# Flow evaporation and boiling in micro- and minichannels

## Onset of Nucleate Boiling in Microchannel Flow

Figure 1: Geometric configuration of the microchannels.

The first onset of the vapor bubble marks the transition from the single-phase flow in the microchannel to two-phase flow, and for this reason, the prediction of onset of nucleate boiling (ONB) is very important. Liu et al. (2005) thoroughly reviewed the existing works on ONB in subcooled boiling. Hsu (1962) proposed the ONB heat flux correlation based on the minimum superheat criterion for pool boiling

${{{q}''}_{ONB}}=\frac{{{k}_{\ell }}{{h}_{\ell v}}{{\rho }_{v}}{{({{T}_{w}}-{{T}_{sat}})}^{2}}}{12.8\sigma {{T}_{sat}}} \qquad \qquad(1)$

The effect of contact angle on the ONB heat flux was considered by Davis and Anderson (1966), who proposed the following correlation

${{{q}''}_{ONB}}=\frac{{{k}_{\ell }}{{h}_{\ell v}}{{\rho }_{v}}{{({{T}_{w}}-{{T}_{sat}})}^{2}}}{8(1+\cos \theta )\sigma {{T}_{sat}}} \qquad \qquad(2)$

Both eqs. (1) and (2) are obtained based on pool boiling, but they are also used to predict ONB in conventional channels. Liu et al. (2005) proposed an analytical model for ONB in microchannels (grooves) as shown in Fig. 1.

It is assumed that (1) the bubble shape is a truncated sphere, (2) the liquid temperature is not affected by the bubble nucleus due to its small size, and (3) the vapor and liquid are in equilibrium under saturation. The condition required for the bubble nucleus to grow is that the liquid temperature at y = yb is greater than the superheat requirement.

The superheat equation for the bubble nucleus can be obtained from equilibrium theory

${{T}_{b}}-{{T}_{sat}}=\frac{2\sigma {{T}_{b}}}{{{h}_{\ell v}}{{\rho }_{v}}{{R}_{b}}} \qquad \qquad(3)$

where Tb is the liquid temperature required for the bubble nucleus to grow and Tsat is the saturation temperature corresponding to the liquid pressure ${{p}_{\ell }}.$

Since the bubble shape is a truncated sphere, we have

${{y}_{b}}={{R}_{b}}(1+\cos \theta ) \qquad \qquad(4)$
${{r}_{c}}={{R}_{b}}\sin \theta \qquad \qquad(5)$

Substituting eq. (4) into eq. (3) and solving for Tb, one obtains

${{T}_{b}}={{{T}_{sat}}}/{\left( 1-\frac{2\sigma C}{{{h}_{\ell v}}{{\rho }_{v}}{{y}_{b}}} \right)}\; \qquad \qquad(6)$

where C = 1 + cosθ.

The liquid temperature near the wall of the microchannel can be assumed as

${{T}_{\ell }}(y)={{T}_{w}}-{{{q}''}_{w}}y/{{k}_{\ell }} \qquad \qquad(7)$

where q''w is the effective wall heat flux at the inner wall of the microchannels.

If a constant heat flux, q'', is applied to the bottom, the liquid temperature at the outlet is related to that at the inlet by

${{T}_{\ell ,out}}={{T}_{\ell ,in}}+\frac{{q}''WL}{{{\rho }_{\ell }}{{c}_{\ell }}{{u}_{0}}(n{{w}_{c}}{{H}_{c}})} \qquad \qquad(8)$

where u0 is the liquid inlet velocity, and n is the number of microchannels. If the convective heat transfer coefficient is uniform along the channel surfaces and flow in the channel is fully-developed, the channel wall temperature can be obtained by

${{T}_{w}}={{T}_{\ell }}+\frac{{{{{q}''}}_{w}}}{\overline{Nu}{{k}_{\ell }}/{{D}_{h}}} \qquad \qquad(9)$

where $\overline{\text{Nu}}$ is the Nusselt number for fully developed flow in a three-sided (bottom and side walls) rectangular channel (Shah and London, 1978)

$\overline{\text{Nu}}=8.235(1-1.883/\alpha +3.767/{{\alpha }^{2}}-5.814/{{\alpha }^{3}}+5.361/{{\alpha }^{4}}-2/{{\alpha }^{5}}) \qquad \qquad(10)$

where α is the aspect ratio of the microchannel.

The effective wall heat flux, q''w, is related to the applied heat flux by

${{{q}''}_{w}}=\left( \frac{\alpha }{1+2\eta \alpha }\frac{{{w}_{c}}+{{w}_{w}}}{{{H}_{c}}} \right){q}'' \qquad \qquad(11)$

where η is the fin efficiency.

