Modelling dynamic systems with differential equations
3.3 Chemical process models
3.3.3 A binary distillation column model
Distillation columns, which are used to separate mixtures of different vapour pressures into al-most pure components, are an important chemical unit operation. The columns are expensive to manufacture, and the running costs are high owing to the high heating requirement. Hence there is much incentive to model them with the aim to operate them efficiently. Schematics of two simple columns are given in Figures3.6and3.9.
Wood and Berry experimentally modelled a binary distillation column in [202] that separated water from ethanol as shown in Fig.3.6. The transfer function model they derived by step testing a real column has been extensively studied, although some authors, such as [185], point out that the practical impact of all these studies has been, in all honesty, minimal. Other distillation column models that are not obtained experimentally are called rigorous models and are based on fundamental physical and chemical relations. Another more recent distillation column model, developed by Shell and used as a test model is discussed in [137].
Columns such as given in Fig.3.6typically have at least 5 control valves, but because the hydro-dynamics and the pressure loops are much faster than the composition hydro-dynamics, we can use the bottoms exit valve to control the level at the bottom of the column, use the distillate exit valve to control the level in the separator, and the condenser cooling water valve to control the column pressure. This leaves the two remaining valves (steam reboiler and reflux) to control the top and bottoms composition.
The Wood-Berry model is written as a matrix of Laplace transforms in deviation variables
y=
where the inputs u1and u2are the reflux and reboiler steam flowrates respectively, the outputs y1 and y2 are the mole fractions of ethanol in the distillate and bottoms, and the disturbance variable, d is the feed flowrate.
It is evident from the transfer function model structure in Eqn.3.24that the plant will be inter-acting and that all the transfer functions have some time delay associated with them, but the
Feed, F, xF
-
?s
6
?
c/w ?
- Distillate, D, xD
6
6 - Bottoms, xB, B
condensor
-Distillation column
Reflux, R
Steam, S
reboiler
-Figure 3.6: Schematic of a distillation column
off-diagonal terms have the slightly larger time delays. Both these characteristics will make con-trolling the column difficult.
In MATLAB, we can construct such a matrix of transfer functions (ignoring the feed) with a matrix of deadtimes as follows:
>> G = tf({12.8, -18.9; 6.6 -19.4}, ...
{ [16.7 1],[21 1]; [10.9 1],[14.4 1]}, ...
'ioDelayMatrix',[1,3; 7,3], ...
4 'InputName',{'Reflux','Steam'}, ...
'OutputName',{'Distillate','bottoms'})
Transfer function from input "Reflux" to output...
12.8
9 Distillate: exp(-1*s) * ---16.7 s + 1 6.6
bottoms: exp(-7*s) *
---14 10.9 s + 1
Transfer function from input "Steam" to output...
-18.9 Distillate: exp(-3*s) *
---19 21 s + 1
-19.4
bottoms: exp(-3*s) * ---14.4 s + 1
Once we have formulated the transfer function model G, we can perform the usual types of analysis such as step tests etc as shown in Fig.3.7. An alternative implementation of the column model in SIMULINKis shown in Fig.3.8.
−20
Figure 3.7: Step responses of the four transfer functions that make up the Wood-Berry binary distillation col-umn model in Eqn.3.24.
A 3-input/3 output matrix of transfer functions model of an 19 plate ethanol/water distillation column model with a variable side stream draw off from [151] is
which provides an alternative model for testing multivariable control schemes. In this model the three outputs are the overhead ethanol mole fraction, the side stream ethanol mole fraction, and the temperature on tray 19, the three inputs are the reflux flow rate, the side stream product rate and the reboiler steam pressure, and the two disturbances are the feed flow rate and feed temperature. This system is sometimes known as the OLMR after the initials of the authors.
Problem 3.2 1. Assuming no disturbances (d = 0), what are the steady state gains of the Wood-Berry column model? (Eqn3.24). Use the final value theorem.
2. Sketch the response fory1andy2for;
(a) a change in reflux flow of+0.02
steam
reflux distillate
bottoms reflux
steam
distillate
bottoms
Wood Berry Column
(a) Overall column mask.
