Turbulent flow heat transfer and friction factor characteristics of shrouded fin arrays with uninterrupted fins

ELSEVIER

Turbulent Flow Heat Transfer and Friction Factor Characteristics of Shrouded Fin Arrays with Uninterrupted Fins S. B. Thombre Department of Mechanical Engineering, Visveswaraya Regional College of Engineering, Nagpur, India S. P. Sukhatme Department of Mechanical Engineering, Indian Institute of Technology, Bombay, India

• Experiments were performed to study the fully developed turbulent flow heat transfer and friction factor characteristics of shrouded, rectangular cross-sectioned longitudinal fin arrays with uninterrupted fins subjected to a uniform heat flux boundary condition at the fin base. The inter-fin spacing was varied from 9.3 to 53 mm, the fin height from 0 to 40 ram, and the fin tip-to-shroud clearance from 0 to 20 mm. The overall height and width of the duct (flow passage) were held constant at 40 mm and 228.6 ram, respectively. The working fluid was air, and tests were conducted for Reynolds numbers ranging from 2000 to 50,000. The equivalent diameter, defined as four times the flow area divided by the wetted perimeter, was taken as the characteristic dimension. In all, seven different fin configurations and the rectangular duct configuration with no fins were investigated. The heat transfer tests showed that the Petukhov-Popov equation is applicable for the rectangular duct configuration with no fins and that the Dittus-Boelter equation is applicable for the configurations with no clearance. The Dittus-Boelter equation is also applicable for fin configurations with clearance if the clearance-to-spacing ratio is small or the spacing-to-fin height ratio is large. Lower values of the Nusselt number were obtained for the fin configuration that did not satisfy the above conditions. As expected, the pressure drop tests showed that the Prandtl equation for evaluating friction factor is applicable for the rectangular duct configuration with no fins as well as for the fin configurations with no clearance. The configurations with clearance yielded friction factor values different from the Prandtl equation. These values have been correlated.

Keywords: fully developed flow, turbulent heat transfer, friction factor, shrouded fin arrays INTRODUCTION The present investigation deals with shrouded, rectangular cross-sectioned longitudinal fin arrays with uninterrupted fins (Fig. 1). It is seen that when air (or any other fluid) flows over a bare fin array (without a shroud), it tends to flow out of the inter-fin spacings into the ambient in a direction at right angles to the main direction of flow, thereby reducing the heat transfer rate. In o r d e r to avoid such outflow and to confine the flow to the inter-fin spacings, a shroud is positioned adjacent to the fin tips. A clearance between the fin tip and the shroud is frequently encountered in practice either by design or by chance. Shrouded fin arrays find applications in solar air heaters and in cooling electronic equipment.

It should be noted that the enhancement achieved in the heat transfer rate due to fins and the shroud is offset by the higher pumping power required to overcome the resistance that they offer to the flowing fluid. The extent of the enhancement is a function of the fin array paramet e r s - t h e inter-fin spacing S, the fin height Hf, and the fin tip-to-shroud clearance C [1-4]. The fully developed heat transfer coefficient is found to decrease with increase in C / H f ratio and to be a weak function of fin height [1, 2]. The effect of S on the heat transfer coefficient has not been studied experimentally so far. A correlation has been derived by Lau and Mahajan [2] to predict the effects of different fin array p a r a m e t e r s on the heat transfer coefficient. However, more experimental data are needed to establish its validity and limitations. As far as friction

Address correspondence to Professor S. P. Sukhatme, Department of Mechanical Engineering, Indian Institute of Technology, Bombay, 400 076, India.

