Kinetics Problems 1 May 2012 Problem 4 Consider the gas phase reaction: A + B → C which is catalyzed by a solid catalyst and follows dual-site (both A and B adsorb) Langmuir-Hinshelwood kinetics. 1) Write a reaction mechanism that accounts for all elementary steps involved in the dual site catalytic cycle. 2) Derive the expression of the reaction rate for A, the limiting reactant, assuming the surface reaction is the rate-determining step. Your expression should be in terms of the partial pressures of each species, equilibrium constants, and temperature-dependent rate constants. You must explicitly and clearly define any equilibrium or rate constants you use. Problem 6 Equilibrium data has been obtained for the reaction: A(g) + 2B(g) = 2C(g) with a stoichiometric feed of A and B. At 200°C and 1 bar, 25 mol % of the species in the reactor was product C. At 300°C and 1 bar, 53.9% C was produced. A) One mole of A and 2 moles of B react at 250°C and 2 bar. Based on the data above, estimate the concentrations at equilibrium. Clearly state any reasonable assumptions that you need to make to solve the problem. B) As a process engineer, you wish to maximize the production of C. As completely as you can, discuss the implications of the following strategies: i. Increase temperature ii. Increase pressure iii. Add an inert to the feed stream
Transcript
Kinetics Problems
1
May 2012
Problem 4
Consider the gas phase reaction: A + B → C which is catalyzed by a solid catalyst and follows dual-site (both A and B adsorb) Langmuir-Hinshelwood kinetics. 1) Write a reaction mechanism that accounts for all elementary steps involved in the dual site catalytic cycle. 2) Derive the expression of the reaction rate for A, the limiting reactant, assuming the surface reaction is the rate-determining step. Your expression should be in terms of the partial pressures of each species, equilibrium constants, and temperature-dependent rate constants. You must explicitly and clearly define any equilibrium or rate constants you use.
Problem 6
Equilibrium data has been obtained for the reaction:
A(g) + 2B(g) = 2C(g)
with a stoichiometric feed of A and B. At 200°C and 1 bar, 25 mol % of the species in the reactor was product C. At 300°C and 1 bar, 53.9% C was produced. A) One mole of A and 2 moles of B react at 250°C and 2 bar. Based on the data above, estimate the concentrations at equilibrium. Clearly state any reasonable assumptions that you need to make to solve the problem. B) As a process engineer, you wish to maximize the production of C. As completely as you can, discuss the implications of the following strategies:
i. Increase temperature ii. Increase pressure iii. Add an inert to the feed stream
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Problem 8
The following data were reported for a gas-phase constant volume decomposition of dimethyl ether at 504°C in a batch reactor. Initially, only (CH3)2O was present. Time(s) 390 777 1195 3155 ∞ Total Pressure (mm Hg) 408 488 562 799 931
It is known that the reaction (CH3)2O → CH4 + H2 + CO is first-order and goes to completion.
A) What is the pressure at the start of reaction? B) From the data provided, determine the reaction rate constant.
January 2012
Problem 3
Consider the following constant-volume liquid-phase complex reactions:
A→B ; (rB) = 1 mol/liter/hr
A→C ; (rC) = 2CA mol/liter/hr
A→D ; (rD) = CA2
mol/liter/hr
The reaction is taking place in an ideal isothermal PFR and the initial concentration of A is 1 M
(no other chemical species are present initially). B is the desired product.
(a) Find the size of the PFR for 90% conversion of A if the flow rate Q =10 L/hr.
(b) What is the minimum and maximum instantaneous selectivity for B? Identify
conditions that correspond to the minimum and maximum instantaneous selectivity.
(c) Your boss thinks that the maximum concentration for B is 0.67 M under the condition
given above. Prove or disprove him. Provide detailed calculations.
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Problem 4
A certain type of chemical sensing device consists of a thin film of a porous inorganic material
grown on an impermeable quartz support. When the top surface of the film (of thickness L) is
initially exposed to a concentration (Co) of an analyte chemical A, the molecules of A diffuse
(with diffusivity D) into the porous film and increase its total mass as a function of time. The
change in mass is measured by vibrating the quartz substrate at high frequencies to determine its
resonance frequency, which depends on the total mass of the film on top of it. The film is
initially empty (it does not contain any molecules of A). The sensor response is considered
reliable when the spatially-averaged concentration of molecule A in the film reaches 50% of its
maximum possible value (Co).
