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McMaster UniversitySolutions for Tutorial 3Modelling of Non-Linear Systems3.1 Isothermal CSTR: The chemical reactor shown in textbook Figure 3.1 andrepeated in the following is considered in this question. The reaction occurring in thereactor is0.5A → Br = -kCA AThe following assumptions are appropriate for the system.(i) the reactor is well mixed,(ii) th isothermal,(iii) density of the liquid in the reactor is constant,(iv) flow rates are constant, and(v) reactor volume is constant.a. Formulate the model for the dynamic response of the concentration of A in thereactor, C (t).Ab. Linearize the equation(s) in (a).c. Solve the linearized equation analytically for a step change in the inletconcentration of A, ∆C .A0d. Sketch the dynamic behavior of C (t).Ae. Discuss how you would evaluate the accuracy of the linearized model.Goal → Variable System→ Balance → DOF → Linear? (or constitutive equation)Again, we apply the standard modelling approach, with a check for linearity.a. Goal: Determine composition of A as a function of time.Variable: C in the reactorASystem: The liquid in the reactor.Balance: Component balance on A.Accumulation = in - out + generation0.5(1) MW()VC | −VC | = MW ∆t(FC − FC − VkC )A A t+∆t A t A A0 A A01/02/01 Copyright © 2000 by Marlin and Yip 1McMaster UniversityDivide by delta time and take the limit to obtaindCA 0.5(2) V = F(C − FC ) − VkCA0 A AdtAre we done? Let’s check the degrees of freedom.DOF = ...

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McMaster University

Solutions forTutorial 3 Modelling of NonLinear Systems

3.1 Isothermal CSTR:The chemical reactor shown in textbook Figure 3.1 and repeated in the following is considered in this question. The reaction occurring in the reactor is

b. c.

d. e.

0.5 rA= kCA

The following assumptions are appropriate for the system. (i) the reactor is well mixed, (ii) the reactor is isothermal, (iii) density of the liquid in the reactor is constant, (iv) flow rates are constant, and (v) reactor volume is constant.

Formulate the model for the dynamic response of the concentration of A in the reactor, CA(t). Linearize the equation(s) in (a). Solve the linearized equation analytically for a step change in the inlet concentration of A,CA0. Sketch the dynamic behavior of CA(t). Discuss how you would evaluate the accuracy of the linearized model.

Goal

Variable

SystemBalance (or constitutive equation)

DOF

Again, we apply the standard modelling approach, with a check for linearity.

a. Goal: Determine composition of A as a function of time.

Variable:CAin the reactor

System:The liquid in the reactor.

Balance:Component balance on A.

(1)

01/02/01

Accumulation



out

0.5 MW(VC |VC |)=MWt FCFCVkC) A A t+t A At A A0 A

generation

Linear?

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Divide by delta time and take the limit to obtain

(2)

dC A 0.5 V=F(CFC )VkC A0 A A dt

Are we done? Let’s check the degrees of freedom.

DOF = 1  1 = 0

Yes!

b. Is the model linear? If we decide to solve the model numerically, we do not have to linearize; in fact, the nonlinear model would be more accurate. However, in this problem we seek theinsightobtained from the approximate, linear model.

All terms involve a constant times a variable (linear) except for the following term, which is linearized using the Taylor series..

(3)

0.5 0.50.5 CC)+0.5 C)(CC)+higher order terms A A s A s A As

This approximation can be substituted into equation 2, and the initial steadystate model subtracted to obtain the following, with C’A= CA CAS.

(4)

dC' A0.5 V=F(C'FC' )Vk(0.5C )C' A0 A As A dt

This linear, first order ordinary differential equation model can be arranged into the standard form, given in the following.

(5)

dC' A +C'=KC' A A0 dt

V with = 0.5 F+0.5VkC As

F K= 0.5 F+0.5VkC As

c. Let’s solve this equation using the Laplace transform method. We can take the Laplace transform of equation (5) to obtain

(6)

(sC' (s)|C' (t) )+C' (s)=KC' (s) A A t=A00 A

Note that equation (6) is general for any function CA0can rearrange this equation(t). We and substitute the Laplace transform of the step change in feed composition (C’A0(s)=CA0/s to give.

