Thermal Diffusion Effects and Optimum Frequencies in Parametric Pumps
Ultimate separation in a closed, direct-thermal mode parametric p u m p is analyzed with respect to limitations imposed by heat and mass transfer. Theoretical evidence is given to show there exists an optim u m frequency for maximum separation.
Recently, a theory was published (Rice, 1973) which indicates there exists an optimum kinematic Peclet number (Ar,w/D = 3) for maximum ultimate separation in the closed, direct thermal mode parametric pump. The theory was derived to show the destructive effect owing to the periodic coupling of velocity and concentration profiles (often called “induced dispersion,” (Horn and Kipp, 1967)). Thermal diffusion effects, along with fluid-particle mass and heat transfer resistances were neglected. The purpose of the current communication is to show the effect of thermal penetration on ultimate separation (infinite time) when interphase resistances are small and a linear equilibrium adsorption/desorption condition exists between solid adsorbent and fluid. Baker and Pigford (1971) have presented a rather complete model for the adsorption process, except that radial diffusion gradients were not included. This model is used in the current work, subject to the conditions mentioned above, and in addition radial diffusion effects are included. In the direct thermal mode (heat addition or extraction through the wall of a packed bed), the temperature gradient along the axis is usually quite small (Rice and Mackenzie, 1973) and will be neglected in the model considered here. However, slow thermal penetration in the radial direction can give rise to a periodic radial concentration profile and this will necessarily affect the ultimate separation. With these considerations, the equations describing the transport of a single solute in the direct-thermal mode closed parametric pump are taken to be: mass balance
thermal balance
uation after a long time (ultimate separation) such that the time-average concentration in the reservoirs attached a t the ends of the parapump are constant, that is
The physical picture is such that during upflow, the jacke t surrounding the parapump bed is made cold, while during downflow the jacket is hot. If the solid isotherm behaves normally, this means solute accumulates in the lower reservoir and is depleted from the upper reservoir. Hence, a periodic velocity and temperature which are purely sinosoidal can be represented by
and
V ( t ) = A--.eiwt U 2
2
(8)
For reasons discussed more fully in a previous work (Rice, 1973), it is assumed the dependent variables are comprised of a steady and an unsteady part A
c(x, Y, t ) = C ( X ) T ( Y , t) =
F+
+ c (Y,t) ?(V, t)
(9 )
(10)
The unsteady parts are purely sinusoidal and are made u p of a function plus its complex conjugate (hence real)
The steady and unsteady parts of the solids composition can be deduced from eq 3 and 4
q = (kC(x)-
adsorption equilibria q = IZC
+ f(T)
(3 )
I t is assumed that mass movement in the radial direction is controlled by molecular diffusion, while mixing along the axis is controlled by a dispersion coefficient D* which depends on amplitude and frequency (Harris and Goren, 1967; Rice and Eagleton, 1970). A specific linear type of equilibrium isotherm is selected such that only the intercept changes with temperature; i e . , the isotherms are parallel. Such a condition exists, for example, in the type 1 isotherm a t high concentration where the adsorption curves a t different temperatures are horizontal. In the current model, the temperature dependence is taken as f ( T ) = -UT
+
b
(4)
Boundary conditions are chosen to reflect the physical sit396
+ A_We-iwt
Ind. Eng. Chem.,
Fundam., Vol. 13, No. 4, 1974
UT +
b)
+
( k E - u?)
(13)
I t is noted that the steady fluid concentration is satisfied by
-a2E =o a x2 hence, from ( 5 ) and (6)
When eq 11, 12, 13, 15, and 8 are inserted in eq 1, there results after introducing dimensionless groups
+ -dC+l + E dE -
-
(- iAPeM)C,l = -
u6(iPeM)T,,(t)- Aw -A‘ 2 L and T + l ( ( must ) satisfy eq 2 so that
(16)
eq 23 and the real parts of the complex Bessel functions are extracted, the following rather simple expression to preduct ultimate separation results The boundary conditions for eq 16 and 17 are
(< = 0 ( s y m m e t r y ) )
dC*i = 0
at;
dC,, = dt
(16.1)
A ~ p e , cos ~ e(e, - eo - 3/4n)
= R
AC
'M,/M,
1 L
- - ATab AR (< = 1 (no flux through wall)) (16.2)
0
and dT+i -
d
