Design and Optimization

Distillation Design: from Whiteboard, to Virtual/Pilot Scale, to Reality

Site Author(s) and University of Waterloo (left) and Chiesa, L. (2005). Colonne distillazione [Online image]. Wikimedia. https://commons.wikimedia.org/wiki/File:Colonne_distillazione.jpg. Licensed under CC BY 3.0. (right).

Distillation column is generally a downstream process in terms of process design, and in some applications, a distillation column could sit in the middle of a chain of continuous unit operation units. As such, designing a distillation column that fits into a plant of multiple process units and is able to fulfil the capacity requirement and performance specifications is imperative. From design and operation perspective, this means determining column diameter and height with the highest possible accuracy so that the distillation column is seamlessly aligned with other process units for the best overall plant operation continuity and performance. An overdesigned or under-designed distillation column, like a speeding or decelerated vehicle in a busy highway disrupting the traffic flow, can compromise the operation and economics of the plant. Hence, the distillation design always calls for the best design methods and practice.

Just as any other chemical engineering process unit design, there is general design heuristics for distillation column design. The diagram in Figure 1 shows general distillation column design scope and heuristics. The short-cut methods, such as the McCabe–Thiele method illustrated on the Distillation Column component page, despite being the oldest method and subject to the constant molol overflow (CMO) assumption, has been still in use, particularly for steady state design purposes and the estimation of limiting or optimum operating conditions such as minimum reflux ratio and optimum feed tray. Rigorous method that involves the rigorous solution of all the common conservations, equilibrium, and summation equations, on the other hand, provides much more accurate design solutions. Nevertheless, both the short-cut methods and rigorous methods are usually based on many assumptions and empirical correlations such as thermodynamic, hydraulic, and efficiency correlations. Hence, the accuracy of any of these design methods would depend on the validity of the assumptions and accuracy of the correlations adopted by the design method.

Figure 1. General distillation column design and design methods. | Description

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Now that the design practice of a distillation column heavily relies on the accuracy of design correlations in the design method, the best design practice has to come down to experiment-based design practice. Not only does a distillation experiment, particularly pilot-scale experiment, measure necessary design parameters such as efficiencies, but also provides essential insights into the accuracy of the thermodynamic models as well as design correlations, as shown in the general design scheme of the experiment-based distillation design in Figure 2. The cyclic design approach, using experimental data and observation to validate the design correlations and methods as well as the final design, is deemed to lead to the most reliable design solution. In the absence of a pilot-scale distillation experiment, distillation design normally relies on many years of knowledge and experience of designers. From engineering learning perspective, a distillation design project, accompanied with a pilot-scale distillation, is the ideal tool to achieve deep understanding of the connection between distillation column design, the underlying theories, and distillation experiment design.

Figure 2. Iterative distillation design scheme based on experimental design and validation. | Description

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As suggested in Figure 1 and 2, an effective learning strategy for implementing the experiment-based distillation design in a laboratory setting calls for understanding the specific distillation design first, instead of the usual experimental studies in the lab courses. With the design specifications, the specific design methods can be devised, and the corresponding theories and solution algorithms can be developed. Then, the roles of the laboratory experiment should be readily identified based on the design methods and theories. Typically, a pilot-scale experiment is needed to measure necessary design parameters and/or generate performance data to validate the design methods and correlations. As well, the experiment normally provides first-hand observation and process insights into the justification of the final design. With all these unique objectives of the laboratory experiment, a well-purposed and more meaningful experimental plan can be put forward, the rest of the design process is to follow through the design cycles in Figure 2 and come to the final design solution. The following details some relevant theories for the experiment-based design cycle and practice scheme.

One of the most essential components for a successful distillation design is the accuracy of vapour-liquid equilibrium (VLE) for the distillation system. Although the experimental measurement of VLE for any system under distillation conditions is desirable, it is mostly impractical due to the broad conditions and cost. As a result, the computation of VLE and validation of the calculated VLE are utterly important not only for binary distillation design but also for the design of complex systems such as azeotropic or pressure swing distillation.

