# Thermal Management with Solid-Fluid Slip Irreversibility Treatment in Conjugate Microdevices

Authors: E. O. B. Ogedengbe
Journal: Journal of Thermodynamics
 (2009)

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## Abstract

A numerical study of the effect of slip flow irreversibility and axial conduction in microdevices with a conjugate heat transfer between unmixed streams is presented. The effects of axial conduction due to parallel flows for thermal management in energy systems are investigated. Silicon substrate containing rectangular microchannels is simulated using a finite volume, staggered coupling of the pressure-velocity fields. The entropy generation transport within the entire system is analyzed and coupled with the solution procedure. The effects of channel size perturbation, Reynolds number, and pressure ratios on the thermal performance and exergy destruction are presented. Comparative analysis of the axial conduction and flow irreversibility between parallel flow on thermal management is studied. A proton exchange membrane (PEM) fuel cell model is used as a quality indicator to access the importance of the exergy-based design method.


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## Body

## 1. Introduction

Increasing demand for performance of MEMS, including microelectronic devices and microfuel
cells, has often
overweighed recent advances in the development of the very large-scale
integration (VLSI) technology. These advances, particularly in electronic
industry, require increase in circuit density and operating speed which
generates more heat than the level of development of microheat sink can handle.
Also, the design of innovative microfuel cell technologies has driven the
ongoing investigations into efficient energy conversion at the near wall
microchannels [1]. Apart from the problem of water flooding at the cathode side
of the PEM fuel cell, inadequate analysis of the exchange of energies at the
near wall of the enclosing channels constitutes unresolved thermal
mismanagement. It is anticipated that future generation of microelectronic
devices and microprocessors will require heat sink capacity in excess of 1000W/cm2 [2].Parallel
microchannels are utilized in the design configuration of fluid flows in
microfuel cells and microheat sinks for electronic packages. This configuration
has been proposed as effective and promising cooling techniques for
microelectronic devices [3]. Consequently, increasing attention has been drawn
to the study of fluid flow and heat transfer characteristics in a single
microchannel [4–6]. Reducing
exergy destruction of fluid motion through microchannels can have beneficial
impact on reducing pressure losses and input power needed for these flow
control methods. Although these effects are insignificant in high-temperature
applications like solid electrolytes fuel cells and compact heat exchangers,
where effective heat recoveries are cogent on microscale treatment of
environmental barrier coatings, irreversibility treatments in microdevices can
provide efficient design of all constituent energy systems. Previous studies by
Ogedengbe et al. [6] have identified the importance of pressure ratios, channel
size perturbation, and entropy production in optimal design of microfluidic
devices. Their studies focused on flow characteristics with irreversibility
effects in a single microchannel. Maxwell’s first-order boundary conditions
were imposed explicitly at the wall of the microchannel. While the interactions
of kinetic and thermal energies at the near-wall were analyzed, the effect of
the second component of the wall-slip, based on axial thermal gradient, was not
investigated since the problem setup involves a single channel flow.This paper
develops a finite volume model of developing thermal boundary layer on conjugate
flow model, using EnerghxFlow solver due to the effect of heat conduction from
the solid wall between microdevices. Analysis on how both pressure differences
and axial thermal conduction affect the slip flow irreversibility and exergy
losses in microchannels will be presented. Second law analysis will be performed and entropy
generation will be determined for fluid motion in microchannels. Results will
be presented and discussed for parallel microchannel arrangement (see Figure1).Figure 1
Schematic of conjugate microchannel system.