(1) Write down the differential equation for the concentration of A at any location in the sensor and
at any time. Your answer must be a dimensionless differential equation with the dimensionless
variables and parameters clearly defined.
(2) Write the initial and boundary conditions accompanying this differential equation.
(3) Two different sensors of this type are available. Sensor I has a film of thickness 10 microns (1
micron = 10-6
m) of a material in which the diffusivity of A is 2×10-11
m2/s, whereas Sensor II
has a film of thickness 20 microns and is made of a material in which the diffusivity of A is
1×10-10
m2/s. Determine which sensor has a faster response.
(4) From the solution of the partial differential equation, it can be shown that the spatially-averaged
concentration of A in the film is given by:
If a reliable response is required in 2 seconds for a particular application, will the faster sensor
identified in Part (3) above be suitable for use? Neglect all terms with n > 1.
2 2
2 2 20
( ) 8 (2 1)1 exp , ( 0, 1, 2 ,...)
(2 1) 4no
C t D n tn
C n L
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Problem 7
You are studying an irreversible reaction between two materials A and B that produces your
desired product P. In order to design a reactor for this system you need to determine the rate law
(i.e. evaluate the reaction orders for A and B) and rate constant for the reaction. Your assistant
goes into the lab and runs a series of batch reactions for this process, i.e. they mix eight solutions
containing varying amounts of A and B (see Table 1 below for the concentrations of A and B at
time zero), quench the reactions after allowing each mixture to reaction for 10 seconds, and
spectroscopically measures the amount of product P in the solution after the 10 second reaction
period.
Table 1: Reaction experiment data. All concentrations given in moles/liter units (i.e. [M]).
Run # [A]0 [B]0 [P] after 10 seconds reaction
1 0.037 0.133 0.0007
2 0.037 0.200 0.0016
3 0.185 0.133 0.0034
4 0.185 0.266 0.0118
5 0.278 0.200 0.0100
6 0.278 0.300 0.0209
7 0.370 0.133 0.0061
8 0.370 0.266 0.0216
Determine an appropriate expression for the rate law (i.e. determine an expression to calculate
d[P]/dt for a given concentration of A and B) and determine the rate constant for this rate law.
Clearly explain how you arrive at your answers to receive full credit.
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May 2011
Problem 3
You are interested in studying a surface reaction in which a reactant molecule (R) in the gas
phase chemisorbs onto a catalytic surface and then combines with another chemisorbed reactant
molecule on the surface in a dimerization reaction to produce a product molecule (P) which can
then desorb from the catalytic surface. It is known that the product molecule once desorbed has
very limited propensity to chemisorb back onto the surface. One can represent such a process
using the following “reaction” scheme description which includes both expressions for
chemisorptions and desorption and the surface reaction.
→ (1)
→ (2)
→ (3)
→ (4)
→ (5)
Here s represents a surface site where a molecule can be chemisorbed on the surface. We will
treat these as “elementary reactions” with the rate constants for the reactions given as shown
above. In a manner similar to Langmuir, we can write expressions for the rates of
chemisorption, desorption, and surface reactions as follows:
[ ]
where [R] is the gas phase concentration of R, qv is the fraction of unoccupied surface sites on
the catalyst, qRs is the fraction of surface sites occupied by R, qPs is the fraction of surface
sites occupied by P unoccupied surface sites on the catalyst.
ANSWER THE FOLLOWING QUESTION:
1. If you assume that the rate limiting step in this sequence is the desorption of the product P
from the catalyst surface, derive an expression for the rate of formation of product P in terms
of only the various reaction rate constants (i.e. k1, k1’, k2, k2’, and k3) and the gas phase
concentration of R (i.e. [R]).
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Problem 4
Tank 1, shown below, initially contains 200 L of a sucrose/water solution with a sucrose (S)
concentration of 0.25 kg S/L. Fresh sucrose solution with a concentration of 0.2 kg S/L is fed to
the tank at a constant rate of 10 L/min. Water is evaporated from Tank 1 at a rate of 4 kg/min.
Concentrated sucrose solution is drained into Tank 2 (which is initially empty) at a rate of 6
L/min. Assume that the contents of both tanks are well mixed and the volume of the sucrose is
negligible (and therefore the density of sucrose solution is the same as pure water).