(7)

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KC A0 C' (s)= A s+1 s

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We can take the inverse Laplace transform using entry 5 in textbook Table 4.1 to give

(8)

t /C' (t)= 1C K e) A A0

d. A typical sketch is given here. We already have experience with the step response to a linear, first order system. We know that

 the output changes immediately after the step is introduced.  the maximum slope appears when the step is introduced  the curve has a smooth (nonoscillatory response)  63% of the change occurs when t =(past the step)  the final steady state is K(input)

C’ A

C’ A0

Time

e. We should always investigate the accuracy of our mathematical models! We can estimate the accuracy of the parameters used based on

Laboratory data used in developing the constitutive model Construction of equipment Accuracy of measurements used to achieve desired values

 Is the rate expression accurate  uncertainty in k V (cross sectional area) V (level) and F (flow)

In addition, we should estimate the error introduced by the linearization. No error is introduced if the process stays exactly at the initial steady state, and the errors generally increase as the process deviates further from the initial steady state. Here, two methods are suggested. (Remember, we do not seek highly accurate models – we seek simple, approximate models for control design, which will be explained shortly.)

1.Evaluate the key parameters over the range of operation. We can evaluate the gain (K) and the time constant () at different values over the range of operation. If these parameters do not change much, the linearization would be deemed accurate.

2.Steadystate prediction. Compare the steadystate output values from the nonlinear model with steadystate output values from the linearized model (Kinput). This method will check the gain only, not the time constant.

01/02/01

C A

0.5 MW(VC |VC |)=MWt FCFCVkC) A A t+t A AA0 A t A

a. We begin by applying our standard method for modelling.

Variable:CAin the reactor

c. d.

System:The liquid in the reactor.

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Develop the dynamic model to predict the concentration of A. Linearize the equation and solve the linearized equation analytically for a step change in the feed flow rate,F. Sketch the dynamic behavior of the effluent concentration, CA(t). Describe the equipment required to maintain the feed flow rate at a desired value.

Balance:Component balance on A.

01/02/01

(1)

Divide by delta time and take the limit to obtain

a. Goal: Determine composition of A as a function of time.

generation

out



Figure 3.1

3.2 Controlling the Reactor Concentration by Feed Flow Rate:The reactor in question 3.1 above is considered again in this question. Component A is pumped to the reactor from the feed tank. The inlet concentration of A, CA0, isconstant, and the feed flow rate varies with time.

Accumulation

F 0

a. b.

C A0

Motivation:Why are we interested in this model? Often, the feed composition cannot be adjusted easily by mixing streams. Therefore, we sometimes adjust the feed flow rate to achieve the desired reaction conversion. (We do not like to do this, because we chan e both the roduction rate and the conversion when we ad ust feed flow rate.

(2)

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dC A 0.5 V=F(CFC )VkC A0 A A dt

Are we done? Let’s check the degrees of freedom.

DOF = 1  1 = 0

Yes!

We see that several terms are nonlinear. In fact, when flow is a variable, we would usually find terms (F)(variable), where “variable” is temperature, compositions, etc. The following terms will be linearized by expanding the Taylor series.

(3)

(4)

(5)

FC(FC )+F C'+C F'+higher order terms A0 A0 s s A0 A0s

FC(FC )+F C'+C F'+higher order terms A A s s A As

0.5 0.50.5 CC)+0.5 C)(CC)+higher A A s A s A As

er terms

Substituting the approximations, subtracting the initial steady state, and rearranging gives the following.

(6)

dC' A +C'=KF' A dt

V with = 0.5 F+0.5VkC s As

(CC ) A0s As K= 0.5 F+0.5VkC s As

We can solve this equation for step change in flow rate by taking the Laplace transform, substituting F’(s) =The result is given inF/s, and taking the inverse Laplace transform. the following equation.

(7)

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t /C' (t)=(F)K 1e) A

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c. The plot and qualitative properties are the same as for other first order systems.

C’ A

F’

Time

Does this make sense?we increase the feed flow, the “space time” in the reactor As decreases. (See Fogler (1999) or other textbook on reaction engineering for a refresher.) When the space time decreases, the conversion decreases, and the concentration of reactant increases.Yes, the model agrees with our qualitative understanding!

d.Equipment is required to control the flow is needed if we are to adjust the flow to achieve the desired reactor operation, e.g., conversion. Any feedback controller requires a sensor and a final element. (See Chapter 2.) The sensor could be any of the sensors described in the Instrumentation Notes. The most common sensor in the process industries is the orifice meter, which measures flow based on the pressure drop around an orifice restriction in a pipe. The final element would be a control valve that can adjust the restriction to flow.

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Pump to supply the “head” for flow

P

Orifice meter

Valve with adjustable stem position

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3.3 Figure 3.2 shows two distillation towers with all sensors, piping, valves and other ancillary equipment. A drawing like this is termed a piping and instrument (P&I) drawing. Select one example of each of the four basic sensors (flow, temperature, pressure and level) and describe an appropriate physical principle for the sensor. Also, select one example of a control valve and select an appropriate body type.