Experimental Thermal and Fluid Science 1995; 10:388-396 © Elsevier Science Inc., 1995 655 Avenue of the Americas, New York, NY 10010

0894-1777/95/$9.50 SSDI 0894-1777(94)00059-H

Shrouded Fin ArTays with Uninterrupted Fins 389 Wd

END CHANI -.p

7

standard orifice meter. The specifications of the experimental setup are given in Table 1. The test section is made in two parts: a body and a cover. The body (Fig. 3) housed three independently controlled electric heater pairs (EP1 to EP3) with six flat heaters (EH1 to (EH6) isolated thermally from one another by means of slotted, rectangular cross-sectioned Teflon bars (TB1 to TB5). A typical fin array module (Fig. 4) is bolted to these heaters with a layer of highconductivity grease in between. Glass wool (GW) and "thermocole" (THC) insulation was provided at the bottom and at the sides to reduce the losses to the surrounding areas. The cover (the shape of an inverted open channel) formed the shroud for the fin array module. A 6-mm-thick Bakelite plate on the inner side provided a smooth surface to the flow. Glass wool insulation was provided to reduce the heat loss from the top. The cover was so dimensioned that, when fitted on the top of the body, it formed a smooth continuous surface with the development section and the exit section. A fin array module was assembled by arranging the fin plates and spacers alternately and holding them together by means of brass nuts and bolts. The fin plates were cut out of 3.2-mm-thick aluminium sheet while the spacers were cut from rectangular cross-sectioned aluminum bars of different dimensions. Various values of S and Hf tested during the experimentation are given in Table 2. More details about the experimental setup are given in [5].

•

q ,-2"

?,

,2;11 7

Figure 1. Shrouded, rectangular cross-sectioned longitudinal fin array.

factor is concerned, fully developed friction factor characteristics for nine different configurations of shrouded fin arrays have been reported by Sparrow and Beckey [4]. However, no equations have been proposed for the same. The experimental data available in the literature discuss the specific configurations for which the work has been carried out, but they do not provide sufficient information regarding the simultaneous effects of all the parameters on the heat transfer and friction factor characteristics. Such information is essential for optimum design of shrouded fin arrays. The main object of the present work is to develop heat transfer and friction factor relationships, after first obtaining extensive experimental data. The fin arrays were subjected to a uniform heat flux boundary condition, and heating was from one side (i.e., the fin base) only. The other three sides were insulated. The fin array parameters S, Hf, and C were varied over a wide range during the experimentation. Air was the working fluid, and the flow was in the turbulent regime. The data generated in this manner were then correlated in terms of the fin array parameters for the heat transfer and friction factor characteristics.

INSTRUMENTATION The air flow rate through the test section was obtained by measuring the pressure drop across a standard ASME orifice meter with a vertical water tube manometer. The desired flow rate was obtained by controlling the opening of a gate valve provided on the delivery side of the blower. The most probable uncertainty in the measurement of mass flow rate was estimated to be + 1.3%. The heat input per heater pair (2Pj) was directly measured with the help of three analog-type wattmeters whose accuracy was checked against a standard voltmeter-ammeter combination. The readings of the wattmeters were found to be within + 3% of the standard values in the range of interest. Temperatures were measured with the help of 34-gage wire calibrated copper constantan thermocouples. There were all in all 26 thermocouples mounted on the fin array surface (18 in one of the central channels and 8 in the other channels) and 12 thermocouples on the shroud. These were essentially evenly distributed amongst the six heater sections. Two thermocouples were used for measuring the inlet temperature of the air and two for the exit temperature after the test section. Besides these, six ther-

E X P E R I M E N T A L SETUP The general arrangement of the experimental setup is shown in Fig. 2. It consists of a development section, a test section, an exit section, and a flow-measuring device connected in series on the suction side of a blower. The air first enters into the development section that ensures a hydrodynamically developed air flow prior to the test section. In the test section, heat is added to the flow through finned surfaces and is absorbed by the flowing air by forced convection. The heated air flows through an air-mixing device in the exit section, a reducer that changes the cross section from a rectangle to a circle, and finally a

REDUCER i~.__.ORIFICE METER

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390

S.B. Thombre and S. P. Sukhatme

Table 1° Specifications of the Experimental Setup Height of test section duct, H a Width of duct, Wd Length of test section Distance between the pressure taps T2-T3 Length of development section Length of exit section Length of pipe upstream of orifice Length of pipe downstream of orifice Longitudinal length of Teflon spacer