(a) Solve for an expression giving the total volume of solution in Tank 1 as a function of time t,
V1(t).
(b) Solve for an expression giving c(t), the concentration of sucrose in Tank 1 as a function of
time t.
(c) If Tank 2 can hold 200 L of liquid, will it overflow before the concentration in Tank 1 gets to
0.3 kg/L? Must show V2 volume as > or < 200 L to get credit.
V2 = _________ L ____ yes, overflows _____ no, does not overflow
Fresh feed10 L/min0.2 kg S/L
4 kg/min water evaporated
Conc. Soln.6 L/minc(t) kg S/L
At t = 0:V1 = 200 Lc = 0.25 kg S/L
Tank 1
Tank 2
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Problem 5
To a well-stirred tank (with initial volume of 400L) containing 40 mol of B a stream of a at 0.04
mol L is added at 1 L/min. An irreversible reaction proceeds as follows.
A+B==C+D
The reaction r = 0.2 CA, (mol/L/min).
Determine: 1. How long will it take to exhaust B?
2. What is the concentration B when the reactor volume doubles to 800 L?
Problem 8 The first order gas phase reaction A B is performed in an adiabatically operated CSTR. A conversion
of 90% is desired. The reaction is operated at 1 atm, and the gas stream consists of 10% A and 90% inerts
(I). Assume ideal gas behavior. The initial temperature is 400 K. The following information is known:
Cp’A = 50 J.mol
-1.K
-1 (assume constant over relevant temperature range)
Cp’B = 50 J.mol
-1.K
-1 (assume constant over relevant temperature range)
Cp’I = 40 J.mol
-1.K
-1 (assume constant over relevant temperature range)
ΔHr = -80 kJ.mol
-1 (assume constant over relevant temperature range)
v0 = 20 L.min
-1
a) Which temperature will the exit stream have?
b) Which reactor volume is required to reach the desired conversion? If you did not find an exit stream
temperature in part a) use 525 K for this part.
c) For safety reasons it is decided to limit the temperature in the reactor to 500K. How do you have to
adjust the feed stream to achieve this? The pressure and volume flow are remaining constant.
Kinetics Problems
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January 2011
Problem 1
The liquid phase reaction A + B → C is observed to be first order in A and zero order in B; the
rate constant is 0.003 min-1
. An aqueous mixture containing A & B (CAo= 0.5 mol/L, CBo = 0.2
mol/l), and containing no C is reacted in a tubular Reactor (PFR). What space time is needed to
achieve 80% conversion of B?
Problem 7
Write the series of elementary reaction steps that is consistent with the following surface reaction
and the surface reaction can be considered to be irreversible. Note, all constants shown in the
rate law may be lumped constants. How many surface sites are involved in the rate-limiting
step?
Kinetics Problems
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May 2010
Problem 2
Problem 5
Kinetics Problems
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Problem 7
Kinetics Problems
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January 2010
Problem 2
Kinetics Problems
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Problem 6
Problem 8
Kinetics Problems
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May 2009
Problem 2
Kinetics Problems
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Problem 4
Kinetics Problems
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Problem 8
You are going to run an exothermic liquid-phase catalytic reaction (A B) in two sequential,
adiabatic, steady-state CSTRs. The reaction is first-order and irreversible. Since the amount of
catalyst available is limited, it is decided that the combined volume of the two reactors will be fixed
at a value Vt. The individual volumes of the two reactors are V1 and V2 respectively. The feed to the
first reactor is at a volume flow rate F and has a molar concentration C0 (moles per unit volume) of
the reactant A. The exit stream from the first reactor has a molar concentration C1 of the reactant and
is fed to the second reactor, which produces a final exit stream of concentration C2 of the reactant.
The liquid-phase reaction produces negligible volume change, and hence the volume flow rate F
remains fixed throughout the system. Since the reactors are adiabatic, they have different
temperatures. Hence, the rate constants in the two reactors are different (k2 > k1).
(1) Determine an equation for the relative concentration of the reactant exiting the system (i.e.,
C2/C0) in terms of the feed volume flow rate, the two rate constants, and the reactor volumes (Vt, and
either V1 or V2).
(2) Determine the fractional volume of the first reactor (i.e., V1/ Vt) that would give rise to the
optimum overall conversion of the reactant. The answer should be only in terms of the feed volume
flow rate, the two rate constants, and the total reactor volume Vt.
