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D. Roeleveld 1 Department of Mechanical & Industrial Engineering, Ryerson University, 350 Victoria Street, Toronto, ON M5B 2K3, Canada e-mail: [email protected] D. Naylor Department of Mechanical & Industrial Engineering, Ryerson University, 350 Victoria Street, Toronto, ON M5B 2K3, Canada W. H. Leong Department of Mechanical & Industrial Engineering, Ryerson University, 350 Victoria Street, Toronto, ON M5B 2K3, Canada Free Convection in Asymmetrically Heated Vertical Channels With Opposing Buoyancy Forces Laser interferometry and flow visualization were used to study free convective heat trans- fer inside a vertical channel. Most studies in the literature have investigated buoyancy forces in a single direction. The study presented here investigated opposing buoyancy forces, where one wall is warmer than the ambient and the other wall is cooler than the ambient. An experimental model of an isothermally, asymmetrically heated vertical chan- nel was constructed. Interferometry provided temperature field visualization and flow vis- ualization was used to obtain the streamlines. Experiments were carried out over a range of aspect ratios between 8.8 and 26.4, using temperature ratios of 0, 0.5, and 0.75. These conditions provide a modified Rayleigh number range of approximately 5 to 1100. In addition, the measured local and average Nusselt number data were compared to nu- merical solutions obtained using ANSYS FLUENT. Air was the fluid of interest. So the Prandtl number was fixed at 0.71. Numerical solutions were obtained for modified Rayleigh num- bers ranging from the laminar fully developed flow regime to the turbulent isolated boundary layer regime. A semi-empirical correlation of the average Nusselt number was developed based on the experimental data. [DOI: 10.1115/1.4026218] Keywords: interferometry, flow visualization, numerical modeling, natural convection, vertical channel 1 Introduction Free convection in a heated vertical channel is a classical heat transfer problem that has been studied extensively in the literature. There are many applications of this problem, such as electronics cooling, simulation of flow in nuclear reactors, and fenestration systems (i.e., windows with blinds). Many of these studies have investigated buoyancy driven flow in a single direction inside an asymmetrically heated vertical channel. The current study investi- gates opposing buoyancy driven flow, where the fluid tends to flow in opposite directions inside the vertical channel. A schematic diagram of the vertical channel geometry is shown in Fig. 1. An open-ended vertical channel is created by two iso- thermal channel walls of height L separated by a channel spacing b. The aspect ratio of the channel is defined as A ¼ L/b. The cold wall has a temperature T C and the hot wall has a temperature T H . The cold wall temperature is set below the ambient temperature T 1 and the hot wall is set above the ambient temperature. Aung [1] defined a temperature ratio as R T ¼ T C T 1 T H T 1 (1) Most existing studies of free convection inside a heated vertical channel have investigated temperature ratios in the range of 0 R T 1 (i.e., T C ,T H > T 1 ). This study will investigate oppos- ing buoyancy forces, where the temperature ratio is in the range of 1 < R T 0 (i.e., T H > T 1 and T C < T 1 ). There have been numerous studies on free convection in iso- thermally heated vertical channels. Elenbaas [2] was one of the first to study heat flow in a symmetrically (i.e., T H ¼ T C ,R T ¼ 1), isothermally heated vertical channel. Using two square plates sep- arated by various channel spacings, he was able to obtain experi- mental data for a wide range of modified Rayleigh numbers. Some analytical work determined that the Nusselt number approaches two asymptotes at the upper and lower modified Ray- leigh numbers. An overall channel average Nusselt number corre- lation was developed using the analytical work and experimental data. Aung et al. [3] studied the conditions of a uniform heat flux and a uniform wall temperature in an asymmetrically heated vertical channel. Numerical solutions were obtained over a wide range of modified Rayleigh numbers and some experimental work was per- formed to verify the results. They showed that, for uniform wall temperatures, a nearly universal curve can be used to relate the Nusselt numbers and the modified Rayleigh numbers for a wide range of temperature ratios. This is the case if the Nusselt number and Rayleigh number are defined by using the appropriate charac- teristic temperature difference. This characteristic temperature difference is defined as DT ¼ T H þ T C 2 T 1 (2) The modified Rayleigh number based on this temperature differ- ence is Ra b=L ð Þ¼ gb DT q 2 b 3 l 2 Pr b L (3) where g is gravity, b is the fluid thermal expansion coefficient, q is the fluid density, l is the fluid dynamic viscosity, and Pr is the 1 Corresponding author. Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received September 11, 2013; final manuscript received November 28, 2013; published online March 7, 2014. Assoc. Editor: Zhixiong Guo. Journal of Heat Transfer JUNE 2014, Vol. 136 / 062502-1 Copyright V C 2014 by ASME Downloaded From: http://asmedigitalcollection.asme.org/ on 06/09/2015 Terms of Use: http://asme.org/terms

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D. Roeleveld1

Department of Mechanical &

Industrial Engineering,

Ryerson University,

350 Victoria Street,

Toronto, ON M5B 2K3, Canada

e-mail: [email protected]

D. NaylorDepartment of Mechanical &

Industrial Engineering,

Ryerson University,

350 Victoria Street,

Toronto, ON M5B 2K3, Canada

W. H. LeongDepartment of Mechanical &

Industrial Engineering,

Ryerson University,

350 Victoria Street,

Toronto, ON M5B 2K3, Canada

Free Convection inAsymmetrically HeatedVertical ChannelsWith Opposing BuoyancyForcesLaser interferometry and flow visualization were used to study free convective heat trans-fer inside a vertical channel. Most studies in the literature have investigated buoyancyforces in a single direction. The study presented here investigated opposing buoyancyforces, where one wall is warmer than the ambient and the other wall is cooler than theambient. An experimental model of an isothermally, asymmetrically heated vertical chan-nel was constructed. Interferometry provided temperature field visualization and flow vis-ualization was used to obtain the streamlines. Experiments were carried out over a rangeof aspect ratios between 8.8 and 26.4, using temperature ratios of 0, �0.5, and �0.75.These conditions provide a modified Rayleigh number range of approximately 5 to 1100.In addition, the measured local and average Nusselt number data were compared to nu-merical solutions obtained using ANSYS FLUENT. Air was the fluid of interest. So the Prandtlnumber was fixed at 0.71. Numerical solutions were obtained for modified Rayleigh num-bers ranging from the laminar fully developed flow regime to the turbulent isolatedboundary layer regime. A semi-empirical correlation of the average Nusselt number wasdeveloped based on the experimental data. [DOI: 10.1115/1.4026218]

Keywords: interferometry, flow visualization, numerical modeling, natural convection,vertical channel

1 Introduction

Free convection in a heated vertical channel is a classical heattransfer problem that has been studied extensively in the literature.There are many applications of this problem, such as electronicscooling, simulation of flow in nuclear reactors, and fenestrationsystems (i.e., windows with blinds). Many of these studies haveinvestigated buoyancy driven flow in a single direction inside anasymmetrically heated vertical channel. The current study investi-gates opposing buoyancy driven flow, where the fluid tends toflow in opposite directions inside the vertical channel.

