# Fluid Flow Effect in Pore/Slots during Evaporation

Khrustalev and Faghri (1996) investigated the influence of the liquid and vapor flows on the heat transfer in evaporation from a liquid-vapor meniscus, which is presented in this subsection. Evaporation of liquids from capillary grooves, pores, and slots takes place as a result of the solid wall superheat. The temperature of the solid-liquid interface, Tw, is higher than that of the vapor-liquid interface. Since the solid wall thermal conductivity is significantly higher than that of the liquid in many technical applications, constant wall temperature is assumed in the present analysis. The solid wall in the vicinity of the interline, which is an apparent liquid-vapor-solid boundary, can be dry or covered with nonevaporating liquid film; in both cases, the temperature of the solid wall or of the nonevaporating liquid free surface is Tw. Since the temperature of the vapor over the evaporating meniscus is lower than Tw, the vapor is affected by the superheated wall and/or liquid such that it also becomes superheated. This means that the temperature of the vapor some distance from the liquid-vapor interface can be higher than Tsat(pv). Note that for comparatively thick liquid films ($\delta \ge 1\text{ }\!\!\mu\!\!\text{ m}$), the liquid-vapor interface temperature is very close to the vapor temperature. This is true because the interfacial thermal resistance is small compared to that of the liquid films. Based on this consideration, the momentum conservation and energy equations for the evaporating liquid and vapor over the liquid-vapor interface are included in the present model of the evaporating extended meniscus.

The evaporation of liquid from the liquid-vapor meniscus interface in a narrow slot as considered here is illustrated by Fig. 9.21(a). Gravitational body forces are negligible in comparison to surface tension forces, due to the small size of the meniscus. It is assumed that the liquid-vapor interface curvature is constant: K = 1/Rmen. This assumption can be justified by the results of Swanson and Peterson (1994), where one-sided formulation of the heat transfer and flow in liquid was used. Actually, the curvature of the liquid-vapor interface of the evaporating non-isothermal meniscus is not exactly constant, especially in the microfilm region. However, the assumption of constant curvature is sufficient to model the heat transfer and flow in the vapor phase over the evaporating meniscus. The evaporating microfilm is the thin-film part of the meniscus where the disjoining pressure is significant and where the temperature of the liquid-vapor interface differs from that of the vapor phase over the given point of the interface. This microfilm is considered using the approach in Section 5.5.4, while the fluid flow and heat transfer in the major part of the domain shown in Fig. 9.21(a) are treated by two-dimensional analysis.

(a) Computational domain (b) Control volume that contains interface, Figure 9.21 Evaporating liquid-vapor meniscus in a slot.

For an incompressible fluid with constant vapor and liquid viscosities and temperature dependent vapor density, the continuity and momentum equations for both the vapor and liquid are:

 $\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0$ (1)
 $\rho \left( u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y} \right)=-\frac{\partial p}{\partial x}+\mu \left( \frac{{{\partial }^{2}}u}{\partial {{x}^{2}}}+\frac{{{\partial }^{2}}u}{\partial {{y}^{2}}} \right)$ (2)
 $\rho \left( u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y} \right)=-\frac{\partial p}{\partial y}+\mu \left( \frac{{{\partial }^{2}}v}{\partial {{x}^{2}}}+\frac{{{\partial }^{2}}v}{\partial {{y}^{2}}} \right)$ (3)

The energy equation for both the vapor and liquid is

 $\rho {{c}_{p}}\left( u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y} \right)=k\left( \frac{{{\partial }^{2}}T}{\partial {{x}^{2}}}+\frac{{{\partial }^{2}}T}{\partial {{y}^{2}}} \right)$ (4)

The wall temperature is assumed constant (except for the control volume enclosing the microfilm).

 ${{\left. T \right|}_{x=0}}={{T}_{w}}$ (5)

The inlet flow temperature is known.

 ${{\left. T \right|}_{y=0}}={{T}_{w}}$ (6)

Nonslip conditions at the wall are assumed.

 ${{\left. u \right|}_{x=0}}={{\left. v \right|}_{x=0}}=0$ (7)

At the inlet, the velocity profile is assumed to be fully developed:

 ${{\left. u \right|}_{y=0}}=0$ (8)
 ${{\left. v \right|}_{y=0}}=\frac{3{{{\dot{m}}}_{\ell ,in}}}{2{{\rho }_{\ell }}}\frac{x}{X_{L}^{3}}(2{{X}_{L}}-x)$ (9)

At the symmetry line, x = XL, u = 0, dv / dx = 0, and dT / dx = 0.

Heat transfer at the interface was considered through analysis of the mass and energy balance for a rectangular control volume containing a liquid-vapor interface [see Fig. 9.21 (b)]. The microfilm region, where the temperature of the liquid-vapor interface changes from Tw at the interline, which is the apparent liquid-vapor-solid boundary, to Tsat at the end of the microfilm, has been enclosed into a special control volume with the dimensions $\Delta x\times \Delta y$. This is the vapor control volume since the microfilm thickness, δ, is much smaller than Δx. This means that in the two-dimensional analysis, the liquid microfilm is represented only by the vapor blowing velocity at the solid wall, ${{\mathbf{V}}_{v,mic}}$, and by the temperature of the wall, Tmic, (this is the mean temperature of the microfilm free surface) in the special control volume. Note that for the liquid control volume located under the microfilm control volume [see Fig. 9.21 (b)], the velocity of the vapor at the upper liquid-vapor face is increased in the numerical procedure by ${{v}_{\ell }}=({{\rho }_{v}}/{{\rho }_{\ell }}){{\left| \mathbf{V} \right|}_{v,mic}}\Delta y/\Delta x$ in order to satisfy conservation of mass. The values ${{\mathbf{V}}_{v,mic}}$ and Tmic are the microfilm characteristics averaged over Δy along its length, which does not exceed Δy.

