The role of pore size in heat transfer of oscillating liquid flow in metal foam

International Journal of Thermal Sciences 145 (2019) 105978

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International Journal of Thermal Sciences journal homepage: www.elsevier.com/locate/ijts

The role of pore size in heat transfer of oscillating liquid flow in metal foam a,*

b

c

Nihad Dukhan , Özer Bağcı , Altay Arbak a b c

T

Department of Mechanical Engineering, University of Detroit Mercy, Detroit, MI 48221, USA Flanders Research Institute for Agriculture, Fisheries and Food, 9000 Ghent, Belgium Makina Fakültesi, Istanbul Technical University, Gümüşsuyu, 34437, Istanbul, Turkey

ARTICLE INFO

ABSTRACT

Keywords: Oscillating Water Metal foam ppi Electronic cooling Regenerator

The huge accessible surface area, high conductivity make metal foam attractive for energy technologies, e.g., reactors, regenerators and Stirling engines. An experiment establishing the impact of pore density of metal foam on heat transfer caused by oscillating water flow through foam is described. Various pertinent parameters for two foams having 10 and 40 pores per inch (ppi), of commercial high-porosity metal foams were obtained in this experiments; and are compared to those of a 20-ppi results from the literature. Experimental runs were conducted at kinetic Reynolds numbers between 1875 and 9366. The wall temperature uniformity and cycleaveraged Nusselt number for the three pore densities were compared to their counterpart for steady-state heat transfer. The pressure drop in the three pore densities was measured and discussed along with the heat transfer results. The effect of pore density was evident in the cycle-average wall temperature and Nusselt number, with the 10-ppi foam outperforming the 20- and 40-ppi foams. The benefit of oscillating flow was seen in producing more uniform wall temperature. It seemed possible for steady-state flow to produce higher heat transfer rates compared to steady-state flow, if the flow displacements were not sufficiently long compared to the length of metal foam.

1. Introduction In addition to having large surface area and good fluid mixing, heat transfer can be further enhanced by employing oscillating flow in porous media. Heat transfer due to oscillating flow in porous media is present in many energy systems, e.g., normal and oscillating heat pipes, regenerators (for Stirling engines and cryocoolers), cooling designs of nuclear power plants and heat exchangers of pulse tubes. Some coolers of nuclear power plants, thermo-acoustic engines, magnetic refrigerators and reverse-flow reactors are other examples of energy devices having oscillating flow and heat transfer. In addition to high transport rates, oscillating heat transfer in porous media provides more uniform temperature distribution on a hot surface. Therefore, oscillating heat-transfer designs can be used to cool modern high-power electronic systems. The reliability and operational speed of these devices depend not only on their average temperature, but also on their temperature uniformity. Habibi et al. [1] investigated cooling of compact circuit boards using oscillating flow in porous media. Heat transfer due to oscillating flow in traditional porous media (packed spheres, granular beds, woven wire meshed and mesh screens) have received considerable attention [2–9]. Open-cell metal foams are a new class of porous media that enjoy high porosity and surface area *

density (specific surface or surface area per unit volume). The permeability of metal foams is relatively high compared to other porous media, while their effective thermal conductivity is very good. The internal structure of metal foams is composed of thin ligaments or struts forming a web of cells and windows (pores). This morphology generates severe mixing of fluids passing through the foams and enhances convection between the solid ligaments of the foam and the fluids. Important information regarding how metal foams are manufactured and their potential applications; as well as fundamentals regarding flow and heat transfer through them, have appeared in Ref. [10]. Unlike other more dense porous media, the entry length in metal foam is significant [11–14], which can further enhance heat transfer in the foam, when subject to oscillating flow. For heat transfer due to oscillating flow in metal foam, there is a limited number of published studies, e.g. Refs. [15–19]. All of these studies employed a gas, mostly air, as the working fluid. Unlike air, liquid flow in metal foam includes dispersion which augments heat transfer [20]. Recently, Bağcı et al. [21] presented heat transfer due to oscillating water flow in metal foam having a single pore density of 20 pores per inch (ppi). This investigation, was followed by another investigation on the same pore density by Dukhan et al. [22] elucidating the effect of flow frequency on heat transfer. To the knowledge of the authors, these two studies

Corresponding author. E-mail address: [email protected] (N. Dukhan).

https://doi.org/10.1016/j.ijthermalsci.2019.105978 Received 24 September 2018; Received in revised form 8 May 2019; Accepted 7 June 2019 1290-0729/ © 2019 Elsevier Masson SAS. All rights reserved.