= = = = = = = = =

40 mm 228.6mm 690 mm 872 mm 1450 mm 730 mm 1800 mm 800 mm 18 mm

mocouples were mounted inside the body structure to monitor body temperatures, and a thermopile with a magnification factor of 5 was provided to measure the rise in air temperature due to heating in the test section. It is estimated that errors in temperature measurement can lead to an uncertainty of +4.1% in the determination of the average temperature difference between the fin array surface and the flowing air. Forty-five heat transfer test runs were performed. Typically, in a heat transfer test it took approximately 5 hours to reach the first steady state and about 2 hours for every subsequent steady state. An overall energy balance check was carried out for each test run by comparing the total heat supplied (]E,_ 1P) with the sum of the total heat loss (Qio~) and the total enthalpy gain of the flowing air (aa). The two values generally agreed with each other within +5%. The pressure drop across the fin arrays was measured with the help of two capacitance-type micromanometers. The values measured ranged from 0.1 to 30 mm of water and were measured with an accuracy of + 1%. All pressure drop tests were conducted under isothermal conditions with no heating.

surfaces; and (2) from the unheated surfaces of the shroud that receive energy by radiation and by conduction through the walls. Most of the heat transfer takes place from the heated base surface and the fin surfaces. One way of calculating the heat transfer coefficient from the fin array to the flowing fluid is to evaluate the rate of heat transfer from the base and fin surfaces and to divide it by the appropriate area and temperature difference. The heat transfer rate can be obtained by subtracting from Qinj the rate of heat transfer to the shroud by radiation and conduction. It is however difficult to calculate the radiation and conduction terms to the required degree of accuracy because (1) exact values of emissivities of the surfaces are not known; (2) the temperature gradient at the shroud-fin array interface is not known; and (3) the shape factor calculations are intricate. In view of these difficulties, a different approach is adopted that eliminates the need for knowing these quantities but that requires a knowledge of the temperatures of all the surfaces. The heat transfer coefficient at any heater section is now given by

Oinj = ( n f -

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+ 2Afcj(Tfj

Taj)

-- Taj ) + Ascj(Tsj

--

TAX)}

(1)

+ 2h 7

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j--

.

.

.

DATA R E D U C T I O N The main parameters of interest are the fully developed value of the heat transfer coefficient and the friction factor. The following procedure is adopted to calculate these parameters.

Heat Transfer Coefficient It can be seen from Fig. 1 that the net heat entering the duct (Qinj) in any heater section is transferred to the flowing air (1) from the heated base surface and the fin

(Ab~j/2)(Tbs ×

+(Asoj/2)(T,j

-

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Ta) +

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- Taj )

(A,~j/2)(T~s./- Tas)

The first term in the equation represents the heat transfer rate from the central channels and the second term that from the end channels (see Fig. 1). Because the equivalent diameters of the central and end channels are different, the heat transfer coefficient in a central channel ( h ) is slightly different from that in an end channel (h~'). It can be shown that

h~ = hj(De/D* )l-n,

(2)

where n is the exponent on the Reynolds number ( = 0.8) in the heat transfer characteristics. The net heat input at any heater section j (Qinj) is taken as the arithmetic mean of the power supplied (1:',) minus the loss (QJossj) and the enthalpy gain by thee flowing air ( Q a ) at that heater section. Because of the uniform heat flux boundary condition, the value of Qaj is the same for all the heater sections and is therefore taken

\ /.

Figure 3. Test section: body (longitudinal section).

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Shrouded Fin Arrays with Uninterrupted Fins

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as one-sixth of the total enthalpy gain. The value of Qllo~! is also taken as one-sixth of the total heat loss Therefore,

(a)

in the steady state the surface temperature was measured at a number of locations on the wall of the duct along with the ambient air temperature. The test was repeated for five power inputs. In the steady state, no heat is supplied to the air inside the test section. Hence, all the power supplied is lost to the surroundings. The value of Qloss is plotted against the temperature difference between the average wall temperature and the surrounding air temperature. A very good fit is obtained using a second-degree polynomial having a small amount of nonlinearity. This equation is used to estimate the heat loss in the main test runs. It is to be noted that the heat loss never exceeded 10% of the heat supplied in the main tests.

