CHEM E 480 Process Dynamics and Control
Contents
Intro to Process Dynamics and Control
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Definitions
- Process - conversion of feed materials to products using chemical and physical operations
- Continuous, batch, semi-batch
- Process dynamics - unsteady-state/transient process behavior
- Start-ups, shutdowns, process disturbances, planned transitions
- Process control - maintain a process at desired operating conditions
- Utilize information flow
- Impacts safety, environment, quality, economics
- Manual control - control strategy implemented by a person
- Automatic control - control strategy automated by computers
- Types of variables
- Set point - desired nominal value of controlled variable
- Controlled variable - variable being regulated and maintained (to be controlled) at set point
- Manipulated variable - variable that can be adjusted
- Disturbance variable - variable that perturbs the system, changes controlled variable, but cannot be directly tuned
- Process - conversion of feed materials to products using chemical and physical operations
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Classification of control strategies
- Feedback control - controlled variable is measured to adjust manipulated variable (disturbance variable not measured)
- Negative feedback - controller forces controlled variable toward set point
- Positive feedback - controller forces controlled variable farther away from set point
- ➕ Correction occurs regardless of source of disturbance
- ➕ Reduces sensitivity of controlled variable to unmeasured disturbance and process changes
- ➖ Correction only happens after controlled variable deviates set point (disturbance has occurred)
- Feedforward control - disturbance variable measured to adjust manipulated variable (controlled variable not measured)
- ➕ Correction happens before controlled variable deviates set point
- ➖ Disturbance variable must be measured or accurately estimated
- ➖ Do not account for unmeasured disturbances
- ➖ Process model is required
- Challenges
- Systems are often nonlinear
- Time delays are common at scale
- Real systems are complex and interconnected
- Iterative design
- Formulate control objective
- Identify variables
- Model process
- Implement control strategy
- Tune controller
- Feedback control - controlled variable is measured to adjust manipulated variable (disturbance variable not measured)
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Types of control processes
- Single-input/single-output (SISO) - one manipulated variables (input) and one controlled variables (output)
- ➕ Easy to model and implement
- ➖ Less control
- Multiple-input/multiple-output (MIMO) - multiple manipulated variables (input) and multiple controlled variables (output)
- ➕ Better control
- ➖ Hard to model and implement
- Balance robustness to disturbance and complexity of model
- Single-input/single-output (SISO) - one manipulated variables (input) and one controlled variables (output)
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Hierarchy of control activities
- Measurement and actuation (< 1 s)
- Safety and environmental/equipment protection (< 1 s)
- Regulatory control (sec - min)
- Multivariable and constraint control (sec - min)
- Real-time optimization (hr - day)
- Planning and scheduling (day - month)
Theoretical Models of Chemical Processes
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Dynamic models
- Theoretical models - developed using principles of physics, chemistry, and biology