International Journal of Thermal Sciences 145 (2019) 105978

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Nomenclature

A Ao Iuni L Nu ppi q” ReD Re t U z

max

Greek

cross-sectional area of test section (m2) Non-dimensional displacement index (for temperature uniformity) length of porous medium (m) local cycle-averaged Nusselt number number of pores per inch of foam heat flux (W.m−2) Reynolds number based on pipe diameter kinetic Reynolds number time (s) average velocity (m/s) maximum flow displacements (mm) coordinate along flow direction, distance from entrance

μ ρ

viscosity (Pa.s) angular frequency (rad/s) density (kg/m3)

Subscripts f i max min w z

employed oscillating flow of water in metal foam for the first time. More recently, Peng et al. [23] simulated oscillating heat transfer of water to compare the performance of regenerators made from spherical particles, wire screens and open-cell metal foam. Results showed that, in general, metal foam was the optimal choice for the matrix of the regenerator due to its high porosity and low pressure drop. For a gamma-type Stirling engine, a regenerator with porosity about 85% provided optimal performance, Gheith et al. [24]. Clearly, high-porosity metal foam is a prime candidate as core material for energy equipment utilizing oscillating flow. In metal foam, the strut (ligament) diameter and length change with changing pore density (pore size), at the same porosity. Changes in ligament dimeter provide different surface area per unit volume and alter the effective thermal conductivity of the foam-two key heat transfer key parameters. Varying the pore density of metal foam while keeping the porosity constant is similar to changing the mesh number of screens, at same the porosity, e.g. Lee et al. [25] and Xiao et al. [26]. Lee et al. [25] experimentally determined that for mesh screens having different mesh numbers with the same porosity (70%) in oscillating flow produced different pressure drop and heat transfer characteristics. Peng et al. [23] stated that foam regenerators can be superior to particle and wire-screen regenerators under the same size and operating conditions, by adjusting either the porosity and/or the pore density. The specific surface area of foam depends on both porosity and pore density, which are independent structural parameters [23]. Indeed, the specific surface area decreases with increase of porosity and decrease of pore density. The purpose of current study is to establish the effect of pore density (or pore size) on heat transfer caused by oscillating water flow in metal foam having the same porosity. In addition to affecting various heat transfer parameters, changing the pore size also affects pressure drop,

fluid inlet maximum minimum wall flow direction, local

transitions among flow regimes and permeability of the foam [27]. Moreover, Arbak et al. [28] has shown that the thermal entry length in open-cell metal foam having the same porosity, was inversely proportional to foam's pore density. As such, it is indeed impossible to predict heat transfer characteristics of oscillating water flow in metal foam based on a single pore density (20 ppi) of Bağcı et al. [29,30]. In the current study, experimental results of oscillating water heat transfer for pore densities 10 and 40 ppi of open-cell high-porosity commercial aluminum foam are obtained. The porosity of these two foams is the same as that of Bağcı et al. [29]. Therefore, the impact of foam's pore size on pertinent heat transfer parameters of oscillating liquid flow can be ascertained using the results of the current study and those in Ref. [29] for 20-ppi foam. Results from this study can enhance fundamental understanding of heat transfer in oscillating liquid flow in metal foam, and can be useful for validation of numerical and simulation work. They can also provide some input to heat transfer and energy designs employing metal foam with oscillating liquid flow. 2. Experiment Fig. 1 is a drawing of a typical test section used in this experiment. The test section was made from 6061-T6 aluminum alloy with commercially-available open-pore aluminum foam fused to the internal surface of the pipe by brazing, Fig. 2. The dip brazing technique is ideal for porous metals and it minimizes thermal contact resistance. The foam completely filled the 325-mm-long pipe, which had an inner diameter of 50.80 mm. The foam was manufactured by ERG Materials and Aerospace [31] under the trade name Duocel®. The manufacture maintains full and accessible morphological and physical properties of this foam on their web site [31]. One test section was filled with 10-ppi foam, while the other with 40-ppi foam. The porosity of each foam was

Fig. 1. Drawing of the experimental test section.

