Stirred Tank Blending Process

Assumptions Perfect MixingDensity of the liquid is constant (that is, it does not change appreciably with composition)Volume is variable

Degrees of Freedom

8 Entitiesr,x,x1,x2,w,w1,w2One parameter r

That leaves seven unknowns (VARIABLES)

If x, V are the controlled variables (OUTPUT)

There are five degrees of freedom.

x1,x2,w1 can be disturbances and w,w2, the manipulated variable.

*

Remember the bar above V and w (discuss..)

Degrees of Freedom
when V and x2 are constants

8 Entitiesr,x,x1,x2,w,w1,w2Two parameters r, VX2 is constant and w=w1+w2

That leaves four unknowns (VARIABLES)

If x is the controlled variables (OUTPUT)

There are three degrees of freedom.

x1, w1 can be disturbances and w2, the manipulated variable.

*

Remember the bar above V and w (discuss..)

Stirred Tank Blending

Governing EquationIf V is assumed constantLet t = V r / w

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Remember the bar above V and w (discuss..)

Stirred Tank Blending Problem

Given V = 2m3 and r = 900 kg/m3Assume that the tank was fed with w1=500kg/min with x1=0.4 and w2=200kg/min with x2=0.75 for a long time. Get the Steady State Value of xCase b. w1 suddenly changed to 400. Plot x vs. t.Case c. w2 suddenly changed to 100 (w1 being 500). Plot x vs. tCase d. with w2=200, w1 suddenly changed to 400, and x1, to 0.6 plot x vs. t

Solution : Stirred Tank Blending

For all cases w1+w2 = 600t = V r / w =(2m3 * 900kg/m3)/600kg/min= 3 min

Stirred Tank Blending : Results

Stirred tank blending normalized plot

The response of the system to changes in

Flowrate w1

Flowrate w2

Simultaneous change in w2

And x1

Electrically Heated Stir Tank

AssumptionsConstant Hold up (V). Inlet flow = Outlet flowPerfect MixingDensity r and Heat Capacity C are constantsHeat losses are negligibleMetal heating element has significant capacityMetal heating element has uniform temperature.

Electrically Heated Stir Tank

Governing Equations

Electrically Heated Stir Tank

Response of an electrically heated stirred tank to a step change

A dynamic model for CSTR

AssumptionsPerfect MixingV is constantr Is constantThe reaction is of first order, that is,Rate r = k CA

Reaction rate r has units of moles per unit volume.

A dynamic model for multistage (3-stage absorption) systems

Note that the individual equations are linear differential equations.They are Coupled as well, that is, x2 appears in the expression for dx1/dt and both x1 and x3 appear in the expression for dx2/dt.

Distributed Parameter Systems

In the models we considered thus far, we examined variables that were a function of time. In the simple double pipe exchanger, that is not the case as temperature varies with position. For rigorous characterization, we will have to employ partial differential equations. Alternately, we can opt for a DISTRIBUTED PARAMETER (as against LUMPED parameter) model, dividing the length in a number of zones, in each of which, T can be assumed to be constant.

Motivation for Studying
Process Dynamics

Improve understanding of the processTrain plant operatorsAs a basis for developing control strategyOptimize operating conditions

Disadvantages of using Theoretical Models

The simplest models are based on a long list of assumptions.A model quickly gets complicated when some of the assumptions are replaced by more realistic equations/conditions.Complex models are often non-linear, not amenable to easy analysis.For the analysis of dynamic response and getting insight for Process Control, we need to work with linear dynamic models.

Arrhenius Equation

Expresses the rate of reaction as a function of temperatureK0 is frequency factor, as it has units of time-1.E is the activation energy and R is the gas constant.Typically, k0 and E are determined by fitting experimental data.Thus, as used, it is a semi-empirical equation.

Classification of Process Models

Theoretical ModelsProvide physical insightCan be extrapolatedAre difficult, expensive to develop.Empirical ModelsEasy to developShould not be used for extrapolationSemi-empirical ModelsAttractive compromise of simplicity and applicability