Introduction to Chemical Reactor Engineering - Problems.pdf

Introduction to Chemical Reactor

Engineering

Problems

c.m. van den bleek a.w. gerritsen

November 2000

(last revised: Sept 17, 2001)

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1 Ideal reactors: basics 1 The decomposition of A takes place according to the following reaction equation A ! 3 B Experiments were carried out in an ideal, isothermal batch reactor of constant volume. The following concentration profile was measured, starting with 75 vol% A and 25 vol% inert.

0

2

4

6

8

0 10 20 30 40

CA [mol/liter]

time [s] Starting with the same feed and at the same temperature, an ideal PFR, an ideal CSTR and an ideal batch reactor (but now at constant pressure) are compared. a For the continuous reactors calculate the space time and for the batch reactor the time at

which the reaction has been completed. b For all three reactors calculate the concentration of A for τs and t = 15 s, respectively. c Now sketch CA as a function of τs and t, respectively for these three reactors and the given

reactor in one graph (clearly indicate the scale division on both axes).

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2 To study the liquid phase reaction A ! products, the following experiments have been carried out in a small experimental ideal CSTR:

τs [s]

CA, 0 [kmol/m3]

CA [kmol/m3]

44 5 4 52 8 6 58 8 5.5 70 8 7 78 8 5 166 5 3 225 1 0.5 300 2 1 468 5 2

a Without determining the order of the reaction, calculate the space time needed to convert

a feed containing 6 kmol/m3 A for two-thirds in an ideal PFR. b What will be the space time needed to achieve the same in an ideal CSTR? 3 In a laboratory-scale isothermal ideal CSTR A is converted into B. For every mole of A one mole of B is formed. A stream of pure A, that has already been converted into B for a small part, is used as the feed of the CSTR: CA,feed = 10 kmol/m3 and CB,feed /CA,feed = 1/99. The graph shows the relation measured between space time and feed conversion:

τs [min]

ξA [-]

0 0.01 0.030 0.2 0.041 0.4 0.049 0.5 0.062 0.6 0.124 0.8

The table shows some of the measured data. a Determine the kinetics for this reaction. b At what space time the maximal rate of disappearance of species A is reached? c If one has the choice between an ideal PFR and an ideal CSTR, which reactor type will

generally be preferred when this process is carried out at a large scale? Why?

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4 In an ideal CSTR the species A, B and C are converted into the products R and S according to the following reaction scheme:

A + C ! R with rR = k1ACAACC B + C ! S with rS = k2ACBACC and k1 = 2.5Ak2

a If the original reaction mixture consists of equimolar amounts of A, B and C, what will be

the fraction of R in the product mixture R and S if the conversion of A is 50%. b As in a, but now if the conversion of C is 50%. 5 In an ideal CSTR R and S are produced from A according to the following reaction scheme: A ! R with k1 = 1012⋅e-10800/T s-1 A ! S with k2 = 108⋅e-7200/T s-1 The temperature of the reactor can only be adjusted to each temperature in the range of 27 to 87 oC.

a Calculate the maximum amount of R that can be produced per mole of A converted. b What is the space time required to produce 99% of the maximum amount of R? c At what temperature should the reactor be set in order to obtain a the maximum production

of S per mole of A fed, at the same space time as under b?

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2 Ideal reactors: in a little more depth 6 The reaction A ! P takes place in a batch reactor with a volume of 2 m3. The density does not change. The kinetics of the reaction are: ! rA = kACA. a After 10 minutes of reaction (t = t1 = 10 minutes) 45.12% of the original amount of A in

the reaction mixture has been converted. Calculate k and give its dimension. From t = t1 onwards the reactor is used as an ideal CSTR: a volume flow rate of φv m3/s pure A is fed to the reactor and a volume flow rate of φv m3/s of reactor fluid is withdrawn from the reactor. b Derive the algebraic equation that describes the conversion of A in the reactor for t > t1,

and which only contains φv as unknown variable. c What value should φv have, so that CA remains constant at the value CA(t1) for t > t1? 7 In a CSTR with volume VR m3 the reaction A ! P takes place without change in density. The rate of reaction is given by: ! rA = kACA. At the start-up of the process the reactor is fed from the bottom with a stream of pure A, with volume flow rate φv and concentration CA,0. Because the reactor is drained by overflow it will take some time before the reactor is completely filled and the reactor fluid is also being drawn off. During this filling of the reactor reaction DOES take place.

a Derive the differential equation that describes the concentration of A in the reactor as a

function of time between t = 0, the moment filling of the reactor starts, and t = t1, the moment the liquid level reaches the brim. Assume that the reactor behaves as an ideal isothermal (batch) reactor and that the values of φv and CA,0 remain constant.

b From this differential equation derive an expression for CA, t1 (the concentration of A in the reactor at time t = t1). The equation must be expressed in terms of k, V , CA and φv, and must not contain t1.

c How does CA change after t = t1? Is CA constant or does it change? In case CA changes: does CA increase or decrease? Why?

8 In the seventies, the management of a nitric acid plant producing 500 t/day, has installed a unit to decolorize its exhaust gas in order to please the people living in the vicinity of the plant. The unit catalytically converts brownish NO2 into colorless NO. Consequently, a colorless gas is vented through the stack into the atmosphere. In the atmosphere, however, the NO reacts with oxygen and produces NO2 again. Usually, mixing of the exhaust gas with the atmosphere is so fast that the amount of newly formed NO2 remains below the visual

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detection limit (approximately 100 ppm), so the failure of this method is not discovered. Sometimes, however, the atmospheric conditions are such that the exhaust gas does not mix with the surroundings but moves away from the chimney as a cone (see sketch).

The radius of the cone is described by: r = 0.01AX, with X = the distance from the stack [m]. Other data: − φv = 1.0A105 m3/h − the NO level in the exhaust gas leaving the stack is 2000 ppm (by volume) − ! rNO = k'AC2

NOACO2, but, as the concentration of oxygen in the stack is very much − higher than the concentration of NO, the following approximation may be used: − ! rNO = kAC2

NO with k = 8.0A104 m3/(kmolAh) (at 25 °C) − the temperature of the exhaust gas is 25 °C − the pressure in the cone is 1 bar − 1 kmol of an ideal gas at 0 oC and 1 bar has a volume of 22.7 m3 − the stack has a height of 100 m − the cone can be considered as an ideal PFR. Assuming the atmospheric conditions are such that the exhaust gas does not mix with its surroundings, calculate the distance at which the failure of the method becomes visible. 9 In an ideal PFR A is converted into products according to a first order irreversible reaction. Because the reaction is strongly exothermic, a reactor is used that consists of a bundle of parallel small-diameter tubes, which are surrounded by a coolant (see scheme). This arrangement enables isothermal operation of the reactor.

productsA

coolant

After some time, a number of tubes become plugged at the outlet. Consequently, these tubes do not longer contribute to normal production. However, some conversion of the reactant will still take place in these tubes, also at steady state, be it only at very small scale.

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a Describe the behavior in a plugged tube by setting up the mass balance for species A and working it out to a differential equation that describes the concentration of A as a function of the distance to the inlet. (The differential equation need not be solved.) A tube has a length of L m and a diameter of d m. The reaction rate constant is k s-1 and the diffusion coefficient is ID m2/s. The flow rate of the reactant stream is φv,0 m3/s, and the inlet concentration CA,0 mol/m3. Assume that the density does not change.

b What are the boundary conditions to solve this differential equation and what is the physical meaning of the boundary condition at the position L?

c Which combination of parameters will determine the concentration profile in the tube? Sketch this profile for a number of values of this combination.

10 The exothermic reaction A ! B only takes place in the liquid phase at the boiling point of the mixture. The reaction is studied in a laboratory, using a glass container of 400 ml, filled with 270 ml pure A. The container is connected to a cooling element and heated by a flame that produces 600 W of heat (see figure).

As boiling causes violent motion, the temperature and composition may be assumed homogeneous. The amount of liquid in the cooler may be neglected. However, in the return line to the container 30 ml of liquid is always present, of which the composition and temperature are the same as in the glass container. The amount of gas in the container, the cooling element and the lines may be neglected. The glass container, the cooling element and the lines are insulated from the surroundings in such a way that all 600 W produced by the flame are absorbed by the glass container and its content. Removal of heat only takes place by the cooling element. The kinetics of the reaction at 87 °C are: !rA = kACA with k = 0.111 min-1 . Other data: − Boiling point mixture = boiling point A = boiling point B = 87 °C − The density of the fluid is independent on ξA and T − The heat of vaporization of the liquid is 360 J/ml, and independent on composition and

temperature. − The heat of reaction is 100 kJ per mole of A converted. − The concentration of pure A in the liquid phase is 12 mol/liter. Determine the volume flow rate φv,cond returning from the cooling element to the glass container.

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11 In a dark room, a balloon is filled with a gas A up to a volume of 1 liter. The pressure in the balloon then is 2 atmosphere. Subsequently, the balloon is placed under a UV-source. This causes the conversion of A according to: hv A ! 3 B Formulate the mass balance of A over the balloon and subsequently determine the volume of the balloon as a function of time. The following statements hold: − The gases behave ideally. − The temperature in the balloon is constant. − The balloon can be considered as an ideal batch reactor, for which the volume is directly

proportional to the pressure. − The reaction is a first order reaction in A; de reaction rate constant is k s-1. 12 The closed circuit of the figure below consists of a differential reactor with volume Vr, a vessel with volume V and a circulation pump. The volume of the pump and the volume of the lines may be neglected. At time t = 0 a certain amount of A is added to the system. Because the reactor is at elevated temperature and the vessel only at room temperature A only reacts in the (isothermal) reactor. The pump recycles the mixture so fast that the reactor behaves as a differential reactor. As the vessel is well mixed, any change in concentration due to the reaction in Vr will be adjusted instantaneously in the rest of the system.

In the reactor the reaction A ! R takes places. The reaction is first order in A and has a reaction rate constant of k s-1. a Give the molar balance over the reactor for species A. b Integrate this molar balance to obtain the concentration of species A as a function of the

time. Note: In a differential reactor the change in concentration is small compared to the concentration itself. Consequently, the rate of reaction is independent on the position in the reactor. 13 The kinetics of the gas-phase reaction A + B ! 2 C are studied at laboratory scale in an ideal PFR. It is already known that the concentration of B does not influence the rate at which A is converted. Despite high exothermicity of the reaction, it is possible to study the system isothermally at laboratory scale by choosing correct experimental conditions.

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To determine the influence of the concentration of A on the rate of reaction of A the following experiments have been carried out:

τs [min]

CA [kmol/m3]

0.21 9 0.45 8 1.02 6 3.21 2

The concentration of A in the feed was 10 kmol/m3 for all experiments. a Determine the value of the reaction rate constant k from these experiments if the rate of

reaction is given by: !rA = kACA. In practice, one uses a semi-batch reactor to control the production of heat. At t = 0 the batch reactor, with a volume of 1 m3, only contains B at reaction temperature, with CB,0 = 10 kmol-/m3. All A required for the reaction is fed to the batch reactor as a small, continuous stream of 1 kmol/min starting at t=0. Some additional cooling enables isothermal operation of the system. The operating temperature is exactly the same as the temperature for the kinetics measurements. b Derive the molar balance for species A. c Determine the concentrations of A and B in the system at t = 5 minutes. 14 Two students study the decomposition of P in boiling water under atmospheric pressure. At these conditions the decomposition reaction is an irreversible reaction with first order kinetics. The students start with 400 ml of a 1.2 M solution of P in water and let the solution boil for 20 minutes in a more or less closed glass vessel. Then the concentration of P in the solution is only 0.441 M. Both students know that the density of the solution is independent on the concentration of P and its decomposition products. They neglect the reactions taking place during heating and cooling of the reaction mixture. They assume (correctly) that the boiling water is ideally stirred. The first student forgets to determine the final volume and assumes that the volume did not change during the experiment. He determines the reaction rate constant k and finds the value a. The second student does determine the volume at the end of the experiment. It appears to be 320 ml. The student makes two assumptions: 1. The change in volume is completely caused by the evaporation of water; neither P nor its

decomposition products influence the volume. 2. The evaporation took place at a constant rate. He also determines the reaction rate constant k and finds the value b. a What is the value of a? b What is the value of b? c Are the two assumptions above required to determine the value of b? Why? 15 At t = 0 the reactor sketched below contains pure gaseous A at a concentration of 5 kmol/m3 and a pressure P0. The reactor is well-stirred, so the concentration and temperature are the same everywhere in the reactor. From t = 0 on A is converted into B according to: A ! 3 B.