2

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provided the test section with a constant heat flux of 15,519 W/m2. For a desired run, the stroke length of the piston was set and the oscillation frequency was slowly increased to an aimed value. The oscillating heat transfer system was allowed to reach a quasi-steady state which took about 2.5 h. A data logger connected to sensors, reported signals to a Keithley 2700 XLINX data acquisition system installed in a computer. For each cycle, a signal came from a clicker that touched a bump on a flywheel, which was part of the oscillation generator. This particular signal allowed calculation of the angular speed. Depending on the run, the sampling rate varied between 10 and 30 readings per second due to device limitations. Several runs were taken in the line frequency range 5 and 15 Hz, for stroke lengths 130 mm, 170 mm and 195 mm. Higher line frequencies caused mechanical vibrations and some buckling of the supporting metal frame of the test system, while lower line frequencies caused overheating of the motoreductor. The minimum sampling rate was 2.42 Hz, which was more than three times higher than the highest oscillation frequency. Therefore the data logger was able to capture the actual frequency of the system and the temperature amplitude. When changing frequency, the system took about 30 min to reach new steady periodic conditions. For various stroke-frequency combinations, static pressure at each side of the test section was measured. Keller models PR-23R and PA21Y piezoresistive transmitters were used. PR-23R had a pressure measurement range up to 100 mbar, whereas the maximum measured value for PA-21Y was 2.5 bars. The specific model was selected depending on the pressure level of a given run. The transmitters provided analog signals between 4 and 20 mA to a data logger at a sampling rate of 25 Hz. The actual static pressures were calculated by using a linear relationship between pressures and current readings. The transmitters had been calibrated by the manufacturer.

Fig. 2. Cross-sectional view of test section.

88.5%. This porosity was found by first weighing both specimens and estimating the masses of the porous cores to be 0.207 and 0.205 kg, respectively. Then those values were divided by the density of 6101-T6 alloy to find the solid volume. The result was divided by the bulk volume of the core, and then subtracted from one to find the porosity as a fraction. For measuring the wall temperature along the test section, a series of 1-mm holes were bored along the aluminum pipe at a depth of 4 mm. The holes were 10 mm apart. Each hole housed the bead of a type-K thermocouple and was plugged with thermal glue. A 1780-Watt surface heater covered the foam-filled pipe. To provide DC power to the heater, two power supplies were used. The test section was insulated by 20-mm thick ceramic fiber sheet. To assess heat loss from the external surface of the insulation, unidirectional steady-state heat transfer runs were performed. A heat loss of approximately 8.2% was calculated. The inlet and outlet water temperatures were measured in Polyethylene tubes connected at the two sides of the test section. Five thermocouples spanning the diameter of the cross section were placed in each tube to measure the water temperatures. The Polyethylene tubes were connected to 32-mm stainless steel pipes, which were coupled to an oscillating flow apparatus via plastic hoses. Two 225-W cooling thermostats (LAUDA Alpha RA 8) removed heat from the steel pipes. A 7.5-kW motoreductor operating at 69.7 rpm drove a double-acting hydraulic cylinder to provide oscillating flow. A variable-speed AC drive operating in the range 7–21 rpm drove the motoreductor. A 32mm piston reciprocated in the cylinder and provided tidal displacements of water. The piston was mechanically-linked to the motoreductor via a crank rod. The maximum stroke of the piston was 200 mm, and it could be adjusted using a flywheel with several settings. In order to completely fill the system with water, vacuum at absolute pressure of 80 mbar was pulled, then filtered water was introduced into the system from the two ends of the piston simultaneously. A purger was used to remove any accidental air bubbles, by careful maneuvering of the whole system, or if needed, by repeated dismantling and reassembling parts of the set-up. Once fully charged, the system was connected to the oscillating flow generator. The water coolers were turned on and were allowed to reach their steady operational state, at which they could provide cooling water at 17 °C, approximately. At this state, the surface heater was turned on and

3. Data reduction For water as an incompressible fluid, continuity dictates that the maximum fluid displacement, xmax, is related to the displacement of the piston through the cross-sectional areas of the piston and the test section:

x max =

2RAp

(1)

A

where R is the flywheel radius and Ap and A are the cross-sectional areas of the piston and metal-foam pipe, respectively. For the three strokes of the current study, the resulting values of xmax are 74.4 mm, 97.2 mm and 111.5 mm. The dimensionless displacement is

Ao =

x max D

(2)

where D is the foam pipe inner diameter. At each displacement, flow frequency which was linearly proportional to the line frequency, changed from 0.116 to 0.348 Hz. The fluid displacement x m changed according to

x m (t ) =

x max (1 2

cos t )

(3)

where angular frequency and t is time. The maximum fluid velocity is calculated as

u max =

(4)

x max /2

The average fluid velocity through the foam is

u (t ) = umax sin

t

(5)

The kinetic Reynolds number is defined as

3

Re =

D2 µ

where

and µ are the density and viscosity of water.