2 bottoms 1 distillate
Transport Delay3
Transport Delay2 Transport Delay1 Transport Delay
−19.4 14.1s+1 Transfer Fcn3
10.9s+1 6.6
Transfer Fcn2
−18.9 21s+1 Transfer Fcn1 16.7s+1
12.8
Transfer Fcn
2 steam 1 reflux
(b) Inside the Wood Berry column mask. Compare this with Eqn.3.24.
Figure 3.8: Wood-Berry column model implemented in SIMULINK
(b) A change in the reflux flow of −0.02 and a change in reboiler steam flow of +0.025 simultaneously.
3. Modify theSIMULINKsimulation to incorporate the feed dynamics.
More rigorous distillation column models
Distillation column models are important to chemical engineers involved in the operation and maintenance of these expensive and complicated units. While the behaviour of the actual multi-component tower is very complicated, models that assume ideal binary systems are often good approximations for many columns. We will deal in mole fractions of the more volatile compo-nent, x for liquid, y for vapour and develop a column model following [130, p69].
A generic simple binary component column model of a distillation column such as shown in Fig.3.9assumes:
1. Equal molar overflow applies (heats of vapourisation of both components are equal, and mixing and sensible heats are negligible.)
2. Liquid level on every tray remains above weir height.
3. Relative volatility and the heat of vapourisation are constant. In fact we assume a constant relative volatility, α. This simplifies the vapour-liquid equilibria (VLE) model to
yn= αxn
1 + (α− 1)xn on tray n with typically α≈ 1.2–2.
4. Vapour phase holdup is negligible and the feed is a saturated liquid at the bubble point.
6
-
?
?
- Distillate, D, xD
-
-tray # N
tray #1 Feed, F, xF
-...
-?
Bottoms, xB, B reboiler
reflux, R, xD
6 6 ...
feed tray, NF
condensor c/w
condensor collector
Figure 3.9: Distillation tower
The N trays are numbered from the bottom to top, (tray 0 is reboiler and tray N + 1 is the con-denser). We will develop separate model equations for the following parts of the column, namely:
Condenser is a total condenser, where the reflux is a liquid and the reflux drum is perfectly mixed.
General tray has liquid flows from above, and vapour flows from below. It is assumed to be perfectly mixed with a variable liquid hold up, but no vapour hold up as it is assumed very fast.
Feed tray Same as a ‘general’ tray, but with an extra (liquid) feed term.
Top & bottom trays Same as a ‘general’ tray, but with one of the liquid (top) or vapour (bottom) flows missing, but recycle/reflux added.
Reboiler in a perfectly mixed thermo-siphon reboiler with hold up Mb.
The total mass balance in the condenser and reflux drum is dMd
dt = V − R − D and the component balance on the top tray
dMdxd
dt = V yNT − (R + D)xD (3.26)
For the general nth tray, that is trays #2 through N − 1, excepting the feed tray, the total mass balance is
dMn
dt = Ln+1− Ln+ Vn−1− Vn
| {z }
≈0
= Ln+1− Ln (3.27)
and the component balance dMnxn
dt = Ln+1xn+1− Lnxn+ V yn−1− V yn (3.28) For the special case of the feed nFth tray,
dMnF
dt = LnF+1− LnF + F, mass balance dMnFxnF
dt = LnF+1xnF+1− LnFxnF + F xF + V ynF−1− V ynF, component balance and for the reboiler and column base
dMB
dt = L1− V − B dMBxB
dt = L1x1− BxB− V xB
In summary the number of variables for the distillation column model are:
VARIABLES
Tray compositions, xn, yn 2NT
Tray liquid flows, Ln NT
Tray liquid holdups, Mn NT
Reflux comp., flow & hold up, xD, R, D, MD 4 Base comp., flow & hold up, xB, yB, V, B, MB 5 Total # of equations 4NT+ 9 and the number of equations are:
EQUATIONS
Tray component balance, NT
Tray mass balance NT
Equilibrium (tray + base) NT + 1
hydraulic NT
Reflux comp. & flow 2 Base comp & flow 2 Total # of equations 4NT + 7
Which leaves two degrees of freedom. From a control point of view we normally fix the boildup rate, ˙Q, and the reflux flow rate (or ratio) with some sort of controller,
R = f (xD), V ∝ ˙Q = f (xB)
Our dynamic model of a binary distillation column is relatively large with two inputs (R, V ), two outputs (xB, xD) and 4N states. Since a typical column has about 20 trays, we will have around n = 44 states. which means 44 ordinary differential equations However the (linearised) Jacobian for this system while a large 44× 44 matrix is sparse. In our case the percentage of non-zero elements or sparsity is≈ 9%.