 ${{T}_{mic}}={\int_{0}^{\Delta y}{{{{{q}''}}_{\delta }}(y){{T}_{\delta }}(y)dy}}/{\int_{0}^{\Delta y}{{{{{q}''}}_{\delta }}(y)dy}}\;$ (10)
 ${{\left| \mathbf{V} \right|}_{v,mic}}={\int_{0}^{\Delta y}{{{{{q}''}}_{\delta }}(y)dy}}/{\left( \Delta y{{h}_{\ell v}}{{\rho }_{v}} \right)}\;$ (11)

where q''δ is the heat flux at the interface, which is equal to that due to heat conduction through the liquid film; eq. (9.240) defines that for the case of a comparatively small heat flux. Note that the interfacial thermal resistance is very important for the microfilm and results in the difference between the interface temperature, Tδ, and that of the vapor over the interface, Tsat. For the rest of the meniscus, the interfacial thermal resistance is negligible in comparison to that of the liquid film, and Tδ ~ Tsat, which is justified by calculations made at the end of the microfilm.

The effective evaporative heat transfer coefficient characterizing the intensity of the evaporative heat transfer is defined as follows:

 ${{h}_{e}}=\frac{{{{\dot{m}}}_{\ell ,in}}{{h}_{\ell v}}}{{{X}_{L}}({{T}_{w}}-{{T}_{sat}})}$ (12)

(a)ΔT=5K (b) ΔT=10K Figure 9.22 Vapor flow over evaporating meniscus. (a)ΔT=5K (b) ΔT=10K Figure 9.23 Liquid flow in the evaporating meniscus.

where Tsat is the vapor temperature over the liquid-vapor interface, which is assumed constant due to saturation conditions and because the curvature of the interface was constant.

The problem was solved using the SIMPLE algorithm (Patankar, 1980) employing the power law discretization scheme and a line-by-line TDMA method. All control volumes in the computational domain were divided into two categories: “liquid” and “vapor,” with corresponding thermophysical properties. The following nonequality was used to identify the vapor control volumes:

 $x>{{X}_{L}}-[R_{\text{men}}^{\text{2}}-{{(y-{{Y}_{\text{men}}}-{{R}_{\text{men}}})}^{2}}]$ (13)
Figure 9.24 Effective heat transfer coefficient versus superheat of water at atmospheric pressure.

Numerical results were obtained for water at the saturation temperature of 373 K for ${{X}_{L}}=50\text{ }\!\!\mu\!\!\text{ m,}$ ${{Y}_{L}}=400\text{ }\!\!\mu\!\!\text{ m}$, ${{Y}_{B}}=100\text{ }\!\!\mu\!\!\text{ m}$, and ${{\theta }_{\text{men}}}={{15}^{\circ }}$. The accommodation coefficient, α, was set equal to unity. Typical vapor velocity fields over the evaporating meniscus are shown in Fig. 9.22 for comparatively small (a) and large (b) temperature drops (ΔT = TwTsat) between the wall and the saturation temperature. The vapor velocity in the microfilm control volume can be as high as 100 m/s for ΔT = 15 K. Such a high-blowing velocity results in a recirculation zone in the vapor near the wall over the interline, as shown in Fig. 9.22(b). Since the velocities in the liquid are much smaller than those in the vapor, they are not shown in Fig. 9.22. For small temperature drops, the vapor flows along the x-coordinate near the heated wall and over the interline. However, for larger temperature drops, the vapor flow changes its direction in the above-mentioned region due to appearance of the recirculation zone. This effect can be important for micro heat pipes and other applications. The liquid flow is shown in Fig. 9.23. The liquid-vapor interface in Fig. 9.23 is represented by the velocity vectors at the vapor side of the interface. The liquid flow pattern did not change with temperature drop, as can be seen from comparison of Fig. 9.23 (a) and 9.23 (b).

The numerical results for the effective heat transfer coefficient he were compared to those obtained using the one-dimensional model (see Section 9.6.1) by Khrustalev and Faghri (1995a). For small ΔT the agreement was very good: the one-dimensional model underestimated the heat transfer coefficients by only about 3%, as shown in Fig. 9.24. For larger temperature drops, however, the difference between the present and the one-dimensional model became 30 % (for ΔT = 10 °C) due to the convective heat transfer.

## References

Khrustalev, D. and Faghri, A., 1995a, “Heat Transfer in the Inverted Meniscus Type Evaporator at High Heat Fluxes,” International Journal of Heat and Mass Transfer, Vol. 38, pp. 3091-3101.

Khrustalev, D.K. and Faghri, A., 1996, “Fluid Flow Effects in Evaporation from Liquid/Vapor Meniscus,” ASME Journal of Heat Transfer, Vol. 118, pp. 725-730.

Patankar, S.V, 1980, Numerical Heat Transfer and Fluid Flow, McGraw-Hill, New York.

Swanson, L.W., and Peterson, G.P., 1994, “Evaporating Extended Meniscus in a V-Shaped Channel,” AIAA Journal of Thermophysics and Heat Transfer, Vol. 8, No. 1, pp. 172-180.