(6)

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4. Uncertainty analysis

ppi foam it is only 122 mm. In other words, the thermal entry length for 10-ppi foam is about 23% longer than for 40-ppi foam. So in oscillating flow, the 10-ppi provides more cooling, and produces lower wall temperature as compared to the 40-ppi foam. Typical periodic behavior of the wall temperature at three locations along the test section is shown in Fig. 4. The instantaneous velocity is plotted for reference. Due to symmetry, the wall temperature at the center z/D = 3.20 does not change with time. The wall temperatures are seen to be in phase with the velocity. For both foams, the amplitude of the wall temperature is higher at 1.62 diameters compared to 0.04 diameters, since the latter is close to the entrance of the foam, where a thermal development region exists. The amplitude of the wall temperature at a given location is lower for the case of 10-ppi foam compared to the 40-ppi, which suggests that the former provides more cooling than the latter. This will be further assessed in conjunction with other heat transfer results given below. The periodic behavior for the two foams was present for other displacements and frequencies, with the period being smaller at higher frequencies. No phase shift is present between the wall temperature and the velocity. Similar trends were observed for 20-ppi foam having the same porosity [29,30]. Other heat transfer studies of oscillating air flow in metal and graphite foams [15,17,18] did not report any phase shift between velocity and wall temperature. For other porous media, however, such phase shifts were present. For example, 3-mm packed steel spheres subjected to oscillating water flow experienced a phase shift of π radians between the velocity and wall temperature [2]. This may be due to the large difference in porosity and thermal conductivity of the steel spheres of [2] and the aluminum foams of the current study. Phase shifts were also reported for heat transfer due to oscillating helium flow in a 75%-porous medium composed of square cylinders [3].

Uncertainty in calculated quantities had contributions from directly measured values that were used in computing each calculated quantity, and is determined according to the method outlined in Ref. [32]. For example, the local cycle-averaged Nusselt number is calculated from

Nu =

qD kf (Tw Ti )

(7)

where q is the heat flux, D is the foam pipe inner diameter, kf is the fluid conductivity, Tw is the local wall temperature and Ti is the inlet fluid temperature. Therefore, the uncertainty in Nu has contributions from measurements errors in q , D , kf , Tw and Ti : Nu

Nu

=±

q

q

2

+

D

D

2

+

kf

kf

2

+

Tw

Tw

2

+

Ti

2

Ti

(8)

where is the uncertainty. The uncertainty in diameter of the metal foam tube was 0.04%. As for the temperature, the uncertainty included error in the reading of ± 1.1 °C or 0.4%, whichever was greater, as provided by the manufacturer of the measuring devices. As for the heat flux, which was calculated based on the voltage and current readings, the following was implemented. The accuracies of the voltage and current digital readouts were 0.1 V and 0.1 Amp, respectively. The uncertainty in the outside surface area of the test section (covered by the heater) was 0.039%. The uncertainty in the heat flux was obtained in a manner similar to Eq. (8), and it happened to be much less than 10%. Nonetheless, an uncertainty in the heat flux of 10% was assumed, as to not underestimate the uncertainty in Nu. The effective thermal conductivity was calculated according to Calmidi, R. L. Mahajan [33], who stated that their method had less than 10% error. Hence, the uncertainty in the effective conductivity kf was taken as 10%. Equation (8) gave an uncertainty of 14.7% in Nu. The uncertainties in the angular frequency and maximum fluid displacement were calculated as 0.43% and 0.51%, respectively, using the same method. The uncertainty in the pressure readings was ± 1% of the full scale of the particular pressure transmitter. Repeatability, hysteresis and linearity were accounted for in the accuracies of the transmitters, according to product catalogues.

5.2. Cycle-averaged wall temperature For certain applications, e.g. cooling of electronic and high-power devices, time-average characteristics are more relevant. The effect of pore density on the cycle-averaged wall temperature is shown in Fig. 5 for the same displacement 1.9 and the same heat flux 15,500 W/m2. Two cases are considered: the relatively low kinetic Reynolds number 1873 (low frequency), part (a), and the high kinetic Reynolds number 5610, part (b). In both cases, the average wall temperature increases starting at the inlet and attains it maximum value at the center of the test section z/D = 3.2. Close to the inlet of the test section there exists a thermal entry region [11–14], which produces higher heat transfer rates from the heated wall, and lowers the wall's temperature. In the thermal entry region, the heat transfer coefficient at the inlet is maximum and decays as the distance from the inlet increases. This is reflected in the increase of the wall temperature as the distance from the entrance increases. Similar behavior was observed for heat transfer due to oscillating water flow in packed spheres [2]. The same was also reported for heat transfer due to oscillating air flow in 20-ppi aluminum

