# Evaporation from Cylindrical Pore under High Heat Flux

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At high heat flux, high velocity vapor flow may create an extended thick liquid film attached to the evaporating meniscus in spite of the capillary pressure drop between the hemispherical meniscus and the nearly flat thick film. Khrustalev and Faghri (1997) analyzed the evaporation of a pure liquid in the vicinity of a hemispherical liquid-vapor meniscus formed within a circular micropore, as shown in Fig. 9.18(b). Since the radius of the pore and the liquid film thickness are of the same order of magnitude, it is necessary to describe the problem in a cylindrical coordinate system, as shown in Fig. 9.25. Both the vapor and the liquid flow along the z-direction. The origin of the coordinate system is fixed with respect to the minimum film thickness. It is assumed that the vapor is at saturation condition. Heat transfer in the liquid film is in the radial direction by conduction only, and convective effects are neglected. The velocity profile in the fluid region is assumed to be fully developed and to behave as a modified laminar-type flow. This accounts for surface tension and mass transfer effects through the boundary conditions, but otherwise a single, axial direction momentum equation completely describes the flow field in the fluid. Combining expressions for the conservation of mass and the conservation of energy in the liquid film, one obtains

$\int_{R-\delta }^{R}{r{{w}_{\ell }}\left( r \right)dr}=\frac{1}{2\pi {{\rho }_{\ell }}}\left( {{{\dot{m}}}_{\ell ,in}}-\frac{q}{{{h}_{\ell v}}} \right)$ (9.264)

where ${{\dot{m}}_{\ell ,in}}$ is the incoming liquid mass flow rate at z = 0 in the liquid film layer and q(z) is the heat flow rate through a given cross-section due to evaporation in the film from z = 0 to z , i.e.,

$q=2\pi R\int_{0}^{z}{{{{{q}''}}_{R}}dz}$ (9.265)

For conduction heat transfer through the liquid film,

${{{q}''}_{R}}={{k}_{\ell }}\frac{{{T}_{w}}-{{T}_{\delta }}}{R\ln \left[ R/(R-\delta ) \right]}$ (9.266)

Drawn from the definition of total heat flow rate, an expression for the gradient of heat flow rate in the z-direction is

$\frac{dq}{dz}=2\pi {{k}_{\ell }}\frac{{{T}_{w}}-{{T}_{\delta }}}{\ln \left[ R/(R-\delta ) \right]}$ (9.267)

The differential equation of the velocity profile from the conservation of momentum in the film is

$\frac{1}{r}\frac{\partial }{\partial r}\left( r\frac{\partial {{w}_{\ell }}}{\partial r} \right)=\frac{1}{{{\mu }_{\ell }}}\frac{d{{p}_{\ell }}}{dz}$ (9.268)

The boundary conditions for this equation are the nonslip condition at the wall and equality of shear stress at the liquid-vapor interface:

$r=R\begin{matrix} , & {{w}_{\ell }}=0 \\ \end{matrix}$ (9.269)
$r=R-\delta \begin{matrix} , & \frac{\partial {{w}_{\ell }}}{\partial r}=\frac{1}{{{\mu }_{\ell }}}\left( -\frac{{{f}_{v}}}{2}{{\rho }_{v}}\bar{w}_{v}^{2}-\frac{d\sigma }{dT}\frac{d{{T}_{\delta }}}{dz} \right)=E \\ \end{matrix}$ (9.270)

where fv is the vapor fraction factor. The solution to eq. (9.268) with eqs. (9.269) and (9.270) as boundary conditions, gives

${{w}_{\ell }}=-\frac{1}{{{\mu }_{\ell }}}\frac{d{{p}_{\ell }}}{dz}\left[ \frac{1}{4}\left( {{R}^{2}}-{{r}^{2}} \right)+\frac{{{\left( R-\delta \right)}^{2}}}{2}\ln \frac{r}{R} \right]+E\left( R-\delta \right)\ln \frac{r}{R}$ (9.271)