A schematic diagram of the vertical channel geometry is shownin Fig. 1. An open-ended vertical channel is created by two iso-thermal channel walls of height L separated by a channel spacingb. The aspect ratio of the channel is defined as A¼L/b. The coldwall has a temperature TC and the hot wall has a temperature TH.The cold wall temperature is set below the ambient temperatureT1 and the hot wall is set above the ambient temperature. Aung[1] defined a temperature ratio as

RT ¼TC � T1TH � T1

(1)

Most existing studies of free convection inside a heated verticalchannel have investigated temperature ratios in the range of0�RT� 1 (i.e., TC, TH>T1). This study will investigate oppos-ing buoyancy forces, where the temperature ratio is in the range of�1<RT� 0 (i.e., TH>T1 and TC<T1).

There have been numerous studies on free convection in iso-thermally heated vertical channels. Elenbaas [2] was one of thefirst to study heat flow in a symmetrically (i.e., TH¼TC, RT¼ 1),isothermally heated vertical channel. Using two square plates sep-arated by various channel spacings, he was able to obtain experi-mental data for a wide range of modified Rayleigh numbers.Some analytical work determined that the Nusselt numberapproaches two asymptotes at the upper and lower modified Ray-leigh numbers. An overall channel average Nusselt number corre-lation was developed using the analytical work and experimentaldata.

Aung et al. [3] studied the conditions of a uniform heat flux anda uniform wall temperature in an asymmetrically heated verticalchannel. Numerical solutions were obtained over a wide range ofmodified Rayleigh numbers and some experimental work was per-formed to verify the results. They showed that, for uniform walltemperatures, a nearly universal curve can be used to relate theNusselt numbers and the modified Rayleigh numbers for a widerange of temperature ratios. This is the case if the Nusselt numberand Rayleigh number are defined by using the appropriate charac-teristic temperature difference. This characteristic temperaturedifference is defined as

DT ¼ TH þ TC

2� T1 (2)

The modified Rayleigh number based on this temperature differ-ence is

Ra b=Lð Þ ¼gb DT�� ��q2b3

l2Pr

b

L(3)

where g is gravity, b is the fluid thermal expansion coefficient, qis the fluid density, l is the fluid dynamic viscosity, and Pr is the

1Corresponding author.Contributed by the Heat Transfer Division of ASME for publication in the

JOURNAL OF HEAT TRANSFER. Manuscript received September 11, 2013; finalmanuscript received November 28, 2013; published online March 7, 2014. Assoc.Editor: Zhixiong Guo.

Journal of Heat Transfer JUNE 2014, Vol. 136 / 062502-1Copyright VC 2014 by ASME

Downloaded From: http://asmedigitalcollection.asme.org/ on 06/09/2015 Terms of Use: http://asme.org/terms

Prandtl number. Note that the absolute value of the characteristictemperature difference is taken because in the context of this pa-per, the modified Rayleigh number is always positive. The overallchannel average Nusselt number is defined as

Nu ¼ qH þ qCð Þb2kDT

(4)

where qH and qC are the average convective heat fluxes at the hotand cold walls in the vertical channel and k is the fluid thermalconductivity. The hot wall heat flux is always positive and thecold wall heat flux is usually negative in the negative temperatureratio cases. All air properties are evaluated at the film temperature,unless otherwise noted. The film temperature is defined as

Tf ¼TH þ TCð Þ=2þ T1

2(5)

Fully developed flow in an asymmetrically heated verticalchannel was studied analytically by Aung [1]. It was found that atlow modified Rayleigh numbers (Ra(b/L) ! 0), the asymptote ofthe Nusselt number varied depending on the temperature ratio.The average Nusselt number at the fully developed limit is

Nufd ¼4R2

T þ 7RT þ 4

90 1þ RTð Þ2Ra b=Lð Þ (6)

Various channel heating configurations were studied by Bar-Cohen and Rohsenow [4] using analytical expressions for the

asymptotes at low and high modified Rayleigh numbers to de-velop correlations. They developed correlations for both isother-mal and isoflux conditions in symmetrically and asymmetricallyheated vertical channels. The new correlations were developed tofit various experimental and numerical data. A review of all thecorrelations was performed by Raithby and Hollands [5]. It wasdetermined that the best correlation for the overall channel aver-age Nusselt number for isothermally heated channel walls is

Nu ¼ Nufd

� ��1:9þ 0:618 Ra b=Lð Þ½ �1=4

!�1:9�1=�1:90@

24 (7)

where Nufd is determined from Eq. (6). There have been manyother studies on asymmetrically, isothermally heated verticalchannels with positive temperature ratios [6–10]. The existingcorrelations from the literature cannot accurately predict the heattransfer of the negative temperature ratio cases.

There are three studies in the literature that investigate oppos-ing buoyancy forces in free convection of an antisymmetrically(i.e., RT¼�1) heated vertical channel. The antisymmetrical caseis where the temperature difference between the hot wall and theambient is the same as the temperature difference between the am-bient and the cold wall (i.e., TH – T1¼T1 – TC). Habib et al.[11] studied turbulent flow in an antisymmetrically (RT¼�1)heated vertical channel with a channel aspect ratio of 3.125. TheRayleigh number was 2.0� 106, with the hot wall 10 �C aboveambient and the cold wall 10 �C below ambient. (The Rayleighnumber was defined based on the channel height and the tempera-ture difference between the hot wall and the ambient temperature.)Velocity profiles of the flow were determined using a laserDoppler anemometer. The results showed a large vortex flow,with the air flowing up the hot wall and down the cold wall similarto flow inside a tall vertical cavity.

Ayinde et al. [12] also investigated turbulent flow in an anti-symmetrically heated vertical channel. Two temperature differen-ces of TH – TC¼ 15 �C and 30 �C were used between the twochannel walls. Two aspect ratios of 6.25 and 12.5 and twoRayleigh numbers of 1.0� 108 and 2.0� 108 were studied. (TheRayleigh number was defined based on the channel height and thetemperature difference between the hot wall and the ambient.) Aparticle image velocimeter was used to determine velocity profilesand a correlation for dimensionless flow rate inside the channelwas developed. The results indicated that the flows entering at thetop and bottom of the channel were mixed with the recirculatedflow inside the channel before exiting the other side of the chan-nel. The results also showed that the flow pattern inside the chan-nel was similar to a sealed tall enclosure. In these twoexperimental studies, velocity field measurements were made forthe antisymmetrical case, in relatively low aspect ratio channels(A< 13). In contrast, the present work investigates the convectiveheat transfer rates in higher aspect ratio channels (typicallyA> 13) at lower Rayleigh numbers and over a wider range of neg-ative temperature ratios.

