The thermal entrance length for laminar flow in ducts varies with the Reynolds number, Prandtl number Pr µc p /k and the type of the boundary condition imposed on the duct wall.. The mi
Trang 1The continuum assumption breaks down, however, whenever the mean free path of the moleculesbecomes the same order of magnitude as the smallest significant dimension of the problem In gas flows,the deviation of the state of the fluid from continuum is represented by the Knudsen number, defined as
Kn ⬅ λ/L The mean free path λ is the average distance traveled by the molecules between successive lisions, and L is the characteristic length scale of the flow The appropriate flow and heat-transfer models
col-depend on the range of the Knudsen number, and a classification of the different gas flow regimes is asfollows [Schaaf and Chambre, 1961]:
103 Kn 101 slip flow
101 Kn 101 transition flow
101 Kn free molecular flow
In the slip-flow regime, the continuum flow model is still valid for the calculation of the flow propertiesaway from solid boundaries However, the boundary conditions have to be modified to account for theincomplete interaction between the gas molecules and the solid boundaries Under normal conditions,
Kn is less than 0.1 for most gas flows in microchannel heat sinks with a characteristic length scale on the
order of 1 µm Therefore, only the slip-flow regime will be discussed, not the transition- or the molecular-flow regime The continuum assumption is of course valid for liquid flows in microchannelheat sinks
The most convenient framework within which heat-transfer problems can be studied is the system, which
is a quantity of matter, not necessarily constant, contained within a boundary The boundary can be ical, partly physical and partly imaginary, or wholly imaginary The physical laws to be discussed arealways stated in terms of a system A control volume is any specific region in space across the boundaries
phys-of which mass, momentum, and energy may flow and within which mass, momentum, and energy age may take place and on which external forces may act The complete definition of a system or a con-trol volume must include at least implicitly the definition of a coordinate system, as the system may bemoving or stationary The characteristic of interest of a system is its state, which is a condition of the sys-tem described by its properties A property of a system can be defined as any quantity that depends onthe state of the system and is independent of the path (i.e., previous history) by which the system arrived
stor-at the given ststor-ate If all the properties of a system remain unchanged, the system is said to be in an librium state
equi-A change in one or more properties of a system necessarily means that a change in the state of the systemhas occurred The path of the succession of states through which the system passes is called the process.When a system in a given initial state goes through a number of different changes of state or processesand finally returns to its initial state, the system has undergone a cycle The properties describe the state
of a system only when it is in equilibrium If no heat transfer takes place between any two systems whenthey are placed in contact with each other, they are said to be in thermal equilibrium Any two systemsare said to have the same temperature if they are in thermal equilibrium with each other Two systemsthat are not in thermal equilibrium have different temperatures, and heat transfer may take place fromone system to the other Therefore, temperature is a property that measures the thermal level of a system.When a substance exists as part liquid and part vapor at a saturation state, its quality is defined as theratio of the mass of vapor to the total mass The quality χ may be considered a property ranging between
0 and 1 Quality has meaning only when the substance is in a saturated state (i.e., at saturated pressureand temperature) The amount of energy that must be transferred in the form of heat to a substance held
at constant pressure so that a phase change occurs is called the latent heat It is the change in enthalpy,which is a property of the substance at the saturated conditions, of the two phases The heat of vaporiza-tion, boiling, is the heat required to completely vaporize a unit mass of saturated liquid
Trang 212.2.4 General Laws
The general laws when referring to an open system (e.g., microchannel heat sink) can be written in either
an integral or a differential form The law of conservation of mass simply states that in the absence of anymass–energy conversion the mass of the system remains constant Thus, in the absence of a source or
sink, Q 0, the rate of change of mass in the control volume (CV) is equal to the mass flux through the
control surface (CS) Newton’s second law of motion states that the net force F acting on a system in an
inertial coordinate system is equal to the time rate of change of the total linear momentum of the system.Similarly, the law of conservation of energy for a control volume states that the rate of change of the total
energy E of the system is equal to the sum of the time rate of change of the energy within the control
vol-ume and the energy flux through the control surface
The first law of thermodynamics, which is a particular statement of conservation of energy, states thatthe rate of change in the total energy of a system undergoing a process is equal to the difference betweenthe rate of heat transfer to the system and the rate of work done by the system The second law of ther-
modynamics leads to the introduction of entropy S as a property of the system It states that the rate of
change in the entropy of the system is either equal to or larger than the rate of heat transfer to the systemdivided by the system temperature during the heat-transfer process Even in cases where entropy calcula-tions are not of interest, the second law of thermodynamics is still important because it is equivalent tostating that heat cannot pass spontaneously from a lower to a higher temperature system
12.2.5 Particular Laws
Fourier’s law of heat conduction, based on the continuum concept, states that the heat flux due to duction in a given direction (i.e., the heat-transfer rate per unit area) within a medium (solid, liquid, orgas) is proportional to temperature gradient in the same direction, namely:
where qⴖ is the heat flux vector, k is the thermal conductivity, and T is the temperature.