The condition for nucleation boiling to occur is ${{T}_{\ell }}({{y}_{b}})\ge {{T}_{b}}$. The necessary condition for ONB can be obtained from eqs. (6) and (7), i.e.,

${{T}_{w}}-\frac{{{{{q}''}}_{w}}}{{{k}_{\ell }}}{{y}_{b}}={{{T}_{sat}}}/{\left( 1-\frac{2\sigma C}{{{h}_{\ell v}}{{\rho }_{v}}{{y}_{b}}} \right)}\; \qquad \qquad(12)$

which can be rearranged as an equation of yb,

$\frac{{{{{q}''}}_{w}}}{{{k}_{\ell }}}y_{b}^{2}-\left( {{T}_{w}}+\frac{2\sigma C}{{{\rho }_{v}}{{h}_{\ell v}}}\frac{{{{{q}''}}_{w}}}{{{k}_{\ell }}} \right){{y}_{b}}+\frac{2\sigma C}{{{h}_{\ell v}}{{\rho }_{v}}{{y}_{b}}}{{T}_{w}}=0 \qquad \qquad(13)$

For the roots of eq. (13) to be real, the following condition must be satisfied

${{\left( {{T}_{w}}+\frac{2\sigma C}{{{\rho }_{v}}{{h}_{\ell v}}}\frac{{{{{q}''}}_{w}}}{{{k}_{\ell }}} \right)}^{2}}-4\frac{{{{{q}''}}_{w}}}{{{k}_{\ell }}}\frac{2\sigma C}{{{h}_{\ell v}}{{\rho }_{v}}{{y}_{b}}}{{T}_{w}}\ge 0 \qquad \qquad(14)$

which can be rearranged to yield the superheat criterion

$\sqrt{{{T}_{w}}}-\sqrt{{{T}_{sat}}}\ge \sqrt{\frac{2\sigma C}{{{\rho }_{v}}{{h}_{\ell v}}}\frac{{{{{q}''}}_{w}}}{{{k}_{\ell }}}} \qquad \qquad(15)$

Equation (15) can be used to determine if nucleate boiling will occur in the microchannels under specified parameters. If the liquid inlet conditions are prescribed but the heat flux is allowed to vary, the ONB heat flux can be obtained by

$\sqrt{{{T}_{\ell ,in}}+\frac{{q}''WL}{{{\rho }_{\ell }}{{c}_{\ell }}{{u}_{0}}(n{{w}_{c}}{{H}_{c}})}+\frac{\left( \frac{\alpha }{1+2\eta \alpha }\frac{{{w}_{c}}+{{w}_{w}}}{{{H}_{c}}} \right){{{{q}''}}_{ONB}}}{{{\overline{Nu}}_{{}}}{{k}_{\ell }}/{{D}_{h}}}}-\sqrt{{{T}_{sat}}}$
$=\sqrt{\frac{2\sigma C}{{{\rho }_{v}}{{h}_{\ell v}}}\frac{\left( \frac{\alpha }{1+2\eta \alpha }\frac{{{w}_{c}}+{{w}_{w}}}{{{H}_{c}}} \right){{{{q}''}}_{ONB}}}{{{k}_{\ell }}}} \qquad \qquad(16)$

If nucleate boiling is to be avoided or delayed for some application, one can increase the liquid inlet velocity u0, or reduce the liquid inlet temperature, ${{T}_{\ell ,in}}$. In this case, eq. (16) can be used to obtain the minimum liquid velocity or maximum liquid inlet velocity to avoid nucleate boiling.

The wall superheat can be obtained from inequality eq. (15) as

${{T}_{w}}-{{T}_{sat}}=\frac{2\sigma C}{{{\rho }_{v}}{{h}_{\ell v}}}\frac{{{{{q}''}}_{w}}}{{{k}_{\ell }}}+2\sqrt{{{T}_{sat}}}\sqrt{\frac{2\sigma C}{{{\rho }_{v}}{{h}_{\ell v}}}\frac{{{{{q}''}}_{w}}}{{{k}_{\ell }}}} \qquad \qquad(17)$

Substituting eq. (11) into eq. (17), the wall superheat becomes

\begin{align} & {{T}_{w}}-{{T}_{sat}}=\frac{2\sigma C}{{{\rho }_{v}}{{h}_{\ell v}}{{k}_{\ell }}}\left( \frac{\alpha }{1+2\eta \alpha }\frac{{{w}_{c}}+{{w}_{w}}}{{{H}_{c}}} \right){q}'' \\ & +2\sqrt{{{T}_{sat}}}\sqrt{\frac{2\sigma C}{{{\rho }_{v}}{{h}_{\ell v}}{{k}_{\ell }}}\left( \frac{\alpha }{1+2\eta \alpha }\frac{{{w}_{c}}+{{w}_{w}}}{{{H}_{c}}} \right){q}''} \\ \end{align} \qquad \qquad(18)

## Flow Evaporation in Minichannel Heat Sink with Axial Grooves

New advanced cooling technologies are needed to meet the demands for dissipating extremely high heat fluxes (over 300 W/cm2) from electronic components. The performance characteristics of a flat high heat flux evaporative mini-channel heat sink with axial capillary grooves on its inner surface was modeled by Khrustalev and Faghri (1996). The small axial grooves allow this heat sink to withstand high heat fluxes with a small pressure drop, low thermal resistance, and a comparatively large effective length. To evaluate the advantages of this design compared to existing designs, the maximum heat flux, thermal resistance, and the pressure drop on the element will be estimated for some independent operational parameters, such as the saturated-vapor outlet temperature Tv,o, the geometry of the grooved surfaces, and the effective lengths of the evaporators under consideration.

