Standing wave design of a four-zone thermal SMB fractionator and concentrator (4-zone TSMB-FC) for linear systems

Abstract

For a dilute feed, a four-zone SMB with temperature gradient (4-zone TSMB-FC) can fractionate and simultaneously concentrate two solutes. Optimization of the operating parameters for this system, which include four zone flow rates and port switching time, is a major challenge because of the five variables. Initial choice of the variables using the triangle theory followed by a systematic search using rate model simulations is time-consuming. In this study, the SWD developed previously for isothermal systems is modified for the first time for a 4-zone TSMB-FC. The optimum operating parameters are determined with the new method. Only linear isotherm systems with significant mass transfer resistances are considered. For a dilute feed, a 4-zone SMB-FC can produce pure desorbent, in addition to extract and raffinate products with high purities. For the separation of p-xylene and toluene with silica gel in the temperature range from 0 to 80 °C, the enrichment factor in the linear region can be as high as 80 fold for an extract purity of 99.99 % and tenfold for raffinate purity of 99.8 %, while withdrawing the desorbent at a desorbent to feed flow rate ratio of 0.53 and 0.78, respectively.

Appendices

Appendix 1: Derivation of SWD equation

The mass balance equations for solutes in the mobile phase and pore phase within zone i shown in the equations below are used in deriving the mathematical model for the Linear, Non-Ideal SWD method (Ma and Wang 1997).

$$\frac{{\partial c_{bi} }}{\partial t} = D_{ax,i,j} \frac{{\partial^{2} c_{bi} }}{{\partial x^{2} }} - \bar{u}_{o,j} \frac{{\partial c_{bi} }}{\partial x} - Pk_{f,i,j} \left( {c_{bi} - c_{i}^{ * } } \right) $$

(13a)

$$\bar{u}_{o,j} = u_{o,j} - u_{port} $$

(13b)

$$\varepsilon_{p} \frac{{\partial c_{i}^{ * } }}{\partial t} + \left( {1 - \varepsilon_{p} } \right)\frac{{\partial q_{i}^{ * } }}{\partial t} = k_{f,i,j} \left( {c_{bi} - c_{i}^{ * } } \right) + u_{port} \varepsilon_{p} \frac{{\partial c_{i}^{ * } }}{\partial x} + \left( {1 - \varepsilon_{p} } \right)u_{port} \frac{{\partial q_{i}^{ * } }}{\partial x} $$

(14)

where c _bi and $c_{i}^{*}$ are the concentration of solute i in the mobile and pore phase, respectively; D _ax,i,j is the axial dispersion coefficient of solute i in zone j; $\overline{u}_{o,j}$ is the interstitial velocity of the mobile phase in the axial direction; u _o,j is the linear velocity in zone j that controls the propagation of the concentration waves relative to the solid phase; P is the bed phase ratio defined by $\frac{{\left( {1 - \varepsilon_{e} } \right)}}{{\varepsilon_{e} }}$; k _f,i,j is the mass transfer coefficient of solute i in zone j; u _port is the port velocity; and ε _p is the intra-particle void fraction.

The steady state solution to (13) and (14) in each zone must satisfy:

$$\left( {D_{ax,\,i,j} + P\frac{{\left( {u_{port} \delta_{i} } \right)^{2} }}{{k_{f,i,\,i} }}} \right)\frac{{d^{2} C_{bi} }}{{dx^{2} }} + \left[ {u_{port} \left( {1 + P\delta_{i} } \right) - u_{o,\,j} } \right]\frac{{dC_{bi} }}{dx} = 0 $$

(15)

where δ _i is the retention factor of solute i defined by δ _i  = ε _p  + (1 − ε _p )K _i ; and K _i is the linear isotherm equilibrium constant. Remember that zones 1 and 2 desorbs solute B and A, respectively, while zones 3 and 4 adsorbs solute B and A, respectively. Using the feed port (x = 0) as the reference point, the steady-state equation for solute A in zone 2 can be solved by rearranging (15) with the following boundary conditions:

$$\left[ {u_{o,\,2} - \left( {1 + P\delta_{A,hot} } \right)u_{port} } \right]\frac{{dC_{b,A} }}{dx} = \left( {D_{ax,\,A,2} + P\frac{{\left( {u_{port} \delta_{A,hot} } \right)^{2} }}{{k_{f,A,\,2} }}} \right)\frac{{d^{2} C_{b,A} }}{{dx^{2} }} $$

(16a)

$$C_{b,A} = 0\, {\text{at x }} = \, - \infty $$

(16b)

$$C_{b,A} = C_{s,A} \, {\text{at x }} = \, 0 $$

(16c)

The solution to (16) is shown in (3b).

