Reboiler dynamics

In this chapter we will derive the dynamic relationships between internal reflux and both vapor rate and external reflux (or feed) as function of tray and column design. The basic tray hydraulic equations are based on the treatment by Van Winkle. First we discuss what happens on an individual tray, and then derive an approximate model for a combination of trays. Vapor flow will be assumed to occur without lags, and heat-storage effects will be assumed to be negligible. A discussion of reboiler dynamics will be deferred to Chapter 15. 

If we are not limited in column-base holdup and can design for reasonably well-damped control, then we can treat reboiler dynamics as ne gible. This says that steam flow responds to the flow controller set point immediately, and that boilup follows steam flow without lag. We may Aen prepare the signal flow diagram of Figure 16.7. Note that = steam flow-meter gain = 12/(Wsr)max- This may be partially reduced to the form of Figure 16.8. From this last illustration we can see some of the loop s characteristics as they ate afiected by reboiler swell and inverse response. 

For tight pressure contol, we should use these models with caution. Most of the tight column pressure controls we have studied have closed-loop resonant frequencies in the range of 0.5-2 cpm. For the upper value one should make at least a rov h check of condenser and reboiler dynamics. It may be of interest that the only applications of tight pressure control we have found are in heat-recovery schemes where the vapor from one column serves as the heating medium for the reboiler of another column, and perhaps furnishes heat to other loads. If the vapor flow must be throttled to each load, constant up- and downstream pressures help good flow control. 

If we treat reboiler dynamics as negligible, and if we assume that steam will be flow or flow-ratio controlled, we may combine the above equations as shown in the preliminary signal flow digram of Figure 17.2. This, in turn, may be reduced to the form of Figure 17.3. 

For the case where the steam valve is manipulated by some variable other than flow or flow ratio, we may need to account for reboiler dynamics to calculate qr- Referring to Figure 15.8, we may make a partial s al flow diagram as shown on Figure 17.4 where ... 

Partial signal flow diagram for reboiler dynamics... 

For columns where the plate dynamics are significantly faster than the reboiler dynamics (due to very small plate holdups and/or wide boiling components), the stiff integrator often fails to find a solution. The solution to this problem is to split the system into two levels (a) the reboiler, where the dynamics are slower, can be represented by differential equations (Equations 4.12-4.13), and (b) the rest of the column can be assumed to be in the quasi-steady state. Thus, the composition changes in the condenser and accumulator (dx /dt), the composition changes on plates (dx j /dt), and the enthalpy changes in the condenser and on plates (Stlo... 

Finally, we will neglect the dynamics of the condenser and the reboiler. In commercial-scale colunms, the dynamic response of these heat exchangers is usually much faster than the response of the column itself In some systems, however, the dynamics of this peripheral equipment are important and must be included in the model... 

Coolant and steam dynamics are negligible in the condenser and reboiler. 

For example, it is important to have large enough holdups in surge vessels, reflux drums, column bases, etc., to provide effective damping of disturbances (a much-used rule of thumb is 5 to 10 minutes). A sufficient excess of heat transfer area must be available in reboilers, condensers, cooling Jackets, etc., to be able to handle the dynamic changes and upsets during operation. The same is true of flow rates of manipulated variables. Measurements and sensors should be located so that they can be used for eflcctive control. 

We would like to compare the closedloop dynamic performance of two types of reboilers. 

Changes in steam flow are achieved by increasing or decreasing the area used for condensing steam in the reboiler. This variable-area flooded reboiler is used in some processes because it permits the use of lower-pressure steam. However, as you will show in your calculations (I hope), the dynamic performance of this configuration is distinctly poorer than direct manipulation of steam flow. 

There is a first-order dynamic lag of t minutes between a change in the signal to the steam valve and vapor boilup. The low base-level override controller pinches the reboiler steam valve over the lower 25 percent of the level transmitter span. 

Figure 11.5a shows a typical implementation of feedforward controller. A distillation column provides the specific example. Steam flow to the reboiler is ratioed to the feed flow rate. The feedforward controller gain is set in the ratio device. The dynamic elements of the feedforward controller are provided by the lead-lag unit. 