Introduced by a pump, the liquid enters the heat sink and, under the influence of an extremely high heat flux, boils on the wall at a short distance from the entrance. While the vapor phase appears to be moving along the z-coordinate, the vapor and liquid cocurrent flows become separated, with the liquid flowing mostly in the grooves. The separation of the phases in a small, horizontal, flat rectangular channel with smooth walls (19.05x3.18 mm2) has been experimentally investigated by Wambsganss et al. (1991). They have observed the annular flow pattern for a water-air mixture starting with an air velocity of about 10 m/s.

Wilmarth and Ishii (1994) examined two test sections with gap widths of 1 and 2 mm with channel widths of 20 and 15 mm, respectively. In their experiments, the annular flow regime was observed with air velocity of about 5 m/s. For the heat sink with a flow channel cross-section of 7×1 mm2, and the heat flux of 100 W/cm2 on both side walls, it can be estimated that the velocity of the vapor reaches 5 m/s at the distance of about 3.3 mm from the entrance (for Tv = 100 °C).

This means that for such a heat sink, the flow boiling length preceding the annular flow pattern would be very small, even with smooth walls. Since surface tension tends to pull the liquid into the grooves, the flow boiling length should be even smaller, and the annular regime transforms into the separated cocurrent two-phase flow for the considered heat sink. Surface tension seems to be a stabilizing factor for the liquid flow in a groove; it predominates over the destabilizing dynamic vapor-liquid interaction when the Weber number is less than unity. An expression for the Weber number – including the empirically-adjusted characteristic size for a capillary-grooved surface with vapor flow over it – has been given by Vasiliev et al. (1984). Their expression is based on the analysis of the experimental data obtained from the gravity-assisted heat pipe experiments with several fluids, including water.

$\text{We}\equiv \frac{{{\rho }_{v}}\bar{w}_{v}^{2}D_{h,\ell }^{2}}{2\pi \sigma W}<1 \qquad \qquad(19)$

where W is the half-width of the groove. Khrustalev and Faghri (1997) confirmed the importance of the entrainment limitation through experimental results obtained for a miniature heat sink with axial grooves. Therefore, with the inequality given by eq. (19) satisfied, the emphasis is on the mode where vapor and liquid flows are separated with the liquid flowing exclusively in the grooves. Since the inlet liquid is supposed to be saturated, and the cross-section of the vapor channel is very small, it is assumed in this analysis that ${{L}_{\ell }}={{L}_{b}}=0$, which means that only the part of the evaporator with separated vapor and liquid flows is modeled.

Another significant feature of this evaporator is that the formation of vapor bubbles in the liquid, which can occur with high heat fluxes, does not obstruct the liquid flow in the grooves (unlike in heat pipes) due to the cocurrency of the liquid and vapor flows. Perturbations in the liquid flow caused by a growing vapor bubble are compensated by the dynamic vapor-liquid interaction during the bubble’s explosion into the vapor space, with the subsequent return of the accelerated liquid droplets to the liquid flow. Moreover, for comparatively moderate heat fluxes, the liquid in the grooves does not boil along most of the effective length, because the liquid superheat is not sufficient for boiling due to the intensive evaporation of the liquid from the surfaces of the thin films attached to the liquid-vapor menisci corners. The analysis also attempts to predict the onset of nucleate boiling using the methodology by Khrustalev and Faghri (1997).

Another essential feature of this evaporator is that the liquid flow in the grooves is strongly influenced by the vapor flow. The vapor-liquid frictional interaction can be the most important factor of those affecting the hydraulic resistance of the liquid flow. Another factor of importance to the liquid flow is the axial gradient of the capillary pressure, due to the variation of the main radius of the liquid-vapor meniscus along the groove.

This variation takes place not far from the entrance region of the evaporator, where the grooves are nearly filled with liquid. Farther from the entrance, starting from the point on z where the meniscus main radius Rmen reaches its minimum value Rmen,min, the effective meniscus height Hmen decreases along the z-coordinate, so that at some point closer to the evaporator outlet, the liquid-vapor meniscus interface nearly touches the bottom of the groove. At this interval along z, the liquid flows in a groove and is influenced by the pressure gradient in the vapor and the vapor-liquid frictional interaction at the interface. However, the surface tension indirectly controls the liquid distribution by collecting the occasional liquid droplets from the walls of the grooves.

With a heat flux close to the maximum, the effective meniscus height, Hmen, can become very small at the outlet of the evaporator; therefore, this region, which can be called the “thin film flow region,” requires a special treatment.