Newton’s law of cooling states that the heat flux from a solid surface to the ambient fluid by
convec-tion q is proporconvec-tional to the temperature difference between the solid surface temperature T w and the
fluid free-stream temperature T∞as follows:
Trang 3In this set of equations, ρ is the density; P is the thermodynamic pressure; B is the body force (e.g., gravity);
µand η are the shear and the bulk viscosity coefficients respectively; c p is the specific heat; θ is the heatsource or sink; and φ is the viscous dissipation given by:
φ 2µ冤冢∂
∂
u x
冣2
冢∂
∂
v y
冣2
冢∂
∂
w z
∂
∂
w y
∂
∂
w x
冣2
冥 (12.6)
where u, v and w are the three components of the velocity vector U in a rectangular coordinate system
(x, y, z) The state of a simple compressible pure substance or of a mixture of gases is defined by two
inde-pendent properties From experimental observations, it has been established that the behavior of gases atlow density is closely given by the ideal-gas equation of state:
where R is the specific gas constant At very low density, all gases and vapors approach ideal-gas behavior;
however, the behavior may deviate substantially from that at higher densities Nevertheless, due to its plicity, the ideal gas equation of state has been widely used in thermodynamic calculations
sim-12.2.7 Size Effects
Length scale is a fundamental quantity that dictates the type of forces or mechanisms governing physicalphenomena Body forces are scaled to the third power of the length scale Surface forces depend on thefirst or the second power of the characteristic length This difference in slopes means that a body forcemust intersect a surface force as a function of the length scale Empirical observations in biological studiesand MEMS show that 1 mm is approximately the order of the demarcation scale [Ho and Tai, 1998] Thecharacteristic scale of microsystems is smaller than 1 mm; therefore, body forces such as gravity can beneglected in most cases, even in liquid flows, in comparison with surface forces The large surface-to-volumeratio is another inherent characteristic of microsystems This ratio is typically inversely proportional tothe smaller length scale of the device cross-section and is about 1 µm in surface-micromachined devices.The large surface-to-volume ratio in microdevices accentuates the role of surface effects
12.2.7.1 Noncontinuum Mechanics
The characteristic length scale of a microchannel (i.e., the hydraulic diameter) is typically on the order of
a few micrometers When gas is the working fluid, the mean free path is about 10 to 100 nm, resulting in
a Knudsen number of about 0.05 Thus, the flow is considered to be in the slip regime, 0.001 Kn 0.1,
where deviations from the state of continuum are relatively small Consequently, the flow is still governed
by Equations (12.3) to (12.5), derived and based on the continuum assumption The rarefaction effect ismodeled through Maxwell’s velocity-slip and Smoluchowski’s temperature-jump boundary conditions[Beskok and Karniadakis, 1994]:
U w and T w are the wall velocity and temperature respectively; U s and T jare the gas flow velocity and
tem-perature at the boundary; n is the direction normal to the solid boundary; γ c p /c vis the ratio of specificheats; and σU and σT are the momentum and energy accommodation coefficients respectively, whichmodel the momentum and energy exchange of the gas molecules impinging on the solid boundary.Experiments with gases over various surfaces show that both coefficients are approximately 1.0 Thisessentially means a diffuse reflection boundary condition, where the impinging molecules are reflected at
a random angle uncorrelated with the incident angle
T
∂U
∂n
Trang 412.2.7.2 Electric Double Layer
Most solid surfaces are likely to carry electrostatic charge (i.e., an electric surface potential) due to brokenbonds and surface charge traps When a liquid containing a small amount of ions is forced through amicrochannel under hydrostatic pressure, the solid-surface charge will attract the counterions in the liq-uid to establish an electric field The arrangement of the electrostatic charges on the solid surface and thebalancing charges in the liquid is called the electric double layer (EDL), as illustrated in Figure 12.2.Counterions are strongly attracted to the surface and form a compact layer, about 0.5 nm thick, of immo-bile counterions at the solid–liquid interface due to the surface electric potential Outside this layer, the ionsare affected less by the electric field and are mobile The distribution of the counterions away from theinterface decays exponentially within the diffuse double layer, with a characteristic length inversely pro-portional to the square root of the ion concentration in the liquid The thickness of the diffuse EDL rangesfrom a few up to several hundreds of nanometers depending on the electric potential of the solid surface,the bulk ionic concentration, and other properties of the liquid Consequently, EDL effects can be neglected
