Problems: Lumped parameter systems
Problem 1
Consider the three storage tanks in series shown by Figure X-1.
1. Write the mathematical model that describes the dynamic behavior of the
process. Assume the density of the fluid constant.
2. Identify the states of the process and the degrees of freedom.
F
Fr
F
F
h2
h1
h3
F3
Figure X-1
Problem 2
Consider the three cooling storage tanks in series shown in Figure X-2.
Ff Tf
F0 T0
F1 T1
F3 T3
h2
h1
wc1 Tc1
F4 T4
h3
wc2 Tc2
wc3 Tc3
Figure X-2
1. Assuming constant fluid properties, write the mathematical model for the
process. Identify the states and the degrees of freedom.
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2. Assume now the following non-isothermal reversible first-order liquid-phase
reaction: A  B
is taking place in the three
above CSTR in series.
Assuming all CSTRs are adiabatic, write down the unsteady state model for the
process if the holdups is kept constant in all reactors.
Problem 3
Consider the well-stirred non-isothermal reactor shown in Figure X-3 below. Write
down the mathematical equations that describe the dynamic behavior of the
fundamental quantities of the process. Consider an exothermic liquid-phase second
order reaction rate in the form of A  B is taking place. The density is assumed to be
constant, develop the necessary equations describing the process dynamic behavior.
F CA1
h1
F CA3
Figure X-3
Problem 4
Consider a two CSTRs in series with an intermediate mixer introducing a second feed
as shown in Figure X-4. A first order irreversible exothermic reaction: A  B is carrier
out in the process. Water at ambient temperature (Tc1i and Tc2i) is used to cool the
reactors. The densities and heat capacities are assumed to be constant and independent
of temperature and concentration. Develop the necessary equations describing the
process dynamic behavior. Note that the mixer has negligible dynamics and that the
inlet feed to CSTR2 has the same temperature as that of the outlet of CSTR1.
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Q1, C1f, T1f
QC1, TC1
Q3, C3, T3
QC1, TC1i
Q2, C2, T2
Mixer
QC2, TC2
CSTR 1
QC2, TC2i
Q4, C4, T4
CSTR 2
Figure X-4
Problem 5:
Consider a thermometer bulb with pocket is immersed in hot fluid of temperature T1 as
shown in Figure X-5 below. The pocket temperature is T2 (assumed uniform through the
thickness) and that for the bulb is T3. Write the model equation that describes how T2
and T3 varies with time.
T1
T2
T3
Figure X-5
Problem 6
Consider the single-effect steam evaporator shown in Figure X-6. A salty water (brine)
with mass fraction Cb0 and mass flow rate B0 is fed to the evaporator where it is heated
with saturated steam with mass flow rate W. The concentrated product comes out of the
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evaporator with mass flow rate B1 and mass fraction Cb1 while the vapor is withdrawn
from the top with mass flow rate V. Develop the dynamic model that describe the
process behavior. Assume the heat supplied by the steam is mainly equal to the heat
used for vaporizing the brine.
Vapor
Steam
Condensate
Feed
Concentrated
liquid
Figure x-6
Problem 7
Consider the process shown in Figure X-7. A stream of pure component A is mixed with
another stream of a mixture of component A and B in an adiabatic well stirred mixing
tank. The effluent is fed into an adiabatic CSTR where the following reaction takes
place: A + B  C. Assume the process is isothermal. Develop the dynamic model for
the process and determine the degrees of freedom assuming constant fluid properties.
Q CA1
Q2 CA2
Q3 CA3
Q4 CA4
Figure X-7
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Problem 8:
Consider the following irreversible first-order liquid-phase reaction:
k1
k2
A 
B 
C
where k1 and k2 are the reaction rate constants in sec-1.
1. Write down the unsteady state model for the process if the reaction takes place
in an isothermal well-mixed CSTR.
2. Write down the unsteady state model for the process if the reaction takes place
in a non-isothermal well-stirred batch reactor, where the heat needed for the
endothermic reaction is supplied through electrical coil.
3. Repeat part 2 but assuming that the reactor is cooled by cold fluid flowing in a
jacket and that the reactor wall has high resistance to heat transfer.
Problem 9:
A perfectly mixed non-isothermal adiabatic reactor carries out a simple first-order
exothermic reaction, A  B in the liquid phase. The product from the reactor is cooled
from the output temperature T to Tc and then introduced to a separation unit where the
un-reacted A is separated from the product B. The feed to the separation unit is split into
two equal parts top product and bottom product. The bottom product from the
separation unit contains 95% of the un-reacted A in the effluent of the reactor and 1% of
B in the same stream. The bottom product which is at Temperature Tc (since the
separation unit is isothermal) is recycled and mixed with the fresh feed of the reactor
and the mixed stream is heated to temperature Tf before being introduced to the reactor.