Under close examination, it is evident that the bottom of a capillary groove rarely is rectangular, a shape that is essential for modeling the “thin film flow region” (but can be practically insignificant for other regions). This depends upon the manufacturing process used. When the electro-discharge machining process (EDM) is used to manufacture grooves on the inner evaporator surface, the groove bottom is rounded, and the liquid cross-section has a very complicated form. This reality should be kept in mind, even though the present analysis and its model are simplified by use of a rectangular strip to approximate the liquid cross-sectional shape in the “thin film flow region.”

Simultaneously, the axial capillary pressure gradient is assumed to be negligible in this region in comparison to the vapor pressure gradient, which is consistent with the description of the previously discussed region with a larger Hmen. With extremely high heat fluxes, dryout of the evaporator can occur for two reasons: (1) hydrodynamic limitations of the fluid transport (the corresponding heat flux is denoted as the maximum, q''max), or (2) the beginning of vapor film boiling (the corresponding heat flux is usually referred to as the critical heat flux, CHF). The present model can predict q''max but CHF should be determined experimentally.

At any axial location, the following mass conservation equation must hold over the cross-section of the evaporator modeled under steady-state conditions:

${{\dot{m}}_{\ell }}(z)={{\dot{m}}_{\ell ,o}}+{{\dot{m}}_{v,o}}-{{\dot{m}}_{v}}(z) \qquad \qquad(20)$

where ${{\dot{m}}_{v}}={{\bar{w}}_{v}}{{\rho }_{v}}{{A}_{v}}$, ${{\dot{m}}_{\ell }}=N{{\bar{w}}_{\ell }}{{\rho }_{\ell }}{{A}_{\ell }}$, N is the number of grooves, and the subscript “o” denotes the evaporator outlet.

For laminar liquid flow in a groove, the momentum conservation equation is

$\frac{d{{p}_{\ell }}}{dz}+{{\rho }_{\ell }}g\sin \phi =-{{f}_{\ell }}\frac{2{{\rho }_{\ell }}\bar{w}_{\ell }^{2}}{{{D}_{h,\ell }}}\equiv -{{f}_{\ell }}\frac{2\dot{m}_{\ell }^{2}}{{{D}_{h,\ell }}{{\rho }_{\ell }}A_{\ell }^{2}{{N}^{2}}} \qquad \qquad(21)$

where φ is the inclination angle. The inertia terms in eq. (21) are neglected, since Khrustalev and Faghri (1994) demonstrated that this simplification is true even when the liquid cross-sectional area varies along the flow.

For vapor flow in a channel with a small Mach number (Ma$\ll$1), the momentum conservation equation is (Faghri, 1995)

$\frac{d}{dz}({{p}_{v}}+{{\rho }_{v}}{{\beta }_{v}}\bar{w}_{v}^{2})+{{\rho }_{v}}g\sin \phi =-{{f}_{v}}\frac{2{{\rho }_{v}}\bar{w}_{v}^{2}}{{{D}_{h,v}}} \qquad \qquad(22)$

The Laplace-Young equation relates the interfacial radius of curvature to the pressure difference between the liquid and vapor. In differential form, the equation is

$\frac{d}{dz}\left( \frac{\sigma }{{{R}_{men}}} \right)=\frac{d{{p}_{v}}}{dz}-\frac{d{{p}_{\ell }}}{dz} \qquad \qquad(23)$

The mass and energy conservation balances relate the vapor mass flow rate with the heat flux at the inside surface of the vapor channel, q''v:

$\frac{d{{{\dot{m}}}_{v}}}{dz}=\frac{2{{{{q}''}}_{v}}(z)}{{{h}_{\ell v}}}({{t}_{t}}-2{{t}_{w2}}) \qquad \qquad(24)$

where ${{{q}''}_{v}}={{h}_{e}}({{\bar{T}}_{w}}-{{T}_{v}})$ he is the local effective heat transfer coefficient describing the evaporative heat transfer from the wall of the evaporator to the saturated vapor, through the liquid films and the fin between the grooves. ${{\bar{T}}_{w}}$ is the local mean wall temperature. Note that q''v = q'' for cases in which axial conduction in the wall is insignificant and where q'' is the heat flux on the outer surfaces of the heat sink.

These conservation equations should be solved simultaneously for the inlet interval of z where Rmen varies and tmen = tg (note that Hmen is not constant). However, for the outlet z-interval at the heat sink outlet, the liquid recedes into the grooves, and tmen varies with z (as does Hmen) while Rmen = Rmen,min. At this “outlet” interval, Hmen is one of the unknown variables, while the pressure gradients in the liquid and vapor are equal: $d{{p}_{\ell }}/dz=d{{p}_{v}}/dz$ [see Eq. (23)]. For the “outlet” interval, taking the equality of the pressure gradients into consideration, the following transcendental equation for Hmen can be obtained from eqs. (20) and (21):

