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Abstract

We adopt the orthogonal collocation finite element method to solve an equilibrium diffusion model of a simulated moving bed chromatography separation process. We propose an optimization strategy based on the improved moving asymptotes algorithm to improve system performance indexes. By the triangle theory, we verify the feasibility of the improved moving asymptotes method. The simulation results illustrate that the improved moving asymptotes method converges quickly and results in uniformly distributed optimal solutions. This work aims to show that the simulated moving bed process can be dynamically controlled and optimized using an optimizing controller based on the method of improved moving asymptotes, facilitating the design and operation of a simulated moving bed.

Keywords

Adsorption separation process simulated moving bed optimizing control improved moving asymptotes algorithm equilibrium diffusion model

Introduction

Simulated moving bed (SMB) separation is a continuous chromatographic separation technique for scale production (Ruthven and Ching, 1989), having the advantages of strong separation ability, small size, and low operation cost. It has a large range of applications in the petroleum chemical, fine chemical, and sugar industries (Qamar et al., 2014; Song et al., 2016), among others. Recently, due to strong coupling and complexity in the SMB separation process, many researchers have turned to modeling and optimizing the SMB separation process to provide theoretical guidance for product development and expand its industrialization and application range (Bentley et al., 2013; Heegeun and Sungyong, 2012; Kostka et al., 2011; Püttmann et al., 2013; Qamar et al., 2014). Commonly used SMB chromatography models (Gomes et al., 2016) include the general rate model (GRM), lumped pore diffusion model (PORM), ideal model (IM),equilibrium dispersive model (EDM), and transport dispersive model (TDM). Compared with other models, the EDM is simpler, more precise, and more practical. Therefore, it has a large range of applications (Schramm et al., 2003).

However, an SMB model is very difficult to solve because of its partial differential equations. Therefore, a typical solution method involves converting the partial differential equations to ordinary differential equations using discrete methods, such as the orthogonal collocation finite element method (Absar et al., 2008; Finlayson, 1980), Galerkin method (Thomée, 2006), or the space-time conservation element and solution element method (Wang et al., 2010). Compared to other methods, the orthogonal collocation finite element method is not only convenient and fast but also flexible and accurate in the processing of stiff problems. There are many operation conditions affecting the chromatographic separation performance in an SMB process, such as switching time, area flow rate, and column size. Therefore, the optimization analysis of SMB is a hot research topic in this area (Bentley et al., 2013; Lim, 2004). Marco Mazzotti first proposed the triangle theory (Mazzotti et al., 1997) under ideal assumptions, ignoring axial dispersion and mass transfer resistance. In recent years, genetic algorithms or particle swarm optimization (PSO) have been used for SMB process design (Aniceto, 2016; Neto et al., 2016). The genetic algorithm requires copy, crossover, and mutation operations, affecting its efficiency. PSO also has shortcomings, including easily falling into local optimal solutions, being prone to instabilities, and exhibiting slow convergence.

In this paper, we first solve the equilibrium diffusion model for an SMB chromatographic separation process using the orthogonal collocation finite element method. Then, based on the moving asymptote method (Svanberg, 1987), we propose a method of improving the moving asymptote (IMA) and apply it to the optimization of purity, yield, and solvent reduction. Finally, we simulate optimization of an SMB chromatographic separation process. The simulation results show that IMA can be applied to an SMB chromatographic separation process and verified by the triangle theory. Compared with other algorithms, the optimization control method adopting IMA is more effective in modeling the chromatography separation process, helping to improve the economic benefits of SMB and guide production.

The mathematical model

The mathematical model of SMB chromatographic separation is composed of a series of single chromatographic column models and node models (Horvath et al., 2010; Lu, 2003). Because the equilibrium diffusion model (EDM) is more practical and quick for simulation, we adopt it as our single column chromatographic model. We consider the effects of molecular diffusion, eddy diffusion, and mass transfer resistance while ignoring the influence of interphase mass transfer resistance. The diffusion of the fluid phase and solid phase reaches an equilibrium state instantaneously; thus, we do not consider it, either.

