Extension of the Equilibrium Stage Model to Include Rigorous Liquid Holdup Calculations for Reactive Distillation

Abstract

An extension of the equilibrium stage model to improve its applicability to reactive distillation is presented. The significant aeration of the liquid holdup on trays leads to the amount of clear liquid present being significantly less than the volume available. Tray hydraulic correlations are incorporated by leveraging the existing inside-out algorithm to rigorously calculate the liquid holdup on distillation trays, turning this parameter into an additional model output and eliminating the need to estimate this parameter beforehand. Application of this extended model shows that the aeration of the liquid holdup cannot be neglected for systems where the reaction kinetics limit the reactive productivity, and leads to column designs where additional reactive trays are needed to provide adequate reactive capacity. The workflow of this model provides a more robust path to obtaining reactive distillation column and tray designs that comply with liquid holdup requirements and tray hydraulic limitations.

1. Introduction

Reactive distillation (RD) is one of the most prominent examples of process intensification (PI) [1], a design philosophy driven by the need and desire to reduce the carbon footprint and increase the safety and profitability of the chemical processing industry. RD integrates the unit operations of reaction and separation into a single unit, potentially simplifying process flowsheets and reducing capital and operational expenditure [2,3]. By continuously removing one of the products from the reacting phase, an equilibrium limited reversible reaction can be partially or fully driven to completion (Le Chatelier’s principle). This in turn reduces or even removes the need to recycle in a process. RD can also be used to solve difficult separation problems, in which the reaction converts a component in the mixture which would otherwise be difficult to separate due to a low volatility difference, e.g., as the result of an azeotrope. Application of RD is not always technically feasible or economically desirable, however, as overlapping temperature and pressure conditions for reaction and separation (related to reaction kinetics, chemical equilibrium and physical equilibrium) are necessary for the integrated process option to outperform the conventional process design with dedicated units and optimized conditions for reaction and separation [4].

From a reaction engineering perspective, the liquid holdup is one of the key design parameters for accommodating reaction kinetics in an RD column. The liquid serves as the reacting phase (homogeneously catalyzed and uncatalyzed systems) or as the source of reactants through contacting with a solid catalyst (heterogeneously catalyzed systems). While a distillation column is strictly speaking a suboptimal choice for providing large liquid holdups (as this runs contrary to design constraints such as limited pressure drop), the combination of in situ reaction and separation has the potential to save costs to such a degree that an RD column becomes a preferable reactive vessel over a dedicated liquid phase reactor in some cases. For a model to be able to identify such opportunities, the modelling of the liquid holdup needs to be taken into account with sufficient accuracy to include its effect on the reaction and separation capacity of the RD column, while also adhering to the limits imposed by the hydraulics of the column internals to maintain separation efficiency. Specialized high liquid holdup tray design was one of the key features of the development of the Eastman methyl acetate process [5], which is the biggest success story of reactive distillation. It is surprising that since then, little has been published on further development of reactive tray design for high liquid holdup applications. This may be due to the esterification and etherification reactions that were investigated being good candidates for heterogeneous catalysis via reactive packings.