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The reaction is zero order in A. The pressure in the reactor is kept at P0 by means of an ideal pressure controller that controls valve S, through which part of the reaction mixture is purged. a Give the mass balance over the reactor for species A. b Integrate the balance to obtain the concentration of A as a function of time.

PC

S

16 In a certain plant product Q is made by decomposition of a stream of pure A (volume flow rate φv, f, CA = 10 kmol/m3) according to: A ! 2Q. The density of mixtures of A and Q is independent on their mixing ratio. The reaction takes place in an ideal CSTR with volume V m3. At the usual process conditions the kinetics of the decomposition are: !rA = kACA with k = 2 h-1. Without purification the concentration of A in the product stream would be 2.612 kmol/m3, which is too high to sell the product directly. Therefore, a separator S is added that separates A and Q; the pure A is recycled. Because S has a limited capacity, only the volume flow φv, 1 is fed to S and separated into pure A and Q. The second stream leaving the reactor, φv, 2, is mixed with the stream of pure Q from separator S. In this way the concentration of A in the product stream has decreased sufficiently to sell it. If φv, 1 = φv, 2 determine: � CA, 1, the concentration of A in the stream from the tank to the separation unit � CA, 2, the concentration of A in the stream from the tank to the point of mixing with pure Q � CA, p, the concentration of A in the product stream after the mixing point.

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17 In an ideal CSTR B is produced from A according to the first order irreversible reaction: A ! B with k = 2 min-1. The reactor has a volume of 1 m3, the feed has a volume flow rate of 0.5 m3/min, with concentration of A of 1 kmol/m3. At a certain time the con-centration of A is doubled instantaneously and permanently. Calculate the time at which the concentration of A has approached its new steady-state value to within 4%. Sketch the inlet concentration of A and the concentration of A in the tank as a function of time (both in the same graph). 18 The ideal CSTR (see figure) with volume V m3 is fed with a stream of pure water; the volume flow rate is φv m3/s. At time t = 0 valve 1 is closed and valve 2 is opened. This results in the same volume flow φv, now with a dissolved species A, being fed to the reactor. The concentration of A in this stream is CA,0. In the reactor the reaction A ! B takes place. This reaction is first order in A and has a reaction rate constant of k s-1.

a Formulate the molar balance of species A in the reactor. b Integrate this balance to obtain the concentration of A as a function of time. c Check the final concentration of A at steady-state. 19 In an adiabatic ideal batch reactor with a volume of 1 m3 B is produced according to the exothermic gas-phase reaction A ! 3 B. The pressure in the system should not exceed 75 bar in relation to the thickness of the reactor wall. Therefore, an ideal pressure-relief valve has been mounted on the reactor, which lets off when the pressure becomes over 75 bar. Calculate at which temperature the valve starts operating, starting from pure A at a pressure of 25 bar and a temperature of 27 oC. It may be assumed that the temperature of the reactor wall equals the temperature of the gas, which behaves as an ideal gas. The heat capacity of the total system (i.e. reactor + content) is 500 J/K, the enthalpy of reaction ∆Hr = - 150 J/mol A. The value of the gas constant R = 8.33⋅10-2 (liter bar)/(mol K).

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20 Six models for a continuous tubular reactor are shown in the figure below. For models 4, 5 and 6 formulate the steady-state molar balance for species A, which is converted into products.

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3 Ideal reactors: multiple reactors 21 A cascade of 3 ideal CSTRs, each with a volume of V liter, is fed with φv liter/s of pure liquid A (concentration CA,0 mol/liter). In the reactors the decomposition of A takes place (first order kinetics, reaction rate constant k s-1). The conversion of the stream leaving the last reactor is ξ3 at steady state. At a certain moment the second reactor starts to leak. The size of the leakage, φv, leak, is approximately 0.1Aφv. Until the leak has been repaired, it is decided to recycle the leaking stream to one of the reactors 1, 2 or 3 using a pump. At steady state these options result in ξ3,1 , ξ3,2 and ξ3,3, respectively, in the effluent leaving the last reactor. Note: The height of the liquid level in the tanks does not change. Determine the order in which the conversions ξ3,1, ξ3,2 and ξ3,3 increase and explain your answer.

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22 A substance A decomposes into B and C in the liquid phase according to a first order reaction with reaction rate constant k1 s-1. P also decomposes according to first order kinetics into Q and R. The reaction rate constant is k2 s-1. The products B and Q react in the liquid phase according to a reaction that is first order in B and first order in Q. The reaction constant is k3 liter/(molAs). Thus: A ! B + C k1

P ! Q + R k2 B + Q ! Z k3

Of the three reaction rate constants it is known that k1 = 2Ak2 and k3 = 3Ak1. The density of the liquid phase appears to be independent on the composition of the reaction mixture. It is desired to convert a stream of 2 liter/s pure A (concentration CA,0 mol/liter) and a stream of 3 liter/s pure P (concentration CP,0) into Z. There are 3 ideal PFRs available, each with a volume of 30 liters. Only two ways of connecting the reactors will be considered (see figure). Determine which two of the following six assertions are correct and explain your answer:

CA,1 < CA,2 CA,1 = CA,2 CA,1 > CA,2 CP,1 < CP,2 CP,1 = CP,2 CP,1 > CP,2

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23 The reaction A ! 2P + Q is used to produce P from a stream of pure A using two identical ideal CSTRs in series:

It is proposed to change the flow scheme to obtain a higher conversion (the other conditions, such as φv,0 , CA,O , V, p and T remain unchanged): First proposal: Second proposal: The motivation for these two proposals is that, by recycling, reaction takes place all over again, resulting in a higher conversion.

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Third proposal: Fourth proposal: The motivation for proposals three and four is that a partial shortcut results in a higher residence time in the reactor, and consequently, to a higher conversion. Which proposal will result in the largest increase in conversion? Why? Does your answer depend on the order of the reaction? 24 To produce Q dimerisation of A can be used. This reaction, 2A ! Q, takes place in the liquid phase without a change in density. At the usual process conditions the kinetics are: !rA = kACA

2 with k = 0.8 m3/(kmolAs). At a certain moment two reactors are available for the partial conversion of a stream of A of 50 liter/s (CA,0 = 0.1 kmol/m3) into Q. These reactors are an ideal CSTR with a volume of 7.5 m3 and an ideal PFR with a volume of 1.875 m3. The following alternatives are considered:

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Give the order in which the conversion increases for the four alternatives (e.g. ξB < ξBT = ξTB < ξT ) and explain. 25 For the following arrangement of ideal reactors

calculate: a The volume flow rate at points 2 and 8. b The concentration of A at points 7 and 13. Data: − φv1 = 18 m3/s − CA,1 = 10 kmol/m3 − The reaction A ! P with kinetics !rA = 2ACA kmol/(m3As) − No change of density due to the reaction. − The streams are divided such that ξA, 13 is as high as possible. 26 A plant produces 100 m3 of radioactive wastewater each hour. Before discharging the wastewater into the sewer its activity has to be reduced to 0.002 percent of its original value. It is intended to feed the wastewater to three large underground basins in series before discharging it into the sewer. Circulation in the basins is provided by large mixers. Determine the volume of the basins if the half-life is 10 hours and the basins may be assumed to be ideally mixed.

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27 For the production of B from A two ideal CSTRs are available, with a volume of 8 and 2 m3, respectively. The feed is a stream (4 m3/min) of A with a concentration of 8 kmol/m3 A. The reaction proceeds according to A ! B. The kinetics were measured in a laboratory in a small ideal CSTR at the same temperature. A fractional value is expected for the reaction order. The following observations were made:

τs [min]

CA,0 [kmol/m3]

CA [kmol/m3]

0.15 10 8 0.43 10 6 1.06 10 4 3.36 10 2 9.00 10 1 0.18 5 4 1.26 5 2 4.00 5 1

What is the maximum possible yield of B in the practical situation? 28 The kinetics of the reaction A ! products are studied in a pilot plant consisting of three ideal CSTRs in series. The operating temperature of each reactor is equal to the temperature of its feed. Where needed intermediate heat exchanger can be used. The heat of reaction will be removed completely by a cooling coil that is present in each reactor. The concentration of A in each reactor is measured. See the sketch for other data. The process is carried out in the liquid phase.

Data: − φv,0 = 2 m3/s − CA,0 = 15 kmol/m3 − V1 = 2 m3 − V2 = 8 m3 − V3 = 8 m3 − !rA = k4 Ae -Ea/RTACA

n − R = 8.3 kJ/(kmolAK) Measured values: − CA,1 = 6.36 kmol/m3 − CA,2 = 0.99 kmol/m3 − CA,3 = 0.23 kmol/m3

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a Determine the order n, the activation energy Ea and the frequency factor k4. Subsequently, it is intended to convert a stream A, with a concentration of 10 kmol/m3 A and a total molar flow of 50 kmol/s A in an industrial installation that consists of one ideal CSTR. The feed is NOT at reactor temperature but at 300 K. This is also the temperature of the cooling water used to remove the excess heat of reaction. Other data: − UAA (for cooling of reactor) = 2.0A104 kJ/(sAK) − ∆Hr,A = !1.0A105 kJ/kmol − cp = 50 kJ/(kmolAK) − cp and ∆Hr,A are independent on temperature and composition b Calculate the temperature at which the conversion is 90 % and the required reactor

volume. 29 The production of B from A according to reaction A ! B is carried out in three ideal CSTRs in series of volume 4, 10 and 20 liter, respectively. The feed is a stream of pure A of 100 liter/min with concentration CA of 10 mol/liter. The reaction rate equation is: !rA = CA

2 (k = known). a Graphically determine the concentration at the exit of the first, second and third tank,

respectively. b Subsequently make a graph of 1/(!rA ) versus CA and in this graph indicate the areas that

are a measure of the volumes of the first, second and third tank, respectively. c In the graph of b indicate the area that represents the volume of a PFR yielding the same

conversion as the three CSTRs in series. d Estimate the volume of the PFR in c from the area ratios and given tank volumes.

Compare this estimated volume with the volume of a PFR that can be calculated from the molar balance for the conversion in a PFR from CA, 0 to CA, 3 (the concentration of A in the third CSTR).

30 Species B is produced from A according to a first order irreversible reaction A ! B. The system used consists of two CSTRs in parallel, as shown in the figure below. The volumes of the CSTRs are V1 and V2, respectively, and V1 > V2.

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a Calculate the volume of a CSTR that converts the molar flow φv,0ACA,0 to the same degree of conversion ξA, out if ξA, 1 = ξA, 2.

b Determine which of the two reactors V1 and V2 has to be in front if the CSTRs are placed in series to yield the highest conversion.

c If only the total volume of the two CSTRs is fixed, but the volume of each separate CSTR can be chosen freely, calculate the optimal ratio V1 / V2 so that the conversion is as high as possible when the two CSTRs are placed in series.