${{\dot{m}}_{\ell ,o}}+{{\dot{m}}_{v,o}}-{{\dot{m}}_{v}}(z)={{\left[ -\frac{{{D}_{h,\ell }}{{\rho }_{\ell }}A_{\ell }^{2}{{N}^{2}}}{2{{f}_{\ell }}}\left( \frac{d{{p}_{v}}}{dz}+{{\rho }_{\ell }}g\sin \phi \right) \right]}^{1/2}} \qquad \qquad(25)$

which should be solved for Hmen at every point on z since ${{D}_{h,\ell }}$, ${{A}_{\ell }}$, ${{f}_{\ell }}$ and dpv / dz depend on Hmen. According to eq. (24), the vapor mass flow rate at the outlet is

${{\dot{m}}_{v,o}}=2\int_{2}^{{{L}_{s}}}{{{{{q}''}}_{v}}(z)({{t}_{t}}-2{{t}_{w2}})\frac{dz}{{{h}_{\ell v}}}} \qquad \qquad(26)$

The coefficients, fv, ${{f}_{\ell }}$, ${{D}_{h,\ell }}$, Dh,v and others appearing in the governing equations, can be found in Khrustalev and Faghri (1996) and will not be repeated here.

For the given temperature of the vapor at the outlet, Tv,o, the boundary conditions for eqs. (22) and (24) at the right-hand end of the evaporator (z = Ls) are

${{p}_{v}}={{p}_{sat}}({{T}_{v,o}}) \qquad \qquad(27)$

${{\dot{m}}_{v}}={{\dot{m}}_{v,o}} \qquad \qquad(28)$

Taking into consideration that ${{h}_{\ell v}}$ varies along the z-coordinate due to the axial vapor temperature gradient, mv,o – estimated initially by eq. (26) at Tv,o – should be adjusted so as to satisfy the following condition:

${{\left. {{{\dot{m}}}_{v}} \right|}_{z=0}}=0 \qquad \qquad(29)$

When predicting the maximum heat flux, the liquid flow rate at the outlet of the evaporator is infinitesimal: ${{\dot{m}}_{\ell ,o}}=\varepsilon {{\dot{m}}_{v,o}}$ where $\varepsilon$ is a very small number set equal to 10 − 3 in the numerical experiments. In this case the heat flux, q'', should be chosen so as to satisfy the following boundary condition at the beginning of the evaporator (nearly planar liquid-vapor interface):

$\frac{1}{{{\left. {{R}_{men}} \right|}_{z=0}}}=\frac{\cos {{\theta }_{men,\min }}}{{{t}_{h}}/2-{{t}_{w1}}-{{t}_{g}}} \qquad \qquad(30)$

which implies that the liquid and vapor flows become separated at z = 0 under the influence of the capillary forces. In cases of a given heat flux below the maximum heat flux, ${{\dot{m}}_{\ell ,o}}$ should be chosen so as to satisfy eq. (30).

The numerical procedure begins at z = Ls and moves toward the left-hand end of the vapor zone. At some point on z – which can be denoted as z1 – the condition tmen = tg is reached, and, starting from this point, eqs. (21) – (24) should be solved for ${{p}_{\ell }},\text{ }{{p}_{v}},\text{ }{{R}_{men}}\text{ and }{{\dot{m}}_{v}}$ with the following boundary conditions (at z = z1):

${{p}_{\ell }}={{\left. {{p}_{v}} \right|}_{z={{z}_{1}}}}-\frac{\sigma }{{{R}_{\text{men,m}in}}} \qquad \qquad(31)$

${{p}_{v}}={{\left. {{p}_{v}} \right|}_{z={{z}_{1}}}} \qquad \qquad(32)$

${{R}_{\text{men}}}={{R}_{\text{men,min}}} \qquad \qquad(33)$

${{\dot{m}}_{v}}={{\left. {{{\dot{m}}}_{v}} \right|}_{z={{z}_{1}}}} \qquad \qquad(34)$

Khrustalev and Faghri (1996) solved the above problem numerically, and an example of the characteristics of the copper-water evaporator for the following operating parameters was presented: W = 0.063 mm, L1 = 0.02 mm, Lt = 80 mm, tg = 0.229 mm, th = 2.58 mm, tt = 7.75 mm, tw1 = 0.5 mm, φ = 0, ${{\theta }_{\text{men,}\min }}={{33}^{\circ }}$, α = 0.05, Tv,o = 353 K and q'' = 65 W/cm2. A grooved structure featuring such a thin fin was reported by Plesch et al. (1991). In most numerical experiments, the overall meniscus height at the outlet, tmen,o was larger than tmen,min, and in all cases Hmen,o was larger than the roughness, Ra. This means that the concept of the “thin film region” did not practically affect the results obtained.

