The continuum assumption breaks down, however, whenever the mean free path of the molecules becomes 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 ϵ λ /L. The mean free path λ is the average distance traveled by the molecules between successive col- lisions, and L is the characteristic length scale of the flow. The appropriate flow and heat-transfer models depend on the range of the Knudsen number, and a classification of the different gas flow regimes is as follows [Schaaf and Chambre, 1961]: Kn Ͻ 10 Ϫ3 continuum flow 10 Ϫ 3 Ͻ Kn Ͻ 10 Ϫ 1 slip flow 10 Ϫ1 Ͻ Kn Ͻ 10 ϩ1 transition flow 10 ϩ1 Ͻ Kn free molecular flow In the slip-flow regime, the continuum flow model is still valid for the calculation of the flow properties away from solid boundaries. However, the boundary conditions have to be modified to account for the incomplete 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 free- molecular-flow regime. The continuum assumption is of course valid for liquid flows in microchannel heat sinks. 12.2.3 Thermodynamic Concepts 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 phys- ical, partly physical and partly imaginary, or wholly imaginary. The physical laws to be discussed are always stated in terms of a system. A control volume is any specific region in space across the boundaries of which mass, momentum, and energy may flow and within which mass, momentum, and energy stor- 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 be moving 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 on the state of the system and is independent of the path (i.e., previous history) by which the system arrived at the given state. If all the properties of a system remain unchanged, the system is said to be in an equi- librium state. A change in one or more properties of a system necessarily means that a change in the state of the system has 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 processes and 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 when they are placed in contact with each other, they are said to be in thermal equilibrium. Any two systems are said to have the same temperature if they are in thermal equilibrium with each other. Two systems that are not in thermal equilibrium have different temperatures, and heat transfer may take place from one 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 the ratio 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 pressure and 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. 12-4 MEMS: Applications © 2006 by Taylor & Francis Group, LLC 12.2.4 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 any mass–energy conversion the mass of the system remains constant. Thus, in the absence of a source or sink, Q ϭ 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 that the rate of change in the total energy of a system undergoing a process is equal to the difference between the 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 system divided 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 to stating 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 con- duction in a given direction (i.e., the heat-transfer rate per unit area) within a medium (solid, liquid, or gas) is proportional to temperature gradient in the same direction, namely: q؆ ϭ Ϫk∇T (12.1) where q؆ 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 qЉ is proportional to the temperature difference between the solid surface temperature T w and the fluid free-stream temperature T ∞ as follows: q؆ ϭ h(T w Ϫ T ∞ ) (12.2) where h is the heat transfer coefficient. 