31 It is planned to make the product R in a reactor system consisting an ideal CSTR and an ideal PFR in parallel (see figure).

Data: − reaction equation A ! R, with reaction rate constant k = 1 kmol/(m3As). − V1 = 2 m3 − V2 = 3 m3 − CA,0 = 1 kmol/m3 − φ v,0 = 10 m3/s Determine into which flows φv, 1 and φv, 2 the original flow φv, 0 has to be divided in order to maximize the total conversion of the system. 32 For the production of P from A two ideal CSTRs with volumes of 3 and 6 m3 and a separation column that separates the mixtures of A and P into the pure species are used (see figure).

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The kinetics of the reaction are: !rA = kACA with k = 0.5 min-1. The feed to the first reactor is a stream pure A: φv = 1.5 m3/min, CA,0 = 12 kmol/m3. A, P and their mixtures may be assumed to have the same density. Determine the concentrations of A and the sizes of the volume flow rates at the following points: � the exit of the CSTR with a volume of 3 m3, � the exit of the CSTR with a volume of 6 m3, � the inlet of the separation column, � the exit of the separation column. 33 In a certain process a stream of 6 m3/h of pure A and concentration CA,0 = 20.6 kmol/m3 is converted for 75% into P by a first order irreversible reaction. The densities of A and P are equal. The reactor is a glass vessel with a volume of 3 m3 equipped with two stirrers (see figure a). This reactor may be assumed to be ideal. At a certain moment the axis between the two stirrers breaks, so the lower stirrer stops working. The exit concentration of A increases to 6 kmol/m3. Because the reactor is made of glass one can see that the upper part is still well mixed, whereas the lower part of the reactor seems to behave more or less as a two PFRs (see figure b).

After estimating the volumes it is decided to regard the resulting reactor as shown in figure c: a combination of an ideal CSTR (volume 2 m3 ) and two ideal PFRs (each with a volume of 0.5 m3 ). Check whether the proposed model also leads to the increased exit concentration of A.

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4 Ideal reactors: a financial question 34 It is intended to produce 10 kmol/s of product B from A in a number of CSTRs in series. Each CSTR has a volume of 0.1 m3. Downstream reactor n unreacted reactant A will be separated from the product and recycled to the inlet of the cascade of CSTRs (see figure). The reaction equation is A ! B with !rA = 2ACA. The feed is pure A and the separation is ideal. What number of tanks, n, is financially most attractive if: − yield of B: $ 40 per kmol B − costs of A: $ 5 per kmol A − reactor costs: $ 4 per m 3 and per s − separation costs: $ 3 per kmol of B to be separated. 35 The government is planning on introducing eco-tax on the emission of carbon monoxide from motor vehicles, namely for the quantity exceeding 2 g/km. For every gram of carbon monoxide exceeding this limit, 0.002 $ should be paid for every kilometer. At an average velocity of 50 km/h the emission of current vehicles is 18 g/km higher than the emission standard. However, afterburners are available, which can be mounted between the cylinder and the exhaust, and in which secondary air is used to combust the CO to CO2 over a supported platinum catalyst (see scheme).

cylinder afterburner exhaustgasoline

air

secondary air The costs of these afterburners are entirely determined by the cost of the required catalyst. Currently the cost is $ 180 per kg. A price-conscious student in chemical reactor engineering wonders, given the high cost of the catalyst, whether it would be sensible to use exactly the amount of catalyst required in his car exhaust to completely comply with the standard. He subsequently calculates whether it would be more attractive to oxidize only part of the CO to CO2 and pay tax for the remaining part, as far as that exceeds the standard. He assumes a car

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usage of 20,000 km/year and for simplification a constant speed of 50 km/h. He further assumes that the reactor can be considered as a homogeneous, isothermal, ideal PFR. Other data: − at a speed of 50 km/h a car produces 800 liter/km of exhaust gas − the secondary combustion air at that speed is 200 liter/km − at these conditions (excess oxygen) the rate of conversion of CO is: − rCO = k⋅CCO with k =

39400 h-1 − the bulk density of the catalyst is 700 kg/m3 − the life of the catalyst corresponds with 20,000 km; then the catalyst must be completely

replaced − the molar mass of CO is 18 g/mol Will the student exceed the standard, and if so, how much eco-tax will he pay each year? 36 In an ideal CSTR A is converted into R. This reaction is first order in A, the reaction rate constant k is 3 h-1. Besides R a product S is formed, which yields the same amount of money per kmol as A costs per kmol. The kinetics for the formation of S are unknown. After the reactor R and S are separated from each other and from unconverted A. The unconverted A is recycled to the reactor (see figure).

R

S

S

A

,

Data: − reaction scheme: A ! R A ! S − volume of reactor = 10 m3 − feed φv,f = 50 m3/h − costs of separation = $4 per kmol A to be separated − cost of A = $ 2.50 per kmol A − yield of R = $ 12.50 per kmol R. Determine the degree of conversion ξΑ for which the plant operates most profitably. 37 In a plant a stream of pure A (φv = 2 m3/s, CA,0 = 0.5 kmol/m3) is available. It is intended to produce product B according to the first order irreversible liquid phase reaction

A ! B (k = 2 s-1). Because small amounts of the highly undesired product C are formed from B and C can not be separated from B, the selling price of B decreases with increasing conversion in the tubular reactor.

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The selling price of B is: $ 15Aexp(!ξΑ) per kmol B. The plant is built as shown in the following scheme:

A

B (+ C)

ideal PFR

idea

l sep

arat

or

Data: − the separation is ideal and costs $ 3 per kmol A to be separated. − the costs of the reactor are $ 2.50 per m3 and per second. − the costs of A are $ 1.50 per kmol. Graphically determine which reactor volume (approximately) will be installed. 38 It is intended to convert a stream of pure A (φv = 1 m3/s, CA,0 = 0.5 kmol/m3) into B according to a second order reaction of which the reaction rate constant has the value 0.5 m3/(kmolAs). The proposed reactor is an ideal PFR. The product mixture will be separated into pure B and pure A. The pure A will be recycled to the reactor. Using the following data calculate the reactor volume that is financially most attractive for such a process. − costs A: $ 0.50 per kmol A − yield B: $ 40 per kmol B − costs reactor: $ 2 per (m3As) − costs separation: $ 4 per kmol A to be separated. What will be the optimal reactor volume if the yield of B is only $ 0.6 per kmol B? 39 It is intended to make the products B and C from pure A (CA,0 = 4 kmol/m3) according to the reaction sequence: A ! B ! 2C. All steps are first order, the reaction rate constants are k1 and k2 and their values are 3 and 2 h-1, respectively. The reactions take place in the liquid phase, for which the density may be assumed to be constant. The reaction is carried out in an ideal CSTR. The effluent of the reactor is separated into the pure species B and C in a separation unit. In order for the separation to behave ideally the ratio of the concentrations C and B after the reactor is chosen to be 4 : 1. The stream A leaving the separation unit can not be used further. The following figure shows a schematic of this process:

23

a Calculate the concentration of C after the reactor as a function of space time τs. b Which τs is used in this process? What are the concentrations of A, B and C in the reactor? c The following financial data are important to the process:

− A costs $ 13 per kmol − C yields $ 20 per kmol − The price of B depends strongly on the production by the process. If the molar flow of

B is expressed in kmol/h the yield of B can, within reasonable limits, be expressed as (40 ! 4Aφmol,B) per kmol B.

− The costs of the reactor are $ 4 per (m3Ah). − The separation costs $ 5 per hour.

Determine the volume flow rate of A that will render this process most profitable. 40 In a batch reactor B is produced from A by the second order reaction A ! B. The reaction rate constant is 1 m3/(kmolAh). Pure A is used with a concentration of 1 kmol/m3. The time needed for cleaning up, loading etc. is 1 hour. The reactor is in operation all day (24 h/day). One reactor load of A costs $ 20.000. If all A is converted into B this will yield $ 30.000. The operating costs of the reactor and the costs for purification of B are negligible compared to the aforementioned costs. Therefore, incomplete conversion reduces the yield proportionally. How many cycles a day will be completed: a If one wants to produce as much B as possible in one day? b If one wants to make as much money as possible, while unconverted A can not be reused? c If one wants to make as much money as possible and unconverted A can be reused, but the

purification costs of A are negligible? 41 In an ideal PFR (V = 5 m3) B is produced from A (CA,0 = 2 kmol/m3) according to the reaction: A ! B. This reaction is exothermic (∆Hr,A = -226.8⋅103 kJ/kmol) and second order in A, with k = 4 kmol/(m3⋅h). In order to be able to operate the process isothermally, the reactor has been equipped with a cooling jacket, in which water (40 bar, 250 oC) is converted into steam (40 bar, 250 oC). After the reactor unconverted A is completely separated from B and recycled to the inlet of the reactor (see figure).

24

A ideal PFR S

water40 bar250 oC

steam40 bar250 oC

Other data: − ∆Hevaporation, water = 2268 kJ/kg (at 40 bar and 250 oC) − yield steam: $ 0.01 per kg (at 40 bar and 250 oC) − costs A: $ 2 per kmol A − yield B: $ 4 per kmol B − separation cost: $ 1 per kmol A to be separated − reactor cost: $ 3 per (m3⋅h) Calculate what amount of B has to be produced per hour in order to operate the installation as economically attractive as possible.

25

5 Ideal reactors: no reaction without heat 42 It is intended to produce 800 kg/s B in an ideal CSTR (V = 0.5 m3). A volume flow of pure A of 2 m3/s at 27 oC is available. The reaction proceeds according to: A ! 2B. The reaction is exothermic, ∆Hr,A is !250 kcal/kg, and at 27oC the reaction rate is 1.0A10-3ACA kmol/(m3As). In order to achieve the desired production, the process has to be carried out at elevated temperature. Therefore, it is intended to install a heat exchange surface in the tank reactor and use condensing high pressure steam (T = 527 oC) as heating medium. Calculate the heat exchange capacity UAA required if: − Ea = 9.95 kcal/mol − R = 2A10-3 kcal/(mol.K) − ρA = ρB = 800 kg/m3 − cp,A = cp,B = 1.25 kcal/(kgAoC) You may assume that ∆Hr,A , ρ and cp are not a function of temperature and that the temperature at the steam side of the heat exchange surface remains constant at 527 oC. 43 The solvent ILAP is produced in a small ideal CSTR (V = 20 liters) from a middle distillate species A. The decomposition reaction is a first order reaction: A ! ILAP + R, with R being the byproduct of the decomposition. The required heat of reaction is partially produced by the reaction itself (∆Hr,A = -100 kcal/kg, independent on temperature) and partly by preheating of the feed (4 m3/h, 27 oC) in a heat exchanger. After the reactor the stream is separated in a distillation column, where 90 % of the produced ILAP is recovered from the product stream. The unreacted A, the remaining ILAP and the byproduct R are used as fuel elsewhere in the plant. The following figure shows a schematic of the process. Determine the heat flow exchanged in the preheater. Data: − net production of ILAP = 5760 t/year − reaction rate constant k = 150 h-1 at 127 oC − activation energy Ea = 8 kcal/mol − gas constant R = 2 cal/(molAK) − densities ρA, ρILAP, ρR = 500 kg/m3 (indep. on T and comp.) − heat capacities cp,A, cp,ILAP, cp,R = 1 kcal/(kgAK) (indep. on T and comp.) − molar mass MILAP = MR − one year continuous = 8000 h