The mathematical description of EDM is as follows

\frac{\partial c_{i}}{\partial t} + F_{a} \frac{\partial q_{i}}{\partial t} + u \frac{\partial c_{i}}{\partial x} = D_{a} \frac{\partial^{2} c_{i}}{\partial x^{2}}

(1)

where initial conditions c_i(0, x) = q_i(0, x) = 0.

Our boundary condition is

1) c_{i} (t, 0) = ψ_{i} (t)

(2)

2) u c_{i} (t, 0) + D \frac{\partial c_{i}}{\partial x} |_{0, t} = u c_{k} (t)

(3)

\frac{\partial c_{i}}{\partial x} |_{x = L} = 0

(4)

where c_i is the concentration of component i in fluid phase, q_i is the concentration of component i in solid phase, F_a is the phase rate, u is superficial velocity, D_a is the axial diffusion coefficient, ψ_i is the inlet concentration of the ith chromatographic column, and c_k(t) is known.

The adsorption isotherm equation is q_i (x,t) = f [c(x,t)].

The equilibrium relationship between the nodes can be obtained by the mass conservation relations

Q_{4} + Q_{D} = Q_{1}, c_{i, 4}^{o u t} Q_{4} = c_{i, 1}^{i n} Q_{1}

(5)

Q_{1} - Q_{E} = Q_{2}, c_{i, 1}^{o u t} = c_{i, 2}^{i n}

(6)

Q_{2} - Q_{F} = Q_{3}, c_{i, 2}^{o u t} Q_{2} + c_{i, F} Q_{F} = c_{i, 3}^{i n} Q_{3}

(7)

Q_{3} - Q_{R} = Q_{4}, c_{i, 3}^{o u t} = c_{i, 4}^{i n}

(8)

where Q_j is the liquid flow rate of zone j (j = 1, 2, 3, 4); subscript i is the index denoting component; subscripts E, F, R, D represent extraction, feed, raffinate, and elution respectively; superscript in and out are inlet and outlet values, respectively.

Model solutions

Orthogonal collocation finite element method

The orthogonal collocation finite element method (Absar et al., 2008; Finlayson, 1980)is a combination of orthogonal collocation and the finite element method that uses piecewise smooth orthogonal polynomials to approximate an unknown function. By discretizing the unknown derivatives, the differential equations to be solved are transformed into algebraic equations whose unknown values are node function values (Lin, 2004).

We carry out the discrete processing of the EDM using the orthogonal collocation finite element method. First, we divide the solution domain of the model into finite elements, and obtain the element stiffness matrix in every finite element using the orthogonal collocation points. Then we combine the element stiffness matrix with our boundary conditions to obtain the whole stiffness matrix, producing a set of nonlinear equations.

We use the subsection finite element method to partition the definition domain, and orthogonalize the finite elements using Lagrange polynomials. This arrangement is shown in Figure 1. Dividing the definition domain into LH = L/h number of finite elements, we configure the collocation points in each finite element, and adopt Lagrange polynomials as interpolation polynomials. If N represents the number of points per finite element internal configuration, there are N(LN + 2) residual error conditions. Because the total number of conditions should be (N + 1)(LH)+1, we need (LH-1) more conditions. To supplement, we keep the first-order derivatives continuous on (LH-1) boundaries of the finite elements. In Figure 1, for 0 = X₁ < X₂ ⋯ < X _LH = 1, h_i is the length of the ith finite element; thus h_i = X_i+₁–X_i. We adopt the average finite element method so that h₁ = h₂ = … = h_LH = h.

Figure 1.

Orthogonal configuration of Lagrange polynomial on the finite element.

