CHAPTER 5

SOLUTION FOR CASE 2

5.1 Introduction

This chapter describes the numerical as well as experimental results of natural con­

vection with surface radiation from planar heat generating element mounted freely in

a vertical channel formed between two parallel plates. To study the effect of surface

radiation, the surface emissivities of the heat source and the side adiabatic plates are

varied. As per the observation in chapter 4, it is proposed to perform three dimensional

numerical simulations using commercial software FLUENT 6.3.

5.2 Experiments conducted

Experimental methodology, procedure and data analysis for the experiments conducted

are described in chapter 3. Experiments are conducted by varying the surface emis­

sivities of the heat source and the side adiabatic plates by coating these surfaces with

different paints. The emissivity of the paint coated surfaces are estimated based on

the procedure described in section 3.4. The spacing between the adiabatic side plates

and the stream-wise location of the heat source inside the channel has been varied to

study its effect on the heat transfer characteristics of the heat source. Thermocouples

are placed at different locations on the adiabatic side plate and the heat source so as to

measure their surface temperature. All the thermocouples are connected to a PC based

data acquisition system. The power input to the heat source is given by regulated DC

power supply system.

5.3 Discussion of Results from Experiments

The important variable geometry related parameters namely, channel aspect ratio has

been varied as 6.25, 5, 4.17, and 3.5, while the stream-wise location of the heat gen­

erating element studied are the top, middle, and the bottom locations of the channel.

The surface emissivities of both the heat source (ch)and the side plate (C8) have been

varied in the range (0.05-0.85). The heat input is varied from 1-40W, as a result volu­

metric heat generation (q*) and the modified Rayleigh number (Ra*) were changed in

the range 3.18-12.59 and 3.2 x 105- 1.6 X 107 respectively.

5.3.1 Effect ofstream-wise locations ofheat generating elementand

channel aspect ratio on heat transfer

The measured surface temperature of the heat generating element for three different

stream-wise locations of the heat generating element such as the bottom, middle and

top for a channel aspect ratio, (AR=6.25) shows that for a given modified Rayleigh

number, and for fixed value of emissivities of the heat source and channel walls, the

temperature of the heat generating element is independent of its stream-wise location.

It must be noted that this aspect ratio corresponds to the minimum channel spacing

possible with the experimental setup. The heat generating element, placed inside the

channel can be regarded as a heated short plate kept in an infinite fluid medium, hence

the physical situation remains identical for different stream-wise locations of the heat

generating element. Consequently, one would expect identical temperature distribution

for all stream-wise locations as observed experimentally. However, the effect of stream­

wise location may be prominent for narrow channel spacings. Table 5.1 shows the tem­

perature of the heat generating element for four different channel aspect ratios. The

results indicate that the temperature slightly increases with channel aspect ratio, but the

difference is not discernible. The maximum deviation is only 1.8f{ for Ra* = 8.1 x 106 ,

which suggest that the influence of channel aspect ratio on the temperature of the heat

generating element is negligible when compared to the average surface temperature of

65

Table 5.1: Measured surface temperature of the heat generating element for differentchannel aspect ratio (Heat source located at the middle of the channel, th =

0.85, t s = 0.05)

Modified Rayleighnumber (Ra*)

3.3 X 105

8.1 X 106

Surface temperature (K) of the heat sourcefor different channel aspect ratio (AR)

AR=6.25 AR=5 AR=4.17 AR=3.57310.9 311.1 311.4 310.7366.7 366.6 366.3 365.1

the heat generating element. This insignificant effect of aspect ratio on temperature is

attributed to the same reason discussed previously for the effect of stream-wise loca­

tion. However, it is worth mentioning that the effect of aspect ratio on heat transfer

characteristics had been reported in previous studies conducted on geometries like par­

allel plate with protruding heat source [9], parallel plates [22], enclosures [24] and side

vented cavities, [30] and [31]. From the preceding discussion, it can be concluded that

for the range of geometrical parameters considered, the effect of channel aspect ratio

and stream-wise location of the heat generating element on heat transfer performance

is negligible. Therefore, further experimental investigations have been performed by

placing the heat generating element at the geometrical center of the channel and for a

channel aspect ratio of AR = 6.25.