- ➕ Physical insight into process behavior
- ➕ Applicable over wide ranges of conditions
- ➖ Expensive and time-consuming to develop
- ➖ Model parameter not readily available
- Empirical models - obtained by fitting experimental data
- ➕ Easy to develop and use
- ➖ Do not extrapolate to conditions beyond range
- Semi-empirical models - numerical values of 1(+) parameters in a theoretical model are calculated from experimental data
- ➕ Incorporate theoretical knowledge
- ➕ Extrapolate over wider range of operating conditions
- ➕ Requires less developmental effort
- Improve process understanding
- Train plant operating personnel
- Develop control strategies for new processes
- Optimize process operating conditions
- Theoretical models - developed using principles of physics, chemistry, and biology
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General modeling principles
- Simplifying assumptions balances model complexity and accuracy
- Higher-order ODEs approximated by first-order processes have solutions with exponential and oscillatory response
- Conservation of mass
- Conservation of energy
- Rate expressions
- Equilibrium expressions
- Normalized response - $x_N(t) = \dfrac{x(t) - x(0)}{x(\infty) - x(0)}$
Laplace Transforms
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Definitions
- Laplace transform - $\mathcal{L}[f(t)] = F(s) = \displaystyle\int_{0}^{\infin} e^{-st}f(t) \ dt$
- Heaviside function (unit step function centered at $c$)
- $u_c(t) = u(t-c) = \begin{cases} 0 & t < c \cr 1 & t \ge c \end{cases}$
- Rectangular pulse function
- $f(t) = \begin{cases} 0 & t<0 \\ h & t \in [0, t_w] \\ 0 & t \ge t_w \end{cases}$
- Impulse (Dirac delta) function
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Properties of Laplace transform
- Laplace transform is linear
- $\mathcal{L}[c_{1}f(t)+c_{2}g(t)] = c_{1}\mathcal{L}[f(t)] + c_{2}\mathcal{L}[g(t)]$
- Laplace transforms of derivatives incorporate initial conditions
- $\mathcal{L}[f'(t)] = sF(s) - f(0)$
- $\mathcal{L}[f''(t)] = s^2F(s)-sf(0)-f'(0)$
- $\mathcal{L}[f^{(n)}(t)] = s^nF(s)-s^{n-1}f(0) - s^{n-2}f^{(1)}(0) - … - f^{(n-1)}(0)$
- Heaviside function has a simple Laplace transforms
- $\mathcal{L}[u_c(t)] = \dfrac{e^{-sc}}{s}$
- Laplace transform is linear
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Translation theorems
- Time domain translation
- $\mathcal{L}[f(t-c)u_{c}(t)] = e^{-sc}\mathcal{L}[f(t)]$
- $\mathcal{L}^{-1}[e^{-sc}\mathcal{L}[f(t)]] = f(t-c)u_c(t)$
- Laplace domain translation
- $\mathcal{L}[e^{ct}f(t)] = F(s-c)$
- $\mathcal{L}^{-1}[F(s-c)] = e^{ct}f(t)$
- Time domain translation
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Solving ODEs with Laplace transform
- Time domain: difficult ODE
- Laplace transform ($t \to s$)
- Laplace domain: easy algebra problem
- Solve the algebra problem
- Laplace domain: solution to algebra problem
- Inverse Laplace transform ($s \to t$)
- Time domain: solution of difficult ODE
- Problem solved
- Time domain: difficult ODE
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Partial fraction decomposition
- Heaviside expansion
- $Y(s) = \dfrac{N(s)}{D(s)} = \dfrac{N(s)}{\displaystyle \prod_{i=1}^n (s+b_i)} = \displaystyle \sum_{i=1}^n \dfrac{\alpha_i}{s + b_i}$
- Heaviside expansion
Fundamental signals
| Inverse L.T. $f(t)$ |
Laplace Transform $F(s)$ |
Inverse L.T. $f(t)$ |
Laplace Transform $F(s)$ |
|---|---|---|---|
| $1$ | $\dfrac{1}{s}$ | $\delta(t)$ | $1$ |