Taking the EDM equation (1) as an example, equation (1) can be converted into

(1 + F_{a} \frac{\partial q_{i}}{\partial c_{i}}) \frac{\partial c_{i}}{\partial t} = D_{a} \frac{\partial^{2} c_{i}}{\partial x^{2}} - u \frac{\partial c_{i}}{\partial x}

(9)

To carry out orthogonal collocation, in each finite element, we take N_p zeros of N_p displacement Legendre polynomials as internal collocation points, also accounting for the two endpoints, 0 and 1. We then construct the Lagrange interpolation polynomials, so the first-order and second-order spatial derivatives $\partial c / \partial x$ and $\partial^{2} c / \partial x^{2}$ are discretized into

\frac{\partial c}{\partial x} |_{c = c_{j i}} = \frac{1}{h} \sum_{k = 1}^{N_{P + 2}} A_{j k} c_{k i} = \frac{1}{h} [A] [C]

(10)

\frac{\partial^{2} c}{\partial x^{2}} |_{c = c_{j i}} = \frac{1}{h^{2}} \sum_{k = 1}^{N_{P + 2}} B_{j k} c_{k i} = \frac{1}{h^{2}} [B] [C]

(11)

where c_ji is the concentration of the j configuration point in the ith finite element, and A_jk and B_jk are matrix elements of matrix [A] and matrix [B], respectively. The connection conditions of the concentration and its derivative at the convergence point between the two adjacent elements are

{[C_{N_{p} + 2}]}_{j} = {[C_{1}]}_{j + 2}

(12)

{[\sum_{k = 1}^{N_{p} + 2} A_{N_{p^{+ 2}}, k c_{k}}]}_{j} = {[\sum_{k = 1}^{N_{p} + 2} A_{1, k c_{k}}]}_{j + 1}

(13)

where j = 1, 2,…, N_p+2. According to equations (10) to (13), we can convert equation (9) into equation (14), then transform the chromatographic equations into ordinary differential equations about time t. Subsequently, we can solve the ordinary differential equations numerically.

\begin{array}{l} \frac{\partial c_{i}}{\partial x} |_{c = c_{j i}} = \frac{D_{a}}{h^{2} (1 + F_{a} H_{i})} \sum_{k = 1}^{N_{P + 2}} B_{j k} c_{k i} - \frac{u}{h (1 + F_{a} H_{i})} \sum_{k = 1}^{N_{P + 2}} A_{j k} c_{k i} \\ = \frac{D_{a}}{h^{2} (1 + F_{a} H_{i})} [B] [C] - \frac{u}{h (1 + F_{a} H_{i})} [A] [C] \end{array}

(14)

Model solution and simulation

We used MATLAB software to numerically solve the mathematical model of the SMB.

In the initialization stage, we set the initial concentration values in each zone, physical parameters, operating parameters, and the isotherm parameters as shown in Tables 1 and 2.

Table 1.

Physical parameters and isotherm parameters.

Column section area S/cm²	0.166
Column length L/cm	15
Column porosity ε	0.68
Column configuration	2–2–2–2
Henry constant of component A H_A	4.976
Henry constant of component B H_B	1.864
Axial diffusion coefficient D/cm²·min⁻¹	0.001

Table 2.

Operation parameters.

Number of components	2
Feed concentration c_F/(g/L)	5
Column porosity ε	0.68
Flow rate of zone 1 Q₁/(mL/min)	1.65
Flow rate of zone 2 Q₂/(mL/min)	0.91
Flow rate of zone 3 Q₃/(mL/min)	1.13
Flow rate of zone 4 Q₄/(mL/min)	0.70
Feed flow rate Q_F/(mL/min)	0.23

When inlets and outlets periodically, the concentration of the components in two adjacent switching cycles remains essentially unchanged and periodic steady state can be achieved. Figure 2 shows the concentration changes over the switching cycle, and Figure 3 illustrates the concentration changes of two components in the extract and the raffinate. From Figures 2 and 3, it can be seen that periodic steady state is achieved after 68 switching cycles. Figure 4 shows the axial concentration distribution of the SMB under periodic steady state.

Figure 2.

Changes of concentration over switching cycle.

Figure 3.

Change in outlet concentration.

Figure 4.

Changes in outlet concentration of components A and B in each column.

Optimization strategy of SMB

Optimizing control approach

SMB is a separation technology combining high productivity and low solvent consumption and playing an important role in chemical industry. During its development, it was found that the most challenging problem is the optimization of SMB process operations. Therefore, we propose an on-line optimal control scheme, integrating optimization and control of the SMB process. The optimizer is based on a repetitive model predictive control and a simple SMB device having few parameters. These include adsorption behavior of the mixture separation and the overall void fraction of the column, both of which are easy to measure. These parameters can also be modified continuously by Kalman filter feedback to detect component concentrations. In this cycle control scheme, measurement, optimization, and control behavior occur in each cycle.

