458.308 Process Control & Design
Jong Min Lee
Lecture 3: Dynamic Simulation & Analysis (part 1)
0 0.5 1 1.5 2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Time
Tank Level (h)
Linear Approximation Exact Nonlinear Solution
Motivation
• Find a steady-state corresponding to given values of inputs (steady-state solution)
• Find how state variables change over time when inputs change from a previous steady state
We have a set of differential equations as a process model.
What can we do with it?
Two Things
1. Steady-State Solution
Vector notation
˙
x = f (x, u, p) y = g (x, u, p)
x : state var. y : output var.
u : input var. p : parameters
Steady-state solution
˙
x = dx
dt = 0 , f (x, u, p) = 0
In this case, all inputs and parameters would be specified → D.O.F. = 0
Consider the stirred tank heater in Lec. 2. For steady-state inlet and outlet flow rates of 100 L/min, a liquid volume of 500 L and, inlet and outlet temperatures of 20℃ and 40℃, respectively, find the steady-state heating rate Q. The liquid density is 1kg/L and heat
capacity is 4 kJ/(kg ℃).
dV
dt = F
iF
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dT
dt = F
V (T
iT ) + Q V ⇢c
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F
i= F = 100 [L/min]
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dV
dT
dt = 0 = 100
500(20 40) + Q 500 · 1 · 4
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A stirred-tank blending process with a constant liquid holdup of 2 m3 is used to blend two streams whose densities are both approximately 900 kg/m3. The density does not change during mixing.
(a) Assume that the process has been operating for a long period of time with flow rates of w1 = 500 kg/min and w2 = 200 kg/min, and feed composition (mass fractions) of x1 = 0.4 and x2 = 0.75. What is the steady-state value of x?
d V ¯
dt = 1
⇢ ( ¯ w
1+ ¯ w
2w) = 0 ¯
<latexit sha1_base64="7twrzhVuJWTGaJVEWWzf8d86XnM=">AAACXnicbZDLSgMxFIbT8V5vVTeCm2ARKmKZEUE3QsGNggsF2wqdUs5kMjY0cyHJKCXkmXwaF27UJxHTaQVtPRD4+P9zcpI/yDiTynXfSs7c/MLi0vJKeXVtfWOzsrXdkmkuCG2SlKfiIQBJOUtoUzHF6UMmKMQBp+1gcDny209USJYm92qY0W4MjwmLGAFlpV7l2o8EEB36AQjdMkaHyuALPFY9o33RT02tcJ9Nz8NH+IdP8PEPH9oJt1epunW3KDwL3gSqaFK3vcqXH6Ykj2miCAcpO56bqa4GoRjh1JT9XNIMyAAeacdiAjGVXV182eADq4Q4SoU9icKF+ntCQyzlMA5sZwyqL6e9kfif18lVdN7VLMlyRRMyXhTlHKsUj/LDIROUKD60AEQw+1ZM+mDTUjblP1tGdwsZSXOAOWQcCK1jv1hIhL4xZRuYNx3PLLRO6p5b9+5Oq43GJLpltIf2UQ156Aw10BW6RU1E0At6Re/oo/TpLDrrzua41SlNZnbQn3J2vwFbVrea</latexit><latexit sha1_base64="7twrzhVuJWTGaJVEWWzf8d86XnM=">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</latexit><latexit sha1_base64="7twrzhVuJWTGaJVEWWzf8d86XnM=">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</latexit><latexit sha1_base64="7twrzhVuJWTGaJVEWWzf8d86XnM=">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</latexit>
0 = ¯
