05 modeling dynamic and static behavior of chemical processes
TRANSCRIPT
-
7/28/2019 05 Modeling Dynamic and Static Behavior of Chemical Processes
1/47
Modeling Dynamic and Static
Behavior of Chemical Processes
Cheng-Liang ChenPSELABORATORY
Department of Chemical EngineeringNational TAIWAN University
-
7/28/2019 05 Modeling Dynamic and Static Behavior of Chemical Processes
2/47
Chen CL 1
State Variables and State Equations
- State Variables:A set of fundamental dependent quantities whose values
will describe the natural state of a given system
(temperature, pressure, flow rate, concentration )
- State Equations:
A set of equations in the state variables above which
will describe how the natural state of a given system
changes with time
-
7/28/2019 05 Modeling Dynamic and Static Behavior of Chemical Processes
3/47
Chen CL 2
Principle of Conservation of A Quantity S
S =
total massmass of individual components
total energy
momentum
-
7/28/2019 05 Modeling Dynamic and Static Behavior of Chemical Processes
4/47
Chen CL 3
accumulation of S
within a system
time period=
flow of S
in the system
time period
flow of S
out the system
time period
+
amount of S generated
within the system
time period
amount of S consumed
within the system
time period
-
7/28/2019 05 Modeling Dynamic and Static Behavior of Chemical Processes
5/47
Chen CL 4
- Total Mass Balance:
d(V)
dt
= i:inlet iFi j:outlet jFj
- Mass Balance on Component A:
dnAdt
= d(cAV)dt
=i:inlet
cAi
Fi
j:outlet
cAj
Fj rV
-
Total Energy Balance:dE
dt=
d(U + K+ P)
dt=i:inlet
iFihi
j:outlet
jFjhjQWs
C C
-
7/28/2019 05 Modeling Dynamic and Static Behavior of Chemical Processes
6/47
Chen CL 5
Mathematical ModelA Stirred Tank Heater
- Mathematical model of a process
= state equations with associated state variables
Ch CL 6
-
7/28/2019 05 Modeling Dynamic and Static Behavior of Chemical Processes
7/47
Chen CL 6
- Total mass in tank: V = Ah
- Total energy of liquid in tank:
E = U + K+ PdU
dt
dH
dt;
dK
dt=
dP
dt= 0
H = Ahcp T Tref- State variables: h, T
- Total mass balance:
d(Ah)
dt= Fi F
=c
A
dh
dt = Fi
F
-
7/28/2019 05 Modeling Dynamic and Static Behavior of Chemical Processes
8/47
Ch CL 8
-
7/28/2019 05 Modeling Dynamic and Static Behavior of Chemical Processes
9/47
Chen CL 8
- Summary: State equations
Adh
dt= Fi F
AhdT
dt= Fi (Ti T) +
Q
cp
-Summary: variables
state variables: h, T
output variables: h, T
disturbances: Ti, Fi
manipulated variables: Q, F
parameters: A,,cp
Ch CL 9
-
7/28/2019 05 Modeling Dynamic and Static Behavior of Chemical Processes
10/47
Chen CL 9
Mathematical ModelA Stirred Tank Heater (cont)
- Assumed initial steady states:
0 = Adh
dt= Fi,s Fs
0 = AhdT
dt= Fi,s (Ti,s Ts) +
Qscp
Chen CL 10
-
7/28/2019 05 Modeling Dynamic and Static Behavior of Chemical Processes
11/47
Chen CL 10
- Temperature response to a step decrease in inlet temperature:
- Dynamic response to a step decrease in inlet flow rate:
Chen CL 11
-
7/28/2019 05 Modeling Dynamic and Static Behavior of Chemical Processes
12/47
Chen CL 11
Additional Element:Transport Rate Equations
- Transport Rate Equations:
To describe rate of mass, energy, and momentum transfer between
a system and its surroundings
- Example: a stirred tank heater
heat supplied by steam:
Q = U At (Tst T)
Chen CL 12
-
7/28/2019 05 Modeling Dynamic and Static Behavior of Chemical Processes
13/47
Chen CL 12
Additional Element:Kinetic Rate Equations
- Kinetic Rate Equations:
To describe rates of chemical reactions taking place in a system
-
Example: a 1st-order reaction in a CSTR reaction rate equation:
r = k0eE/RTc
A
Chen CL 13
-