Substituting eq. (9.271) into eq. (9.264), the following equation for liquid pressure in terms of the mass flow rate and heat flow rate is obtained:

$\frac{d{{p}_{\ell }}}{dz}=\frac{{{\mu }_{\ell }}\left[ \frac{1}{2\pi {{\rho }_{\ell }}}\left( \frac{q}{{{h}_{\ell v}}}-{{{\dot{m}}}_{\ell ,in}} \right)+E\left( R-\delta \right)F \right]}{\left[ \frac{{{R}^{4}}}{16}+\frac{{{\left( R-\delta \right)}^{2}}}{2}\left( F+\frac{{{\left( R-\delta \right)}^{2}}}{8}-\frac{{{R}^{2}}}{4} \right) \right]}$ (9.272)

where

$F=\frac{{{\left( R-\delta \right)}^{2}}}{2}\left( \ln \frac{R}{R-\delta }+\frac{1}{2} \right)-\frac{{{R}^{2}}}{4}$ (9.273)

The pressure difference in the vapor and liquid phases due to capillary and disjoining pressure is

${{p}_{v}}-{{p}_{\ell }}=\sigma \left\{ \frac{{{d}^{2}}\delta }{d{{z}^{2}}}{{\left[ 1+{{\left( \frac{d\delta }{dz} \right)}^{2}} \right]}^{-\frac{3}{2}}}+\frac{1}{R-\delta }\cos \left( \arctan \frac{d\delta }{dz} \right) \right\}-{{p}_{d}}$ (9.274)

where the two terms in the braces represent the two radii of curvature in the microchannel. Equation (9.274) can be rewritten as the following two ordinary differential equations:

$\frac{d\delta }{dz}={\delta }'$ (9.275)
$\frac{d{\delta }'}{dz}={{\left( 1+{{{{\delta }'}}^{2}} \right)}^{3/2}}\left[ \frac{{{p}_{v}}-{{p}_{\ell }}+{{p}_{d}}}{\sigma }-\frac{\cos (\arctan {\delta }')}{R-\delta } \right]$ (9.276)

The vapor flow is assumed to be compressible and quasi-one-dimensional (Faghri, 1995):

\begin{align} & \frac{d{{p}_{v}}}{dz}=\frac{1}{{{A}_{v}}}\left[ \frac{d}{dz}\left( -{{\beta }_{v}}{{\rho }_{v}}\bar{w}_{v}^{2}{{A}_{v}} \right)-{{f}_{v}}\rho \bar{w}_{v}^{2}\left( R-\delta \right) \right. \\ & \begin{matrix} {} & {} & \left. -2\pi \left( R-\delta \right){{\rho }_{v}}v_{v,\delta }^{2}\sin \left( \arctan \frac{d\delta }{dz} \right) \right] \\ \end{matrix} \\ \end{align} (9.277)

where βv = 1.33 for small Reynolds numbers. The blowing velocity due to evaporation is

${{v}_{v,\delta }}=\frac{dq}{dz}\frac{1}{2\pi \left( R-\delta \right){{\rho }_{v}}{{h}_{\ell v}}}$ (9.278)

The vapor is assumed to behave as an ideal gas.

${{\rho }_{v}}=\frac{{{p}_{v}}}{{{R}_{g}}{{T}_{v}}}$ (9.279)

The density gradient in the z-direction is then

$\frac{d{{\rho }_{v}}}{dz}=\frac{1}{{{R}_{g}}}\left( \frac{d{{p}_{v}}}{dz}\frac{1}{{{T}_{v}}}-\frac{{{p}_{v}}}{T_{v}^{2}}\frac{d{{T}_{v}}}{dz} \right)$ (9.280)

The temperature and pressure at saturation are related by the Clausius-Clapeyron equation –

$\frac{d{{T}_{v}}}{dz}=\frac{d{{p}_{v}}}{dz}\frac{{{R}_{g}}T_{v}^{2}}{{{p}_{v}}{{h}_{\ell v}}}$ (9.281)