## 2. Thermophysical Model

Figure1 represents the schematic of
conjugate microchannel system, comprising the flow of primary and secondary
fluid streams in parallel. For a fuel cell system, the fuel flows through the
primary channels while the air flows through the secondary channel streams. But,
in a microheat sink system, the primary and secondary channels carry the cold
and hot fluid streams, respectively. The schematic of the microheat sink is
investigated in this paper, where the size falls within the regime of flow with
slip boundary condition (see Table 1). Three-dimensional effect is neglected as
symmetry assumption is reasonable with the scale of the flow field. Both the
hot and cold streams, as shown in Figure 2, are modeled using the
thermophysical properties of nitrogen. The solid material is modeled with the
thermophysical properties of silicon. Capillarity, interface, microconvection,
and other transport effects through the interconnection or porous material
require microscale modeling which cannot be captured under continuum
assumption. A hybrid numerical scheme based on both continuum and lattice Boltzmann
(for simulation of flows in porous medium) assumptions can be developed, for
relevant investigation of previously neglected model kinetics, that are capable
of improving the efficiency heat recovery energy systems, especially processes
involving phase change and crystal growth [7].Table 1
Model
thermophysical properties.
Flow parametersValueLength of microchannels(μm)2560Square size of microchannels(μm)1.0Dynamic viscosity of gas(Ns/m2)0.0000164Density of gas(kg/m3)1.2498Density of solid(kg/m3)998.2Specific gas constant of gas(J/KgK)296.8Specific gas constant of solid(J/KgK)390+0.9TOutlet pressure (Pa)100 800.0Pressure ratio,Pin/Pout1.34,3.00Figure 2
Flow arrangement in parallel microchannels.One major
controversy with slip flow condition at the near wall in microchannels (when
the Knudsen number is between 0.001 and 0.1) is that the mechanism of exchange
between kinetic and thermal energies is variedly treated. Since entropy
production encompasses both friction and thermal irreversibilities, it provides
key insight to enhance heat exchange without sacrificing excessive pressure
losses that contribute to higher pumping power.In addition
to convincingly evidence recently published on this issue [6], this paper introduces
the effect of axial conduction due to exchange of thermal energy across the
separating wall of parallel microchannels.


## 3. Coupled Heat-Mass Model

The computational
domain encompasses both the silicon-modeled solid and two-unmixed stream of gas
flowing in parallel direction. The steady-state governing equations represent
the continuity equations, momentum equations, energy equations for both solid
and gas streams. The equations for the gas streams (where subscripti=1 or 2 and represents upper hot
stream or lower cold stream) are as follows: (1)∂(ρi)∂t+∇⋅(ρiui)=0,(2)∂(ρiui)∂t+∇⋅(ρiviui)=−∂pi∂x+∇⋅(μi∇ui),(3)∂(ρivi)∂t+∇⋅(ρivivi)=−∂pi∂y+∇⋅(μi∇vi),(4)∂(ρicPi)∂t+∇⋅(uiTi)=∇⋅(ki∇Ti). The energy equation of the
solid can be written as (5)∂(ρscPs)∂t+∇⋅(ks∇Ts)=0. Maxwell’s first-order slip velocity [8] will be used for boundary
conditions at the walls of the microchannel. This boundary model incorporates
two coefficients, involving both velocity and temperature gradients at the
wall, that is, (6)ugas−uwall=ξ1∂u∂y|wall+ξ2∂T∂x|wall, where ξ1=λ(2−σ)/σ and ξ2=3μ/4ρTgas.Coupled
conjugate wall boundary conditions [9] (7)Ts=Ti,−ks(∂Ts∂n)=−ki(∂Ti∂n) are imposed
at the solid-fluid interface. All other wall surfaces are treated as adiabatic.The numerical treatments of advection terms in (2)–(4) represent a
significant consideration in the design of energy systems, since these efforts
control the product design time and the adequacy of the flow-solver.
EnerghxFlow is based on the non-inverted skew upwind scheme (NISUS) [10] and
second-order approximations. The finite volume discretization of NISUS
introduces the concept of non-inverted computation of convected variables into
the SIMPLEC formulation. Equations (1)–(5), after
integration with the control volume, result in the net flows across a control volume boundary which becomes the sums of
integrated flux terms across all control volume edges. The differential
conservation balance can be simplified with the following standard summation
form: (8)aPϕP=∑anbϕnb+b,  where a,ϕ, and b refer to the finite volume
coefficient, scalar variable (such as the nodal
velocity in the u-momentum equation), and the source term, respectively.
Detailed formulation of the
finite volume-based NISUS is documented elsewhere [4]. Figures 3 and 4 present
the computational time-saving advantage and accuracy, respectively, of NISUS
over other advection schemes.Figure 3
Computational time saving advantage of NISUS.Figure 4
Accuracy of NISUS over previous advection schemes.The inversion of a local12×12
matrix (for a 3D hexahedral control volume element) refers to the upwind
coefficient matrix that needs to be inverted in order to represent the
integration point variables in terms of nodal variables alone. Significantly,
the time-gain pattern presents an increasing propagation with further
refinement and larger problem size. In term of accuracy, the predicted results are
compared against other schemes (i.e., upwind, hybrid, power-law, and Muir-Baliga) between the range 0<Pe≤104. The numerical error of the
proposed scheme is maintained at an average of about 0.0058 between 0<Pe≤103, while showing excellent
accuracy and numerical stability in comparison with other schemes. It should be
noted that the test case (i.e., radial heat in a rotating hollow spheres)
involves flows with recirculation and has been demonstrated as applicable to the design
of a radial lubrication systems [11].The
modeling of the heat-mass equations governing the energy system, as described
above, has not included the quality of the available quantities of energies.
The design of energy systems without consideration of the transport of entropy
is based on the First law of thermodynamics, while the second law analysis only
guarantees a quality-based design. Section4 presents the exergetic
considerations and analysis of conjugate microdevices.


