A “2.5-D” modeling approach for single-phase flow and heat transfer in manifold microchannels

International Journal of Heat and Mass Transfer 126 (2018) 317–330

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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt

A ‘‘2.5-D” modeling approach for single-phase flow and heat transfer in manifold microchannels Raphael Mandel, Amir Shooshtari, Michael Ohadi ⇑ Small and Smart Thermal Systems Laboratory, Department of Mechanical Engineering, University of Maryland, College Park, MD 20742, USA

a r t i c l e

i n f o

Article history: Received 2 October 2017 Received in revised form 23 April 2018 Accepted 26 April 2018

Keywords: Single-phase Liquid cooling Manifold-microchannels Laminar flow Developing flow Reduced-order modeling 2.5-D

a b s t r a c t A reduced-order ‘‘2.5-D” computational fluid dynamics (CFD) modeling approach for single-phase flow and heat transfer in manifold-microchannel heat exchangers was developed, and found to exhibit an order-of-magnitude reduced computational cost compared to a full 3-D simulation. Unlike previous approaches that neglect the convective terms in the momentum equations and assume fully developed flow, in the present work, the inertial terms in the momentum equations were retained, and a userdefined-scalar was used to calculate flow distance so that developing flow could be assumed. The 2.5D model was then compared to a full 3-D CFD simulation, and was shown to be accurate as long as inertia is low enough to prevent the onset of secondary flows. The governing dimensionless parameters were defined, and the effect of each dimensionless parameter was investigated via parametric studies. Finally, a multi-dimensional parametric study was performed to determine the dimensionless parameter that governs the accuracy of the 2.5-D approach. In the end, it was determined that as long as dimensionless length is above 0.1, pressure drop can be predicted to within an average error of
7% for any fluid, and heat transfer can be predicted to within an average error of 6% for water and air. Ó 2018 Elsevier Ltd. All rights reserved.

1. Introduction The ubiquity of heat exchangers and their potential to affect system efficiency has made heat exchanger design of critical interest. Due to their high surface area to volume ratios, microchannel heat exchangers are capable of transferring a given amount of heat in a compact and lightweight design. However, their small hydraulic diameters create large pressure drops and require high pumping power, which can reduce system efficiency. One way to minimize this effect is to divide the microgrooves into a system of parallel microchannels, thereby reducing both the flow length and the flow rate through each channel. Such a system, known as a manifold-microchannel system, is shown in Fig. 1. Due to the simultaneous reduction in both flow rate and flow length with each division, the pressure drop and pumping power tend to decrease proportional to the number of divisions squared [1]. Thus, microchannels with smaller hydraulic diameters can achieve the same pressure drop and pumping power as minichannels as long as the number of divisions is increased accordingly. In addition, due to the short flow lengths, manifold-microchannels ⇑ Corresponding author at: Center for Environmental Energy Engineering (CEEE), 4164C Glenn L. Martin Hall, Department of Mechanical Engineering, University of Maryland, College Park, MD, 20742, USA. E-mail address: [email protected] (M. Ohadi). https://doi.org/10.1016/j.ijheatmasstransfer.2018.04.145 0017-9310/Ó 2018 Elsevier Ltd. All rights reserved.

can take advantage of thermally-developing flow, where a thin boundary layer results in a higher local heat transfer coefficient. Manifold-microchannels have been extensively simulated in the literature, beginning with Harpole and Eninger [2]. They created a 2-D computational fluid dynamics (CFD) model, that neglected inertia and assumed fully developed flow and Nusselt numbers to simulate the effect of friction and heat transfer in the third dimension. They also simulated conjugate conduction using similarly defined source terms in the energy equation. Copeland et al. [3] used 3-D CFD to parametrically analyze the effects of the various geometric variables associated with manifoldmicrochannels. They neglected the effects of conjugate conduction (i.e. the solid domain), assuming instead an isothermal or isoflux boundary condition on the solid–liquid interface. Since then, numerous three-dimensional numerical studies have been conducted [4–6], including multi-objective optimization studies [1,7–10]. However, since no correlations exist to predict the pressure drop and heat transfer in manifold-microchannels, CFD is required to predict their performance. While conventional heat exchangers can be designed in a matter of hours using widely available correlations, manifold-microchannel heat exchangers require days to run the necessary CFD. Thus, the primary objective of this work was to develop a computationally-efficient, reduced-order model

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Nomenclature

Dh

area, [m2] aspect ratio ðhch =wch Þ, [–] Poiseuille number ratio, [–] Nusselt number ratio, [–] specific heat, [J/kg K] heat capacity rate ratio (C min =C max Þ, [–] parallel plates hydraulic diameter (2wch ), [m]
 hydraulic diameter w2wchþhhch , [m]

F fRe h IR k Lþ L Lch Lin Lman Lout hch _ m N Nu NTU P Pr q00 q000 Q Q max

source term vector in momentum equations, [N/m3] Poiseuille number, [–] heat transfer coefficient, [W/m2-K] inlet ratio (Lin =Lch Þ, [–] thermal conductivity of the fluid, [W/m K] hydrodynamic dimensionless length, [–] thermal dimensionless length, [–] length of microchannel, [m] length of manifold inlet, [m] length of manifold wall, [m] length of manifold outlet, [m] height of channel, [m] mass flow rate, [kg/s] number of nodes, [–] Nusselt number, [–] number of transfer units, [–] pressure, [Pa] Prandtl number, [–] wall heat flux, [W/m2] source term in energy equation, [W/m3] total heat, [W] maximum possible heat, [W]