The antisymmetrical case (RT¼�1) is a special case that hasbeen recently investigated by Roeleveld et al. [13]. Similar to thecurrent study, flow visualization and laser interferometry wereused to study free convection in an open-ended vertical channel.The heat transfer rates were also determined using laser interfer-ometry and a numerical model was developed to solve over awide range of Rayleigh numbers. The Rayleigh number wasdefined based on the channel spacing and the temperature differ-ence between the hot and cold walls. The special nature of theRT¼�1 case can be illustrated by noting that the modified Ray-leigh and Nusselt numbers will always be zero using the defini-tions in the current paper, Eqs. (3) and (4). The flow- andtemperature-field were found to have similarities to that of a tallsealed enclosure. It was determined that the average convectiveheat transfer of the antisymmetrical case can be approximated

Fig. 1 Schematic of the problem geometry

062502-2 / Vol. 136, JUNE 2014 Transactions of the ASME

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using an existing correlation from the literature for a tall sealedenclosure.

In the current study, an experimental model was constructed tobe used for laser interferometry and flow visualization. Flow visu-alization was used to determine the streamlines and laser interfer-ometry was used to determine the isotherms. Experimental datawere obtained for three different temperature ratios (RT¼ 0, �0.5and �0.75) at four different aspect ratios (A¼ 26.4, 17.6, 13.2,and 8.8). These conditions provide a modified Rayleigh numberrange of 4.7<Ra(b/L)< 1084. The temperature field data werethen analyzed to determine the local and average Nusselt num-bers. General purpose computational fluid dynamics (CFD) soft-ware ANSYS FLUENT [14] was used to develop a numerical model offree convection inside an asymmetrically, isothermally heatedvertical channel. Both laminar and turbulent numerical predictionswere validated against the experimental data. Additional numeri-cal solutions were obtained for RT¼�0.5 and �0.75 over a widerange of modified Rayleigh numbers (0.1�Ra(b/L)� 104).

In the literature, most of the studies on free convection inside aheated vertical channel with buoyancy forces in a single directionhave examined cases where these buoyancy forces are in the op-posite direction of the gravity vector (i.e., TH, TC>T1). Thesestudies can also be applied to cases where the buoyancy forces arein the same direction of the gravity vector (i.e., TH, TC<T1), butthere is an inherent problem when using these correlations fornegative temperatures relative to the ambient. The problem lies incalculating the temperature ratio and the modified Rayleigh num-ber. For example, if the cold wall is �20 �C, the hot wall is�10 �C, and the ambient is 0 �C, then the temperature ratio isRT¼ 2. This temperature ratio is outside the range of 0�RT� 1of most existing studies in the literature. This can be overcome byunderstanding that every case of free convection in an open-endedvertical channel has an equivalent case with the buoyancy forcesin the opposite direction. Figure 2 shows a schematic of twoequivalent cases, one with negative buoyancy forces (RT¼ 2) andthe other with positive buoyancy forces (RT¼ 0.5) relative to thegravity vector. This figure shows that a case with RT¼ 2 is equiv-alent to a case with RT¼ 0.5 assuming the air properties are con-stant. So if the temperature ratio is outside the range of existingstudies (0�RT� 1), it can be modified by

R�T ¼ RTð Þ�1(8)

to convert into an equivalent temperature ratio inside the range ofexisting studies. This modified temperature ratio can be used withthe existing definitions and correlations. Using this same

approach, the entire range of possible cases for free convectioninside a heated vertical channel can be reduced to �1�RT� 1and Eq. (8) can be used if the temperature ratio falls outside thisrange.

Another concern in this paper is that when studying a case thatis outside the �1�RT� 1 range, the characteristic temperaturedifference can become negative. Using the same example fromabove, the characteristic temperature difference is DT¼�15 �C.The traditional definition of modified Rayleigh number based onthe surface to ambient temperature difference gives a negativevalue. In order to circumvent this problem, the absolute value ofthe characteristic temperature difference is, therefore, used in Eq.(3). It should be noted that the characteristic temperature differ-ence can be negative in the Nusselt number definition in Eq. (4),so absolute values are not required in this equation.

2 Experimental Apparatus and Methodology

2.1 Apparatus. A vertical channel was constructed to be usedwith a Mach-Zehnder Interferometer (MZI) and flow visualiza-tion. Two aluminum plates were mounted vertically, with an ad-justable channel spacing in order to study various aspect ratios.The aspect ratio was adjusted in order to study various modifiedRayleigh numbers. The aluminum plates had a width ofW¼ 355 mm and a length of L¼ 264 mm. The plates wereroughly 38 mm thick, with beveled edges filled with polystyreneat the top and bottom of the channel. Two constant temperaturewater baths controlled the temperatures of the two aluminumplates, by running water from the baths through grooves machinedinto the back of each plate. One wall was cooled and the otherwall was heated creating the opposing buoyancy forces betweenthe plates. The cold wall was typically cooled 7.5 �C below theambient room temperature except in the RT¼ 0 case whereTC¼T1. This temperature was set such that it would not fallbelow the dew point of the ambient air and create condensation onthe surface of the cold wall. The hot wall was typically heated 10to 15 �C above the ambient temperature depending on the temper-ature ratio being studied. The temperature differences between thetwo channel walls were between 15 and 22.5 �C. These tempera-ture differences were used to obtain sufficient interference fringesin the output of the interferometer. The thermocouples and ther-mopiles were Type T constructed out of copper and constantanwire with special limits of error. The thermopiles were made with12 junctions, 6 in each surface of interest. One thermopile moni-tored the temperature difference between the cold and hot walls ofthe vertical channel and another monitored the temperature differ-ence between the hot wall and the ambient. Six thermocoupleswere used as a reference temperature for the two thermopiles inthe cold wall. The thermocouples and thermopiles were uniformlyspaced and embedded inside the channel walls to within approxi-mately 1.6 mm (1/16 in.) of the surface. The thermocouples werecalibrated with thermometers traceable to national standards andthe thermopiles were checked against the NIST standard tables.The plates were measured to be isothermal to within 0.2 �C. Inorder to prevent air entrainment, optical windows or acrylic panelswere mounted to the sides of the experimental model. The experi-mental model was placed in a “smoke room” during the experi-ments where no drafts or outside ventilation could disturb the flowpattern inside the vertical channel.