∂ ˙x
|{z}∂x
44×44
... ∂ ˙x
|{z}∂u
44×2
The structure of the A and B matrices are, using thespycommand, given in Fig.3.10.
0 5 10 15 20 25 30 35 40 45
0
5
10
15
20
25
30
35
40
45
States & input, ([holdup, comp*holdup| [R V]) Jacobian of the Binary distiilation column model
ODE equations, (zdot=)
Figure 3.10: The incidence of the Jacobian and B matrix for the ideal binary distillation column model. Over 90% of the elements in the matrix are zero.
There are many other examples of distillation column models around (e.g. [140, pp 459]) This model has about 40 trays, and assumes a binary mixture at constant pressure and constant rela-tive volatility.
Simulation of the distillation column
The simple nonlinear dynamic simulation of the binary distillation column model can be used in a number of ways including investigating the openloop response, interactions and quantify the extent of the nonlinearities. It can be used to develop simple linear approximate transfer function models, or we could pose “What if?” type questions such as quantifying the response given feed disturbances.
An openloop simulation of a distillation column gives some idea of the dominant time constants and the possible nonlinearities. Fig.3.11 shows an example of one such simulation where we step change:
1. the reflux from R = 128 to R = 128 + 0.2, and 2. the reboiler from V = 178 to V = 178 + 0.2
0 10 20 30
Figure 3.11: Open loop response of the distillation column control
From Fig.3.11we can note that the open loop step results are overdamped, and that steady-state gains are very similar in magnitude. Furthermore the response looks very like a 2× 2 matrix of second order overdamped transfer functions,
xD(s)
So it is natural to wonder at this point if it is possible to approximate this response with a low-order model rather than requiring the entire 44 states and associated nonlinear differential equa-tions.
Controlled simulation of a distillation column
A closed loop simulation of the 20 tray binary distillation column with a feed disturbance from xF = 0.5 to xF = 0.54 at t = 10 minutes is given in Fig.3.12.
We can look more closely in Fig.3.13at the distillate and base compositions to see if they really are in control, x⋆b = 0.02, x⋆d= 0.98.
Distillation columns are well known to interact, and these interactions cause difficulties in tuning.
We will simulate in Fig.3.14a step change in the distillate setpoint from x⋆D = 0.98 to x⋆D= 0.985 at t = 10 min and a step change in bottoms concentration at 150 minutes. The interactions are evident in the base composition transients owing to the changing distillate composition and visa versa. These interactions can be minimised by either tightly tuning one of the loops so consequently leaving the other ‘loose’, or by using a steady-state or dynamic decoupler, or even a multivariable controller.
0 0.2 0.4 0.6 0.8 1
Liquid concentration
Binary distillation column model
124 126 128 130
R
0 20 40 60 80 100
175 180 185
V
time (min)
Figure 3.12: A closed loop simulation of a binary distillation column given a feed dis-turbance. The upper plot shows the con-centration on all the 20 trays, the lower plots shows the manipulated variables (re-flux and boil-up) response.
0.975 0.98 0.985 0.99
Distillate
0 20 40 60 80 100
0.01 0.02 0.03 0.04
Bottoms
time [mins]
Figure 3.13: Detail of the distillate and base concentrations from a closed loop simula-tion of a binary distillasimula-tion column given a feed disturbance. (See also Fig.3.12.)