12.2.6 Governing Equations The integral form of the conservation laws is useful for the analysis of the gross behavior of the flow field. However, detailed point-by-point knowledge of the flow field can be obtained only from the equations of fluid motion in differential form. Microchannel heat sinks typically incorporate arrays of elongated microchannels varying in cross-sectional shape; therefore, it is most convenient to use the governing equa- tions derived either in a rectangular or cylindrical coordinate system.The governing equations for forced con- vection heat transfer in differential form include conservation of mass, momentum, and energy as follows: ϩ ρ (∇ и U) ϭ 0 (12.3) ρ ϭ Ϫ∇P ϩ ρ B ϩ µ ∇ 2 U ϩ ( µ ϩ η ) ∇ (∇ и U) (12.4) ρ c p ϭ k∇ 2 T ϩ ϩ φ ϩ θ (12.5) DP ᎏ Dt DT ᎏ Dt DU ᎏ Dt D ρ ᎏ Dt Microchannel Heat Sinks 12-5 © 2006 by Taylor & Francis Group, LLC In this set of equations, ρ is the density; P is the thermodynamic pressure; B is the body force (e.g., gravity); µ and η are the shear and the bulk viscosity coefficients respectively; c p is the specific heat; θ is the heat source or sink; and φ is the viscous dissipation given by: φ ϭ 2 µ ΄ ᎏ ∂ ∂ u x ᎏ 2 ϩ ᎏ ∂ ∂ v y ᎏ 2 ϩ ᎏ ∂ ∂ w z ᎏ 2 ϩ ᎏ 1 2 ᎏ ᎏ ∂ ∂ u y ᎏ ϩ ᎏ ∂ ∂ x v ᎏ 2 ϩ ᎏ 1 2 ᎏ ᎏ ∂ ∂ v z ᎏ ϩ ᎏ ∂ ∂ w y ᎏ 2 ϩ ᎏ 1 2 ᎏ ᎏ ∂ ∂ u z ᎏ ϩ ᎏ ∂ ∂ w x ᎏ 2 ΅ (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 at low density is closely given by the ideal-gas equation of state: P ϭ ρ RT (12.7) 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 sim- plicity, the ideal gas equation of state has been widely used in thermodynamic calculations. 12.2.7 Size Effects Length scale is a fundamental quantity that dictates the type of forces or mechanisms governing physical phenomena. Body forces are scaled to the third power of the length scale. Surface forces depend on the first or the second power of the characteristic length. This difference in slopes means that a body force must intersect a surface force as a function of the length scale. Empirical observations in biological studies and MEMS show that 1 mm is approximately the order of the demarcation scale [Ho and Tai, 1998]. The characteristic scale of microsystems is smaller than 1 mm; therefore, body forces such as gravity can be neglected in most cases, even in liquid flows, in comparison with surface forces. The large surface-to-volume ratio is another inherent characteristic of microsystems. This ratio is typically inversely proportional to the 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 Ͻ Kn Ͻ 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 is modeled through Maxwell’s velocity-slip and Smoluchowski’s temperature-jump boundary conditions [Beskok and Karniadakis, 1994]: U s Ϫ U w ϭ ᎏ 2 Ϫ σ U σ U ᎏ λ Έ w (12.8a) T j Ϫ T w ϭ λ Έ w (12.8b) U w and T w are the wall velocity and temperature respectively; U s and T j are the gas flow velocity and tem- perature at the boundary; n is the direction normal to the solid boundary; γ ϭ c p /c v is the ratio of specific heats; and σ U and σ T are the momentum and energy accommodation coefficients respectively, which model 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. This essentially means a diffuse reflection boundary condition, where the impinging molecules are reflected at a random angle uncorrelated with the incident angle. ∂T ᎏ ∂n k ᎏ µ c p 2 γ ᎏ γ ϩ 1 2 Ϫ σ T ᎏ σ T ∂U ᎏ ∂n 12-6 MEMS: Applications © 2006 by Taylor & Francis Group, LLC 12.2.7.2 Electric Double Layer Most solid surfaces are likely to carry electrostatic charge (i.e., an electric surface potential) due to broken bonds and surface charge traps. When a liquid containing a small amount of ions is forced through a microchannel 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 the balancing 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 ions are affected less by the electric field and are mobile. The distribution of the counterions away from the interface 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 ranges from 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 charac- teristic size of the channel, and its effect on the fluid flow and heat transfer may not be negligible. 