2.2 Flow Visualization. Flow visualization was conducted toobserve the streamlines between the two vertical channel walls. Aplane laser sheet produced by a cylindrical lens introduced fromthe top of the vertical channel was used to illuminate the smoke ina cross section of the model. The smoke was sulfuric acid aerosol,generated using a Dr€ager tube, which was blown through hoses tothe experimental model from outside the smoke room. The smokewas introduced slowly into the top and bottom of the verticalchannel and allowed to settle for a few seconds before an imagewas captured. The flow was observed to be steady for the two

Fig. 2 Schematic of two equivalent vertical channel cases withdifferent temperature ratios; one with negative buoyancy forcesand one with positive buoyancy forces relative to the gravityvector

Journal of Heat Transfer JUNE 2014, Vol. 136 / 062502-3

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cases that were investigated. So a still image was taken with aDSLR camera to show the streamlines inside the channel.

2.3 Laser Interferometry. A 200 mm diameter beam Mach-Zehnder Interferometer was used to obtain temperature fieldmeasurements. A 15 mW helium-neon laser was used in a stand-ard MZI as described in Goldstein [15]. The interferometer hastwo settings: infinite fringe mode and finite fringe mode. The infi-nite fringe setting was used for temperature field visualization,because a line of constant fringe shift is an isotherm in this mode.The finite fringe mode was used for the heat transfer measure-ments. For steady cases, the interferogram was photographed witha high resolution (39 megapixels) still image camera and for theunsteady cases, a high speed digital movie camera was used.

The local Nusselt numbers on each channel wall were deter-mined using the gradient measurement technique as described byPoulad et al. [16]. The temperature gradient at the cold surfacewas calculated using

@T

@x

����x¼0

¼ Rk0T2S

WPG

@e@x

����x¼0

(9)

where R is the gas constant, k0 is the wavelength of the laser lightin a vacuum, Ts is the absolute surface temperature (in this case,Ts¼TC), P is the absolute pressure and G is the Gladstone-Daleconstant. (It should be noted that R¼ 287 J/kg K for air,k0¼ 632.8 nm for a helium-neon laser, and G¼ 0.226� 10�3 m3/kg for a helium-neon laser in air.) The fringe gradient normal tothe measurement surface, @e=@xjx¼0, was extracted from the finitefringe interferograms using a custom MATLAB [17] image proc-essing code developed by Poulad et al. [16]. A scan of pixel inten-sity was taken along a horizontal line at the y-location of interest.A nonlinear regression technique similar to Slepicka and Cha [18]was used to unwrap the phase from the pixel intensity data. Thepixel intensity near the surface was expressed as

IðxÞ ¼ Iavg þ F sin D x� x1ð Þ þ /½ � (10)

where Iavg is the average pixel intensity, F is the amplitude, D isthe amplitude of spatial fringe intensity, / is the phase shift, andx1 is the location of the first pixel in the scan. Equation (10) is fitto the extracted pixel intensity data by adjusting the four constantsiteratively until the sum squared error between the data and I(x)was minimized. The fringe shift gradient at the surface was calcu-lated from

@e@x

����x¼0

¼ D

2p(11)

For the hot surface at x¼ b, the temperature and fringe gradientswere similarly calculated using Eqs. (9) and (11), respectively.Using the temperature gradient, the local heat fluxes on the coldand hot walls of the channel were calculated using

qy;C ¼ �ks@T

@x

����x¼0

and qy;H ¼ ks@T

@x

����x¼b

(12)

where ks is the thermal conductivity of the air at the surface tem-perature. The hot wall heat flux is always positive and the coldwall heat flux is typically negative in this study. The local Nusseltnumber was defined as

Nuy;C ¼qy;Cb

DTkand Nuy;H ¼

qy;Hb

DTk(13)

where k is the fluid thermal conductivity calculated at the filmtemperature (Eq. (5)). The temperature difference was calculated

from the thermopiles and thermocouple readings. The overallchannel average Nusselt number was calculated from

Nu ¼ 1

2L

ðL

0

Nuy;CdyþðL

0

Nuy;Hdy

� �(14)

where the integrals were evaluated using the trapezoidal rule.When studying the unsteady cases, the temperature field was

three-dimensional and time dependent. The output of the MZI istwo-dimensional. So the temperature field was beam-averaged inthe z-direction. A sequence of interferograms was recorded at 30fps for 20 s using a high-speed digital movie camera. The time-averaged heat flux became nearly stationary after 20 s. Each inter-ferogram was analyzed using the same custom MATLAB imageprocessing code developed by Poulad et al. [16] in order to obtainthe instantaneous fringe gradient. The instantaneous heat fluxes ofboth the cold and hot walls were determined and then integratedin order to obtain the time-averaged heat fluxes. The time-averaged heat fluxes were then substituted into Eqs. (13) and (14)in order to determine the local and average Nusselt numbers.

As with any experiment, there is uncertainty in the results. Adetailed error analysis was conducted based on the Kline andMcClintock [19] method. Consider an experimental result Y cal-culated from n independent variables a1, a2,…,an. Each variablehas a total uncertainty of da1, da2,…,dan. If the uncertainty ineach variable were given the same odds, then the uncertainty ofthe result dY at these odds is

dY ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXn

i¼1

@Y

@aidai

� �2

vuut (15)

Each variable has two components of uncertainty: bias error andrandom error. The two types of uncertainty were combined usinga root sum square method with a 95% confidence level. Table 1shows the typical values of the variables and their total uncertain-ties. The uncertainty in the air property data was estimated fromthe property data scatter plots presented by Touloukian et al. [20],Touloukian and Makita [21], and Touloukian et al. [22]. Theuncertainty in the fringe gradient is the largest source of error inthese experiments. This is due to the regression algorithm used inthe custom MATLAB image processing code and the opticalimperfections in the MZI. The estimated uncertainty in the modi-fied Rayleigh number is 65%. The error in the local Nusselt num-bers is estimated at 612%. The integration process averages outsome of the random error in the local Nusselt number data. Forthis reason, the overall channel average Nusselt number is moreaccurate than the individual local Nusselt number data and theestimated uncertainty is 69%. Further information on the uncer-tainty analysis is given in Roeleveld [23].