The modelling of RD is performed using the equilibrium (EQ) or non-equilibrium (NEQ) stage models. These models are covered extensively by Taylor and Krishna [2]. The NEQ stage model directly obtains the liquid holdup from tray hydraulic relations and is already available in commercial software, for instance in Aspen Plus. This model is more fundamentally sound in the description of mass transfer processes and their effects on chemical reactions, but also brings with it the requirement of providing additional parameters for mass transfer coefficients, interfacial areas and, when the Maxwell-Stefan theory is used, additional diffusion coefficients. While correlations for certain internals are available, the development and/or publishing of these correlations tends to lag behind the availability of new internals. Consequently, the majority of (initial) RD column design studies still rely on the EQ stage model. The addition of tray hydraulic relations to the EQ model has been explored before to more accurately describe the pressure profile throughout the RD column, thereby evaluating the reaction kinetics, chemical equilibrium and vapor–liquid equilibrium at their correct pressure and temperature [6]. The weir height used to evaluate the hydraulics of the trays are also used to calculate the liquid holdup calculation in this work. This can be contrasted to the workflow of commercial flow sheeting software implementations of the EQ model (e.g., Aspen Plus), where the liquid holdup is specified as a constant value per stage to be estimated separately by the user, which may happen independently of considering the tray weir height, diameter and other tray parameters. Liquid holdup (for defining a tray reaction rate) is often estimated by taking the geometric volume available on a tray (bubbling area x weir height) [6,7,8,9]. However, correlations used in conventional distillation predict the liquid fraction of the froth (clear liquid height) to be only 10–40% of the weir height under typical distillation conditions, depending on tray type, design and vapor and liquid traffic on the trays [10]. Basing the liquid holdup estimate on the geometrical volume on the tray has two clear shortcomings: (1) the amount of liquid holdup is overestimated as there is a significant volumetric vapor fraction in the froth and this may lead to column designs that cannot produce the desired conversion capacity when built, (2) the effect of the nonuniform distribution of the liquid holdup that sets in on a uniform set of reactive trays on the performance of the column cannot be assumed to be trivial. Not including this feature in an RD model may at best lead to missing an opportunity for process optimization and at worst result in an underperforming RD column. Adding this consideration by external means to the EQ model will require iterative calculation procedures as the bubbling area will depend on the calculated column profiles (through the vapor velocity) which in turn depend on the liquid holdup, for which you need to know a bubbling area to make an estimate. This has been achieved, for instance, by coupling CFD tools to Aspen Plus [11]. However, such an approach can be expected to be prohibitive for optimization due to the computational load.

In this work, we therefore propose a generalized extension of the EQ model to include the prediction of stage liquid holdups from tray hydraulic correlations for computation of the stage reaction rates. The tray type and dimensions replace the liquid holdup as model input, and the non-uniformity of the liquid holdup is accounted for by resolving the tray hydraulics on each tray using the local compositions, temperatures, and flow rates. The liquid holdup then becomes an additional output of the model. This formulation enables the evaluation of the hydraulic stability criteria of the tray design without the need for an additional iterative loop to match the liquid holdup input of the conventional distillation model to the calculated liquid holdup from the post-processing of converged simulations. In the following sections of this article, we first provide the equations that are added to the conventional (reactive) distillation model to rigorously calculate the liquid holdup, together with the structure of the inside-out algorithm that incorporates these holdup equations. Ideally, the improved predictive capabilities of this model would be shown by application to an experimental data set. A literature survey, however, did not yield any suitable data sets for which a liquid phase reaction was performed in an RD column where trays were used as reactive internals and the dimensions of these trays were also reported. Instead, the results of applying this model to the column design of Luyben [12] are shown to showcase how this method applies to the evaluation of the column and tray design.

2. Model Formulation

The steady state equilibrium (referring to the phase equilibrium only) stage model used in this work follows the same formulation as that of Naphtali and Sandholm [13]. This formulation uses algebraic mass balances, heat balances, vapor–liquid equilibrium relations and implicit molar summation relations to describe the exchange of mass and energy at each stage, collectively known as the MESH equations (Equations (1)–(3)). A schematic overview of this model with the conventional method of including reaction kinetics is shown in Figure 1a.

$$M_{i,j}=0=\left(1+U_j\right)v_{i,j}+\left(1+W_j\right)l_{i,j}-v_{i,j+1}-l_{i,j-1}-f_{i,j}-M_{hold,j}{R}_{i,j}$$

(1)

$$E_{i,j}=0=K_{i,j}{l}_{i,j}\frac{V_j}{L_j}-v_{i,j}$$

(2)

$$H_j=0=\left(1+U_j\right)h_j^V{V}_j+\left(1+W_j\right)h_j^L{L}_j-h_{j+1}^V{V}_{j+1}-h_{j+1}^L{L}_{j-1}-h_j^F{F}_j-Q_j$$

(3)

Here index i refers to the component (1 to N_C) and j to the stage number (1 to N_stages, top to bottom), v_i,j and l_i,j are the vapor and liquid component flowrates (mol/s), V_j and L_j are the total vapor and liquid flowrates (mol/s), F_j is the feed flow rate (mol/s), K_i,j is the vapor–liquid partitioning coefficient, h_j is the mixture enthalpy (J/mol), $Q_j$ is heat duty (J/s), U_j and W_j are the vapor and liquid side-draw factors, $M_{hold,j}$ is the liquid holdup (m³) and $R_{i,j}$ is the overall component reaction rate (mol/m³ s). The heat of the reaction is accounted for implicitly in Equation (3) by considering the component enthalpies from their reference state. Note that this formulation does not use the molar summation (S) equations as the molar fractions are eliminated by using component molar flow rates.

