CENG0017 - Process Dynamics and Control - Dynamic Mathematical Model - Single Tray System - Engineering Assignment Help

Assignment Task:

CENG0017: Process Dynamics and Control Engineering Assignment Help

Problem Description
We would like to build a dynamic model to describe the dynamics of the liquid phase in the distillation tray shown in Figure 1a. The liquid from one tray goes over a weir and cascades down to the next tray through a downcomer. The volumetric flow rate q(t) of the liquid to the next tray (expressed in [m3/s]) can be calculated from the following empirical expression:

where lw is the weir length and h(t) is the height of the liquid level on the tray.

We assume that:
The total area of the tray is At = 1.20 m2 and the weir length is lw = 0.950 m;
 The density of the liquid ρ does not change significantly from tray to tray;
 At nominal steady-state conditions the inlet liquid flowrate is qi(t) = 60 m3/s.

We want to study the dynamic behaviour of the system by developing a dynamic model, deriving transfer functions and block diagrams. Finally, we want to simulate the dynamic model of the tray with the use of gPROMS ModelBuilder.

Things to Do:
1. Develop a dynamic mathematical model for the tray, neglecting the energy balance and including only the material balance of the liquid phase. Indicate the time-dependent terms and the nonlinear terms in the balance model.

2. Linearise the model around the steady state operation and write the equations in terms of deviation variables.

3. Determine the transfer functions linking the liquid level h(t) and the outlet liquid flowrate q(t) to the inlet flowrate qi(t).

4. What is the order expressed by each transfer function? Calculate the values of the relevant transfer function coefficients indicating their dimensions and sketch the block diagram for the system, including all the relevant transfer functions for the system.

5. Now we want to extend the study to the rectifying section of a distillation column with N trays operating at total reflux (Figure 1b). What will be the transfer function linking the inlet flow rate to the first tray (tray 1) to the liquid level in the last tray (tray N)? Determine the overall order of the transfer function. We assume that all the trays have the same geometry and that the same functional relationships (1) can be used to describe the outlet volumetric flowrate in each tray.

6. Provide a set of suitable initial conditions (i.e. conditions of the state variables for t = 0) for solving the single tray model. Justify your choice of the appropriate numerical values.
7. Simulate the single tray system using gPROMS:

a. Compare and plot the response of the linear model and the response of the nonlinear model when a 30% step and a 150% step increase on the nominal inlet flowrate qi(t) is applied. In the graphs include the responses h(t) and q(t) and the step variation on the forcing function qi(t).

b. Is the response to the perturbation stable? Is the linearised model adequate to represent the response of the system? Motivate the answers.