5. Results and discussion In the presentation of the results, the effect of pore density will be elucidated as much as possible. This will be done with the help of selected plots of pertinent heat transfer parameters many of which have practical relevance. Conciseness and summarizing will be implemented in order to allow the extraction of sound conclusions as much as possible. The results for the 20-ppi foam [29,30] will be incorporated in some of the plots in order to clearly show the impact of pore density on heat transfer parameters and on pressure drop. 5.1. Symmetry and periodic behavior of wall temperature The wall temperature along the test section for 10- and 40-ppi foams is shown in Fig. 3. The non-dimensional displacement is 2.2 and kinetic Reynolds number is 1837. The heat flux was 15,500 W/m2 for this run. Clearly, Fig. 3 displays symmetry of the two sides of the test section. The wall temperature attains its maximum value at the midpoint and gradually decreases to a minimum at the two entrances of the test section. The symmetry provides some assurance that the test-set up produced reasonable data. The thermal entry length plays a major role in oscillating flow heat transfer. (There are two entry lengths-one on each inlet of the test section). Furthermore, this entry phenomenon is repeated frequently depending on the frequency. Inspection of Table 1 reveals that the thermal entry length for the 10-ppi foam is 150 mm, while for the 40-

Fig. 3. Wall temperature for A0 = 2.2 and Reω = 1873 at q″ = 8866 W/m2. 4

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Table 1 Morphological, flow and thermal parameters of open-cell aluminum foams. Pore Per Inch (ppi)

Porosity (%)

Ligament Diametera (mm)

Surface Area Densityb (m2/m3)

Permeability,c K × 108 (m2)

Thermal Entry Lengthd (mm)

Effective Thermal Conductivitye (W/m.K)

10 20 40

88.5 87.6 88.5

0.43 0.35 0.20

913 1455 2077

11.27 13.15 5.10

150 129 122

9.077 10.109 9.085

a b c d e

Adopted from Bhattacharya et al. [34]. Calculated as in Dukhan and Petal [35]; Appendix. Adopted from Bağcı and Dukhan [27]; Appendix. Adopted from Arbak et al. [28]; Appendix. Calculated as in Ref. [33]; Appendix.

Fig. 4. Periodic behavior of wall temperature for A0 = 1.9, Reω = 1873: (a) 10ppi and (b) 40-ppi foam.

and 60-ppi copper foams [18] and in 40-ppi aluminum and 45-ppi graphite foams [19]. Comparing parts (a) and (b) of Fig. 5 indicates that higher flow frequency generally lowers the wall temperature for all pore densities. Higher frequency means higher average velocity through the foam, which enhances convection. The noticeable increase in the wall temperature in the vicinity of the inlet at the higher frequency, part (b) of Fig. 5, can be attributed to an overall rise in the bulk temperature due to improved axial conduction. As for the effect of pore density, the cycle-averaged wall temperature is higher for higher pore density. Indicating that foams with lower pore density produce better heat transfer when subjected to oscillating water flow, strictly within the range of flow displacements and frequencies of the current study. This is counter to what would be expected based on heat transfer due to unidirectional steady-state water flow [28] in the same types of foams. In the steady-state case, the 40ppi foam produced the lowest wall temperature followed by the 20-ppi and then the 10-ppi foam. Many effects are present in oscillating flow that are absent in unidirectional steady-state flow. For example, within one cycle of oscillating flow, the flow reverses its direction which leads to a vastly different flow and temperature fields. To further explain the above trends, some of the morphological, hydrodynamic and thermal properties of the three foam samples listed in Table 1 are considered. The effective thermal conductivities of the three foams are within 10%

Fig. 5. Cycle-averaged wall temperature for three pore densities at Ao = 1.9 and (a) Reω = 1873, (b) Reω = 5610.