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 potential ψ 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]: ϭ ψ (12.9) where ε is the dielectric constant of the medium, and ε 0 is 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 n 0 is the ionic con- centration. The characteristic thickness of the EDL is the Debye length given by k d Ϫ1 ϭ ( εε 0 k b T/2 n 0 ζ 2 e 2 ) 1/2 . For the boundary conditions when ψ ϭ 0 at the midpoint, y ϭ 0, and ψ ϭ ξ on both walls, y ϭ ϮH/2, the solution is ψ ϭ |sin h(k d y)| (12.10) where ξ is the electric potential at the boundary between the diffuse double layer and the compact layer. ξ ᎏᎏ sin h(k d H/2) 2n 0 ζ 2 e 2 ᎏ εε 0 k b T d 2 ψ ᎏ dy 2 Microchannel Heat Sinks 12-7 Ψ Diffuse double layer Compact layer Channel wall Diffuse double layer Co-ions Counter-ions FIGURE 12.2 Electric double layer (EDL) at the channel wall. © 2006 by Taylor & Francis Group, LLC 12.2.7.3 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 in nonpolar 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 couple- stress tensor. The most important effect of couple stresses is to introduce a size-dependent effect that is not predicted by the classical nonpolar theories [Stokes, 1984]. In micropolar fluids, rigid particles contained in a small volume can rotate about the center of the vol- ume element described by the microrotation vector. This local rotation of the particles is in addition to the 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 of microinertia 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. The modified momentum, angular momentum, and energy equations are ρ ϭ ∇ и τ ϩ ρ f (12.11) ρ I ϭ ∇ и σ ϩ ρ g ϩ τ x (12.12) ρ c p ϭ k∇ 2 T ϩ τ : (∇U) ϩ σ : (∇ ΩΩ ) Ϫ τ x и ΩΩ (12.13) where ΩΩ is 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 τ ji U i,j , the scalar product of τ and ∇U. If σ ϭ 0 and g ϭ ΩΩ ϭ 0, then the stress tensor 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, a cylinder, or a sphere are examples of external flows, and they are not within the scope of this article. Only internal flows, in either liquid or gas phase, within microducts will be discussed, with an emphasis on size effects, 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 laminar regime, flow structure is characterized by smooth motion in laminae, or layers. The flow in the turbulent regime is characterized by random three-dimensional motions of fluid particles superimposed on the mean motion. These turbulent fluctuations enhance the convective heat transfer dramatically. However, turbulent flow occurs in practice only as long as the Reynolds number, Re ϭ ρ U 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 cr for var- 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 Re cr ϭ Х 2300 (12.14) ρ U m D h ᎏ µ DT ᎏ Dt D ΩΩ ᎏ Dt DU ᎏ Dt 12-8 MEMS: Applications © 2006 by Taylor & Francis Group, LLC where U m is the mean flow velocity and D h ϭ 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 length with a hydraulic diameter of about 10 µm. The mean velocity for gas flow under a pressure drop of about 0.5 MPa is less than 100 m/s, and the corresponding Reynolds number is less than 100. The Reynolds number 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 the duct. The boundary layer fills the entire duct gradually, as sketched in Figure 12.3. The region where the velocity 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 h in