A AR C fRe C Nu Cp Cr D *

ch

ch

Re Rech S T T wall U V * V VR W wch x y z

inlet, parallel plates Reynolds number, ½qV in D=l microchannel Reynolds number, ½qV ch Dh =l flow length, [m] fluid temperature, [K] wall temperature, [K] velocity in x-direction, [m/s] velocity in y-direction, [m/s] velocity vector, [m/s] velocity ratio, [–] velocity in z-direction, [m/s] width of channel, [m] coordinate direction, [m] coordinate direction, [m] coordinate direction, [m]

Greek symbols e effectiveness, [–] p scalar, [–] l dynamic viscosity, [Pa s] q density, [kg/m3] s wall shear stress, [Pa] Subscripts/superscripts app apparent area- or mass-average av e ch microchannel fd fully developed in microchannel inlet out microchannel outlet w microchannel fin wall

capable of simulating single-phase, laminar flow and heat transfer in manifold-microchannels accurately, such that numerous simulations can be performed quickly. In addition, the model also provides insights into governing physical phenomena in manifoldmicrochannels by allowing physical phenomena to be isolated, resulting in an improved understanding of manifoldmicrochannel flow phenomena. 2. 2.5-D model A ‘‘2.5-D” model of the manifold-microchannel flow configuration was created. Unlike Harpole and Eninger’s model [2], which neglects inertia and assumes fully developed flow, the present model includes the effects of inertia and models developing flow by assuming that the flow develops along a streamline as if it were in a straight channel. The assumption of developing flow is equivalent to assuming a boundary layer profile in the third dimension— hence, our coining of the term ‘‘2.5-D”.

Fig. 1. Manifold-microchannel system.

2.1. Domain 2.2. Assumptions Due to the symmetries present in manifold microchannel arrays, the domain for the manifold-microchannel simulations can be simplified to the unit-cell shown in Fig. 2(a) [1,7–14]. The definitions of the geometric variables are given in Fig. 2(b). The flow path is shown in Fig. 2(a). Flow enters from the top left in the velocity-inlet. The flow then impinges on the top of the microchannel fin and enters the microchannel, where the fluid absorbs heat. The fluid then turns upward and flows around the fin tip, and leaves through the pressure outlet.

The following assumptions and simplifications were made in the 2.5-D model: (1) Steady-state, laminar, incompressible flow, with negligible effects of gravity on momentum and viscous dissipation on temperature (2) Constant fluid properties (3) Constant wall temperature

R. Mandel et al. / International Journal of Heat and Mass Transfer 126 (2018) 317–330

319

Fig. 2. Manifold-Microchannel domain (a) Full domain, (b) Simplified domain.

(4) Negligible effect of inlet/outlet contraction/expansion on pressure drop and heat transfer in the microchannel (5) The flow is thermally and hydrodynamically developing, and develops along a streamline as if it were in a straight channel (6) Source terms are used to simulate the effects of pressure drop and heat transfer due to the fin and are derived from the solutions for one-dimensional flow between parallel plates Since the bottom and top walls are included in the 2.5-D model, the effect of these walls on pressure drop and heat transfer is already included in the model. Accordingly, the only missing effect is the shear stress and heat transfer on the microchannel fin. For high aspect ratio channels, it is reasonable to make the approximation that the boundary layers normal to the fin are not affected by the top and bottom walls, and hence, the momentum and energy source terms should be derived from the boundary layer profiles present for one-dimensional flow between parallel plates. However, accuracy is expected to reduce for low aspect ratio channels, and therefore, the 2.5-D model will be tested with low aspect ratio channels in order to quantify this error. While material properties could be a function of temperature in both the 2.5-D and 3-D models, constant material properties were assumed to simplify the model and to assess the best-case scenario for the accuracy of the 2.5-D model. In addition, in the 3-D model, the Nusselt number on the fin will always fall between constant temperature and constant heat flux, the value of which will result from solution of the conjugate conduction problem. However, in the 2.5-D model, the Nusselt number must be taken from developing flow correlations, and either a constant temperature or constant heat flux correlation must be selected a priori. Since constant temperature predicts lower values, and Nusselt numbers in heat exchangers tend to be closer to constant temperature, a constant wall temperature boundary condition was selected. In addition, it is assumed that if the average Nusselt number on the fin is known, the total thermal resistance can be computed by assuming 1-D conduction through the base and fin. Thus, conduction in the solid domain need not be included in the simulation.

In addition, it is reasonable to assume that the effect of the contraction and expansion at the inlet and outlet of the microchannel does not affect the pressure drop or heat transfer in the microchannel domain, and the pressure drop due to contraction and expansion at the inlet and outlet to the microchannel can be accounted for using standard correlations for contraction and expansion loss coefficients, such as those recommended by [15,16]. Thus, the contraction and expansion at the inlet and outlet of the microchannel need not be included in the domain, but rather, can be included externally using standard correlations for contraction and expansion pressure loss coefficients. Given these assumptions and simplifications, the solid domain can be eliminated, and the fluid domain simplified from the one shown in Fig. 2(a) to the one shown in Fig. 2(b). 2.3. Governing equations In the 2.5-D model, the velocity component in the z-direction is assumed to be negligible, and the wall shear stress are simulated using source terms. Therefore, the governing Navier-Stokes equations reduce to *

*

rV ¼ 0 *

* *

ð1Þ *

*

*

*

q V  r V ¼  r P þ lr2 V þ F

ð2Þ

where U, V, and P, are assumed to be averaged values in the z* * * ! direction, r denotes ð @ i ; @ j Þ, and F is the volumetric momen@x @y tum source term intended to simulate the effect of shear stress on the fin. Similarly, the energy equation reduces to *

*

*

qC p V  r T ¼ kr2 T þ q000

ð3Þ

where T is the averaged fluid temperature in the z-direction, and q0 is a volumetric energy source term intended to simulate the effect of heat transfer between the fluid and the fin.