5.3.2 Heat transfer rates

In a combined mode of energy transport involVing surface radiation and natural convec­

tion, the parameters such as emissivity and Rayleigh number playa vital role. Firstly,

the influence of surface emissivity on the heat transfer rate were examined. Figure

5.1 illustrates the influence of surface emissivity of the heat generating element on the

relative contribution of natural convection and surface radiation heat transfer to the to-

tal heat transfer rate. The curves are drawn for the cases with the lowest emissivity

of channel walls, t s = 0.05. As seen from this Figure. the convective heat transfer

66

.,................ ,....0" ~ 7....•.. Ra =1.14 >< 100.25

1.00 r-~~;o;-.-----------.-.

0.75 ~

0.50

•......0.00 !--.....:......I------'----'---'---'-"""'t

........................... ' .•.... Ra' =8.1 x 106

•.........0.00 I-_......JL.....I-~----'__--'-__'-_-'-......j

1.00 I-------,.---~-----

0.75 ~

ri 0.50

a0.25

1.00 I--=--~----0--0----%-Q-,--..-..-.-..-.-Q-,----l

~~ ~ ~---~

1.0

Ra' =6.58 x 10'

0.2 0.4 0.6 0.8Emissivity of heat source, (~)

...................,.',.',.'.. ',.'.. '.,'",•....0.00 L....:.......--l.._~--'-~_.L..-~--J.__--J

0.0

0.50

0.25

Figure 5,1: Variation of convective and radiative heat transfer with emissivity of heatsource for different modified Rayleigh number (Es = 0.05)

67

0.8o ell = 0.05

.-. 0.7 ... e h = 0.55

g: .6h =0.75

a • E h = 0.85-- 0.6...

'* •Ie 0.5... ••-m • •• • • • • • •• •J: 0.4 •c:: .A •••0 ...... • • • • • • • •

16 ......... ... ... ... .A ... ... ...=c 0.3~1llc::

0.20:0::0~LL

0.1

poooo 0 o 0 0 0 0 0 0

0.03.2E+005 3.3E+OO6 6.4E+OO6 9.6E+006 1.7E+OO7

Modified Rayleigh number (Ra*j

Figure 5.2: Variation of fractional radiation heat transfer with modified Rayleigh num­ber (Es = 0.05)

decreases with the increase of surface emissivity of heat source, but at the same time

radiative heat transport from the heat source increases. This trend is expected, as the

largest temperature gradient between the heat source and the fluid occurs at lowest sur­

face emissivity for which the natural convection is the dominant mode of heat transfer.

But, as surface emissivity increases the temperature of heat generating element drops,

thereby reducing the convection transport. This is because, the strength of convection

current is determined by the buoyancy force which depends on the fluid temperature

differential and isobaric volume expansivity. To provide further insight into the rela­

tive strength of surface radiation and natural convection in the total heat exchange, the

fractional radiative heat transfer as a function of modified Rayleigh number is plotted

in Fig. 5.2. Evidently, for high surface emissivities of the heat generating element, the

contribution of surface radiation is found to be maximum at the lowest value of modi­

fied Rayleigh number encountered in the respective cases. Stated differently, the surface

radiation is seen to playa significant role as compared to natural convection at very low

68

thermal loads. It can be seen from Fig. 5.2 that the maximum radiation accounts for

about 54% of the total heat removed for the highest emissivity considered in this study.

At low values of modified Rayleigh number, the buoyant force becomes weaker and

the resulting slow fluid circulation reduces the natural convection heat transfer coeffi­

cient. This also brings out the fact that for high surface emissivities, radiation effects

are predominant even for a moderate temperature differential, typically encountered

during the operation of electronic components. Therefore, surface radiation effects has

to be taken into account for the accurate estimation of heat transfer rates even at low

thermal loads. This observation is also in accordance with the results reported in [22].

Figure 5.2 also reveals that the radiative heat transfer increases with increasing surface

emissivity for a fixed value of modified Rayleigh number. However, for a fixed surface

emissivity of the heat generating element the relative radiation contribution decreases

with increasing modified Rayleigh number, reaches a minimum for a Rayleigh number

of approximately Ra* = 2.5 x 106 , and thereafter both natural convection and surface

radiation are in competing mode, contributing almost equally to the total heat transfer.

It may be noted that for a heat generating element with low emissivity (Elt = 0.05),

natural convection is the dominant mode of heat transfer contributing to about 96% of

the total heat transferred. Figure 5.3 shows the influence of surface emissivity of heat

source and side plate on convective and radiative heat transfer. From Fig. 5.2 and Fig.