laminar flow in a duct is given by Shah and Bhatti (1987): ᎏ D L h h ᎏ ϭ 0.056 Re (12.15) The region beyond the entrance region is referred to as the hydrodynamically fully developed region. In this region, the boundary layer completely fills the duct and the velocity profile becomes invariant with the axial coordinate. If the walls of the duct are heated (or cooled), a thermal boundary layer will also develop along the inner surfaces 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 t is then the duct length required for the developing flow to reach fully developed condition. The thermal entrance length for laminar flow in ducts varies with the Reynolds number, Prandtl number (Pr ϭ µ c p /k) and the type of the boundary condition imposed on the duct wall. It is approximately given by: Х 0.05 Re Pr (12.16) 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 Prandtl number is on the order of 1. Thus, both the hydrodynamic and thermal entrance lengths are less than 5 times the hydraulic diameter. Because the length of microchannels is typically two orders of magnitude larger than the hydraulic diameter, both entrance lengths are less than 5% of the microchannel length and can be neglected. L t ᎏ D h Microchannel Heat Sinks 12-9 L t L h Fully developed flow x y y z UUU ∞ T ∞ T T Simultaneously developing flow (Pr >1) FIGURE 12.3 Hydrodynamically and thermally developing flow, followed by hydrodynamically and thermally fully developed flow. © 2006 by Taylor & Francis Group, LLC 12.3.3 Governing Equations Representing the flow in rectangular ducts as flow between two parallel plates, the two-dimensional gov- erning equations can be simplified as follows (Sadik and Yaman, 1995): Continuity: ϩ ϭ 0 (12.17) x-momentum: ϩ ϭ Ϫ ϩ µ ϩ ϩ ϩ (12.18) y-momentum: ϩ ϭ Ϫ ϩ µ ϩ ϩ ϩ (12.19) Energy: u ϩ v ϭ ϩ ϩ ΄ 2 ϩ 2 ϩ ϩ 2 ΅ (12.20) 12.3.4 Fully Developed Gas Flow Forced Convection Analytical solution of Equations (12.17) to (12.20) is not available. Some solutions can be obtained upon further simplification of the mathematical model. Indeed, incompressible gas flows in macroducts with different 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 of second order compared to the compressibility effect (van den Berg et al., 1993). The simplest system for demonstration of the rarefaction and compressibility effects is the two-dimensional flow between paral- lel plates separated by a distance H, with L being the channel length (L/H ϾϾ 1). If MaKn ϽϽ 1, all stream- wise derivatives can be ignored except the pressure gradient, which is the driving force. The Mach number, Ma ϭ U/a, is the ratio between the fluid speed and the speed of sound a. In such a case, the momentum equation reduces to: Ϫ ϩ µ ϭ 0 (12.21) with the symmetry condition at the channel centerline, y ϭ 0, and the slip boundary conditions at the walls, y ϭ ϮH/2, as follows: ϭ 0 @ y ϭ 0 (12.22) u ϭ Ϫ λ Έ yϭH/2 @ y ϭ ϮH/2 (12.23) Integration of Equation (12.21) twice with respect to y, assuming P ϭ P(x), yields the following velocity profile [Arkilic et al., 1997]: u(y) ϭ Ϫ ΄ 1 Ϫ 2 ϩ 4Kn(x) ΅ (12.24) y ᎏ H/2 dP ᎏ dx H 2 ᎏ 8 µ du ᎏ dy du ᎏ dy d 2 u ᎏ dy 2 dP ᎏ dx ∂v ᎏ ∂x ∂u ᎏ ∂y 1 ᎏ 2 ∂v ᎏ ∂y ∂u ᎏ ∂x 2 µ ᎏ ρ c p ∂ 2 T ᎏ ∂y 2 ∂ 2 T ᎏ ∂x 2 k ᎏ ρ c p ∂T ᎏ ∂y ∂T ᎏ ∂x ∂v ᎏ ∂y ∂u ᎏ ∂x ∂ ᎏ ∂y µ ᎏ 3 ∂ 2 v ᎏ ∂y 2 ∂ 2 v ᎏ ∂x 2 ∂P ᎏ ∂y ∂( ρ vv) ᎏ ∂y ∂( ρ uv) ᎏ ∂x ∂v ᎏ ∂y ∂u ᎏ ∂x ∂ ᎏ ∂x µ ᎏ 3 ∂ 2 u ᎏ ∂y 2 ∂ 2 u ᎏ ∂x 2 ∂P ᎏ ∂x ∂( ρ vu) ᎏ ∂y ∂( ρ uu) ᎏ ∂x ∂( ρ v) ᎏ ∂y ∂( ρ u) ᎏ ∂x 12-10 MEMS: Applications © 2006 by Taylor & Francis Group, LLC where Kn(x) ϭ λ (x)/H. The streamwise pressure distribution P(x) calculated based on the same model is given by: ϭ Ϫ6Kn o ϩ Ί 6 K n o ϩ 2 Ϫ ΄ Ϫ 1 ϩ 1 2K n o Ϫ 1 ΅ (12.25) where P i is the inlet pressure, P