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Phase-Change Transpiration Cooling in a Porous Medium: Determination of the Liquid/Two-Phase/Vapor Interfaces as a Problem of Eigenvalues

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Abstract

In this work, we carry out a theoretical analysis of the transpiration cooling with a liquid coolant phase change in a porous medium. The evaporation of the liquid inside the porous medium causes the appearance of three regions: a liquid region, a two-phase region and a vapor region. This kind of physical problem has been widely studied in the specialized literature and the main contributions are based on the separated phase model or by using the two-phase mixture model. Here, we propose a new model that permits to numerically determine the thickness of the three regions that are formed inside of the porous medium. To analyze this phenomenon, the governing equations were appropriately nondimensionalized, where suitable Péclet numbers for each region appear such that these numbers represent eigenvalues for the mathematical model, and when they are obtained, the relative position of the interfaces between each region is found.

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Abbreviations

a :

Parameter defined in Eq. (14)

b :

Parameter defined in Eq. (14)

Bo :

Bond number, \(K\rho _{\mathrm{l}}g{/}\sigma \)

\(c_{\mathrm{pl}}\) :

Effective specific heat of liquid

\(c_{\mathrm{pv}}\) :

Effective specific heat of vapor

Ca :

Capillary number, \(\dot{m}_{0}\nu _{\mathrm{l}} / \sigma \)

Da :

Darcy number, \((\varepsilon K)^{1/2}/\hbox {H}\)

\(d_{\mathrm{p}}\) :

Pore diameter

g:

Acceleration of gravity

H :

Thickness of the porous matrix

\(h_{\mathrm{lv}}\) :

Latent heat of evaporation

J(s):

Leverett’s function

\(Ja_{\mathrm{lv}}\) :

Jakob number for the two-phase region, \(c_{\mathrm{pl}} T_{\mathrm{sat}} /h_{\mathrm{lv}}\)

Ja :

Jakob number, \(c_{\mathrm{pl}} (T_{\mathrm{sat}}-T_{0}) /h_{\mathrm{lv}}\)

K :

Absolute permeability

\(K_{\mathrm{rl}}\) :

Relative permeability of the liquid phase

\(K_{\mathrm{rv}}\) :

Relative permeability of the vapor phase

\(k_{\mathrm{lv}}\) :

Effective thermal conductivity in the two-phase region

\(k_{\mathrm{l}}\) :

Thermal conductivity of the liquid phase

\(k_{\mathrm{v}}\) :

Thermal conductivity of the vapor phase

\(k_{\mathrm{s}}\) :

Thermal conductivity of the solid

\(k_{\alpha e}\) :

Effective thermal conductivity in the liquid and vapor regions

\(k^{*}\) :

Dimensionless effective thermal conductivity

\(L_{\mathrm{l}}\) :

Unknown length of the liquid region

\(\bar{L}_{\mathrm{l}}\) :

Dimensionless length of the liquid region, \(L_{\mathrm{l}}/\hbox {H}\)

\(L_{\mathrm{lv}}\) :

Unknown length of the two-phase region

\(\bar{L}_{\mathrm{lv}}\) :

Dimensionless length of the two-phase region, \(L_{\mathrm{lv}}/\hbox {H}\)

\(L_{\mathrm{v}}\) :

Unknown length of the vapor region

\(\bar{L}_{\mathrm{v}}\) :

Dimensionless length of the vapor region, \(L_{\mathrm{v}}/\hbox {H}\)

Le :

Modified capillary Lewis number

\(\dot{m}_{0}\) :

Coolant flow rate

\(\dot{m}_{\mathrm{l}}\) :

Mass flux of liquid in the two-phase region

\(\dot{m}_{\mathrm{v}}\) :

Mass flux of vapor in the two-phase region

M :

Molar mass

\(p_{\mathrm{l}}\) :

Pressure of the liquid phase

\(p_{\mathrm{v}}\) :

Pressure of the vapor phase

\(Pe_{\mathrm{l}}\) :

Péclet number of liquid region

\(Pe_{\mathrm{v}}\) :

Péclet number of vapor region

\(Pe_{\mathrm{c}}\) :

Péclet number of the system

\((Pe_{\mathrm{v}})_{i+1}\) :

Péclet number used in the iteration process

\((Pe_\mathrm{v})_{i}\) :

Initial guessed Péclet number

\(\Delta Pe_\mathrm{v}\) :

Increment to the corrected Péclet number

\(p_{\mathrm{c}}\) :

Capillary pressure

\(q''\) :

Heat flux

\(q''_{\mathrm{v}}\) :

Heat flux defined in Eq. (24)

Q :