5.3, it is clear that increase of surface emissivity of the side plates results in greater

interaction of surface radiation.

5.3.3 Heat transfer correlations

The experimental design consists of a total of ninety eight experiments corresponding

to different values of heat input, surface emissivity of the heat generating element and

adiabatic side plates for a channel of AR=6.25. Since the height of the heat generating

element is small, no appreciable variation in the local temperature of the heat source

along the height (also the flow direction) is observed. The average convective Nusselt

number for the heat transfer process is evaluated using Eq. (3.12) and the total Nusselt

69

0.980t---------------,

0.736 ................. : .0.492 0-- A

0.248 eh=O.85,e.=0.85,relatlveNu,---0- eh = 0.85, e. = 0.85. relative Nu c

.......... eh = 0,05. e. = 0.05, relative Nu,~ eh = 0,05. e. = 0.05. relative Nu c

3.2E+006 6.1 E+006 9.0E+006Modified Rayleigh number (Ra~)

0.248 f­

0.0051-k=A===""~==Cio:::==~==~-l

3.3E+005

c 0.005o~ 0.9801-----------------1.c~ 0.736 .o~0.492 0-::::l v---o----<>----o------<>------o6°,248 ........ "h = 0,75, e. = 0.75. relative Nu,... ~"h = 0.75, e. = 0.75, relative Nu c~ O,0051---~---'---~--..L------lEE0.9801------------------;-'i 0.736 .(fj

~ 0.492~ 0.248 D-u--O------{::::::J_----O-~-_{~J_--_I-n::Ji eh = 0.55, ",=0.55, relative NUi

-0- "h=0,55, ".=0.55. relative Nu c~ 0,0051--------"-----.....;".------'---\

0.980 r-*'··~..,.......,..........~.........., ..........~~........~.........., ........~~........___j

0.736 -

0.492 -

Figure 5.3: Variation of relative Nusselt number with modified Rayleigh number

70

number, due to the collective contribution of convective and radiative modes is evalu­

ated using the Eq. (3.14). A correlation for the average convective Nusselt number for

the present conjugate heat transfer problem is developed as:

(5.l)

The agreement of the experimental data with the proposed correlation is repre­

sented in the form of a parity plot shown in Fig. 5.4. The negative exponent in the term

consisting of emissivity of heat source indicates that emissivity of heat source weakens

the convective heat transport. This is clearly reflected in the experimental observation

reported in Table 5.2. However, it is interesting to note that the strength of convection is

increased with the emissivity of the adiabatic side plates. The reason for this observa­

tion is clear from the numerically simulated velocity and temperature profiles shown in

Figs. 5.5 and 5.6 respectively. The Figures suggest that the increase of convective heat

transport with the emissivity of the adiabatic side plates is attributed to the modification

37

__ 33..::L.o.£ 29

u;:,Z

~ 25.cE;:,

.: 21Q)IJ)IJ)

i 17Q)

>'';::;

~ 13>cou 9

Correlation coefficien t =0.95 +1 O~.'O

Std error = =0.02

-10%

9 13 17 21 25 29 33 37

Convective Nusselt number, Nuc(Exptl.)

Figure 5.4: Parity plot of average convective Nusselt number

71

0.25

0.20

0.15-III- 0.10E-Ql....ctl 0.05C.Ql

"0 0.00III'tl 0.25r::ctlQl(,) 0.20~

::s0III.... 0.15ctlQl

.r::r:: 0,10QlQl;;:.... 0.05Ql

.Q

0.ctl 0.00O'lQl 0.25.r::....IIIIII 0.200~

(,)ctl>- 0.15....'u0

0.10Ql>>- 0.05

Ra" = 81 x 1013

/Ra·=6.58 x 105

-- ell = 0.85, e s = 0.858 11 = 0,0, l!s =0,0

._ •• , 1:11

= 0.85. 1:. = 0.0

o.00 1!..-_-'--_...1...-_-'-_--L._---'-_----L_----'_~

0.000 0.004 0,008 0.012 0.016Spacing between the heat source and one side plate(m)

Figure 5.5: Variation of fluid velocity across the gap between heat source and one sideplate measured at the mid-height of the heat source