The fundamental basis for the VLE calculation is rooted in the thermodynamic condition that the fugacity of each species in vapour phase and liquid phase must be equal under equilibrium,

$${\phi_i}{y_i}P = {\gamma_i}{x_i}P_i^o$$

where $P$ is the total pressure of the system, $P_i^o$ is the vapour pressure of pure species $i$, $y_i$ is the vapour composition that is in equilibrium with liquid composition $x_i$, $\phi_i$ is the fugacity coefficient that accounts for the non-ideality of the gas phase, and $\phi_i$ is about unity if gas phase is an ideal gas (i.e., P <10 atm), $\gamma _i$ is the activity coefficient that accounts for the non-ideality of liquid phase. There are many correlations such as equation of states, Wilson equation, NRTL model that can be used to model the non-ideality of liquid phase. Once a correlation is validated with experimental data, it can then be used to compute and predict VLE under any other operation conditions. As an example, the Wilson equation, generally valid for both polar and non-polar systems, is given for the activity coefficients of multicomponent systems by,

$$ln{\gamma _k} = - ln\left[ {\sum\limits_{j = 1}^C {{x_j}{G_{kj}}} } \right] + 1 - \sum\limits_{i = 1}^C {\frac{{{x_i}{G_{ik}}}}{{\sum\limits_{j = 1}^C {{x_j}{G_{ij}}} }}} $$

where $\gamma _k$ is the activity coefficient of species $k$ in a system with $C$ components, $G_{kj}$ or $G_{ik}$ are binary molecular interaction coefficient. The equation is reduced for binary system to,

$$ln{\gamma _1} = - ln({x_1} + {x_2}{G_{12}}) + {x_2}\left( {\frac{{{G_{12}}}}{{{x_1} + {x_2}{G_{12}}}} - \frac{{{G_{21}}}}{{{x_2} + {x_1}{G_{21}}}}} \right)$$

$$ln{\gamma _2} = - ln({x_2} + {x_1}{G_{21}}) - {x_1}\left( {\frac{{{G_{12}}}}{{{x_1} + {x_2}{G_{12}}}} - \frac{{{G_{21}}}}{{{x_2} + {x_1}{G_{21}}}}} \right)$$

The binary interaction coefficients can be obtained from experimental data or estimated using the thermodynamic relationships,

$${G_{12}} = \frac{{{v_2}}}{{{v_1}}}exp( - \frac{{{\alpha _{12}}}}{{RT}})$$

$${G_{21}} = \frac{{{v_1}}}{{{v_2}}}exp( - \frac{{{\alpha _{21}}}}{{RT}})$$

where $v_1$ and $v_2$ are the molar volume of the pure liquid at the absolute temperature $T$, $R$ is molar gas constant, and $\alpha_{12}$ and $\alpha_{21}$ are parameter which characterizes molecular interaction.

2.1. Column Height Design

Once a reliable VLE correlation is established, a distillation design comes down to the determination of column height or the number of stages required for the design specifications as well as the determination of the required column diameter.

The actual number of stages required for a specified separation sensibly depends on the following design factors:

• Design method (short-cut vs. rigorous method)
• Tray or column efficiency

If the assumptions for the short-cut methods are justified for the system in design, the short-cut method such as the McCabe-Thiele method for binary distillation summarized on the Distillation Column component page can be used for the binary system. Alternatively, the Fenske-Underwood-Gilliland (FUG) shortcut method for multicomponent distillation summarized in Table 1 below also becomes handy for both binary and multicomponent systems, at least for preliminary design.

