocity near the walls. This cause to have a relatively higher heat transfer for θ = 0° in comparison with θ = 15°. With increasing the partition angle from θ = 0° to θ = 15°, some of air passes through distance between the partition and heated wall. Therefore, the previous symmetry is breaking down and two strong rotational flows in the cavity are replaced by one bigger circulation (see Fig. 18). For this case, the partition is positioned such that it neither prevents the crossing of the flow from one part to the other part of the cavity, nor it lets the boundary layer develop freely. Therefore, the average velocity near the heated wall and, as a consequence, the average Nusselt number decreases. As it can be seen in Figs. 17 and 18, the maximum velocity for θ = 0° and θ = 15° is 4.38 × 10−2 and 5.83 × 10−2 respectively. It means that the maximum velocity for θ = 15° is higher than =0°. But this high local velocity value is extended to higher length of the heated wall in θ = 0° and then the average velocity near the heated wall for θ = 0° is higher than θ = 15° and as a consequence the average Nusselt number for θ = 0° is higher than θ = 0°. Increasing the partition angle from θ = 15° to θ = 45°, will increase the gap between the partition and walls. This allows the air to circulate much easier and to create one strong rotational flow. Consequently, air velocity near the hot wall is increased, and therefore there would be an enhancement in heat transfer. As it can be seen in Fig. 19, when the partition angle isθ = 45°, air can pass through the gap easily and create a

Fig. 17 a Numerical velocity vectors and (b) experimental isotherms inside cavity for partition angle ofθ = 0°at Ra = 4.5 × 105

(a)

(b)

Heat Mass Transfer Fig. 18 a Numerical velocity vectors and (b) experimental isotherms inside cavity for partition angle of θ = 15°at Ra = 4.5 × 105

(a) strong rotational flow inside the cavity. This makes the largest value of the local velocity which is 6.21 × 10−2 that is higher than the maximum local velocity for θ = 0° and θ = 15° and also other degrees of the partition except θ = 90°. It leads to having higher average velocity near the heated wall and consequently the biggest value of the average Nusselt number for θ = 45°among other cases. The same phenomenon, in the opposite sense, happens by further increasing the partition angle. The partition will start to hinder the flow circulation. Hence the average Nusselt number decreases. This time, the adiabatic partition separates the heated wall from the cold one, thereby, weakening the flow circulation comparing with θ < 45°. Although the maximum local velocity for θ = 90° is 6.29 × 10−2 that is higher than all other degrees of the partition but as it can be seen in Fig. 20a, it happens in a very small area between the partition and the top and down insulated walls of the cavity. Therefore it has a very small effect on the heat transfer from the heated wall. In fact the local and average velocity near the heated wall is very less than the other degrees. Consequently, the average Nusselt number has its minimum value at partition angle of θ = 90° (corresponds to Fig. 20). It is observed that the average Nusselt number at θ = 90° is less than 20% of average Nusselt number at θ = 45°, for low Rayleigh numbers and is about 50% for high Rayleigh numbers.

(b) Finally two mechanisms responsible for flow and thermal modifications are identified at a given Rayleigh number. The first one is due to the blockage effect of the partition (similar to that reported by Shi and Khodadadi [18]), whereas the other is due to the partition orientation. For large blockage ratios, partition orientation does not produce any significant change in the heat transfer rate across the cavity. Because these arrangements does not greatly disturb the flow field [32]. For instance the heat transfer coefficient is not varying significantly when comparing the θ = 30° and θ = 60° cases (see Figs. 15 and 16). However, for lower blockage ratios the flow field and heat transfer are reduced at higher partition angles. This suggests that the diminished heating mechanism caused by partition orientation outweighs the blockage effect for low blockage ratios. For instance the heat transfer coefficient for θ = 75° is lower than that when θ = 15°. 4.4 Correlation As it is noticed in Figs. 15 and 16, the average Nusselt number has a unique pattern while changing Rayleigh number or partition angle. Therefore, it might be prudent to express the average Nusselt number as a function of these variables. Hence, the experimental data are used to develop a generic correlation that takes the variation of both Rayleigh number

Fig. 19 a Numerical velocity vectors and (b) experimental isotherms inside cavity for partition angle of θ = 45°at Ra = 4.5 × 105

(a)

(b)

Heat Mass Transfer Fig. 20 a Numerical velocity vectors and (b) experimental isotherms inside cavity for partition angle of θ = 90°at Ra = 4.5 × 105

(a) and partition angle into account. To do so, the following polynomial curve fitting is performed: Num ¼ C0 þ C1 θ þ C2 θ2 þ C3 θ3 þ C4 θ4 :

ð9Þ

ave , and polynomial constants are respecHere, Num ¼ Nu Ra0:4 tively C0 = 7.197875 × 10−2, C1 = − 1.963386 × 10−3, C2 = 1.339572 × 10 − 4 , C 3 = − 2.391943 × 10 − 6 , and C 4 = 1.212256 × 10−8. Figure 21 illustrates the correlation given by equation (10), along with all the experimental measurements. It is seen that the proposed correlation represents the wide range of experimental analysis fairly well. The standard deviation for the curve fitting of equation (9) is about ±3.97%.

(b) investigated by numerical and laboratory experiments. Computations were carried out by an FVM code and experiments were performed using a Mach-Zehnder interferometer. While the latter is used to calculate the magnitude of both local and average Nusselt number, the former was utilizes to investigate the velocity contours and to describe the heat transfer behavior. The effects of the angel of the adiabatic partition and Rayleigh number on the heat transfer from the heated wall were investigated. The current study were performed for various Rayleigh numbers ranging from1.5 × 105 to 4.5 × 105 and various partition angle ranging from θ = 0° to θ = 90°. All the experimental results are presented with a single correlation which gives the average Nusselt number as a function of Rayleigh number and partition angle of θ. The following results were obtained:

5 Conclusions – Laminar free convection heat transfer in partitioned cavities with adiabatic horizontal and isothermally vertical walls was –

0.14 Ra = 1.5× 105 5 Ra = 2.0× 10 5 Ra = 3.0× 10 5 Ra = 4.0× 10 Ra = 4.5× 105 Correlation, Eq. (9)

0.12 0.1

–

0.08

Num

–

0.06

–

0.04

–

0.02 0

0

10

20

30

40

50

60

70

80

Fig. 21 Correlation of Nusselt number, based on experimental data

90

–

Local Nusselt number has a local maximum value at the bottom of the heated wall and decreases by increasing the distance from the bottom. Local Nusselt number has an unexpectedly increase at the middle of the heated wall, y = 25 mmfor θ = 0° and at y = 31 mm for θ = 15°. Since the blockage effect it can be concluded that unexpected increase in local Nusselt number would be observed at any angle less than 15 degs. Both local and average Nusselt numbers increase by an increment in the Rayleigh number regardless of the partition angle. At each Rayleigh number the maximum average Nusselt number occurs at θ = 45° and the minimum average Nusselt number occurs at θ = 90°. At each Rayleigh number there is a relative minimum for average Nusselt number at θ = 15°. Average Nusselt number at θ = 90° is less than 20% of that whenθ = 45°for low Rayleigh numbers and is about 50% smaller for high Rayleigh numbers. Two dominant mechanisms responsible for flow and thermal modifications caused by blockage ratio and partition angle are identified.

Heat Mass Transfer

– –

For large blockage ratios partition orientation does not produce significant changes in the heat transfer rate across the cavity. For low blockage ratios partition orientation found to have a significant effect on the flow field and heat transfer.

References 1.

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