72

0.0160,012

•

0.008

-

Ra'=114 x107

..···0·..·· Z:n-{).{}'~5=o.o

,•• ..:.- •• 8",=0.1l:5, 8 s ='O.{J• llh=0.85,8§=0.85

...,.0.. ,.. llh ~ 0.0, 8 1 = 0.0,•• 06 ••• !lM'" 0.85, l!s~' (LO

• l:h=0.85,lls=O.85

.....0..... eh = 0.0, 8: 5 =0.0••• "" ••• 8:h=0.85., 'a:s=O.O

• Bh = 0.85,1::. = 0,85

0.004

400

380

- 360~-<U.... 340(l:J

0-(l) 320~en"'0 300s::(l:J

(l) 400<J'-=0 380en-(l:J 360(l)

.t:s::(l) 340(l)

3:-(Ll 320.c0-(l:J 3000)

(l).t: 400-I/)en0 380'-<J(l:J

(l) 360'-=-(l:J 340'-(l)0-E 320(l)

I-

3000.000

Spacing between the heat source and one side plate(m)

Figure 5.6: Variation of fluid temperature across the gap between heat source and oneside plate measured at the mid-height of the heat source

73

Table 5.2: Variation of surface temperature, convective and total Nusselt number withemissivity of heat generating element corresponding to C8 = 0.05 and Ra* =

8.1 x 106

Emissivity of Surface temperatureheat Source of heat source

(ch) (K)0.05 374.70.55 371.70.75 367.00.85 366.7

Convective Nusseltnumber(Nuc)

23.718.618.016.6

Total Nusseltnumber(NUt)24.727.929.130.0

of the thermal and flow field under the influence of surface radiation. The fluid tem-

perature profiles indicate that for a given heat input, the temperature of the heat source

decreases with the increase of its surface emissivity. This reduction in temperature of

the heat source at high values of emissivity is primarily due to larger heat energy radi­

ated to the cold surfaces as compared to that radiated from the heat source at low and

zero emissivity cases. Consequently, to satisfy the energy balance the radiative heat

flux leaving the heated surface needs to be absorbed by the cold surfaces. At zero and

very low values of emissivity of the adiabatic side plates, the incident radiation reflected

from the side plates enable it to maintain low temperatures. Under this situation, the

emitted radiation from the heated surface and the reflected radiation from the cold adia-

batic side plates are expected to be absorbed by the ambient; the third body considered

in the radiative heat transfer analysis. As emissivity of the side plate is increased, the

incident radiation will be absorbed and re-emitted. The process of absorption and emis­

sion continues, and this eventually results in the increase of temperature of the side plate

when thermal equilibrium is established. Thus, as shown in Fig. 5.6, the temperature

gradient on the adiabatic side plate is strikingly different from the case with no surface

radiation and consequently heat transfer is affected by the surface radiation. Hence, it

appears that at high values of emissivity of the heat source and adiabatic side plates, a

radiation induced boundary layer is developed along the length of the side plates. In

other words, with high emissivity of both heated and unheated surfaces, the unheated

surface becomes an active surface and part of the input heat to the heat generating ele­

ment is convected to the fluid from the unheated surface. This observation is reflected

74

by the positive exponent of emissivity of the side plates in Eg. (5.1).

In a conjugate heat transfer problem like the one considered here, an accurate predic­

tion of total heat transfer rate is of great importance from an engineering view point.

So, to report the overall heat transfer characteristics of the heat generating element, the

experimental data of the total Nusselt number is expressed as a function of modified

Rayleigh number, emissivity of the heat generating element and adiabatic side plates.

A multiple regression analysis results in the folloWing correlation for the average total

Nusselt number.

(5.2)

Figure 5.7 illustrates the trend of predicted values of the total Nusselt number obtained

from Eg. (5.2) with the experimental values. It can be seen that the distribution of

data points about the parity line is unbiased and the data points are distributed within

42

38

--...034

(.)-­..:::I:-:30(jj..cE~ 26::Gltil

gj 22:zS{: 18

14

Correlation coefficient = 0.97

Std error = ±0.04+10%

10 "'""'''''"-.L.-.......L-''''--...L..............L..._...I...-..I-.............................I

10 14 18 22 26 30 34 38 42

Total Nusselt number, NUt (Exptl.)