## 4. Exergetic System Analysis

The second law equation describes the state of
irreversibility within the boundaries of energy systems. Due consideration of
this law in addition to the first law can provide better estimate of the
quality of available heat energy recoverable via the serving stream of the
conjugate micro devices. These two laws of thermodynamics can be written as [12] (9)dEdt=Q˙O+∑i=lnQ˙i−W˙+∑inm˙ht+∑outm˙ht,S˙gen=dSdt−Q˙OTO−∑i=lnQ˙iTi−W˙+∑inm˙ht+∑outm˙ht. By eliminating Q˙O from (9), the work rate output can
be maximized as (10)W˙=−ddt[E−TOS]+∑i=ln(1−TOTi)Q˙i+∑inm˙(ht−TOs)−∑outm˙(ht−TOs)−TOS˙gen. Since S˙gen cannot be negative, the maximum possible work
from the system is obtained at the minimum value of TOS˙gen,
known as the lost available work or Gouy-Stodola theorem. In order to
understand the application of this theorem to conjugate heat transfer design,
it will be useful to comprehend the process of entropy generation via the
interaction of the streams with the walls.System optimization demands exergy analysis for all
energy systems, where power or refrigeration effect is operational. In this
case and as it applies to heat transfer design, the first law which deals with
the conservation of energy will not be adequate in order to capture the heat
and work interaction through the conjugate system.From the first law,(11)m˙dh=q′dx. Moreover, assuming steady-state condition with no work
and heat loss or gain from the environment, the second law states that (12)dS˙gen=dQ˙T±∇T+m˙ds≥0,foreachside, while the ± sign denotes either the hot or cold stream of
the heat exchanger. Now, the canonical entropy relationship states that (13)dhdx=Tdsdx+1ρdpdx. Therefore, entropy generation term (after linking (11)–(13)) is (14)S˙gen′=dSgendx=q′ΔTT2(1+τ)+m˙ρT(−dpdx), where τ=ΔT/T, the dimensionless temperature difference.
This equation reveals the two basic components of
entropy generation, including the temperature gradient term and the pressure
gradient term. Since the heat transfer gradient is directly
proportional to the temperature gradient, it implies that the entropy
generation rate for the thermal component is proportional to the square of the
dimensionless temperature difference τ,
and this term plays a vital role in the minimization of the generation of
entropy within the energy system. The importance of entropy generation to the
design of energy systems can be significant even with microdevices, especially
by incorporating the thermal kinetic energy influence of slip at the near wall
at microscale.The balance
of entropy for a control volume can be written as [6] (15)∂(ρs)∂t+∇⋅(ρvs)=−∇⋅(qT)+S˙gen′, where S˙gen′ refers to the entropy production rate (per
unit volume). When this entropy production rate is multiplied by a suitable
reference temperature, it represents the local rate of exergy destruction, X˙d′′′,
per unit volume. Using the Gibbs equation, it can be shown that the rate of
exergy destruction for microchannel flows can be written directly in terms of
both the velocity and temperature gradients as follows: (16)X˙d′′′=μ(∂u∂y+∂v∂x)2+2μ((∂u∂x)2+(∂v∂y)2)+μ((∂T∂x)2+(∂T∂y)2).  Thus, wall slip in (6) affects the velocity and
temperature profiles and resulting exergy destruction in (16). This frictional
dissipation of kinetic energy leads to pressure losses in microchannels, which
