Evaporation from Inverted Meniscus in Porous Media
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Evaporators that are capable of withstanding highheat fluxes, for example larger than 100 W/cm^{2}, are of great interest for electronic component cooling systems. The most promising evaporator design is the socalled “inverted meniscus” type. To predict the critical heat flux and effective heat transfer coefficients in the evaporator, a mathematical model has been developed by Khrustalev and Faghri (1995a). The model includes the following interconnected problems, which are treated simultaneously in the frames of the numerical analysis: (1) heat transfer during evaporation from a pore, (2) heat transfer and vapor flow in the dry region of a porous structure with a stable side boundary, the location of which depends on the operational conditions, and (3) heat conduction in a solid fin (or wall) with a nonuniform heat sink on the side surfaces. Heat transfer during evaporation from a pore has been presented in Section 9.6.1. Heat transfer and vapor flow in the dry region, as well as heat conduction in the solid fin, will be presented in this subsection.
Schematics of two configurations for the characteristic elements of inverted meniscus evaporators are shown in Fig. 9.30. In the first configuration [Figs. 9.30 (a) and (b)], a heated triangular fin is inserted into the porous plate and sintered with it in order to provide good thermal contact. In the second configuration [Fig. 9.30 (c)], the heated wall is flat. With small heat flux, evaporation of the liquid, which saturates the porous element, can take place exclusively from the surface of the porous body into the vapor channel as shown in Fig. 9.30(a). Extremely high heat fluxes are significantly more interesting for industrial applications, and the existence of a stable vapor blanket along the heated solid surface inside the uniform porous structure was anticipated, as shown in Figs. 9.30(b) and (c). Evaporation takes place into the dry region of the porous structure at the liquidvapor interface, the location of which shifts depending on the operational conditions. Heat from the heated surface is conducted through the porous element’s dry region to the interface. The varying curvatures of the menisci along the liquidvapor interface create a capillary pressure gradient, which in turn drives the vapor flow. While the vapor flow takes place in a comparatively narrow porous passage, the liquid with the same total mass and flow rate (steady state) is filtered perpendicularly through the entire porous element to the liquidvapor interface. The pressure gradient in the liquid along this interface is negligible compared to that in vapor. This assumption can be justified by the case where the maximum pressure drop in liquid over the wetted region of the characteristic element under consideration is negligible compared to the pressure drop in vapor in the dry region. This assumption allows for onedimensional approximation description of the heat transfer in the vapor blanket.
The thickness of the vapor blanket, δ_{v}, increases with the heat flux, which can lead to increasing thermal resistance in the element. In the situation where δ_{v}x=0 = 0 is of the same order of magnitude as the minimum thickness of the porous element [see Fig. 9.30 (b)], the vapor can penetrate into the liquid channels and obstruct the liquid supply of the evaporator, eventually resulting in dryout. The value of the heat flux at which the dryout takes place can be considered as critical.
The operating parameters of the evaporator depend upon the heat and mass circulation in the entire system (for example in a heat pipe), with the evaporator in consideration. In the present model, the physical situation for the characteristic element is determined by three parameters: (1) the pressure in the liquid near the interface, , (2) the temperature of the solid surface, T_{0}, at x = 0, and (3) the liquidvapor meniscus radius at the end of the vapor blanket (x = L_{vδ}), R_{men},o. Note that the superheat of the fin exists at the following condition:

where the subscript “sat” denotes the normal saturation temperature corresponding to a pressure () and R_{men,min} = R_{p} / cosθ_{men,min} (R_{p} is pore radius). Inequality (1) characterizes how the value of the solidliquid superheat, Rmen,o (curvature radius of meniscus at outlet), is related to the fluid circulation in the entire device. For the case of the evaporator with the forced liquid supply it can be set , because the pressure drop in the liquid is not due to the capillary pressure in this case.
Since it can be anticipated that the temperature drops in metallic fin or wall are much smaller than those across the dry zone of the porous structure, because kw keff, the heat conduction in the solid fin or wall is considered using a lD approach. For the case of the flat wall [Fig. 9.30(c)] it means that dTw/dy is not included in the consideration. The heat con¬duction in the triangular metallic fin is described by the following equation [Fig. 9.30(b)], which was obtained as a result of energy balance on a differential element consideration:

where T_{s} is the local temperature of the porous struc¬ture at the liquidvapor interface location. Similarly, the heat conduction equation for the wall in Fig. 9.30 (c) is