in macrochannel flow In microchannels, however, the EDL thickness is often comparable to the teristic size of the channel, and its effect on the fluid flow and heat transfer may not be negligible
charac-Consider a liquid between two parallel plates, separated by a distance H, containing positive and negative
ions in contact with a planar, positively charged surface The surface bears a uniform electrostatic potentialψ
0, which decreases with the distance from the surface The electrostatic potential ψ at any point near the surface is approximately governed by the Debye–Huckle linear approximation [Mohiuddin Mala
et al., 1997]:
where ε is the dielectric constant of the medium, and ε0is the permittivity of vacuum; ζ is the valence of
negative and positive ions; e is the electron charge; k b is the Boltzmann constant; and n0is the ionic
con-centration The characteristic thickness of the EDL is the Debye length given by k d1 (εε0k b T/2 n0ζ2e2)1/2
For the boundary conditions when ψ 0 at the midpoint, y 0, and ψ ξ on both walls, y H/2,
Diffuse double layer
Co-ions Counter-ions
FIGURE 12.2 Electric double layer (EDL) at the channel wall.
Trang 512.2.7.3 Polar Mechanics
In classical nonpolar mechanics, the mechanical action of one part of a body on another is assumed to
be equivalent to a force distribution only However, in polar mechanics, the mechanical action is assumed
to be equivalent to not only a force but also a moment distribution Thus, the state of stress at a point innonpolar mechanics is defined by a symmetric second-order tensor, which has six independent components
On the other hand, in polar mechanics, the state of stress is determined by a stress tensor and a stress tensor The most important effect of couple stresses is to introduce a size-dependent effect that isnot predicted by the classical nonpolar theories [Stokes, 1984]
couple-In micropolar fluids, rigid particles contained in a small volume can rotate about the center of the ume element described by the microrotation vector This local rotation of the particles is in addition tothe usual rigid body motion of the entire volume element In micropolar fluid theory, the laws of classi-cal continuum mechanics are augmented with additional equations that account for conservation ofmicroinertia moments Physically, micropolar fluids represent fluids consisting of rigid, randomly ori-ented particles suspended in a viscous medium, where the deformation of the particles is ignored Themodified momentum, angular momentum, and energy equations are
ρc p k∇2T τ : (∇U) σ : (∇ΩΩ) τx ΩΩ (12.13)where ΩΩis the microrotation vector and I is the associated microinertia coefficient; f and g are the body
and couple force vectors, respectively, per unit mass; τ and σ are the stress and couple-stress tensors; τ : (∇U)
is the dyadic notation for τjiUi,j, the scalar product of τ and ∇U If σ 0 and g ΩΩ 0, then the stresstensor t reduces to the classical symmetric stress tensor, and the governing equations reduce to the classi-cal model [Lukaszewicz, 1999]
12.3 Single-Phase Convective Heat Transfer in Microducts
Flows completely bounded by solid surfaces are called internal flows and include flows through ducts, pipes,nozzles, diffusers, etc External flows are flows over bodies in an unbounded fluid Flows over a plate, acylinder, or a sphere are examples of external flows, and they are not within the scope of this article Onlyinternal flows, in either liquid or gas phase, within microducts will be discussed, with an emphasis on sizeeffects, which may potentially lead to behavior that is different than similar flows in macroducts
12.3.1 Flow Structure
Viscous flow regimes are classified as laminar or turbulent on the basis of flow structure In the laminarregime, flow structure is characterized by smooth motion in laminae, or layers The flow in the turbulentregime is characterized by random three-dimensional motions of fluid particles superimposed on themean motion These turbulent fluctuations enhance the convective heat transfer dramatically However,
turbulent flow occurs in practice only as long as the Reynolds number, Re ρU m D h/µ, is greater than a
critical value, Re cr The critical Reynolds number depends on the duct inlet conditions, surface roughness,
vibrations imposed on the duct walls, and the geometry of the duct cross-section Values of Re crfor ious duct cross-section shapes have been tabulated elsewhere [Bhatti and Shah, 1987] In practical appli-cations, though, the critical Reynolds number is estimated to be
Trang 6where U m is the mean flow velocity and D h 4A/S is the hydraulic diameter, with A and S being the
