1.5: Industrial Process Models
- Page ID
- 24389
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Industrial processes comprise exchange of chemical, electrical or mechanical energy in the manufacturing of industrial products. An industrial process model in its simplified form is represented by a first-order lag with a dead-time that represents the time delay between the application of input and the appearance of the process output.
Let \(\tau\) represent the time constant associated with an industrial process, \(\tau _ d\) represent the dead-time, and \(K\) represent the process dc gain; then, simplified industrial process dynamics are represented by the following delay-differential equation:
\[\tau \frac{ dy(t)}{ dt} +y(t)=Ku(t-t_{d} ) \nonumber \]
Application of the Laplace transform produces the following first-order-plus-dead-time (FOPDT) model of an industrial process:
\[G(s)=\frac{Ke^{-\tau _ d s} }{\tau s+1} \nonumber \]
where the process parameters \(\{ K,\; \tau ,\; \tau _ d \}\), can be identified from the process response to inputs. An rational process model is obtained by using a Taylor series approximation of the delay term, \(e^{-\tau _ d s}\). Typical such approximations include:
\[ e^{-\tau_d s} \simeq 1-\tau _ d s, \ \ \ e^{-\tau _ d s} = \frac{1}{e^{\tau _d s} } \simeq \frac{1}{1+\tau _d s}, \ \ \ e^{-\tau _ds}=\frac{e^{-\tau _ds/2}}{e^{\tau _ds/2}}\simeq \frac{1-\tau _ds/2}{1+\tau_ds/2} \nonumber \]
The last expression is termed as first-order Pade’ approximation and is often preferred. Higher order approximations can also be used.
The process parameters of a stirred-tank bioreactor are given as: \(\{ K,\; \tau ,\; \tau _ d \} =\left\{20,0.5,1\right\}\). The transfer function model of the process is formed as:
\[G(s)=\frac{20e^{-s} }{0.5s+1}.\]
By using a first-order Pade’ approximation, a rational transfer function model of the industrial process with delay is obtained as:
\[G(s)=\frac{20\left(1-0.5s\right)}{\left(0.5s+1\right)^{2} }.\]
The step response of the bioreactor transfer function with Pade’ approximation shows an undershoot due to the presence of right half-plane (RHP) zero in the transfer function.