Table 1. Summary of the Fenske-Underwood-Gilliland (FUG) shortcut method for multicomponent and binary systems

	Multicomponent system	Binary System	Usage of the models
Fenske Equation	${N_m} = \frac{{ln\left[ {{{\left( {\frac{{{x_{LK}}}}{{{x_{HK}}}}} \right)}_D}{{\left( {\frac{{{x_{HK}}}}{{{x_{LK}}}}} \right)}_B}} \right]}}{{ln{\alpha _{LH}}}}$	${N_m} = \frac{{ln\left[ {\left( {\frac{{{x_D}}}{{1 - {x_D}}}} \right)\left( {\frac{{1 - {x_B}}}{{{x_B}}}} \right)} \right]}}{{ln\alpha }}$ or ${x_B} = \frac{{{x_D}}}{{{x_D} + {\alpha ^{{N_m}}}(1 - {x_D})}}$	Determine minimum number of stages $(N_m)$ using the design composition of low key and high key components in distillate $(D)$ and bottom $(B)$ based on the average relative volatility $(\alpha)$.
Underwood Equation	$\sum\limits_i {\frac{{{\alpha _i}{x_{i,F}}}}{{{\alpha _i} - \theta }}} = 1 - q$ $\sum\limits_i {\frac{{{\alpha _i}{x_{i,D}}}}{{{\alpha _i} - \theta }}} = {R_m} + 1$	$\frac{{\alpha {x_F}}}{{\alpha - \theta }} + \frac{{1 - {x_F}}}{{1 - \theta }} = 1 - q$ $\frac{{\alpha {x_D}}}{{\alpha - \theta }} + \frac{{1 - {x_D}}}{{1 - \theta }} = {R_m} + 1$	Determine minimum reflux ratio $(R_m)$ based on feed condition and distillate compositions of all species, where $\theta$ is a parameter related to feed quality $(q)$.
Gilliland Equation	$\frac{{N - {N_m}}}{{N + 1}} = 0.75\left[ {1 - {{\left( {\frac{{R - {R_m}}}{{R + 1}}} \right)}^{0.5668}}} \right]$	$\frac{{N - {N_m}}}{{N + 1}} = 0.75\left[ {1 - {{\left( {\frac{{R - {R_m}}}{{R + 1}}} \right)}^{0.5668}}} \right]$	Determine the number of stages $(N)$ required for the specified separation

As indicated in the McCabe-Thiele method in the Distillation Column component page and the FUG method, the shortcut method are clearly suitable for quick calculations of the number of equilibrium stages, minimum reflux ratio, and optimum feed tray, which are otherwise not directly available from the rigorous simulation. This means that the shortcut method can be used to establish an initial design approximation, particularly for non-ideal systems such as alcohol distillation. More accurate design of the distillation column has to rely on rigorous simulation or solving the MESH equations (material balances, equilibrium relationships, summation equations, and heat/enthalpy balances). This design approach from short-cut method to rigorous design simulation reduces the number of iterations through direct rigorous simulation. If the system under design such as alcohol systems exhibits low efficiency or drastic efficiency change along the column, then non-equilibrium needs to be considered, and validating the design under pilot scale operation or through simulations is critically important for the expected performance of the designed column.

It is important to note that the two short-cut methods above differ not only in the way they are implemented for design (stage-wise graphic calculations versus solving equations), but also their respective assumptions. The primary assumption for the McCabe-Thiele method is the constant molol overflow (CMO). For the FUG method, the constant average relative volatility is another major assumption in addition to the CMO assumptions. Therefore, the primary assumptions of each short-cut method for specific design system must be checked and verified before using the method for design calculation. As an example, the average relative volatility is typically evaluated based on the geometric average of the relative volatility from top and bottom compositions, and the assumption is generally valid for ideal systems. But for polar systems such as alcohol systems, the average volatility, geometric or arithmetic average of top, bottom, and feed, should be tested and validated using experimental data to determine the most accurate average value for column design.

The MESH equations for a distillation system of components and trays are summarized below.