26

44 Using of a feed A (φv,0 = 1 m3/s, CA,0 = 4 kmol/m3, T = 20 °C) product P is produced in an ideal CSTR (V = 10 m3). The reaction A ! P (k = 0.9 s-1) is exothermic. The reaction temperature is 320 °C. Cooling of the reaction mixture takes place by using a heat transfer surface installed inside the reactor: UAA = 1.2A103 kJ/(KAs). Sometimes the feed has a 25% lower concentration of A. With an extra heat exchanger this feed is preheated to 110 oC to maintain the original temperature and conversion (but not CA,out). Calculate the average temperature of the cooling medium of the reactor, the conversion reached and the amount of heat released per kmol reacted A, if it is further known that the heat capacity (5 kJ/(kgAK)) and density (800 kg/m3) of the mixture are independent on temperature and composition. 45 In an ideal CSTR a first order irreversible exothermic reaction is carried out: A ! products (density is constant). Data: − V = 0.2 m3 − τs = 0.2 s − ρ = 1000 kg/m3 − cp = 3000 J/(kgAK) − ∆Hr,A = !281000 J/mol − Ea = 22500 J/mol − k = 1 s-1 (at 27 oC) − R = 8.33 J/(molAK) Density, heat capacity and reaction enthalpy are independent on temperature and composition. a What temperature is needed to reach a conversion of 80%? b What (theoretical) maximum concentration of A in the feed is allowed so that stable

operation of the reactor is still possible at this conversion? c If the temperature of the feed stream can only be controlled between 27 and 127 °C, what

is the maximum allowable concentration of A in the feed? 46 In an adiabatic ideal batch reactor B is produced from A by the first order irreversible exothermic reaction A ! B with k = 10⋅e(-1000/T) h-1, with T expressed in K. At t = 0 the reactor is loaded with pure A and the temperature is 200 oC. The maximum adiabatic temperature rise is 100 oC. The reactor does not absorb any heat. Graphically determine the time required for obtaining a conversion of 40%. What is the temperature in the reactor for this conversion? 47 A small production unit is built for the conversion of raw material A in product B by a first order irreversible reaction. The reactant stream (pure A, φv = 5 m3/h, T = 20 oC), prior to being fed to the reactor (ideal CSTR, V = 5 m3), is preheated in a feed-effluent heat exchanger with the hot product stream. Subsequently, the product B is separated from unconverted A in a separation step. The unconverted A can not be reused (see figure). The temperature in the reactor is chosen such that 1 kg/s B is produced. To this end, additional heat is supplied by a steam coil inside the reactor. In order to obtain an optimal separation, the feed to the separator must be at 80 oC.

27

B

S

A

steam

A

Data: − cpA = cpB = 2 kJ/(kg⋅K), independent on T − ρA = ρB = 800 kg/m3, independent on T − ∆Hr,A = − 100 kJ/kg The following relation exists between k and T:

0

1

2

3

4

5

1.8 2.0 2.2 2.4

ln (k) with k [h-1]

1000/T with T [K] Calculate: a The reaction temperature. b The heat exchange capacity U⋅A of the heat exchanger. c The amount of heat that has to be supplied by the steam coil.

28

48 The production of P from A is carried out in an ideal CSTR with a volume of 4 m3 at a reaction temperature of 275 oC. The reactor is equipped with a cooling coil with a heat transfer area of 3 m2. In order to save cooling water, it was decided to use the cold feed as coolant inside the coil in stead of cooling water. The process scheme is as follows: The feed flowing through the cooling coil is 2 m3/h pure A with a concentration of 50 kmol/m3 and a temperature of TF oC. For the reaction A ! P the following data are available: ! rA = kACA with k = 0.5 h-1 at 275 oC, Ea = 40 MJ/(kmol A) and ∆Hr,A = - 50 MJ/kmol, independent on temperature. Other data: − ρ = 1250 kg/m3 independent on temperature and composition − cp = 10 kJ/(kgAK) independent on temperature and composition − R = 8.3 kJ/(kmolAK) For the calculation of the heat transfer to the cooling coil the following simplified equation may be used:

φheat = UAAA(Treactor - (T0+TF)/2), with U = 10 MJ/(m2AhAK) a What are the values of T0 and TF at steady state? b Is this steady state stable or unstable with respect to disturbances in the heat flows? c At a certain moment the coil becomes plugged and it is decided to feed the feed stream A

directly into the reactor and not use the cooling coil anymore. Qualitatively show how the temperature in the reactor will change in the new situation (TF remains the same).

49 The exothermic decomposition of A into 2B + C takes place in the liquid phase without change of reaction volume. The reaction is studied in an ideal PFR, which is also ideally insulated from its surroundings: there is no heat loss whatsoever through the wall. Regularly distributed along the length of the tube thermocouples have been placed to study the temperature profile in the reactor. The following results are obtained in one of the experiments:

Location thermocouple [% of length of reactor]

Temperature [oC]

0 100 20 110 40 123 60 137 80 153 100 169

29

Other data: − φv = 10 liter/h − R = 8.3 J/(molAK) − VR = 2 liter − Ea = unknown − CA,0 = 20 mol/liter

− k4 = 1.1A105 h-1 − -rA = k4Aexp(-Ea/[RT])ACA − ∆Hr,A = − 24 kJ/mol − cp = 4 kJ/(kgAK) − ρ = 1.2 kg/liter

∆Hr,A, cp and ρ are independent on temperature and composition of the reaction mixture. a Determine the conversion ξa at the outlet of the reactor. At a certain moment the insulation of the first half of the reactor falls off. From this moment on heat is transferred to the surroundings (Tsur. = 27 oC) and the conversion becomes ξb. b Is the conversion ξb higher, equal to or lower than the original conversion ξa? Why? Does this answer depend on the value of Ea? Why? c Give a well-considered estimation of the value of Ea. 50 At an organic chemistry laboratory you have to carry out a synthesis starting from n-onsensol using a 3 liters container, a stirrer and a heating plate. The container is insulated from the surroundings. The heating element is only used to bring the n-onsensol (2 kg) to its reaction temperature (500 K). When this temperature has been reached, the heating element is removed and replaced by insulation. Up to 500 K the n-onsensol did not react! The conversion of n-onsensol is an exothermic process (∆Tadiab. = 200 K) and first order in n-onsensol. Within the temperature range considered the following approximation is valid:

k = 1 + 0.001AT h-1 (T expressed in K) The reaction is stopped when 90% of the n-onsensol has been converted. The used container used is made out of teflon, which means that the temperature must not exceed 560 K. To prevent a temperature of over 560 K you have several ice cubes (80 g a piece), which you can throw into the container. You may assume that the ice is at 273 K and that it melts instantaneously and is vaporized from the solution (heat of melting of ice = 335 kJ/kg, cp,water = 4.2 kJ/(kgAK), heat of vaporization of water = 2250 kJ/kg). The heat capacity of the n-onsensol-mixture is 3 kJ/(kgAK), while the container does not absorb any heat. a How many ice cubes did you need to add to the mixture before the n-onsensol has been

converted for 90 %? b After what time (in minutes) do you have to add the first ice cube at the latest? c Should you add all the ice cubes at once, or is there a better alternative? d Why would a pyrex container be preferable for this experiment? Note:

( )( )x'b'aln

'bb'a'ab

'bbxdx

x'b'abxa

2 +−+=++

∫

( )( )

++

−=

++∫ bxax'b'aln

b'a'ab1

x'b'abxadx

( )( ) ( ) ( )

+−+

−=

++∫ x'b'aln'b'abxaln

ba

b'a'ab1

x'b'abxaxdx

30

51 The production of P from pure A takes place according to the first order irreversible gas phase reaction: A ! P with kinetics ! rA = k0⋅exp(!Ea/RT)⋅CA in which − k0 = 2.0⋅107 s-1 − Ea = 8.3⋅104 kJ/kmol − R = 8.3 kJ/(kmol⋅K) The reaction is exothermic, ∆Hr,A = − 2.0⋅105 kJ/kmol. In order to control the reaction, a reactor consisting of a bundle of tubes, each with a diameter of 4 cm and a length of 1 m, is used (see figure).

T0

Tc

Tc

0 Lx

The bundle is cooled by a cooling medium, which enters the cooling inlet with a temperature Tc = 450 K. The heat transfer coefficient U is 0.025 kJ/(m2⋅K⋅s). The flow of the cooling medium is such that heating of the cooling medium can be neglected. The inlet temperature T0 of the reactant flow is also 450 K for each of the tubes. The tubes can be considered as ideal PFRs. Per tube 1.25⋅10-4 kg/s A is fed, with a density of 2.5 kg/m3. The molar mass of A is 100 kg/kmol. The heat capacity of both the feed and the products (and their mixtures) is 1.5 kJ/(kg⋅K). ∆Hr,A, ρ, and cp are independent on temperature. In the tubes an axial temperature profile will exist. a Formulate a mass balance and a heat balance for one of the tubes. b Make the resulting differential equation dimensionless by using the following substitutions: ξ = (CA,0 − CA)/CA,0; t = (T − T0)/T0 and z = x/L Solving of this set of equations is not possible analytically. Numerically, however, this is no problem, for instance by dividing the reactor length in 10 slices and calculating ∆t and ∆ξ over each slice, using the initial conditions of that slice and the balance equations above, but then written as differential equations. c In this way calculate the temperature and the degree of conversion after the first and

second slice of a PFR that has been divided in 10 slices.

31

6 Ideal reactors: with a solid catalyst 52 In a laboratory, research takes place on the heterogeneously catalyzed decomposition of A. The reaction kinetics of this decomposition appear to be ! rA,s = ksACA , with ks dependent on the temperature according to the Arrhenius equation. The used catalyst consists of spherical particles with an internal surface area of 62 m2/g. During all experiments the decomposition takes place isothermally and in the same reactor. The inlet concentration of A is always 10 mol/m3. The diffusion coefficient ID does not depend on temperature. Calculate the value of the effectiveness factor E for each experiment (1-8), using the results listed below.

Experiment [-]

Temp. [oC]

φmol A,0 [mol/h]

Weight cat. [g]

Diam.cat. [mm]

Conversion [-]

1 127 1 400 8 4.62A10-4 2 127 1 400 1 à 2 4.62A10-4 3 127 2 800 8 4.62A10-4 4 177 10 400 8 2.98A10-3 5 177 10 400 1 à 2 2.98A10-3 6 177 20 800 8 2.98A10-3 7 350 1000 400 8 3.44A10-2 8 350 2000 800 8 3.44A10-2

53 The temperature dependence of a first order gas phase reaction is being examined on two spherical catalysts with different average particle diameter and internal surface area. The catalytic activity per unit internal surface area, ks, the effective diffusion coefficient, IDeff, the porosity in the catalyst bed, εbed, and the apparent density of the catalyst particles, ρp, ARE equal for both catalysts. Furthermore, IDeff is independent on temperature. In a series of isothermal experiments, the following values of the amount of reactant converted per unit mass of catalyst are measured. Temp [oC] 200 250 260 270 280 290 300 350 Catalyst 1 4.325 228.2 434.7 766.9 1241 1853 2607 10690 Catalyst 2 2.161 110.0 206.6 353.1 552.4 802.7 1113 4536 Rate ratio 2.001 2.074 2.104 2.172 2.247 2.308 2.342 2.357 a What is the ratio of the particle diameters dp,1/dp,2? b What is the ratio of the specific internal surface areas Sv,1/Sv,2? 54 Two materials can be used for drying a gas stream of 100 ml/min (at 27 oC and 1 bar) containing 1 vol% water: mole sieve 5A and γ-Al2O3. The adsorption of water on these materials can be considered as a surface reaction with reaction kinetics: ! r = kAC a Determine which drying material results in the lowest water content if the gas stream is

fed to an ideal PFR (empty volume 250 ml) filled with one of the drying materials. b What water content can be achieved? The following data are known:

32

symbol units Mol. sieve Alumina average pore diameter dpo Å 5 1040 internal surface area S m2/g 200 200 particle diameter dp mm 8 2 particle density ρp kg/m3 1200 1500 bed porosity εbed - 0.6 0.6 effective diffusion coeff. water in the particles

IDeff m2/s 2.4A10-8 3.6A10-6

adsorption rate constant ks m3/(m2As) 1.0A10-8 3.0A10-11

55 The heterogeneously catalyzed first order surface reaction A ! P is studied in an ideal PFR (empty volume 250 ml) which has been filled completely with 200 g porous, spherical catalyst. Catalyst data:

− Internal surface area S 200 m2/g − Apparent density of a catalyst particle ρp 1600 kg/m3 − Porosity of the catalyst bed εbed 0.5 - − Diameter of a catalyst particle dp 4 mm − Effective diffusion coeff. in cat. particle IDeff 0.3A10-9 m2/s − Kinetics: !rA,s = ksACA,s with ks = ks,4Ae(-Ea/RT) and T in K.

