Experimental study on the forced convective flow in a channel with heated blocks in tandem

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Experimental study on the forced convective ¯ow in a channel with

heated blocks in tandem

Yau-Ming Chen

*

_{, Kun-Chieh Wang}

Department of Mechanical Engineering, National Taiwan University, Taipei, Taiwan 10764, ROC Received 7 February 1996; received in revised form 5 July 1996; accepted 11 September 1997

Abstract

This study experimentally examines the forced convective ¯ow over two sequentially heated blocks mounted on one principal wall of a channel. The experiments, involving mass transfer, were carried out via the naphthalene sublimation technique (NST). By virtue of the analogy between heat and mass transfer, the results can then be converted to determine the heat transfer. In the experiments, the block spacings were set at 2, 4, 6, 8, 12, 16, and 22 and the Reynolds numbers were set at 1300 and 104_{which correspond to the}

laminar and the turbulent convective ¯ow cases, respectively. Results show that the Sherwood number increases or decreases mono-tonically along the block surfaces in the laminar convection cases; while the hump and sharp increase in the Sherwood number can be found in the turbulent convection cases. This is attributed to the reattachment of the separating bubble and the ¯ow impinge-ment, respectively. Comparison between the experimental and numerical results is made and the eect of the block spacing on heat transfer is discussed. Ó 1998 Elsevier Science Inc. All rights reserved.

Keywords: Forced convection; Naphthalene sublimation technique; Reattachment; Separating bubble; Flow impingement

1. Introduction

The heat transfer and ¯ow characteristics in a chan-nel or duct with heated blocks have in the past been the subject of numerous investigations. This is partly be-cause of its relevance in a multitude of engineering appli-cations and partly because these simple geometries lend themselves well as a proving stone on which theoretical or numerical models can be tested. A better understand-ing of the detailed ¯ow and thermal characteristics of discrete heated protrusions in forced convection situa-tions is therefore highly desirable from both practical and fundamental points of view. However, the existing literature which shows an insight into the detailed heat transfer and ¯ow behavior is still limited. Previous stud-ies pertinent to this work are brie¯y summarized below. Considering the cold ¯ow predictions, many inv-estigations of turbulent recirculating ¯ows in the past have been concerned principally with testing turbulence models, particularly the k± model [1±4]. Generally speaking, the k± model was found to be a viable alter-native to the more complex schemes involving higher

or-der turbulence closures. Because of its successful performance, as well as its relative simplicity and adapt-ability, this model is, arguably, the most popular one used in numerical predictions of turbulent ¯ow and heat transfer today. Besides, it is noted that, in the predic-tions of turbulent convective heat transfer problems, the accuracy of calculation results primarily depend up-on the ability to analyze the velocity and the tempera-ture ®elds in the near wall region. Jones and Launder [5] suggested that, for accuracy and width of applicabil-ity, a ®ne-grid, low-Reynolds-number treatment should be employed near the wall in place of wall functions, de-spite the attractive simplicity of the latter approach.

Heat transfer predictions for discrete heated protru-sions in forced convection situations have also been treated to some extent in the literature. In the laminar forced convection studies, Davalath and Bayazitoglu [6] numerically studied the conjugate heat transfer for the two-dimensional developing ¯ow over a sequence of three rectangular heat generating blocks placed in a parallel plate channel. Similar studies were conducted by Zebib and Wo [7]. Hsieh and Huang [8] studied the separated laminar forced convection near surface-mounted ribs numerically and gave a correlating equa-tion of average Nusselt number as a funcequa-tion of the

*_{Corresponding author. Tel.: +886 2 3634701; fax: +886 2 3631755.}

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Reynolds number and the rib width. As to the turbulent forced convective ¯ows, only a few studies could be found. Knight and Crawford [9] made a simulation of convective heat transfer in channels with periodically varying cross-sectional area. Wietrzak and Poulikakos [10] carried out a numerical study on the turbulent forced convective cooling of a heated rectangular block. In their work, a low-Reynolds-number version of the k± model was used to predict thermal and ¯ow ®elds. Unfortunately, there was no experimental data for com-parison with their numerical predictions.

In the past studies of the ¯ow ®eld measurement, a large proportion of the available experimental work was restricted to the measurement of integral quantities, such as the length of the recirculation region or the mean pressure coecient. Recently, detailed measure-ments of the ¯ow ®eld have been carried out by some workers [11±14] who used the Laser±Doppler velocime-ter to obtain the velocity distributions and other related quantities.

Similar to the large numbers of studies on the integral quantities found in the ¯ow ®eld measurements, there were also many published papers studying the average heat transfer coecient for blocks in tandem or prob-lems of the same kind [15±20]. It is worth noting that the traditional technique for measuring the local heat ¯ux and wall temperature distributions with heat ¯ux gages and thermocouples is not only dicult but also in-ecient in studies with such a con®guration. Sparrow et al. [21±24] made serial investigations on the heat transfer and pressure drop characteristics of arrays of lar modules deployed along one wall of a ¯at rectangu-lar duct via the naphthalene sublimation technique. In their studies, however, only module-averaged heat/mass transfer coecients were obtained.