The holdup term appearing in Equation (1) is normally assigned a constant value for each equilibrium stage in commercial flow sheeting software. Here, we introduce a methodology for calculating the holdup rigorously as a function of the tray geometry, temperature, flow rates and compositions (Figure 1b):

$$M_{hold}^{exact}=f(\overline{geo},T,P,L,V,x,y)$$

(4)

where $\overline{geo}$ is the set of tray dimensions that is relevant for the used holdup correlation e.g., weir height, weir length, bubbling area etc., $T$ is the temperature (K), $P$ is the pressure (Pa), x and y are the liquid and vapor mole fractions.

The exact form of Equation (4) depends on the type of internal considered. Using sieve trays as an example, the tray liquid holdup can be determined as the product of clear liquid height and bubbling area:

$$M_{hold}=A_b\times h_c$$

(5)

where $A_b$ is the tray bubbling area (m²) and $h_c$ is the clear liquid height (m). The clear liquid height takes into account the aeration of the froth held by the weir and the liquid height flowing over the weir using the Francis weir equation;

$$h_c={\Phi }_e\left[h_w+C_l{\left(\frac{Q_L}{L_w{\Phi }_e}\right)}^{2/3}\right]$$

(6)

where $h_w$ is the weir height (m), $C_l$ is a Francis weir coefficient, $Q_L$ is the liquid load over the weir (m³/s) and $L_w$ is the weir length (m). The effective froth density ${\Phi }_e$ accounts for the aeration of the froth. Several empirical models are available for sieve trays, notably Colwell’s [14] and Agrawal, Bennett and Cook’s (ABC) [15] methods, both containing an inverse relation between ${\Phi }_e$ and the superficial gas velocity $U_a$. The ABC method is given as:

$${\Phi }_e=e^{-C_4{K}_S}$$

(7)

$$K_S=U_a{\left(\frac{{\rho }_V}{{\rho }_L-{\rho }_V}\right)}^{C_5}$$

(8)

where $C_4$ and $C_5$ are correlation specific coefficients. $U_a$ is the vapor velocity through the bubbling area (m/s) and ${\rho }_V$, ${\rho }_L$ are vapor and liquid densities, respectively, (kg/m³). Equations (5)–(8) indicate the dependencies of Equation (4) on the tray geometry (through A_b, h_w and L_w), stage temperature (through ${\rho }_V$ and ${\rho }_L$, depending on the thermodynamic models used), liquid flow rate (through $q_L$) and the vapor flow rate (through $U_a=\frac{V}{{\rho }_m^V}/A_b$). Incorporating such a liquid holdup model directly may cause the mass balance Equation (1) to become excessively non-linear, making convergence difficult. To alleviate this, the inside-out method [16] is applied to use a simplified form of Equation (4) for each equilibrium stage when solving the MESH equations. The froth density ${\Phi }_e$, via $U_a$, is the term that varies most strongly between iterations of the solution of the MESH equations. Therefore, the functional form of Equation (9) is chosen to serve as the ‘simple’ correlation to be used when solving the MESH equations and assuming a fixed tray geometry with $a_j$ and $b_j$ finetuned for each iteration.

$$M_{hold,j}={A_b\times e}^{a_j-b_j\times V_j}$$

(9)

This reduction in parameters is supported by a sensitivity analysis shown in Appendix A.