Figure 3.2 Distillation process (from Woods,Process Design and Engineering Practice, Prentice Hall, 1995)

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Before, discussing sensor principles, let’s clarify an important factor in the selection. Sensors can be installed for two different applications

Local displayThe plant personnel near the process equipment need to – determine some of the key variable values. They do not have access to the process control computers. Therefore, we install some sensors that give a visual display of the variable values.

Transmitted for remote display and controlMany sensor signals are – transmitted to the control system for display, data storage, and control calculations. Typically, the control equipment is located in a centralized building. Advantages for the centralized location include physical protection from accidents, collect of much data for thorough monitoring, and lower cost. Occasionally, the control equipment is distributed around the plant; this layout is appropriate if the personnel usually work locally at the equipment.

After we have selected the location for the variable to be used, we must indicate this on the drawing. We use a simple convention.

If the circle showing the instrument has a line across the middle, the variable is transmitted to the control system, usually the centralized control room. If the circle has no line, the variable is displayed locally. You can find examples of both in Figure 3.2.

Why is this difference important? Some Othersensors provide only a local display. sensors can provide either a local or centralized display. We must match the sensor principle to the application. Naturally, we cannot select a physical principle that only provides a local display for a variable that is used remotely.

Let’s consider each of the four typical variables and select a sensor for remote use in the control system and a sensor for local display. The descriptions are from the Instrumentation Notes.

Temperature

Filled system for local display, T7 on cooling water to C8 condenser:A fluid expands with increasing temperature and exerts a varying pressure on the containing vessel. When the vessel is similar to a bourbon tube, the varying pressure causes a deformation that changes the position detected to determine the temperature. This provides a rugged, low cost sensor that is often used for local displays

Thermocouple transmitted for remote use, T5 for tray temperature control:When the junctions o two dissimilar metals are at different temperatures, an electromotive force (emf) is developed. The cold unction, referred to as the reference, is maintained at a known temperature, and the measuring junctio is located where the temperature is to be determined. The temperature difference can be determined from the measured emf. The relationship between temperature difference and emf has been determined for several commonly used combinations of metals; the mildly nonlinear relationships are available in tabular form alon with ol nomial e uations relatin emf to tem erature

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Flow

Positive displacement for local display, no example on drawing:(Note, essentially every flow sensor can be used for remote display.) In these sensors, the fluid is separated into individual volumetric elements and the number of elements (revolutions) per unit time is measured. These sensors provide high accuracy over a large range. An example is a wet test meter.

Orifice plate for remote use, F1 for fee flow control to C8:An orifice plate is a restriction with an opening smaller than the pipe diameter which is inserted in the pipe; the typical orifice plate has a concentric, sharp edged opening, as shown in Figure 1. Because of the smaller area the fluid velocity increases, causing a corresponding decrease in pressure. The flow rate can be calculated from the measured pressure drop across the orifice plate, P1P3. The orifice plate is the most commonly used flow sensor, but it creates a rather large nonrecoverable pressure due to the turbulence around the plate, l a in t hi h n r n m ti

Pressure

Bourbon for local display, P3 at exit of pump F26:A bourbon tube is a curved, hollow tube with the process pressure applied to the fluid in the tube. The pressure in the tube causes the tube to deform or uncoil. The pressure can be determined from the mechanical displacement of the pointer connected to the Borden tube. Typical shapes for the tube are “C” (normally for local display), spiral and helical.

Resistive, strain gauge for remote use, P10 for control of C8 overhead pressure:The electrical resistance of a metal wire depends on the strain applied to the wire. A diaphragm is typically constructed of two flexible disks, and when a pressure is applied to one face of the diaphragm, the position of the disk face changes due to deformation. Deflection of the diaphragm due to the applied pressure causes strain in the wire, and the electrical resistance can be measured and related to ressure.

Level

Sight glass of local display,L1 on V29 feed vessel: A vertical tube is connected at its top and bottom to the vessel. The liquid in the vessel can flow freely between the vessel and the tube, so that the liquid level in the tube is essentially the same as in the vessel. Since the tube is transparent (usually made o glass), a person can see the level.

Differential pressure for remote use, L2 for control of the liquid in C8 bottoms:The difference in pressures between to points in a vessel depends on the fluids between these two points. If the difference in densities between the fluids is significant, which is certainly true for a vapor and liquid and can be true for two different liquids, the difference in pressure can be used to determine the interface level between the fluids. Usually, a seal liquid is used in the two connecting pipes (legs) to prevent plugging

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