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However, one additional equation is needed to simulate developing flow: the flow length from the entrance of the channel. The equation for flow length, S, is given by

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi * * q V  r S ¼ q U2 þ V 2

ð4Þ

The equation for flow length can be derived easily as follows: by definition, any field scalar, S, must satisfy the identity,

dS ¼

@S @S dx þ dy @x @y

ð5Þ

In addition, the formula for the distance the flow travels over some time step, dt, is

dS ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi U 2 þ V 2 dt

ð6Þ

Since dt ¼ dx ¼ dy , Eq. (6) can be re-written as U V

U dx ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dS 2 U þ V2

ð7Þ

V dy ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dS 2 U þ V2

ð8Þ

*

F ¼ rP ¼ 

1 fRe * lV 8 w2ch

ð9Þ

where fRe is the product of the Darcy friction factor and Reynolds number, known as the Poiseuille number. It is often convenient to define the Poiseuille number ratio, cfRe , as 1

cfRe ¼ fRe=ðfReÞfd

ð10Þ

so that Eq. (9) can be re-written as *

1

F ¼ cfRe

ðfReÞfd l * V 8 w2ch

ð11Þ

1 ðfReÞfd

where ¼ 96, and is the fully developed Poiseuille number for parallel plates (AR ¼ 1), and the Poiseuille number ratio, cfRe , will be provided by a developing flow correlation. Similarly, the volumetric energy source term can be written as follows [2]:

k q000 ¼ cNu Nu1 ðT wall  TÞ fd w2ch

ð12Þ

where Nu1 fd ¼ 7:54 and is the fully developed Nusselt number for flow between isothermal parallel plates (AR ¼ 1), and the Nusselt number ratio, cNu , defined as

cNu ¼

Nu=Nu1 fd ;

At this point, the 2.5-D model is still incomplete, as Eqs. (11) and (12), which are used to provide source terms to Eqs. (2) and (3), require terms that come from developing flow correlations. These missing terms will be provided in this section. Many correlations exist in the literature for Poiseuille number for developing flows between parallel plates [17–22]. Due to its simplicity and validity over the entire range of dimensionless lengths, the correlation proposed by Muzychka and Yovanovich [17] for Poiseuille number ratio, was used here. The correlation was derived using the method of Churchill and Usagi [23], and is of the form

ðfReÞapp ¼ ðfReÞfd

ð14Þ

S=D Re

ð15Þ

where S is the distance that the fluid has travelled down the streamline, and D and Re are a characteristic hydraulic diameter and Reynolds number, respectively. Similarly, many correlations exist in the literature for Nusselt number in thermally developing flow [21,22,24–26]. The correlation of Stephan [25], as described by Muzychka and Yovanovich [21], was selected due to its validity for 0:1 6 Pr 6 1000. This correlation is of the form 1:14

Nuapp ð155L Þ ¼1þ 0:64 Nufd 1 þ 0:0358Pr0:17 ðL Þ

ð16Þ

where L is the thermal dimensionless length defined as

L ¼

S=D RePr

ð17Þ

and Pr is the fluid Prandtl number. It is worth noting that the correlations shown in this section provide the apparent Poiseuille number or Nusselt number from the entrance of the channel to any given point downstream. In the 2.5-D model, we require local values. Accordingly, these correlations must be manipulated to provide local values. The average value of any continuous function, gðuÞ, from the origin to a point u is given by the definition

gðuÞ ¼

1

u

Z

0

u

gðuÞdu

ð18Þ

This equation can be manipulated to give local value:

gðuÞ ¼

d ðugðuÞÞ du

ð19Þ

Thus, Eqs. (14) and (16) can be differentiated in accordance with Eq. (19) to obtain the local forms of the correlations:

 cfRe ¼

0:14333 pffiffiffiffiffiffi

2

þ1 fRe 2Lþ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ¼ s 2 ðfReÞfd 0:14333 pffiffiffiffi þ1 þ

ð20Þ

L

ð13Þ

will be provided by a developing flow correlation. Eqs. (1)–(4) were solved using Fluent’s laminar, double precision, pressure-based solver to a tolerance of 1e-4 for continuity, momentum, and flow length, and a tolerance of 1e-13 for energy.

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 0:14333 pffiffiffiffiffi þ1 Lþ

where Lþ is the local hydrodynamic dimensionless length, defined as

Lþ ¼

After multiplying Eq. (5) by q, one can substitute in Eqs. (7) and (8) and arrive at Eq. (4). The form of the partial differential equation given in Eq. (4) is a convection equation with a source term equal to the magnitude of the mass velocity. This equation can be solved—along with the Navier-Stokes equations—using a commercial CFD software package such as Fluent by defining it as a user-defined-scalar (UDS). Fluent can solve Eqs. (1)–(4) if the momentum, energy, and flow length source terms are specified using user-defined-functions. The momentum source term can be derived using Darcy’s law for parallel plates [2]: *

2.4. Developing flow correlations

1:14

cNu ¼

0:64

Nu ð155L Þ ð0:0179Pr 0:17 ðL Þ  0:14Þ ¼1þ 0:17  0:64 2 Nufd Þ ð1 þ 0:0358Pr ðL Þ

ð21Þ

which matches the local form of the correlation derived by Shah and Bhatti, as described in [26]. The values of cfRe and cNu predicted

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material properties for the fluids are given in Table 1 and were obtained using the database found in EES for an operational temperature of 30 °C [27]. In addition, these working fluids represent fluids with a wide range of Prandtl numbers from 0:73 6 Pr 6 5:5.