The brief literature survey made above clearly indi-cates the lack of experimental studies on the laminar and the turbulent convective ¯ow over heated blocks in a channel. Even the most basic information such as the local velocity, the turbulent kinetic energy, and the heat transfer coecient around the heated blocks is lacking. Therefore, the present study attempts to exper-imentally characterize the laminar and the turbulent convective ¯ows over protruding heated blocks in tan-dem in a smooth channel. Detailed measurement of heat/mass transfer coecients were performed using the NST (Naphthalene sublimation technique). Numer-ical results including both the laminar and the turbulent conditions were also conducted for comparison and un-derstanding of the ¯uid ¯ow and heat transfer mecha-nisms.

2. Experimental apparatus and measuring technique The apparatus used in this study is comprised of three major components: the test blocks, the wind tun-nel, and the automated data acquisition system. This ap-paratus was designed to measure the mass transfer coecient from the surfaces of the blocks.

2.1. The test blocks

Each test block was made of chromium coated deep cooled steel with dimensions of 216 mm ´ 10 mm ´ 16 mm. Details of the block geometry can be found in [25]. A recess about 2 mm deep was machined into the outer surfaces of the block in the region where naphtha-lene cover was desired. The naphthanaphtha-lene layer extended about 150 mm near the central portion of the block. The rest of the area adjacent to the ends was left uncoated. The test block served as a part of the four-piece mold. The other pieces of the mold were fabricated to match the test block precisely and its surface was polished to a mirror-like ®nish. Six thermocouples were used for temperature measurements on the test block. They were installed at the central region of the block and located about 1 mm below the outer surface of the naphthalene layer. Three thermocouples were positioned along the block and the other three were located around its cir-cumference. These thermocouples monitored the tem-perature during the wind tunnel runs as well as during the casting processes.

2.2. The wind tunnel

The experiments were performed in a suction-type wind tunnel as shown in Fig. 1(a). The entrance section of the tunnel contained a honey-comb ¯ow straightener and dampening screens before a 10 to 1 contraction sec-tion. This was followed by a 4 m long ´ 220 mm wide ´ 20 mm high rectangular test section. The ®rst test block was located 2.6 m downstream from the chan-nel inlet, which corresponds to about 71 hydraulic diam-eters. Each test block was inserted in a 10 mm wide ´ 11 mm high slot on the bottom surface of the test section. The naphthalene covered portion extended from 2 mm below the tunnel bottom wall to 5 mm above it.

The bi-dimensional characteristic of the oncoming ¯ow was con®rmed in more than 85% of the spanwise direction. Flow stability in long duration operations was within 1.5%. The laser Doppler velocimeter was comprised of a 3 W Argon laser and a transmitting op-tical arrangement based on an acousto-optic cell. The arrangement is sketched in Fig. 1(b). The optical char-acteristics of the velocimeter can be found in [25]. The light scattered from the seeding particles was collected in a backward scattering con®guration. The Doppler signal was processed by a frequency counter (TSI 1990C) interfaced to a personal computer. The optical components were arranged on a milling table which al-lowed translation in three directions with accuracy bet-ter than 0.1 mm. The corresponding errors were estimated to be within 2.8% in the velocity measure-ment, 5% in the turbulence intensity measuremeasure-ment, and 3.6% in the Reynolds number measurement. 2.3. Automated data acquisition system

The automated data acquisition system was used to measure the local sublimation depth. This system

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con-sists of a depth gauge with a signal conditioner, a digital multimeter, two stepping motors, a motor controller unit, and a personal computer. A linear variable dier-ential transformer (LVDT) gauge, having a 2 mm oper-ation range and 0.1 lm resolution, was connected to a signal conditioner. A digital multimeter acquired the sig-nal from the sigsig-nal conditioner and sent it to the person-al computer for data reduction and storage. The personal computer and the motor controller unit con-trolled the stepping motors to position the LVDT sen-sor. The stepping motors were able to move the sensor in 1 lm increments.

2.4. Data reduction

The detailed experimental procedures can be found in [25]. The local mass transfer coecient is de®ned as hm_q lsbqs

nwÿ qnb; 1

where lsbis the rate of change in naphthalene thickness

due to sublimation, qs is the density of the solid

naph-thalene, q_nwis the naphthalene vapor density on the

sur-face, and qnb is the bulk naphthalene vapor density in

the channel. The measurement of lsbis as follows. Before Fig. 1. Wind tunnel and ®ber-LDV system: (a) Wind tunnel; (b) Fiber- LDV system.