Figure 5.7: Parity plot of total Nusselt number

75

Table 5.3: Variation of surface temperature, convective and total Nusselt number withemissivity of adiabatic side plate corresponding to Ch = 0.85 and Ra* =8.1 x 106

Emissivity of Average surfaceadiabatic side plate temperature of

(1:8) heat source (K)0.05 366.70.55 359.5

0.75 359.20.85 359.1

Convective Nusselt

number(Nue)

16.619.3

19.619.7

Total Nusselt Relative contribution of

number convective Nusselt number(NUt) (%)30.1 44.832.6 40.7

32.9 40.433.0 40.3

an error band of ±10%. It is apparent that the total Nusselt number is an increasing

function of both modified Rayleigh number and emissivity of the heat source. Also,

the exponents of the governing parameters of buoyancy driven convection as well as

surface radiation indicate the strength of two distinct heat transfer modes in the com­

bined process of thermal energy transport. The dependence of total heat transfer on

emissivity of the side plate is only minimal. In a conjugate heat transfer problem in­

volving surface radiation and natural convection, the influence of radiative heat transfer

is controlled by the surface emissivity and as a consequence of which the total Nusselt

number is an increasing function of the emissivity. The exponent of modified Rayleigh

number is larger in Eq. (5.1) compared to Eq. (5.2). This larger value of exponent

in the convective Nusselt number correlation is an indication of strong dependence of

natural convection on Rayleigh number even in the presence of other modes of heat

transfer. However, in the conjugate problem considered here, the extent to which the

natural convection is affected by the surface radiation is found to be a strong function of

the emissivity of the surface of heat source. For this reason, the exponent of Rayleigh

number in the total Nusselt number correlation is expected to have a lower value as

compared to the exponent of Rayleigh number in the convective Nusselt number cor­

relation. Experimental data shown in Table 5.3 gives a quantitative idea of effect of

emissivity of the side plate on heat transfer characteristics for a given value of emis­

sivity of the heat source and heat input. It is clear that the convective Nusselt number

slightly increases with emissivity of the side plate, but the relative contribution to the

total heat transfer decreases due to increase of the radiative Nusselt number. Since the

76

o

Correlation coefficient = 0.996

Std error = ± 0.04

0.9

-:- 0.8...oU~ 0.7

e:::l10 0.6~c.E 0.5CllI-III~ 0.4

§.~ 0.3Cll

Eis 0.2

0.1

0.0 ~~'--,--------,'--'-------.J~'-------.J'-----'---J~-l~--l~-----'-~-----'-

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Dimensionless Temperature,e (Exptl.)

Figure 5.8: Parity plot of dimensionless temperature of the heat generating element

total Nusselt number also increases, the effect of increase of emissivity of the side plate

is to decrease the temperature of the heat source. In fact, the proposed Nusselt number

correlations are expected to mirror these experimental observations in a realistic way.

In many practical design problems, particularly in cooling of electronic application de­

vices, the primary requirement is the accurate prediction of the operating temperature

corresponding to a given thermal load. With this in view, a correlation for dimension­

less temperature of the heat generating element in terms of dimensionless volumetric

heat generation, and emissivities of the heat generating element and the adiabatic side

plates has been developed as

e (5.3)

The parity plot shown in Fig. 5.8 suggests that the predicted values agree well with

the experimental data. Equation (5.3) demonstrates that the dimensionless temperature

77

decreases with both fh and f s , while the effect of fh is strikingly visible. As temperature

is a direct function of volumetric heat generation (q*), a stronger exponent of q* is jus­

tifiable. Furthermore, a value close to unity signifies that (j is monotonically increasing

with q*.

5.4 Numerical Analysis

The numerical analysis of the problem has been performed using FLUENT 6.3. The

physical geometry is three dimensional consisting of a fluid zone between the parallel

plates, extended regions at the inlet, exit and sides of the channel and a solid zone rep­

resenting the heat generating element. The fluid medium (air) is assumed to be incom­

pressible with constant thermophysical properties. However, the body force arising due

to density variation imparted by the temperature gradient is accounted by an additional

source term in the vertical direction of the momentum equation and this source term is

modeled using Boussinesq approximation [64]. In-order to model surface radiation the

fluid medium is treated as transparent for radiation exchange.

5.4.1 Governing Equations

The governing equations presented in section 4.2.1 are for the two dimensional problem.