The reactor is fed with a stream of pure A (φv = 500 ml/min, CA,0 = 0.01 mol/liter). In the first experiment the temperature in the reactor is 400 K and a degree of conversion of 0.6321 is reached. In the second experiment the temperature is 500 K and a degree of conversion of 0.8647 is reached. Determine Ea/R for the following two �extremes�: a In both experiments the concentration of A in the porous catalyst hardly depends on the

distance to the external surface of the particle. b In both experiments the concentration of A in the porous catalyst strongly depends on the

distance to the external surface of the particle. Assume the densities of A, P and their mixtures and the diffusion coefficient to be independent on temperature and composition. Neglect concentration gradients in the film surrounding the catalyst particles. 56 A small vessel with a volume of 50 ml is more or less filled with a packed bed of solid, spherical catalyst K. It behaves like an ideal PFR. The decomposition of a gas stream pure of A, flowing through this reactor is determined by first order reaction kinetics: !rA,s = ksACA,s, with:

!rA,s = decomposition rate of A per unit catalyst surface ks = reaction rate constant = 2A10-6 m3/(m2As) CA,s = concentration of A at the catalyst surface [mol/m3]

Data for the catalyst K: − ρ = density of the catalyst particle = 1000 kg/m3 − Sv = specific internal surface area = 4.0A108 m2/m3 − Sg = specific internal surface area = 400 m2/g − ID = effective diffusion coefficient of A in catalyst = 2.0A10-6 m2/s

33

In the first experiment the reactor is completely filled with particles with a diameter of 3 mm, which are loosely packed; the porosity of the bed is 0.60. In this situation a conversion of A of 38.1% is obtained. a Determine the volume flow rate φv through the reactor. The catalyst bed is then �tapped�. The catalyst bed in the reactor now resembles a �close-packing� of particles and the porosity has decreased to 0.52. No additional catalyst has been added to the bed. b Determine the conversion of A that is obtained in the new steady state. After this experiment the catalyst particles of 3 mm are replaced by an equal mass of particles with a diameter of 0.03 mm, which are also �tapped�: the porosity of the bed again is 0.52. c Determine the steady-state conversion of A in this situation. 57 Many industrial catalysts are strongly poisoned by traces of oxygen in the feed stream, even at ppm level. When a stream of hydrogen contains traces of oxygen, these traces can be removed by passing the stream over platinum at room temperature prior to feeding it to a reactor. In this way the oxygen is converted completely into water. This water is often less harmful or can be removed with mole sieves. In a certain process, a �sponge� of platinum wire is used to remove traces of oxygen from a hydrogen stream of 200 ml/s. The �sponge� weighs 10 g and is homogeneous distributed in a space of 50 ml. The diameter of the non-porous wires of this �sponge� is 0.2 mm; the density of the platinum wire is 2.0A104 kg/m3. Unfortunately, the �sponge� does not operate very well: only 63.2 % of the oxygen is removed. It is decided to replace the �sponge� with porous spheres, which are homogeneously loaded with 0.1 wt% platinum. For each sphere:

− diameter = 5 mm − weight = 10 mg − Pt-surface = 0.131 m2 (ks, wire = ks, sphere)

The space of 50 ml is homogeneously filled with 200 of these spheres. Determine by how much the oxygen conversion increases or decreases in the new situation. Assume that the reaction at the platinum surface is first order in oxygen and zero order in hydrogen. The effective diffusion coefficient of oxygen in the spheres is 2.0A10-6 m2/s.

34

7 Non-ideal reactors: residence time distribution 58 In a commercial installation the product B is produced from A by the first order irreversible liquid phase reaction: A ! B. The reactor used behaves non-ideally. From experiments it appears that the age distribution of volume elements leaving the reactor can be described by: E(t) = E(0) ! 0.1At as is also shown in the following figure.

a What is the value of E(0)? b What is the conversion obtained in the commercial reactor, given that in an ideal CSTR

with the same average residence time distribution as in the commercial installation a conversion of 60% is obtained?

c If reaction A ! B is not a first, but a second order reaction, what can be concluded about the conversion based only on the residence time distribution? Is the conversion less, equal to or higher than the conversion observed in reality? Motivate your answer.

59 A suspension polymerization takes place in a continuous reaction system. A suspension of small droplets of monomer in oil (volume flow rate 1 m3/min) is fed to the reactor. The reaction rate of the monomer is: !rmonomer = 0.1AC2

monomer kmol/(m3Amin) The system can be regarded as fully segregated. The initial monomer concentration in the droplets is 1 kmol/m3. From residence time distribution measurements it can be concluded that at the flow rate of 1 m3/min none of the volume elements in the reaction system for longer than 10 minutes, while for t < 10 minutes E(t) is constant. Calculate the conversion at the outlet of the reactor and the reactor volume.

35

60 During an investigation of the first order irreversible reaction A ! P in an ideal CSTR it has been determined that with a space time τs a conversion of A of 40% is obtained. Subsequently, the reaction is carried out at the same space time (and temperature) in a non-ideal PFR. Of this reactor it is known that a pulse of marker M injected at t = 0 at the inlet of the reactor (space time τs) results in the following concentration profile at the outlet of the reactor:

a Sketch and formulate the relationship between E(t) and t for this non-ideal PFR. b Which space time τs is used? c What conversion is obtained with the non-ideal PFR? 61 A restaurant wants to install a continuous electric fryer for its French fries. The fryer is continuously fed with a mixture of fresh oil and unfried fries (as a �constant mixture�), and outgoing oil and baked fries are separated by a conveyor belt with holes in it, as schematically shown in the following figure.

fryer

oil

oil

unfriedFrench fries

friedFrench fries

From representative tracer experiments, in which at time t = 0 a pulse of 1 mole of colored oil is injected at the entrance of the fryer, the following concentration profile at the exit of the fryer has been measured:

36

a What is the volume of the fryer? It is known from experiments with a normal batch electric fryer that for the mass balance of the system the following equation holds: d(Br)/dt = kA(1-Br) with Br = the degree of browning of the French fries. And: t = 0: Br = 0

t = 3 min: Br = 0.5 t = 4: Br = 1 (burnt fries)

b What is the average degree of browning for the continuous fryer? Assume that the residence time distributions of oil and fries in the fryer are the same.

62 In a continuous non-ideal reactor B is produced from A by the first order irreversible liquid phase reaction A ! B. Measurements of the residence time distribution resulted in the following concentration profile at the exit of the reactor, as response to a pulse marker M at the inlet of the reactor:

Time [min] 0 1 2 3 4 5 6 7 8 9 10 C [mol/liter] 0 0 2 3 6 7 5 2 0 0 0

From kinetic measurements in an ideal CSTR it was already known that with a space time equal to the average residence time in the non-ideal reactor the conversion is 90%. Which conversion can be expected (approximately) in the real system?

37

63 A new reactor has been installed for the production of B by the first order irreversible reaction 2 A ! B. Because it is presumed that the reactor is not ideal, the non-ideal behavior is investigated using a pulse marker. At the exit of the reactor the following marker concentrations are measured, as response to a pulse at t = 0 at the inlet:

Time [min] 10 20 30 40 50 60 70 80 C [mol/liter] 0 3 5 5 4 2 1 0

Based on these measurements, which conversion can be expected in the new reactor, if it is known that in an ideal CSTR at the space time chosen for the new reactor, a conversion of 80% is achieved? 64 A flow of pure A (φv = 12 m3/min, CA,0 = 2 kmol/m3) is converted into B in two continuous ideal PFRs in parallel. The reaction is a second order irreversible liquid phase reaction A ! B with reaction rate constant k = 0.5 m3/(kmolAmin). The reactor system is shown in the following figure:

a Calculate the conversion of the system. b Sketch the E(t) versus t curve of the system. Indicate the scale divisions and units of the

axes! c Sketch the F(t) versus t curve of the system. Indicate the scale divisions and units of the

axes! d Is the conversion calculated on the basis of the E(t) curve larger, smaller or the same as the

conversion calculated for question a? Explain the answer. 65 Sketch the E(t) curve for the following systems (five separate sketches!): a An ideal PFR (volume 2 liters, feed 1 liter/min.). b An ideal CSTR (volume 2 liters, feed 1 liter/min). c An ideal PFR (volume 2 liters) followed by an ideal CSTR (volume 2 liters). The feed has

a volume flow rate of 1 liter/min. d An ideal CSTR (volume 2 liters) followed by an ideal PFR (volume 2 liters). The feed has

a volume flow rate of 1 liter/min. e An ideal CSTR (volume 2 liters) in parallel with an ideal PFR (volume 2 liters). The feed

(volume flow rate 1 liter/min) is equally divided over the reactors: 0.5 liter/min to the CSTR and 0.5 liter/min to the PFR. After the two reactors the streams are mixed together. Give the E(t) curve of the complete system, NOT of each individual reactor.

Use a linear scale division in each figure (add a sketch of the corresponding reactor system) and indicate: − the location of the origin: E = 0, t = 0

− the location of the point: E = 0, t = 1 min − the location of the point: E = 1 min-1, t = 0

38

66 a Sketch and formulate the E(t) versus t curve for a short-circuited ideal CSTR and a short-circuited continuous ideal PFR, respectively (see figure):

b Calculate the concentration at the exit of the short-circuited ideal for a first order

irreversible reaction, CSTR on the basis of E(t). 67 Sketch the E(t) curves for each of the reactor systems shown below. The CSTR symbol stands for an ideal CSTR and the PFR symbol an ideal continuous PFR. The squares indicate the volume flow rates expressed in liters per second. The circles indicate the reactor volumes in liters. For each reactor system make a separate sketch. Show the scale divisions of the axes and the size of the areas below the curves. If possible, give the values of the points of intersection with the axes. Also, indicate all other relevant information (DO NOT derive mathematical equations for E(t)!). a)

e)

b)

f)

c)

g)

d)

h)

39

68 a Sketch and formulate the relation between E(t) and t for a continuous ideal PFR (space

time τ1) and an ideal CSTR (space time τ2). b Sketch and formulate the relation between E(t) and t for a series connection of the reactors

mentioned in a. c Calculate the conversion obtained in this reactor system on the basis of the relation for E(t)

found in b for an irreversible reaction with first order kinetics. d What data are missing for the calculation of the answer to question c the case of a reaction

with second order kinetics? Note: mxmx e

m1dxe =∫ nmxnmx e

m1dxe ++ =∫

mx3

22mx2 e

m2mx2xmdxex +−=∫

mx2

mx em

12mxdxxe −=∫

69 In an ideal CSTR (see figure) B is produced from A by a first order irreversible liquid phase reaction A ! B (k = 9 min-1.). The CSTR has a volume of 10 m3 and the feed is 5 m3/min, with CA = 1 kmol/m3.