(5)

Friction F a c t o r

Twj = AbcjTbj + AscjT~j + 2AfcjTfj A totallj

(6)

The fully developed value of the friction factor is given by

Atotal2j = Abe j + As~ j + 2Arc j + Ass j

(7)

Tw~j = Ab*sTbj + As*jTsj + 2AfcjTfj + AssjTs~j

(8)

ainj = ({Pj - (Oloss/6)} + {thaCp(Tae - Tai)/6})/2. (3) Substituting Eqs. (2) and (3) in Eq. (1), we get hi= ({Pj - (QlosJ6)} + {rhaCp(Ta~ - Tai)/6})/2

(nf- l)Atotallj(Twj-Taj)

+(Oe//O*e)l-nAtotal2j(Twej-Taj)

where Atotalli

=

Abe j + Asc j + 2Afc j

Atotal2j A set of preliminary tests were performed to determine the heat loss from the test section (Qlos~)- The following procedure was adopted. Configuration 1 (corresponding to a smooth rectangular duct with no fins) was assembled, and the test section was closed at both ends so that no air could flow in or out. A certain amount of energy was supplied to the heaters, and

f=

4LtsV2pa Ahnet '

(9)

where Ahne t = Ah m Ah c _ Ah c _ Ahs. (10) Ah e and Ah c are obtained using equations given in [6], and Ah s is obtained experimentally. These corrections are found to be at most 15% of Ahm, the measured pressure drop across the fin array. A n estimate of errors in the determination of the heat transfer coefficient and friction factor due to uncertainty in the measurements is made. The most probable values are 5.1 and 3.3%, respectively.

Table 2. Various Configurations Investigated

Configuration Number 1 2 3 4 5 6 7 8

Configuration Code 1-09-40-690 1-53-40-690 ) 1-09-35-690/ 1-09-20-690[ 1-25-20-690~ 1-53-35-690[ 1-53-20-690]

Classification

S(mm)

Hf(mm)

Smooth rectangular duct with no fins No clearance 9.3 40 53.0 40 configurations 9.3 35 9.3 20 Configurations 25.4 20 with clearance 53.0 35 53.0 20

C(mm)

C/ S

S / Hf

00 00 05 20 20 05 20

0 0 0.54 2.15 0.79 0.09 0.38

0.23 1.33 0.27 0.46 1.27 1.51 2.65

392

S.B. Thombre and S. P. Sukhatme 30(: 200 Re D = 1 6 5 0 0

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Figure 5. Axial variation of heat transfer coefficient for smooth rectangular duct with no fins.

,.-,3O z

H E A T T R A N S F E R RESULTS 10

Axial Variation of Heat Transfer Coefficient A representative axial variation of the regionally averaged heat transfer coefficients obtained for the smooth rectangular duct configuration with no fins is shown in Fig. 5. As expected, the value of the heat transfer coefficient decreases rapidly at the beginning and then attains a constant fully developed value. The value for the last heater section is usually observed to be about 15 to 20% higher than the fully developed value. This is to be attributed to an end effect. This general shape has been observed for all the configurations investigated, with the constant fully developed value being attained by the third or fourth heater section. The heater section by which the fully developed value is attained depends on the Reynolds number and equivalent diameter of the configuration. For this reason, the fully developed value is taken as the arithmetic mean of h4 and h 5, that is, the heat transfer coefficients at heater sections (4) and (5), respectively. Fully Developed Values The fully developed heat transfer characteristics for configuration 1 (smooth rectangular duct with no fins), configuration 2 and 3 (shrouded fin arrays with no clearance), and configurations 4-8 (shrouded fin arrays with clearance) are shown in Figs. 6, 7, and 8, respectively. The 300

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Figure 7. Heat transfer characteristics for fin arrays without fin tip-to-shroud clearance.

equivalent diameter of the respective configurations is taken as the characteristic dimension in the Reynolds and the Nusselt numbers, and the properties of air are taken at the local bulk mean temperature. The two well-known equations, the Dittus-Boelter (D-B) equation and the Petukhov-Popov (P-P) equation [7] are also plotted in these figures for comparison. Kays equation (Nu D = 0.0158 Re~ 8 [8]), developed analytically for flow through infinitely long parallel plates with one plate heated and the other adiabatic, is also plotted in Fig. 6. The Dittus-Boelter equation and the Petukhov-Popov equation are reproduced because they are frequently re-

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Figure 6. Heat transfer characteristics for smooth rectangular duct with no fins.