Since the analysis presented in this chapter is three dimensional, the relevant equations

are presented in the divergence free form in the following

Conservation ofmass

V.u = 0

Conservation ofmomentum

1(u.\7)u = --\7P + v\72u + B

p

78

(5.4)

(5.5)

where B is the body force vector defined by [0, -g{3(T - Too), O]T

Conservation ofenergy

(a) Fluid domain

(b) Solid domain

1/1

(lis\72Ts + gc = 0. p p

5.4.2 Boundary conditions

(5.6)

(5.7)

No slip for velocity components and zero normal pressure gradient are the boundary

conditions used on all solid walls. The velocity components at the inlet and exit of the

extended domain not known a priori, are obtained using appropriate boundary condi­

tion for pressure. The stagnation pressure at the inlet of the domain is equated to the

atmospheric pressure, whereas static pressure at the exit of the domain is taken as at­

mospheric pressure. The emissivity of the surfaces, used for the numerical simulations,

are same as those obtained in section 3.4. In general the mathematical representation of

the boundary conditions can be described as follows

On the side walls

Inlet of extended domain

Pressure Po = Patml Temperature T = Too

Exit of extended domain

79

Pressure Pst = Patm, aT = 0al1

5.4.3 Computational domain and grid

Initially, a 3D computational domain without an extended domain has been created

for the numerical solutions. The computational domain is discretized using structured

non-uniform hexahedral cells. Fine grids have been employed over all the solid-fluid

interfaces to resolve the gradient of field variables. Numerical simulations have been

carried out for the same heat input, used in experimental investigation. The heat input

has been modeled as a volumetric heat generation term, specified in Eg. 3.17. The

results show that the simulated temperature over-predicts the measured temperature of

the heat source. This observation has become the motivation to adopt an extended

domain approach for the numerical simulations. In view of this, numerical simulations

have been carried out by extending the domains of the channel by 25%, 50%, 100% of

the channel height and the results of the studies are summarized in Table 5.4.

It can be seen that the numerical simulations performed with extended domains

50% and 100% of channel lengths provide close results. From this observation, we

have chosen 50% extended domain for numerical studies. With 50% extended domain,

a grid sensitivity study has been conducted to arrive at an optimum grid for numerical

simulations. A summary of the results of grid sensitivity study is given in Table 5.5. The

results are found to be insensitive to grid beyond the one with 192750 cells, consisting

of 594295 faces and 208964 nodes. The final computational domain along with the

Table 5.4: Summary of domain independence study (Eh=0.85,E8 =0.05, AR=6.25)

Modilled Rayleigh Measured surface Numerically predicted surface temperature of heat source (K)number temperature of for different percentage of extended size of the domains

heat source

(Ra·) (K) 0% 25% 50% 100%.

3.3 x 105 310.9 312.8 312.2 311.6 311.68.1 x lOG 366.7 373.3 371.1 369.5 369.4

80

Table 5.5: Summary of grid sensitivity study (Eh = 0.85, Es = 0.05, AR=6.25, Ra* =8.1 x 106)

Number of Measured surface Numerically predicted Temperature at different y-Iocationscells temperature of the surface temperature on the adiabatic plate

heat source of the heat source (y is in mm)

(K) (K) Y = 50 y = 100 Y = 125 Y = 150 Y = 200

36640 376.7 303.9 304.6 305.7 306.5 307.994200 370.3 304.0 304.9 305.9 306.6 308.1132000 369.7 304.1 305.4 306.2 306.8 308.1192750 366.7 368.4 304.2 305.1 306.2 306.8 308.2241680 368.3 304.2 305.2 306.2 306.8 308.2

I~Y

~-L..:X

_--- Pressure outlet

Extended domainat channel exit

---Adiabatic wall

Heatgeneratingelement

Extended domainat channel inlet

-~--_ Pressure inlet

Figure 5.9: Computational domain along with the grids used for the 3D numerical sim­ulations

81

a.One fourth height of the heat source

c.Three fourth height of the heat source

b.Half height of the heat ource

d.Top edge of the heat source

Figure 5.10: Temperature field plotted along the dimensionless horizontal distance andchannel span at different levels along the height of the heat source.(Eh =0.85, lOs = 0.05, Ra* = 8.1 x 106

)

82

grid used is shown in Fig. 5.9. Variations of the predicted temperature cOlTesponding

to different locations along the height of the heat source are shown in Fig. 5.10. The

plots undoubtedly indicate that the temperature variation along the z-direction (span

of the heat generating element) is almost uniform except close to the supports. Stated

differently, the temperature field is nearly two dimensional in the x-y plane.