cross-section area and the wetted perimeter respectively Microchannels are typically larger than 1000 µm in lengthwith a hydraulic diameter of about 10 µm The mean velocity for gas flow under a pressure drop of about0.5 MPa is less than 100 m/s, and the corresponding Reynolds number is less than 100 The Reynoldsnumber for liquid flow will be even smaller due to the much higher viscous forces Thus, in most appli-cations, the flow in microchannels is expected to be laminar Turbulent flow may develop in short chan-nels with large hydraulic diameter under high-pressure drop and therefore will not be discussed here
12.3.2 Entrance Length
When a viscous fluid flows in a duct, a velocity boundary layer develops along the inside surfaces of theduct The boundary layer fills the entire duct gradually, as sketched in Figure 12.3 The region where thevelocity profile is developing is called the hydrodynamics entrance region, and its extent is the hydrody-
namic entrance length An estimate of the magnitude of the hydrodynamic entrance length L hin laminarflow in a duct is given by Shah and Bhatti (1987):
D
L h h
The region beyond the entrance region is referred to as the hydrodynamically fully developed region Inthis region, the boundary layer completely fills the duct and the velocity profile becomes invariant withthe axial coordinate
If the walls of the duct are heated (or cooled), a thermal boundary layer will also develop along the innersurfaces of the duct, shown in Figure 12.3 At a certain location downstream from the inlet, the flow becomes
fully developed thermally The thermal entrance length L tis then the duct length required for the developingflow to reach fully developed condition The thermal entrance length for laminar flow in ducts varies with
the Reynolds number, Prandtl number (Pr µc p /k) and the type of the boundary condition imposed on
the duct wall It is approximately given by:
More accurate discussion on thermal entrance length in ducts under various laminar flow conditions can
be found elsewhere [e.g., Shah and Bhatti, 1987]
In most practical applications of microchannels, the Reynolds number is less than 100 while the Prandtlnumber is on the order of 1 Thus, both the hydrodynamic and thermal entrance lengths are less than 5times the hydraulic diameter Because the length of microchannels is typically two orders of magnitudelarger than the hydraulic diameter, both entrance lengths are less than 5% of the microchannel length andcan be neglected
x
z
U U
Simultaneously developing flow (Pr >1)
FIGURE 12.3 Hydrodynamically and thermally developing flow, followed by hydrodynamically and thermally fully developed flow.
Trang 712.3.4 Fully Developed Gas Flow Forced Convection
Analytical solution of Equations (12.17) to (12.20) is not available Some solutions can be obtained uponfurther simplification of the mathematical model Indeed, incompressible gas flows in macroducts withdifferent cross-sections subjected to a variety of boundary conditions are available [Shah and Bhatti, 1987].However, the important features of gas flow in microducts are mainly due to rarefaction and compress-ibility effects Two more effects due to acceleration and nonparabolic velocity profile were found to be ofsecond order compared to the compressibility effect (van den Berg et al., 1993) The simplest system fordemonstration of the rarefaction and compressibility effects is the two-dimensional flow between paral-
wise derivatives can be ignored except the pressure gradient, which is the driving force The Mach
number, Ma U/a, is the ratio between the fluid speed and the speed of sound a In such a case, the
momentum equation reduces to:
Integration of Equation (12.21) twice with respect to y, assuming P P(x), yields the following velocity
profile [Arkilic et al., 1997]:
Trang 8where Kn(x) λ(x)/H The streamwise pressure distribution P(x) calculated based on the same model is
given by:
6Kn o冪 冢 莦6K莦n莦o莦莦莦莦 冣2
莦 莦莦 冤 冢 莦莦 莦莦1冣 莦莦 莦12K莦n莦o莦莦 冢 莦莦1冣冥 莦 冢 莦莦 冣 莦 (12.25)
where P i is the inlet pressure, P o the outlet pressure, and Kn ois the outlet Knudsen number It is difficult to
verify experimentally the cross-stream velocity distribution u(y) within a microchannel However, detailed
pressure measurements have been reported [Liu et al., 1993; Pong et al., 1994] A picture of a microchannelintegrated with pressure sensors for such experiments is shown in Figure 12.4a Indeed, the calculatedpressure distributions based on Equation (12.25) were found to be in a close agreement with the measured
values as shown in Figure 12.4b [Li et al., 2000] Furthermore, the mass flow rate Q mas a function of the
inlet and outlet conditions is obtained by integrating the velocity profile with respect to x and y as follows:
P
P o
i
where W is the width of the channel This simple equation was found to yield accurate results for three
different working gasses: nitrogen, helium, and argon, with ambient temperatures ranging from 20 to60°C, as demonstrated in Figure 12.5[Jiang et al., 1999a]
Microchannel
Inlet/outlet hole
0 5 10 15 20 25 30 35 40
butions (Reprinted by permission of Elsevier Science from Li, X et al [2000] “Gas Flow in Constriction Microdevices,”
Sensors and Actuators A, 83, pp 277–83.)