1000

10000

100000

Re D

Figure 8. Heat transfer characteristics for fin arrays with fin tip-to-shroud clearance.

Shrouded Fin Arrays with Uninterrupted Fins 393 ferred to in this paper: Nu o = 0.023 Re~ s Pr °'4 NUD

=

1.07

+

( f / 2 ) Re D Pr 12.7[Pr 2/3 - 1]v/f/2 "

(11) (12)

The friction factor f appearing in Eq. (12) is obtained from the Prandtl equation: 1

= 2 lOgl0 [ R e o g ~ ]

-- 0.8.

(13)

It can be seen from Fig. 6 that the experimental data for the smooth rectangular duct with no fins agree well with both the D-B and P-P equations, the agreement with the P-P equation being slightly better. The data lie above the P-P equation and below the D-B equation with a maximum deviation of + 5 % and - 1 0 % , respectively. In contrast to these equations, good agreement is not obtained with the Kays equation. The deviation in this case ranges from about + 30 to + 60%. The disagreement of the data with the Kays equation is not an unexpected result because it was developed for infinitely long parallel plates (aspect ratio a = 0), whereas the present data are for a = 0.175. In addition, in-the present situation a certain amount of convective heat transfer (5 to 10%) is also taking place from the side and top walls of the duct, which again was not considered while developing the Kays equation. It may be worth noting here that the experimental data obtained by Garcia and Sparrow [9] on a smooth rectangular duct with the larger side coated with naphthalene (similar to a situation of heat transfer in a rectangular duct with heating from one side only) also agree well with the P-P equation, thus strengthening the validity of the present experimental data. The experimental data for "no-clearance" configurations (Fig. 7) are also in good agreement with the D-B and P-P equations. However, now the agreement is slightly better with the D-B equation. It is therefore recommended that the D-B equation be used for evaluating the heat transfer characteristics of configurations without fin tip-to-shroud clearance if the heating is primarily from one side. Figure 8 displays the interesting information that the fully developed heat transfer characteristics for the configurations "with clearance" tested are also in good agreement with the D-B equation (_+ 5%), except for one configuration. The exception is configuration 1-09-20-690, which has yielded values about 25% lower than the D-B equation. In a configuration with clearance, the fluid prefers to bypass the space between the fins by flowing through the clearance. Thus, the average velocity through the clearance is more than the average velocity through the space between the fins. The difference in velocities increases as the clearance increases a n d / o r the spacing decreases. The difference in velocities is not very significant for configurations 4, 6, 7, and 8. Only in configuration 5 is the difference significant enough to cause a lowering of the heat transfer coefficient. From a practical point of view, configurations 4, 6, 7, and 8 would be preferred to configuration 5. In order to differentiate between the configurations on a nondimensional basis, consider the values of (C/S) and (S/Hf) for the configurations (Table 1). An approximate criterion that emerges from an inspection of the values is that the value of (C/S) should be less than