5.4.4 Numerical solution procedure

FLUENT solver uses control volume approach to solve the governing differential

equations offluid flow and heat transfer [65]. The segregated steady state solver option

has been selected for the present computation. A second order upwind scheme is used

to interpolate the unknown cell interface values required for modeling the convection

tellliS. The coupling between velocity and pressure is resolved by selecting the SIMPLE

(Semi-Implicit method for pressure linked equations) option and radiation heat transfer

by the discrete ordinate model. In using discrete ordinate model the optical thickness

is set equal to zero, thereby the fluid medium is assumed as non-participating. Non­

linearity of the equations requires the solution to be progressed in a controlled manner

with the use ofrelaxation factors. The under relaxation factors used in the present study

are 0.3 for pressure, 0.7 for momentum, 0.9 for energy and 0.9 for intensity of radiation.

Convergence ofthe solution is checked by examining the residues ofdiscretized conser­

vation equations of mass, momentum, energy and intensity of radiation. The iteration

is terminated only when the maximum of all the residues reaches less than 1 x 10-6 to

assure conservation of quantities.

5.4.5 Discussion of results from Numerical analysis

Numerical simulations have been performed for all the values of variable parameters,

used in the experimental investigations with the heat generating element placed at the

geometrical center of the channel and for a channel aspect ratio of AR = 6.25. Figure

5.1 0 shows the temperature field plotted across the gap between the heat generating

83

40 .---------------------,

35 ...'"'"

:i '" ~

Z '" l8! l8!

..: 30 '" l8!Q)

'"l8!

.c'"

l8!E ~

:::J l8!c • NUl experimental... 25 ~

"ai l8! NUl numerical(/)(/) t:::JZ i~

20~

~

15

10 Ll----''-----'-_.........._.l...-........_.....L..._"''''--------J_---'------I

3.2E+005 3.7E+006 7.0E+006 1.0E+007 1.4E+007 1.7E+007

Modified Rayleigh number, Ra*

Figure 5.11: Comparison between the experimental and numerical values of total Nus­selt number (ch = 0.85, Cs = 0.05)

element andone side wall corresponding to four span-wise locations. A close examina­

tion of the sub-figures reveals that the heat generating element appears as an isothermal

body, which is in full agreement with the experimental observation reported earlier.

Furthelmore, these figures communicate that the vmiation of thennal field is predom­

inant in the x-y plane, which means that the heat generating element can be regarded

as a planar heat source. Figure 5.11 presents a comparison of experimentally and nu­

merically estimated values of total Nusselt number as a function of modified Rayleigh

number for the case Ch = 0.85 and Cs = 0.05. As can be seen, the experimental and

numerical results compare well for low values ofRayleigh numbers, while the deviation

between the experimental and numerical values progresses with Rayleigh number, the

maximum deviation is found to be 16.6% at Ra* = 1.6x 107. The reason for this trend

84

Table 5.6: Mass flow rate through the channel for the case with Ra*AR=6.25

Emissivity ofheat generating element (Eh)

0.050.85

% increase of rrif

Es = 00.002410.00236

-2.07

Mass flow rate, kglsfor different Es

Es = 0.05 Es = 0.550.00256 0.002760.00303 0.00398

18.35 44.20

Es = 0.850.002770.00403

45.48

can be traced to the fact that the unaccountable heat losses, which is inherent to any

experimental setup, increases with increase of temperature. Revisiting the Figs. 5.5 and

5.6, it can be seen that in the absence of radiation (Eh = 0.0 and Es = 0.0), the boundary

layer formed adjacent to the heat source is similar to natural convection boundary layer

over a vertical isothennal plate kept in an infinite fluid medium. With increasing sur­

face emissivity of the walls, the fluid velocity profiles in Fig. 5.5 tend to shift towards

the channel side wall which is indicative of flow field redistribution effected by surface

radiation. Clearly, the flow field redistribution under the influence of surface radiation

is seen to be affecting the mass flow rate through the channel. Table 5.6 gives the mass

flow rate through the channel computed for different values of the surface emissivity of

the heat generating element and channel walls. For a given value of emissivity of the

heat generating element, mass flow rate increase with increase of emissivity of the side

walls. It may be noted that for low emissive surface of the heat generating element,