In addition, the steady-state equation for solute B in zone 3 can also be solved using the following boundary conditions:

$$\left[ {u_{o,\,3} - \left( {1 + P\delta_{B,cold} } \right)u_{port} } \right]\frac{{dC_{b,B} }}{dx} = \left( {D_{ax,\,B,3} + P\frac{{\left( {u_{port} \delta_{B,cold} } \right)^{2} }}{{k_{f,B,\,3} }}} \right)\frac{{d^{2} C_{b,B} }}{{dx^{2} }} $$

(17a)

$$C_{b,B} = 0\, {\text{at x }} = \infty $$

(17b)

$$C_{b,B} = C_{s,B} \,{\text{at x }} = \, 0 $$

(17c)

The solution to (17) is shown in (3c). The velocities of zones 1 and 4 can also be derived using a similar approach where zones 1 and 4 will be solved using solute B and A, respectively. The overall zone velocity equation results are shown in (3).

Appendix 2: Derivation of decay coefficients of the SWD equation

The decay coefficients, $\beta_{j}^{i}$, for the SWD equations (3) is defined by the natural log of the ratio of the highest and lowest concentration of solute i in zone j. However, since the concentrations of the solutes inside the columns in each zone is unknown a priori, it is necessary to estimate the decay coefficients for the standing wave concentrations in each zone. One approach to do this is to link the concentration wave to the yield of each component through a simple mass balance of the inlet and outlet ports. The decay coefficient equations for each zone based on the concentration waves are as follows;

$$\beta_{1}^{B} = \ln \left( {\frac{{c_{E,B} }}{{c_{D,B} }}} \right) \quad \beta_{2}^{A} = \ln \left( {\frac{{c_{Fb,A} }}{{c_{E,A} }}} \right) $$

(18a,b)

$$\beta_{3}^{B} = \ln \left( {\frac{{c_{FP,B} }}{{c_{R,B} }}} \right) \quad \beta_{4}^{A} = \ln \left( {\frac{{c_{R,A} }}{{c_{D,A} }}} \right) $$

(18c,d)

where c _k,i is the concentration of solute i in the k port where the subscripts F, E, R, and D represents the feed, extract, raffinate, and desorbent ports while c _FP,i is the concentration of solute i after the feed port.

The yield of each component is defined by the ratio of the amount product produced in the outlet port and the amount of feed entering the system.

$$Y_{A} = \frac{{c_{R,A} Q_{R} }}{{Fc_{F,A} }} = 1 - \frac{{c_{E,A} Q_{E} }}{{Fc_{F,A} }} \quad Y_{B} = \frac{{c_{E,B} Q_{E} }}{{Fc_{F,B} }} = 1 - \frac{{c_{R,B} Q_{R} }}{{Fc_{F,B} }} $$

(19a,b)

where Q _R  = Q ₃  − Q ₄ and Q _E  = Q ₁  − Q ₂ . By substituting Q _R and Q _E , and rearranging (B2) and solving for c _E,B and c _R,A , the following equations are obtained.

$$c_{E,B} = \frac{{Fc_{F,B} Y_{B} }}{{Q_{1} - Q_{2} }} \quad c_{R,A} = \frac{{Fc_{F,A} Y_{A} }}{{Q_{3} - Q_{4} }} $$

(20a,b)

Next, the mass balance needed for solute A and B around both the raffinate and extract port are:

Solute A:

$$Q_{2} c_{Fb,A} + Fc_{F.A} = Q_{3} c_{R.A} \quad Q_{4} c_{D,A} = Q_{1} c_{E,A} $$

(21a,b)

Solute B:

$$Q_{3} c_{FP,B} = Q_{2} c_{E.B} + Fc_{F.B} \quad Q_{1} c_{D,B} = Q_{4} c_{R,B} $$

(22a,b)

The four equations are used to express the four unknown concentrations, c _Fb,A , c _FP,B , c _D,A , and c _D,B . Rearrangement of (B4) and (B5), and substituting these equations along with (B3) into (B1), the overall decay coefficient expression in terms of yield and zone flow rates are obtained and shown in (4).