Staehle Controls Exam 1 Review
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CHE 06405: Process Dynamics and Control
Exam #1: Practice Exam Problem
A pure liquid evaporator that generates vapor at the mass flow rate Fout by using a steam-‐heated heat exchanger coil to vaporize a liquid stream (temperature Tin) flowing in at mass flow rate Fin, is a classic example of how mass and energy interact in process systems. If T is the temperature of the liquid in the vessel, typical material and energy balances yield nonlinear mathematical model equations which, when linearized around the nominal operating values T*, T*in, F*in, and F*out (as you will learn to do in a few weeks), produce the following approximate transfer function model:
(1)
where the deviation variables y and u are defined as follows: y = Fout – F*out (2) u = Fin – F*in (3) and θ is the normalized “nominal temperature load” defined by (4) The indicated parameters are related to the process physics at the nominal operating conditions around which the modeling equations have been linearized. Specifically, τ1 is the effective (material) residence time in the vessel; τ2 is the effective thermal time constant; τ3 is the vessel thermal capacity relative to that of the steam coil; β is a dimensionless constant related to the pressure in the vessel.
1. What is the steady state gain for this process? Discuss succinctly why this value makes sense from the physics of the problem.
2. For a process with specific parameter values τ1 = 2.5 min; τ2 = 1.25 min; τ3 = 5 min; and θ = 0.5, sketch the system response to a unit step increase in inlet flow rate for the following three values of the dimensionless parameter β: 32, 8, and 2.
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y(s) =−τ3θs+1
τ1τ3β
&
' (
)
* + s2 +
τ1τ3τ2β
&
' (
)
* + s+1
u(s)
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θ =T* −Tin
*
T*
3. On the other hand, it can be shown that, under some special conditions, the linearized model for a multi-‐phase separator (where the liquid mixture contains more than one component) is given by:
(5) where, in terms of deviations from their respective nominal steady state values, y is the bottom concentration of light material; u is the feed flow rate. The parameters τC and τT are, respectively, the “concentration” and “thermal” time constants, while K is a process parameter interaction between mass and energy. Derive an expression for the response, y(t) to a unit step change. Sketch this response, showing all aspects that you consider to be important and characteristic of this response.
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y(s) =Ks
(τCs+1)(τT s+1)u(s)