Monotonic reformulation and bound tightening for global optimization of ideal multi-component distillation columns

Abstract

This paper addresses the problem of determining cost-minimal process designs for ideal multi-component distillation columns. The special case of binary distillation was considered in the former work (Ballerstein et al. in Optim Eng 16(2):409–440, 2015. https://doi.org/10.1007/s11081-014-9267-5). Therein, a problem-specific bound-tightening strategy based on monotonic mole fraction profiles of single components was developed to solve the corresponding mixed-integer nonlinear problems globally. In the multi-component setting, the mole fraction profiles of single components may not be monotonic. Therefore the bound-tightening strategy from the binary case cannot be applied directly. In this follow-up paper, a model reformulation for ideal multi-component distillation columns is presented. The reformulation is achieved by suitable aggregations of the involved components. Proofs are given showing that mole fraction profiles of aggregated components are monotonic. This property is then used to adapt the bound-tightening strategy from the binary case to the proposed model reformulation. Computational results are provided that indicate the usefulness of both the model reformulation and the adapted bound tightening technique for deterministic global optimization of ideal multi-component distillation column designs.

Appendices

Appendix 1: Proof of Corollary 3

Consider System (29) with parameters $\alpha _1 \ge \alpha _2 \ge \dots \ge \alpha _n>0$ and variables $(Y,\nu _{\rm r})$. We show that applying the transformation rules (30) to $(Y,\nu _{\rm r})$ and $\alpha _1,\ldots ,\alpha _n$ leads to System (31) with parameters $\hat{\alpha }_1 \ge \hat{\alpha }_2 \ge \dots \ge \hat{\alpha }_n > 0$ and with variables $\hat{X}$, $\hat{\nu }_{\rm s}$ whose feasible solutions satisfy the conditions of Theorem 1 (and Corollary 2). Note that feasible solutions to System (29) are in one-to-one correspondence to solutions feasible to (31) via the transformation rules (30). As we obtain from Corollary 2 that, for every feasible solution $(\hat{X}, \hat{\nu }_{\rm s})$, the sequences $\{\hat{X}_{k,\hat{l}_{\rm s}}\}_{\hat{l}_{\rm s}=1}^{\hat{u}+1}$ with $k=1,\ldots ,n$ are nondecreasing, the corresponding sequences $\{Y_{k,l_{\rm r}}\}_{l_{\rm r}=1}^{u+1}$, $k=1,\ldots ,n$ are nonincreasing.

The first part of System (29) is given as a combination of the mass balance equations (14) and the inverted phase equilibrium equations (28).

$$\begin{aligned} {Y}_{k,l_{\rm{r}}+1} = \nu _{\rm{r}}\, {X}_{k,l_{\rm{r}}}+(1-\nu _{\rm{r}})\, Y_{k,1} \quad \text{and} \quad X_{k,l_{\rm r}}&= \frac{\sum _{j=1}^{k}\alpha _j^{-1}(Y_{j,l_{\rm r}}-Y_{j-1,l_{\rm r}})}{\sum _{j=1}^{n}\alpha _j^{-1}(Y_{j,l_{\rm r}}-Y_{j-1,l_{\rm r}})}, \end{aligned}$$

for $k=1,\ldots ,n$ and $l_{\rm r}=1,\ldots ,u$. We apply the transformation rules (30) to each constraint, separately. Recap that after the transformation, the components appear in reverse order. To indicate that, we introduce a new index $m := n-k$.

For all $k=1,\ldots ,n$ and all $l=1,\ldots ,u$, we obtain

$$\begin{aligned} Y_{k,l+1} = \nu _{\rm r} X_{k,l} + (1-\nu _{\rm r}) Y_{k,1}\quad \Leftrightarrow \quad&(1-Y_{k,l+1}) = 1-\Big (\nu _{\rm r} X_{k,l} + (1-\nu _{\rm r}) Y_{k,1}\Big )\\ \Leftrightarrow \quad&(1-Y_{k,l+1}) = \nu _{\rm r} (1-X_{k,l}) + (1-\nu _{\rm r}) (1-Y_{k,1}). \end{aligned}$$

Thus, the transformation rules (30) yield

$$\begin{aligned} \hat{X}_{k,l+1} = \hat{\nu }_{\rm s} \hat{Y}_{k,l} + (1-\hat{\nu }_{\rm s}) \hat{X}_{k,1}, \qquad m = 0,\dots ,n, \quad l =1,\ldots ,u. \end{aligned}$$

