Dynamics of sorption

The process of sorption of an adsorbate (vapors, gases or dissolved substance) by a solid body, accompanied by adsorption and absorption, i.e. surface and volume sorption, respectively. The dynamics of the sorption is determined by the adsorption rates, by the external and internal diffusion of the adsorbate, and is described by a system of differential equations of diffusional transfer of matter, taking into account the kinetics of adsorption. Sorption takes place in most cases under non-isothermal conditions, since it is accompanied by the emittance of adsorption heat and by capillary condensation, and, as a result, processes of mass transfer (diffusion) are accompanied by heat transfer (heat exchange), i.e. can be described by a system of differential equations of mass and heat transfer. If a mixture of gases and vapors or a mixture of dissolved substances constitutes the adsorbent, the molecular mass transfer and heat transfer are described by Onsager's system of equations [2].

In the case of a two-component mixture, the system of differential equations of heat and mass transfer — the solution of which, under certain boundary conditions, determines the dynamics of the sorption — has the form

$$ \tag{1 } \rho \frac{d \rho _ {10} }{d \tau } = \ \mathop{\rm div} \left [ D \rho \nabla \rho _ {10} + \frac{k _ {T} }{T} \nabla T \ \right ] + v I _ {1} ( \rho _ {10} , T ), $$

$$ \tag{2 } c _ {p} \rho \frac{d T }{d \tau } = \mathop{\rm div} ( \lambda \nabla T ) + \mathop{\rm div} ( D \rho Q ^ {*} \nabla \rho _ {10} ) + $$

$$ +( h _ {1} - h _ {2} ) I _ {1} + ( c _ {p1} + c _ {p2} ) j _ {1} \nabla T, $$

where $ \rho _ {10} $ is the relative density of component "1" ; $ \rho _ {10} = \rho _ {1} / \rho $; $ \rho = \rho _ {1} + \rho _ {2} $; $ D $ is the diffusion coefficient; $ T $ is the temperature; $ \tau $ is the time; $ c _ {p} $ is the specific isobaric heat capacity; $ \lambda $ is the heat conductivity coefficient; $ Q ^ {*} $ is the isothermic heat transfer; $ k _ {T} $ is the thermal-diffusion constant; $ h $ is the specific enthalpy; $ j _ {1} $ is the diffusion flow of mass of the component "1" ; $ d / d \tau $ is the complete or the substantial derivative, equal to

$$ \frac{d}{d \tau } = \frac \partial {\partial \tau } + v \cdot \nabla \rho $$

(where $ v $ is the rate of the motion of the adsorbate flow); and $ I _ {1} ( \rho _ {10} , T ) $ is the capacity of the source of the mass substance, which depends on the kinetics of the adsorption conversion of phases (these, in turn, are usually functions of the concentration $ \rho _ {10} $ and the temperature $ T $).

The rate of motion $ v $ of the absorbate flow is obtained by solving the Navier–Stokes equations. The boundary conditions are determined by the nature and the physical mechanism of interaction between the surface of the solid and the surrounding medium (adsorbate). Here, the rate of mass transfer is determined by the external diffusion of the adsorbate towards the surface of the body and by the kinetics of the adsorption. Two boundary cases are usually considered:

1) the mass exchange is determined by diffusion;

2) the concentration on the surface of the solid depends only on the rate of adsorption.

In the case of vapor sorption by bodies with capillary pores, solutions of the set of differential equations (1), (2), as applied to bodies of very simple shapes, have been obtained [1].

The process of desorption of water vapour by porous bodies constitutes a stage in a process of drying. In this case the dynamics of the sorption is approximately calculated with the aid of the following equations of mass and heat transfer:

$$ q ( \tau ) = \rho _ {0} R _ {v} r \frac{d \overline{u}\; }{d \tau } ( 1 + \mathop{\rm Rb} ), $$

$$ - \frac{d \overline{u}\; }{d \tau } = \kappa N ( \overline{u}\; - {\overline{u}\; } _ {p} ) , $$

where $ d \overline{u}\; / d \tau $ is the rate of desorption; $ r $ is the specific heat of the sorption; $ \rho _ {0} $ is the density of the dry body; $ q ( \tau ) $ is the heat flow on the surface of the body; $ \overline{u}\; $ is the average humidity content (relative to the concentration) of the body; $ {\overline{u}\; } _ {p} $ is the equilibrium humidity content; $ \kappa $ is the relative drying coefficient; $ N $ is the rate of drying in the first period (the period of constant rate); and Rb is the Rehbinder number.

References

[1]	A.V. Lykov, Yu.A. Mikhailov, "The theory of heat and mass transfer" , Moscow-Leningrad (1963) (In Russian)
[2]	S.R. de Groot, P. Mazur, "Non-equilibrium thermodynamics" , North-Holland (1962)
[3]	A.V. Lykov, "The theory of drying" , Moscow (1968) (In Russian)
[4]	D.A. Frank-Kamenetskii, "Diffusion and heat transfer in chemical kinematics" , Moscow (1967) (In Russian)
[5]	A.N. [A.N. Tikhonov] Tichonoff, A.A. Samarskii, "Differentialgleichungen der mathematischen Physik" , Deutsch. Verlag Wissenschaft. (1959) (Translated from Russian)

References

[a1]	R. Aris, "The mathematical theory of diffusion and reaction in permeable catalysts" , 1–2 , Oxford Univ. Press (1975)
[a2]	M. Jacob, "Heat transfer" , I-II , Wiley (1955)
[a3]	H.S. Carslaw, "Introduction to the mathematical theory of the conduction of heat in solids" , Macmillan (1921)