of each other, and their effect on heat transfer is likely minimal. It is clear that the 10-ppi foam has a larger pore diameter and higher permeability indicating easier flow that can sweep more heat. The ligament diameter of the 10-ppi foam is the largest among the three foams, which means possibly more metal contact with the solid wall and increased conduction from the wall. A key phenomenon in oscillating heat transfer is thermal entrance, in which convection heat transfer is maximum. Inspection of Table 1 reveals that the thermal entry length for the 10-ppi foam is 150 mm, for 20-ppi foam 129 mm and for the 40-ppi foam it is only 122 mm. In other words, the thermal entry length for 10-ppi foam is about 23% longer than for 40-ppi foam and 16.3% longer than for the 20-ppi foam. So in oscillating flow, the 10-ppi provides more cooling and lower wall temperature. As a matter of fact, the wall temperature curves are stacked according to the thermal entry length of a given foam: lower 5

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N. Dukhan, et al.

wall temperature for longer thermal entry length. The effect of thermal entry is seen to dominate the effect of surface area.

seen to produce higher Nu in all pore densities. Higher frequency implies higher mean fluid velocity, which promotes convection and produces higher Nusselt numbers. In addition, the transverse-dispersion thermal conductivity increases with frequency for a given displacement [36], which increases heat transfer from the heated wall. The pore density is seen to have an effect at the low and high frequencies: the 10-ppi foam produces higher Nusselt numbers (higher heat transfer), with the 20- and 40-ppi foam being close to each other. This influence of pore density is seen to be more pronounced at the higher oscillating frequency, Fig. 8 part (b). In order to have an explanation for these observations, one has to look at key parameters affecting oscillating heat transfer in porous media and relate them to the pore density, as much as possible. These parameters include Darcy number (K/R2 where K is the permeability and R is the radius of the porous medium), the ratio of the effective thermal conductivity of the porous medium to that of the fluid keff / kf [37] and porosity [6]. In the

5.3. Temperature uniformity Temperature uniformity of a heated substrate surface is of practical importance in systems sensitive to temperature variations, e.g. some electronic devices. One advantage of oscillating flow heat transfer is that it can provide uniform wall temperatures. As a means of assessing the level of wall temperature variations, the temperature uniformity index Iuni [18,19] is defined as

Iuni =

Tmax Tmin Tmax

(9)

where Tmax and Tmin are the wall maximum and minimum temperatures. As such the index is basically a non-dimensional maximum temperature difference. Fig. 6 compares the uniformity index for steady and oscillating flows for each of the pore densities. In order to compare steady and oscillating flows, Reynolds number ReD was calculated as

ReD =

UD µ

(10)

According to Fu et al. [19], U is the average velocity for steady flow or the time-averaged maximum velocity for oscillating flow:

U=

2umax

(11)

Due to experimental limitations, the Reynolds number for the steady-state and oscillating heat transfer data is the same only over a limited range. It can be seen in all parts of Fig. 6 that, it is conceivable that more temperature uniformity can be achieved with steady-state liquid flow in all pore densities of metal foam. This is especially true at low frequencies (or kinetic Reynolds numbers). This finding is opposite to what has been reported for oscillating air flow [18,19]. The reason for this is the fact that in the current study considerably short fluid displacements, compared to the length of the test section, were employed. The short displacements did not sweep significant amounts of heat in order to produce more uniform wall temperatures. Fig. 7 compares the temperature uniformity index for the three pore densities at three displacements: part (a) is for Ao = 1.5, part (b) is for Ao = 1.9 and part (c) is for Ao = 2.2. It can be stated that, in general, higher frequency (higher kinetic Reynolds number) produces lower uniformity index, or more uniform wall temperature, for all pore densities. Also, at the same frequency, longer fluid displacements produce a more-uniform temperature distribution for a given pore density. This is so due to the presence of a longer penetration zone of fresh fluid when the displacement is long. This fresh fluid can absorb more heat from the heated wall. In parts (b) and (c), the missing points are a result of the fact that at longer displacements, high kinetic Reynolds numbers could not be achieved due to safety concerns. As for the effect of pore density on temperature uniformity, at the short displacement, the 20-ppi foam seems to produce better uniformity, with the 10- and 40-ppi foams producing practically the same uniformity index, part (a) of Fig. 7. At the medium and long displacements, there is no difference between the three pore densities in term of temperature uniformity, parts (b) and (c) of Fig. 7. 5.4. Cycle-averaged Nusselt number The local cycle-averaged Nusselt number for the three pore densities is presented in Fig. 8 for displacement 1.9; part (a) is for Reω = 1873 representing low oscillating frequency and part (b) is for Reω = 5619 representing high frequency. For all pore densities and for both frequencies, Nu decreases gradually with distance from the inlet reaching a minimum at the midpoint of the test section. This trend is attributed to the presence of a thermal entrance region. The higher frequency is