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January 2009
Problem 3
Kinetics Problems
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Problem 6
Kinetics Problems
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May 2008 Problem 4 The following data were reported for a gas-phase constant volume decomposition of dimethyl ether at 504 C in a batch reactor. Initially, only (CH3)2O was present.
Time (s) 390 777 1195 3155 ∞ Total Pressure 408 488 799 931
Assuming that the reaction (CH3)2)O → CH4 + H2 + CO is irreversible and goes to completion, determine the reaction order and the reaction rate constant (with units).
Problem 7
Enzyme is retained in a continuous flow bioreactor (Figure) using an ultrafiltration membrane.
Active enzyme in the reactor deactivates following first-order kinetics with rate constant kd. In
order to maintain uniform effluent product quality, active enzyme is continuously added at a rate
FIN to the reactor in order to maintain a constant concentration [Ea] of active enzyme in the
reactor. However, a gel layer of enzyme accumulates on the membrane, adding resistance to
solution flow. To compensate for this, the pressure drop across the membrane is continuously
adjusted to maintain a constant flux J of the reaction mixture through the membrane. However,
the desired flux J cannot be achieved if the total concentration of enzyme [Etot] (active +
inactive) exceeds a maximum value given by
[Emax] = [Es] exp(-J/ke)
where [Es] is a saturation concentration of enzyme and ke is an enzyme mass transfer coefficient.
a) Assuming that all the enzyme in the reactor is initially active, determine the total
enzyme concentration [Etot] in the reactor as a function of time.
b) The reactor must be shut down and cleaned when [Etot] exceeds [Emax] and the flux
falls below J. Determine the maximum reactor operating time tmax as a function of J.
FIN
[kg active enzyme/day]
Semipermeable
membrane
FOUT [kg/day] = 0Ea [kg/m3]
Ei [kg/m3]
FIN
[kg active enzyme/day]
Semipermeable
membrane
FOUT [kg/day] = 0Ea [kg/m3]
Ei [kg/m3]
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January 2008
Problem 4 An irreversible zero order homogeneous gas reaction, A → r R, proceeds at an elevated temperature in an isothermal constant volume reactor, starting with an equimolar mixture of A and an inert gas I. The initial total pressure of the reactor is 2.0 atm, which rises to 3.0 atm after 10 minutes of reaction.
If the same reaction is run in a constant pressure reactor at the same temperature with the same feed composition and initial pressure, what is the % change in reactor volume after 10 minutes?
Problem 7
An automotive exhaust clean-up system is tested by feeding a stream containing dilute amounts
of NO2 in an inert gas. Tests are run in a small experimental reactor that operates essentially as a
CSTR. Three different concentrations of NO2 are tested at different temperatures. The plot below
gives the fractional decomposition of NO2 fed versus the ratio of reactor volume V (m3) to the
NO2 feed rate, FNO2 (gmol/hr) at different feed concentrations of NO2 (in parts per million by
volume). Each data point in the figure below represents a steady state run.
What can you deduce from these data about the reaction order, rate constant, and activation energy?
Kinetics Problems
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May 2007
Problem 1 A liquid reaction, A + 2 B → C, is taking place in a batch reactor under isothermal and constant volume conditions. The initial concentration of B, CB0 = 2CA0.
The reaction rate can be represented as ( - rA ) = k CA5/4
CB3/4
. a.) Derive an equation that shows how the concentration CA is dependent on reaction time, t.;
In other words, you are to derive [CA = Function of (t, k, CA0)]. Show all steps. b.) Using the equation derived, calculate the reaction time required to achieve 85% conversion
of A for the following set of conditions: CA0=1 M, k=0.01 L mol-1
s-1
.
Problem 6
An irreversible second order liquid-phase reaction is being run in a plug flow reactor. The feed rate is 20 L/min at a reactant concentration of 10 mols/L; final conversion of reactant is 90%. If 2/3 of the product stream leaving the PFR is recycled to the reactor entrance, and if the fresh feed concentration and rate are unchanged, what is the overall percent conversion of reactant in the product now leaving the system?