Parameter defined in Eq. (38)

R :

Universal gas constant

s :

Dimensionless liquid saturation, \((s_{\mathrm{l}}-s_{\mathrm{li}})/(1-s_{\mathrm{li}})\)

\(s_{\mathrm{l}}\) :

Liquid saturation

\(s_{\mathrm{li}}\) :

Irreducible liquid saturation

T :

Absolute temperature

\(T_{\mathrm{sat}}\) :

Saturation temperature

\(T_{\mathrm{top}}\) :

Temperature at the upper surface

\(T_{\mathrm{l}}\) :

Temperature of the liquid region

\(T_{\mathrm{lv}}\) :

Temperature of the two-phase region

\(T_{\mathrm{v}}\) :

Temperature of the vapor region

\(T_{0}\) :

Temperature of the coolant flow at the bottom of the porous matrix

\(T_{\mathrm{lv}}\) :

Temperature in the two-phase region

\(T_{\mathrm{int}}\) :

Temperature of the interface between the two-phase and vapor region

\(\Delta T_{\mathrm{v}}\) :

Temperature drop in the vapor region

\(u_{\mathrm{l}}\) :

Superficial velocity of the liquid phase

\(u_{\mathrm{v}}\) :

Superficial velocity of the vapor phase

y :

Transversal coordinate at the system

Y :

Dimensionless transversal coordinate of the vapor region

z :

Dimensionless transversal coordinate of the two-phase region

\(\alpha \) :

Represents the liquid or vapor phases, \(\alpha = \hbox {l, v}\)

\(\bar{\alpha }\) :

Ratio of vapor to liquid density

\(\beta \) :

Dimensionless parameter defined in Eq. (43)

\(\gamma \) :

Dimensionless parameter defined in Eq. (31)

\(\bar{\gamma }\) :

Dimensionless parameter defined in Eq. (31)

\(\varepsilon \) :

Porosity of the porous medium

\(\eta \) :

Dimensionless transversal coordinate for the liquid region

\(\theta _{\mathrm{l}}\) :

Dimensionless temperature for the liquid region

\(\theta _{\mathrm{lv}}\) :

Dimensionless temperature for the two-phase region

\(\theta _{\mathrm{v}}\) :

Dimensionless temperature for the vapor region

\(\mu _{\mathrm{l}}\) :

Dynamic viscosity of the fluid

\(\nu _{\mathrm{l}}\) :

Kinematic viscosity for the liquid phase

\(\nu _{\mathrm{v}}\) :

Kinematic viscosity for the vapor phase

\(\bar{\nu }\) :

Ratio of kinematic viscosities of liquid and vapor

\(\Omega \) :

Dimensionless parameter defined in Eq. (31)

\(\xi \) :

Dimensionless parameter defined in Eq. (38)

\(\rho _{\mathrm{l}}\) :

Density of the liquid phase

\(\rho _{\mathrm{v}}\) :

Density of the vapor phase

\(\sigma \) :

Surface tension

\(\phi \) :

Dimensionless parameter defined in Eq. (38)

c:

Capilar

e:

Effective

int:

Refers to the interface between the two-phase region and vapor region

0:

Coolant conditions

l:

Conditions at the liquid phase

lv:

Conditions at the two-phase region

r:

Relative

s:

Conditions at the solid phase

sat:

Conditions of saturation

top:

Refers to the upper surface

v:

Conditions at vapor phase

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Acknowledgments

This work has been supported by the research grants no. CB-2013/220900 of Consejo Nacional de Ciencia y Tecnología (CONACyT) and 20150919 of SIP-IPN at Mexico. M. Peralta acknowledges the support of UNAM program of doctoral fellowship, and to the CONACyT program for a postdoctoral fellowship (12525) granted to spend a year as a postdoctoral researcher at SEPI-ESIME IPN. The authors also wish to express their thanks to the very competent Reviewers for the valuable comments and suggestions

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Authors and Affiliations

  1. SEPI-ESIME Azcapotzalco, Instituto Politécnico Nacional, 02250, Mexico, D.F., Mexico

    O. Bautista

  2. Departamento de Termofluidos, Facultad de Ingeniería, UNAM, 04510, Mexico, D.F., Mexico

    M. Peralta & F. Méndez

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  1. M. Peralta
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  2. F. Méndez
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Peralta, M., Méndez, F. & Bautista, O. Phase-Change Transpiration Cooling in a Porous Medium: Determination of the Liquid/Two-Phase/Vapor Interfaces as a Problem of Eigenvalues. Transp Porous Med 112, 167–187 (2016). https://doi.org/10.1007/s11242-016-0637-7

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