ORIGINAL

A correlation for free convection heat transfer from vertical wavysurfaces

M. Ashjaee Æ M. Amiri Æ J. Rostami

Received: 22 July 2006 / Accepted: 1 December 2006 / Published online: 16 January 2007� Springer-Verlag 2007

Abstract Free convection heat transfer along an iso-

thermal vertical wavy surface was studied experimen-

tally and numerically. A Mach-Zehnder Interfer

ometer was used in the experiment to determine the

local heat transfer coefficients. Experiments were done

for three different amplitude–wavelength ratios of

a = 0.05, 0.1, 0.2 and the Rayleigh numbers ranging

from Ral = 2.9 · 105 to 5.8 · 105. A finite-volume

based code was developed to verify the experimental

study and obtain the results for all the amplitude–

wavelength ratios between a = 0 to 0.2. It is found

that the numerical results agree well with the experi-

mental data. Results indicate that the frequency of the

local heat transfer rate is the same as that of the wavy

surface. The average heat transfer coefficient decreases

as the amplitude–wavelength ratio increases and there

is a significant difference between the average heat

transfer coefficients of the surface with a = 0.2 and

those surfaces with a = 0.05 and 0.1. The experi-

mental data are correlated with a single equation which

gives the local Nusselt number along the wavy surface

as a function of the amplitude–wavelength ratio and

the Rayleigh number.

List of symbolsA surface area (m2)

a amplitude of the wavy surface (m\right)

cp specific heat (J kg–1 K– 1)

c1, c2 parameters in Eq. (15)

Grx local Grashof number

g gravitational acceleration (m s– 2)

hx local heat transfer coefficient (W m– 2 K– 1)

h average heat transfer coefficient (W m– 2 K– 1)

htot total convective heat transfer coefficient,

h �A� �

J Jacobean of the coordinate transformation

k thermal conductivity of air (W m– 1 K– 1)

L total curve length of the wavy surface (m)

l wavelength of the wavy surface (m\right)

N specific refractivity (m3 kg– 1)

Nux local Nusselt number,hxl=k

p pressure

q parameter defined in Eq. (12)

Pr Prandtl number

R parameter in Eq. (6)

Ro gas constant (J kg– 1 K–1)

Ral Rayleigh number,gb Tw � T1ð Þl3�am

r distance normal to the wavy surface (m)

s curve length along the wavy surface (m)

T temperature

u, v velocity components

W width of the test section (m)

x, y coordinates

z parameter in Eq. (6)

Greek symbolsa amplitude–wavelength ratio

b volumetric thermal expansion coefficient (K–1)

e fringe shift

n, g non-orthogonal curvilinear coordinates

d thermal boundary layer thickness (m)

k laser wave length (m)

M. Ashjaee (&) � M. Amiri � J. RostamiDepartment of Mechanical Engineering,University of Tehran, Tehran 11365-4563, Irane-mail: ashjaee@ut.ac.ir

123

Heat Mass Transfer (2007) 44:101–111

DOI 10.1007/s00231-006-0221-8

l dynamic viscosity (kg m– 1 s– 1)

m kinematic viscosity (m2 s–1)

Superscripts

^ dimensional quantities

Subscripts

¥ ambient condition

ref reference condition

tot total value

w at the plate surface

1 Introduction

The study of free convection heat transfer from vertical

surfaces with complex geometry has received consid-

erable attention due to its practical applications.

Roughened surfaces are encountered in many appli-

cations such as flat plate solar collectors, electronic

cooling and flat plate evaporators in refrigerators.

Cooling of the electronic devices is of great impor-

tance, because the operating temperature strongly af-

fects the components performance and reliability.

Climate control within building interiors is another

important application. Most of the above applications

fall between the amplitude–wavelength ratio range of

a = 0 to 0.2 and Rayleigh number between

Ral = 2.9 · 105 to 5.8 · 105. The sinusoidal wavy

surface encompasses all other roughened surfaces and

can be viewed as an approximation to many practical

geometries for which free convection heat transfer is of

interest. Yao [32] used a transformation method to

numerically evaluate heat transfer for a vertical sinu-

soidal surface at uniform temperature. He showed that

the frequency of the local heat transfer coefficient was

twice that of the wavy surface. Also the numerical re-

sults showed a decrease in the Nusselt number when

compared to a flat plate. Bhavnani and Bergles [4]

studied natural convection from a vertical isothermal

wavy surface experimentally at a constant Rayleigh

number. Their results differed from those of Yao [32]

with respect to the amplitude and frequency depen-

dency of the local heat transfer rate. Their experi-

mental data indicates that the period of the local heat

transfer coefficient is about the same as that of the

wavy surface. This difference could be attributed to

numerical solution of Yao [32] where some approxi-

mations were considered in the governing equations.