## Boiling Heat Transfer in a Miniature Axially-Grooved Rectangular Channel

Different configurations of miniature heat sinks are needed for the varied applications. The most important characteristics of miniature heat sinks are listed below (Khrustalev and Faghri, 1997):

1. Heated length and external thickness

2. Uniformity of temperature over the surface

3. Effective heat transfer coefficients (or thermal resistance)

4. Maximum attainable heat fluxes on the surface

5. Pressure drop and mass flow rate

6. Stability of operation

7. Operating temperature versus absolute pressure range

Obviously, heat sinks with low thermal resistance, and those of sufficient length and capability to withstand high heat fluxes, would be more practical for various technical applications. For this reason, it would be useful to enhance the inner surfaces of the miniature heat sinks. However, the mechanisms of heat transfer and those leading to dryout phenomena in two-phase miniature channels with enhanced surfaces are not completely understood, due to their complexity, especially for the case of discrete, in-line sources addressed by Maddox and Mudawar (1989) and Gersey and Mudawar (1993). While axially-grooved evaporators have been used in heat pipes (Plesch et al., 1991; Faghri, 1995), Khrustalev and Faghri (1997) attempted to foster a better understanding of the performance of miniature two-phase heat sinks in a minichannel with forced convection.

The goal of the experimental investigation presented by Khrustalev and Faghri (1997) was to obtain and analyze the performance characteristics of a flat copper-water miniature two-phase heat sink with small axial grooves on the inner walls. These capillary grooves provide significant heat transfer enhancement during evaporation and boiling of the liquid, and accelerate rewetting of the inner surface after occasional local dryout. Moreover, since the liquid can flow along the axial grooves under the influence of the capillary pressure and the pressure gradient along the flow channel, the heated length can be larger for the considered configuration than it would be for miniature channels with smooth walls.

The heat pipe-type (that is, pure evaporative) regime has been modeled by Khrustalev and Faghri (1996; see the preceding subsubsection) for moderate heat loads (${q}''\le 65\text{ W/c}{{\text{m}}^{\text{2}}}$). Because surface tension tends to pull the liquid into the grooves, the flow boiling length should be even smaller than that for the case with smooth walls, so that the annular regime transforms into separated cocurrent two-phase flow. Surface tension seems to be a stabilizing factor for liquid flow in a groove, and it predominates over the destabilizing dynamic vapor-liquid interaction when the Weber number is small. As mentioned above, the liquid flows in the grooves under the influence of the capillary pressure gradient and the vapor pressure gradient, which can be expressed by the Laplace-Young equation in differential form.

A test module was manufactured from a plate of oxygen-free copper using an electro-discharge machining process (EDM). The module consisted of a rectangular cross-section channel with an axially grooved inner surface on the side walls and six thick-film resistors soldered onto the outer surfaces (three resistors per side). The test section had the following dimensions: tt = 7.7 mm, th = 2.6 mm, Lt = 50 mm, and tw1 = tw2 = 0.5 mm. There were 27 grooves on each of the two inner surfaces of the side. The half-width of each groove, W, was about 0.11 mm. The ends of the test module were inserted into copper tubes with crimped ends and soldered so that 44 mm of the section length was available for installation of the resistors. The surface area of a resistor soldered onto the section wall was 6x12 mm2. The gap between the resistors was about 1 mm, so that the maximum effective heated length was about 40 mm. As a result of the manufacturing process, the test section had two longitudinal fins with cross sections of 11 mm2 along the centerlines of the 2.6 mm wide lateral walls. Nine copper-constantan thermocouple beads were soldered directly to the test section 5 mm apart, in the corner between the section wall and the fin. This allowed for the observation of local fluctuations in the wall temperature along the heat sink. The thermocouple beads were covered with a high-temperature, low-thermal conductivity cement layer. The test section and the connecting copper tubes were thoroughly insulated to prevent heat losses into the environment. Heat losses from the heat sink were estimated to be less than 1% of the total heat load.

The heat flux on the wall of the heat sink was restricted by dryout, which led to a drastic increase in the heated wall temperature. Khrustalev and Faghri (1997) noted that the entrainment of the liquid in the grooves by the high-velocity vapor flow is one of the most important mechanisms for the comparatively long miniature channel. The entrainment of the liquid begins when the dynamic impact of the vapor flow is comparable with the stabilizing effect of the capillary pressure. The latter is usually described by the condition We = 1, where the Weber number is defined in eq. (19). Therefore, for uniform heat flux, and neglecting the effect of subcooling, it follows from eq. (19) and the condition We = 1 that

${{{q}''}_{crit}}=\frac{{{(\pi \sigma W{{\rho }_{v}}/2)}^{1/2}}}{{{L}_{eff}}{{D}_{h,\ell }}}{{h}_{\ell v}}[{{t}_{h}}-2({{t}_{w1}}+{{t}_{g}})] \qquad \qquad(35)$

Table 1 Comparison of the experimental q''crit with the prediction by eq. (35).

 Effective heated length (mm) po,sat(kPa) q''crit,W/cm2 Experimental q''crit,W/cm2 eq.(35) 40 101 84 87 26 101 108 134 13 101 130 269 13 20 – 138

assuming that the grooves are filled with liquid prior to the beginning of entrainment. Table 1 compares the critical heat fluxes predicted by eq. (35) and those obtained experimentally. For the short length of 13 mm, eq. (35) failed to predict the experimental value of q''crit because in that case, dryout occurred due to the traditionally recognized mechanism whereby the intense vapor bubble formation began on the heated surface before the entrainment could do so. For larger heated lengths, however, the prediction was accurate. The predicted critical heat flux increased with the saturation pressure. To better determine the critical mechanisms inducing dryout, more experimental data with different geometries of the grooved surfaces are needed.