During operation of the CSTR the middle blades of the stirrer break off. Experiments show that the tank does not behave as an ideal CSTR anymore, but can be regarded as two ideal CSTRs in series, each with a volume of 5 m3. a Will the stirrer be repaired? Motivate the answer. b Derive a relation for E(t) for the new situation. c Sketch E(t) versus t for the new situation. d Calculate the conversion obtained in the reactor for the new situation. Do this on the basis

of E(t). Comment 1. The first order linear differential equation:

( ) ( ) 0xQyxPdxdy =+⋅+

in which P(x) and Q(x) are arbitrary functions of x, has the general solution: { }∫ +⋅⋅= − ttanconsdxeQey zz with ∫= Pdxz

Comment 2: b

a

mx2

b

amx e

m1mxdxxe −=∫

40

70 a Prove that, if two ideal CSTRs of equal volume V are connected in parallel as shown in

the figure below, the E(t) curve of the system is given by:

( ) ( ) ( )

⋅φ−⋅

⋅φ+φφ

+

⋅φ−⋅

⋅φ+φφ

=V

texp

VVt

expV

tE 2

21

221

21

21

(All φ�s are volume flow rates.)

b Explain (in words) what can be proven with this equation for the case that the two volume flow rates, φ1 and φ2, are equal (φ1 = φ2).

41

8 Non-ideal Reactors: the dispersion alternative 71 A hydrocarbon fraction is converted by a first order reaction (k = 1.06 s-1) in a tubular reactor. The flow rate of the feed (ρ = 800 kg/m3, η = 5.0A10-4 NAs/m2) is 15 liters/s. The reactor, which has a length L of 1 m and an internal diameter d of 30 cm, behaves non-ideally. However, the system can be described with the axial dispersion model. The mass balance that can be derived using this model has been solved analytically by Wehner and Wilhelm1. Bischoff and Levenspiel2 have converted this solution into graph that is easy to use. In this graph (1-ξ) is plotted on the �x-axis� and the ratio of the volumes of the real reactor and an ideal PFR required to obtain a certain conversion is plotted on the �y-axis� (see Figure 1). Comment: The material properties do not change due to the reaction. The process is isothermal.

1- ξ

volume ratio

Determine the conversion in the reactor using the dispersion coefficient obtained by using Figure 2, taken from Levenspiel3.

1 Chem. Eng. Sci. 6, 89 (1956). 2 Ind. Eng. Chem. 51, 1431 (1959) and 53, 313 (1961). 3 Ind. Eng. Chem. 50, 343 (1958).

42

D/vd

Re Figure 2 72 A chemical plant in the south of The Netherlands produces a waste stream (5.66 m3/h , ρ = 800 kg/m3, η = 10.0A10-3 NAs/m2), which contains a radioactive hydrocarbon. The radioactive decay (a first order process!) has a half-life of 2.6A103 s. To be able to discharge the waste stream into the Maas (a river nearby) safely, the radioactivity of the material has to be reduced to 2% of its original value. It is intended to transport the waste stream through an old mineshaft to provide a sufficiently long space time for reducing the radioactivity to the desired level before discharging the stream into the Maas. The mineshafts can be assumed to be more or less tubular with a diameter of 1 m. The behavior of the mineshafts will not be ideal, but can be described with the axial dispersion model. Bischoff and Levenspiel4 have developed a convenient graph representing the reactor design equation, see Figure 1 of this chapter. The dispersion coefficient D for this model can be determined by D/(v⋅d) = 3ARe, with v the linear liquid velocity and d the diameter of the tube. Determine, by trial and error, the approximate length of the mineshafts required for safe discharge of the waste stream into the Maas.

4 see footnote 2

43

73 In a real tubular reactor, volume 6 m3, a stream of pure A (φv, 0 = 3 m3/s) is converted into product B by a first order reaction (k = 2.5 s-1). It appears that the behavior of the real tubular reactor can be described by a model consisting of a cascade of four ideal CSTRs, all with the same volume. a Sketch (in one figure) the E(t) versus t curves of an ideal CSTR, a continuous ideal PFR

and the real reactor. b On the basis of the �four CSTRs in series� model calculate the conversion in the real

reactor. c The behavior of the real reactor can also be described by the dispersion model. On the

basis of the conversion calculated in b and Figure 1 (problem 71), what is the dispersion coefficient, given that the cross section of the reactor is 1 m2?

(Note: Figure 1 gives the solution of the reactor design equation based on the dispersion model for a first order reaction.)

44

9 Various Problems 74 The following data apply to the CO shift reaction: inlet temperature T0 627 K inlet molar flow φmol, tot 9280 kmol/h pressure P 30 bar inlet composition (molar fractions)

132.0y

003.0y

344.0y

048.0y

394.0y

079.0y

2

4

2

2

2

N

CH

H

CO

OH

CO

HTS reactor: diameter D 3.66 m length L1 6.10 m bed density Fe2O3/Cr2O3 ρbed,1 1114.1 kg/m3

LTS reactor: diameter D 3.66 m length L2 4.27 m bed density CuO/ZnO ρbed,2 1424.6 kg/m3

steam

water

ZnO

L1

L2

0

x

0

x

T1

T2

T0

HTSFe2O3/Cr2O3

LTSCu/ZnO

T3

Shift reaction: CO + H2O º CO2 + H2 Keq: 422 < T [K] < 589 Keq = exp (4800/T − 4.72) 589 < T [K] < 867 Keq = exp (4577.8/T − 4.33) 867 < T [K] Keq = exp (4084/T − 4.3765) cp: 34.2 J/(mol⋅K) independent on temperature and composition ∆Hr: − 38400 J/mol independent on temperature

− rCO: )h(kgconverted/ CO kmol 66.23

1K

yyyyk cat

bedeq

HCOOHCO

222

⋅ρ⋅

⋅

⋅−⋅⋅ψ

ψ: HTS P > 20 bar ψ = 4 LTS P > 24.8 bar ψ = 4.33 k: HTS k = exp(15.95 − 4900/T) h-1 LTS k = exp(12.88 − 1856/T) h-1 a Consider the HTS shift reactor as a pseudo-homogeneous ideal continuous PFR. Although

originally designed as an insulated reactor, in practice it appears that some heat exchange with the surroundings (300 K) takes place via the insulation layer (U = 100 W/(m⋅K), independent on temperature).

Calculate and sketch the temperature profile over the reactor. b What would have been the outlet temperature if the insulation had been ideal?

45

75 Until recently a stream of water that contained toxic species T was drained into the sewer (φv = 45 liters/min, CT = 1A10-3 kmol/m3). Environmentally conscious as they are, engineers decided to feed this stream to a CSTR (V = 900 liters), in which the toxic species T is converted into the environmentally friendly species F using 5 kg catalyst particles:

K H2O + T ! F with: !rT,s = ks.CT kmol/(m2As) (water is present in excess) If CT is expressed in kmol/m3 the value of ks is 5A10-9 m3/(m2As) According to the producer of catalyst K:

diameter mm 0.5 catalyst surface m2/g 200 particle density kg/m3 1500 bed density kg/m3 900 bed porosity - 0.4 diff.coeff. T in particles m2/s 4A10-8 mean pore diameter nm 7

The catalyst particles are retained inside the reactor by a sieve placed at the entrance of the outlet pipe. a Calculate the steady-state concentration of the toxic species T at the outlet of the CSTR. Now that the CSTR has been installed and is operating properly at steady state, the engineers decide to drain another vessel containing the same species T into the CSTR. The drain is connected with the reactor by valve A.

At a certain moment the valve is opened and immediately a constant stream (φv = 30 liter/min, CT = 2.2A10-3 kmol/m3) starts flowing into the CSTR. The liquid volume in the reactor does not change. After 4 minutes the vessel is empty. b Calculate the concentration of toxic species T in the CSTR at t = 4 min.

46

76 a Water flows through a small CSTR (V = 5 liters, φv = 10 liter/min), see scheme below.

H2O

A

At time t = 0 the flow is instantaneously switched from water to the reactant A, also with a volume flow rate of 10 liter/min. In the reactor A is converted into B by the equilibrium reaction k1 2 A º B k2 with − rA = k1⋅CA

2 − k2⋅CB and k1 = 16 liter/(mol⋅min); k2 = 3 min-1. The reactant feed does not contain B, the concentration of A is 0.125 mol/liter, and the feed is at reaction temperature. We are interested in the concentration profile of A during the first 5 minutes. Formulate the required molar balance(s), make these dimensionless with respect to concentration and subsequently calculate the concentration profile using appropriate software. Construct a table with the (dimensionless) concentration of A, at least at time t = 0.5, 1, 2 and 5 minutes. Sketch the concentration profile of A.

b Subsequently, it is intended to repeat the �switch exercise� for a system of two CSTRs in series, each with half of the original volume under a, see figure below.

A

H2O

Because it is assumed that the �distance from equilibrium� is still large, the results will be

predicted using only the forward reaction: − rA = k1⋅CA

2 Formulate the required balances for the determination of the concentration of A at the exit

of the second CSTR as a function of time. Again make these dimensionless with respect to concentration. Formulate the boundary condition(s). Work out the differential equation(s) and boundary condition(s) as far as possible. It is not necessary to perform the actual calculation.

47

77 To an ideal CSTR (volume 5⋅10-3 m3) a flow of A is fed for which holds: − φv = 1.5⋅10-4 m3/s − CA = 125 mol/m3 − CB = 0 mol/m3 − ρ = 1100 kg/m3 − cp = 4000 J/(kg⋅K) − T = 350 K In the reactor A is converted into B by an equilibrium reaction for which the rate equation is: − rA = k1⋅CA

2 − k2⋅CB with: − k1 = 5.3⋅1021e-22000/T m3/(mol⋅s) and T [K] − k2 = 2.0⋅1019e-24000/T s-1 and T [K] − ∆Hr,A = − 60000 J/mol The tank is equipped with a heating coil, but is further ideally insulated. Water with a temperature of 350 K flows through the coil; U⋅A = 5 W/K. It is assumed that under these conditions initially: assumption #1 The conversion of A may be neglected. assumption #2 Only the forward reaction is of importance. At time t = 0 the heating coil is instantaneously and permanently switched from water to condensing steam at 450 K. This results in an instantaneous and permanent change of U⋅A to 200 W/K. We are interested in the concentration profile of A during the first 2 minutes. assumption #3 Assume that still only the forward reaction is of importance. a Formulate the molar balances and heat balance and make them dimensionless with respect

to concentration and temperature. b Calculate the concentration profile using appropriate software. Calculate (with a precision

of 5 digits) the concentration of A and the temperature at time t = 30, 60, 90 and 120 seconds.

c Finally, check whether the assumptions #1, #2 and #3 are allowed. 78 Sketch the E(t)-curve and the F(t)-curve for each of the reactor systems shown in the figure below. The tank symbol denotes an ideal CSTR. The tube symbol denotes an ideal continuous PFR. The squares indicate the volume flow rates (in liters/s), the circles the volumes (in liters). Give an E(t)-curve for each system separately with the scale divisions of the axes. Also indicate the sizes of the areas below the curves. Give an F(t)-curve for each system separately with the scale divisions of the axes. Indicate, if possible, the values of the points of intersection of the curves with the axes and other relevant information. For each sketch state whether it concerns an E(t)-curve or an F(t)-curve and also state the problem number. Mathematical derivations are not expected!

ste

A + A

wa

48

1) 5) 2) 6) 3) 7) 4) 8) 79 Two ideal CSTRs V1 and V2 are connected to each other with a pipe. In this pipe there is a valve, which is controlled by a pressure controller connected to reactor V1 (see figure).