Table 1 Thermophysical properties of materials used in the simulations. Fluid T

q Cp

l

k Pr

Temperature [°C] Density [kg/m3] Specific heat [J/kg K] Viscosity [mPa s] Thermal conductivity [W/m K] Prandtl Number [–]

Air

Water

30 1.164 1005 18.72 0.02588 0.7270

30 995.6 4183 797.7 0.6029 5.5345

2.6. Dimensionless variables For manifold microchannel flow, there are five dimensionless variables: aspect ratio, AR, inlet ratio, IR, velocity ratio, VR, and  hydrodynamic (Lþ ch ) and thermal (Lch ) dimensionless lengths. Aspect ratio, AR, is defined as the ratio of the height of the channel, hch , to the width of the channel, wch :

by the correlations in Eqs. (20) and (21) can be used to evaluate Eqs. (11) and (12). It is worth noting that the Reynolds number, Re, and diameter, D, have not yet been defined. As described above, the boundary layer development is assumed to occur as though it is between parallel plates, regardless of the actual aspect ratio of the microchannel. Thus, the hydraulic diameter used here should be computed for parallel plates, as shown in Eq. (24). However, with respect to Reynolds number, it is unclear which characteristic velocity to use: the velocity through the cross section of the channel, V ch , or the velocity at the inlet to channel, V in . The relationship between these two velocities can be derived from continuity:

V in Lin =2 ¼ V ch hch

AR ¼ hch =wch

The inlet ratio, IR, is defined as the ratio of the width of the inlet of the channel, Lin , to the total length of the channel, Lch :

IR ¼ Lin =Lch

VR ¼

It is worth noting that even though these velocities differ by a constant, since the Reynolds number is used to compute dimensionless length using Eqs. (15) and (17), and dimensionless length are then used to compute Poiseuille and Nusselt numbers using Eqs. (20) and (21), the correct velocity must be used. Since the flow begins to develop when it first enters the channel, it stands to reason that the velocity at the inlet of the channel should be used when computing Reynolds number and dimensionless length. Thus, the dimensionless lengths defined in Eqs. (15) and (17) should be computed using a Reynolds number based on the inlet velocity, V in :

V in 2hch ¼ V ch Lin

ð27Þ

The hydrodynamic and thermal dimensionless lengths, respectively, are defined as

Lch =Dh Rech

ð28Þ

Lch ¼ Lþch =Pr

ð29Þ

Lþch ¼

where Reynolds number, Rech , and hydraulic diameter, Dh , are defined as

ð23Þ

Rech ¼

ð24Þ

Dh ¼

where the corresponding hydraulic diameter, D, is defined as

D ¼ 2wch

ð26Þ

It is worth noting that in the present work, the width of the inlet is always assumed to be equal to the width of the outlet, i.e. Lin ¼ Lout . The velocity ratio is defined as the ratio of the velocity at the microchannel inlet, V in , to velocity through the channel cross section, V ch :

ð22Þ

Re ¼ qV in D=l

ð25Þ

qV ch Dh l

ð30Þ

4A 2wch hch ¼ P wch þ hch

ð31Þ

With the working fluid selected and these five dimensionless parameters specified, the geometric and operational conditions are fully defined from Eqs. (25)–(31).

2.5. Boundary conditions The boundary conditions for the domain are given in Fig. 2. A velocity-inlet is located at the top of the inlet volume, and a pressure-outlet is located at the top of the outlet volume. At the velocity-inlet, a uniform inward velocity, V in , is specified, along with a liquid temperature, T in , and an initial value of flow length, Sin , which is set equal to zero. The static pressure on the pressure outlet is set equal to zero. All source terms in the inlet and outlet domains are set equal to zero, and walls in this domain are set to adiabatic. A constant wall temperature, T wall , is applied to the bottom wall, fin wall, and top wall defined in Fig. 2(b). The working fluids were selected to be water and air, due to their ubiquity in heat exchangers applications of all types. The

2.7. Performance metrics To assess performance of a given design and mesh independence, some performance metrics need to be defined. The performance metrics were defined to allow for easy comparison between manifold-microchannels and straight microchannels, so that the relative performance of manifold-microchannels can be easily observed. In addition, by comparing the performance metrics predicted by the 2.5-D model to those predicted by the 3-D model, the performance metrics can be used to quantify the accuracy of the 2.5-D model.

Table 2 Test case geometry and operational conditions. wch [mm]

hch [mm]

Lin [mm]

Lout [mm]

Lch [mm]

AR [–]

IR [–]

VR [–]

Lþ ch [–]

0.1

1

2

2

5

10

0.4

1

0.025–1

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ve The average Poiseuille number ratio on the fin, C afRe , was computed directly in Fluent using area-averaging [28]:

ve C afRe

RR

fin ¼ RR

cfRe dA

fin

dA

ð32Þ

 in Next, the average microchannel inlet and outlet pressure, P  and Pout , were computed directly in Fluent using area-averaging,  in  P  out . The apparent and the pressure drop computed as DP ¼ P Poiseuille number and Poiseuille number ratio can be computed as