For the effective heat transfer coefficients measured, the values were comparatively high (up to 100,000 W/m-K) and of the same order of magnitude as those predicted by the evaporative model by Khrustalev and Faghri (1996). The effective heat transfer coefficients did not increase with the heat flux for ${q}''\le 45\text{ W/c}{{\text{m}}^{\text{2}}}$. This means that with moderate heat flux, the impact of evaporation on heat transfer prevailed over the impact of boiling across the heated length. Actually, the heat sink worked in the mixed boiling-evaporation regime. With small mass flow rates (even with a vapor quality less than 0.3), instabilities and dryout occurred with heat fluxes less than 50 W/cm2. This may have been because the grooves were too shallow to provide the necessary mass flow rate across the entire heated length. With larger mass flow rates, operation was stable due to higher pressure gradients in the channel. The critical heat flux decreased with the heated length. Since the total heat load was proportional to the heated length, it was assumed that dryout was induced mainly by the entrainment of the liquid in the grooves with the vapor flow. However, the mechanisms of dryout in this heat sink should be further investigated.

## Boiling in Micro/Miniature Channels

For two-phase flow in a microchannel, there is no stratified flow; therefore, the orientation of the channel has little effect on the flow pattern. The flow patterns for boiling in microchannels in order of appearance from the inlet are: bubbly flow, elongated bubbles, annular, mist, and flows with partial dryout. Since the diameter of the channel is so small, the bubbly flow regime is very short, as the bubbles grow to the size of the channel very quickly. Once the bubbles grow to the size of the channel, the flow pattern becomes an elongated bubble flow, followed by an annular flow regime. Cyclic dryout is also possible in the elongated bubble flow regime, which is also referred to as the confined bubble regime (Vlasie et al., 2004).

Thome et al. (2004) developed a three-zone flow boiling model for the evaporation of elongated bubbles in microchannels. The vapor bubble is quickly nucleated and grows to the size of the channel to become an elongated bubble. At any given location in the microchannel that is passed by a liquid slug, an elongated bubble then follows, leaving liquid film behind. Evaporation of the thin film makes the elongated bubble grow longer and move forward. If the thin film dries out before the next liquid slug arrives, a vapor plug (dry zone) passes. This cycle repeats itself so any given point in the microchannel is passed by liquid slugs, elongated bubbles and vapor slugs. Thome et al. (2004) and Dupont and Thome (2004) developed a three-zone heat transfer model based on a homogeneous flow assumption, i.e., the liquid and vapor were assumed to have the same velocity. The model illustrated the importance of strong cyclic variation in the heat transfer coefficient as well as the strong dependency of the heat transfer on bubble frequency, the minimum film thickness at dryout, and the liquid film’s formation thickness. They concluded that the heat transfer coefficient in the elongated film region is several orders of magnitude larger than that in the liquid slug region, while heat transfer in the dry zone is nearly negligible. The time-averaged overall heat transfer coefficient strongly depends on the relative lengths of these three zones.

Vlasie et al. (2004) thoroughly reviewed the available empirical correlations for heat transfer in flow boiling in small-diameter channels. Lazarek and Black (1982) studied flow boiling of R-113 in 123 and 246 mm long circular vertical tubes with internal diameters of 3.1 mm. They recommended the following correlation:

$\text{N}{{\text{u}}_{\text{D}}}=30\operatorname{Re}_{D}^{0.857}\text{B}{{\text{o}}^{0.714}} \qquad \qquad(36)$

where

$\text{Bo}=\frac{{{q}''}}{{{h}_{\ell v}}\dot{{m}''}} \qquad \qquad(37)$

is the boiling number. Equation (36) is obtained by analyzing experimental data performed in the following ranges: 14k W/m2 < q'' < 380 kW/m2, 125 kg/m2-s $<\dot{{m}''}<725\text{ kg/}{{\text{m}}^{\text{2}}}\text{-s}$, and 130 kPa < p < 140 kPa. In arriving at eq. (36), the effect of quality on heat transfer is neglected.

Kew and Cornwell (1997) performed experiments of boiling of R-141 in a 500 mm-long, small-diameter tube (D = 1.39 to 3.69 mm). While the results for 3.69 and 2.87 mm tubes follow trends similar to those observed in conventionally sized tubes, the results for the 1.39 mm tube depart from these trends significantly. To account for the effect of quality on boiling heat transfer, Kew and Cornwell (1997) proposed a modified Lazerek and Black correlation as follows:

$\text{N}{{\text{u}}_{\text{D}}}=30\operatorname{Re}_{D}^{0.857}\text{B}{{\text{o}}^{0.714}}{{(1-x)}^{-0.143}} \qquad \qquad(38)$

Despite the fact that there can be three different flow regimes – isolated bubble, confined bubble, and annular flow – in flow boiling in small-diameter tubes, nucleate boiling correlations, such as eqs. (36) and (38), often yield better results.