Table 1 Measured variables and their estimated totaluncertainties

Parameter Symbol Measured value Total uncertainty

Channel height L 264.2 mm 60.3 mmChannel width W 355.1 mm 63.2 mmChannel spacing b 10 mm – 30 mm 60.1 mmSpecific heat cp 60.8%Dynamic viscosity l 60.5%Thermal conductivity k, ks 61%Pressure P 747.6 mm Hg 60.2 mm HgSurface temperature TS 288.0 K – 310.0 K 60.35 KTemperature difference DT 1.25 K – 7.5 K 60.10 K

Fringe gradient

@e@x

����x¼0;x¼b 200 m�1 – 1500 m�1 610.2%

062502-4 / Vol. 136, JUNE 2014 Transactions of the ASME

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3 Numerical Solution

A numerical model was developed to determine the convectiveheat transfer. The flow was assumed to be two-dimensional andincompressible. At first a steady laminar solution was sought. Butconvergence could not be obtained over the entire modified Ray-leigh number range of interest. For the higher modified Rayleighnumbers, a standard k-e turbulence model was used with enhancedwall functions. Walsh and Leong [24] have shown that theenhanced wall functions give better results for free convection ona vertical wall. The standard governing equations used are givenin Versteeg and Malalasekera [25]. A control-volume formulationwith a second-order upwind scheme for evaluation of the convec-tive terms, the PRESTO option [26], and the SIMPLEC algorithm[27] were used in the numerical model. The fluid properties wereassumed to be constant and viscous dissipation was neglected inthe energy equation. The Boussinesq approximation was used toaccount for variation in fluid density due to temperature variation.For this study, the Prandtl number was fixed at 0.71, since thefluid of interest was air.

The domain of the numerical model and the boundary condi-tions are shown in Fig. 3. No slip and impermeability conditionswere applied to the walls. An adiabatic boundary condition wasused on the horizontal surfaces near the inlet and outlet of thechannel. This is the typical treatment from the literature (Nayloret al. [8]). Above and below the channel, two “pre-entry” plenumswere added to set the outflow conditions with higher accuracybecause the air temperature will approach ambient temperature faraway from the channel. A pressure outlet condition was applied tothe semicircular inlet and outlet boundaries, where the fluid veloc-ity was set normal to the boundary and the pressure defect was setto zero. A pressure outlet condition was needed in order to allowair to enter and exit the domain at either end of the channel due tothe opposing buoyancy forces.

A grid sensitivity study was conducted to ensure that the nu-merical solution was grid independent. Both grid density and far-field boundary conditions were tested. Using the same size do-main, three different grid densities were tested. A grid of approxi-mately 50,000 nodes was determined to be sufficient. The averageNusselt numbers are estimated to be grid independent to betterthan 0.7% using the Richardson extrapolation [28]. By adjustingthe radius B shown in Fig. 3, the sensitivity of the results to thefar field boundary location was tested. The radius B¼ 5�b was

determined to be sufficient for the current simulations. Roeleveld[23] has further details on the grid sensitivity study.

Aung [1] solved fully developed flow solutions analytically forlaminar free convection inside an asymmetrically heated verticalchannel. This analytical solution presents equations for both thedimensionless velocity and temperature profiles of the fully devel-oped flow. Figure 4 shows that the dimensionless velocity profilesof the analytical solution and the numerical model at Ra(b/L)¼ 0.5 and y/L¼ 0.5 compare favorably. Using this analyticalsolution and setting dV/dx¼ 0 at x¼ 0, it was determined thatreverse flow starts at approximately RT¼�0.5 in the fully devel-oped flow regime. At such low modified Rayleigh numbers, theheat transfers by pure conduction, so the temperature profile is alinear function of distance between the two channel walls. The an-alytical work was developed for asymmetrically heated verticalchannels with positive temperature ratios (0�RT� 1), but thisgraph shows that this analytical solution also applies for negativetemperature ratios (�1<RT< 0). Further numerical validationagainst the current experimental data will be presented in Sec. 4.

4 Experimental Results

4.1 Flow Visualization. Flow visualization was conductedfor two different temperature ratios of RT¼�0.5 and RT¼�0.75at an aspect ratio of A¼ 17.6. Figure 5 shows the streamlines forRT¼�0.5 at A¼ 17.6 with Ra(b/L)¼ 67.5. A sketch is includedin Fig. 5(b) to show the observed streamlines. The flow wasobserved to be steady. In this case, on the hot wall side of thechannel, air flowed in the bottom of the channel and out the top ofthe channel. On the cold wall side of the channel, air entered fromthe top and flowed to near the bottom of the channel. A separationpoint was located at approximately y/L¼ 0.07 where the air flow-ing down the cold wall reversed direction with the air flowing upthe hot wall inside the channel. Figure 5(a) shows the separationpoint with a line of smoke, which separates the air flowing up thehot wall and the air flowing up the channel from the cold wall af-ter reversing direction. There was also some smoke separating theopposing flow from the cold wall in the upper half of the channelwhere there was slow air flow. The flow pattern in this case wassimilar to what was observed by Sparrow et al. [29] when study-ing a vertical channel with the hot wall isothermally heated andthe cold wall unheated (RT¼ 0). They observed that in a channelwith an aspect ratio of 15.2 at Ra(b/L)¼ 5270, some ambient airflowed down about 25% of the cold wall and then re-circulatedwith the air flowing up the hot wall.

Figure 5(c) shows the streamlines from the numerical model. Itcan be seen that the experimental and numerically predictedstreamline patterns agree qualitatively, with the separation pointlocated in a similar location. An area of interest is at the top of thechannel, where the air is exiting the channel. The experimental

Fig. 3 Boundary conditions and computational domain

Fig. 4 Comparison of dimensionless velocity profiles at vari-ous temperature ratios for fully developed flow at y/L 5 0.5 andRa(b/L) 5 0.5 obtained from the numerical model and the analyt-ical solution by Aung [1]

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visualization using smoke shows that the plume bends abruptlyaway from the centerline of the channel towards the hot wall sideas it exits the channel. The numerical results show the plume exit-ing straight upwards, which differs from the flow visualization.

Figure 6 shows the case of RT¼�0.75 at A¼ 17.6, with Ra(b/L)¼ 23.1. Figure 6(b) shows a sketch of the observed flow pat-tern. Again the flow was observed to be steady. Similar to the pre-vious case, air flowed up the hot wall side of the channel anddown the cold wall side of the channel. But in this case, there wasa closed recirculation cell in the center of the channel. Near thecold wall, the air flowed out the bottom of the channel and therewas no separation point. It should also be noted that the cell wasoff center from the vertical centerline of the channel due to theupwards buoyancy force from the hot wall being stronger than thedownwards buoyancy force from the cold wall. As the tempera-ture ratio was decreased, less air flowed out the top of the channel,but more air flowed out the bottom of the channel due to theincreasing negative buoyancy force. The numerical streamlinesare shown in Fig. 6(c). Again, the experimental and numericallypredicted streamline patterns agree qualitatively. Also, the plumebends towards the hot wall as it exits the channel, unlike the nu-merical prediction.