ðfReÞapp ¼

C app fRe ¼

2DPD2h lV ch Lch

ðfReÞapp AR

ðfReÞfd

ð33Þ

ð34Þ

AR

where ðfReÞfd is the fully developed Darcy Poiseuille number for the given microchannel aspect ratio. The fully developed Poiseuille numbers for varying aspect ratios were taken from [24]. Even though the flow length for manifold microchannels can be longer or shorter than the length of the channel, Lch , the length of the channel was used here to facilitate an easy comparison between manifold microchannels and straight microchannels. In addition, it is worth noting that in general, the area-averaged Poiseuille number given by Eq. (32) will not be equal to the apparent Poiseuille number given by Eq. (34) due to reasons that will be discussed later. Similarly, the average Nusselt number ratio on the fin, C aNuv e , can be computed using area-averaging:

C aNuv e

RR

fin ¼ RR

cNu dA

fin

dA

ð35Þ

Next, the total heat transfer rate, Q , was computed in Fluent by summing the integrals of the wall heat flux on the three walls in the microchannel volume. The heat exchanger effectiveness can then be computed from the definition,

e¼

Q Q ¼ _ p ðT wall  T in Þ Q max mC

ð36Þ

_ is the microchannel mass flow rate and is related to the where m microchannel velocity by

_ ¼ qwch hch V ch m

ð37Þ

The heat exchanger NTU can be computed using the standard min e  NTU relationship for constant temperature (C r ¼ CCmax ¼ 0):

NTU ¼  lnð1  eÞ

ð38Þ

The apparent wall heat transfer coefficient, happ , can then be computed from the definition of NTU:

happ ¼

_ p mC NTU Awall

ð39Þ

where the wall area, Awall , is given by

Awall ¼ 2ðwch þ hch ÞLch

ð40Þ

It is worth noting that the wall area defined here includes the microchannel-inlet and outlet area, defined in Fig. 2(b), which are not, strictly speaking, walls. Nevertheless, since the definition is arbitrary and used for comparison purposes only, the wall area was defined as though the channels were straight to facilitate an easy comparison between straight microchannels and manifoldmicrochannels. The apparent Nusselt number, Nuapp , and Nusselt number ratio can be computed from

Fig. 3. Test case (a) apparent Poiseuille number ratio, (b) average Poiseuille number ratio, (c) apparent Nusselt number, and (d) average Nusselt number vs. dimensionless length. Points are from 3-D model; lines are from 2.5-D model.

R. Mandel et al. / International Journal of Heat and Mass Transfer 126 (2018) 317–330

323

Fig. 4. Plots of local Poiseuille number ratio (left) and local Nusselt number ratio (right) vs. flow length at dimensionless lengths of (a) 1, (b) 0.1, and (c) 0.02. Circles correspond to nodal values computed for the 3-D simulation using Eqs. (45) and (46), while the line corresponds to the values predicted by the appropriate correlations appearing in Eqs. (20) and (21).

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Nuapp ¼ happ Dh =k C app Nu ¼

ð41Þ

Nuapp

ð42Þ

NuAR fd

where NuAR fd is the fully developed Nusselt number for the given microchannel aspect ratio. The fully developed Nusselt numbers for constant temperature walls at varying aspect ratios were taken from [24]. Once again, it is worth noting that the area-averaged Nusselt number given by Eq. (35) will not, in general, be equal to the apparent Nusselt number given by Eq. (42) due to the effects of heat transfer on the top and bottom walls, which are not included in the average, as well as the effect of intra-microchannel flow maldistribution. These effects will be discussed in detail in a later section. 3. Test case It is first informative to construct a test case to understand the trends in manifold-microchannels. Table 2 summarizes the geometry used in the test case. For simplicity, only water was used for the test case. 3.1. Effect of dimensionless length The test geometry was simulated using the 2.5-D and 3-D models over a wide range of dimensionless lengths, and the performance metrics computed. The results are given in Fig. 3. For large dimensionless lengths, both the apparent and areaaveraged Poiseuille number ratios were observed to approach a constant value, as shown in Fig. 3(a)–(b), respectively. However, while the area-averaged Poiseuille number was observed to approach a value of unity—which is expected, due to the presence of hydrodynamically fully developed flow—the apparent Poiseuille number approaches a value below unity. The reason for this will be discussed in a later section. In addition, as dimensionless length decreases, both Poiseuille number ratios were observed to increase, which is expected due to developing flow. Similar trends were observed for apparent and area-averaged Nusselt number ratios, as shown in Fig. 3(c)–(d), respectively. In addition, Fig. 3 shows that for large and moderate dimensionless lengths, the 2.5-D model (lines) provides accurate predictions of the 3-D model (points). However, for small dimensionless lengths, significant

errors in the prediction of apparent and area-averaged Poiseuille and Nusselt number ratios begin to accumulate. 3.2. 3-D to 2-D data reduction To understand why the 2.5-D model failed to accurately predict Poiseuille number and Nusselt number ratios at small dimensionless lengths, a more detailed analysis was needed. The field variables from the converged 3-D simulations were exported for advanced post-processing in Matlab. For each node, the spatial ^ V; ^ W), ^ static pressure ^; ^z), velocity components (U; coordinates (^ x; y