Trang 9The microchannel flow temperature distribution and heat flux depend on the boundary conditions,and extensive analytical work has been conducted (Harley et al., 1995; Beskok et al., 1996) However,closed-formed analytical solutions in general are still not available Numerical simulations of Equations(12.17) to (12.20) were carried out for constant wall temperature and constant heat flux boundary con-ditions by Kavehpour et al (1997), and the results are summarized in Figure 12.6 The heat transfer ratefrom the wall to the gas flow decreases while the entrance length increases due to the rarefaction effect(i.e., increasing Knudsen number) This may not be a universal result, however, as the slip flow conditionsinclude two competing effects [Zohar et al., 1994] The velocity slip at the wall increases the flow rate, thusenhancing the cooling efficiency On the other hand, the temperature jump at the boundary acts as a bar-rier to the flow of heat to the gas, thus reducing the cooling efficiency The net result of these effectsdepends on the specific material properties and specific geometry of the system.
0 1 2 3 4 5 6
FIGURE 12.5 Slip flow effect on microchannel mass flow rate as a function of the total pressure drop for
various working gases (a) and wall temperatures (b) (Reprinted with permission from Jiang, L et al [1999]
“Fabrication and Characterization of a Microsystem for Microscale Heat Transfer Study” J Micromech Microeng., 9,
pp 422–28.)
Trang 10A microchannel integrated with suspended temperature sensors was constructed (Figure 12.7a) for
an initial attempt to experimentally assess the slip-flow effects on heat transfer in microchannels [Jiang
et al., 1999a] The resulting temperature distributions along the microchannel are shown in Figure 12.7bfor different wall temperatures and pressure drops In all cases, the temperature along the channel is almostuniform and equal to the wall temperature, and no cooling effect has been observed Indeed, on the one
Kn = 0.00
Kn = 0.03
Kn = 0.10
0.10 10
100
FIGURE 12.6 Numerical simulations of the effect of the inlet Knudsen number Kn i on the Nusselt number Nu along
a microchannel for uniform wall temperature (Nu T ): (a) and heat flux (Nu H), (b) boundary conditions (Reprinted by
permission of Taylor & Francis, Inc., from Kavehpour, H.P et al [1997] “Effects of Compressibility and Rarefaction
on Gaseous Flows in Microchannels,” Numerical Heat Transfer A, 32, pp 677–96.)
Trang 11hand, the slip flow effects are small, but on the other hand, the sensitivity of the experimental system isnot sufficient Thus, experiments with higher resolution and greater sensitivity are required to accuratelyverify the weak slip flow effects on the temperature and the heat-transfer coefficient predicted by theo-retical analyses and numerical simulations.
