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CHEM-E7130
Process Modeling
Exercise
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Exercises 1&2, 3&4 and 5&6 are related. Start with one of the packages and then continue to the others.
You may select the order based on your interests.
1. a) Derive an equation which describes cooling of an object which is a good thermal conductor.
Assume uniform heat transfer outside the surface of the object.
b) Spherical iron ball with diameter of 1 cm is initially at 100 oC temperature. Air at temperature 20 oC
flows outside it. Calculate temperature of the ball after one minute, when the heat transfer coefficient
between the ball and the flowing air is 50 W/m2K.
c) Calculate the same thing numerically directly from the time dependent energy balance by using ten
second time steps.
2. Stirred tank with 2300 kg of liquid is being heated from initial temperature of 15°C to a final
temperature of 120°C. Saturated steam at 350 kPa(abs) is being fed to the jacket of the tank. Heat
capacity of the liquid is 2.0 kJ/kg K. Heat transfer area is 5m2 and overall heat transfer coefficient 300
W/m2K.
a) How long does the heating take in this batch operation
b) What is steam mass flow in the beginning and in the end?
c) If the vessel would be fully mixed and operated in a continuous mode, how big much cold flow could
be heated?
d) How much of the feed could be heated with a countercurrent heat exchanger of similar area? 2
3. Derive an equation describing concentration changes in a batch reactor with first order irreversible
reaction to the reactant.
4. Derive an equation for concentration changes in a steady state isothermal plug flow reactor, and
compare it to the previous result. Calculate concentration changes with Excel, and study the effect of
the step size and reaction kinetic parameter.
5. a) Derive compositions at the vapor-liquid interface according to the two film mass transfer theory.
Solve the compositions also numerically from the flux equations using Excel.
b) Derive an equation for the overall mass transfer coefficient in the same situation
6. Calculate instantaneous mass transfer rate (g/s) for a spherical water droplet falling in completely dry
air at atmospheric pressure at terminal velocity of 6 m/s. Temperature of the droplet is 20 oC.
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Tips
Always start from the general balance equation ACCUMULATION = IN – OUT + GENERATION
Always write balances for the extensive variables describing the state of the system. Extensive variable
is such that depends on the size of the system (mass, amount of moles, total energy). After this, select
proper physical models to describe the terms in the balances. Also think which of the balance equation
term (IN, OUT or GENERATION) each rate equation describes.
If you wish, you can divide all the balance equation terms with a system size dependent variable (such
as total volume), so that you end up with an intensive variable (temperature, concentration etc.), but do
this only after writing the balance equations.
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Tips
Always start from the general balance equation ACCUMULATION = IN – OUT + GENERATION
Always write balances for the extensive variables describing the state of the system. Extensive variable
is such that depends on the size of the system (mass, amount of moles, total energy). After this, select
proper physical models to describe the terms in the balances. Also think which of the balance equation
term (IN, OUT or GENERATION) each rate equation describes.
If you wish, you can divide all the balance equation terms with a system size dependent variable (such
as total volume), so that you end up with an intensive variable (temperature, concentration etc.), but do
this only after writing the balance equations.
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Some useful equations for this exercise:
Heat energy (dimension J, no phase changes)
E  c p mT  Tref 
(1.)
Heat transfer (J/s = W)
Q  hA T  Tout 
(2.)
Energy for phase change (J), (evaporation or condensation)
E  m
(3.)
Power for phase change (W), (evaporation or condensation)

Qm
(4.)
Logarithmic mean temperature difference in countercurrent heat exchangers
TLM 
T1  T2
 T 
ln  1 
 T2 
(5.)
First order reaction rate (mol/m3s)
r  kc
Mass transfer flux (diffusion + convection, mol/m2s)
ND
(6.)
dc
dc
 xN tot  D  cv
dz
dz
(7.)
Mass transfer flux according to the film model (with mass transfer coefficients)
N  k v cV  cIV   k L cIL  cL 
(8.)
Distribution coefficient in ideal vapor-liquid systems at equilibrium
y p0
K 
x p
(9.)
Correlation for mass transfer coefficient outside a spherical object
Sh  2  0,552 Re1 / 2 Sc1 / 3
(10.)
 x1  a 
dx

ln
x x  a   x 0  a 
0
x1
One perhaps useful integration formula:
(11.)
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