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Gas Capture in a Cavity with Porous Walls

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Abstract

A configuration like an upside-down bell made of porous material is considered which is initially dry but then subjected to a rising pool of liquid. As liquid touches the rim of the bell, capillary transport is initiated. Starting with a vertical wicking phase, the imbibing liquid will eventually reach the ceiling of the bell and switch over to horizonal wicking. At the end of the horizontal wicking, the cavity inside the porous bell is enclosed by liquid and the gas inside it is captured. We present a model to describe the capillary transport in the bell for both Cartesian and cylindrical geometry. As far as possible, we derive analytical solutions to the normalized differential equations that describe the problem. Beyond analytical solutions, we use Runge–Kutta shooting method to obtain numerical results. We calculate the normalized closure time to capture the gas, the amount of captured gas, and reflect on the pressure development in the gas chamber.

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Abbreviations

\(a\) :

Wicking constant, relates surface pressure to viscous stress (\(\mathrm {m^2/s}\))

\(b\) :

Wicking constant, relates hydrostatic pressure to viscous stress (\(\mathrm {m^2/s}\))

\(\mathrm {Bo_\theta }\) :

Type of Bond number, relates hydrostatic to surface pressure

\(\mathrm {Bo_c}\) :

Critical Bond number corresponding to \(X_0\)

\(d\) :

Thickness of the porous walls of the cavity (\(\mathrm {m}\))

\(\mathrm {Fo}\) :

Forchheimer number, relates convective to viscous pressure loss

\(g\) :

Earth gravity constant, \(g=9.81~\mathrm {m/s^2}\)

\(H\) :

Height of the porous walls of the cavity (\(\mathrm {m}\))

\(\Delta h\) :

Hydrostatic height between wicking front and liquid reservoir (\(\mathrm {m}\))

\(K\) :

Hydraulic permeability of the porous medium (\(\mathrm {m^2}\))

\(L\) :

Combined vertical and horizontal length (\(\mathrm {m}\))

\(L_\sigma \) :

Capillary length (\(\mathrm {m}\))

\(\Delta p\) :

Pressure difference between front and back side of the cavity ceiling (\(\mathrm {Pa}\))

\(Q\) :

Liquid flow rate (\(\mathrm {m^3/s}\))

\(r\) :

Radial coordinate (\(\mathrm {m}\))

\(r^*\) :

Dimensionless radial position of the wicking front

\(R\) :

Radial position of the wicking front (\(\mathrm {m}\))

\(R_0\) :

Radius of the porous walls of the cavity (\(\mathrm {m}\))

\(R_s\) :

Static radius of the pores (\(\mathrm {m}\))

\(S\) :

Saturation of the pores with liquid, between 0 and 1

\(t\) :

Time (\(\mathrm {s}\))

\(t^*\) :

Dimensionless time

\(t_d\) :

Delay time until liquid contacts the cavity (\(\mathrm {s}\))

\(u\) :

Rising velocity of the liquid (\(\mathrm {m/s}\))

\(v_\bot \) :

Velocity of gas cross flow through the cavity ceiling (\(\mathrm {m/s}\))

\(\Delta V_\mathrm{{jump}}\) :

Volume between outer and inner edge profile at liquid contact (\(\mathrm {m^3}\))

\(W\) :

Width of the porous walls of the cavity (Cartesian) (\(\mathrm {m}\))

\(x\) :

Horizontal coordinate (Cartesian) (\(\mathrm {m}\))

\(x^*\) :

Dimensionless horizontal position of the wicking front

\(X\) :

Horizontal position of the wicking front (\(\mathrm {m}\))

\(X_0\) :

Aspect ratio, wicking height to total wicking length

\(z\) :

Vertical coordinate (\(\mathrm {m}\))

\(z^*\) :

Dimensionless vertical position of the wicking front

\(Z\) :

Vertical position of the wicking front (\(\mathrm {m}\))

\(\Delta z\) :

Height difference due to wall adhesion (\(\mathrm {m}\))

\(\phi \) :

Porosity of the cavity

\(\Gamma \) :

Aspect ratio, radial to total wicking length

\(\mu \) :

Dynamic viscosity of the liquid (\(\mathrm {kg/(m~s)}\))

\(\theta \) :

Contact angle

\(\rho \) :

Liquid density (\(\mathrm {kg/m^3}\))

\(\sigma \) :

Surface tension (\(\mathrm {N/m}\))

\(\Pi _\mathrm{{u}}\) :

Dimensionless parameter, relating wicking time to pool rise time

\(\Pi _\mathrm{{c}}\) :

critical \(\Pi _\mathrm{{u}}\) corresponding to \(X_0\)

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Acknowledgments

The funding of the research project by the German Federal Ministry of Economics and Technology (BMWi) through the German Aerospace Center (DLR) under grant number 50 RL 0921 is gratefully acknowledged.

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  1. Center of Applied Space Technology and Microgravity (ZARM), University of Bremen, Am Fallturm 01 , 28359 , Bremen, Germany

    Michael Conrath & Yulia Smiyukha

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  1. Michael Conrath
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Correspondence to Michael Conrath.

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Conrath, M., Smiyukha, Y. Gas Capture in a Cavity with Porous Walls. Transp Porous Med 103, 131–153 (2014). https://doi.org/10.1007/s11242-014-0292-9

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