Figure 5 shows the structure of the SMB optimizer.

Figure 5.

‘Cycle to cycle’ control scheme of the SMB process.

Principles of IMA

The moving asymptotes (MA) method is a convex function-based optimization method proposed by Svanberg in 1987. Since then, this method has attracted much interest. It uses convex functions and variable-separable subproblems to approximate the objective and constraints of the original problem. By solving these subproblems, we can obtain an optimal solution. In this method, the approximation function is replaced by the first-order derivative of the current iteration point. This optimization method is flexible and general. It can be used to handle not only variable element sizes but also all kinds of constraints, provided that the derivatives of the constraint functions with respect to the design variables can be calculated (analytically or numerically). Thus, it is able to handle general nonlinear programming optimization problems. Because obtaining the analytical derivative of our approximation function is difficult, we adopt a discrete information feedback control system and modify the algorithm (MIMA). MIMA replaces the derivative with a more easily obtained and less costly difference scheme. In addition, MIMA is easy to implement and use. MIMA can optimize and solve multi-objective problems by changing an implicit problem into subproblems. The optimization problem is as follows

M I N f_{0} (X) X \in R^{n}

(15)

S . T . f_{i} (X) \leq f_{i}^{*} i = 1, 2, …, m

(16)

x_{j a} \leq x_{j} \leq x_{j b} j = 1, 2, …, n

(17)

where f₀(X) is the objective function, X = (x₁, x₂,…, x_n) ^T is the design variable, f_i(X) ≤ f_i* is a behavior constraint, and x_ja ≤ x_j ≤x_jb is a technical constraint.

The first step to solving the problem is to construct the subproblems. The solution of the original problem is approximated by that of the subproblems. The detailed algorithm procedure is as follows:

Step 1: Let k = 0, choose the initial point x⁰.

Step 2: Give the iteration point x^k, and calculate f_i (X^k) and ▽f_i(X^k), i = 1, 2,…,m

Step 3: Use the established subproblem to approximate the original problem, replacing f_i(X) by f_i^k(X), using the data calculated in step 2.

Step 4: Solve subproblem. The optimal solution of the subproblem is taken as the next iterative point, i.e. k = k + 1, and we return to step 2.

Construction of subproblem

When using IMA for solving optimization problems, the key point is to construct and solve the subproblem. In the kth iterative of the algorithm, the subproblem is constructed as follows

M I N f_{0}^{k} (X) X \in R^{n}

(18)

S . T . f_{i}^{k} (X) \leq f_{*}^{k} i = 1, \dots, m

(19)

x_{j}^{α} \leq x_{j} \leq x_{j}^{β} j = 1, \dots, n

(20)

where i = 0, 1,…, m.

f_{i}^{k} (X) = r_{i}^{k} + \sum_{j = 1}^{n} (\frac{p_{i j}^{k}}{U_{j}^{k} - x_{j}^{k}} + \frac{q_{i j}^{k}}{x_{j}^{k} - L_{j}^{k}})

(21)

r_{i}^{k} = f_{i} (X^{k}) - \sum_{j = 1}^{n} (\frac{p_{i j}^{k}}{U_{j}^{k} - x_{j}^{k}} + \frac{q_{i j}^{k}}{x_{j}^{k} - L_{j}^{k}})

(22)

p_{i j}^{k} = {\begin{array}{l} (U_{j}^{k} - x_{j}^{k}) \frac{f_{i + 1}^{k} (x) - f_{i}^{k} (x)}{△ x} & \frac{f_{i + 1}^{k} (x) - f_{i}^{k} (x)}{△ x} > 0 \\ 0 & \frac{f_{i + 1}^{k} (x) - f_{i}^{k} (x)}{△ x} \leq 0 \end{array}