o the outlet pressure, and Kn o is 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 microchannel integrated with pressure sensors for such experiments is shown in Figure 12.4a. Indeed, the calculated pressure 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 m as a function of the inlet and outlet conditions is obtained by integrating the velocity profile with respect to x and y as follows: Q m ϭ ΄ ᎏ P P o i ᎏ 2 Ϫ 1 ϩ 12Kn o ᎏ P P o i ᎏ Ϫ 1 ΅ (12.26) 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 to 60°C, as demonstrated in Figure 12.5 [Jiang et al., 1999a]. H 3 WP o 2 ᎏ 24 µ RTL x ᎏ L P i ᎏ P o P i 2 ᎏ P o 2 P i ᎏ P o P(x) ᎏ P o Microchannel Heat Sinks 12-11 (a) Pressure microsensors Microchannel Inlet/outlet hole 0 5 10 15 20 25 30 35 40 0 500 1000 1500 2000 2500 3000 3500 4000 4500 ∆P (Psi) ∆P = 35.15 psi 26.15 psi 16.35 psi (b) X (µm) FIGURE 12.4 Slip flow effect on a microchannel flow: (a) microchannel, 40 µm wide, integrated with pressure microsensors; (b) acomparison between calculated (dash lines) and measured (symbols) streamwise pressure distri- 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.) © 2006 by Taylor & Francis Group, LLC The 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 rate from 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 conditions include two competing effects [Zohar et al., 1994]. The velocity slip at the wall increases the flow rate, thus enhancing 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 effects depends on the specific material properties and specific geometry of the system. 12-12 MEMS: Applications 0 1 2 3 4 5 6 0 50 100 150 200 250 300 350 400 450 p i - p o (kPa) Tw = 20°C, Exp. Tw = 20°C, Cal. Tw = 40°C, Exp. Tw = 40°C, Cal. Tw = 60°C, Exp. Tw = 60°C, Cal. Gas: Nitrogen (b) Channel size: 5000 mm*40 µm*1.4 µm Q m (µg/min) 0 1 2 3 4 5 6 7 8 0 100 200 300 40 0 p i - p o (kPa) Argon, Exp. Argon, Cal. Helium, Exp. Helium, Cal. Nitrogen, Exp. Nitrogen, Cal. T = 20°C (a) Channel size: 4000 mm*40 µm*1.4 µm Q m (µg/min) 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.) © 2006 by Taylor & Francis Group, LLC A 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 12.7b for different wall temperatures and pressure drops. In all cases, the temperature along the channel is almost uniform and equal to the wall temperature, and no cooling effect has been observed. Indeed, on the one Microchannel Heat Sinks 12-13 Kn = 0.00 Kn = 0.03 Kn = 0.10 0.10 10 100 1.00 x/D h (a) Nu T Kn = 0.00 Kn = 0.03 Kn = 0.10 0.10 1.00 x/D h (b) 10 100 Nu H 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.) © 2006 by Taylor & Francis Group, LLC [...]... transport aircraft (e.g., the U.S National Aerospace Plane) provides new challenges for researchers in the field of flow control The three books by Gad- el- Hak et al (1998), Gad- el- Hak (20 00) and Gad- el- Hak and Tsai (20 05) provide an up-to-date overview of the subject of flow control In this chapter, following a description of the unifying principles of flow control, we focus on the concept of targeted... for a two-dimensional duct flow can be reduced to [Mohiuddin Mala et al., 1997]: d 2u dP Es d 2 µ ᎏ Ϫ ᎏᎏ Ϫ εε0 ᎏᎏ ᎏ ϭ 0 dy 2 dx L dy 2 © 20 06 by Taylor & Francis Group, LLC ( 12. 27) Microchannel Heat Sinks 1 2- 1 5 60 50 = 163 .2, = 0 = 40.8, = 0 = 40.8, = 50 Nu 40 30 20 10 0 . 25 0 .50 1.00 x/Dh FIGURE 12. 8 Electric double layer effect on the variation of the local Nusselt number Nu along the channel length... mini- and microchannel curves, which was explained as a result © 20 06 by Taylor & Francis Group, LLC 1 2- 2 2 MEMS: Applications of the large difference in L/D ratio (L and D being the channel length and diameter respectively): 3.94 for the minichannel and 19.6 for the microchannel Consequently, they proposed the following CHF correlation: L qmp ᎏ ϭ 0.16 WeϪ0.19 ᎏ D Ghfg Ϫ0 .54 ( 12. 