The boundary conditions for equations (2) and (3) are


For the second configuration, q''_{0} is the uniform heat flux at the outer surface of the heated part of the flat wall. The value of q''_{0} and the functions δ_{v}(x) and T_{s}(x) should be given by the results of the vapor flow and heat transfer in the dry region solution considered below.
The local heat flux resulting from heat conduction across the dry region of the porous structure from the solid surface to the liquidvapor interface where evap¬oration takes place is

which is valid for the case and . Hence, the mean velocity of the vapor flow for a given x along the solid surface is (the mass and energy conservation balances)

where is the mean vapor velocity along the x¬coordinate. The modified Darcy’s equations for the vapor flow in both directions through a porous structure where the value of 0.55 is used for a dimensionless formdrag constant are


where u_{v} and v_{v} are the areaaveraged vapor velocities. The corresponding continuity equation is

The Darcy’s equation is semi¬empirical and describes the flow with the uniform velocity profile; therefore, it is assumed in the present analysis that u_{v} does not depend on y. Taking the definitions of the mean vapor pressure () and axial vel¬ocity () for a given x into consideration and integrating eq. (8) over y, the following equation can be obtained for the gradient of the mean vapor pressure along the xcoor¬dinate

Since the situation when is considered, the vapor pressure drop across the vapor blanket is much smaller than that along the xcoordinate. At the solid fin (or wall) surface , and at the liquidvapor interface where ζ = cos[arctan(dδ_{v}dx)] is the cosine of the angle between the y coordinate and the normal to the liquidvapor interface and v_{vδ} is the blowing velocity (normal to the liquidvapor interface) :

Equation (12) implies that the total amount of energy transferred from the heated solid surface to the liquidvapor interface by the heat conduction across the dry porous zone is spent on vaporization of the liquid. Since the axial velocity profile is nearly uniform, it follows from equation (10) that v_{v} = v_{vδ}ζy / δ_{v}. Integrating eq. (9) twice over y for a given x and implementing the definition of the difference between the vapor pressure near the liquidvapor interface, p_{vδ}, and the mean vapor pressure of the vapor flow, , for a given x can be estimated as follows

Combining eqs. (7) and (11), we have for the vapor filtration flow pressure gradient along the xcoordinate:

The boundary condition for eq. (14) follows from eqs. (9.248), (12) and (13)

The local heat flux at the liquidvapor interface due to the evap¬oration of the liquid is:

Combining eqs. (6) and (16) because of the steady state situation, the expression for the local temperature of the porous structure at the liquidvapor interface location is:

Substituting eqs. (12) and (13) into eq. (9.248) and differentiating it, the following equation for the radius of the meniscus curvature can be obtained

with the boundary condition

where C_{0} should be chosen from the constitutive con¬dition for the minimum value of the meniscus radius along the liquidvapor interface

Now, the condition of the liquidvapor interface mechanical equilibrium should be considered in order to find its location or δ_{v}(x). In the analysis by Solov’ev and Kovalev (1987) it was assumed that which is not quite satisfactory for several reasons. For example, in a hypothetical situation in which the definite start point along the xcoordinate is x_{1}, and there is no evaporation from the liquidvapor interface, , a condition which is not satisfied by the discussed expression. Wulz and Embacher (1990) have modeled the vapor flow in the uniform zone of the dry porous structure. The thickness of the vapor zone was determined as 0.1 mm at q''_{max} = 17500 W/m2 by comparing the calculated temperature difference between the fin top and the phase boundary with the value determined by the experiment using the simpler vapor zone model. Chung and Catton (1993) have considered the problem of steam injection into a slow water flow through porous media, where the interface location was also unknown. They have found “…that the interface can be idealized as a stream line as far as the momentum equations are concerned.” Here the concept of a streamline is used indirectly as explained below. For the asymptotic case, integrating Euler's equation along a streamline gives

where the terms containing correspond to the inertia effects due to acceleration of fluid. In the present analysis, the vapor flow through a porous medium is described by Darcy’s momentum equation. However, it is assumed that since the velocity profile of the vapor flow along the xcoordinate, u_{v}, is nearly uniform, eq. (21) can be used for the description of the inertia effects at the liquidvapor interface due to acceleration of the vapor. The liquidvapor interface can be stable provided it has the shape that eliminates the influence of the inertia effects due to acceleration of the vapor flow on the vapor pressure near this interface. While the steadystate situation is analyzed, the liquid pressure along the interface is constant, and the pressure losses in the vapor flow in both directions due to friction and solid obstacles are compensated by the capillary pressure, the vapor pressure gradient along the stable interface due to these inertia effects should be equal to zero. Since the velocity profile of the vapor flow along the xcoordinate is nearly uniform, it follows from eq. (21) that