Parameter a_j will be such that $e^{a_j}$ is of the same order of magnitude as the weir height. Parameter b_j contains the sensitivity of the clear liquid height to the vapor flow rate and at the same time absorbs the smaller effects of temperature, L/V and the fluid densities between iterations. The sign of b_j will be positive for trays (inverse relationship of clear liquid height and vapor flow rate) and negative for packings (direct relationship between clear liquid height and vapor flow rate) [17].

The use of Equation (9) as the liquid holdup relation makes the proposed model flexible in accepting user-specified correlations for clear liquid height, without having to provide specific analytical derivatives for a solution by the Newton–Raphson method. Stage specific values for the constants $a_j,b_j$ are obtained by perturbation of $V_j$ in the outer loop of the inside-out method. A schematic overview of this extended equilibrium stage model is given in Figure 2. The simplified models for the K-values and excess enthalpy are also shown and are part of the original implementations of the inside-out method. The solution of the MESH equations in the inner loop is achieved with the Newton–Raphson method once the sum of squares of the function vector becomes less than the tolerance (N_stages × 10⁻¹⁰):

$${\left|\stackrel{-}{F}\right|}^2\le {eps}_{inner}$$

(10)

where ${\left|\stackrel{-}{F}\right|}^2$ is the sum of the squared residuals of the entire vector of equations for the column. Convergence of the outer loop is achieved by direct substitution of the simple correlation parameters until the relative change in the holdup and K-values between iterations falls below the tolerance (1 × 10⁻⁸):

$${\left|\stackrel{-}{SR}\right|}^2={\sum }_{j=1}^{N_{stages}}\left[\frac{{\left(M_{hold,j}^{k+1}-M_{hold,j}^k\right)}^2}{M_{hold,j}^k}\right]+{\sum }_{j=1}^{N_{stages}}{\sum }_{i=1}^{N_c}\left[\frac{{\left(K_{i,j}^{k+1}-K_{i,j}^k\right)}^2}{K_{i,j}^k}\right]\le {eps}_{outer}$$

(11)

where ${\left|\overline{SR}\right|}^2$ is the sum of the squared residuals of the stage holdups and vapor–liquid distribution coefficients. The residuals are calculated over the current (k+1) values, obtained by applying the rigorous correlations with the latest converged T, L, V, x, y profiles from the converged inner loop, and the previous (k) values, obtained from the simple correlations with the set of constants that went into the preceding inner loop.

The described equilibrium stage model, together with the relevant physical property methods, was implemented in MATLAB as a custom standalone RD column model. The MESH equations of the inner loop are solved with a custom implementation of the Newton–Raphson method for nonlinear equations.

3. Results and Discussion

The proposed model inherently deals with some of the assumptions that are made regarding the liquid holdup on trays and provides a straightforward path to evaluating realistic tray designs as well. Luyben formulated a column design for the generic quaternary liquid phase reaction $A+B\leftrightarrow C+D$ and investigated the effects of various parameters, including the liquid holdup [12]. This case will be used in this work to show the differences in several design aspects of the column and trays when using both models. This reference case consists of an RD column with five stripping, five rectifying and nine reactive stages. The liquid phase reaction kinetics on a reactive tray are described by Equation (12):

$$R_A=M_{hold,m}\left(k_f{x}_A{x}_B-k_b{x}_C{x}_D\right)$$

(12)

where $M_{hold,m}$ is the tray liquid holdup in mol, henceforth obtained by multiplying $M_{hold}$ by the molar liquid density.

The physiochemical and column design parameters for this case are given in

Table 1. Physiochemical and design parameters of the reference case, parameter values adapted from Luyben [12]. *Added in this work for the tray hydraulic correlations.

Parameter	Value
ΔvapH (KJ/mol)	29
kf, kb (mol/molholdup∙S)     $k=k_0\mathit{exp}\left(\frac{-E_A}{RT}\right)$	forward   backward $k_{0,f}$: 7.6559 $\times$ 1015  $k_{0,b}$: 3.7722 $\times$ 1021 $E_{A,f}:126$   $E_{A,b}:168$
MW (G/MOL)	50
P (bar)	8
Vapor pressure constants     $ln\left(P^{sat}\left(bar\right)\right)=A_j-\frac{B_j}{T\left(K\right)}$	Aj     Bj
A	12.34     3862
B	11.65     3862
C	13.0     3862
D	10.96     3862
αAD, αBD, αCD	4, 2, 8
ρL (kg/m3)	800
ρV (kg/m3)	ideal gas law
γ (mN/m) *	30
μ (cP) *	0.5
Nstages	19 (21 with total condenser and reboiler)
Nrx	9
Nrect, Nstrip	5
Mhold,m	1000 (per reactive tray)
hc	0.12 m
Dc	0.8 m

.