Figure 3. Materials and enthalpy flow on Tray j. V=vapour molar flow, L=liquid molar flow, x=liquid mole fraction, y=vapour mole fraction, h is the enthalpy of the liquid, H is the enthalpy of vapour, F=feed, and M= liquid mole holdup on the tray. | Description

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The MESH equation for any tray j based on Figure 3 are given by,

Overall Materials Balance:

$$\frac{{d({M_j})}}{{dt}} = {L_{j - 1}} - {L_j} - {V_j} + {V_{j + 1}} + F \qquad\qquad (j = 1,.....,N)$$

Component Material Balance:

$$\frac{{d({M_j}{x_{i,j}})}}{{dt}} = {L_{j - 1}}{x_{i,j - 1}} - {L_j}{x_{i,j}} - {V_j}{y_{i,j}} + {V_{j + 1}}{y_{i,j + 1}} + F{x_{i,F}} \qquad\qquad j = 1,.....,N$$

Equilibrium Relationship:

$${P_j} = \sum\limits_{i = 1}^c {{x_{i,j}}{\gamma _{i,j}}{K_{i,j}}} \qquad\qquad j = 1,.....,N$$

$$y_{i,j}^* = {x_{i,j}}{\gamma _{i,j}}{K_{i,j}}$$

$$E_{i,j}^{MV} = \frac{{{y_{i,j}} - {y_{i,j + 1}}}}{{y_{i,j}^* - {y_{i,j + 1}}}}$$

Energy Balance:

$$\frac{{d({M_j}{h_j})}}{{dt}} = {L_{j - 1}}{h_{j - 1}} - {L_j}{h_j} - {V_j}{H_j} + {V_{j + 1}}{H_{j + 1}} + F{h_F} = 0 \qquad\qquad j = 1,.....,N$$

$${h_j} = \sum\limits_i {{x_{i,j}}(\Delta H_i^f + \int_{{T_o}}^{{T_{bj}}} {{c_{pi}}} dT)}$$

$${H_j} = \sum\limits_i {{y_{i,j}}[\Delta H_i^f + \int_{{T_o}}^{{T_{dp}}} {{c_{pi}}} dT + {\lambda _i}({T_{dp}})}$$

Summation Equations:

$$\sum\limits_i^c {{x_{i,j}}} = 1$$ and
$$\sum\limits_i^c {{y_{i,j}}} = 1$$

Pressure Drop Equation:

$${P_{j + 1}} = {P_j} + {\rho _L}g{h_T}$$

Where $P_j$ is the pressure on Tray $j$, $K_{i,j}$ is the equilibrium constant of species $i$ on Tray $j$, and $E_{i,j}^{MV}$ is Murphree vapour tray efficiency of species $i$ on Tray $j$.

All the equations (at least 4*C*N equations) need to be solved numerically for unsteady-state and steady-state performance with proper initial conditions that need to come from the knowledge of the system, or from the shortcut calculations. If steady-state simulation or design is desired, then the accumulation terms of the above equations become zero, all the process equations become a series of non-linear equations that can be more easily solved based on experimental conditions (e.g. steam flow rate or boil up rate) for simulations and the results from the short-cut calculations.

2.2. Column Diameter Design

The design of column diameter relies on hydrodynamics of distillation tray, or more specifically flooding point. Since the vapour flow velocity has more pronounced effects on tray hydraulics such as pressure drop, froth height, and flooding point, vapour flow rate, typically determined from the above column height design, is used to design the column diameter based on the well-known Fair’s design method, in which a capacity factor is universally correlated with flow parameter and tray spacing as summarized in the Distillation Column component page.

It is important to note that column diameter may also be needed for the column height design, especially when the rigorous method involves the calculation of tray efficiency from column diameters. In this case, the design will involve iterations between column diameter design and height design until all the design specifications are met. It is also important to select a proper tray spacing for the column diameter design because it is a very sensitive factor affecting column diameter. A proper tray spacing would be a value that minimize the entrainment on the tray. Obviously, the observation from the pilot scale experiment would provide direct and first-hand verification of the sufficient tray spacing.