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experiments, the thickness of the naphthalene cover was measured. After the experiments, the thickness of the naphthalene cover was measured at the same location. The depth of the naphthalene cover due to free convec-tion subtracted from the dierence between these two thicknesses is the net sublimation thickness due to forced convection. The lsbis obtained from this net sub-limation thickness divided by the measurement time.

The naphthalene vapor density (qnw) can be determined

using the ideal gas law p qRT together with the empir-ical equation of Ambrose et al. [26]. The results are ex-pressed in terms of the Sherwood number as

Sh hmh=D, where h is the height of the block and D

is the mass diusion coecient of naphthalene in the air, which can be determined from Mack's measurement data [27], and as also suggested by Goldstein et al. [28]. The positioning and surface elevation readings using the computer controlled measurement system were found to be very repeatable. Uncertainty analysis according to Kline and McClintock [29] revealed that the estimated errors of the Sherwood numbers were within 7% in the entire range of our measurements.

3. Theoretical treatment 3.1. Governing equations

The con®guration of the problem under consider-ation is depicted in Fig. 2. In the laminar convective ¯ow cases, the ¯uid enters with a parabolic pro®le from one end and leaves at the other end of the plates carry-ing the heat dissipated by the blocks. The governcarry-ing equations used are the steady two-dimensional Navier± Stokes equations for the incompressible elliptic ¯ow. It is worth noting that, in the energy equation,

U@T @X V @T @Y as af 1 RePr @2_T @X2 @2_T @Y2 ; 2

as=af means the ratio of thermal diusivity of a solid

block to that of the ¯uid. In the ¯uid domain, as=af is set to one. In the solid region, as=af is set to a very large value. This makes the terms on the left-hand side of Eq. (2), which represent the thermal convection eects, be neglected and Eq. (2) becomes a simple conduction equation.

In the turbulent convective ¯ow cases, the ¯ow is considered to be steady, incompressible, and two dimen-sional. The time-averaged Navier±Stokes equation in conjunction with the k±, low-Reynolds-number closure model proposed by Jones and Launder [5] are used in

this study. And, the treatment of as=af in the energy

equation is the same as that in the laminar case. 3.2. Boundary conditions

In this study, the inlet conditions of the axial velocity in the laminar convective ¯ow cases; and the axial veloc-ity, the turbulent kinetic energy, and the dissipation rate in the turbulent convective ¯ow cases were obtained by theoretical calculations. This is dierent from the given semi-empirical or measured inlet conditions which are commonly used in the literature. The adoption of the calculated inlet conditions can reduce the ambiguity coming from the dierent speci®cations for the inlet pro-®les of U, k and . In the laminar convection cases, the inlet axial velocity is obtained by solving the fully devel-oped forms of the Navier±Stokes equations. The inlet temperature is assumed uniform. In the turbulent con-vection cases, we ®rst consider an uniform ¯ow in a long smooth parallel channel. Assuming the outlet conditions for U, k and are fully developed, the governing equa-tions can be simpli®ed to the fully developed forms. In solving the simpli®ed equations in each iteration, the pressure gradient is set to be able to adjust itself to ®t the mass conservation and enable the near wall velocity to satisfy the log-law distribution. After about 200±350 iterations (depending on the Reynolds number), the ful-ly developed pro®les for U, k and are acquired. The calculated inlet pro®les of these variables are shown as

an example for Re 104 _{in Fig. 3. For comparison,}

the axial velocity (U), obtained by LDV, is also includ-ed. Good agreement can be found between them.

Fluid on the channel walls and block surfaces satis-®es the no-slip boundary conditions. Both the upper and the lower walls of the channel were kept adiabatic. The block bases were maintained at a constant

temper-ature TB. Hydrodynamically and thermally fully

devel-oped convective ¯ow are assumed at the outlet of the channel. At the ¯uid±block interface,

ks kf @T @n s @T_@n f ; Ts Tf: 3

In Eq. (3), the thermal conductivity of block ksis set to a very large value to ensure constant wall temperature on the block surface. The distance between the inlet and the ®rst block (L1) is set as 3H and the distance between the second block and the outlet (L2) is chosen as 27H in both the laminar and the turbulent convective ¯ow Fig. 2. Physical model and con®guration of ¯ow ®eld.

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cases. The chosen distance L2 is considered to be

ade-quate, since no noticeable change for Lr2and Nu could

be found in the numerical tests as L2is higher than 27H. 3.3. Numerical scheme

The governing equations and boundary conditions were made discrete by a control ± volume, ®nite ± dier-ence method. The numerical solution was obtained by using the SIMPLER algorithm [30]. The convective and diusive ¯uxes were approximated using the pow-er-law scheme which is a more judicious approximation than either a hybrid or an upwind scheme. The discrete conservation equations were solved by the TDMA and the line-by-line iteration method. The use of these schemes was necessary to obtain a faster convergence rate. The under-relaxation factor was carefully chosen to prevent large variations in the source terms. The so-lution was assumed to be converged when the following criterion was satis®ed.

