Figure 11.5. Multi-Pass Sequential Steady State (MPSSS) operation. [Mujtaba, 1997]d
Figure 11.5 illustrates the operating sequence for a binary mixture. Successive passes (multi-pass) are used sequentially using the same column to separate only component. The purity of the distillate product remains the same for each pass but the distillate rate varies (this in turn varies the reflux ratio). This strategy is similar to a time sequenced reflux ratio operation for individual cuts in CBD operation. The total recovery of a component (say A) is calculated from the accumulated amount of distillate from all the passes.
11.2.3. Single Separation Duty in Continuous Columns 126.96.36.199. Performance Measure
Mujtaba (1997) used the maximum distillate problem to compare the performances of the two types of distillation columns (CBD and continuous). With the amount of initial charge and the feed flow rate fixed in a continuous column, the operation time (pass time) also becomes fixed. The performance measure using maximum distillate problem allows fixing of the operation time. Other types of optimisation problems such as minimum time or maximum profit problems (presented in the previous chapters) are not suitable for the purpose of comparing the performances of these two types of distillation columns as the operation time in these optimisation problems is not fixed but is relaxed and optimised.
Also Mujtaba (1997) considered the separation of binary mixtures into one distillate product of specified purity. The objectives were to find out whether it was possible to replace conventional dynamic operation of batch columns by steady state operation using continuous columns for a comparable recovery, energy consumption, operation time, productivity, etc. and to obtain optimal operating policy in terms of reflux ratio. The following strategy was considered to compare the performances of the two types of operations:
Given an initial feed mixture, say Bs (kmol)-
(a) choose a feed flow rate (say Fh kmol/hr) for the steady state operation using a continuous column.
(b) determine the operation time (?/ = Bj/Fi) for the continuous column
(c) for the operation time tj, obtain maximum achievable amount of distillate Dj of specified purity using (i) a CBD column (Figure 11.1) and (ii) a continuous column (Figure 11.2).
(d) compare the performances in terms of total recovery, energy consumption,
188.8.131.52. Optimisation Problem Formulation and Solution
The optimisation problem for a single cut (in CBD) or for a single pass (in a continuous column) can be stated as follows:
the column configuration (including feed location for continuous column), the feed mixture, vapour boilup rate, feed rate (for continuous column), cut time (or pass time) and a separation task (e.g. obtain product of specified purity)
the optimal reflux ratio profile for the operation so as to maximise: the amount of distillate product subject to:
equality and inequality constraints.
Mathematically the problem (problem OP) can be written as:
OP Max Dp
R,ts s.t. Model Equations (equality constraints)
XD ^ (inequality constraints)
Linear bounds on reflux ratio (inequality constraints)
where, R is the reflux ratio value, Dp is the total amount of distillate produced (kmol) over the operation time tp and xkD is the purity of distillate achieved
(molefraction) and xkD is the specified product purity for component k. ts [0, tp] is the switching time from one reflux ratio value to another in a single cut multiple reflux ratio operation (explained in detail in the example section) using CBD operation. For a single cut single reflux ratio operation in CBD or in continuous column operation (SPSS or MPSSS) the cut time or the pass time is equal to tp and therefore there is no switching time to be optimised.
For CBD operation, the problem presented above results in a non-linear dynamic optimisation problem, which is solved using the technique in Mujtaba and Macchietto (1993, 1996) as outlined in earlier chapters. For continuous column operation the problem OP results in a non-linear steady state optimisation problem which is solved using the computer software SRQPDV1.1 due to Chen (1988).
Piecewise constant reflux ratio levels (which are optimised) are assumed with a finite number of intervals (or passes) over the total time period concerned for CBD and continuous column operations respectively. Figure 11.6 illustrates the computation sequence for dynamic optimisation of CBD operation for a cut or steady state optimisation of continuous column operation for a pass using one reflux ratio level to be optimised within the specified time period of tp. For each iteration of the OPTIMISER, dynamic optimisation requires full integration of the model equations from t = 0 to t = tp. This requires the solution of the DAEs at each integration step (the number of steps depends on the type of numerical method employed). On the other hand, for continuous column operation, each iteration of the OPTIMISER requires the solution of AEs only once. This explains why the solution of dynamic optimisation problem is computationally much more expensive than that of steady state optimisation problem (as already mentioned in the earlier section). Mujtaba (1997) used an efficient DAE integrator based on Gear (1971) and Morison (1984) which uses variable step length to minimise the number of integration steps.
Both the steady state and dynamic column models (for CBD only) used by Mujtaba (1997) are based on the assumptions of constant relative volatility and equimolal overflow and include detailed plate-to-plate calculations. This will allow a direct comparison between CBD and continuous column operation. The continuous column model is presented in section 4.3.1 and the CBD model (Type III) is presented in section 4.2.3. Some of the modelling assumptions, for example, constant molar holdup, constant pressure, equimolal overflow, etc., can be relaxed, if needed, by replacing them with more realistic assumptions and therefore by adding the relevant equations (as presented in Chapter 4).
The case study is taken from Mujtaba (1997). Here, a simple binary mixture is considered. The separation task is defined as: Obtain distillate product with purity of 0.90 molefraction in component 1. The column configuration and input data are given in Table 11.1.
Table 11.1. Column Configuration and Input Data. [Adopted from Mujtaba, 1997]
Number of plates, N=20 Feed location, NF = bottom stage
No. of components, nc = 2 Total feed, Bh kmol = 10.0
Initial feed composition, xB1 = <0.6,0.4> Relative volatility = <2.0, 1.0>
Vapour boilup rate, V, kmol/hr = 5.0 xD , molefraction = 0.90
Feed rate, Fp, kmol/hr = variable
For pass 1: feed composition, xF1 = xBj
To allow a direct comparison with CBD operation, the feed is introduced at the bottom of the column so that a same number of stages (in the rectification section) are available in both cases. Therefore, the model equations given in SS.9 and SS.10 (Chapter 4, section 4.3.1) will have the following form:
The variable L' and therefore the equation SS.6 (presented in Chapter 6) will be redundant. Note the variables are defined in Chapter 4.
The steady state optimisation problem is solved for different feed flow rates. The maximum achievable distillate rate, optimum reflux ratio (internal), total amount of distillate, pass time and recovery of key component (e.g. component 1) for the first pass are summarised in Table 11.2. For any pass p, the pass time (tp, hr), total amount of distillate (Dp, kmol) and recovery of key component (Rep) are calculated using:
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