They Moulic and Yao [21] studied the mixed convec-

tion along a wavy surface to explain this difference.

They reported that this difference could be attributed

to the effect of an induced forced flow on the natural

convection along the wavy surface. Such a flow is

commonly induced by a heated object in a room with

cool walls. This induced flow acts as a forced convec-

tion to the heated wavy surface in the experiment. The

forced-convection component of heat transfer contains

two harmonic. The amplitude of the first harmonic is

proportional to the amplitude of the wavy surface; the

second harmonic is proportional to the square of this

amplitude.

A vast amount of literature about convection along a

sinusoidal wavy surface is available for different heat-

ing conditions and various kinds of fluids [ 1, 5, 6, 9–12,

14–20, 24–26, 29–31]. Finally Yao [33] has investigated

the problem of free convection along a vertical complex

wavy surface numerically. The major conclusion of his

numerical study was that the total heat transfer rate for

a wavy surface of any kind is, in general, greater than

that of the corresponding flat plate, and may be a

function of the ratios of amplitude to wavelength of the

surface. On the other hand, he reported that the aver-

age heat transfer rate per unit of wetted surface for a

wavy surface is less than that of a flat plate.

Extensive studies on natural convection heat trans-

fer from the wavy surfaces reveal the importance of

this problem in nature and industries. Since most of the

previous studies were numerical, an experimental

study which encompasses the effect of parameters such

as waviness of the surface and the Rayleigh number on

the free convection heat transfer was considered nec-

essary. The present work aims to obtain a correlating

equation for free convection heat transfer as a function

of amplitude–wavelength ratio and Rayleigh number

which has not been worked on yet. Therefore, the local

Nusselt numbers along the wavy surface are deter-

mined and presented with an experimental-based cor-

relation as a function of the amplitude–wavelength

ratio and the Rayleigh number.

2 Experimental setup and procedure

2.1 Test section

The test sections were made of aluminum plates

shaped into the sinusoidal wavy surface by means of

wire-cut machinery. A schematic of the test section is

shown in Fig. 1a. The height of the test sections was

102 Heat Mass Transfer (2007) 44:101–111

123

152.4 mm, with the wavelength, (l), of 50.8 mm. Three

test sections with amplitudes, (a), of 2.54, 5.08, and

10.16 mm were made. This resulted in amplitude–

wavelength ratios of 0.05, 0.1 and 0.2 respectively.

Width of each test section was 127 mm, resulting in the

same projected area. Nine holes with 2 mm diameter

were drilled into the rear surface of the plates till

reaching 1 mm far from the front surface. These holes

were in three rows, located 30, 63.5, and 97 mm far

from one lateral edge. Thermocouples of type K were

placed and fixed in these holes using epoxy adhesive. A

schematic of the plate and the location of the holes are

shown in Fig. 2. All the temperatures were monitored

continuously in a PC by a ‘‘TESTO 177’’ data logger. A

strip heater assembly was mounted on the rear surface

of the plate. Then the rear of the heater assembly was

insulated using 30 mm thick glass wool, compacted

between two wood fiber plates. The thermal conduc-

tivity of the fiber wood plates was 0:05 W m�2 K�1� �

[2]. The electrical power supplied to the heater was

controlled by a variable transformer, therefore

obtaining different surface temperature and Rayleigh

number. For each electrical power input supplied to

the heater it took about 4 h to achieve the steady state

condition. At the steady state condition the difference

in surface temperature readings for nine junctions were

about 0.4�C. The pressure and the relative humidity of

the laboratory were recorded during all the experi-

ments. The room air temperature was measured at two

different vertical locations about 70 cm away from our

test section and they both indicated the same value.