30) where qmp is the. .. important first to verify the hydrodynamic effects Indeed, it has been suggested in a few reports that theoretical calculations based on the classical model did not agree with experimental © 20 06 by Taylor & Francis Group, LLC 1 2- 1 6 MEMS: Applications 4 .50 Nu 4 . 25 4.00 Γ=1 Γ=3 Γ =5 3. 75 3 .50 0 5 10 ky /my 15 20 FIGURE 12. 9 Micropolar fluid effect on the Nusselt number Nu as a function of the viscosity ratio,... “Microchannel Fluid Behavior Using Micropolar Fluid Theory,” Proc 11th Int Workshop on Micro Electro Mechanical Systems (MEMS ’98), pp 54 4 54 9, IEEE, Piscataway, NJ © 20 06 by Taylor & Francis Group, LLC Microchannel Heat Sinks 1 2- 2 9 Peles, Y.P., and Haber, S (20 00) “A Steady State, One-Dimensional Model for Boiling Two-Phase Flow in Triangular Micro-Channel,” Int J Multiphase Flow 26 , pp 10 95 11 15 Peles,... distribution along the microchannel heat sink centerline (Reprinted with permission from Jiang, L et al [1999] “Phase Change in Micro-Channel Heat Sinks with Integrated Temperature Sensors,” J MEMS 8, 358 – 65 © 1999 /20 00 IEEE.) © 20 06 by Taylor & Francis Group, LLC 1 2- 2 0 MEMS: Applications 25 0 T 20 0 Plateau 150 T (°C) q 100 0 . 25 mL/min (80 kPa), device I 1.1 mL/min (160 kPa), device II 50 1.8 mL/min ( 320 kPa),... ( 85 kPa), device III 1.8 mL/min (50 kPa), device IV 0 0 5 10 (a) 15 q (W) 20 25 30 75 70 65 T (°C) 60 55 50 45 40 35 1 (b) 10 100 1000 2 q (W/cm ) FIGURE 12. 14 Boiling curves of device temperature as a function of the input power for microchannels with: (a) water and Dh ϭ 40 µm or 80 µm [Jiang et al., 1999b], and (b) R-113 and Dh ϭ 51 0 µm [Bowers and Mudawar, 1994] (Reprinted with permission from Elsevier... respectively, and σij is the tensor of viscous tension The superscript b represents either vapor (b ϭ 1) or liquid (b ϭ 2) When the interface surface is expressed by a function, x ϭ f(y, z), the general radii of curvature ri are found from the equation: A2r 2 ϩ A1ri ϩ A0 ϭ 0 i © 20 06 by Taylor & Francis Group, LLC ( 12. 34) Microchannel Heat Sinks 1 2- 2 7 y Liquid Vapor x FIGURE 12. 20 Physical model of forced... Huang, Y., Bau, H.H., and Zemel, J.N (19 95) “Gas Flow in Micro-Channels,” J Fluid Mech 28 4, pp 25 7–74 Ho, C .M., and Tai, Y.C (1998) “Micro-Electro-Mechanical-Systems (MEMS) and Fluid Flows,” Ann Rev Fluid Mech 30, pp 57 0–6 12 Jacobi, A.M (1989) “Flow and Heat Transfer in Microchannels Using a Microcontinuum Approach,” J Heat Transfer 111, pp 1083– 85 Jiang, L., Wang, Y., Wong, M., and Zohar, Y (1999a) “Fabrication... approached The curves in Figure 12. 14a suggest that the saturated nucleate boiling does not develop in such microducts due to size effect, which could be verified by flow visualization of the boiling pattern 300 0 mL/min (0 kPa) 1.1 mL/min (160 kPa) 25 0 0 . 25 mL/min (80 kPa) 1.8 mL/min ( 320 kPa) T (°C) 20 0 150 100 50 Device I, Dl water, q = 3.6 W 0 0 5 10 x (mm) 15 20 FIGURE 12. 13 Flow rate effect on the temperature . important parameter associated with the CHF condition is the corresponding 1 2- 2 0 MEMS: Applications 75 70 65 60 55 50 45 40 35 1 10 100 1000 T (°C) q (W/cm 2 ) (b) 25 0 20 0 150 100 50 0 5 10 15. pattern. Microchannel Heat Sinks 1 2- 1 9 0 50 100 150 20 0 25 0 300 0 10 15 20 x (mm) T (°C) 0 mL/min (0 kPa) 0 . 25 mL/min (80 kPa) 1.1 mL/min (160 kPa) 1.8 mL/min ( 320 kPa) Device I, Dl water, q = 3.6 W 5 FIGURE. increase in the pressure drop. However, 1 2- 1 6 MEMS: Applications 0 5 10 15 20 3 .50 3. 75 4.00 4 . 25 4 .50 Nu k y /m y Γ = 1 Γ = 3 Γ = 5 FIGURE 12. 9 Micropolar fluid effect on the Nusselt number Nu as