Pm

j1

Pn

i1j/k1i;j ÿ /ki;jj

P_m

j1

P_n

i1j/ki;jj

6 n; 4

where k 1 and k respectively denote values in two con-secutive iterations, / represents the dependent variables of U, V, P, k, , and T; and n is a prescribed error. Here, we select n 10ÿ4 _{for p U, V, and k; and n 10}ÿ5_for / T . A nonuniform grid was used in both the vertical and the horizontal directions. Tests of various nonuni-form grid systems for the laminar convection and the turbulent convection cases were made for 63 21; 75 27; and87 33; and140 50; 160 70; and180 90, respectively. And, the choices of the grid distribu-tions of 75 27and160 70 are sucient to provide the grid-independent average Nusselt number for overall faces of the blocks in the laminar and the turbulent

con-vection cases, respectively. Further increasing the grid densities to 87 33 and 180 90 led to deviations of the overall average Nusselt number with the formers within 1% and 5%, respectively. The calculations for the laminar and the turbulent convective ¯ows were per-formed on the PC and the HP computer system (9000 se-ries Model 735), respectively. Typically, it takes about 1000±1500 and 2500±3000 iterations to reach conver-gence in the laminar and turbulent cases, respectively. The numerical code for the turbulent convective ¯ow cases was tested against previous studies for turbulent ¯ow over a backward facing step [31]. The calculated re-attachment length behind the step by this code is 6.5h (h is the step height) and it is in good agreement with the experimental results as documented in [32]. The heat transfer results are presented in terms of the local Nu-sselt number at the block surface, de®ned as

Nu @h@n ÿ

w

hw 5

where n is the direction normal to the block surface. The local temperature gradient in Eq. (5) was evaluated by using a three-point Taylor series expansion method. 4. Results and discussion

In the experiments, the block spacing s changes from 2 to 24 for both the laminar and the turbulent convective ¯ow cases. For comparison with numerical results, the test conditions for the base cases are at Re 1300 and

s/h 4 and Re 104 _{and s/h 4 for the laminar and}

the turbulent ¯ows, respectively.

4.1. The streamline and isotherm patterns

For the laminar convection cases, the calculated ¯ow patterns for Re 1300 and s/h 4 are shown in Fig. 4(a). It is seen that there occur separating bubbles on the front and rear faces of the blocks. There is no seperation phenomenon found on the top faces of the blocks. Isotherm patterns shown in Fig. 4(b) demon-strate that dense isotherms occur near the front and the top faces of the blocks, especially near the leading and trailing corners. Better heat transfer can be expected there. On the contrary, sparse isotherms appear near the rear faces of the blocks, indicating worse heat transfer from the blocks there. For the turbulent convection

cases, the calculated ¯ow patterns for Re 104 _{and s/}

h 4 are demonstrated in Fig. 5(a). Due to the obstruc-tion of the ®rst block, the streamline begins de¯ecobstruc-tion at about 4h upstream from the ®rst block and thus the ®rst separating bubble forms. This bubble is located at the bottom-front corner of the ®rst block, and it has a length of 1.4h on the lower wall of the channel and a height of 0.54h on the front face of the ®rst block. As the ¯uid turns upward into the narrow gap between the top face of the ®rst block and the upper wall of the channel, the ¯uid is drastically accelerated due to the contraction eect. Meanwhile, because of the large Fig. 3. Inlet pro®les of velocity, turbulent kinetic energy, and

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increasing adverse pressure gradient created by the ac-celerating ¯uid, the near-wall ¯uid cannot aord to de-velop, and thus a small separation bubble is induced. This separating bubble starts at the front corner of the ®rst block, extends along the top face of the ®rst block, and then reattaches to it. This top separating bubble has a reattachment length of 1.1h. Accompanied by the ¯uid ¯ow out from the tunnel above the ®rst block, the ¯uid will directly impinge on the front face of the second block, and there forms a big recirculation region be-tween blocks. It is noted that, on the top face of the

sec-ond block, there is no separating bubble found. Behind the second block, the static pressure increases rapidly due to the expansion eect. The accelerating ¯uid can-not aord enough axial momentum to overcome the pressure lift in the transverse direction. Thus, a big recir-culation forms behind the second block. The reattach-ment length of this separation region is about 5h and the ¯uid in it circulates in the clockwise direction. Fur-ther downstream, the ¯uid redevelops and the ¯ow grad-ually recovers into the unidirectional condition to the outlet of the channel. It requires about a length of 15h to attain this fully developed condition. Although the sizes of the existing separating bubbles are all smaller for the turbulent cases than that for the laminar cases, the eects of them on the heat transfer behavior are to-tally dierent. This will be discussed later. Isotherm pat-terns in Fig. 5(b) also indicate similar facts about heat transfer as that found in the laminar case. However, the distribution of more concentrating isotherms around the blocks indicates a much higher heat transfer from the blocks for turbulent convection cases than for lami-nar ones.