7/28/2019 05 Modeling Dynamic and Static Behavior of Chemical Processes
14/47
Chen CL 13
Additional Element:Reaction and Phase Equilibrium Relationships
- Reaction and Phase Equilibrium Relationships:To describe equilibrium situations reached during a chemical
reaction or by two or more phases
- Example: a flash drum temperature of liquid phase= temperature of vapor phase
pressure of liquid phase
= pressure of vapor phase
chemical potential of component i
in liquid phase =
chemical potential of component i
in vapor phase
Chen CL 14
-
7/28/2019 05 Modeling Dynamic and Static Behavior of Chemical Processes
15/47
Chen CL 14
Additional Element:Equations of States
- Equations of States:
To describe the relationshipamong intensive variables
describing the
thermodynamic state
of a system
- Example: a flash drum
Ideal gas law for vapor phase:
pVvapor
= (moles of A + moles of B)RT
= mass of A + mass of Baverage MW
RT
=mass of A + mass of B
yA
MA
+ yB
MB
RT
vapor =mass of A + mass of B
Vvapor
= [yA
MA
+ yB
MB
] p
RT
liquid
= (T, xA
)
Chen CL 15
-
7/28/2019 05 Modeling Dynamic and Static Behavior of Chemical Processes
16/47
Chen CL 15
Dead Time
-Dead Time:
Whenever an input variable of a system changes
there is a time interval (short or long) during which
no effect is obsrved on outputs of the system
- dead time, transportation lag, pure delay,
distance-velocity lag
Chen CL 16
-
7/28/2019 05 Modeling Dynamic and Static Behavior of Chemical Processes
17/47
Chen CL 16
- Example: liquid through a pipe
A: temperature of inlet changes
B: temperature of outlet response
dead time: d
d =volume of pipe
volumetric flow rate=
A L
A Uav=
L
Uav
Tout(t) = Tin(t d)
-
7/28/2019 05 Modeling Dynamic and Static Behavior of Chemical Processes
18/47
-
7/28/2019 05 Modeling Dynamic and Static Behavior of Chemical Processes
19/47
Chen CL 19
-
7/28/2019 05 Modeling Dynamic and Static Behavior of Chemical Processes
20/47
Chen CL 19
- Total Mass Balance:
d(V)
dt= iFi F 0
=c= dVdt
= Fi F
- Mass Balance on Component A:
(r: rate of reaction per unit volume)
dnA
dt=
d(cA
V)
dt= c
AiFi cAF rV
Vdc
A
dt
+ cA
dV
dt=FiF= c
AiFi cAF k0e
E/RTcA
V
dc
A
dt=
FiV
cAi c
A
k0e
E/RTcA
Chen CL 20
-
7/28/2019 05 Modeling Dynamic and Static Behavior of Chemical Processes
21/47
Chen CL 20
- Total Energy Balance:total energy E = U + K+ P = U H(T, n
A, n
B) (enthalpy)
dE
dt
=dU
dt
dH
dt
= iFihi(Ti) F h(T)Q (1)
alsodH
dt=
H
TV cp
dT
dt+
H
nA
HA(T)
dnA
dt+
H
nB
HA(T)
dnB
dt
note
dnA
dt =
d(cA
V)
dt = cAiFi cAF rVdn
B
dt=
d(cB
V)
dt= c
BiFi
=0
cB
F + rV
dH
dt= V cp
dT
dt+ H
A
cAi
Fi cAF rV
+ HB cBF rV (2)
Chen CL 21
-
7/28/2019 05 Modeling Dynamic and Static Behavior of Chemical Processes
22/47
(1) = (2) V cpdT
dt =
HA cAiFi cAF rV HB cBF rV+Fi ihi(Ti)
ihi(T) cAi
HA(T)
+icpi(TiT)
F h(T) cAHA(T)+c
BHB(T)
Q
= HA
cAi
Fi (i)
+ HA
cA
F (ii)
+HA
rV
+ HB
cB
F (iii)
HB
rV + FicAiHA (i)
+Fiicpi(Ti T) F cAHA (ii)F c
BH
B (iii)Q
=
HA H
B
Hr
rV + Fiicpi(Ti T)Q
=i,cp=cpi= VdT
dt= Fi(Ti T) +
(Hr)
cp JrV
Q
cp
Chen CL 22
-
7/28/2019 05 Modeling Dynamic and Static Behavior of Chemical Processes
23/47
- Summaries:
state var.s: V, cA
, T
state eqn.s:dV
dt = Fi F
dcA
dt=
FiV
cAi c
A
k0e
E/RTcA
dT
dt =
Fi
V (Ti
T) + Jk0e
E/RT
cA Q
cpV
output var.s: V, cA
, T
input var.s: cAi
, Fi, Ti, Q , F
manip. var.s: Q, F
disturbances: cAi
, Fi, Ti
const. par.s: , cp, (Hr), k0, E , R
Chen CL 23
-
7/28/2019 05 Modeling Dynamic and Static Behavior of Chemical Processes
24/47
Additional Examples of Mathematical Modeling
An Ideal Binary Distillation Column
Chen CL 24
-
7/28/2019 05 Modeling Dynamic and Static Behavior of Chemical Processes
25/47
- Assumptions:
constant vapor holdup:
equal molar heats of vaporization for A and B
negligible heat loss constant relative volativility
100% tray efficiency
V = V1 = = VNyi =
xi1 + ( 1)xi
neglect dynamics of condenser and reboiler
neglect momentum balance for each trayleaving liquid = Li = f(Mi), i = 1, , N
liquid holdup = Mi
Chen CL 25
-
7/28/2019 05 Modeling Dynamic and Static Behavior of Chemical Processes
26/47
- State Equations (1): feed tray (i = f)
total mass:dMf
dt= Ff + Lf+1 + Vf1 Lf Vf
= Ff + Lf+1 Lf
comp A:d(Mfxf)
dt= Ffcf + Lf+1xf+1 + Vf1yf1 Lfxf Vfyf
Chen CL 26
-
7/28/2019 05 Modeling Dynamic and Static Behavior of Chemical Processes
27/47
- State Equations (2): top tray (i = N)
total mass:dM
N
dt= F
R+ V
N1 L
N V
N
= FR LN
comp A:d(M
NxN
)
dt= F
RxD
+ VN1
yN1
LN
xN V
NyN
Chen CL 27
-
7/28/2019 05 Modeling Dynamic and Static Behavior of Chemical Processes
28/47
- State Equations (3): bottom tray (i = 1)
total mass:dM1
dt= L2 L1 + V V1
= L2 L1
comp A:d(M1x1)
dt= L2x2 + V yB L1x1 V1y1
Chen CL 28
-
7/28/2019 05 Modeling Dynamic and Static Behavior of Chemical Processes
29/47
- State Equations (4): ith tray (i = 2, , N 1; i = f)
total mass:dMi
dt= Li+1 Li + Vi1 Vi
= Li+1 Li
comp A: d(Mixi)
dt= Li+1xi+1 Lixi + Vi1yi1 Viyi
Chen CL 29
-
7/28/2019 05 Modeling Dynamic and Static Behavior of Chemical Processes
30/47
- State Equations (5): reflux drum
total mass:dM
RD
dt= V
N F
R F
D
comp A: d(MRDxD)dt
= VN
yN (F
R+ F
D)x
D
Chen CL 30
-
7/28/2019 05 Modeling Dynamic and Static Behavior of Chemical Processes
31/47
- State Equations (6): column base
total mass:dM
B
dt= L1 V FB
comp A:d(M
BxB
)
dt= L1x1 V yB FBxB
Chen CL 31
-
7/28/2019 05 Modeling Dynamic and Static Behavior of Chemical Processes
32/47
- Relationships:
equilibrium relationships:
yi =xi
1 + ( 1)xi i = 1, , f, , N; B
hydraulic relationships: (Francis weir formula)
Li = f(Mi) i = 1, , f, , N
- State Variables:
liquid holdups:
M1, M2, , Mf, , MN; MRD, MB
liquid concentrations:
x1, x2, , xf, , xN; xD, xB
Chen CL 32
-
7/28/2019 05 Modeling Dynamic and Static Behavior of Chemical Processes
33/47
- Summaries:
2N + 4 nonlinear differential equations (state eqn.s)
2N + 1 algebraic equations (equilibrium and hydraulic)
example: N = 20 trays
2N + 4 = 2(20) + 4 = 44 nonlinear diff. eqn.s
2N + 1 = 2(20) + 1 = 41 algebraic equations
Chen CL 33
-
7/28/2019 05 Modeling Dynamic and Static Behavior of Chemical Processes
34/47
Modeling Considerationsfor Control Purposes
- State-variables model
input-output model (convenient for control)
- Degrees of freedom ( df) inherent in the process extent of control problem to be solved
Chen CL 34
-
7/28/2019 05 Modeling Dynamic and Static Behavior of Chemical Processes
35/47
- Input-Output Model:
output = f(input variables)
yi = f(m1, , mk; d1, , dt) i = 1, , m
Chen CL 35
-
7/28/2019 05 Modeling Dynamic and Static Behavior of Chemical Processes
36/47
- Example: Input-Output Model for CSTR
Assumptions: Fi = F dV/dt = 0
Chen CL 36
-
7/28/2019 05 Modeling Dynamic and Static Behavior of Chemical Processes
37/47
Total Energy Balance:
VdT
dt= Fi(Ti T) +
Q
cp
Q = U At(Tst T)
dT
dt+
FiV
+U AtV cp
a1/+K
T =FiV1/
Ti +U AtV cp K
Tst
dT
dt
+ aT = 1Ti + KTst
SS: 0 + aTs =1Ti,s + KTst,s
d(T Ts)
dt+ a (T Ts)
T
= 1 (Ti Ti,s) T
i,s
+K(Tst Tst,s) T
st
dT
dt + aT
=1T
i + KT
st
T
(t) = c1eat +
t0
1
T
i + KT
st
dt
initial: T
(t = 0) = 0 c1 = 0
T
(t) = t
0 1
T
i + KT
st dt
Chen CL 37
-
7/28/2019 05 Modeling Dynamic and Static Behavior of Chemical Processes
38/47
Block Diagram: inputs (T
i(t), T
st(t)) output (T
(t))
This example: output variables = state variables
Chen CL 38
Di ill i bl bl !