Luyben arrives at a value of 1000 mol for the tray liquid holdup, based on a column diameter of 0.8 m and an assumed (clear) liquid height of 0.12 m, together with a liquid density of 800 kg/m³ and a molecular weight of 50 g/mol for each of the compounds:

$$M_{hold,m}=\frac{h_c\times \frac{1}{4}\pi D_c^2\times {\rho }_L}{M_w}=\frac{0.12m\times \frac{1}{4}\pi \times {\left(0.8 \mathrm{m}\right)}^2\times 800 \mathrm{kg}/{\mathrm{m}}^3}{0.05\mathrm{kg}/\mathrm{mol}}=965 \mathrm{mol}\approx 1000 \mathrm{mol}$$

(13)

It is unclear how the downcomer area is considered in this case, however, the clear liquid in the downcomer backup (the height of the froth in the downcomer) will be at least twice the tray clear liquid height and the downcomer area is typically around 10% of the tray area (per downcomer, so 20% in total). Taking the tray holdup to be the product of the column cross sectional area and clear liquid height is then a reasonable approximation in this context. The following calculations have been performed with this same assumption, as the feature of interest here is the inclusion of the froth density in the liquid holdup model and not so much the extent to which the downcomer backup is modelled.

Assuming the clear liquid height to be equal to the weir height of the tray significantly overestimates the amount of unaerated liquid present on trays. The difference between this assumption and the clear liquid height calculated by pressure-drop correlations, taking into account the aeration of the liquid, for sieve trays is shown in Figure 3a. The reflux ratio required to maintain the 95% product purity for the reference case increases from 2.65 to 2.78 based on our calculations, as seen in Figure 3b. This increase is necessary to compensate the lower clear liquid height predicted by the rigorous liquid holdup model. The intrinsic kinetics (equivalent to roughly 0.9 h residence time in a CSTR to approach 95% of chemical equilibrium) are sufficiently fast to enable the designed column to meet the required productivity with a limited reflux ration increase, despite the lower than anticipated liquid holdup.

It is anticipated that more significant differences in column design will arise when the kinetics become slower. To test this, the number of reactive stages at the minimized reflux ratio was determined with and without considering the froth density (keeping all other parameters at their original values). This was evaluated at slower kinetics relative to the reference case, shown in Figure 4. Unlike for conventional distillation (where the minimal efflux ratio occurs at an infinite number of distillation stages), for RD the reflux ratio reaches a minimum at a finite number of reactive stages, since the total consumption of either reactant before reaching the feed stage of the other reactant is detrimental to the conversion process. This has also been shown for this column by Luyben [12]. These results were obtained by fixing the number of rectifying and stripping stages (five each and also a total condenser and reboiler) and varying the number of reactive stages until the minimum in the reflux ratio to obtain 95% product purity was obtained. It is seen in Figure 4 that as the relative reaction rate decreases, the difference in the number of reactive stages at this minimized reflux becomes larger. For a forward reaction rate (k_f) of 20% of that for the base case (k_f,0), at an unchanged equilibrium constant, more than twice the number of reactive stages would be required for the liquid holdup to be considered properly (according to our methodology), indicating the importance of accounting for the hydraulics, and resulting gas and liquid holdup in that case. The total (reactive) liquid holdup converges towards the same value for both models at k_f/k_f,0 = 0.2, indicating that the column design here is strongly sensitive to the amount of liquid holdup available, whereas the ratio of the two liquid holdups is a factor 2 for the base case kinetics, shown in Figure 4c. This indicates that the productivity was indeed not limited by the kinetics at the reference case values but is becoming limited towards k_f/k_f,0 = 0.2. It can be concluded that it is important to consider the froth density of the liquid holdup, especially at slower kinetics.