4.2 Laser Interferometry. A MZI was used to obtain inter-ferograms for three different temperature ratios (RT¼ 0, �0.5,

and �0.75) at four different aspect ratios (A¼ 26.4, 17.6, 13.2,and 8.8). This provided data over a modified Rayleigh numberrange of 4.7<Ra(b/L)< 1084. Figure 7 shows three infinitefringe interferograms at a channel aspect ratio of A¼ 17.6 at thethree different RT values. There are six horizontal pins that arevisible on each channel wall. These small pins were used for laseralignment and image locating purposes and have no significantimpact on the convection. All of these three cases were observedto be steady. At RT¼ 0, a developing thermal boundary layer canbe seen on the hot wall over the full length of the channel. As RT

becomes negative, it can be seen that the boundary layer nature ofthe temperature field quickly diminishes and the heat transfer inthe center region of the channel becomes increasingly conductiondominated. This is evident from the more evenly spaced isothermsin the interferograms. Similar to what was observed with flow

Fig. 5 (a) Flow visualization, (b) sketch of the observed flowpatterns, and (c) numerical solution streamlines for RT 5 20.5,A 5 17.6 and Ra(b/L) 5 67.5

Fig. 6 (a) Flow visualization, (b) sketch of the observed flowpatterns, and (c) numerical solution streamlines for RT 5 20.75,A 5 17.6 and Ra(b/L) 5 23.1

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visualization, Fig. 7 shows that at the top of the channel, theplume of warm air bends abruptly away from the centerline of thechannel.

Figure 8 shows two interferograms at RT¼�0.5 at two differ-ent aspect ratios. These two interferograms along with Fig. 7(b)show the effect of modified Rayleigh number on the temperaturefield. In Fig. 8(a), the modified Rayleigh number is low (near thefully developed limit) and the temperature field appears to bealmost pure conduction over much of the channel length. In Fig.8(c), the modified Rayleigh number is much higher and thermalboundary layers can be seen on both walls. It can be seen that thefringes are wavy in the center region near the top of the channel.In fact, the temperature field was observed to be unsteady at thislocation for this case. It is interesting to note that compared to achannel with unidirectional buoyancy, the flow in a channel withstrongly opposing buoyancy forces becomes unsteady at a muchlower modified Rayleigh number. The experimental results ofAung et al. [3] and Sparrow et al. [29] show that a vertical channelwith positive temperature ratios (RT> 0) remains steady and lami-nar up to Ra(b/L) 104, whereas the current flows are unstable atRa(b/L)¼ 214 for RT¼�0.5 and Ra(b/L)¼ 76.6 for RT¼�0.75.

Figures 8(b) and 8(d) provide the ability to compare betweenthe numerically predicted isotherms and the interferograms forthese two cases. Overall, there is good agreement between thesteady laminar numerical isotherms and the interferograms. How-ever, some disagreement can be seen at the top of the channel.Once again, the plume of air leaving the channel bends towardsthe hot wall side in the interferograms, which is not predicted

numerically. The interferograms in Figs. 7 and 8 show that thehorizontal surfaces at the entrance and exit region of the channelare not perfectly adiabatic, as they were assumed to be in the nu-merical model. A test was conducted to check if this difference inthe temperature boundary condition on the horizontal surfaceswas the cause of the poorly predicted outlet flows. Using a userdefined function in ANSYS FLUENT, numerical solutions wereobtained with a variable temperature boundary condition on thesesurfaces for comparison. This variable temperature boundary con-dition was based on the measured surface temperature distributionfrom the interferograms. When comparing the numerical resultswith adiabatic and variable temperature boundary conditions, theplume still exited the channel straight vertically and the differencebetween the heat transfer rates was about 1%. Based on this test-ing, it appears that other issues associated with the imperfect na-ture of the outflow conditions on the semicircular boundaries areproducing the discrepancy.

Figure 9 shows a comparison between the experimentally meas-ured and numerically predicted local Nusselt number distributionsfor RT¼�0.5 with A¼ 26.4 and Ra(b/L)¼ 12.3. These resultscorrespond to the interferogram in Fig. 8(a). For comparison, theNusselt numbers corresponding to one-dimensional pure conduc-tion between the walls were added to this graph. Figure 9 showsthat the local heat transfer rate corresponds to essentially pureconduction for the top three quarters of the channel. It can also beseen that the laminar CFD solution shows good agreement withthe experimental data except for on the hot wall near the bottomof the channel. For the bottom 25% of the hot wall the CFD pre-diction was 15% higher than the experimental data. This slightdifference is due to the nonadiabatic horizontal walls in the

Fig. 8 Infinite fringe interferograms and numerically predictedisotherms at RT 5 20.5 for different aspect ratios

Fig. 7 Infinite fringe interferograms at A 5 17.6 for differenttemperature ratios

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experimental model. Air was being preheated as it flowed into thechannel causing the measured heat flux to be slightly lower on thehot wall near the entrance.

The flow was observed to be unsteady for RT¼�0.5 at A¼ 8.8and Ra(b/L)¼ 1085, so an unsteady interferometric analysis wasperformed with a high speed digital movie camera. Figure 10shows typical instantaneous heat fluxes for this case. These meas-urements were made at y/L¼ 0.5 on both the hot and cold wallsover a 20 s interval at a frame rate of 30 fps. The RMS fluctuationis 10% or less for these cases and the peak to peak fluctuation isbetween 20 and 30% about the mean. This graph also shows therunning time-averaged heat flux becomes nearly stationary after20 s. The local Nusselt numbers for RT¼�0.5 at A¼ 8.8 andRa(b/L)¼ 1084 are shown in Fig. 11. Two CFD solutions, oneassuming laminar steady conditions and the other using a turbu-lence model, were added to the graph for comparison. It can beseen that the turbulent CFD solution shows better agreement thanthe steady laminar solution with the experimental data near thetop of the channel. Again, there was preheating of the air at thebottom of the channel. An area of interest is near the top of thechannel, where both the hot and cold walls show some discrep-ancy with the numerical solutions. As previously discussed, this ismost likely due to the plume bending to the hot wall side of thechannel when it exits the top of the channel. This draws airtoward the hot wall as it leaves the top of the channel, increasingthe heat transfer, while drawing air away from the cold wall,decreasing its heat transfer. Some of the error of the cold wallcould also be from some precooling of the air as it enters the topof the channel.