^ and temperature (T) ^ were exported to a text file. In addition, (P), on the microchannel wall, the shear stress, sw , and heat flux, q00w , were also exported to a text file. These files were then loaded into Matlab, and the 3-D Cartesian mesh was reconstructed. Variables in the microchannel domain were then averaged in the z-direction using an appropriate averaging scheme. Line-averaging was computed from [28]:

p ðx; yÞ ¼

R w2ch

p^ ðx; y; zÞdl

0

R w2ch 0

ffi

dl

Nz 1 X p^ ðx; y; zÞDz wch =2 1

ð43Þ

^ is a three-dimensional scalar, and p  is a two-dimensional where p ^ in the z-direction. Similarly, mass scalar of averaged values of p averaging was computed using the velocity as a weighting parameter [28]:

p ðx; yÞ ¼

R w2ch

*

p^ ðx; y; zÞjv  j

0

R

wch 2

0

*

jv  j

Nz X

p^ ðjujDyDz þ jv jDxDz þ jwjDxDyÞ

ffi

1 Nz X

ð44Þ

ðjujDyDz þ jv jDxDz þ jwjDxDyÞ

1

Area-averaging was used to compute the average values for the  V;  W),   while massvelocity components (U; and pressure (P),  averaging was used to compute the average liquid temperature (T). Since the sharp turns and stagnation regions in manifoldmicrochannels can lead to an increase in pressure along a streamline if Eq. (9) is manipulated to provide an expression for Poiseuille number, negative values of Poiseuille number can be obtained under certain conditions. Accordingly, rather than using pressure,

Fig. 5. Vector plots of inlet symmetry plane and outlet symmetry plane at dimensionless lengths of (a) 1, (b) 0.1, and (c) 0.02.

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Fig. 6. Effect of aspect ratio on (a) apparent Poiseuille number ratio, (b) average Poiseuille number ratio, (c) apparent Nusselt number, and (d) average Nusselt number vs. dimensionless length. Points are from 3-D model; lines are from 2.5-D model.

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Fig. 7. Effect of inlet ratio on (a) apparent Poiseuille number ratio, (b) average Poiseuille number ratio, (c) apparent Nusselt number, and (d) average Nusselt number vs. dimensionless length. Points are from 3-D model; lines are from 2.5-D model.

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Table 3 Range of parameters tested in multi-dimensional parametric study. wch [mm]

AR [–]

IR [–]

VR [–]

Rech [–]

Lþ ch [–]

0.1

1–20

0.2–0.8

0.25–4

5–2000

0.005–5

which can lead to unrealistic values, wall shear stress was used to compute local Poiseuille number, according to the formula,

fRe ¼

16wch

l

sw

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 þ V 2 U

ð45Þ

The local Nusselt number was calculated using

Nu ¼

2wch q00w =k T wall  T

ð46Þ

The local Poiseuille and Nusselt number ratios were then computed by dividing by the fully developed value for parallel plates, as defined in Eqs. (10) and (13). The local Poiseuille and Nusselt numbers computed from the 3-D simulations using Eqs. (45) and (46) can be directly compared to those used in the 2.5-D simulations using Eqs. (20) and (21). First, the local Poiseuille and Nusselt number ratios are plotted versus flow length for various dimensionless lengths, and the results are shown in Fig. 4. The circles correspond to nodal values computed for the 3-D simulation using Eqs. (45) and (46), while the line corresponds to the values predicted by the correlations appearing Eqs. (20) and (21), for Poiseuille and Nusselt numbers, respectively. For large dimensionless lengths, the flow is hydrodynamically fully developed, and the Poiseuille number ratio was observed to be close to unity, as expected (Fig. 4(a)). However, a significant spread is observed, with some points having Poiseuille number ratios as low as 0.9 and as high as 1.3. Nevertheless, most grid points fit well within +/- 5% of the fully developed value, so it should come as no surprise that the 2.5-D model accurately predicts the apparent and area-averaged Poiseuille number ratios, as previously shown in Fig. 3(a) and (b). Regarding Nusselt number ratio, values larger than one are observed in the entrance region since the flow is thermally developing, and the correlation adequately predicts this trend, as shown in Fig. 4(a). However, just as for Poiseuille number ratio, there is a significant spread in the observed Nusselt number ratios in the developing region. Nevertheless, the vast majority of points deviate by only a small percentage from the correlation, and so, it is should not be surprising that the 2.5-D model can accurately predict apparent and areaaveraged Nusselt number ratio, as previously shown in Fig. 3(c) and (d). As dimensionless length increases, a small hydrodynamically developing region is observed and captured by the correlation, as shown in Fig. 4(b). However, more notably, the spread of Poiseuille number ratio in the fully developed region is observed to increase, with most points falling approximately 25% larger than the fully developed value. In addition, the number of grid points that do not follow the correlation increases, despite the flow being fully developed. A similar trend is observed for Nusselt number ratio. Nevertheless, for small decreases in dimensionless length, the 2.5-D model can still accurately predict apparent and areaaveraged Poiseuille number and Nusselt number ratios, as previously shown in Fig. 3. As dimensionless length decreases further, the correlation completely breaks down. While the flow is hydrodynamically developing, the Poiseuille number ratio does not appear to follow the correlation. In fact, Poiseuille number ratios do not even decrease monotonically with flow length as predicted from the correlation. A similar trend is observed for Nusselt number ratio. Instead,

Fig. 8. Modified Poiseuille number vs. dimensionless length. Points are from 3-D model; lines are from 2.5-D model.