(23)

q_{i j}^{k} = {\begin{array}{l} - (x_{j}^{k} - L_{j}^{k}) \frac{f_{i + 1}^{k} (x) - f_{i}^{k} (x)}{△ x} & \frac{f_{i + 1}^{k} (x) - f_{i}^{k} (x)}{△ x} < 0 \\ 0 & \frac{f_{i + 1}^{k} (x) - f_{i}^{k} (x)}{△ x} \geq 0 \end{array}

(24)

where

\frac{f_{_{i + 1}}^{k} (x) - f_{i}^{k} (x)}{△ x}

is the value at x = x^k which changes with the iteration point x_j^k. At x = x^k, we have f_i^k(x^k) = f_i(x^k),

\frac{f_{i + 1}^{k} (x) - f_{i}^{k} (x)}{△ x} = \frac{f_{_{i + 1}} (x) - f_{i} (x)}{△ x}

(i = 1, 2,…, m; j = 1, 2,…, n).

In the subproblem, if x_j^k is close to L_j^k or U_j^k, the value of f_i^k will increase sharply, making x_j^k = L_j^k or x_j^k = U_j^k asymptotes. The solution after each iteration is maintained between L_j^k and U_j^k, which change between iterations; thus, L_j^k and U_j^k are called MA.

The second partial derivative of f_i^k at X = X^k is

\frac{f_{i + 1}^{k} - 2_{f}^{i k} + f_{i - 1}^{k + 1}}{△ x^{2}} = \frac{2 p_{i j}^{k}}{{(U_{j}^{k} - x_{j})}^{3}} + \frac{2 q_{i j}^{k}}{{(x_{j} - L_{j}^{k})}^{3}}

(25)

Equation (25) is simplified as equation (26).

\frac{f_{i + 1}^{k} - 2_{f}^{i k} + f_{i - 1}^{k + 1}}{△ x^{2}} = {\begin{matrix} \frac{2 (\frac{f_{_{i + 1}} (x) - f_{i} (x)}{△ x})}{U_{j}^{k} - x_{j}^{k}} & and \frac{f_{_{i + 1}} (x) - f_{i} (x)}{△ x} > 0 \\ \frac{- 2 (\frac{f_{_{i + 1}} (x) - f_{i} (x)}{△ x})}{x_{j}^{k} - L_{j}^{k}} & and \frac{f_{_{i + 1}} (x) - f_{i} (x)}{△ x} < 0 \end{matrix}

(26)

Thus, since p_ij^k ≥ 0 and q_ij^k ≥ 0, f_i^k is a convex function. In addition, the next iterative point always exists between the lower bound L_j^k and upper bound U_j^k. Therefore, we can obtain a good approximation by regulating the boundary values. When we compare IMA with other optimization algorithms, we see that it has some advantages including easily obtained solutions, low computational cost, and subproblems constructed with convexity and separable independent variables.

Solution of subproblem

In IMA, the subproblem we constructed also needs to be solved; here we express it with the Lagrange function.

For equations (18) to (20), we omit the number of iterations k, and present the simplified form as follows

M I N r_{0} + \sum_{j = 1}^{n} (\frac{p_{0 j}}{U_{j} - x_{j}} + \frac{q_{0 j}}{x_{j} - L_{j}})

(27)

S . T . \sum_{j = 1}^{n} (\frac{p_{i j}}{U_{j} - x_{j}} + \frac{q_{i j}}{x_{j} - L_{j}}) \leq b_{i} i = 1, …, m

(28)

α_{j} \leq x_{j} \leq β_{j} j = 1, …, n

(29)

where

α_{j} = \max {x_{j}^{a}, α_{j}}, β_{j} = \min {x_{j}^{a}, β_{j}}, L_{j} < α_{j} \leq β_{j} < U_{J}

, and

b_{i} = f_{i} -^{*} r_{i}

The subproblem is a convex function with separable variables, which can be solved by the dual method. The subproblem is constructed as a Lagrange function, which is

l (x, y) = f_{0}^{k} (X) + \sum_{i = 1}^{m} y_{i} f_{i}^{k} (X)

(30)

Substituting subproblem (27) into (30), we have

l (x, y) = r_{0} - Y^{T} B + \sum_{j = 1}^{n} (\frac{p_{0 j} + Y^{T} P_{j}}{U_{j} - x_{j}} + \frac{q_{0 j} + Y^{T} Q_{j}}{x_{j} - L_{j}})