one or that the value of (S/Hf) should be more than one for the D-B equation to be valid. Test runs with more configurations would be needed in order to state a more accurate criterion. Following the suggestion of Lau and Mahajan [2], the results obtained in this investigation have also been analyzed with heating diameter (Dhd) as the characteristic dimension. It is seen that the data for all the configurations (except configuration 1-09-20-690) fall within _ 10% of the line: N U D h = 0.017 "'cOhl~-°s3. (14) This indicates that the heating diameter can also be used as a characteristic dimension in the Reynolds and the Nusselt numbers calculations. However, the equivalent diameter (D e) is a better choice because the agreement with the D-B equation is better (within +_5%). Sparrow and Kadle [1] have also obtained data for fin configurations with and without clearance. In order to facilitate a direct comparison, their data in the form Nu s v e r s u s / ~ L (obtained for Hf/S = 3.75, Hf/S 7.5, and for various clearances) is replotted in the form N u D v e r s u s Re D in Fig. 9. The best-fit equations from [1] are used while making this transformation. It can be seen from the upper panel of this figure that the data for S/Hf = 3.75 and C/Hf = 0 agree closely with the P-P equation and that slightly lower values are obtained as the clearance (C/Hf) increases. For clearances equal to 8.3, 16.5, and 33.6% of the fin height, the decrease observed in the Nusselt number is 3.7, 3.8, and 7.2% of the "no-clearance" case, respectively. A similar trend is also seen for Hy/S = 7.5. In this case however, the data for C/Hf = 0 lie 12 to 15% below the P-P equation. The decrease observed in the Nusselt number for clearances equal to 8.3, 16.5, and 33.6% of the fin height is 0.8, 3.8, and 12.4%, respectively. The underlined configuration values satisfy the proposed constraints C/S < 1 o r S/Hf > 1. It can be seen that the decrease in the Nusselt number for these configurations is small and that the equation for the "no-clearance" case =

100 C/Hf • -0.000

C/s 0.000

"~'" ~,,TJ-----__~

.-o.o83

o.3~1~

0o165 0"618J • - 0"336 1.260 - - - - - DpB EQ.

.,f~ ~

~

S/H~-- 0"270 :~/ Mf = U'Z ;V Hf/S=3"75

~

C/Hf

0.083 B-0.165 • - 0 . 336 -----DB

C/S

z

"-]..~.~J

..~.;/~

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z

"~:~..~'"

100

50 , - o.ooo o . o o o l ,

50

,

10

5

~/.';" .~7.~/~//S/Hf = 0.133 ~ / / Hf/S--7.5

IO

5

I 1000

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I I I I,I 10000 ReD

I

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I I ' ' I 100000

Figure 9. Analysis of data of Sparrow and Kadle [1].

394

S.B. Thombre and S. P. Sukhatme

can well be used to evaluate their heat transfer performances without much loss in accuracy. Thus, the results of Sparrow and Kadle [1] are similar to those of the present investigation. The common feature between the results of the two investigations is that the heat transfer characteristics of all configurations that satisfy the condition C / S < 1 or S / H f > 1 can be represented by a single equation. However, the two investigations differ slightly by the equations to be used. The present study proposes the D-B equation for any combination of the fin array parameters satisfying the mentioned constraint, whereas Sparrow and Kadle's data suggest the use of the P-P equation for H f / S = 3.75 and an equation that yields values of Nusselt numbers 12 to 15% less than those obtained from the P-P equation for H f / S = 7.5. F R I C T I O N F A C T O R RESULTS The friction factor characteristics for the smooth rectangular duct with no fins, and the "no-clearance" configurations are shown in Fig. 10. The well-known Prandtl equation is also plotted for comparison. For the case of the smooth rectangular duct, an extra pressure tap T 1 is provided 55 cm upstream from the inlet of the test section as shown in Fig. 2. This is to check that the section is long enough to establish a fully developed flow prior to the test section. The friction factor results obtained with the pressure drop measured across taps 7"1 and T3 are also shown in Fig. 10. Tap T 1 was inoperative for the other configurations investigated. It can be seen from Fig. 10 that all the data are in very good agreement ( + 5 % ) with the Prandtl equation for Re o > 6000. For lower values of ReD, the values of friction factor fall slightly below the Prandtl equation. This is probably because the flow is in the transition region. It can also be seen that the data for the smooth rectangular duct as obtained with taps T1-T 3 and taps T2-T 3 do not differ much ( + 3 % ) , indicating that a hydrodynamically developed flow is established even before it |

0.03

I

i ! !