Fig. 6. Temperature uniformity for steady and oscillating flows for: (a) 10-ppi, (b) 20-ppi and (c) for 40-ppi foam. 6

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Fig. 8. Cycle-averaged Nusselt number at Ao = 1.9 for (a) Reω = 1873, (b) Reω = 5619.

dissimilar behaviors can be further explained as follows. In Ref. [38], significantly longer displacements and higher frequencies were used. For the current study the frequency ranged from 0.116 to 0.348 Hz while in Ref. [38] it ranged from 1 to 9 Hz. The three displacements of the current study were 74.4 mm, 97.2 mm and 111.5 mm and the length of the foam test section was 305 mm. This means that the fluid only swept a small section of the foam. In Ref. [38], a single displacement of 68 mm was used and the foam length was only 50 mm. 5.5. Comparison to steady-state heat transfer

Fig. 7. Temperature uniformity for three pore densities at: (a) Ao = 1.5, (b) Ao = 1.9 and (c) Ao = 2.2.

Fig. 9 compares heat transfer results of the current study to those of

current study, only water was used and the porosity of the three foams was practically the same. At the same porosity, changes in pore density cause minor changes in the effective thermal conductivity, Table 1. The pore density dictates the surface area per unit volume and the ligament dimeter and generates different entry regions [28]. As shown in Table 1, the thermal entry length in the 10-ppi foam is 150 mm, while this length is 129 mm and 122 mm for the 20- and 40-ppi foams, respectively. It is clear that the Nusselt numbers for the three forms are strongly linked to the entry length. Meaning, longer thermal entry regions correspond to higher Nusselt numbers. Actually, the thermal entry length explains the close Nusselt numbers for the 20- and 40-ppi in Fig. 8. The impact of pore density on heat transfer for the oscillating flow of water of the current study is different from what had been reported in Ref. [38] for oscillating air flow heat transfer. For the same porosity, higher pore densities produced higher heat transfer rates in Ref. [38]. Aside from using a different fluid (different Prandtl number), the

Fig. 9. Length-averaged Nusselt number for current study and steady-state heat transfer form [28]. 7

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differences between the current study and that of [39]. The foam in Ref. [39] was 40 ppi and had a porosity of 91%. Also, the case investigated by Ref. [39] was that of an aluminum foam heat sink heated from the bottom only a constant heat flux between 8.5 and 13.8 W/cm2. The flow frequency was ranged from 0.04 to 0.1 Hz. The heat sink measured 37.5 mm by 37.5 mm by 12.7 mm, which is much smaller than the samples investigated by the current authors. 5.6. Pressure drop Fig. 10 compares the maximum pressure drop for the three pore densities. The pressure drop for the 10-ppi foam is lower than those for the 20- and 40-ppi foams for all the fluid displacements of the current study. The 10-ppi foam has the least surface area density and the largest pore or window size among the three types of foam, Table 1. The small surface area corresponds to low viscous frictional loss, while the large pore size corresponds to high permeability. For the three foams, the pressure drop increases with Reynolds number due to higher mean velocities. Taking this pressure drop into account, and considering the heat transfer behavior of the cycle-averaged Nusselt number and wall temperature, one can conclude that the 10-ppi foam performs the best among the three pore densities: it provides the highest heat transfer and lowest wall temperature while requiring the minimum pressure drop. 6. Conclusion The effect of pore density on characteristics of heat transfer generated by oscillating water flow in commercial open-cell metal foam was investigated experimentally. Strictly within the ranges of flow displacements and flow frequencies covered by the current study, the following concluding remarks can be offered:

• It is conceivable that more temperature uniformity can be achieved • • •

Fig. 10. Pressure drop results for the three pore densities for: (a) Ao = 1.5, (b) Ao = 1.9 and (c) Ao = 2.2.

steady-state results in the same three foams [28]. The length-averaged Nusselt number for steady-state flow is higher than its counterpart for oscillating flow for the three pore densities. One would expect the opposite to be true. It is most likely that the reason of the poor heat transfer performance of the oscillating flow of the current study is due to the rather short displacements and low frequencies employed. These short displacements were not long enough for fresh fluid to sweep more heat from the heated wall, while the low frequency hindered the realization of heat transfer benefits of oscillating flow, i.e, the number of flow reversals and entry lengths per cycle as well as the high average flow speeds. It is worth mentioning that Bayomy and Saghir [39] reposted and enhancement of 14% in the average Nusselt number for pulsating water flow in metal foam, as compared to steady-state case. There are few