Note that eq. (22) is not used for the fluid flow in the porous medium but describes the inertia effects at the adjustable liquidvapor interface in which the momentum equations for the vapor flow in the porous medium are concerned. Equation (22) is necessary in order to find the equilibrium location of the liquidvapor boundary. Substituting eqs. (7) and (12) into eq. (22) and differentiating it after some rearrangements gives the equation for the vapor blanket thickness, δ_{v}

where all of the terms containing (T_{w} − T_{s}) can be calculated in the numerical procedure using the func¬tions T_{w}(x) and T_{s}(x) determined at the previous iter¬ation. The secondorder differential equation (9.331) should be solved with the two boundary conditions for the variables δ_{v} and dδ_{v} / dx. The first boundary condition is

where C_{1} should be chosen from the constitutive condition as the value of which satisfies the following boundary condition

The second boundary condition is due to the sym¬metry of the considered element (Fig. 9.29). Since at the point x = 0, dTw/dx = 0, dTs/dx = 0, and dv_{vδ} / dx = 0 due to the physical reasons, it follows from equation (12)

Thus we have six main variables (or unknown func¬tions): p_{vδ}, R_{men}, 'T_{s}, v_{vδ} and δ_{v} which should be found from the six equations: (9.248), (12), (14), (17), (18), and (23). These six equations should be solved along with those presented in the previous sections for variables h_{e,p}(x) and Tw(x). Note that the value q''_{0}which is needed for eq. (9.311) can now be found as:

For the first configuration, q''_{0} is the heat flux in the solid fin corresponding to the porous structurevapor channel plane. The heat flux on the outer surface of the evaporator (and the corresponding effective heat transfer coefficient) can be recalculated taking the geometry of the evaporator into consideration. Although the vapor leaving the dry zone of the porous structure is superheated, it is convenient to relate the local effective heat transfer coefficient to the vapor saturation temperature, because . Thus, the local effective heat transfer coefficient cor¬responding to the point x = L_{vb} (outlet of the vapor flow) is defined as:

Khrustalev and Faghri (1995a) modeled heat transfer during evaporation from a pore (see Section 9.6.2), heat conduction in the solid fin or wall, vapor flow and heat transfer in the dry region of the porous structure. Numerical solution was performed and the results were obtained with constant thermal physical properties corresponding to the saturation temperature . The numerical results were obtained for the first configuration [Fig. 9.30(b)] for the case of a miniature evaporator: t_{pen} = 0.2 mm, α = 0.05,k_{eff} = 10 W/mK, k_{w} = 438 W/mK, and Since the longitudinal circulation of the fluid in the heat pipe was not considered, the numerical results were obtained for several fixed Rmen,o. The temperatures of the fin surface and the porous structure at the liquidvapor interface, as well as local heat flux across the vapor blanket, are shown in Fig. 9.31. It can be seen that the temperature drops at the solid heated surface were significantly smaller than those corresponding to the porous skeleton at the liquidvapor interface. The real superheat of the liquid, T_{s} − T_{v}, which could initiate the boiling, was significantly smaller than the superheat of the heated solid surface: T_{s} − T_{v} < T_{w} − T_{v}. The local heat fluxes across the dry zone had their maximums at the point x = 0, as can be seen in Fig. 9.31(b).
For extremely high heat fluxes, q''_{loc}, when the temperature drop (T_{s} − T_{v}) is large, boiling of the liquid at the liquidvapor interface can occur. This can cause instabilities at the liquidvapor interface. However, boiling of the liquid does not necessarily result in dryout for an evaporator of this type. Since q''_{o} > q''_{loc} due to L_{vb} > W, the triangular geometry of the solid fin helps to postpone boiling; this may make the fin more desirable than the flat wall in the second configuration shown in Fig. 9.30 (c). In other words, the triangular geometry of the fin provides a higher value for the heat flux on the outer surface of the evaporator, which corresponds to the beginning of liquid boiling at the liquidvapor interface.
References
Chung, M., and Catton, I., 1993, “Steam Injection into a Slow Water Flow through Porous Media,” ASME Journal of Heat Transfer, Vol. 115, pp. 734743
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. 30913101.
Wulz, H., and Embacher, E., 1990, “Capillary Pumped Loops for Space Applications Experimental and Theoretical Studies on the Performance of Capillary Evaporator Designs,” Proceedings of the AIAA/ASME 5th Joint Thermophysics and Heat Transfer Conference, Seattle, WA, AIAA Paper No. 901739.