i

i

reaches tap T I. Because the Prandtl equation is a wellestablished equation in the literature, the good agreement with it is a validation of the techniques used for measuring pressure drop and flow rate in the present setup. The friction factor characteristics for configurations "with clearance" are shown in Fig. 11. The data for Hf = 35 mm are plotted in the upper half, and those for Hf = 20 mm are in the lower half. The Prandtl equation is also plotted for comparison. It can be seen that the data for the same fin height irrespective of the spacing fall together, with the data for H f = 35 mm lying below and most of the data for H i = 20 mm (at higher Reynolds numbers) lying above the Prandtl equation. This indicates that the equivalent-diameter concept helps to correlate the variation in the inter-fin spacing but not the fin height. The higher or the lower value of the friction factor for configurations "with clearance" as compared to the Prandtl equation ("no-clearance" configuration) is related to the definition of the friction factor in the present study and is not a direct measure of pressure drop across the fin arrays. The net pressure drop for configurations "with clearance" is always less than that for "no-clearance" configurations if other parameters are kept constant. A comparison of the data obtained by Sparrow and Beckey [4] for two configurations, S / H d = 0.25, H f / H d = 0.875 and S / H d = 1, H f / H d = 0 . 8 7 5 , with the present data for two similar configurations, 1-09-35-690 ( S / H d = 0.233, H f / H d = 0.875) and 1-53-35-690 ( S / H a = 1.38, H f / H d = 0.870), is shown in Fig. 12. The present data are found to lie a little (15 to 20%) below that of Sparrow and Beckey. No reason can be ascribed for this disagreement. Finally the data for the "with clearance" configurations have been correlated by the following equations: ReD < 150001 Re D > 15000 J '

f = M/Re'~ f = N/Re~

(15)

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0.001

G-1-09-35-690 o-1 - 53- 35- 690 SMOOTH RECTANGULAR DUCT WITH NO FINS o-TAPS T2-T 3 X-TAPS T1 - T3 e-1-09-40 -690 u - 1 - 5 3 - 4 0 - 690

0.02 Hf =20ram 0,01

0.001

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0,000~ 0.00(

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Figure lO. Friction factor characteristics for rectangular duct with no fins and for fin arrrays without fin tip-to-shroud clearance.

a-1-25-20-690 o-1- 53-20-690

AN ~TL E Q ~ PR D ,

0.003 0.002 1000

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i

I

i

i

t

i

I [

10000 ReD

I00000

Figure 11. Friction factor characteristics for fin arrays with fin tip-to-shroud clearance.

Shrouded Fin Arrays with Uninterrupted Fins 395 0.08

i

i

• .

0.06

•

o

a •&

0.04 0-03 0.02

i i i

a% PRESENT SIHd o - 0-233 a - 1.380

WORK HflHd 0.875 =0-870,

!

•

o

2

i

SPARROW AND BECKEY[4]" S/Hd Hf / Hd • - 0-?,5 0.875 • - 1-0 0.875

o A .

. j.

4

• "o ao

~ .

6

8

~

0

Ao a

a.

0

ReoxlO 3

Figure 12. Comparison of the present friction factor results with those of Sparrow and Beckey [4].

M

= 0.040[2.058

-

(c//nf)

0"313]

m = 0.075[3.40 - (C/Hf) °'711] N = 0.033[ 1.394 - (C/Hf) °'4°8] n = 0.13811.435 - (C/Hf) °773] Equation (15) satisfies the data within -I-5% for Reynolds numbers ranging from 3000 to 50,000. PRACTICAL SIGNIFICANCE The practical significance of the present work lies in the fact that for the first time, specific equations/correlations have been recommended for predicting the heat transfer and friction factor characteristics of shrouded fin arrays with uninterrupted fins. These correlations cover fin configurations with no clearance and also those configurations with clearance which are likely to be used in practice. CONCLUSIONS Based on the heat transfer tests, the Petukhov-Popov equation is found to be valid for the smooth rectangular duct with no fins and heating on one face. On the other hand, the D-B equation is found to be applicable for all the fin configurations with no clearance. It is also found to be applicable for configurations with clearance if the clearance-to-spacing ratio is small or the spacing-to-fin height ratio is large. The Prandtl equation for calculating the friction factor is found to be applicable to the smooth rectangular duct with no fins and shrouded fin arrays without clearance. However, for configurations "with clearance," values different from those of the Prandtl equation are obtained. These values have been correlated. The conclusions are based on experiments conducted with air as the working fluid. They are therefore valid only for fluids having Prandtl numbers close to the value for air. NOMENCLATURE A

