10.34: numerical methods applied to chemical engineering · the first integral can be handled with...
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10.34: Numerical Methods Applied to
Chemical Engineering
Lecture 18: Differential Algebraic Equations
1
Quiz 1 Results• Mean: 77
Standard deviation: 12
2
0
5
10
15
20
<50 50-59 60-69 70-79 80-89 >89
•
Recap
• Numerical integration
• Implicit methods for ODE-IVPs
3
4
• Improper integrals:
• Of the sort:
• Can be split into two domains of integration
• The first integral can be handled with ODE-IVP methods or polynomial interpolation
• The second must be handled separately through either:
• transformation onto a finite domain
• or substitution of an asymptotic approximation
• This same idea applies to integrable singularities as well.
Recap
Z 1
t0
f(⌧)d⌧
Z 1
t0
f(⌧)d⌧ =
Z tf
t0
f(⌧)d⌧ +
Z 1
tf
f(⌧)d⌧
5
• Improper integrals:
• Example:
Recap
Z tf
0
cos ⌧p⌧
d⌧
⇡Z t0
0
1� ⌧2/2p⌧
d⌧ +
Z tf
t0
cos ⌧p⌧
d⌧
⇡ 2t1/20 � 1
5
t5/20 +
Z tf
t0
cos ⌧p⌧
d⌧
6
• Example:
• Use implicit Euler to solve:
Recap
dx
dt
= �x, x(0) = x0
Give a closed form formula for the numerical solution
Recap• Example:
• Use implicit Euler to solve:
dx
• Let:
• Stability:
7
dt
= �x, x(0) = x0
xk = x(k�t)
xk+1 = xk +�t�xk+1
xk+1 =1
1��t�
xk
xk =
✓1
1��t�
◆k
x0
|1��t�| � 1 ) (1��tRe�)2 + (�tIm�)2 � 1
�t�
Recap• Example:
• Use implicit Euler to solve:
dx
• Numerical solution:
• Exact solution:
• Stability and accuracy do not correlate! 8
dt
= �x, x(0) = x0xk+1 = xk +�t�xk+1
xk+1 =1
1��t�
xk
xk =
✓1
1��t�
◆k
x0
xk = x0ek��t
�t�
Differential Algebraic Equations• Problems of the sort:
f
✓dx
x,
• Called “well-posed” when x, f 2 R
N
• Example: stirred tank 1
9
dt, t
◆= 0, x(0) = x0
c1(t) c2(t)dc2dt
=Q
V(c1(t)� c2(t))
c1(t) = �(t)
c1(0) = �(0), c2(0) = c0
Solution:c1(t) = �(t)c2(t) = c2(0)e
�(Q/V )t
+Q
V
Z t
0�(t0)e�(Q/V )(t�t0)dt0
( ) ( )
cc
Differential Algebraic Equations• Problems of the sort:
f
✓dx
x,
• Called “well-posed” when x, f 2 R
N
• Example: stirred tank 1
10
dt, t
◆= 0, x(0) = x0
c1(t) c2(t)✓
dc2dt � Q
V (c1(t)� c2(t))c1(t)� �(t)
◆= 0
✓c1(0)c2(0)
◆=
✓�(0)c0
◆
Differential Algebraic Equations• Problems of the sort:
• Called “well-posed” when x,
• Example: stirred tank 2
11
f 2 R
N
f
✓x,
dx
dt, t
◆= 0, x(0) = x0
c1(t) c2(t)dc2dt
=Q
V(c1(t)� c2(t))
Solution:
c2(t) = �(t)
c1(t) = c0, c2(0) = �(0)
c2(t) = �(t)
c1(t) = �(t) +V
Q�
12
• Common in dynamic simulations involving conservation, constraints, or equilibria.
• Conservation of total:
• energy
• mass
• momentum
• particle number
• atomic species
• charge
• Models of reaction networks utilizing the pseudo-steady-state approximation.
• Models of control neglecting controller dynamics.
Differential Algebraic Equations
13
• Problems of the sort:
• is called the “mass matrix”
• Stirred tank example 1:
• Semi-explicit form:
• When mass matrix is full rank these problems can be solved by applying typical ODE-IVP techniques to:
Semi-explicit DAEsM
dx
dt= f(x, t), x(0) = x0
M
✓dc2dt � Q
V (c1(t)� c2(t))c1(t)� �(t)
◆= 0
✓1 00 0
◆d
dtc =
✓QV �Q
V�1 0
◆c+
✓0
�(t)
◆
dx
dt= M
�1f(x, t)
Semi-explicit DAEs• Write stirred tank example 2 in semi-explicit form:
14
c1(t) c2(t) dc2dt
=Q
V(c1(t)� c2(t))
c2(t) = �(t)
15
• Problems of the sort:
• is called the “mass matrix”
• Many semi-explicit DAEs can be written in the simplified form:
• where
• are called the differential states
• are called the algebraic states
Semi-explicit DAEsM
dx
dt= f(x, t), x(0) = x0
M
dx
dt= f(x,y, t)
0 = g(x,y, t)
x
y
x(0) = x0
0 = g(x0,y(0), t)
Fully implicit DAEs• Example: mass-spring system
• Conservation of energy:
16
E =1
2m(x)2 +
1
2kx
2
f(x, x, t) =1
2m(x)2 +
1
2kx
2 � E = 0
x = a cos(!t)
has a solution:
1
2
ma2!2sin
2(!t) +
1
2
ka2 cos2(!t) = E
! =
rk
m, a =
rE
k
17
• Many problems contain non-linearities with respect to differentials of the state variables.