4.2. Comparison between calculation and LDV measure-ment

The calculated U and k in the channel with two sur-face-mounted blocks versus Y =H coordinates at various axial stations are plotted in Fig. 6(a) and (b), respective-ly. For comparison, the measured U is also depicted. It is worth noting that the isotropic assumption has been

employed to obtain the measured k 1:5u02_since

de-tailed measurements of the turbulence intensity were only made for the axial component. In general, it is seen that the computed results agree well with the measured ones. The negative velocity near the wall around X ÿ0:3H indicates the existence of the small separa-tion region at the concave corner of the abrupt contrac-tion formed by the ®rst block. The calculated distance between the separation point and the ®rst block is 1:4h0:35H. However, this cannot be con®rmed be-cause the seperation zone is too small and too close to the wall to be measured by LDV. Fig. 6(a) indicates that velocity pro®les upstream from the ®rst block are faith-fully predicted. Over and behind the ®rst and the second block, the calculated axial velocity pro®les start to devi-ate from those measured. The same phenomenon was also indicated by Durst and Rastogi [1]. The calculated negative velocity adjacent to the top wall of the ®rst block indicates the occurrence of ¯ow separation at the convex corner of the abrupt contraction. The dis-tances between the separation point and the reattach-ment point are 1.1h and 1.2h according to the computation and the experiment, respectively. In addi-tion, both calculated and measured results show the ve-locity overshoot near the top face, which comes from the eects of the abrupt contraction. This velocity overshoot is gradually diminishing from X 0:1H to X 0:5H due to the viscous eect. Downstream behind the ®rst block, the ®nding of negative and zero velocities be-Fig. 5. The streamline and isotherm patterns for Re 104_{and s/h 4:}

(a) The streamline pattern; (b) The isotherm pattern.

Fig. 4. The streamline and isotherm patterns for Re 1300 and s/h 4: (a) The streamline pattern; (b) The isotherm pattern.

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tween two blocks indicates the existence of a big recircu-lation region and the dividing separation line. Upon the top face of the second block, there is no negative veloc-ity found, indicating the absence of the separating bub-ble. The experimental results also show the same fact. Further downstream behind the second block, the veloc-ity pro®les look like that of a backward-facing-step ¯ow. The calculated and measured reattachment lengths are 5:23h and 6:3h, respectively. The numerical results seem to overpredict the velocity in the downstream recircula-tion region behind the second block. After the reattach-ment point, the ¯ow starts to redevelop and a new developing shear layer is gradually spreading into the old shear layer which originated from the second block. As X P 22H, the local axial velocity tends to distribute fully developedly till the outlet. The above observations may suggest that the low-Reynolds-number model should be improved in order to achieve a better predi-tion for separated ¯ows.

As to the turbulent kinetic energy distribution (k), from Fig. 6(b), it is seen that k has two local peak values (symmetrical to the centerline) near the channel walls in the inlet section. The peak value of k near the lower wall of the channel gradually increases in the axial direction, and its position gradually shifts towards the front corner of the ®rst block where maximum k occurs. Upon the top face of the ®rst block, the positions of the local max-imums at every section above the ®rst block are located just upon the top face. The values of these local maxi-mums are much larger than those before the ®rst block. Between blocks, the local maximums at dierent axial sections all occur around the separation line. On the top face of the second block, it is found that (1) all peak values for k are smaller than that on the top face of the ®rst block, and (2) the pro®les of k are much smoother than that on the top face of the ®rst block. Downstream behind the second block, the local maximums at

dier-ent axial sections all occur in the neighborhood of the dividing separation line, just as that behind the ®rst block. After about X 17H, k tends to distribute sym-metrically to the centerline, indicating the recovery of the fully developed condition. Overall, the degree of tur-bulence enhancement is high in the regions around the blocks' corners, the reattachment points, and the separa-tion lines.

Although the quantitative agreement between the calculated and the measured values is not expected, the qualitative agreement is worth noting. The discrep-ancy between the calculated and measured k occurs mainly in the near wall regions, around the separation lines, and in the separation regions. The calculated peak positions for k are located higher and happen earlier than the measured ones. It may be partly due to the ef-fect of anisotropy in these regions and partly due to the inadequacy of the turbulence model used in the predic-tion.