C Cp

Oe Dhd

f Hd /-/, h Ah c Ah e

where

Acf

Acre

heat transfer surface area, m 2 cross-sectional area of a central channel, m 2

Ah m Ah s Ahnet tts rha nf Nu P

e~f Pale PH

Pr

aa Qin Qloss Re S T tf

P

w~ X

O~

p

cross-sectional area of an end channel, m 2 fin tip-to-shroud clearance, m specific heat of air, J / k g . °C equivalent diameter, 4 Acf/Pcf, m equivalent diameter of end channel, 4 Acre/Poee, m heating diameter, 4 Acf/P H, m fanning friction factor, Eq. (9), dimensionless duct height, m fin height, m heat transfer coefficient, W / m 2. K pressure drop due to sudden contraction, m of water column pressure drop due to sudden expansion, m o f water column measured pressure drop across the arrays, m of water column pressure drop occurring in the free zone before and after the fin array module, m of water column net pressure drop across the fin arrays, m of water column length of test section, m mass flow rate through fin array, k g / s number of fins Nusselt number, dimensionless power input, W wetted perimeter (central channel), m wetted perimeter (end channel), m heating perimeter, m Wd for smooth rectangular duct with no fins S + 2 H d for fin arrays without tip clearance S + 2Hf -t- tf for fin arrays with tip clearance Prandtl number, dimensionless enthalpy gain by flowing air, rhaCp(Tae - Tai), W net heat input, W total heat loss from the test section, W Reynolds number, dimensionless inter-fin spacing, m temperature, °C fin thickness, m average velocity, tha/[ pa{(nf - 1)A~f + 2A~f~}], m/s duct width, m distance from the leading edge of the fin array in flow direction, mm

Greek Symbols aspect ratio, H d / W d , dimensionless density, k g / m 3 Subscripts

a ae ai b bc be D Dh

air air exit air inlet base base central channel base end channel equivalent diameter D e as characteristic dimension heating diameter Dhd as characteristic dimension

396

S . B . T h o m b r e and S. P. Sukhatme f fc j s sc se ss w

fin fin, central c h a n n e l at j t h h e a t e r section shroud shroud, central c h a n n e l shroud, e n d c h a n n e l s h r o u d sides water REFERENCES

1. Sparrow, E. M., and Kadle, D. S., Effect of Tip to Shroud Clearance on Turbulent Heat Transfer from a Shrouded Longitudinal Fin Array, ASME J. Heat Transfer 108(3), 519-524, 1986. 2. Lau, K. S., and Mahajan, R. L., Convective Heat Transfer from Longitudinal Fin Arrays in the Entry Region of Turbulent Flow, Presented at the winter annual meeting, Chicago, ASME Paper No. 88 WA-EEP 1, 1988. 3. Lau, K. S., and Mahajan, R. L., Effects of Tip Clearance and Fin Density on the Performance of Heat Sinks for VLSI Packages,

4.

5.

6. 7. 8. 9.

IEEE Trans. Comp. Hybrids Manufact. Technol. 12(4), 757-765, 1989. Sparrow, E. M., and Beckey, T. J., Pressure Drop Characteristics for a Shrouded Longitudinal Fin Array with Tip Clearance, ASME J. Heat Transfer 103(2), 393-395, 1981. Thombre, S. B., Experimental Studies on Turbulent Heat Transfer from Shrouded Fin Arrays, Ph.D. Thesis, Energy Systems Engrg., Ind. Inst. Tech., Bombay, India 1993. Shepherd, D. G., Elements of Fluid Mechanics, 1st ed., Harcourt, Brace and World, New York, 1965. Sukhatme, S. P., A Text Book on Heat Transfer, 3rd ed., Orient Longman, Bombay, 1989. Sukhatme, S. P., Solar Energy Principles of Thermal Collection and Storage, 2nd ed., Tata-McGraw Hill, New Delhi, 1985. Garcia, A., and Sparrow, E. M., Turbulent Heat Transfer Downstream of a Contraction-Related, Forward Facing Step in a Duct, ASME J. Heat Transfer 109(3), 621-626, 1987.

Received May 17, 1994; accepted June 10, 1994