| $S(t) \equiv u_0(t)$ | $\dfrac{1}{s}$ | $S(t-c) \equiv u_{c}(t)$ | $\dfrac{e^{-sc}}{s}$ |
| $e^{-at}$ | $\dfrac{1}{s+a}$ | $t$ | $\dfrac{1}{s^2}$ |
| $\dfrac{1}{a}e^{-t/a}$ | $\dfrac{1}{as+1}$ | $t^{n}$ | $\dfrac{n!}{s^{n+1}}$ |
| $1 - e^{t/a}$ | $\dfrac{1}{s(as+1)}$ | $\sqrt{t}$ | $\dfrac{\sqrt{\pi}}{2s^{3/2}}$ |
| $e^{-at+b}$ | $\dfrac{e^b}{a+s}$ | $te^{-at}$ | $\dfrac{1}{(a+s)^2}$ |
| $t^ne^{-at}$ | $\dfrac{n!}{(a+s)^{n+1}}$ |
Oscillatory signals
| Inverse L.T. $f(t)$ |
Laplace Transform $F(s)$ |
Inverse L.T. $f(t)$ |
Laplace Transform $F(s)$ |
|---|---|---|---|
| $\sin(at)$ | $\dfrac{a}{s^{2}+a^{2}}$ | $t\sin(at)$ | $\dfrac{2as}{(s^{2}+a^{2})^{2}}$ |
| $\cos(at)$ | $\dfrac{s}{s^{2}+a^{2}}$ | $t\cos(at)$ | $\dfrac{s^{2}-a^{2}}{(s^{2}+a^{2})^{2}}$ |
| $\sin(at)-at\cos(at)$ | $\dfrac{2a^{3}}{(s^{2}+a^{2})^{2}}$ | $\cos(at)-at\sin(at)$ | $\dfrac{s(s^{2}-a^{2})}{(s^{2}+a^{2})^{2}}$ |
| $\sin(at)+at\cos(at)$ | $\dfrac{2as^{2}}{(s^{2}+a^{2})^{2}}$ | $\cos(at)+at\sin(at)$ | $\dfrac{s(s^{2}+3a^{2})}{(s^{2}+a^{2})^{2}}$ |
| $\sinh(at)$ | $\dfrac{a}{s^{2}-a^{2}}$ | $\sin(at+b)$ | $\dfrac{s\sin(b)+a\cos(b)}{s^2+a^2}$ |
| $\cosh(at)$ | $\dfrac{s}{s^{2}-a^{2}}$ | $\cos(at+b)$ | $\dfrac{s\cos(b)-a\sin(b)}{s^2+a^2}$ |
| $e^{at}\sin(bt)$ | $\dfrac{b}{(s-a)^2+b^2}$ | $e^{at}\sinh(bt)$ | $\dfrac{b}{(s-a)^2-b^2}$ |
| $e^{at}\cos(bt)$ | $\dfrac{s-a}{(s-a)^2+b^2}$ | $e^{at}\cosh(bt)$ | $\dfrac{s-a}{(s-a)^2-b^2}$ |
Convenient inverse signals
| Inverse L.T. $f(t)$ |
Laplace Transform $F(s)$ |
Comment |
|---|---|---|
| $\dfrac{t^{n-1} e^{-b t}}{(n-1) !}(n>0)$ | $\dfrac{1}{(s+b)^{n}}$ | |
| $\dfrac{1}{\tau^{n}(n-1) !} t^{n-1} e^{-t / \tau}$ | $\dfrac{1}{(\tau s+1)^{n}}$ | |
| $\dfrac{1}{b_{1}-b_{2}}\left(e^{-b_{2} t}-e^{-b_{1} t}\right)$ | $\dfrac{1}{\left(s+b_{1}\right)\left(s+b_{2}\right)}$ | |
| $\dfrac{1}{\tau_{1}-\tau_{2}}\left(e^{-t / \tau_{1}}-e^{-t / \tau_{2}}\right)$ | $\dfrac{1}{\left(\tau_{1} s+1\right)\left(\tau_{2} s+1\right)}$ | |
| $\dfrac{b_{3}-b_{1}}{b_{2}-b_{1}} e^{-b_{1} t}+\dfrac{b_{3}-b_{2}}{b_{1}-b_{2}} e^{-b_{2} t}$ | $\dfrac{s+b_{3}}{\left(s+b_{1}\right)\left(s+b_{2}\right)}$ | |
| $\dfrac{1}{\tau_{1}} \dfrac{\tau_{1}-\tau_{3}}{\tau_{1}-\tau_{2}} e^{-t / \tau_{1}}+\dfrac{1}{\tau_{2}} \dfrac{\tau_{2}-\tau_{3}}{\tau_{2}-\tau_{1}} e^{-t / \tau_{2}}$ | $\dfrac{\tau_{3} s+1}{\left(\tau_{1} s+1\right)\left(\tau_{2} s+1\right)}$ | |
| $\dfrac{1}{\tau \sqrt{1-\zeta^{2}}} e^{-\zeta t / \tau} \sin \left(\sqrt{1-\zeta^{2}} t / \tau\right)$ | $\dfrac{1}{\tau^{2} s^{2}+2 \zeta \tau s+1}$ | $(0 \leq\vert\zeta\vert<1)$ |
| $1+\dfrac{1}{\tau_{2}-\tau_{1}}\left(\tau_{1} e^{-t / \tau_{1}}-\tau_{2} e^{-t / \tau_{2}}\right)$ | $\dfrac{1}{s\left(\tau_{1} s+1\right)\left(\tau_{2} s+1\right)}$ | $(\tau_1 \not= \tau_2)$ |
| $1-\dfrac{1}{\sqrt{1-\zeta^{2}}} e^{-\zeta t / \tau} \sin \left[\sqrt{1-\zeta^{2}} t / \tau+\psi\right]$ | $\dfrac{1}{s\left(\tau^{2} s^{2}+2 \zeta \tau s+1\right)}$ | $\footnotesize\begin{aligned}\psi=\tan ^{-1} \dfrac{\sqrt{1-\zeta^{2}}}{\zeta}\end{aligned} \newline (0 \leq\vert\zeta\vert<1)$ |
| $\begin{aligned}1-e^{-\zeta t / \tau}\left[\cos \left(\sqrt{1-\zeta^{2}} t / \tau\right)+\dfrac{\zeta}{\sqrt{1-\zeta^{2}}} \sin \left(\sqrt{1-\zeta^{2}} t / \tau\right)\right]\end{aligned}$ | $\dfrac{1}{s\left(\tau^{2} s^{2}+2 \zeta \tau s+1\right)}$ | $(0 \leq\vert\zeta\vert<1)$ |
| $1+\dfrac{\tau_{3}-\tau_{1}}{\tau_{1}-\tau_{2}} e^{-t / \tau_{1}}+\dfrac{\tau_{3}-\tau_{2}}{\tau_{2}-\tau_{1}} e^{-t / \tau_{2}}$ | $\dfrac{\tau_{3} s+1}{s\left(\tau_{1} s+1\right)\left(\tau_{2} s+1\right)}$ | $(\tau_1 \not= \tau_2)$ |