2.2 Interferometer and data reduction procedure

A 100 mm diameter beam Mach-Zehnder Interfer-

ometer (MZI) was used to measure the heat transfer

coefficients. A schematic of the MZI used in this study

is shown in Fig. 3. In this instrument, light from a

10 mW He–Ne laser splits into two beams of approxi-

mately equal intensity. One beam passes through the

undisturbed ambient air, and the other beam passes

through the disturbed medium adjacent to the wavy

surface. Because of variation of index of refraction

with temperature, the two light beams are no longer in

phase when they are recombined. This phase shift

produces an interference pattern in the optical output

of the MZI, which can be captured with a CCD cam-

era. When the test beam and reference beam are par-

allel upon recombination, the constructive and

destructive interference fringes correspond directly to

the isotherms in the flow field. All the interferograms

were digitized with a ‘‘Panasonic WV-CP410’’ 1/3¢¢CCD camera which was connected to a PC. Since the

height of the test section was larger than the diameter

of the optical beam, therefore a vertical movement of

Fig. 1 a Schematic of the test section, b schematic of the physicalmodel and computational domain

Fig. 2 Schematic of the wavy plate and locations of thethermocouple junctions

Heat Mass Transfer (2007) 44:101–111 103

123

the test section was provided by a laboratory jack

which was installed on the optical table (Fig. 1a). At

each vertical location on the wavy surface three in-

terferograms were recorded for assurance of the

experiment repeatability and data reduction. Interfer-

ograms of the wavy surfaces of different amplitude–

wavelength ratios and Rayleigh numbers are shown in

Fig. 4.

A code has been developed to calculate the tem-

perature of each fringe as well as its perpendicular

distance from the wavy surface at 60 different locations

for each recorded interferogram. A sample interfero-

gram is shown in Fig. 5. A thermocouple was placed

over the last fringe completely far from the wavy sur-

face to measure its temperature. The order number of

this fringe is set to zero and is considered as the ref-

erence fringe. Then by decreasing the normal distance,

r; towards the wavy surface, order number of the

fringes increases and reaches the highest value at the

surface (see Fig. 5). Subsequently, the temperature of

each fringe (isotherm) can be calculated from the

classical optical relation, explained by Hauf and Gri-

gull [8] for an ideal gas:

T ¼ 3p1NWTref

3Wp1N � 2kR�Trefeð1Þ

where R� ¼ 287 J kg�1 K�1; N ¼ 1:503� 10�4 m3 kg

and Tref is the reference temperature (ambient

temperature). Then, the digital recording of the

interferogram intensity enables to locate the position

of each fringe with respect to its normal distance from

the surface. Figure 6 shows the variation of the

intensity at one point along the wavy surface for the

sample interferogram of Fig. 5. Each maximum and

minimum corresponds to a bright and dark fringe,

respectively. To evaluate the temperature gradient at

the surface, the variation of the fringe shift e with

respect to normal distance was measured. Then by

differentiating Eq. (1) with respect to normal distance

the temperature gradient at the surface can be

calculated as follows:

dT

dr¼ dT

de� dedr

ð2Þ

The error in fringe shift due to the end effects can be

obtained from the equation explained by Eckert and

Goldstein [7] as follows:

De ¼ 2de3W

ð3Þ

The fringe temperatures and the temperature gradient

at the surface are then recalculated incorporating the

error term.

The local heat transfer coefficient can be obtained as

follows:

hx ¼ �kdT

dr

�����r¼0

� 1

ðTw � T1Þð4Þ

Therefore

Nux ¼hxx

k¼ � x

Tw � T1

� � � dT

dr

�����r¼0

ð5Þ

where x is the vertical distance along the wavy surface

shown in Fig. 1b.

Fig. 3 Schematic of theMach-ZehnderInterferometer (MZI) used inthe experimental study

104 Heat Mass Transfer (2007) 44:101–111

123

The average and total heat transfer coefficient is

then calculated from:

h ¼ 1

L

ZL

0

hs � ds ð6Þ

where s is the curve length along the wavy surface

measured from the leading edge (see Fig. 1b).

In order to check the accuracy of the experiments

and data reduction method, the local heat transfer

coefficients for an isothermal vertical flat plate is

measured and compared with other researchers [3. 22].

The value of the average heat transfer coefficient of the

present work is 1.42% higher than that of Bhavnani [3]

and 4.24% lower than the correlation of Ostrach [22]

for Ral = 468316. Temperature difference in the work

of Bhavnani and present study was 29.8 and 30.1�C

respectively. Also comparisons were made between the

experimental results of present work and the work of

Bhavnani [ 3] for wavy surfaces with a = 0.05 and

a = 0.1. The differences of 4.8 and 2.82% in average

heat transfer coefficients for wavy surfaces with

a = 0.05 and a = 0.1 were observed respectively.