(35)

For the inverted phase equilibrium equations, we further derive that

$$\begin{aligned} \hat{Y}_{n-k,l}&= 1 - X_{k,l} = 1-\frac{\sum _{j=1}^{k}\alpha _j^{-1}(Y_{j,l}-Y_{j-1,l})}{\sum _{j=1}^{n}\alpha _j^{-1}(Y_{j,l}-Y_{j-1,l})} = \frac{\sum _{j=k+1}^{n}\alpha _j^{-1}(Y_{j,l}-Y_{j-1,l})}{\sum _{j=1}^{n}\alpha _j^{-1}(Y_{j,l}-Y_{j-1,l})} \\&= \frac{\sum _{j=k+1}^{n}\alpha _j^{-1}(1-\hat{X}_{n-j,l}-\big (1-\hat{X}_{n-(j-1),l})\big )}{\sum _{j=1}^{n}\alpha _j^{-1}(1-\hat{X}_{n-j,l}-\big (1-\hat{X}_{n-(j-1),l})\big )} \\&= \frac{\sum _{j=k+1}^{n}\alpha _j^{-1}(\hat{X}_{n-(j-1),l}-\hat{X}_{n-j,l})}{\sum _{j=1}^{n}\alpha _j^{-1}(\hat{X}_{n-(j-1),l}-\hat{X}_{n-j,l_{\rm s}})} = \frac{\sum _{j=k+1}^{n}\hat{\alpha }_{n+1-j}(\hat{X}_{n+1-j,l}-\hat{X}_{n-j,l})}{\sum _{j=1}^{n}\hat{\alpha }_{n+1-j}(\hat{X}_{n+1-j,l}-\hat{X}_{n-j,l})} \\ \end{aligned}$$

holds for every $k=1,\ldots ,n$ and for every $l=1,\ldots ,u$. By an index shift $p:=n+1-j$, we derive the equivalence to the non-inverted phase equilibrium equations (16).

$$\begin{array}{rll} & \hat{Y}_{n-k,l} = \frac{\sum _{p=1}^{n-k}\beta _{p}(\hat{X}_{p,l}-\hat{X}_{p-1,l})}{\sum _{p=1}^{n}\beta _{p}(\hat{X}_{p,l}-\hat{X}_{p-1,l})}, & n= 0,\ldots ,n \\ && l = 1,\ldots ,u+1 \\ \Leftrightarrow & \hat{Y}_{m,l} = \frac{\sum _{p=1}^{m}\beta _{p}(\hat{X}_{p,l}-\hat{X}_{p-1,l})}{\sum _{p=1}^{n+1}\beta _{p}(\hat{X}_{p,l}-\hat{X}_{p-1,l})},& m = 0,\ldots ,n \\ &&l = 1,\ldots ,u+1 \\ \end{array}$$

(36)

Combining the Eqs. (35) and (36), we obtain the first line from System (31).

The second line of System (31) results from the following relation.

$$\begin{aligned} Y_{k+1,l}\ge Y_{k,l},\;\; k = 0,\ldots ,n-1 \quad \Leftrightarrow \quad \hat{X}_{m-1,l} \le \hat{X}_{m,l},\; m = 1,\ldots ,n. \end{aligned}$$

The third line of System (31) trivially holds.

It remains to argue that the transformed constant relative volatilities $\hat{\alpha }_m = \alpha ^{-1}_{(n+1)-m}$, $m=1,\ldots ,n$, are strictly positive and monotonically nondecreasing in the new ordering of the components. This, however, holds as $\alpha _1 \ge \dots \ge \alpha _{n}> 0$ implies that

$$\begin{aligned} 0 < \alpha ^{-1}_{1} \equiv \hat{\alpha }_{n} \le \dots \le \alpha ^{-1}_{n} \equiv \hat{\alpha }_{1}. \end{aligned}$$

Now, we can conclude that any feasible solution to System (31) satisfies the conditions of Theorem 1 and Corollary 2. $\square$

Appendix 2: Proof of Lemma 6 and Lemma 7

We only give a proof for Lemma 6 dealing with the stripping section. The correctness of Lemma 7 can be shown in a similar way.

In what follows, the superscript “strip” is omitted.