Problem 7
The following reaction is carried out in an adiabatic plug flow reactor:
A + B C + D
Calculate the maximum conversion that may be achieved if the feed enters at 300 K. The feed
stream contains A & B only. The following data are available:
ΔHr = -120 kJ/mole
K (Eqbm. Const.) = 500,000 at 50 C
F A0 = F B0 = 10 moles/min
CPA = CPB = CPC = CPD = 100 J/mol-K
Kinetics Problems
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January 2007
Problem 1 Equilibrium data has been obtained for the reaction: A(g) + 2B(g) = 2C(g) with a stoichiometric feed of A and B. At 200
oC and 1 bar, 25 mol % of the species in the
reactor was product C. At 300 oC and 1 bar, 54% C was produced.
a. One mole of A and 2 moles of B react at 250
oC and 2 bar. Based on the data above,
estimate the concentrations at equilibrium. Clearly state any assumptions that you make.
b. Discuss the implications of the following strategies on maximizing the production of C: i. Increase temperature ii. Increase preesure iii. Add an inert to the feed stream
Problem 4
The liquid-phase reaction A + B → C is conducted in a flow reactor. Two reactors are available,
an 800 dm3 PFR operating at 300 K and a 200 dm
3 CSTR operating at 350 K. The two feed
streams to the reactor mix to form a single feed stream that is equimolar in A and B, with a total
volumetric flowrate of 10 dm3/min. Which of the two reactors will give us the highest
conversion? At 300 K, k = 0.07 dm3/mol-min, E = 85,000 J/mol-K, CA0B = CB0B = 2 mol/dm
3,
vA0 = vB0 = 0.5*v0 = 5 dm3/min.
Problem 8 For a liquid autocatalytic reaction, A + B + P → P + P (P is the product), the rate expression can be represented as (-rA) = k CA CB CP. Design a flow reactor configuration that requires smallest reactor volume and find this minimal reactor volume. (Configuration: a single s (i) CSTR; (ii) a single PFR; (iii) RSTR then PFR; (iv) PFR then CSTR. Think carefully about the possibilities. Do not simply calculate volumes for all possible configurations. Data: volumetric flow rate v= 10 liters/min: Desired fractional conversion fA = 0.8, k=0.04 L
2Mol
-2Min
-1 . CA0 = CB0 = 1 M; C P0 = 0 M.
Kinetics Problems
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May 2006
Problem 4 You can apply the principles of reaction kinetics to any type of rate process, not just
petrochemical reactions in industrial reactors as traditionally studied. For example, let’s consider
the area of pharmacokinetics, the study of how drugs are ingested, distributed, eliminated and
reacted within the body. In this problem, we will model how long one must wait to drive after
drinking a tall martini. In most states, the legal level of intoxication corresponds to about 1 g
ethanol per liter of body fluid. The ingestion of ethanol into the bloodstream and the elimination
of ethanol can be modeled as a series reaction. The rate of absorption into the bloodstream from
the gastrointestinal tract is a first order reaction with a specific rate constant of 10 h-1
. The rate at
which ethanol is broken down in the bloodstream is limited by the regeneration of a coenzyme.
Consequently, this step can be modeled as a zero order reaction with a specific rate of 0.192 g/h-
L of body fluid.
(a) Write the series of reactions (all irreversible) that must be evaluated and label each
step with its appropriate rate constant.
(b) Calculate how long a person would have to wait to drive after rapidly drinking 2 tall
martinis?
Ethanol volume in a tall martini = 40g
Volume of body fluid = 40L
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Problem 8 Consider the reversible reaction
BA
with forward rate k1 and backward rate k2. Suppose you are given the volume V, and the initial
concentration CA(0) and CB(0).
A. Derive the time-dependent balance equations for CA(t) and CB(t).
B. What are the steady-state concentrations of CA and CB? How many independent equations do
you have? How many unknowns are there, and what are they? Can you solve for the steady-
state concentrations uniquely? Why or why not?
C. The eigenvalues of this dynamic balance indicate the stability of CA and CB. Write the
dynamic balance as a matrix differential equation, with dx/dt and x terms, where x is a
column vector consisting CA and CB.
D. Compute the eigenvalues of the system. (For matrix A, the eigenvalues of A are found by
setting the determinant of (A- qual to zero, where I is the identity matrix.)
E. Eigenvalues with imaginary parts indicate oscillatory behavior. Under what rates k1 and k2
would you expect to see oscillations in the concentrations?
F. Real eigenvalues that are negative indicate unstable behavior and exponential growth. Under
what rates k1 and k2 would you expect to see exponential growth?