PC

V1 V2

V1 = 0.1 m3, V2 = 0.4 m3; the pressure controller is set at 5 bar. Both reactors are and remain isohtermal at reaction temperature. At time t = 0 V1 contains gaseous pure A, at a pressure of exactly 5 bar and with a concentration of 1.34⋅10-2 kmol/m3. At this time V2 is vacuum. From t = 0 on the following reaction proceeds: A ! 2 B

49

This reaction is first order in A. The reaction rate constant is 0.02 min-1. This process occurs in both V1 and V2. Gas can only flow from V1 to V2 through the valve but not vice versa, due to the presence of the pressure controller. a What will be the final pressure in each of the reactors? b In one figure, sketch the concentration profiles of A and B in V1 as a function of time. In

another figure, sketch the concentration profiles of A and B in V2 as a function of time. c Introduce the following molar fractions: A1 = NA1/NA0; A2 = NA2/NA0; B1 = NB1/NA0; B2 = NB2/NA0, with NA0 = mol A in V1 at t = 0; NA1 = mol A in V1 at t = t; NA2 = mol A in V2 at t = t; NB1 = mol B in V1 at t = t; NB2 = mol B in V2 at t = t;

Derive the differential equation(s) with which A1 and A2 are described as a function of time. Indicate the boundary condition(s).

d Calculate A2 at t = 25 minutes (4 decimal places, not rounded). e What is the value of B2 at that time (t = 25 minutes)? 80 A stream of pure A (volume flow rate 4 m3/h, concentration of A: 12 kmol/m3) is split into five equal streams, see flow scheme below. The six reactors are ideal PFRs, each with a volume of 2 m3. Reactors 4 and 5 have a recycle stream with recycle ratio R; R = 3 in both cases. As usual, R is defined as the ratio of the volume flow of the recycle stream to the inlet of the reactor and the volume flow rate of the flow that leaves the reactor + recycle. In the reactors the following reaction takes place: 2A ! P, with kinetics:

!rA = kACA

2 and k = 0.3 m3/(kmolAh)

50

Determine the steady-state concentration of A in the stream leaving the system depicted in the flow scheme. Give the complete calculation. Note: The separator S is ideal: it separates mixtures of A and P into pure streams. Pumps and piping have no volume. All streams have an equal density. 81 The product n-examinol is produced batch wise in a vessel with a volume of 2 m3, starting from aversion-to-study acid (asta). The kinetics of this gas-phase process are:

!rA,asta = kACasta2

The reactor is ideally stirred and adiabatic. The heat capacity of the reactor itself may be neglected. Other data: − k = k0Aexp(-Ea/(RAT)) with k0 = 90m3 /(kmol.min) and Ea/R = 5000 K − ∆Hr,asta = !6000 KJ/kmol , independent on temperature − ρAcp = 12 kJ/(m3AK) , independent on temperature and composition − Casta, 0 = 0.6 kmol/m3 The conversion starts at t = 0 min; At that time the temperature of the mixture is 200 oC. a Determine the time required for a conversion of 0.4 and 0.8, respectively (4 decimal

places, not rounded). Also give the corresponding temperatures. b Sketch the heat production as a function of the time. c Calculate the heat production at t = 200 min and t = 300 min. 82 The gas-phase conversion of A into B is accelerated by the very porous catalyst K. Although the catalyst particles are spherical with a radius of 8 mm, from a comparison of various theoretical models it has been found that the experimental results are best described by the �porous slab model�; as �maximum penetration depth� 1/6 of the diameter should be used. In a certain isothermal steady-state experiment, the concentration of A in the bulk gas surrounding the particles is 0.6 kmol/m3. The kinetics of the reaction at these conditions are: !rA,s = ksACA,s

2 with ks = 1A10-7 m4/(kmolAs) if CA is expressed in kmol/m3. The following is known of catalyst K: ρp 1500 kg/m3

porous material Sg 200 m2/g ρbed 900 kg/m3, with εbed = 0.4 Deff 8A10-6 m2/s a Estimate the effectiveness factor E. b Derive the differential equation that describes the change in concentration of A in catalyst

K using the �porous slab model�. Also indicate the boundary condition(s). c Determine (4 decimal places, not rounded) the value of E numerically.

51

83 Which of the answers below are correct? Further comments are not necessary. a In an ideal batch reactor with constant volume the following concentration profile of A as

a function of time t is measured for the reaction A ! P:

The reaction order is: a1 zero; a2 one; a3 two;

b With equal conversion, the total selectivity for R in the following system is:

ü R rR = k1ACA A with

ú S rS = k2ACA2

b1 higher in a continuous ideal PFR than in an ideal CSTR. b2 lower in a continuous ideal PFR than in an ideal CSTR. b3 equal in a continuous ideal PFR and an ideal CSTR.

c For the conversion of A into P and Q (reaction equation(s) and kinetics unknown) we can either use an ideal continuous PFR or an ideal CSTR of the same volume. c1 Under equal conditions the production of P and Q in the PFR is larger than in the

CSTR. c2 Assertion c1 only holds if all reactions have zero order kinetics. c3 Assertion c1 only holds under specific reaction conditions and/or specific kinetics.

d For the conversion of A into P and Q (reaction equation(s) and kinetics unknown) a

system consisting of two reactors in series is used. Only the PFRs P1 and P2 and the CSTRs T1 and T2 are available. All reactors have the same volume.

d1 It can not be said beforehand whether P1 followed by T1 or T1 followed by P1 will produce the highest conversion of species A..

d2 Regardless of kinetics, P1 followed by P2, will produce a higher conversion of A than T1 followed by T2.

d3 Regardless of kinetics, P1 followed by T1 will produce a higher conversion of A than T1 followed by T2.

52

e In an ideal continuous PFR P is produced from a stream of pure A by a first order irreversible reaction. Part of the product mixture is recycled, without previous separation, to the inlet of the reactor and mixed with the constant stream F.

e1 The production of P is HIGHER than without recycle. e2 The production of P is LOWER than without recycle. e3 The production of P is equal to the production without recycle.

f In a tubular reactor, filled with the spherical porous catalyst K, the catalyzed first order irreversible reaction A ! P takes place. The activation energy Ea of the reaction is positive and the effective diffusion coefficient in the porous particles does not change with temperature. Two almost identical experiments are compared. Only the radius of the catalyst particles and the temperature are different. For experiment 1: R = 2 mm and T = 100 0C. For experiment 2: R = 4 mm and T = 80 0C.

f1 regardless of Ea, the conversion of A in experiment 1 is higher. f2 regardless of Ea, the conversion of A in experiment 2 is higher. f3 the value of Ea determines for which experiment the conversion of A is highest.

g The equation: !rA = !(dCA/dt): g1 is the definition of the rate of conversion of A. g2 is the molar balance of species A for an ideal batch reactor with constant volume.

g3 holds as a long as the system is not (yet) at steady-state. h For the liquid phase reaction A + 2B ! P the molar balance is:

h1 (CA, 0 ! CA) = (CB, 0 ! CB) h2 (CA, 0 ! CA) = 2A(CB, 0 ! CB) h3 2A(CA, 0 ! CA) = (CB, 0 ! CB)

i Two continuous ideal PFRs with volume V1 and V2, respectively, and V1 > V2, are connected in series. In the reactors a second order irreversible reaction takes place. i1 The highest overall conversion is obtained with V1 in front of V2. i2 The highest overall conversion is obtained with V2 in front of V1. i3 The highest overall conversion is independent on the sequence of the reactors.

84 For the conversion of A a system of four ideal CSTRs is used, see scheme. All reactors have a liquid volume of 3 m3. The liquid flow rates are expressed in m3/min, the concentrations of A in mol/m3. In each reactor the following reaction takes place: A ! 2P with: !rA = kpA(CA)n mol/(m3

liquidAmin) and kp = 3.0 min-1

Reactors 1 and 3 contain 6.818 kg of porous catalyst K each. This catalyst is retained in the reactors. Catalyst K catalyzes the following reaction: A ! 3Q with: !rA,s = ksACA,s mol/(msurface

2Amin) and ks = 4.0A10-5 m/min (CA in mol/m3) The manufacturer of K in addition has supplied the following data: − mean pore diameter 10 nm − active surface area 150 m2/g − porosity of particle 0.4 - − density of particle 1400 kg/m3 − diameter of particle 6 mm − porosity of bed 0.35 - − bed density 910 kg/m3 − eff.diff.coeff.of A in particle 8A10-6 m2/s

53

For the effectiveness factor of K the value 0.22 is used. Valve X is closed. a Determine the steady-state concentration of A in each reactor. b Determine the steady-state concentration of A in the stream leaving the system. At a certain moment valve X is opened. A stream of 3 m3/min flows out of reactor 1 into reactor 4. The catalyst stays in reactor 1. c Determine the steady-state concentration of A in the stream leaving the system. Finally: d Determine whether or not it is justified to use the value 0.22 for the effectiveness factor. 85 The spherical porous catalyst with radius Rsphere has been partly poisoned. Hence, in the sphere two zones can be distinguished: − a poisoned shell with the original porosity but completely − a non-poisoned center with the original porosity and activity. The boundary between these two zones is at radius Ractive (Ractive < Rsphere). At the active catalyst surface a first order irreversible reaction takes place. Derive the differential equation(s) that describe the concentration profile of species A in the sphere as a function of the radius, with the corresponding boundary conditions. Also give a complete description of ALL symbols used and their units. 86 In an ideal CSTR operating in the steady-state, the endothermic reaction A ! 2B takes place. The heat of reaction is ∆Hr,A J/kmol. The feed to the reactor has a volume flow rate φv

m3/s, a concentration CA,0 kmol/m3, density ρ kg/m3, heat capacity cp J/(kg⋅K) and a temperature T0 K. The tank has a volume V m3 and is completely surrounded by a heat-exchange surface with a �heat transfer coefficient times area� of UAA J/(s.K). At the outside of this surface steam condenses at a temperature of Tcond K. The steady-state reactor temperature (�operating point�) can be determined by plotting in a graph both heat consumption and heat production. Determine the equations for the heat consumption and heat production curves as a function of temperature T in the reactor. Sketch these curves. Set the intersection of the axes at:

54

consumption = 0, production = 0, T = 0. Notice the value of the heat consumption at extremely high temperatures. Why is the �operating point� always stable in such a system? (Assume that the heat of reaction, density and heat capacities do NOT depend on temperature and/or composition.) 87 A problem during the course �chemical reactor engineering� concerned sketching the E(t)-curve for a system of interconnected ideal reactors. Amongst the answers were the following four sketches: 1) 3) 2) 4) Answer 1) appears to be the correct answer. a Sketch a possible system of interconnected ideal reactors resulting in sketch 1. b What systems of interconnected reactors correspond to the other sketches? c Sketch the corresponding F-curve for each system. Denote PFRs in your answer as P1, P2, ... and CSTRs as T1, T2, ..., with respective volumes

VP1, VP2, ...,VT1, VT2, .... Denote the various volume flow rates as φ1, φ2, .... d Sketch Figure 1 yourself and put relevant information in it (points of intersection, areas,

etc.), expressed in VP1, VP2, ...,VT1, VT2,�, φ1, φ2, .... e If the first order irreversible reaction A ! products, with rate constant k, takes place in

this system of reactors, what is the conversion at the outlet of the system (again expressed in VP1, VP2, ...,VT1, VT2,�, φ1, φ2, ....)?