Determining the optimal number of (reactive) stages and reflux would require the use of a cost function and an iterative optimization loop built around the presented model, which is outside the scope of this discussion. However, the differences shown for the number of reactive stages at the minimized reflux ratio give a first indication of how optimized column designs diverge between the two models towards slower kinetics.

3.1. Feasibility of Tray Designs

Another aspect of the tray design is considering whether stable operation can be achieved from a hydraulics perspective. For this model system, achieving the assumed liquid height of 0.12 m (on average throughout the reactive section) would require a weir height of 0.275 m on sieve trays (see Figure 3b). This weir height is higher than normally encountered in distillation and will require a tray spacing that is also higher than the typically used 400–600 mm [10] to accommodate the tray hydraulics. The non-reactive trays have no liquid holdup requirements and can easily be designed at a weir height that satisfy the entrainment and weeping criteria for stable operation. Figure 5 shows the criteria for stable tray operation i.e., the downcomer backup, fractional entrainment and weeping for this column (calculated using available literature correlations [10,18,19], see Appendix B). When the clear liquid height is increased on a given tray, both the downcomers connecting it to the previous and the following trays need to be increased in height to accommodate the increase in downcomer backup, as dictated by the pressure balance around the downcomers (see Appendix B). Hence, an increase in clear liquid height on a tray is paid for twice in the height of the column shell. The fractional weeping will always exceed the limit of 0.1 [20] for this column when the weir height is higher than 0.1 m, as seen in Figure 5b. The weeping can be counteracted somewhat by increasing the vapor pressure drop through altering the tray design i.e., lowering the tray open area, reducing the hole size, or increasing the tray thickness. Alternatively, a different tray type i.e., valve or bubble-cap can be considered. Nonetheless, the pressure-drop contributions associated with these tray designs will still lead to the downcomer backup to exceed the typical 400–600 mm tray spacing known from conventional distillation when trying to provide 0.12 m of clear liquid height. This evaluation of the tray hydraulics shows that assigning an estimated value to the liquid holdup opens up the possibility of arriving at a tray design that is not possible from a hydraulics perspective when applying typical tray design values from conventional distillation. From a workflow point of view, it would make sense then that the RD model uses the dimensions of the internals directly as an input to calculate a corresponding liquid holdup rather than requiring an estimated value for the holdup. This avoids the need for an external iteration loop to match a tray design to an initial holdup input.

3.2. Algorithm Features

The choice to adopt the inside-out algorithm was made to include tray hydraulic correlations in the solution of the MESH equations. Here, we show how the liquid holdup calculations influence the solution procedure with calculation examples and briefly discuss the reasoning for this choice compared to existing implementations using an equation-oriented approach [6]. The MESH equations for steady-state distillation modelling form a set of nonlinear algebraic equations that are typically solved by Newton–Raphson methods. In the inside-out algorithm, this is applicable for the inner loop, where the outer loop convergence can be achieved with direct substitution of the new values, although Newton–Raphson methods are also applicable if the system has highly nonlinear properties.

Table 2. Convergence characteristics of several test cases. All cases are for Luyben’s column with real component properties for cases 3 and 4. T-dependent properties are taken as ideal component properties of the methyl acetate system (A = methanol, B = acetic acid, C = methyl acetate, D = water), Antoine and enthalpy (C_p, ${∆}_{f}H$) parameters and correlation coefficients were retrieved from the Aspen Plus databank. The number of values given for the iterations represents the number of outer loop iterations.

Test Case	Inner Loop Iterations	Mesh Evaluations	Time to Solve (s)
Constant properties (relative volatility, enthalpy) + constant liquid holdup	16	23	0.7
Constant properties (relative volatility, enthalpy) + rigorous liquid holdup	13, 2, 1	22	0.7
T dependent properties (relative volatility, enthalpy) + constant liquid holdup	24, 45, 10, 17, 1	143	1.4
T dependent properties (relative volatility, enthalpy) + rigorous liquid holdup	27, 13, 10, 16, 18, 1	113	1.4

summarizes the convergence characteristics of several test cases to show how the solution procedure is affected by the incremental addition of nonlinear properties. The inner loop iterations are the solution of the MESH equations, the outer loop iterations (given by the number of values for the inner loop iterations) resolve the rigorous physical property and holdup models.