Write the steady state mass and energy balances for the whole process assuming
constant physical properties and heat of reaction. Discuss also the degree of freedom of
the resulting model. The process is depicted in figure X-8
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V= 0.5 F
CSTR
F, Tf
C A, C B , T c
Separator
F
Fo, To ,CAo
L = 0.5 F
Figure X-8
Problem 10:
Consider a biological reactor with recycle usually used for wastewater treatment as
depicted in figure X-9 below. Substrate and biomass are fed to the reactor with
concentrations Sf and Xf. The effluent form the well-mixed reactor is settled in a clarifier
and a portion of the concentrated sludge is returned to the reactor with flow rate Qr and
Xr. If the reaction of substrate in the clarifier is negligible, the recycle stream would
contain the same substrate concentration as the effluent from the reactor. Sludge is
withdrawn directly from the reactor with a fraction W. Assuming constant holdup
develop the dynamic model for the process. Assume the rate of disappearance of S is
given by: r = mS/(K + S) and the rate of generation of X by r/Y where Y is constant.
W
X
S
Q
Xf
Sf
Reactor
V
Settler
X
S
Qr , Xr , S
Figure X-9
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Q-W
Xt
S
Problem 11
Consider the two phase reactor and a condenser with recycle. Gaseous A and liquid B
enters the reactor at flow rates FA and FB respectively. Gas A diffuses into the liquid
phase where it reacts with B producing C. The latter diffuses into the vapor phase where
B is nonvolatile. The vapor phase is fed to the condenser where the un-reacted A is
cooled and the condensate is recycled back to the reactor. The product C is withdrawn
with the vapor leaving the condenser. For the given information develop the dynamic
model for the process shown in Figure X-10. Consider all flows are in moles.
F1v, yA1, yc1
T2
P2
T1, P1
FA, TA
FB, TB
NA
NA
F2L,xA2
NC
Q1
A + B = 2C
F1L, xA1, xB1, T1
Figure X-10
Problem 12:
Develop the mathematical model for the triple-effect evaporator system shown in figure
X-11 below. Assume boiling point elevations are negligible and that the effect of
composition on liquid enthalpy is neglected.
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V1=F-L1
V2=L1-L2
Vapor
V3=L2-L3
Vapor
Vapor
F (feed)
Vo
Vo
V1
L1
V2
L2
Thick Liquor
Thick Liquor
L3
Thick Liquor
Figure X-11
Problem 13
Liquid vaporizer as depicted in Figure X-12 below is one of the important processing
units in a chemical plant. A liquid feed, enters the vaporizer at specific flow rate F0
density 0 and temperature T0. Inside the vessel, the liquid is vaporized by continuous
heating using hot oil. The mass rate of vaporization is wn. The formed vapor is
withdrawn continuously from the top of the vessel. We would like to develop a
mathematical model to describe the process. Unlike the adiabatic flash operation, the
temperature and pressure in the two phases are different. Correspondingly, the volume
of the two phases varies with time.
Fv
Pv , Vv , v
wv
Fo , To , o
PL, VL, L
Q
Figure X-12
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Problems: Distributed Parameter Systems
Problem 14
At time t = 0 a billet of mass M, surface area A, and Temperature To is dropped into a
tank of water at Temperature Tw. Assume that the heat-transfer coefficient between the
billet and water is h. If the mass of water is large enough that its temperature is virtually
constant, determine the equation describing the variation of billet temperature with time
for the following two cases:
1. The thermal conductivity of the metal in the billet is sufficiently high that the
billet is of uniform temperature.
2. The thermal conductivity of the metal is low and the billet radius is large enough
that the temperature inside the billet is not uniform.
Problem 15
(a)Consider a wall made up of stacked layers of various materials each with different
thermal conductivity usually used for insulation as shown by figure X-13a. Assume the
inner side is at high temperature Tb and the outer side is at the ambient temperature Ta.
Derive the temperature profile through the wall.
Tb
Ta
x
Figure X-13a
(b) Repeat the above development for a composite cylindrical wall shown by figure X13b.
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r
Tb
Ta
Figure X-13b
Problem 16
A chemical reaction is being carried out in a fixed bed flow reactor. The reaction zone
is filled with catalyst pellets. Assume plug flow and the reactor wall is well insulated
such that the temperature is uniform in the radial direction. If the fluid enters the reactor
with temperature T1 and superficial velocity v and that thermal energy per unit volume
(S) is produced inside the reactor due to chemical reaction, derive the steady-state
temperature axial distribution. The unit is depicted in figure X-14.
Insulating wall
Reactant
r
z=0
Catalyst particles
Product
z
Reaction zone
z=L
Figure X-14
Problem 17
A first order irreversible reaction is taking place in a tubular reactor. The reactor can be
considered to be isothermal and radial dispersion is not to be neglected. Using Fick’s
law with effective diffusion coefficient to describe the axial and radial dispersion,
derive the steady state mass balance equation and its boundary conditions.
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Problem 18
(a) Derive the unsteady state mass balance equation for the diffusion of species A
through a spherical porous pellet.
(b) Derive the steady state mass balance equation for the diffusion of species A through
a spherical porous pellet where it converts to B according to the first-order reaction:
AB.
Problem 19
Species A dissolves in liquid B and diffuses into stagnant liquid phase contained in a
cylindrical tank where it undergoes a homogeneous irreversible second-order chemical
reaction: A+B  C. Derive the mass balance equation for component A. Assume that
the depth of liquid is so large compared to the radius of the tank, thus, the radial
distribution can be neglected. The process is shown schematically in figure X-15.
Gas A
z=0
Liquid B
Dz
z=L
Figure X-15
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