Proof

(Lemma 6) We interpret the aggregated phase equilibrium equations (16)

$$\begin{aligned} Y_{k,l_{\rm s}}(\varvec{X}) = \frac{\sum _{j=1}^{k}\alpha _j(X_{j,l_{\rm s}}-X_{j-1,l_{\rm s}})}{\sum _{j=1}^{n}\alpha _j(X_{j,l_{\rm s}}-X_{j-1,l_{\rm s}})} \end{aligned}$$

as functions in the liquid phase concentration variables. For all $k,q=1,\dots ,n$ and for all $l_{\rm s} = 1,\ldots ,u+1$, we consider the partial derivatives $\frac {\partial Y_{k,l_{\rm{s}}}(\varvec{X})}{\partial X_{q,l_{\rm{s}}}}$ where we distinguish the three cases $q\le k-1$, $q=k$ and $q\ge k+1$. To simplify the notation, we introduce the constant $\alpha _{n+1} := 0$.

For $q\le k-1$, we obtain

$$\begin{aligned}&\frac{\partial Y_{k,l_{\rm s}}(\varvec{X})}{\partial X_{q,l_{\rm s}}} \\ =&\frac{(\alpha _q - \alpha _{q+1}) \sum _{j=1}^{n}\alpha _j(X_{j,l_{\rm s}}-X_{j-1,l_{\rm s}})}{\big (\sum _{j=1}^{n}\alpha _j(X_{j,l_{\rm s}}-X_{j-1,l_{\rm s}})\big )^2} - \frac{\sum _{j=1}^{k}\alpha _j(X_{j,l_{\rm s}}-X_{j-1,l_{\rm s}}) (\alpha _q - \alpha _{q+1})}{\big (\sum _{j=1}^{n}\alpha _j(X_{j,l_{\rm s}}-X_{j-1,l_{\rm s}})\big )^2} \\ =&\frac{(\alpha _q - \alpha _{q+1}) \sum _{j=k+1}^{n}\alpha _j(X_{j,l_{\rm s}}-X_{j-1,l_{\rm s}}) }{\big (\sum _{j=1}^{n}\alpha _j(X_{j,l_{\rm s}}-X_{j-1,l_{\rm s}})\big )^2}. \end{aligned}$$

As $(\alpha _q - \alpha _{q+1}) \ge 0$ holds, this derivative is non-negative for all $k=1,\ldots ,n$ and all $l_{\rm s}=1,\ldots ,u+1$.

For $q= k$ we obtain

$$\begin{aligned} \frac{\partial Y_{q,l_{\rm s}}(\varvec{X})}{\partial X_{q,l_{\rm s}}}&= \frac{\alpha _q \sum _{j=1}^{n}\alpha _j(X_{j,l_{\rm s}}-X_{j-1,l_{\rm s}}) - \sum _{j=1}^{q}\alpha _j(X_{j,l_{\rm s}}-X_{j-1,l_{\rm s}}) (\alpha _q - \alpha _{q+1})}{\big (\sum _{j=1}^{n}\alpha _j(X_{j,l_{\rm s}}-X_{j-1,l_{\rm s}})\big )^2} \\&= \frac{\alpha _q \sum _{j=q+1}^{n}\alpha _j(X_{j,l_{\rm s}}-X_{j-1,l_{\rm s}}) + \alpha _{q+1} \sum _{j=1}^{q}\alpha _j(X_{j,l_{\rm s}}-X_{j-1,l_{\rm s}}) }{\big (\sum _{j=1}^{n}\alpha _j(X_{j,l_{\rm s}}-X_{j-1,l_{\rm s}})\big )^2}, \end{aligned}$$

(37)

which is also non-negative for all $q=1,\ldots ,n$ and all $l_{\rm s}=1,\ldots ,u+1$.

For $q\ge k+1$ we obtain

$$\begin{aligned} \frac{\partial Y_{k,l_{\rm s}}(\varvec{X})}{\partial X_{q,l_{\rm s}}}&= \frac{ -\sum _{j=1}^{k}\alpha _j(X_{j,l_{\rm s}}-X_{j-1,l_{\rm s}}) (\alpha _q - \alpha _{q+1})}{\big (\sum _{j=1}^{n}\alpha _j(X_{j,l_{\rm s}}-X_{j-1,l_{\rm s}})\big )^2}, \end{aligned}$$

(38)

which is non-positive for all $k=1,\ldots ,n$ and all $l_{\rm s}=1,\ldots ,u+1$ due to $(\alpha _q - \alpha _{q+1}) \ge 0$.