4.3. Comparison between calculation and NST measure-ment

In the experimental studies what we have obtained is the mass transfer coecient instead of the heat transfer coecient. In order to compare the experimental results with the theoretical ones, the analogy between the heat and mass transfer must be used. The Nusselt number distribution can be obtained via the following equation:

Nu Sh Pr_Sc

n

6 where the power index n, according to Igarashi [33], is approximately equal to 1/3. This value has also been used by many other investigators (Goldstein [28]). For laminar convection cases, The Nusselt number distribu-tions along the block surfaces for Re 1300 and s/h 4 Fig. 6. Distributions of velocity and turbulent kinetic energy along the channel for Re 104_{and s/h 4: (a) distribution of velocity; (b) distribution}

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are shown in Fig. 7. In this ®gure, the X0 _coordinate used includes the vertical surfaces of the blocks. Good agreement can be found between the experimental and predicted results. The maximum deviation is less than 9%. The Nusselt number shows a monotonically increas-ing or decreasincreas-ing distribution on every face of the two blocks. Due to the strong accelerating and wash-down eects at the front and rear corners of each block, the Nusselt number appears as large and small peaks there, respectively. Besides, the appearance of the separating bubbles on the front face of the ®rst block, between blocks, and behind the rear face of the second block seems to have no apparent contribution to the heat transfer from the blocks.

The local Nusselt number distributions for the turbu-lent convection case are plotted in Fig. 8. For compari-son, the numerical results are also included. As can be seen, good agreement can be found, especially on the front and top surfaces of the two blocks. However, dis-crepancy does occur on the rear faces of the two blocks. It is worth noting that the discrepancy between heat and mass transfer results on the rear face has also been re-ported in [25]. In this work, because the agreement be-tween the prediction and experiment on the front and top faces of the two blocks is quite good, the dierence occurring on the rear faces might be either attributed to the low-Reynolds-number model used or the analogy equation. The above observations suggest that better understanding of the ¯ow and heat transfer characteris-tics in a separating and reattaching ¯ow behind the block is essential. From Fig. 8, the Nusselt number humps are found on every face of the two blocks except the front face of the second block where a sharp increase in Nusselt number exists. In comparison to the stream-line patterns in Fig. 4, it is seen that (1) for the ®rst block, the reattachment of the separating bubble and

the Nusselt number hump both occur at about 0.5h and 1.1h on the front and top face, respectively; (2) be-tween the two blocks, there exists larger recirculating ve-locity near the middle section on the rear face of the ®rst block, in which the Nusselt number hump happens near-by. The ¯uid coming from the tunnel gap near the top wall of the ®rst block directly impinges on the front face of the second block. This impingement causes a sharp increase in the Nusselt number. It is noted that the im-pingement and the Nusselt number hump are both lo-cated near 0.9h on the front face of the second block; (3) for the second block, there is no separating bubble found on the top face. The Nusselt number decreases monotonically along the top face. Behind the block, there exists a big recirculation region. The Nusselt num-ber hump occurs around the middle section where larger recirculating velocity exists nearby, just as that on the rear face of the ®rst block.

Overall, the ¯ow impingement and the ¯uid reattach-ment are responsible for the occurrence of the sharp in-crease and hump of the Nusselt number.

4.4. Local Sherwood number distribution 4.4.1. Laminar convective ¯ow

The Sherwood number distribution along the sur-faces of the two blocks for the laminar convection cases (Re 1300, s/h 2±24) are shown in Fig. 9. All Sher-wood number distributions show monotonically increas-ing or decreasincreas-ing characteristics on every face of the blocks.

The ®rst block: On the front face, the Sherwood num-ber drastically increases from the front bottom to the maximum at the leading corner. Due to the strong accel-eration eect above the reattachment point (about 0.64h), the slope of the Sherwood number is larger than Fig. 8. Comparison between predicted and measured Nusselt number distributions for Re 104_{and s/h 4.}

Fig. 7. Comparison between predicted and measured Nusselt number distributions for Re 1300 and s/h 4.

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that under it. In the neighborhood of the front turning corner, the large contraction eect brings about the oc-currence of the strongest accelerating ¯ow and the Sher-wood number attains maximum there. Along the top face of the block, the Sherwood number decreases monotonically due to the gradually growing thermal boundary layer. Near the rear corner of the block, the wash-down eect of the ¯uid induces a slight climb of the Sherwood number. On the rear face of the block, the heat/mass transfer is governed by the cellular motion between blocks. Since the ¯uid is totally restricted and ¯ows slowly in this area, the Sherwood number decreas-es ®rst and keeps at a small constant value, which is much smaller than that on the other two faces. The in-crease in the block spacing seems to have no in¯uence on the distribution of the Sherwood number.

The second block: On every face of the second block, the Sherwood number increases or decreases monotoni-cally, which is similar to but smaller than that of the ®rst

block. The smaller values of the Sherwood number are due to less ¯uid momentum and weaker convective ef-fects around the second block than the ®rst block. In-creasing the block spacing only makes a slight increase in the Sherwood number on the front face of the second block.

4.4.2. Turbulent convective ¯ow

The Sherwood number distributions along the sur-faces of the blocks under dierent block spacings for

Re 104 _{are shown in Fig. 10. The major dierences}

in the turbulent convection from the laminar one are the occurrence of the hump and the sharp increase in the Sherwood number which are attributed to the reat-tachment of separating bubbles and ¯uid impingement, respectively.

The ®rst block: Along the front face, the Sherwood number increases from the concave corner to the convex corner with a hump in the near-middle section.