At this point, seven expressions – eqs. (9.267), (9.272), (9.276), (9.277), (9.278), (9.280), and (9.281) – involving seven unknown variables – q, δ, δ', ${{p}_{\ell }}$, pv, ρv, and Tv – completely describe the stated assumptions. Boundary conditions are required for each of the seven first-order differential equations at z = 0:

q = 0 (9.282)
${{p}_{\ell }}={{p}_{v,in}}-\frac{2\sigma }{R-{{\delta }_{in}}}+{{p}_{d}}$ (9.283)
δ = δin (9.284)
δ' = 0 (9.285)
pv = pv,in = psat(Tv,in) (9.286)
${{\rho }_{v,in}}=\frac{{{p}_{v,in}}}{{{R}_{g}}{{T}_{v,in}}}$ (9.287)
Tv = Tv,in (9.288)

As can be seen from Fig. 9.25, the liquid film ends with a microfilm; therefore, its thickness at z = Lδ equals the nonevaporating film thickness δ0, determined by eq. (9.244). The interfacial temperature Tδ is required to obtain the gradient of the heat transfer rate along the z-direction, as indicated by eq. (9.266). It can be determined by solving eqs. (9.241) and (9.242) simultaneously. The mass flow rate of the liquid at z = 0, ${{\dot{m}}_{\ell ,in}}$, can be determined by the following boundary condition at the end of the microfilm:

${{\dot{m}}_{\ell ,in}}=\frac{q({{L}_{\delta }})}{{{h}_{\ell v}}}$ (9.289)

The problem described by seven ordinary differential equations – eqs. (9.267), (9.272), (9.276), (9.277), (9.278), (9.280) and (9.281) – with corresponding boundary conditions – eqs. (9.282) – (9.288) – can be solved using a Runge-Kutta method in conjunction with the shooting method to satisfy the constitutive equation, eq. (9.286). More detailed information about the numerical procedure can be found in Khrustalev and Faghri (1997).

Numerical solution is performed for water evaporation from a cylindrical pore with an inner diameter of 20 $\text{ }\!\!\mu\!\!\text{ m}$ and wall temperature of 388 K. The accommodation coefficient α is set equal to unity. Fig. 9.26(a) shows the variation in liquid film thickness along the z-direction. The liquid film thickness initially increases slowly along the z-coordinate, and then rapidly decreases approaching the microfilm region at the end of the film. The film is concave at z=0, where it attaches to the hemispherical meniscus; however, it is convex over most of its length. This is due to the pressure gradient in the liquid and vapor along the film. Figure 9.26(b) demonstrates that the vapor pressure is decreasing and the liquid pressure is increasing along the z-coordinate. The liquid flows in the film due mainly to frictional vapor-liquid interaction at the interface.

Figure 9.27(a) demonstrates that the rate of evaporation decreases towards the thickest part of the liquid film and reaches its maximum in the microfilm region. However, the microfilm region at the end of the liquid film does not make any significant contribution to the total mass flow rate at the pore outlet, because (a) the length of the microfilm region is much smaller than in the case where a hemispherical meniscus ends directly with a microfilm; and (b) disjoining pressure causes the liquid-vapor interfacial temperature Tδ to sharply increase at the end of the film, as shown in Fig. 9.27 (b). Khrustalev and Faghri’s (1997) work demonstrated, for the first time, that high-vapor velocities during the evaporation of pure liquids in micropores could allow a thick liquid film to attach to a hemispherical meniscus. In addition to the evaporation of pure liquid in a circular channel discussed in this section, Coquard et al. (2005) investigated evaporation in a capillary tube with a square cross-section. The evaporation rate is much higher than in a circular tube because of the liquid flow along the corner induced by capillary force. They focused on slow evaporation controlled by mass transfer and identified three regimes: the capillary regime, capillary-viscous regime, and capillary-gravity regime.