Tran et al. (1996) studied nucleate boiling of R-12 in a circular channel with diameter of 2.46 mm and rectangular channel of 1.70×4.06 mm2 and proposed the following correlation

$\text{h}=8.4\times {{10}^{5}}{{\text{(B}{{\text{o}}^{2}}\text{W}{{\text{e}}_{\ell }})}^{0.3}}{{({{\rho }_{\ell }}/{{\rho }_{v}})}^{-0.4}} \qquad \qquad(39)$

where the boiling number, Bo, is defined in eq. (37) and the Weber number is

$\text{W}{{\text{e}}_{\ell }}=\frac{{{{\dot{{m}''}}}^{2}}D}{{{\rho }_{\ell }}\sigma } \qquad \qquad(40)$

Equation (39) is obtained within the following range: 3.6 kW/m2 < q'' < 129 kW/m2, 44 kg/m2-s$<\dot{{m}''}<832\text{ kg/}{{\text{m}}^{\text{2}}}\text{-s}$, and 510 kPa < p < 820 kPa.

Yu et al. (2002) experimentally investigated flow boiling of water in a horizontal tube of 2.98-mm inner diameter at a system pressure of 200 kPa. The inlet temperature is 80 °C and the mass flux ranges from 50 to 200 kg/m2-s. They proposed the following correlations

$h=6.4\times {{10}^{6}}{{\text{(B}{{\text{o}}^{2}}\text{W}{{\text{e}}_{\ell }})}^{0.27}}{{({{\rho }_{\ell }}/{{\rho }_{v}})}^{-0.2}} \qquad \qquad(41)$

To enhance boiling heat transfer on the conventional surface, one can use special surface geometry to promote nucleate boiling by reducing the boiling inception superheat, increasing heat transfer coefficient, and increasing the critical heat flux. The early studies focused on application of rough surface to promote nucleation. However, the effect of roughness will diminish due to aging effects. In order to overcome the aging effect, one can utilize the microchannels with re-entrant cavities. (Koşar et al., 2005).

For nucleate boiling in five parallel rectangular channels with a hydraulic diameter of Dh = 0.233mm, Koşar et al. (2005) proposed the following correlation

$h=1.068{{{q}''}^{0.64}} \qquad \qquad(42)$

For convective boiling heat transfer in the same geometric configuration, they suggested:

$\text{h}=4.068\times {{10}^{4}}\operatorname{Re}_{\ell o}^{0.857}{{(1-{{x}_{e}})}^{0.8}}{{\left( \frac{1-{{x}_{e}}}{{{x}_{e}}} \right)}^{0.02}} \qquad \qquad(43)$

where xe is the quality at exit.

Qu and Mudawar (2004) studied the CHF phenomena in microchannels and found that when there is a significant compressible volume upstream of the heating section, an oscillating flow may lead to CHF. Another phenomenon observed by Qu and Mudawar (2004) that causes CHF is excursive instability caused by backflow of the vapor. The vapor flows backward from the individual microchannels into the upstream shallow plenum and forms a thick intermittent vapor layer. This layer breaks up into many small vapor bubbles, which propagate further upstream even deeper in the plenum.

There, they mix with the incoming liquid. Bergles and Kandlikar (2005) suggested that all CHF data for parallel microchannels are subject to the parallel channel/excursive instability. The incoming flow is simply absorbed by the compressible volume, and the stagnated flow in the headed channel may lead to CHF before the flow through the channel is resumed. Therefore, the upstream compressible volume leads to an oscillating flow in the headed channel, which causes the CHF.

Qu and Mudawar (2004) correlated 42 data points from their experimental results for water in microchannels with ${{D}_{h}}=380\text{ }\!\!\mu\!\!\text{ m}$, as well as data for R-113 in circular tube with D = 510 μm and 2.54 mm from their previous study (Bowers and Mudawar, 1994). They recommended the following empirical correlation for critical heat flux in microchannels:

$\frac{{{{{q}''}}_{\max }}}{\dot{{m}''}{{h}_{\ell v}}}=33.43{{\left( \frac{{{\rho }_{v}}}{{{\rho }_{\ell }}} \right)}^{1.11}}\text{W}{{\text{e}}^{-0.21}}{{\left( \frac{L}{{{D}_{h}}} \right)}^{-0.36}} \qquad \qquad(44)$

where

$\text{We}=\frac{\dot{{m}''}L}{\sigma {{\rho }_{\ell }}} \qquad \qquad(45)$

is the Weber number based on the channel heated length L.

It should be pointed out that the CHF obtained from eq. (44) represents a hydrodynamic CHF, which is caused by excursive instability. It is different from the CHF obtained from a stabilized single channel, in which case the CHF is caused by the usual dryout mechanism.

As noted above, boiling heat transfer in micro- and minichannels has attracted significant interest in recent years. However, the complex natures of convective flow boiling and two-phase flow in micro- and minichannels are still not well understood. Additional semi-empirical correlations that are physically based and cover a wide range of conditions in terms of size, fluid properties and flow conditions are needed.

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