Under the above modeling assumptions, the dynamic model of the reactor-column-recycle system consists of the material balance for the total molar holdup of the reactor, condenser, and reboiler, and component-wise balances for the reactant A and product Pi in the reactor, condenser, reboiler, and column trays, having a total of 2N + 9 differential equations. Specifically,... 

Following a similar procedure to the one employed above, it is easy to verify that we obtain a model that approximates the fast dynamics of the system in Figure 4.2, in the form of Equation (4.20). Also, it can be verified that only 2N + 8 of the 2N + 9 steady-state constraints that correspond to the fast dynamics are independent. After controlling the reactor holdup Mr, the distillate holdup Md, and the reboiler holdup MB with proportional controllers using respectively F, D, and B as manipulated inputs, the matrix Lb (x) is nonsingular, and hence the coordinate change... 

Subsequently, we used Aspen Dynamics for time-domain simulations. A basic control system was implemented with the sole purpose of stabilizing the (open-loop unstable) column dynamics. Specifically, the liquid levels in the reboiler and condenser are controlled using, respectively, the bottoms product flow rate and the distillate flow rate and two proportional controllers, while the total pressure in the column is controlled with the condenser heat duty and a PI controller (Figure 7.4). A controller for product purity was not implemented. 

Dynamic simulations were aimed at capturing the multiple-time-scale behavior revealed by the theoretical developments presented above. Figures 7.5 and 7.6 show the evolution of the mole fraction of n-butane and of the temperature on selected column stages for a small step change in the reboiler duty. Visual inspection of the plots indicates that the temperatures exhibit a fast transient,... 

Unlike continuous distillation, batch distillation is inherently an unsteady state process. Dynamics in continuous distillation are usually in the form of relatively small upsets from steady state operation, whereas in batch distillation individual species can completely disappear from the column, first from the reboiler (in the case of CBD columns) and then from the entire column. Therefore the model describing a batch column is always dynamic in nature and results in a system of Ordinary Differential Equations (ODEs) or a coupled system of Differential and Algebraic Equations (DAEs) (model types III, IV and V). 

The optimal control of a process can be defined as a control sequence in time, which when applied to the process over a specified control interval, will cause it to operate in some optimal manner. The criterion for optimality is defined in terms of an objective function and constraints and the process is characterised by a dynamic model. The optimality criterion in batch distillation may have a number of forms, maximising a profit function, maximising the amount of product, minimising the batch time, etc. subject to any constraints on the system. The most common constraints in batch distillation are on the amount and on the purity of the product at the end of the process or at some intermediate point in time. The most common control variable of the process is the reflux ratio for a conventional column and reboil ratio for an inverted column and both for an MVC column. 

Dynamic optimisation of this type of periodic operation was first attempted and reported in the literature by Mayur et al. (1970), who considered the initial charge to the reboiler as a fresh feed stock mixed with the recycled off-cut material from the previous distillation task. Each batch cycle is then operated in two distillation tasks. During the Task 1, a quantity of overhead distillate meeting the light product specification is collected. The residue is further distilled off in Task 2 until it meets the bottom product specification. The overhead during Task 2 meets neither specifications (but the composition is usually kept close to the that of the initial charge for thermodynamic reasons) and is recycled as part of the charge for the next batch. As the batch cycle is repeated a quasi-steady state mode of operation is attained which is characterised by the identical amount and composition of the recycle (from the previous batch) and the off-cut (from the current batch). Luyben (1988) indicates that the quasi-steady state mode is achieved after three or four such cycles. 

Referring to Figure 8.2 and given a batch charge (BO, xB0)> a desired amount of distillate DI of specified purity x D1 and final bottom product B2 of specified purity x b2 Mujtaba (1989) determined the amount and composition of the off-cut (Rl, x R1) and the reflux rate policy r(t) which minimised the overall distillation time. In this formulation instead of optimising Rl, xR1) the mixed charge to the reboiler (Bc, xBC) was optimised and at the end of the solution the optimal (Rl, x RI) was evaluated from the overall balance around the mixer in Figure 8.2. The dynamic optimisation problem is formulated as ... 

As the mismatches of the state variables of a dynamic system (i.e. instant distillate and reboiler compositions in batch distillation) are dynamic in behaviour, they have to be treated as such and not as static processes. To develop them from first principles would be very difficult due to their non-linear dynamic behaviour and it would also be difficult to quantify them in terms of the original state variables. 

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