Accord-Fig. 10. Eect of the block spacing on the Sherwood number distribu-tion for Re 104_.

Fig. 9. Eect of the block spacing on the Sherwood number distribu-tion for Re 1300.

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ing to previous discussions, the occurrence of the Sher-wood number hump primarily comes from the eect of the ¯ow reattachment of the separating bubble. Exami-nation on the maps at dierent block spacings indicates that the hump positions as well as the magnitude of the Sherwood number seems to be little aected by the block spacing. Along the top face, all the Sherwood number curves show the same features for dierent block spac-ings. The maximum occurs at the leading corner where ¯uid turns sharply. Downstream there is a low heat transfer rate in the recirculation region, followed by a steep rise to the local maximum near the reattachment point. The Sherwood number hump occurs at the point of reattachment. The reattachment causes a local aug-mentation of the Sherwood number by a factor of about two, as compared to the local minimum on the same face. Increasing the block spacing from 4 to 22 will lead to about 1/5 fold decrease in the maximum, and about 1/ 5 fold decrease in the minimum, and 0.5h shift to the right for the Sherwood number hump. On the rear face, the Sherwood number is much less than that on the oth-er two faces. But, thoth-ere still exists a hump at the middle position due to the recirculating and the ¯uctuating ¯ow there. The Sherwood number increases slightly with the increasing block spacing.

The Second block: On the top face, the Sherwood number distributions show a great dierence between s=h P 12 and s=h 6 8. When s=h 6 8, the ¯uid coming from the contraction gap upon the top face of the ®rst block ¯ows along the separation line and impinges on the front face of the second block. This eect leads to a sharp increase in the Sherwood number. Above the impingement point, the Sherwood number slightly de-creases because of the slow buer ¯uid near this point and then drastically increases to the maximum at the front leading corner because of the strong acceleration eect. As s=h P 8, larger block spacing brings about the occurrence of the Sherwood number hump instead of the sharp increase in the Sherwood number. The dis-tribution of the Sherwood number is qualitatively like that on the front face of the ®rst block. Further increase in the block spacing seems to have no in¯uence on the Sherwood number. On the top face, as s/h 4, the block spacing is too small to form a separating bubble. With-out the eect of ¯uctuation and reattachment of the sep-arating bubble, the Sherwood number decreases monotonically as that in the laminar convection cases. As s=h P 6, greater convection eect brings about the occurrence of the top separating bubble. The reattach-ment of this separating bubble causes the Sherwood number hump, just like that of the ®rst block. The hump position gradually shifts to the right and the Sherwood number gradually increases its value as the block spac-ing increases. It is noted that, for the second block, the hump positions of the Sherwood number on the

top face for s=h P 12 all concentrate near X0_{=h 10:2.}

This means that the top separating bubble of the second block has a constant size as s=h P 12. For the ®rst block, the hump positions of the Sherwood number on the top

face scatter around X0_{=h 1:5±2:0, indicating that the}

separating bubble changes with the block spacing. Obvi-ously, the ¯uid reattachment and ¯ow impingement greatly in¯uence the occurrence, the location, and even the magnitude of the Sherwood number hump in the turbulent convection cases. Comparing the Sherwood number distribution along the top faces of the ®rst and the second blocks under s=h P 12, one ®nds that they are about the same order but the hump position for the last is located further downstream from the lead-ing corner. On the rear face, the distribution of the Sher-wood number is qualitatively and quantitatively like that of the ®rst block. The Sherwood number slightly in-creases with increasing block spacing.

4.5. Average Sherwood number

The average Sherwood number on every face of the block can be obtained by integrating the local values along that face. The results of the average Sherwood number versus the block spacing for the laminar and the turbulent convective ¯ow cases are shown in Fig. 11 and discussed below.

Laminar convection cases: From Fig. 11, for

Re 1300, it is seen that Sh1 is almost independent of

s/h. Sh2 and Sht slightly increases with increasing s/h.

Changing s/h from 2 to 24 results in an increase of Sh2

from the minimum of 3.53 at s/h 2 to the maximum of 5.56 at s/h 24, about a 1.57 fold increase. For Sht, it increases with increasing s/h from the minimum of 5 at s/h 2 to the maximum of 6.17 at s/h 24, about a 1.24 fold increase. Obviously, larger block spacing

Fig. 11. The average Sherwood number versus the block spacing for Re 1300 and Re 104_.

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brings about better heat/mass transfer from the blocks.

An example shows that, for s/h 8 and s/h 16, Sh1

are about 1.4 and 1.26 folds of Sh2, respectively.