(31)

The following equation can be obtained by simplifying equation (31).

l (x, y) = r_{0} - Y^{T} B + \sum_{j = 1}^{n} l_{j} (x_{j}, Y)

(32)

$B = {(b_{1}, …, b_{m})}^{T}, P_{j} = {(P_{1 j}, …, P_{m j})}^{T}, Q_{j} = {(Q_{1 j}, …, Q_{m j})}^{T} and Y = {(y_{1}, …, y_{m})}^{T}$ where Y is the Lagrange function multiplier

l_{j} (x_{j}, y) = \frac{p_{0 j} + Y^{T} P_{j}}{U_{j} - x_{j}} + \frac{q_{0 j} + Y^{T} Q_{j}}{x_{j} - L_{j}}

(33)

If y_i ≥ 0, the dual objective function can be constructed as

w (Y) = r_{0} - Y^{T} B + \sum_{j = 1}^{n} w_{j} (Y)

(34)

where

w_{j} (Y) = \min_{x_{j}} {l_{j} (x_{j}, Y); α_{j} \leq x_{j} \leq β_{j}}

The minimum value of x_j is determined by Y; it can be expressed as x_j(Y). If y_i ≥ 0, then p₀ _j + Y^TP_j ≥ 0 and q₀ _j + Y^TQ_j ≥ 0. Hence, l_j(x_j,Y) is a convex function.

Therefore, assuming that there is at least one positive term in p₀ _j + Y^TP_j and q₀ _j + Y^TQ_j, and the first-order derivative of l_j(x_j,Y) on x_j is

l_{j}^{'} (x_{j}, Y) = \frac{p_{0 j} + Y^{T} P_{j}}{{(U_{j} - x_{j})}^{2}} - \frac{q_{0 j} + Y^{T} Q_{j}}{{(x_{j} - L_{j})}^{2}}

(35)

The second-order derivative of l_j(x_j,Y) on x_j is

{l^{″}}_{j} (x_{j}, Y) = \frac{2 (p_{0 j} + Y^{T} P_{j})}{{(U_{j} - x_{j})}^{3}} + \frac{2 (q_{0 j} + Y^{T} Q_{j})}{{(x_{j} - L_{j})}^{3}}

(36)

The second-order derivative of l_j(x_j,Y) on x_j is positive, so the first-order derivative of l_j(x_j,Y) on x_j is an increasing function. The conclusions about the minimum value of x_j(Y) are as follows:

If ${l^{'}}_{j} (α_{j}, Y) \geq 0$ , then $x_{j} (Y) = α_{j}$ ;

If ${l^{'}}_{j} (β_{j}, Y) \leq 0$ , then $x_{j} (Y) = β_{j}$ ;

If ${l^{'}}_{j} (α_{j}, Y) < 0$ and ${l^{'}}_{j} (β_{j}, Y) > 0$ , then $x_{j} (Y)$ has a unique solution. The form of the solution is shown in below.

x_{j} (Y) = \frac{{(p_{0 j} + Y^{T} P_{j})}^{1 / 2} L_{j} + {(q_{0 j} + Y^{T} Q_{j})}^{1 / 2} U_{j}}{{(p_{0 j} + Y^{T} P_{j})}^{1 / 2} + {(q_{0 j} + Y^{T} Q_{j})}^{1 / 2}}

(37)

Substituting x_j(Y) into the following function w(Y), we have

w (Y) = r_{0} - Y^{T} B + \sum_{j = 1}^{n} (\frac{p_{0 j} + Y^{T} P_{j}}{U_{j} - x_{j} (Y)} + \frac{q_{0 j} + Y^{T} Q_{j}}{x_{j} (Y) - L_{j}})

(38)

Hence, we obtain the first-order partial derivative of $w (Y)$ on y_i

\frac{\partial w}{\partial y_{i}} = - b_{i} + \sum_{j = 1}^{n} (\frac{p_{i j}}{U_{j} - x_{j} (Y)} + \frac{q_{i j}}{x_{j} (Y) - L_{j}})

(39)

Then, the dual problem of the subproblem is equivalent to the maximum value of the dual function w(Y) when y_i ≥ 0.