<latexit sha1_base64="aXNa2gTrCPcxOu/0tR0Ij/cVDC8=">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</latexit><latexit sha1_base64="aXNa2gTrCPcxOu/0tR0Ij/cVDC8=">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</latexit><latexit sha1_base64="aXNa2gTrCPcxOu/0tR0Ij/cVDC8=">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</latexit><latexit sha1_base64="aXNa2gTrCPcxOu/0tR0Ij/cVDC8=">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</latexit>w
1+ ¯ w
2w ¯
¯
w
<latexit sha1_base64="AFZQKmNWuD8XD5KJdvBJYvInlnM=">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</latexit><latexit sha1_base64="AFZQKmNWuD8XD5KJdvBJYvInlnM=">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</latexit><latexit sha1_base64="AFZQKmNWuD8XD5KJdvBJYvInlnM=">AAACRnicbZDLSgMxFIbP1Futt6pLN8EiCELJFKVuCgU3LlxUsFpoSzmTZjSYuZBklDLMc/g0btVX8CXciTsxbQdR64GEj//ckt+LpdCG0lenMDe/sLhUXC6trK6tb5Q3ty51lCjG2yySkep4qLkUIW8bYSTvxIpj4El+5d2ejPNXd1xpEYUXZhTzfoDXofAFQ2OlQdnteajS+4w0SE4Dlxx8c83qR5RapWbvBqlTOihXaJVOgsyCm0MF8mgNyp+9YcSSgIeGSdS669LY9FNURjDJs1Iv0TxGdovXvGsxxIDrfjr5Wkb2rDIkfqTsCQ2ZqD87Ugy0HgWerQzQ3Oi/ubH4X66bGP+4n4owTgwP2XSRn0hiIjL2iQyF4szIkQVkSti3EnaDCpmxbv7aMp6ttK+zPSIxlsh4lfQmC5lKz7KSNcz9a88sXNaqLq2654eVZjO3rgg7sAv74EIdmnAKLWgDgwd4hCd4dl6cN+fd+ZiWFpy8Zxt+RQG+AInxrZ4=</latexit><latexit sha1_base64="AFZQKmNWuD8XD5KJdvBJYvInlnM=">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</latexit>= ¯ w
1+ ¯ w
2= 500 + 200 = 700
d¯ x
dt = w ¯
1V ¯ ⇢ (¯ x
1x) + ¯ w ¯
2V ¯ ⇢ (¯ x
2x) = 0 ¯
<latexit sha1_base64="/47kfTC56m6/UeqmMAN9UPUMCSI=">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</latexit><latexit sha1_base64="/47kfTC56m6/UeqmMAN9UPUMCSI=">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</latexit><latexit sha1_base64="/47kfTC56m6/UeqmMAN9UPUMCSI=">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</latexit><latexit sha1_base64="/47kfTC56m6/UeqmMAN9UPUMCSI=">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</latexit>
¯
x = w ¯
1x ¯
1+ ¯ w
2x ¯
2¯
w
1+ ¯ w
2= 500 ⇥ 0.4 + 200 ⇥ 0.75
700 = 0.5
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Matlab: fsolve
1. Write the function (e.g., blending_ss.m) describing the steady-state equations.
2. Use ‘fsolve’ as x = fsolve(‘blending_ss’, x0) where
x0 is an initial guess.
2. Dynamic Solution: Integration
A stirred-tank blending process with a constant liquid holdup of 2 m3 is used to blend two streams whose densities are both approximately 900 kg/m3. The density does not change during mixing.
(b) Suppose that w1 changes suddenly from 500 to 400 kg/min and remains at the new value. Determine an expression for x(t) and plot it.
500
400
Time
Blending Process
0.5
Time
?
Matlab: ode45
1. Write the function (e.g., blending_dyn.m) describing the odes
2. Use ‘ode45’ as [t, x] = ode45(‘blending_dyn’, [t0 tf], x0) in the command window where t0
(usually 0) is the initial time, tf is the final time, and x0 is the initial point (or initial condition)
3. You should get the same graph with Figure 2.2 (b)
ode45 is based on the explicit (4,5) - Runge-Kutta method
0.504 0.506 0.508 0.51 0.512 0.514 0.516 0.518
x