Poiseuille and Nusselt numbers are observed to first decrease, as the flow becomes developed, but then rapidly increase at a flow length of approximately 1 mm. Since the channel is 1 mm tall, this corresponds to the bottom of the channel, where the flow impinges on the bottom surface and turns to the right. Thereafter, the Poiseuille and Nusselt numbers remain level but elevated, until the flow impinges on the outlet symmetry, and turns upward toward the outlet. Poiseuille and Nusselt number ratios once again increase as the flow changes direction towards the microchannel outlet. These results can be explained by observing vector plots on the inlet and outlet symmetries of the domain defined in Fig. 2. For large dimensionless lengths, the flow is organized and maintains a fully developed velocity profile as it impinges on the microchannel bottom and outlet symmetry, as shown in Fig. 5(a). As dimensionless length decreases, the flow impinges more intensely on the bottom of the channel and the outlet symmetry plane, causing the flow to stagnate and begin to reverse direction, as shown in Fig. 5 (b). As dimensionless length decreases further, inertia dominates and causes the flow to reverse direction in the stagnation zones, forming secondary flows at the bottom of the channel and on the outlet symmetry plane, as shown in Fig. 5(c). Thus, for large dimensionless lengths, the flow can be seen as essentially one-dimensional, and accordingly, the correlation— and therefore the 2.5-D model—accurately predicts both local and apparent Poiseuille and Nusselt number ratios. However, for lower dimensionless lengths, inertia is strong enough to cause secondary flows, which disturb the boundary layers and create a fully three-dimensional flow. It is no surprise, therefore, that under these conditions, the correlation—and therefore the 2.5-D model—fails to accurately predict local Poiseuille and Nusselt numbers.

4. Parametric studies To determine the effects of the geometric variables on hydrodynamic and thermal performance, as well as the accuracy of the 2.5D modeling approach, a parametric study was performed around the design studied in the test case.

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4.1. Effect of aspect ratio First, the effect of aspect ratio was tested. Aspect ratio was varied between 1 and 20, while the other variables given in Table 2 were held constant. The results are given in Fig. 6. For large dimensionless lengths, the flow is fully developed, and the apparent and area-averaged Poiseuille and Nusselt number ratios approach a constant value, as expected. However, for area-averaged Poiseuille number at low aspect ratios, errors approaching 20% were observed between the 2.5-D model and 3-D model predictions, as shown in Fig. 6(b). As mentioned earlier, since the source term in the momentum equations was derived for parallel plates, the accuracy of the model for low aspect ratios was questionable. Nevertheless, despite the large errors in area-averaged Poiseuille number ratio at low aspect ratios, the apparent Poiseuille number ratios predictions of the 2.5-D model are significantly more accurate, with errors near 10%. This is because as aspect ratio decreases, the percent of the wall shear stress and heat transfer that occurs on the bottom and top walls increases, and therefore, any errors in the prediction of shear stress or heat transfer on the fin will be diluted. For the extreme case of an aspect ratio of one, the fin contributes only 50% of the wall shear stress and heat transfer, and therefore, the effect on apparent Poiseuille and Nusselt number ratios will be reduced by that factor, which has been demonstrated by the error values given. It is worth noting that while the 2.5-D model wasn’t tested with aspect ratios below unity, its accuracy under these conditions is expected to improve, due to the decreased percentage of shear stress and heat transfer occurring on the microchannel fin. In fact, in the extreme case of an aspect ratio of zero, the flow becomes fully two-dimensional, and a 3-D simulation is not needed. Interestingly, for large dimensionless lengths in low aspect ratio channels, the error between the 2.5-D and 3-D predictions of areaaveraged and apparent Nusselt numbers was only 3% and 6%, respectively. Thus, the assumption that heat transfer on the fin is nearly that of parallel plates is accurate even at low aspect ratios. Lastly, for all aspect ratios, as dimensionless length increases, apparent and area-averaged Poiseuille and Nusselt number ratios begin to increase, and for dimensionless lengths below 0.1, errors begin to accumulate between the 2.5-D and 3-D predictions due to the onset of secondary flows, as described above. 4.2. Effect of inlet ratio

Fig. 9. Effect of velocity ratio on (a) apparent Poiseuille number ratio, (b) average Poiseuille number ratio, (c) apparent Nusselt number, and (d) average Nusselt number vs. dimensionless length. Points are from 3-D model; lines are from 2.5-D model.

Next, the effect of inlet ratio was tested. Inlet ratio was varied between 0.2 and 0.8, while the other variables given in Table 2 were held constant. The results are given in Fig. 7. For large dimensionless lengths, the flow is fully developed, and the apparent and area-averaged Poiseuille and Nusselt number ratios approach a constant value, as described above. However, as inlet ratio is increased, the apparent Poiseuille and Nusselt number ratios was observed to decrease, as shown in Fig. 7(a) and (c), respectively, even though the area-averaged Poiseuille and Nusselt number ratios are unaffected by inlet ratio, as shown in Fig. 7(b) and (d), respectively. This is because as inlet ratio increases, the flow length through the microchannel decreases, reducing the pressure drop and heat transfer. Thus, since apparent Poiseuille and Nusselt numbers are computed using the nominal flow length, Lch , they appear to decrease. However, if the proper flow length were used, it stands to reason that no change in Poiseuille or Nusselt number ratios with changing inlet ratio would be observed. A modified Poiseuille number and Poiseuille number ratio were therefore defined as 0

ðfReÞapp ¼

2DPD2h lV ch Saoutv e

ð47Þ

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R. Mandel et al. / International Journal of Heat and Mass Transfer 126 (2018) 317–330

Fig. 10. Comparison between the 3-D model and the ‘‘2.5-D” model for (a) Poiseuille number ratio, (b) Nusselt number ratio (water), and (c) Nusselt number ratio (air). Circles þ correspond to Lþ ch P 0:1; squares correspond to Lch < 0:1.