• If is full rank, then the DAE can be represented as an equivalent ODE:
Fully implicit DAEs
f (x, x, t) = 0, x(0) = x0
@f
@x
����t,x
df =@f
@x
����t,x
dx+@f
@x
����t,x
dx+@f
@t
����x,x
dt = 0
dx
dt= �
@f
@x
����t,x
!�1 @f
@x
����t,x
dx
dt+
@f
@t
����x,x
!= 0
↓
Fully implicit DAEs• Example: mass-spring system
• Conservation of energy:
18
E =1
2m(x)2 +
1
2kx
2
f(x, x, t) =1
2m(x)2 +
1
2kx
2 � E = 0
transform to ODE
↓
@f
@x
= mx,
@f
@x
= kx,
@f
@t
= 0
x = � k
m
x
Fully implicit DAEs• Example: molecular dynamics
• Conservation of energy:
• Simplectic integrators used to integrate equations of motion while exerting control over error in the total energy.
19
E =1
2mkxk22 +
↓0 = x · (mx+rV (x))
f(x,x, t) =1
2mkxk22 + V (x)� E
df =@f
@x
����t,x
dx+@f
@x
����t,x
dx+@f
@t
����x,x
dt = 0
↓
@f
@x= mx
V (x)
20
• Consider the semi-explicit DAE:
• Consider applying the forward Euler approximation:
• Determining the algebraic states always requires solution of a nonlinear equation.
• DAE simulation methods are inherently implicit.
Numerical Simulation of DAEs
dx
dt= f(x,y, t)
0 = g(x,y, t)
x(0) = x0
x(t+�t)� x(t) = �tf(x(t),y(t), t)
0 = g(x(t),y(t), t)
21
• Consider the fully implicit DAE:
• No way in general to avoid solving systems of nonlinear equations.
• Backward difference approximations for are substituted and time marching solutions determined.
• Example, backward Euler approximation:
Numerical Simulation of DAEs
f(x, x, t) = 0, x(0) = x0
x
f
✓x(tk),
x(tk)� x(tk�1)
tk � tk�1, tk
◆= 0
f
✓x(tk+1),
x(tk+1)� x(tk)
tk+1 � tk, tk+1
◆= 0
solve:
solve:
for:
for:
x(tk)
x(tk+1)
22
• Consider the semi-explicit DAE:
• No way in general to avoid solving systems of nonlinear equations.
• Backward difference approximations for are substituted and time marching solutions determined.
• Example, backward Euler approximation:
Numerical Simulation of DAEs
x
solve:
dx
dt= f(x,y, t)
x(0) = x0
0 = g(x,y, t)
0 =x(tk)� x(tk�1)
tk � tk�1� f(x(tk),y(tk), tk)
0 = g(x(tk),y(tk), tk)
for:x(tk)
23
• How suitable are such approaches?
• Consider stirred tank example 1:
Numerical Simulation of DAEs
dc2dt
=Q
V(c1(t)� c2(t))
c1(t) = �(t)
c1(tk) = �(tk)
c2(tk) =1
1 + QV (tk � tk�1)
✓c2(tk�1) +
Q
V(tk � tk�1)c1(tk)
◆
Apply backward Euler method:
+O((tk � tk�1)2)
24
• How suitable are such approaches?
• Consider stirred tank example 2:
Numerical Simulation of DAEs
dc2dt
=Q
V(c1(t)� c2(t))
c2(t) = �(t)
c2(tk) = �(tk)
c1(tk) = c2(tk) +V
Q
✓c2(tk)� c2(tk�1)
tk � tk�1
◆
Apply backward Euler method:
+O(tk � tk�1)
25
• How suitable are such approaches?
• Consider the system of DAEs:
Numerical Simulation of DAEs
What is the exact solution?Apply backward Euler method:
c2 = c1(t)
c3 = c2(t)
0 = c3(t)� �(t)
c3(tk) = �(tk)
c2(tk) =c3(tk)� c3(tk�1)
tk � tk�1
c1(tk) =c2(tk)� c2(tk�1)
tk � tk�1
+O(tk � tk�1)
+O(1) !
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10.34 Numerical Methods Applied to Chemical EngineeringFall 2015
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