Table 2 shows a comparison of the overall channel averageNusselt numbers of the RT¼ 0 case for two different aspect ratios.The numerically predicted overall channel average Nusselt num-bers are within 4% of the experimental data. Since the RT¼ 0case has been previously studied in the literature, the correlationof Raithby and Hollands [5], Eq. (7), was also included in Table2. The correlation is within about 3% of the experimental data.The overall channel average Nusselt numbers for both the experi-mental data and numerical solution are compared in Table 3 forRT¼�0.5 and RT¼�0.75. The numerical predictions presentedin this table were all solved with the steady laminar model. TheNusselt numbers agree quite well for A¼ 13.2, even though thesecases were observed to be unsteady experimentally. Overall, thenumerically predicted overall channel average Nusselt numbersare within 7% of the experimental data. Table 4 shows a compari-son between the results of the experimental data, the steady lami-nar numerical model and the steady turbulence model with theenhanced wall functions at the higher modified Rayleigh numbers.It should be noted that these three cases were all observed to beunsteady experimentally. This table shows that at A¼ 13.2, forRT¼�0.5, the laminar and turbulent predictions give similarresults. But for RT¼�0.75, the turbulent solution shows betteragreement with the experimental data. At A¼ 8.8, Table 4 showsthat the turbulence model has better agreement with the experi-mental data as the modified Rayleigh number was increased.

5 Numerical Results

The numerical model was used to conduct a parametric studyfor the following range of parameters:

A ¼ 50; RT ¼ � 0:75; � 0:5; 0:1 < Ra b=Lð Þ < 104 (16)

The overall channel average Nusselt numbers were determinedusing the numerical solutions. Both laminar and turbulent CFD

Fig. 10 Instantaneous and running time-averaged local heatfluxes for RT 5 20.5 with A 5 8.8 at y/L 5 0.5

Table 2 Comparison of the overall channel average Nusseltnumbers from the experiment, numerical predictions and theRaithby and Hollands [5] correlation for RT 5 0

Aspectratio

ModifiedRayleighnumber

Experimentaldata

Numericalsolution

Raithby andHollands [5]

A Ra(b/L) (Nu)EXP (Nu)LAM Difference (Nu)COR Difference26.4 24.8 0.82 0.79 �3.7% 0.85 3.4%17.6 128 1.93 1.99 3.1% 1.93 0%

Fig. 9 Graph of local Nusselt number versus nondimensionalvertical distance for RT 5 20.5, A 5 26.4, and Ra(b/L) 5 12.3

Fig. 11 Graph of local Nusselt number versus nondimensionalvertical distance for RT 5 20.5, A 5 8.8, and Ra(b/L) 5 1084

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models were used. Figures 12 and 13 show numerical solutionsfor RT¼�0.5 and RT¼�0.75, respectively, at an aspect ratio of50. Figure 12 shows that laminar results were obtained over therange 0.1<Ra(b/L)< 100 and turbulent results over the range50<Ra(b/L)< 104. Figure 13 shows laminar results over0.1<Ra(b/L)< 20 and turbulent results over 10<Ra(b/L)< 104.The air flowing down the cold wall mixes with the air flowing upthe hot wall in the center region of the channel. This shear layercaused by the opposing buoyancy forces creates instability insidethe vertical channel which requires the use of a turbulence modelto obtain a solution at higher modified Rayleigh numbers. Based

on the inability to obtain convergence of a steady laminar model,it seems that instability inside the channel occurs at a much lowermodified Rayleigh number than when studying unidirectionalbuoyancy forces. This is compared to other studies in the litera-ture, where the flow remains steady and laminar to Ra 104 forpositive temperature ratios [3, 29].

Four experimental data points are shown in Fig. 12 and threeexperimental data points are shown in Fig. 13 for comparison.Both the laminar and turbulent solutions show good agreementwith the experimental data points. The average Nusselt numberfor fully developed flow that was developed by Aung [1] given inEq. (6) was added to these graphs for comparison. This analyticalasymptote was developed for positive RT values. But as can beseen in Figs. 12 and 13, the numerical solutions for negative RT

values also follow this asymptote. This asymptote will be used asa starting point for developing a correlation for the average Nus-selt number data.

6 Semi-Empirical Correlation

A semi-empirical correlation was developed using the experi-mental Nusselt number data and applying the correlation methodof Churchill and Usagi [30]. A new correlation was developedbecause the existing correlations in the literature were unable toaccurately predict the heat transfer of the negative temperature ra-tio cases. It can be seen in Figs. 12 and 13 that the Nusselt numberapproaches asymptotes at low and high modified Rayleigh num-bers. As discussed Sec. 5, at low modified Rayleigh numbers thedata follows the fully developed flow Nusselt number asymptotethat was developed by Aung [1]. This asymptote is given in Eq.(6). At high modified Rayleigh numbers, the Nusselt number fol-lows the vertical isothermal flat plate asymptote. An expressionfor the vertical isothermal flat plate Nusselt number asymptotewas determined to be

NuIP ¼ C2

1þ RT

� �nþ1

1þ RTj jnRTð Þ Raðb=LÞ½ �n (17)

This expression was obtained by combining the standard expres-sion for the average Nusselt number for free convection from twovertical isothermal plates into an overall Nusselt number for thechannel (see Appendix). These two asymptotes are then used withthe correlation method of Churchill and Usagi [30] as follows:

Nu ¼ Nufd

� �mþ NuIPð Þm 1

m (18)

Table 4 Comparison of the laminar and turbulent numerical models with the experimental data at aspect ratios of 13.2 and 8.8 forRT 520.5 and 20.75

Temperature ratio Aspect ratio Modified Rayleigh number Experimental data Laminar CFD solution Turbulent CFD solution with EWF

RT A Ra(b/L) (Nu)EXP (Nu)LAM Difference (Nu)TUR Difference�0.75 13.2 76.6 3.62 3.85 6.4% 3.66 1.1%�0.5 13.2 214 3.51 3.53 0.6% 3.48 �0.9%�0.5 8.8 1084 5.42 4.31 �20.5% 5.69 5.0%

Table 3 Comparison of the overall channel average Nusselt numbers from the experiment and numerical predictions for RT 5 20.5and 20.75

RT¼�0.5 RT¼�0.75

Aspect Ratio Modified Rayleigh number Experimental data Numerical solution Modified Rayleigh number Experimental data Numerical solution

A Ra(b/L) (Nu)EXP (Nu)LAM Difference Ra(b/L) (Nu)EXP (Nu)LAM Difference26.4 12.3 0.67 0.70 3.6% 4.7 0.68 0.66 �3.2%17.6 66.6 2.19 2.27 3.7% 22.9 2.17 2.27 4.6%13.2 214 3.51 3.53 0.6% 76.6 3.62 3.85 6.4%