(Eh = 0.05) the increase ofmass flow rate with an increase in the emissivity of side wall

is marginal, the increase is estimated as 14.9% for an increase of Es from 0.0 to 0.85. On

the other hand, with highly emissive surface ofthe heat generating element (Eh = 0.85)

the effect of increasing the emissivity of channel walls on the mass flow rate is strikingly

noticeable, the maximum increase in the mass flow rate is 70.8%. This considerable en­

hancement in the mass flow rate suggests that the average convective Nusselt number

increases due to surface radiation interaction. However, it is important to note that with

the increase of emissivity of the heat source from Eh = 0.05 - 0.85, the mass flow rate

is found to decrease by 2.07%. This is because the convection could be suppressed by a

drop in temperature of the heat source with highly emissive surface. Figure 5.12 shows

85

0.00 '-_........._--1.__""'---_---'-_---'__....1...-_---'-_-'

0.000 0.004 0.008 0.012 0.016

Spacing between the heat source and one side plate(m)

._..._.... Top

-Middle----·Bottom

Ra'=8.1 X 106

...-~Ear 0.30 ....------------'---------------,-.!!!c.Q)

~ 0.25"0s::ccQ)

f:::J 0.20otJ)

1UQ)

.c:s:: 0.15Q)

iQ).cgo 0.100)

Q).c:-VI~ 0.05I­(Jcc~'uo~>-

Figure 5.12: Variation of fluid velocity, across the gap between the heat source and oneside wall, drawn at three different heights of the heat source obtained from3D numerical simulations (th = t s = 0.85, Ra* = 8.1 x 106

)

the variation of fluid velocity across the gap between the heat generating element and

one side wall of the channel, plotted at three different heights of the heat generating

element, viz. bottom, middle and top. The velocity distribution is nearly uniform in

the gap c1oseto the leading edge of the heat source with almost symmetric boundary

layers on either side. From middle height on-wards, the fluid accelerates relatively more

over the surface of the heat generating element. Consequently the velocity distribution

becomes skewed with increase of height. However the boundary layer thickness over

the surface of the heat generating element is found to be nearly same. As seen in many

numerical studies reported earlier [7], [9] and [20], 2D numerical analysis of the prob­

lem was started with a view to reduce computational efforts. It is interesting to note that

the temperature field predicted by the 2D numerical simulation shows close agreement

with 3D numerical prediction. However, the velocity distribution shown in Fig. 5.12

86

.-...- .... Top-Middle----. Bottom

-.!!J.EQ) 0.30 r-----------------------,10 . ,-........."Q. ( ).:-:.....,....(1) I ../' ",~•....;;.~~---_

"C 'I .. ..."iii 0.25 t i ~ "',CI> ,'.f . '""' " .....c: , J , , .•O· I: ".

"C If ,e'i .co I, ~.Q) 0.20 I i "\e ! ~~ I ~o : . "-f/I .f \. \.... \, \m0.15 ! \.c ( '.~ \Q) i \i 0.10 ! \Co ico '.~ \~ 0.05 Ra':: 8.1 x 106 \f/Ie i(JCO~ 0.00 '--_-'-__'--_----'-__'---_-'-_---J__-'--_~

g 0.000 0.004 0.008 0.012 0.016

~ Spacing between the heat source and one side plate(m)>-

Figure 5.13: Variation of fluid velocity at three different stream-wise locations on theheat source obtained from 2D numerical simulations (Ch = Cs = 0.85,Ra* = 8.1 x 106)

and 5.13 plotted across the gap between the heat source and side wall, obtained from

3D and 2D simulations are distinctly different. It is clear from Fig. 5.12 that the average

velocity increases with increase of height of the heat source, which is due to the possi­

ble effect of flow entrainment into the rising plume from the end faces of the channel.

On the other hand for 2D numerical simulation (Fig. 5.13), velocity magnitude remains

same at different heights of the heat source. This observation shows that 2D numerical

simulation for the present problem is inadequate for the accurate simulation of the flow

field.

87

5.5 Summary

The details of experiments and numerical simulations conducted to investigate coupled

laminar natural convection with surface radiation heat transfer from a short planar heat

generating element mounted freely between two thermally insulated vertical parallel

plates along with the results obtained are explained. Heat transfer correlations devel­

oped from experimental data are presented and discussed. The results of numerical

simulations have been compared with experiments. The flow field modifications due to

surface radiation is also presented.

A logical extension to the present research is to investigate the heat dissipation char­

acteristics of multiple heat sources, since multiple heat sources are encountered in a .

number of electronic equipment that are undergoing miniaturization in their develop­

ments. So the problem of natural convection from free standing planar heat sources has

been selected to investigate next.

88