January 2006
Problem 2 You are a member of a research team of industrial chemists discussing plans to operate an ammonia processing plant:
N2 (g) + 3H2 (g) = 2NH3 (g)
a). The plant operates close to 700 K, at which temperature Kp is 1.00 x 10-4
atm-2
, and uses the
stoichiometric ratio 1:3 of N2:H2. At equilibrium, the partial pressure of NH3 is 50 atm. Calculate
the partial pressure of each reactant and the total pressure under these conditions.
b). One of your team members suggests that the plant could produce the same
amount of ammonia if the reactants were in a 1:6 ratio of N2:H2 and could do so at a
lower total pressure. This would lower the operating costs and increase profitability.
Assuming the same operating temperature and equilibrium partial pressure of ammonia (50 atm), is the team member correct?
Kinetics Problems
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Problem 6 A first order, irreversible liquid phase reaction is to be carried out in a 40 L CSTR, followed in series by a plug flow reactor. A feed solution containing 4 mol / L of reactant is to be fed to the CSTR at a rate of 2 L / min. A second feed solution containing 2 mol / L is to be combined at a rate of 3 L / min with the intermediate product from the first reactor and the mixture fed to the PFR. Assuming constant density and a rate constant of 0.1 min
-1, how large should the PFR be to
achieve 50 % overall conversion of the reactant mixture fed? Problem 8 You are in charge of the CSTR (cross sectional area AR and total internal volume VR) unit
pictured below. The reactor is used to perform the first order catalyzed conversion of reactant
A to product B. Reactant A is fed to the reactor as a liquid solution at a concentration CA,in (the
density of the A solution is A and it has a heat capacity CPA) and at a volumetric flowrate of Fin
and temperature Tin. The reactor is filled with very small solid metal catalyst pellets
(diameter=Dc) that have a total volume of VC, a total surface area of AC, a density C, and a heat
capacity of CPC. It is critical that the temperature of the catalyst and the product stream be
controlled to maintain the desired product specifications. In order to design a proper control
system for this reactor, you are assigned the task of building a computational model of the
system.
Given Assumptions:
1. Assume that the catalytic reaction can be described as
BAk
catalyst2
Fin, Tin
Fout, Tout
h
Kinetics Problems
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2. The rate constant k for the reaction can be described by an Arrhenius relationship which is a function of the catalyst temperature (TC).
3. The reaction is exothermic and the heat of reaction of the catalytic process on a molar basis
is given as Hrxn. Also assume that since the catalyst particles are metal, and the reaction occurs on the catalyst surface, that the energy released from the reaction is transferred only to the catalyst particles first and not to the liquid.
4. Assume that the each of the catalyst pellets are small enough that there is no internal spatial temperature variation in each pellet, i.e. the catalyst temperature (TC) does not have spatial dependence in each pellet. The pellets are prevented from flowing out of the reactor by a screen on the exit stream.
5. Assume that any heat transfer involving the pellets can be described using an overall heat transfer coefficient UC.
6. Assume that the product B has approximately the same density and heat capacity as the reactant A.
7. The tank is a gravity draining tank and thus the flow rate of liquid from the tank can be calculated from the height of liquid in the tank (h) as:
hCF Vout
8. Assume that the reactor is well insulated. 9. The flow rate (Fin), temperature (Tin), and concentration (CA,in) can all vary with time.
Answer the following questions:
1. Write down (do not solve) the important dynamic mass balance equation(s) that can be used to solve for the concentrations of materials in the reactor and the mass of material in the system in terms of state variables and input variables to the system.
2. Write down (do not solve) the important dynamic energy balance equation(s) for this system that could be used to solve for the temperatures of the catalyst and fluid in the reactor as a function of time.
3. The system is operating at steady state when suddenly the temperature of the feed stream decreases substantially as shown in the attached graph. Sketch on the same graph what you would expect the catalyst pellet temperature and outlet stream temperature to do as a function of time. Be sure to place the starting temperature (i.e. before t=0) for the catalyst and outlet stream in the correct relative position to the inlet stream temperature in creating the plot. You can briefly describe (no more than 3 to 4 sentences) your choice for plot shapes and locations if you like.
HINT: This looks complicated. It is not. Just write mass and energy balances, using
dimensional analysis to determine how to construct the important terms in each equation.