R. Mandel et al. / International Journal of Heat and Mass Transfer 126 (2018) 317–330 0

C 0fRe ¼

ðfReÞapp

ð48Þ

AR ðfReÞfd

av e where Sout is the actual average flow length through the microchannel, which can be computed directly in Fluent using mass-averaging [28]:

ve Saout

RR ¼ RR

out

*

*

*

*

SðV  dAÞ

out

V  dA

ð49Þ

A graph of the modified Poiseuille number versus dimensionless length is shown in Fig. 8. The results indicate that when the actual flow length is computed, the Poiseuille number ratio is nearly unaffected by changing inlet ratio for large dimensionless lengths. Moreover, for large dimensionless lengths, the Poiseuille number ratio was observed to approach a value of unity, indicating that the lower-than-expected values of the Poiseuille number ratio are due to a mismatch between the actual average flow length and the one used to compute the Poiseuille number. It is worth noting that from Fig. 7(a) and Fig. 8, the 2.5-D model accurately predicts the apparent Poiseuille and Nusselt number ratios, and therefore, this effect is captured by the 2.5-D model. Lastly, for all inlet ratios, as dimensionless length increases, apparent and area-averaged Poiseuille and Nusselt number ratios begin to increase, and for dimensionless lengths below 0.1, errors begin to accumulate between the 2.5-D and 3-D predictions due to the onset of secondary flows, as described above. 4.3. Effect of velocity ratio Lastly, the effect of velocity ratio was tested. Velocity ratio was varied between 0.25 and 4, while the other variables given in Table 2 were held constant. The results are given in Fig. 9. For large dimensionless lengths, the flow is fully developed, and the apparent and area-averaged Poiseuille and Nusselt number ratios approach a constant value, as described above. However, as inlet ratio is increased, the apparent Poiseuille number ratio increases by a factor significantly larger than the area-average Poiseuille number ratio increases, as shown in Fig. 9(a) and (b), respectively. This increase is due to a larger velocity ratio, which corresponds to a higher velocity at the microchannel inlet and therefore higher pressure losses as the fluid is forced to turn at higher speeds. This effect is captured by the 2.5-D model, which accurately predicts apparent Poiseuille number for large dimensionless lengths, as shown in Fig. 9(a). Lastly, for all velocity ratios, as dimensionless length increases, apparent and area-averaged Poiseuille and Nusselt number ratios begin to increase, as expected, and for dimensionless lengths below 0.1, errors begin to accumulate between the 2.5-D and 3-D predictions due to the onset of secondary flows, as described above. 5. Multi-dimensional parametric study With the effect of each dimensionless variable determined, the next step was to establish the accuracy of the model for a wide range of all the dimensionless parameters. The dimensionless parameters were simultaneously varied between the values given in Table 3. Whenever a combination of dimensionless variables resulted in a Reynolds number greater than 2000, the simulation was skipped, since it was assumed to be in the transitional or turbulent regime. In addition, it was noted that some simulations did not converge, and those simulations have not been included in the data presented in this section. The values for Poiseuille and Nusselt numbers for the 2.5-D model are plotted versus the values from the 3-D model, and the

329

results are shown in Fig. 10. The circles indicate that the dimensionless length is above 0.1, while the squares indicate that the dimensionless length is below 0.1. The results indicate that the model is accurate for both water and air as long as dimensionless length is greater than 0.1. The mean error for apparent Poiseuille number for any fluid is 7.1%. Similarly, the mean error for Nusselt number is 5.9% and 5.6% for water and air, respectively. Since the dimensionless length can be computed without performing a simulation, the accuracy of the model can be known a priori without performing a 3-D simulation. Accordingly, this 2.5D model can be used for optimization as long as dimensionless length is constrained to be above 0.1. 6. Conclusions This paper presented a ‘‘2.5-D” modeling approach for singlephase flow and heat transfer in manifold-microchannels with an order-of-magnitude reduced computational cost compared to a full 3-D simulation. The reduced-order model was compared to the full 3-D simulation with excellent results under a wide range of dimensionless conditions. It was determined that: (1) The assumption that the fully developed shear stress on the fin wall is that of parallel plates is a surprisingly accurate assumption even for low aspect ratio channels, producing an error of only 10% for the extreme case of an aspect ratio of 1; similarly, the assumption that fully developed Nusselt number on the fin wall is the same as that of parallel plates is accurate to within 6%. (2) For large dimensionless lengths, boundary layer development in manifold-microchannels occurs in an organized fashion irrespective of microchannel aspect ratio, and if the distance from the inlet is known, the local Poiseuille and Nusselt numbers can be computed from standard correlations. (3) In manifold-microchannels, the apparent Poiseuille and Nusselt numbers can fall below the fully developed value for a straight channel with the same aspect ratio, especially for large inlet ratios; this is due to a mismatch between the actual flow length and that used to compute the Poiseuille and Nusselt numbers. The 2.5-D model is capable of accurately predicting this effect. (4) The sharp turns in manifold-microchannels can lead to large pressure losses, especially at large velocity ratios, and the 2.5-D model is capable of accurately predicting these losses. (5) The 2.5-D model provides accurate predictions of apparent Poiseuille and Nusselt numbers as long as the flow conditions do not produce secondary flows, which disturb the boundary layer and violate the 2-D assumptions of the model. (6) For dimensionless lengths above 0.1, the 2.5-D model can predict pressure drop to within an average error of
7% for any fluid, and Nusselt number to within an average error of 6% for water and air. Conflict of interest The authors declared that there is no conflict of interest. Acknowledgements The authors would like to acknowledge the financial support of this project by Defense Advanced Research Projects Agency (DARPA) Intrachip/Interchip Enhanced Cooling (ICECool) Fundamentals Program under Contract No. HR0011-13-2-0012.

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The content of this work does not necessarily reflect the views of the Government, and no official endorsement should be inferred.

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