With any experimental investigation there is always

a certain degree of unavoidable uncertainty. An

uncertainty analysis has been performed using the

method of Kline and McClintock [13]. This method

uses relative uncertainty in various primary experi-

mental measurements to estimate the uncertainty of

the final result. If the result of an experiment, R, as-

Fig. 4 Composite interferograms of the wavy surfaces withdifferent amplitude–wavelength ratio and Rayleigh number

Fig. 5 Fringe shift evaluation from a sample interferogram

Fig. 6 Fringe intensity variation of Fig. 5 along the normal lineto the wavy surface

Heat Mass Transfer (2007) 44:101–111 105

123

sumed to be calculated from M independent parame-

ters, z1; z2; :::::; zM; then the uncertainty propagated

into the result, dRð Þ is:

dR ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXM

i¼1

@R

@zidzi

� 2

vuut ð7Þ

where dz1; dz2; :::::; dzM are the uncertainties of the

independent parameters. Attention should be drawn to

relative magnitude of the uncertainties in this root-

sum-squared technique. Very little would be accom-

plished in reducing the uncertainty of any of the

smaller variables as the square of the larger variables

dominated the total uncertainty in the final result. In

this experiment, the accuracy of reading the isotherms

distance from the plate surface dominated the uncer-

tainty. Details of the uncertainty analysis procedure

are not presented here. Performing the error analysis

for all the experimental data shows that the measured

uncertainty for the heat transfer coefficient is

5.6 ± 2%.

3 Numerical study

The interferograms give information only about the

temperature distributions. To find the velocity field and

to further verify the experimental results, numerical

simulation is also performed.

A wavy surface is shown schematically in Fig. 1b.

The wavy surface can be represented by the function:

y ¼ a sin2p x

l

� ð8Þ

The origin of the coordinate system is placed at the

leading edge of the wavy surface. The surface is kept

at uniform temperature Tw which is greater than

ambient temperature T1: The u; v are the velocity

components in the x and y directions, respectively.

The flow is assumed to be Newtonian with constant

properties, incompressible, laminar and two dimen-

sional and the buoyancy effect is considered using the

Boussinesq approximation. The dimensionless trans-

formed governing equations in the general non-

orthogonal curvilinear coordinate framework with

n; gð Þ as the independent variables can be written as

follows [28]:

Continuity equation:

1

J

@uc

@nþ @vc

@g

� ¼ 0 ð9Þ

Momentum equations:

@

@nucuð Þ þ @

@gvcuð Þ ¼ @

@nJq11

@u

@n

� þ @

@gJq22

@u

@g

�

� J pnnx þ pggx

� �þ @

@nJq12

@u

@g

�

þ @

@gJq12

@u

@n

� þ J �Gr � T ð10Þ

@

@nucvð Þ þ @

@gvcvð Þ ¼ @

@nJq11

@v

@n

� þ @

@gJq22

@v

@g

�

� J pnny þ pggy

� �þ @

@nJq12

@v

@g

�

þ @

@gJq12

@v

@n

� ð11Þ

Energy equation:

@

@nucTð Þ þ @

@gvcTð Þ ¼ @

@nJq11

Pr

@T

@n

� þ @

@gJq22

Pr

@T

@g

�

þ @

@nJq12

Pr

@T

@g

� þ @

@gJq12

Pr

@T

@n

�

ð12Þ

where

J ¼ xnyg� xgyn; uc ¼ J nxuþ nyv� �

; vc ¼ J gxuþ gyv� �

;

q11 ¼ x2gþ y2

g

� �.J2; q12 ¼ xnxgþ ynyg

� ��J2;

q22 ¼ x2nþ y2

n

� �.J2;

Jnx ny

gx gy

" #

¼yg �xg

�yn xn

�ð13Þ

The employed dimensionless variables are defined as

follows:

x¼ x=l; y¼ y=l; u¼ ul=m; v¼ vl=m; p¼ l2p� ��

qm2� �

;

T ¼ T � T1

� �.Tw� T1

� �;

Gr ¼ gb Tw � T1

� �l3.

m2; Pr ¼ lcp

�k ð14Þ

Moreover, the appropriate boundary conditions for the

problem are: at the wavy surface, u = v = 0, T = 1;

matching with quiescent free stream, u = v = 0,

T = 0.

