Minimum reflux calculation for multicomponent distillation in multi-feed, multi-product columns: Algorithms and examples

TL;DR

The paper tackles the challenge of determining the minimum reflux condition for general MFMP distillation columns separating multicomponent mixtures. Building on a previously developed shortcut model, it presents an algorithmic framework that either solves a MINLP (via BARON) or, when product distributions are specified, executes a deterministic procedure to identify the controlling feed or sidedraw and compute the minimum reboiler duty. The approach yields results that align closely with Aspen Plus simulations and reveals several counterintuitive design insights, such as the impact of feed placement and the limitations of column decomposition. It further demonstrates that a sidedraw can control minimum reflux even when fed as saturated liquids, addressing a gap in traditional distillation design heuristics. Collectively, the work provides a fast, rigorous, and generalizable tool for synthesizing energy-efficient MFMP columns, with broad implications for distillation design and optimization under decarbonization goals.

Abstract

In this work, we present the first algorithm for identifying the minimum reboiler vapor duty requirement for a general multi-feed, multi-product (MFMP) distillation column separating ideal multicomponent mixtures. This algorithm incorporates our latest advancement in developing the first shortcut model for MFMP columns. We demonstrate the accuracy and efficiency of this algorithm through case studies. The results obtained from these case studies also provide valuable insights on optimal design of MFMP columns. Many of these insights are against the existing design guidelines and heuristics. For example, placing a colder saturated feed stream above a hotter saturated feed stream sometimes leads to higher energy requirement. Furthermore, decomposing a general MFMP column into individual simple columns may lead to incorrect estimation of the minimum reflux ratio for the MFMP column. Thus, the algorithm presented here offers a fast, accurate, and automated approach to synthesize new, energy-efficient, and cost-effective MFMP columns.

Figures (13)

• Figure 1: An example MFMP column with three feed streams and two sidedraw product streams and a detailed illustration of liquid and vapor flows within $\mathop{\mathrm{SEC}}\nolimits_3$, in which the variables in bold are component flow vectors (e.g., $\mathbf{d}^{\sec_3} = (d_1^{\sec_3}, \dots, d_c^{\sec_3})$ for a $c$-component system). The column section is numbered from top (1) to bottom ($N_{\mathop{\mathrm{SEC}}\nolimits}=5$). The definitions of variables and parameters used here and for the rest of this paper are summarized in Appendices \ref{['appendixA']} and \ref{['appendixB']}. We follow the convention that $\mathbf{v}_{\mathop{\mathrm{F}}\nolimits_j}$, $\mathbf{l}_{\mathop{\mathrm{F}}\nolimits_j}$, and $\mathbf{f}_{\mathop{\mathrm{F}}\nolimits_j} \geq \mathbf{0}$, whereas $\mathbf{v}_{\mathop{\mathrm{W}}\nolimits_j}$, $\mathbf{l}_{\mathop{\mathrm{W}}\nolimits_j}$, and $\mathbf{f}_{\mathop{\mathrm{W}}\nolimits_j} \leq \mathbf{0}$.
• Figure 2: Roots of Equation \ref{['eqn_characteristic']} for a five-component illustrative example where $(d_1, d_2, d_3, d_4, d_5) = (-0.4, 0.1, 0.2, 0.3, 0.2)$ for section $\mathop{\mathrm{TOP_F}}\nolimits_j$, $(d_1, d_2, d_3, d_4, d_5) = (-0.4, 0.1, 0.2, 0.3, 0.2)$ for feed $\mathop{\mathrm{TOP_F}}\nolimits_j$, and $(d_1, d_2, d_3, d_4, d_5) = (-0.5, -0.4, -0.3, 0.2, 0.1)$ for section $\mathop{\mathrm{BOT_F}}\nolimits_j$. The relative volatilities are $(\alpha_1,\alpha_2,\alpha_3,\alpha_4,\alpha_5) = (1,2,3,4,5)$. The section vapor flow $V$ is set to be 8 and $\mathop{\mathrm{F}}\nolimits_j$ is a saturated liquid. In this case, the pinch roots $\gamma_p^{\mathop{\mathrm{TOP_F}}\nolimits_j} \in (\alpha_1,\alpha_2)$ and $\gamma_p^{\mathop{\mathrm{BOT_F}}\nolimits_j} \in (\alpha_3,\alpha_4)$.
• Figure 3: The relationship between $\mu_i$ and $K_i$ variables for the example illustrated in Figure \ref{['fig_shortcutdemo']}. The green arrows show how the (binary) coefficients in Equation \ref{['eqn_feasibility_K_feed']} are constructed. For this example, $K_2^{\mathop{\mathrm{TOP_F}}\nolimits_j} - K_1^{\mathop{\mathrm{BOT_F}}\nolimits_j} = K_3^{\mathop{\mathrm{TOP_F}}\nolimits_j} - K_2^{\mathop{\mathrm{BOT_F}}\nolimits_j} = K_4^{\mathop{\mathrm{TOP_F}}\nolimits_j} - K_3^{\mathop{\mathrm{BOT_F}}\nolimits_j} = 1$. Therefore, $\mathcal{I}_{\mathop{\mathrm{F}}\nolimits_j} = \{2,3,4\}$ according to Equation \ref{['eqn_Ifeed']}.
• Figure 4: A two-feed column with no sidedraw product stream.
• Figure 5: The pinch simplices at the minimum reflux condition obtained using Algorithms \ref{['algo1']} through \ref{['algo3']}. Hereafter, $X_1,\, X_2,\, X_3$ represent pure n-octane, n-heptane, and n-hexane, respectively. The colors of the pinch simplices match with those in Figure \ref{['fig_case1']}. The blue dots are the actual liquid composition profile of this two-feed column simulated in Aspen Plus as a RadFrac column. By setting up appropriate Design Specs in Aspen Plus to simulate the MFMP containing 150 equilibrium stages, we obtain a minimum reflux ratio of $R_{\min}=2.145$ from Aspen Plus. The exact pinches compositions in $\mathop{\mathrm{SEC}}\nolimits_1$ through $\mathop{\mathrm{SEC}}\nolimits_3$ are $Z_2$ (associated with pinch root $\gamma_p^{\mathop{\mathrm{SEC}}\nolimits_1} = \gamma_2^{\mathop{\mathrm{SEC}}\nolimits_1} \in (\alpha_1, \alpha_2)$), $Z_3$ (associated with pinch root $\gamma_p^{\mathop{\mathrm{SEC}}\nolimits_2} = \gamma_3^{\mathop{\mathrm{SEC}}\nolimits_2} \in (\alpha_2, \alpha_3)$), and $Z_3$ ($\gamma_p^{\mathop{\mathrm{SEC}}\nolimits_3} = \gamma_3^{\mathop{\mathrm{SEC}}\nolimits_3} \in (\alpha_3, \alpha_3+\delta)$), respectively. Therefore, $\mu_2^{\mathop{\mathrm{SEC}}\nolimits_1} = \mu_3^{\mathop{\mathrm{SEC}}\nolimits_2} = \mu_4^{\mathop{\mathrm{SEC}}\nolimits_3} = 1$.