Comparing these four cases, it can be concluded that the majority of the nonlinearity comes from including T-dependent component properties, which describe the vapor–liquid equilibrium and enthalpy. The rigorous holdup calculations add a few outer loop iterations; however, the converged results from previous inner loops are already close to the final result, as evidenced by the low number of subsequent inner loop iterations for case 2.

Newton–Raphson methods require the derivatives of the MESH equations to the temperature, flow rates and compositions. These derivatives can be obtained either numerically or analytically, with analytical derivatives being preferred for better convergence and computational speed (see Appendix C), especially in the context of optimization where many designs of the same RD column are evaluated. Analytical derivatives for tray holdup correlations are cumbersome to obtain; we give an example of such a derivative in Appendix C. The established inside-out algorithm in this work only samples the rigorous holdup correlations in the outside loop and does not require the derivatives of these correlations. The simplified form of the holdup correlation that is used while solving the MESH equations has a straightforward derivative that maintains its shape regardless of the holdup correlation used (provided the main dependence is the vapor velocity). In light of these considerations, we make the following recommendations; if the target application is the design/optimization of a single case, then the equation-oriented approach is the most straightforward to implement with a manageable penalty of calculation time. For a general simulation tool that handles multiple holdup correlations and is to be used for design/optimization of multiple cases, the presented inside-out structure has a higher upfront development time cost but lower calculation times afterwards. As it stands, this model accounts for the liquid holdup on the bubbling area of the reactive trays, but does not yet account for the liquid holdup in the downcomer. The model therefore gives a conservative estimate of the reactive capacity on the trays. A further improvement could be made by modelling the downcomer as a plug flow reactor using a n-CSTR in series approach. This would involve adding n-repetitions of the mass and energy balances (Equations (1) and (3), ignoring the vapor terms) in between the sets of tray equations. Such an approach would require the availability of a correlation that gives the (de)aeration of the froth flowing through the downcomer. This extension has only been tested for the equilibrium stage model. The presented algorithm should be applicable to the non-equilibrium stage model, however, the convergence behavior for such a combination needs to be investigated.

4. Conclusions

In this work, we present an extension of the equilibrium stage model to include rigorous calculation of the tray liquid holdup. The inside-out method, which was originally formulated for rapid and stable thermodynamic calculations in distillation models, was utilized to incorporate holdup calculations in a generalized manner. This formulation replaces the stage independent holdup/residence time as a necessary model input with the tray geometry and turns the holdup into an additional model output. This extension leads to a more realistic and more accurate model description of the liquid and gas holdups and consequently to a more reliable description of the reactant conversion and product purities as a function of the number of (reactive) stages and reflux ratio. The clear liquid height that is calculated with this newly developed model is significantly lower than the clear liquid height under the assumption that it equals the weir height. Using the realistic clear liquid height assures proper representation of the RD column performance, avoiding overprediction, such as too high conversion or product quality, which could be predicted under the assumption that the clear liquid height is equal to the weir height. The difference between the design methods is most significant when the reaction kinetics are slow and limit the reactive productivity. In those cases, it can be the difference between a performing or underperforming design. For cases where the reaction kinetics are less limiting, a change in reflux ratio may be sufficient to remediate the lower-than-expected conversion level, provided that this higher reflux ratio can be met within constraints of the original column design. Additionally, in contrast to the methodology, assuming that the clear liquid height is equal to the weir height, incorporation of the rigorous tray holdup calculation directly shows whether the obtained liquid holdup (as required for the intended conversion and product purity) leads to a feasible tray design and fulfils weeping and flooding criteria. The developed model thus provides a more robust workflow towards obtaining a realistic RD column design, where the actual, stage dependent, liquid holdup is taken into account for the tray reaction rate and tray hydraulics.