depend on thermal convection and streamwise temperature gradients outlined in (6).A finite volume method was developed based on
conjugate conduction of thermal energy across the wall boundary separating the
two fluid streams. By integrating each conservation equation over the discrete
control volume, the algebraic equations are obtained in terms of nodal
variables. An iterative solver was used to solve these resulting algebraic
equations.Using the resulting velocity field solutions, exergy
losses are calculated based on (16). Performing spatial differencing of (16), the exergy destruction rate per unit volume at node P becomes (17)X˙d′′′=μ(uN−uSΔy+vE−vWΔx)2+2μ((uE−uWΔx)2+(vN−vSΔy)2)+μ((TE−TWΔx)2+(TN−TSΔy)2), where Δx and Δy refer to the grid spacing in the x and y directions,
respectively. All terms in (17) remain positive since they are sums of squared
terms. Physically, mechanical energy is irreversibly dissipated to internal
energy through fluid friction, which produces entropy and requires additional
input power to overcome pressure losses through the microchannel. Reducing
entropy production from thermofluid irreversibilities in microfluidic devices
can provide useful benefits like improving the efficiency of energy utilization
within the overall system and pinpointing the locations, causes and types of
thermodynamic losses. In Section 5, numerical results from microchannel flow
simulations will be presented and discussed.


## 5. Thermal Management in PEM Fuel Cell

The proton exchange membrane fuel cell (PEMFC) is an alternative
energy-conversion device with the potential for significantly increased energy
density [13]. While the problem of waste-water management in PEMFCs is
receiving increasing attention, thermal management creates a destructive load
in the design of portable power systems. It has been predicted that the rate of heat removal will
become a critical parameter in the operation of fuel cells [14]. The PEMFC
schematic shown in Figure 5 includes the heat generation from the catalyst
layer at the cathode of the PEMFC. The interconnection material represents the
lumped model for the material assembly, including the graphite plate, gas
channels, electrodes, and the membrane. Detailed modeling of the transfer of
heat across this material involves the derivation of the effective thermal
conductivity and inclusion of the heat generation term in the energy equation.
The half reaction at the anode releases both protons, which conduct through the
membrane, and electrons which are transferred to the cathode through an
external circuit. The complete reaction of both protons and electrons with
oxygen (from air) in the catalyst layer of the cathode is exothermic,
representing the source of the destructive heat load.Figure 5
Exploded view of PEMFC's MEA.The control
of the heat generated from the exothermic reaction at the cathode of the
membrane electrolyte assembly (MEA) is significant to the performance of
PEMFCs. Cooling plate with microchannels adopted recently for the purpose of
regulating the flow of water vapor at the MEA; as both water vapor starvation
and water-flooding reduce the output current density of fuel cells. Detail
study of the design of cooling plate for thermal management in PEMFCs is
documented elsewhere [15].


