LECTURE #2
Last Lecture
• Modeling and simulation
• System Classifications
• Models Classifications
• Models: steady and dynamic, Lumped
and distributed, Deterministic and
stochastic
• Model Building
• Process: batch, continuous
Conservation Law: Momentum
• linear momentum (p) of a mass (m)
moving with velocity (v) is
p = mv
• Rate of momentum in + Rate of Generation of
momentum = Rate of momentum out + Rate of
Accumulation of momentum
• Newton's second law
d ( mv )
= F
dt
Conservation Laws: Energy
• Rate of energy in + Rate of Generation of energy =
Rate of energy out + Rate of accumulation of energy
±amount of energy exchanged with the surrounding
Transport rates: Mass
• Bulk and diffusion flow :nAu= jAu + rAvu
• Total flux = diffusive flux + bulk flux
• Diffusive Flux: Fick's law for binary
mixture
dwA dr A
j Au = − rD AB j Au = − D AB
du du
dx A dC A
J Au = −CD AB J Au = − D AB
du du
Transport rates: Momentum
• Total flux pyx of the x-component in the
y-direction is the sum of the convection
term and diffusive term tyx
pyx = tyx + (rvx)vy
v x
t yx = −
y
Transport rates: Energy
• total energy flux eu (J/s.m2) of a fluid at
constant pressure flowing with a
velocity vu in the u-direction can be
expressed as:
• eu = qu + (rCpT)vu
• Heat Diffusion, conduction q:
T
qu = − k
u
One-dimensional Transport
laws for molecular diffusion
Transport Law Flux Transport gradient
Type property
Mass Fick's JAu D dC A
du
Heat Fourrier qu k dT
du
Momentum Newton tux dvx
du
Macroscopic Transfer Rates
• gradient is the difference between the
bulk properties, i.e. concentration or
temperature in two medium in contact
J A = K C A
q = U T
Thermodynamic relations
• Equation of state
PV = nRT
• Enthalpies
~
h = C p (T − Tref )
~
H = C p (T − Tref ) +
• Internal energy
• u=CV (T-Tref)
•
Phase equilibrium
• yAi = F (xAi) Gas-phase mixture
of A in gas G
Liquid -phase solution
of A in liquid L
• Henry's law yAG yAi
xAi
• Raoult's law NA
xAL
yii P = xi i Pi S
Interface
Figure 1.7: Equlibrium at the interface
Chemical kinetics
The overall rate R in moles/m3s of a
chemical reaction is defined by:
1 dni
R=
V dt
Simple Reaction: A+B → C
R= k CA CB
−E
k = ko e RT
Degree of Freedom
• For a processing system described by a set of Ne independent
equations and Nv variables, the degree of freedom f is
• F = Nv − Ne
• f = 0. The system is exactly determined (specified) system .
Thus, the set of balance equation has a finite number of
solutions (one solution for linear systems)
• f < 0. The system is over-determined (over-specified) by f
equations. f equations have to be removed for the system to
have a solution.
• f > 0. The system is under-determined (under-specified) by f
equations. The set of equation, hence, has infinite number of
solution .
Model solution
• It would be ideal to be able to solve the model
analytically, that is to get closed forms of the state
variables in term of the independent variables.
• Unfortunately this seldom occurs for chemical
processes. why? The reason is that the vast majority
of chemical processes are nonlinear.
• Solution of process models is usually carried out
numerically
Validation
• Model verification (validation) is the last
and the most important step of model
building. Reliability of the obtained
model depends heavily on faithfully
passing this test. Implementation of the
model without validation may lead to
erroneous and misleading results.
Examples of Mathematical Models for
Chemical Processes
Lumped Parameter Systems
• Liquid Storage Tank
• Our objective is to develop a model for
the variations of the tank holdup, i.e.
volume of the tank
Ff
rf V
Fo
ro
Liquid Storage Tank Assumptions
• Perfectly mixed (Lumped) →
density of the effluent is the
same as that of tank content.
• Isothermal
Ff
rf V
Fo
ro
Liquid Storage Tank Model
Rate of mass accumulation = Rate of mass in - rate of mass out
m t +t − m t = r f Ff t − ro Fo t
m −m t
lim t +t
= r f Ff − ro Fo
t
dm d ( rV )
= = r f Ff − r o Fo Ff
rf V
dt dt Fo
ro
Liquid Storage Tank Model
• Under isothermal conditions we assume that the density of the
liquid is constant.
dV
= F f − Fo
dt
dL
A = F f − Fo
dt
Liquid Storage Tank Model
Degree of Freedom
• Parameter of constant values: A
• Variables which values can be externally fixed
(Forced variable): Ff
• Remaining variables: L and Fo
• Number of equations: 1
• Number of remaining variables – Number of
equations = 2 – 1 = 1
Fo = L
Isothermal CSTR
• Our objective is to develop a model for the variation
of the volume of the reactor and the concentration of
species A and B.
• a liquid phase chemical reactions taking place:
k
A⎯
⎯→ B
Ff
rf
CAf V Fo
CBf ro
CAo
CBo
Isothermal CSTR: Assumptions
• Perfectly mixed
• Isothermal
• The reaction is assumed to
be irreversible and of first
order. Ff
rf
CAf V Fo
CBf ro
CAo
CBo
Isothermal CSTR: Model
▪ Component balance
▪ Flow of moles of A in: Ff CAf
▪ Flow of moles of A out:Fo CAo
▪ Rate of accumulation: dn d (VC )
A
=
dt dt
▪ Rate of generation: rV
▪ r= k CA
where r (moles/m3s) is the rate of reaction.
Isothermal CSTR: Model
d (VC A )
= F f C Af − Fo C A − rV
dt
d (VC A ) d (C A ) d (V )
=V + CA = F f C Af − Fo C A − rV
dt dt dt
d (C A )
V = F f ( C Af − C A ) − kC AV
dt
Isothermal CSTR: Degree of Freedom
• Parameter of constant values: A
• (Forced variable): Ff and CAf
• Remaining variables: V, Fo, and CA
• Number of equations: 2
• The degree of freedom is
f= 3 − 2 =1
The extra relation is obtained by the relation
between the effluent flow Fo and the level in
open loop