Fig. 12 Overall channel average Nusselt number variation withmodified Rayleigh number for RT 5 20.5

Fig. 13 Overall channel average Nusselt number variation withmodified Rayleigh number for RT 5 20.75

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It was determined that C¼ 0.311, n¼ 0.249, and m¼�1.51.These constants have been adjusted to minimize the sum-squaredpercent error between the experimental data and the semi-empirical correlation. Figure 14 shows the correlation and the ex-perimental data plotted on the same graph. Since there was no ex-perimental data available at high modified Rayleigh numbers,some numerical predictions were used as can be seen in Fig. 14.This was necessary in order to adjust the constants C and n prop-erly for the upper asymptote. The maximum error between thedata and the correlation is 610% and the standard deviation is64%. It should be noted that this correlation has been developedto be applicable for temperature ratios in the range �1<RT� 1.For positive temperature ratios, it is comparable to the correlationof Raithby and Hollands [5]. Note that this correlation does notapply for the antisymmetrical case RT¼�1. This special case hasbeen recently studied by Roeleveld et al. [13].

7 Conclusions

Free convective heat transfer rates in an asymmetrically, isother-mally heated vertical channel with opposing buoyancy forces weredetermined using laser interferometry. Some flow visualization wasalso conducted to determine the streamlines inside the channel. Asthe temperature ratio was decreased, the opposing buoyancy forcescaused the flow to become unstable at lower modified Rayleighnumbers. Both steady and turbulent numerical models were devel-oped to obtain solutions for a wide range of modified Rayleighnumbers at various temperature ratios. The opposing buoyancyforces produced instability inside the channel and in general, a k-eturbulence model with enhanced wall functions showed goodagreement for this type of flow at higher modified Rayleigh num-bers. A correlation for the overall channel average Nusselt numberhas been obtained based on the asymptotic behavior at high andlow values of modified Rayleigh number.

Acknowledgment

This work was funded in part by the Canadian Solar BuildingsResearch Network under the Strategic Network Grants Programof the Natural Sciences and Engineering Research Councilof Canada.

Nomenclature

A ¼ aspect ratioB ¼ channel inlet and outlet domain size, mb ¼ channel spacing, mD ¼ amplitude of spatial fringe intensityF ¼ rate of change of phase shift, m�1

g ¼ gravity, m/s2

G ¼ Gladstone Dale constant, m3/kgI ¼ pixel intensity

Iavg ¼ average pixel intensityk ¼ fluid thermal conductivity at film temperature, W/m�K

ks ¼ fluid thermal conductivity at surface temperature, W/m�K

L ¼ channel height, mNu ¼ overall channel average Nusselt number

Nufd ¼ fully developed flow Nusselt numberNuIP ¼ vertical isothermal flat plate Nusselt numberNuy ¼ local Nusselt number

P ¼ pressure, PaPr ¼ Prandtl numberq ¼ convective heat flux, W/m2

qy ¼ local convective heat flux, W/m2

R ¼ gas constant, J/kg�KRa(b/L) ¼ modified Rayleigh number

RT ¼ temperature ratioT ¼ temperature, K

DT ¼ characteristic temperature difference, KW ¼ channel width, m

u, v ¼ fluid velocity in x, y-direction, m/sV ¼ dimensionless y-velocity, ðb2vq Pr=LlRaðb=LÞÞ

x, y ¼ Cartesian coordinate system, mx1 ¼ location of first pixel, m

Greek Symbols

b ¼ fluid thermal expansion coefficient, K�1

e ¼ fringe shiftk0 ¼ laser light wavelength in a vacuum, ml ¼ fluid dynamic viscosity, N�s/m2

q ¼ fluid density, kg/m3

/ ¼ phase shift

Subscripts

C ¼ cold wallH ¼ hot walls ¼ surface1¼ ambient

Appendix

An isolated flat plate asymptote was developed for use in thesemi-empirical correlation at high Rayleigh numbers. If the channelis comprised of two isolated flat plates, the heat flux of each plate is

qH ¼ hH TH � T1ð Þ and qC ¼ hC TC � T1ð Þ (A1)

where the Nusselt numbers are of the form

NuH ¼hHL

k¼ CRan

L; TH�T1j j and NuC ¼hCL

k¼ CRan

L; TC�T1j j

(A2)

Note that the channel height L is the characteristic length in Eq.(A2). Since TH�T1 is always positive, the absolute value can bedropped. This is substituted into Eq. (4):

Nu ¼ hH TH � T1ð Þ þ hC TC � T1ð Þð Þb2kDT

¼ 1

2CRan

L;TH�T1

TH � T1

DT

� �þ CRan

L; TC�T1j jTC � T1

DT

� �� �b

L

(A3)

The Rayleigh numbers need to be converted to the common char-acteristic temperature difference, DT:

Fig. 14 Comparison of the semi-empirical correlation forRT 5 20.5 and RT 5 20.75 with the experimental and numericaldata

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RanL;TH�T1

¼ RanL;DT

TH � T1

DT

� �n

and RanL; TC�T1j j

¼ RanL;DT

TC � T1j jDT

� �n

(A4)

This Rayleigh number can then be converted to the modified Ray-leigh number based on the characteristic length, b:

RanL;DT¼ Ran

b;DT

b

L

� �n L

b

� �4n

(A5)

Substituting these into Eq. (A3)

Nu ¼ C

��Rab;DT

b

L

�n�TH � T1

DT

�nþ1

þ�

Rab;DT

b

L

�n

��jTC � T1j

DT

�n�TC � T1

DT

���L

b

�4n�1

(A6)

where

TH�T1

DT

� �nþ1

¼ 2 TH�T1ð ÞTH�T1ð Þþ TC�T1ð Þ

� �nþ1

¼ 2

1þRT

� �nþ1

and

TC�T1j jDT

� �n TC�T1ð ÞDT

� �¼ 2 TC�T1j j

TH�T1ð Þþ TC�T1ð Þ

� �n

� 2 TC�T1ð ÞTH�T1ð Þþ TC�T1ð Þ

� �¼ 2

1þRT

� �nþ1

RTj jnRT (A7)

Substituting these results back into Eq. (A6) and rearranging

NuIP ¼ C Að Þ4n�1 2

1þ RT

� �nþ1

1þ RTj jnRTð Þ Raðb=LÞ½ �n (A8)

It was determined that the effect of aspect ratio on the upper as-ymptote is small (since n 0.25, (A)4n�1 1), so the term (A)4n�1

can be neglected. Therefore the upper asymptote becomes

NuIP ¼ C2

1þ RT

� �nþ1

1þ RTj jnRTð Þ Raðb=LÞ½ �n (17)

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