## 6. Results and Discussion

Numerical predictions of nitrogen gas flow in microchannels are presented in this
section. Numerical simulations for the primary (hot) gas stream with various
grid configurations are studied and grid-independent results are presented in
this section. Figure6
validated the predicted velocity profile within a channel by
comparing results with analytical results. A slip scale of about 0.3 is
observed at the pressure ratio of 1.34 and the profile agrees with analytical
results, showing error of about 1% at the near wall of the channel. The
reduction of the maximum velocity for slip profile appears to be due to the
thermal kinetic energy exchange at the boundary of the microchannels. Slip
irreversibility effects also include the demand of computational resources at
the region of flow, where the greatest gradients of flow variables are
experienced, as shown in Figure 7.Figure 6
Velocity profile validation with slip boundary conditions.Figure 7
Computational effects of slip irreversibility.The axial conduction temperature gradient has an influence on the
slip boundary condition. However, it can be shown that entropy production from
fluid friction depends on both velocity and temperature gradients within the
flow field. Figure8 predicted the effect of the axial conduction of thermal
variable and the momentum accommodation coefficient along the length of the
microdevice. According to (6), the wall slip velocity is dependent on both the
momentum accommodation coefficient and the axial temperature gradient. This dependence directly impacts friction
with an unusual thermal mechanism of entropy production in micro- and nanoscale
channels. This phenomenon may relate to previous studies by Wang et al. [9], which indicate that thermal energy may be temporarily converted to kinetic
motion in a cohesive manner within very small scale systems. It, therefore, implies
that wall temperature gradients affect entropy production in the fluid stream.
Efforts to fully comprehend this mechanism physically require a statistically
based balance of momentum and energy equations for intermolecular motion near
the wall. As channel sizes diminishes to micro- and nanoscales, the number of
molecule interactions within the channel decreases.Figure 8
Effect of axial conduction and momentum accommodation coefficient.Interaction between gas and solid phase molecules can influence the
extent of energy transfer and accommodation in gas-surface collisions at micro-
or nanoscales. The likelihood of random molecular motions within the wall (in
the form of internal energy) being aligned in the direction of reflected motion
of impacting fluid molecules rises when the channel size diminishes. In other
words, when fewer wall molecules are considered in smaller channels, their
alignment toward the gas velocity vector may contribute an effective temporary
rise of kinetic energy. In this way, local temperature gradients affect the
wall slip in (6) and subsequently entropy production in the fluid stream, when
formulating the second law. Thermal energy exchange affects the wall slip,
thereby potentially reducing frictional entropy production. This has important
implications for operation and efficiency of microfluidic devices.Figure9 shows the effects of grid refinement on the
predicted entropy production. The entropy production remains nearly constant
over a range of grid spacings, which suggests that the results are essentially
grid independent.Figure 9
Grid independency based on entropy production.


## 7. Conclusions

Near-wall irreversibility effect on the solid-fluid interface of conjugate microdevice
has been analyzed. Exergetic design procedure of microfluidic devices based on second
law was coupled with the Navier-Stokes model, developed and packaged as a
robust EnerghxFlow solver. The present model is capable of simulating slip
irreversibility effects toward the design of a microheat sink. However, and
especially for a microfuel cell and other devices with porous interconnection, a
complete treatment of microscale phenomena like microconvection within the
solid-fluid interface will require the development of hybrid formulation based
on both continuum assumption and Lattice-Boltzmann technique. Sensitivity
analysis of the effect of the momentum accommodation coefficient and the axial
conduction due to the conjugate transfer of heat across the solid-fluid
interface has been presented. The magnitude of the momentum accommodation
coefficient is inversely proportional to the wall-slip velocity and plays a
significant role in the dynamics of the thermal kinetic exchange at the near wall.



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*Source: 176495-2009-02-02.xml*