Another important aspect of distillation design is the optimization of the distillation operation variables associated with the design, especially with respect to reflux ratio. As indicated previously, distillation is an energy intensive process and reflux ratio is an operation variable that sensitively affects column design (height and diameter) and reboiler and condenser duties. This means that the capital and operation costs of a distillation column can heavily hinge on the reflux ratio used for the design. As such, optimizing distillation design with respect to reflux ratio is an integral part of distillation design and process sustainability.

With the design specifications and a selected reflux ratio above minimum reflux ratio, the vapour flow in the column will be specified, then the column diameter can be designed based on the Fair’s method, and the number of stages required, or column height can be designed based on the short-cut methods and/or rigorous methods summarized above. It should be mentioned that the FUG short-cut equations in Table 4.1 can be very handy for the optimization because of its simplicity and numerical nature, but it is also important to validate the method before using it for column design, because both Fenske and Underwood equation need average volatility which is normally evaluated in the form of geometric or arithmetic average of distillate, bottom, or even feed, and a calculated average volatility may vary considerably for non-ideal systems. As well, with the determined vapour flow, the reboiler duties and condenser duties can be readily evaluated, and they can be used to design heat exchangers for reboiler and condenser as well as evaluate the operation cost. There are many methods to evaluate capital cost of distillation equipment once designed and Table 2 lists some annualized cost equations for the major cost components of a distillation column (Kaymak & Luyben). All these cost components can then be built into an objective function of total annualized cost to be solved numerically for optimum reflux ratio. Alternatively, the annualized capital cost, operation cost, and total cost can be plotted against reflux ratio, and the optimum reflux ratio can be determined graphically from the minima of the total cost, as shown in Figure 4.

Table 2. Samples of annualized cost equations for distillation equipment and operation. Refer to the original reference for symbols (Kaymak & Luyben).

Cost component	Capital cost and operating cost	Notes
Column	${C_{column}}(US\$ ) = (\dfrac{{M\& S}}{{280}}) \cdot 937.636 \cdot {D_c^{1.066}} \cdot {H_c^{0.802}} \cdot (2.18 + {F_{c)}}$ $H_c = (N_a - 1) \cdot T_s + 6$ ${F_c} = {F_p} + {F_m}$	$D_c$: column diameter (m) $H_c$: total column height (m) $T_s$: tray spacing (m) $N_a$: actual number of trays $F_c$: factor for pressure $(F_p)$ and material $(F_m)$.
Trays	${C_{tray}}(US\$ ) = (\dfrac{{M\& S}}{{280}}) \cdot 97.243 \cdot {D_c^{1.55}} \cdot {H_T}\cdot F_{c}$ ${H_T} = (N_a - 1) \cdot T_S$ ${F_c} = F_p + F_m$	$H_T$: total height of trays (m)
Heat exchangers	${C_{HE}}(US{\rm{\$ ) = (}}\dfrac{{M\& S}}{{280}}) \cdot 474.688 \cdot {A_o^{0.65}} \cdot (2.29 + {F_c})$ ${F_c} = ({F_d} + {F_p}){F_m}$	$A_o$: total heat transfer areas for the heat exchanger (m2)
Steam	${C_{steam}} = \left( \dfrac{Q_R}{\lambda_s \varepsilon_s} \right) \cdot C_s \cdot {AOH}$	$Q_R$: rebolier duty (J/h) $\lambda_s$: heat of vapourization of the steam (J/kg) $\varepsilon_s$: steam efficiency ( ~0.6) $C_s$: steam price (~ \$0.02/kg) $ AOH $: annual operation hours (~8000)
Colling water	$C_{steam} = \left( \dfrac{Q_c}{\Delta T_W \cdot C_p} \right) \cdot C_W \cdot AOH$	$Q_c$: condenser duty (J/h) $\Delta T_W$: cooling water temperature change $C_p$: heat capacity of cooling water ($J/kg \cdot K$) $C_W$: cooling water price (~\$0.03 /kg)

Figure 4. Distillation design optimization based on annualized cost analysis. | Description

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