3.1 Solution procedure

The governing equations are solved using the finite

volume technique in a semi-staggered grid. In this

106 Heat Mass Transfer (2007) 44:101–111

123

arrangement velocity components u, v and tempera-

ture are located at the same nodal positions which are

staggered in relation to pressure nodes. The HYBRID

differencing [27] is used for the approximation of the

convective terms. SIMPLE algorithm [23] is employed

for velocity and pressure coupling. The discretized

equations are solved using the line by line method. The

grid points have been generated solving the Poisson

equations for x, y and are distributed in a nonuniform

manner with higher concentration of grids close to the

wavy surface. To ensure that the results of the

numerical study are independent of the computational

grid, a grid sensitivity analysis was carried out. The

details of the grid independence study for a wavy sur-

face with amplitude–wavelength ratio of a = 0.1 are

given in Table 1. A convergence criterion of 10–5 has

been imposed on all the three variables u, v, T. To

further check the adequacy of the numerical scheme

used in this work, the results for the free convection

heat transfer along a flat plate first obtained. The dif-

ference in the computed average heat transfer coeffi-

cient and the results of the work of Bhavnani [3],

Ostrach [22] and present experimental work are 4.47,

1.36, and 3.01% respectively. To demonstrate the

accuracy of the numerical result for the wavy surfaces,

the average heat transfer coefficient of the wavy sur-

face with a = 0.1 for Ral = 295979 is compared with

the interferometric data. Only 1% difference between

results was observed. Figure 7a shows the numerical

and experimental results of the local heat transfer

coefficients for the wavy surface with a = 0.1. The

results of Bhavnani [3] are also presented for the

comparison. The isotherms from experimental and

numerical results for a = 0.1 and Ral = 295979 are

also depicted in Fig. 7b.

4 Results and discussion

Figure 8a shows the experimental results for variation

of the local heat transfer coefficient along the wavy

surfaces of a = 0.05, 0.1 and 0.2 for a constant Ray-

leigh number of Ral = 5.59 · 105. The periodic

nature of the local heat transfer coefficient is seen

clearly and the periodicity is equal to the wavelength of

the sinusoidal surface. It can be seen that the locations

of the crest and trough of the local heat transfer

coefficients for all these surfaces of different a are the

same, but are shifted slightly upstream of the crests and

troughs of the wavy surface. The magnitude and the

variation of the local heat transfer coefficient depend

on the local slope of the surface wave. Figure 8b and c

show the velocity vectors and the velocity gradient

distribution along the wavy surface with a = 0.1 and

Ral = 5.59 · 105 respectively. It can be seen from

these figures that for the portions of the wavy surface

facing downward the flow is approaching the surface

and the velocity gradient is large therefore the local

heat transfer rate is large; but for the portions of the

Table 1 Grid sensitivity analysis in the present study, a = 0.1,Ral = 435168

Grid size �h Percentagechange (abs)

300 · 100 5.2077 –150 · 50 5.2301 0.475 · 25 5.3164 2.136 · 12 5.5703 7.0

Fig. 7 a Experimental and numerical results of local heattransfer coefficient for surface with a = 0.1 at Ral = 295979.b Comparison of isotherms around the wavy surface witha = 0.1 at Ral = 295979

Heat Mass Transfer (2007) 44:101–111 107

123

surface facing upward the flow is leaving the surface

therefore velocity gradient and the local heat transfer

coefficient is small. For all the three wavy surfaces, the

amplitude of the heat transfer coefficient gradually

decreases approaching the downstream. This is due to

an increase in the thermal boundary layer thickness.

Also Fig. 8a shows that the fluctuations in the local

heat transfer coefficient increases as the amplitude–

wavelength ratio increases. An interesting phenomena

in the Fig. 8a is the appearance of the second harmonic

in the local heat transfer coefficient of the wavy surface

with a = 0.2 which cannot be observed for the cases

of a = 0.05 and a = 0.1. This result is in agreement

with the numerical work of Moulic and Yao [21]. They

reported that the second harmonic becomes prominent

at large values of a.

Figure 9 shows the variation of the local Nusselt

number for the cases considered in the Fig. 8a. It can

be observed that the local Nusselt number is almost

independent of the waviness of the surface until the

location of the first crest of the wavy surface. After the

first crest the fluctuation in the local Nusselt number of

the surface with a = 0.2 is significantly larger than

that of the surfaces with a = 0.05 and a = 0.1. As the

amplitude–wavelength ratio increases, the values of the

local Nusselt number at troughs significantly decrease.

This effect will be better observed when the average

Nusselt numbers are plotted for different amplitude–

wavelength ratios. Figure 10 shows the variation of

average heat transfer coefficient h plotted against the

Rayleigh number Ral for different amplitude–wave-

length ratio. The values of the average heat transfer

coefficient of the surface with a = 0.2 is profoundly

smaller than those of the surfaces with a = 0.05,

a = 0.1 and flat plate. This is attributed to a great

decrease in the local heat transfer coefficient at the

troughs where the flow is leaving the surface and the

velocity gradient is small. Therefore as the a increases,

the average heat transfer coefficient decreases. For a

specific wavy surface, the value of h increases with the

increase of the Rayleigh number.