Sh1=Sh2is decreasing with s/h from 1.5 to 1.3 and holds at 1.2 as s=h P 16. For a given Re and s=h say Re 1300 and s/h 4, the front face of the ®rst block has the larg-est average Sherwood number, followed by the top face of the ®rst block, the front face of the second block, the rear face of the second block, and the rear face of the ®rst block. For the total heat/mass transfer amount from the blocks, the front, the top, and the rear faces of the ®rst block and those of the second block contrib-ute sequentially about 24%, 40%, 1%, 7%, 27%, and 1%. From the point of view of better heat transfer from the blocks, the Reynolds number and the block spacing should be designed as large as possible.

Turbulent convection cases: From Fig. 11, for Re 104_{, it is seen that Sh}

1 rapidly falls down ®rst for

s=h 4±6, slightly increases for s=h 6±8, decreases a little for s=h 8±16, and then linearly increases with in-creasing s/h as s=h P 16. It is noted that the variations of

Shtand Sh2with s/h are somewhat dierent from that of

Sh1. As s=h 6 8, the distributions of Sh1, Sh2, and Shtare similar to each other. But, as s=h > 8, Sh2 and Sht in-crease with increasing s/h. Generally speaking, larger block spacing brings about better heat/mass transfer from the blocks. For the ®rst or the second block, at any ®xed block spacing, the average Sherwood number on the front face is in general greater than those on the top and the rear faces. An example shows that, for

s/h 8 and s/h 16, Sh1 is about 1.2 and 1.02 folds of

Sh2, respectively. For Re 104 and s/h 4, the front

face of the ®rst block has the largest average Sherwood number, followed by the front face of the second block, the top face of the second block, the top face of the ®rst block, the rear face of the second block, and the rear face of the ®rst block. For the total heat/mass transfer amount from the blocks, the above faces contribute se-quentially about 17%, 32%, 8%, 13%, 24%, and 7%. From the point of view of better heat transfer from the blocks, the Reynolds number and the block spacing should be designed as large as possible.

5. Practical signi®cance

The measured local information for ¯uid ¯ow and heat transfer are of help both in the veri®cation of the numerical codes and in better understanding of the ¯ow and thermal characteristics in such a con®guration, which might be applicable to the analysis and design of devices or equipment, such as electronic circuit boards.

6. Conclusions

The laminar and the turbulent forced convective ¯ows over heated blocks in tandem in an adiabatic

chan-nel have been experimentally and numerically studied. The major conclusions are described as follows: 1. In the laminar convection cases, separating bubbles

seem to have no apparent eect on the heat transfer from blocks. The Nusselt number increases or de-creases monotonically along the surfaces of the blocks.

2. In the turbulent convection cases, the reattachment of the separating bubbles and the ¯ow impingement are responsible for the occurrence of the hump and the sharp increase of Nusselt number distribution, respec-tively.

3. In the turbulent convection cases, Detailed compari-son of the numerical solution shows good agreement with the LDV measurements near the front and the top recirculation regions. Deviation can be found in the recirculation regions between blocks and behind the second block. It is suggested that the low-Rey-nolds-number model should be improved in order to achieve a better prediction for separated ¯ows. 4. Comparison between the calculated and the measured

Nusselt number distributions show good agreement on all faces of the blocks in the laminar cases and on the front and the top faces of the blocks in the tur-bulent cases. Deviation can be found on the rear faces of the blocks in the turbulent cases, for which further study is highly needed.

5. The eect of the block spacing on the local heat trans-fer distribution is remarkable, especially in the turbu-lent ¯ow cases. Increasing the block spacing lead to a decrease ®rst and then an increase later in the average Nusselt number in the turbulent ¯ow, which is dier-ent from the cases in the laminar one.

Nomenclature

C1 model constant

C2 model constant

D mass transfer coecient, m2_/s

H channel height, m

h block height, convective heat transfer coecient;

m, J/s m2 _K

k turbulent kinetic energy, J

kf thermal conductivity of ¯uid, J/s m K

ks thermal conductivity of solid block, J/s m K

L characteristic length, m

Lc channel length, m

Lr reattachment length, m

Nu local Nusselt number, Eq. (5)

Re Reynolds number, U0H=m

s block spacing, m

T mean temperature, K or °C

T0 inlet uniform temperature K, °C

TB temperature at block base, K or °C

U mean horizontal velocity component, m/s

U0 uniform mean horizontal velocity component, m/s

V mean vertical velocity component, m/s

w block width, m

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Acknowledgements

Financial support for this work was provided by the National Science Council of the Republic of China un-der contract No. NSC 82-0401-E-002-133.

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X0 _{horizontal coordinate including block's surfaces,}

m

Y vertical coordinate, m

Greek letters

af thermal pseudo-diusivity of ¯uid, m2/s

as thermal pseudo-diusivity of solid, m2/s

h mean dimensionless temperature, T ÿ T0=TBÿ T0

dissipation rate of turbulent kinetic energy, m2_/s

Subscripts

in inlet

f front face of the block

n direction normal to the block surface

r rear face of the block

t overall faces of blocks

u top face of the block

w wall

Superscripts

0 _{the ®rst block}

00 _{the second block}

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