SMB optimization strategy

Maximizing productivity

We apply IMA to an SMB operation optimization process, with productivity P_r as the goal function, and purities of component A (strong adsorption component) and B (weak adsorption component) as the constraint conditions. The specific problem can be described as follows

M a X P r = \frac{Q_{E} c_{E, A}^{*} + Q_{R} c_{R, B}^{*}}{\sum n_{j} V}

(40)

S . T . X_{A} = \frac{c_{E, A}^{*}}{c_{E, A}^{*} + c_{E, B}^{*}} \geq 98 %

(41)

X_{B} = \frac{c_{R, B}^{*}}{c_{R, A}^{*} + c_{R, B}^{*}} \geq 98 %

(42)

where superscript * indicates the equilibrium value.

The physical parameters of the SMB are shown in Table. 1. The selection of initial points is as follows: Q_R = 2.5 mL/min, Q_E = 2.5 mL/min, k = 0, and the initial values of the asymptotes are L_j = 0 and U_j = 5. The average concentration in equations (40) to (42) is replaced by the online concentration detection value. The optimization results are shown in Table. 3.

Table 3.

Comparison of results of optimization.

Parameters	IMA	PSO
Q_R (mL/min)	0.43	0.45
Q_E (mL/min)	0.75	0.73
t* (min)	4.17	4.20
Productivity (g/L*h)	3.538	3.506
Extract purity (%)	99.4	98.6
Raffinate purity (%)	99.8	98.9

IMA: improving the moving asymptote; PSO: particle swarm optimization.

Table 3 shows the results of optimization using IMA and PSO, respectively. The simulation results show that the two optimization algorithms can both obtain satisfactory productivity by optimizing Q_R, Q_E, and switching time t* so that the product purity meets requirements. In particular, optimization with IMA is better than that with PSO in terms of product purity and productivity, primarily because their search modes are different. In the same number of iterations, PSO does not converge as well, and does not synchronously exploit and explore the design space, whereas IMA converges faster and is better in fine search ability.

Maximizing extraction and raffinate purity

In an SMB, high product purity is the most basic requirement. In the optimization problem, feed flow rate Q_F and elution flow rate Q_D are fixed values. On this basis, the purity of the extraction and raffinate is maximized.

We took the flow rate of zone 2 as one of the control variables, and the switching time t as another*. The mathematical model of the optimization problem is

M a X P_{1} = X_{E} (Q_{2}, t^{*})

(43)

M a X P_{2} = X_{R} (Q_{2}, t^{*})

(44)

The optimization problem also ensures that the purity of the product reaches a set limit, given as

X_{A} \geq 98 % and

(45)

X_{B} \geq 98 %

(46)

The physical parameters of our SMB are shown in Table. 1. Decision variables are as follows: 3 min ≤ t *≤ 5 min; 0.3 mL/min ≤ Q₂ ≤ 1.5 mL/min; Q_F = 0.23 mL/min; Q_D = 0.95 mL/min; Q₁ = 1.65 mL/min. Figure 6 shows the distribution of Pareto (Wu et al., 2006) optimal solution sets from both IMA and PSO. The simulation results show that both optimization algorithms can converge to the front of the Pareto optimal solution set.

Figure 6.

Optimal solution of IMA and PSO. IMA: improving the moving asymptote; PSO: particle swarm optimization.

In the process of optimization and convergence, the number of Pareto optimal set frontiers of IMA, represented by the solid black square, is less than those of PSO, represented by the hollow circle. This shows that IMA has higher search efficiency and faster convergence than PSO. Finally, they converge to the upper right side of the image with slightly higher extraction purity achieved by IMA.

To ensure the accuracy of IMAs in SMB optimization, we use the triangle theory for verification. Based on balance theory, m₂ and m₃ are fixed values under ideal conditions. However, due to diffusion and mass transfer resistance in actual separation processes, m₂ and m₃ move from high to low with the Pareto optimal solution.