12.3.5 Fully Developed Liquid Flow Forced Convection
Liquid flow is considered to be incompressible even in microducts because the distance between the ecules is much smaller than the characteristic scale of the flow Hence, no rarefaction effect is encoun-tered, and the classical model in Equation (12.21) should be valid Again, in such a case, extensive dataare readily available [Shah and Bhatti, 1987] However, two unique features of liquid flow in microducts,polarity and EDL, could affect the flow behavior
mol-The characteristic length scale of the electric double layer is inversely proportional to the square root
of the ion concentration in the liquid For example, in pure water the scale is about 1 µm, while in 1 mole
of NaCl solution the EDL length scale is only 0.3 nm Thus, in microducts, liquid flow with low ionic
con-centration and the associated heat transfer can be affected by the presence of the EDL The x-momentum
and energy equations for a two-dimensional duct flow can be reduced to [Mohiuddin Mala et al., 1997]:
d
P x
FIGURE 12.7 Slip-flow effect on microchannel flow: (a) microchannel integrated with suspended temperature sensors; (b) measured streamwise temperature distributions for different ambient temperature and pressure drop (Reprinted
with permission from Jiang, L et al [1999] “Fabrication and Characterization of a Microsystem for Microscale Heat
Transfer Study,” J Micromech Microeng 9, pp 422–28.)
Trang 12ρc p冢u ∂
∂
T x
冣2
(12.28)
where E s is the steaming potential and L is the duct length Equation (12.27) was solved analytically, and
Equation (12.28) was solved numerically for constant wall temperature boundary condition for a giveninlet liquid temperature The results showed that both the temperature gradient at the wall and the dif-ference between the wall and the bulk temperature decrease with downstream distance The value of the
temperature gradient decreases much faster, resulting in a decreasing Nusselt number, Nu hD h /k, along
the channel, as plotted in Figure 12.8 However, with no double layer effects (i.e., ξ 0) a higher
heat-transfer rate (higher Nu) is obtained The EDL results in a reduced flow velocity (higher apparent viscosity),
thus decreasing the heat-transfer rate
In order to evaluate micropolar effects on microchannel heat transfer, Jacobi (1989) considered the steadyfully developed laminar flow in a cylindrical microtube with uniform heat flux, for which the energyequation is given by:
where r is the radial coordinate Both the velocity and temperature radial distributions were analytically
estimated Based on the temperature field, the heat-transfer rate was calculated and the results are shown
inFigure 12.9 for different values of Γ, a length scale that depends on the viscosity coefficients of the
micropolar fluid The Nusselt number is smaller than the classical value of Nu 4.3636 by as much as 7%
for this micropolar flow Although the micropolar fluid theory has been applied to many situations, however,the drawback to these analyses is still the unknown viscosity coefficients
Clearly, the EDL and micropolar fluid effects on liquid forced convection in microducts are indirect;namely, the velocity is modified due to these effects and, as a consequence, the heat-transfer rate isaffected Thus, it is important first to verify the hydrodynamic effects Indeed, it has been suggested in afew reports that theoretical calculations based on the classical model did not agree with experimental
= 163.2, = 0
= 40.8, = 0
= 40.8, = 50
FIGURE 12.8 Electric double layer effect on the variation of the local Nusselt number Nu along the channel length.
(Reprinted by permission of Elsevier Science from Mohiuddin Mala, G et al [1997] “Heat Transfer and Fluid Flow in
Microchannels,” Int J Heat Mass Transfer 40, pp 3079–88.)