Figure 11 shows the total convective heat transfer

rate from the wavy surfaces. The total convective heat

transfer rate is defined as follows:

htot ¼ h �A ð15Þ

Total heat transfer rate is more important factor in

designing a heat transfer surface than its average rate. It

is seen that as the amplitude–wavelength ratio increases

the total convective heat transfer rate increases. This is

due to the increase in the heat transfer area.

For the purpose of fast calculating the heat transfer

rate from a heated surface, using a closed form formula

is of great importance. Hence, results of the

experimental investigation are used for developing a

Fig. 8 a Experimental results for local heat transfer coefficientfor different amplitude–wavelength ratios. b Plot of velocity forthe wavy surface with a = 0.1 and Ral = 5.59 · 105. cVelocity gradient distribution along the wavy surface witha = 0.1 and Ral = 5.59 · 105 (us is the velocity componentin direction of s coordinate)

108 Heat Mass Transfer (2007) 44:101–111

123

correlation which gives the variation of the Nusselt

number with amplitude–wavelength ratio a and the

Rayleigh number Ral. The equation which fits the data

well is given as:

Nux ¼ 1:35 x0:76 ec1�x sin 2pxþ 3:8ð Þ þ c2 Ra0:253l

� ð16Þ

where

c1 ¼ 0:3298 Ln aþ 1:1929 ð16aÞ

c2 ¼e�13 a�0:075ð Þ2

3:52ð16bÞ

It should be noted that this equation is correlated only

for the amplitude–wavelength ratio a ranging from 0.05

to 0.2 and the Rayleigh number from

Ral = 2.9 · 105 to 5.8 · 105. The length of each

wavy surface is equal to three wavelengths and the

leading edge of the surfaces faced upwards. Also the

Prandtl number was assumed to be con-

stant(Pr = 0.71).

For the limit of a fi 0 (isothermal vertical flat

plate) Eq. (16) reduces to the following:

Nux ¼ 1:35 x0:76 0:264 Ra0:253l

� �x 6¼ 0ð Þ ð17Þ

by substituting the definition of the Ral and the value

of l = 0.0508 m in the Eq. (17), the Nux can be

expressed as following :

Nux ¼ 0:327 Gr0:253x x 6¼ 0; Pr ¼ 0:71ð Þ ð18Þ

which is in good agreement with the Ostrach [22]

boundary layer analytical equation:

Nux ¼ 0:355 Gr0:25x Pr ¼ 0:71ð Þ ð19Þ

Figure 12 represents the correlation given by Eq. (16)

graphically along with the experimental data. The max-

imum error in curve fitting occurs at the locations near

the leading edge. However the average error in curve

fitting is about ± 7%. Also, in order to verify the accuracy

Fig. 9 Local Nusselt number for different amplitude–wave-length ratios

Fig. 10 Average heat transfer coefficient as a function of theRayleigh number

Fig. 11 Total convective heat transfer coefficient as a function ofthe Rayleigh number

Heat Mass Transfer (2007) 44:101–111 109

123

of the Eq. (16), the numerical results for the wavy sur-

face with a = 0.025, 0.075 and 0.1, which their experi-

mental results are not available, are obtained and

compared with this equation. Equation (16) fits the

numerical results with the acceptable degree of accuracy.

5 Conclusions

The buoyancy driven flow along an isothermal vertical

wavy surface was investigated experimentally and

numerically. Experiments were done using MZI. A fi-

nite-volume based code was developed for the

numerical study. The amplitude–wavelength ratio aand the Rayleigh number Ral were important param-

eters of this study. The frequency of the local heat

transfer coefficient was the same as that of the wavy

surface. The average heat transfer coefficient de-

creased as the amplitude–wavelength ratio increased

and the reduction of the average heat transfer coeffi-

cient for the wavy surface with a = 0.2 was significant.

The total convective heat transfer rate increased as the

amplitude–wavelength ratio increased. This was due to

the increase in the heat transfer area. Finally the

experimental data were correlated with a single equa-

tion which gave the local Nusselt number along the

surface as a function of the amplitude–wavelength ra-

tio and the Rayleigh number.

References

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3. Bhavnani SH (1987) Interferometric study of natural con-vection heat transfer from a vertical flat plate with transverseroughness elements. PhD Thesis, Iowa State University

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