-
7/28/2019 05 Modeling Dynamic and Static Behavior of Chemical Processes
39/47
Distillation: output variables = state variables!
State variables:
liquid holdups:
M1, M2, , Mf, , MN; MRD, MB
liquid concentrations:
x1, x2, , xf, , xN; xD, xB
Output variables:
distillate rate and composition: FD
, xD
bottom rate and composition: FB, xB
Chen CL 39
DOF D f F d
-
7/28/2019 05 Modeling Dynamic and Static Behavior of Chemical Processes
40/47
DOF: Degree of Freedom
-
Degrees of Freedom (DOF):# of independent variables that must be specified in order to define
a process completely
DOF = (# Var.s) (# Indep. Eq.s)
Chen CL 40
E l i d k h
-
7/28/2019 05 Modeling Dynamic and Static Behavior of Chemical Processes
41/47
- Example: stirred tank heater
mathematical model: # of eq.s = 2
Adhdt
= Fi F
AhdT
dt= Fi (Ti T) +
Q
cp
# of variables = 6 (h, Ti, T , F , F i, Q) DOF = 6 - 2 = 4
specify Ti, Fi, F , Q h(t), T(t)
in order to specify a process completely
the # ofDoF
should be zero
Chen CL 41
- E l bi di ill i l
-
7/28/2019 05 Modeling Dynamic and Static Behavior of Chemical Processes
42/47
- Example: binary distillation column
DOF = (4N + 11) (4N + 5) = 6
Chen CL 42
D f F d f A P
-
7/28/2019 05 Modeling Dynamic and Static Behavior of Chemical Processes
43/47
Degrees of Freedom of A Process
-
f = DOF = V E = (# Var.s) (# Indep. Eq.s)
- Case 1: DOF = 0
unique values of the V variables
the process is exactly specified
- Case 2: DOF > 0
multiple solutions result from the E equations
can specify arbitrarily f of the V variables
the process is underspecified by f equations
- Case 3: DOF < 0
no solution to the E equations
the process is overspecified by f equations
Chen CL 43
DOF d P C t ll
-
7/28/2019 05 Modeling Dynamic and Static Behavior of Chemical Processes
44/47
DOF and Process Controllers
-
An under-specified process with DOF = f > 0
- Q: how to reduce DOF to zero
to specify system completely with unique behavior ?
from external world: disturbances
to add control loops
- Control loop:
additional equation between MV and CV
additional variable: set-point
same: DOF
difference: specify MV specify set-point
Chen CL 44
- E l ti d t k h t ith t t l l
-
7/28/2019 05 Modeling Dynamic and Static Behavior of Chemical Processes
45/47
- Example: stirred tank heater with two control loops
DOF = 4 DOF = 0 if we specify
Ti, Fi from external world ( disturbances) set-points of the two controllers
Chen CL 45
- E l bi di till ti l ( DOF 6)
-
7/28/2019 05 Modeling Dynamic and Static Behavior of Chemical Processes
46/47
- Example: binary distillation column ( DOF = 6)
specification of disturbances (external world):
feed rate (Ff) and feed composition (cf)
DOF = 6 DOF = 4
specification of control objectives ( set-points):
(I) for products:
xD
: distillate composition
xB
: bottom stream composition
(II) for operational feasibility:
MRD: liquid holdup in reflux drum M
B: liquid holdup at base of column
four control loops
DOF = 6 DOF = 4 DOF = 0
Chen CL 46
-
7/28/2019 05 Modeling Dynamic and Static Behavior of Chemical Processes
47/47
Note: other alternative control objectives
(1) keep at desired FD
, xD
, MRD
, MB
(2) keep at desired FB
, xB
, MRD
, MB