This shows that the phase equilibrium equations are component-wise monotonic. Therefore, we can apply simple interval arithmetic, again, leading to the following lower and upper bounds on the vapor phase concentration variables $Y_{k,l_{\rm s}+1}$:

$$\begin{aligned} Y_{k,l_{\rm s}}^{\rm{lo}} = \frac{\sum _{j=1}^{k}\alpha _j(X_{j,l_{\rm s}}^{a_k}-X_{j-1,l_{\rm s}}^{a_k})}{\sum _{j=1}^{n}\alpha _j(X_{j,l_{\rm s}}^{a_k}-X_{j-1,l_{\rm s}}^{a_k})} \quad \text{and} \quad Y_{k,l_{\rm s}}^{\rm{up}} = \frac{\sum _{j=1}^{k}\alpha _j(X_{j,l_{\rm s}}^{b_k}-X_{j-1,l_{\rm s}}^{b_k})}{\sum _{j=1}^{n}\alpha _j(X_{j,l_{\rm s}}^{b_k}-X_{j-1,l_{\rm s}}^{b_k})}, \end{aligned}$$

where for $j= 1,\ldots ,n$

$$\begin{aligned} X_{j,l_{\rm{s}}}^{a_k} {{:}{=}} \left\{ \begin{array}{cl} X_{j,l_{\rm{s}}}^{\rm{lo}}, &{} {\text{if }} j\le k,\\ X_{j,l_{\rm{s}}}^{\rm{up}}, &{} {\text{if }} j> k, \end{array} \right. \quad {\text{and}} \quad X_{j,l_{\rm{s}}}^{b_k} = \left\{ \begin{array}{rl} X_{j,l_{\rm{s}}}^{\rm{up}}, &{} {\text{if }} j\le k,\\ X_{j,l_{\text{s}}}^{\rm{lo}} , &{} {\text{if }} j> k, \end{array} \right. \end{aligned}$$

(39)

We remark that the upper bound $Y_{k,l_{\rm{s}}}^{\text{up}}$ on $Y_{k,l_{\rm s}}$ is not tight when $X_{k,l_{\rm s}}^{\rm{up}} > X_{k^\prime ,l_{\rm s}}^{\rm{lo}}$ holds for some $k^\prime > k$. In those cases, we can compute an improved upper bound on $Y_{k,l_{\rm s}}$ by finding the maximum of

$$\begin{aligned} Y_{k,l_{\rm s}}(\varvec{X}) = \frac{\sum _{j=1}^{k}\alpha _j(X_{j,l_{\rm s}}-X_{j-1,l_{\rm s}})}{\sum _{j=1}^{n}\alpha _j(X_{j,l_{\rm s}}-X_{j-1,l_{\rm s}})} \end{aligned}$$

restricted to $X_{k',l_{\rm s}}^{\rm{lo}}\le X_{k,l_{\rm s}} \le X_{k',l_{\rm s}} \le X_{k,l_{\rm s}}^{\rm{up}}$. As $\frac {\partial Y_{k,l_{\rm s}}(\varvec{X})}{\partial X_{k,l_{\rm s}}} \ge 0$ and $\frac {\partial Y_{k,l_{\rm s}}(\varvec{X})}{\partial X_{k^\prime ,l_{\rm s}}} \le 0$, it follows that $X_{k,l_{\rm s}}= X_{k',l_{\rm s}}$ must hold for a solution on that the maximum is attained. A comparison of the Eq. (37) and (38) gives rise to the following relation:

$$\begin{aligned} \frac{\partial Y_{k,l_{\rm s}}(\varvec{X})}{\partial X_{k,l_{\rm s}}} + \sum _{j = k+1}^n \frac{\partial Y_{k,l_{\rm s}}(\varvec{X})}{\partial X_{j,l_{\rm s}}} \ge 0. \end{aligned}$$

This shows that the maximum is attained when $X_{k,l_{\rm s}} = X_{k',l_{\rm s}}= X_{k,l_{\rm s}}^{\rm{up}}$. Hence, we can replace the definition of $X_{j,l_{\rm{s}}}^{b_k}$ in Eq. (39) by

$$\begin{aligned} X_{j,l_{\rm{s}}}^{b_k} = \left\{ \begin{array}{ll} X_{j,l_{\rm{s}}}^{\rm{up}}, &{} \text{if } j\le k,\\ \max \big \{ X_{k,l_{\rm s}}^{\rm{up}}, X_{j,l_{\rm s}}^{\rm{lo}}\big \} , &{} \text{if } j> k. \end{array} \right. \end{aligned}$$

This completes the proof. $\square$