The corresponding positions of operation points after optimization on the m₂–m₃ plane are shown in Figure 7. We can see from the figure that the optimization results from the proposed algorithm remain in the complete separation region, and that the calculation results satisfy the triangle theory.

Figure 7.

m₂–m₃ plane of IMA and PSO. IMA: improving the moving asymptote; PSO: particle swarm optimization.

Maximizing productivity and minimizing solvent consumption

From an economic point of view, SMB solvent consumption and productivity are important indicators. We set the flow rate of feed Q_F and the flow rate of zone 1 Q₁ to minimize solvent consumption and maximize productivity. Because Q_F and Q₁ are given, three of the operation parameters Q_D, Q_R, Q_E, Q₂, Q₃ and Q₄ are independent, and we take Q_D, Q_R, Q_E, and t* as control variables. Therefore, the mathematical description of this optimization is as follows

M a X P_{1} = P r (Q_{D}, Q_{R}, Q_{E}, t^{*})

(47)

M i n P_{1} = S C (Q_{D}, Q_{R}, Q_{E}, t^{*})

(48)

The purity of the product fulfills equations (45) and (46).

Decision variables and fixed parameters:

3 min ≤ t* ≤ 5 min; 0.5 mL/min ≤ Q_D ≤ 5 mL/min; Q_F = 0.23 mL/min; 0.2 mL/min ≤ Q_R ≤ 2 mL/min; 0.3 mL/min ≤ Q_E ≤ 3 mL/min; Q₁ = 1.65 mL/min.

Figure 8 shows the Pareto optimal solution sets distribution of both IMA and PSO. We see that the two algorithms can converge to the upper left side of the image, near solvent consumption of 1 L/g, and productivity near 3.5 g/L-h. However, IMA has a small number of optimal solution sets, showing that the method converges quickly.

Figure 8.

Optimal solution distribution of IMA and PSO. IMA: improving the moving asymptote; PSO: particle swarm optimization.

Figure 9.

m₂–m₃ plane of IMA and PSO. IMA: improving the moving asymptote; PSO: particle swarm optimization.

Figure 9 shows the position of optimized operating points of IMA and PSO in the m₂–m₃ plane, all in the complete separation region. However, IMA is closer to the best operation point W.

The distributions of decision variables of IMA and PSO are shown in Figures 10 and 11, respectively. It can be seen from the figures that both algorithms can converge to the optimal solution set front, showing that they are applicable and robust. However, IMA converges more quickly, and produces more uniformly distributed control solutions.

Figure 10.

Control variable of IMA. IMA: improving the moving asymptote.

Figure 11.

Control variable of PSO. PSO: particle swarm optimization.

Conclusions

The modeling of SMB systems is an open problem due to strong coupling and complex mechanisms. The uncertainty in SMB systems is typically significant, so their performance can be improved by adopting robust feedback control algorithms. The online optimization approach developed in this study is able to find optimal operating schemes for a broad class of SMB processes.

We adopt a method combining orthogonal collocation with the finite element method to solve the equilibrium diffusion mathematical model of an SMB chromatographic separation process. With our improved MA algorithm, extract purity, raffinate purity, productivity, and solvent consumption are optimized and simulated. Its feasibility is verified by triangle theory.

To verify the applicability and effectiveness of the improved moving asymptote algorithm, we simulate the optimization strategy and algorithm and compare the simulation results with the PSO algorithm. These comparisons show that the improved moving asymptote algorithm requires less computation, converges more quickly, and produces well-distributed optimal solutions. This SMB optimization strategy can be used for SMB separation process design and operation guidance.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

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Optimizing control of adsorption separation processes based on the improved moving asymptotes algorithm

Abstract

Keywords

Introduction

The mathematical model

Model solutions

Orthogonal collocation finite element method

Model solution and simulation

Optimization strategy of SMB

Optimizing control approach

Principles of IMA

Construction of subproblem

Solution of subproblem

SMB optimization strategy

Maximizing productivity

Maximizing extraction and raffinate purity

Maximizing productivity and minimizing solvent consumption

Conclusions

Footnotes

Declaration of Conflicting Interests

Funding

References