Trang 13measurements of liquid flow properties in microchannels [Pfahler et al., 1990; Peng et al., 1994; Peng andPeterson, 1996] An experimental study of water flow in a microchannel with a cross-section area of
600 µm 30 µm was carried out specifically to evaluate micropolar effects by Papautsky et al (1998) Theyconcluded that micropolar fluid theory provides a better approximation of the experimental data thanthe classical theory However, a close examination of the results shows that the difference between the results
of the two theories is smaller than the difference between the experimental data and the predictions ofeither theory
In carefully conducted experiments of water flow through a suspended microchannel (Figure 12.10a) with
a cross-section area of 20 µm 2 µm and under a pressure drop of up to 500 psi, none of these effectshas been observed [Wu et al., 1998] The slight mismatch between theory and experiment was found to
be a result of the bulging effect of the channel roof under the high pressure The deformation of the nel roof can be measured accurately Once the corrected cross-section area has been accounted for ade-quately in the calculations, the classical theory results agree well with the experimental measurements asevident in Figure 12.10b However, more research work is required to verify these observations becausethese discrepancies may have to do more with experimental errors rather than true size effects
chan-12.4 Two-Phase Convective Heat Transfer in Microducts
Micro heat sinks have been constructed as micro heat exchangers for cooling of thermal microsystemsdeveloped and investigated either experimentally or theoretically It is a common finding that the coolingrates in such microchannel heat exchangers should increase significantly due to a decrease in the convectiveresistance to heat transport caused by a drastic reduction in the thickness of the thermal boundary layers The potentially high heat-dissipation capacity of such a micro heat sink is based on the large heat-transfer-surface-to-volume ratio of the microchannel heat exchanger In order to increase the heat fluxfrom a microchannel with single-phase flow while maintaining practical limits on surface temperature,
it is necessary to increase the heat-transfer coefficient by either increasing the flow rate or decreasing the hydraulic diameter Both are accompanied by a large increase in the pressure drop However,
3.50 3.75 4.00 4.25 4.50
ky/m y
Trang 14forced-convection flow with phase change can achieve a very high heat-removal rate for a constant flowrate while maintaining a relatively constant surface temperature determined by the saturation properties
of the cooling fluid The advantage of using two-phase over single-phase micro heat sinks is clear phase heat sinks compensate for high heat flux by a large streamwise increase in both coolant and heatsink temperature Two-phase heat sinks, in contrast, utilize latent heat exchange, which maintains stream-wise uniformity both in the coolant and the heat sink temperature at a level set by the coolant saturationtemperature Therefore, it is expected that two-phase heat transfer may lead to significantly more efficientheat transfer, and a two-phase micro heat exchanger would be the most promising approach for cooling inmicrosystems [Stanley et al., 1995]
Single-Heat transfer during boiling of a liquid in free convection is essentially determined by the differencebetween the heating-surface and boiling temperatures, the properties of the liquid, and the properties ofthe heating surface Thus, the heat-transfer coefficient can be represented by a simple empirical correla-
tion of the form h ∝ q m During boiling in forced convection, however, the flow velocities of the vaporand liquid phases and the phase distribution play additional roles Consequently, the mass flow rate and
the quality are additional limiting factors, giving rise to a correlation of the form h ∝ q m Q n
m f(χ) Forced
convection boiling is complex not only due to the coexistence of two separate phases having differentproperties but also especially to the existence of a highly convoluted vapor–liquid interface resulting in avariety of flow patterns Typical patterns that have experimentally been observed in macroducts, such as
0 0 0.1 0.2 0.3 0.4 0.5 0.6
FIGURE 12.10 Microchannel liquid flow: (a) microchannel integrated with temperature sensors on the channel roof; (b) a comparison between liquid flow rate measurements as a function of the pressure drop and theoretical cal-
culations based on classical and bulging models (Reprinted with permission from Wu, P et al [1998] “A Suspended
Microchannel with Integrated Temperature Sensors for High-Pressure Flow Studies,” in Proc 11th Int Workshop on
Micro Electro Mechanical Systems (MEMS ’98), pp 87–92 © 1998/2000 IEEE.)
Trang 15bubble, slug, churn, annular, and drop flow, are sketched in Figure 12.11 Accordingly, flow pattern mapshave been suggested in which the duct orientation on heat-transfer boiling is significant due to gravityeffects [Stephan, 1992].
12.4.1 Boiling Curves
Forced convection boiling is attractive because it ensures low device temperature for high power tion or, alternatively, it allows higher power dissipation for a given device temperature Measurements ofeither the inner wall or the fluid bulk temperature distributions along a microduct under forced convec-tion boiling are not available yet, due to the difficulty in integrating sensors at the desired locations.However, measurements of the surface temperature of a microchannel heat sink device have been reported[Jiang et al., 1999b] A picture of the integrated microsystem consisting of an array of microducts, a localmicroheater, and an array of temperature microsensors is shown in Figure 12.12 The 35 diamond-shapedmicroducts, each with a hydraulic diameter of about 40 µm, are buried between two bonded silicon
dissipa- phase vapor
phase liquid
Single-Drop flow
Annular flow
Slug flow
Plug flow
Bubbly flow
Wall temperature
Bulk fluid temperature
FIGURE 12.11 Wall and mean fluid temperature, flow patterns, and the accompanying heat-transfer ranges in a
typical heated duct (Reprinted with permission from Stephan, K [1992] Heat Transfer in Condensation and Boiling,
Springer-Verlag, Berlin.)
Flow direction
Outlet hole Inlet hole