• • •

8

using steady-state liquid flow in all pore densities of metal foam rather than oscillating flow at low frequencies (or kinetic Reynolds numbers). This is opposite to what has been reported for oscillating air flow. For oscillating water flow heat transfer in metal foam, the cycleaveraged wall temperature is higher for higher pore density. Indicating that foams with lower pore density produce better heat transfer. This is counter to what is expected. The pore density has a minor effect on wall temperature uniformity at the short displacement, and no effect at medium and long displacements. The pore density of metal foam had an impact on the cycle-averaged Nusselt number: the 10-ppi foam produced higher Nusselt numbers (higher heat transfer rates), with the 20- and 40-ppi foam practically having the same Nusselt numbers. This effect was more pronounced at higher oscillating frequency. This influence was directly related to the thermal entry length. For the low frequencies and short fluid displacements encountered in this study, no enhancements of heat transfer were obtained; and the oscillating flow heat transfer rates were lower than their steadystate counterpart. Heat transfer data due to oscillating flow of liquid in metal foam cannot be deduced from data for high-frequency long-displacement oscillating air flows in metal foam. The pressure drop for the 10-ppi was generally the lowest among the pore densities. Since the cycle-averaged Nusselt number of 10-ppi foam was the highest, one can conclude that the 10-ppi foam performs the best among the three pore densities in the Reynolds number and displacement range encountered in this study.

International Journal of Thermal Sciences 145 (2019) 105978

N. Dukhan, et al.

Appendix

• Surface Area Calculations As was outlined by Dukhan and Petal [35], the surface area was calculated from correlations provided by the manufacturer, who company uses a combination of techniques including optical microscopy and geometrical approximation of the shape of the cells. The measurements are further verified by using the multipoint BET method by the adsorption of krypton gas at cryogenic temperatures. BET theory is a well-known rule for the physical adsorption of gas molecules on a solid surface. These measurements are repeatable and produce results that are within 10% error margin. The company's data sets were collected using a large number of 10-, 20- and 40-ppi foam samples and then averaged and correlated, respectively, according to:

m2 = 349.15 ln(1 m3

) + 1667.99

m2 = 442.20 ln(1 m3

) + 2378.62

m2

= 694.57 ln(1

m3

(A1) (A2)

) + 3579.99

(A3)

• Effective Thermal Conductivity Calculations The effective thermal conductivity was calculated based on this formula given by Ref. [33]: r

2 3

ke =

()

(

kf + 1 +

+

3 2

kf +

4r 3 3

b L

b L

b L

( ) (k b L

s

) +

kf )

+ ks

kf

(1 kf +

2 3

() ( ) (k k ) r)

b L

b L

s

f

3 3 2

kf +

4r 3 3

1

b L

( ) (k b L

s

kf )

(A4)

where

b = L

r+

(2 r (1 + ) )/3 ( 2 r (1 + ) )

r 2 + 4*(1 2 3

)*

3 2

4 3

4 3

(A5)

For this formulae and the foam samples used in the current paper, the following values were used solid conductivity ks = 218 W/m.K fluid conductivity kf = 0.02881 W/m.K correlation constant r was given by Ref. [33] as 0.09 porosity ε given in Table 1. These equations are widely used and have been compared to experimental values of the effective thermal conductivity and produced excellent agreement [33].

• The Permeability The permeability was determined from measurements of the steady-state pressure drop for various average flow velocities according to Forchheimer Equations:

p µ = u+ L K

F 2 u K

(A6)

where Δp is the static pressure drop, L is the length of the porous medium in the flow direction, μ is the fluid viscosity, and the superficial velocity u is calculated by dividing the mass flow rate through the porous medium by the cross-sectional flow area and the density of the fluid ρ. The permeability of the porous medium is K and the form-and-inertia drag coefficient is F. The steady-state experiment for obtaining the permeability was indeed conducted and published by Bağcı and Dukhan [27]. The reader is referred to Ref. [27] for more details.

• The Thermal Entry Length The thermal entry length for the three foam samples was obtained experimentally by the current authors, Arbak et al. [28], using steady-state flow of water. This thermal entry length was determined by considering the slope of the wall temperature: then end of the thermal entry length was established as the point at which the slope of the wall temperature became constant, and independent of the distance from the entrance to the foam samples. The full experiment and results are given in Ref. [28].

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International Journal of Thermal Sciences 145 (2019) 105978

N. Dukhan, et al.

References

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