55

10 Exams Problems 88 to 90 form the exam �Chemical Reactor Engineering 1 of January 1 1984. 88 In an ideal CSTR a mass flow A (φm kg/s) is converted into products by an exothermic first order irreversible reaction (− rA = k0⋅e-Ea/RT⋅CA, ∆Hr,A J/mol). In order to be able to operate the reactor, the heat released is removed by heat exchange (overall heat transfer coefficient is U W/(m2.K), independent on temperature) with the feed stream in a concentric heat exchanger around the reactor (see figure). The system is insulated (regarding heat) from the surroundings.

The flow in the heat exchanger may be regarded as ideal plug flow. The heat capacity for the feed, the products and their mixtures is: cp = a + bT J/(kg.K). a What are the balances describing the system? b Formulate both heat balances. Indicate with a sketch over which part exactly each of the

balances is made. Convert the balances to (differential) equations, in which − rA and cp have been substituted.

c What are the boundary conditions required to solve this system of equations? 89 Residence time distribution measurements have been performed on a real continuous tubular reactor. The following E(t) versus t curve has been determined from these experiments.

56

Although Gaussian, this E-curve may be approximated by two straight line segments making angles with the positive time axis of 45o and 135o, respectively, see figure (time in minutes). a Calculate the mean residence time. b Tabulate and sketch he F-curve for this reactor. At the calculated mean residence time in this reactor a flow of A, dispersed in water, is converted into B by a second order irreversible reaction (k = 0.5 m3/(kmol⋅min). The concentration of A in the emulsion droplets at the inlet of the reactor is 4 kmol/m3. c Calculate the conversion of the fraction with a residence time between 0.9 and 1.2

minutes.

Note: ( )1axaedxxe 2

axax −=∫

( )bxalnba

bx

bxaxdx

2 +−=+∫

( )

( )

+++=

+∫ bxaabxaln

b1

bxaxdx

22

90=93 A particular porous catalyst is available as spheres of seven different diameters. This diameter varies in steps of a factor 10, from 0.1 µm up to 100 mm. Of each of these spheres the conversion of a flow of pure A is known in an ideal PFR with a fixed mass of catalyst. Unfortunately, the data have been shuffled, so only the seven conversion levels are known but not the corresponding particle diameters. The seven conversion levels are: 0.8647; 0.8647; 0.0326; 0.2701; 0.5432; 0.8115; 0.8647. It is known that one of these conversion levels is incorrect, while the other six are correct! a Determine which conversion level is incorrect and what its value should be. Of the same catalyst two similar series of spherical particles with different diameters are available, but these have been used for years in an industrial reactor. As a result, the active surface has been poisoned for 50 %. In the first series homogeneous poisoning has occurred, so that the exterior and the center of the catalyst have been poisoned to the same extent. b Determine the conversion in the smallest and the largest spheres of this series at the

standard conditions that were also used in problem a. In the catalyst particles of the second series poisoning progressed from the exterior, so that in the spheres two zones can be distinguished: − a poisoned outer shell of the catalyst particle that is completely inactive and − a non-poisoned center with the original activity. c Derive the differential equation describing the concentration profile of A in the spheres as

a function of the radius. State the boundary condition(s). Notes: � The reaction is first order in A. � All unmentioned parameters may be assumed the same within a series and between series. � Use the porous-sphere model in your calculations.

57

Problems 91 to 93 form the exam �Chemical Reactor Engineering 1 of June 4 1985. 91 See 48 92 At t = 0 a step change of tracer concentration from C = 0 to C = C0 is applied to the feed of a reactor with unknown flow behavior. At the outlet of the reactor the following F-curve is measured (see figure). a Sketch the E-curve of this unknown reactor (supply all relevant information in the

sketch!). b Which combination of ideal reactors has the same E-curve as the unknown reactor? c What is the space time of the unknown reactor? d If at this space time (see c) the reaction A ! R, with rR = 4CA kmol/(m3⋅min), takes

place in the unknown reactor, what will be the conversion of A? Note: If you do not know the answer to c, use τs = 0.5 min. 93 See 90

58

Problems 94 to 96 form the exam �Chemical Reactor Engineering 1 of January 29 1986 and August 18 1987. 94 It is desired to convert a stream pure A (CA,0 = 5 kmol/m3, φv,0 = 0.5 m3/min) to the products P and Q in an unknown type of reactor by the following first order irreversible reaction:

A ! P + Q, with kinetics: !rA = 0.2ACA [kmol/(m3Amin)] The following E(t)-curve is known from relevant residence time distribution measurements (see figure). a What is the volume of the unknown reactor? b Before starting the desired process, an up-down step change of tracer concentration is

applied to the reactor feed (see figure). Sketch the tracer concentration profile in the outlet of the reactor. Supply relevant information accurately at the axes.

59

c On the basis of the residence time distribution curve, calculate the conversion in the outlet of the reactor for the first order process described above.

d Would an ideal PFR with the same volume and at the same conditions produce a lower, identical or higher conversion? Explain your answer.

e Repeat d but now for a second order reaction. 95 In a CSTR (V = 3 m3) a stream of A (φv,0 = 1.5 m3/min, CA, 0 = 5 kmol/m3) is converted into products by a first order irreversible reaction (k = 1.5 min-1). a Calculate the concentration CA, out , assuming an ideal CSTR. Observing the flow profile in the (glass) reactor it seems that the reactor might also be described by two PFRs in �parallel�, with the indicated volumes (see figure). b Assuming that both PFRs are ideal, what should be the ratio φv, 1 / φv, 0 to obtain the same

concentration CA,out as for the assumption of the ideal CSTR? c Answer question b for a reaction with an order different from 1. Explain your answer(s). 96 An ideal batch reactor is completely filled with a mixture of a porous material B and a liquid that contains species A. The pores of material B are completely filled by the liquid. The temperature of the reactor walls (the �round wall�, the bottom and the cover) is kept constant by heat exchange. At the given conditions, A decomposes at the catalytic surface of material B producing C: B A ! 2C

60

Data: Reactor wall temperature 420 K heat transfer area 3 m2 heat transfer coefficient 1.2 kW/(m2AK) heat capacity 0 Liquid volume in reactor 0.5 m3 concentration of A unknown mol/m3 temperature unknown K density 800 kg/m3 heat capacity 2 kJ/(kgAK) Porous material B total amount in reactor 300 kg diameter of spheres 0.5 mm internal catalytic surface 120 m2/g mean pore diameter 100 Å bed density 900 kg/m3 bed porosity 0.4 particle density 1500 kg/m3 particle porosity 0.5 heat capacity 0.9 kJ/(kgAK) effective diffusion

coefficient of A 9A10-9 m2/s Reaction reaction rate !rA = ks.CA mol/(m2As) influence temperature ks = k0.exp(-Ea/RT) units rate constant k0 = 10-3 units relative activation energy Ea/R = 8000 K heat of reaction ∆Hr,A = !20 kJ/mol The heat of reaction, densities and heat capacities are independent on temperature. Assume that the temperature of B equals the temperature of the liquid in the reactor. At a certain time t = 0:

CA,0 = 11000 mol/m3; T = 400 K a Estimate the volume of the reactor. b Determine the units of k0 en ks . c Determine to what extent the concentration of A changes in the pores of B. d At what time the liquid in the tank reactor reaches the highest temperature?

Calculate that temperature. e Determine the concentration of A in the liquid and the temperature at t =20 min (1200 s).

61

Problems 97 to 99 form the exam �Chemical Reactor Engineering 1 of January 28 1987. 97 Besides the �porous-sphere model� and the �porous-slab model� another model is available to describe a catalyst: the �truncated-cone model�. In this model all active catalyst surface is present on walls of the pores, which are shaped like truncated cones. These pores have: − parallel axes, − axes perpendicular to the external surface, − the same largest diameter Dmax and smallest diameter Dmin, − the same depth L. It can be easily derived that the diameter D of a pore as a function of the distance to the external surface can be described by:

( )LxDDDD minmaxmax ⋅−−=

Because Dmax − Dmin equals L (see further on) also: D = Dmax � x. the active surface area of the catalyst consists of the �round� surface of the truncated cones. This surface area is described by:

( ) ( )2minmax

2minmax DD25.0LDD5.0Area −⋅+⋅+⋅π⋅=

Substituting the data for a single pore yields: Area = 1.0537⋅10-7 m2. The model above is used to describe the steady-state conversion of the gas A into the gas B. This conversion only takes place at the active catalyst surface. a Derive the differential equation describing the concentration of A as a function of the

distance to the external surface x. Start with the formulation (in words) of the molar balance of A over a slice of catalyst perpendicular to the axis of the pores. Sketch the direction of the molar flows.

b Determine the boundary condition(s) of this differential equation and explain. c Determine the value of the effectiveness factor E if: L = 0.2 mm Dmax = 0.25 mm Dmin = 0.05 mm Dfree gas phase = 2⋅10-5 m2/s CA,bulk gas = 60 mol/m3

CB,bulk gas = 30 mol/m3

ks = 0.5 m/s − rA,s = ks⋅CA mol/(m2⋅s)

62

98 In an adiabatic near ideal PFR containing a porous catalyst, a gas stream containing species A (φv = 4.5A10-6 m3/s, CA = 2 mol/m3) is partially converted into the desired product P. The goal is to obtain the highest possible steady-state conversion ξ of A. Table 1 shows the possible values of the process variables. Table 2 shows characteristics of the two catalysts available. Table 3 summarizes additional data. The density and heat capacity of the gas may be assumed to be independent on temperature and composition at all conditions. The same is true for the heat of reaction. a Explain your choice of the variables in Table 1. b Determine the obtained steady-state conversion of A. c Determine the temperature at the outlet of the reactor. Table 1 Process variables. Feed temperature T0 500 or 600 K Reactor diameter Dreactor 30 or 100 mm Catalyst type - 1 or 2 - Amount of catalyst W 300 or 400 g Catalyst particle diameter dpart 3 or 5 mm Table 2 Data on catalysts 1 and catalyst 2. Type 1 2 Rate constant at T = 4 ks, 4 6.0A104 1.0A1012 m/s Reduced activation energy Ea/R 2.0A104 3.0A104 K Bed porosity εbed 0.3 0.35 - Particle porosity εpart 0.4 0.45 - Bed density ρbed 770 780 kg/m3 Particle density ρpart 1100 1200 kg/m3 Active surface area of particles Sg 300 200 m2/g Effective diffusion coefficient Deff 2.1A10-8 2.2A10-8 m2/s Table 3 Other data. Density of gas ρgas 1.2 kg/m3 Heat capacity of gas cp 1.3 kJ/(kgAK) Heat of reaction ∆Hr,A 0.0 kJ/kmol 99 In an installation continuous stirred tank reactor is used, which has been designed badly (see figure). Firstly, the inlet and outlet of the reactor are so close to each other that a fraction φ of the entering volume flow leaves the system directly. Secondly, the height of the tank is so much larger than its diameter that, despite the mounted double stirrer, the tank can not be considered as ideally mixed. It rather appears to be divided into two compartments, I and II, both ideally mixed and with exchange between them. The volume of compartment I is a fraction b of the total tank volume VT, the volume of compartment II is the rest: VT = 10 m3, b = 0.9 and φ = 0.2. Transfer between both compartments is described using the transfer coefficient Kv, which indicates the amount of liquid transferred [m3] per minute per unit of reactor volume [m3

reactor]. For the transfer the following equation holds: ( ) )m/(minm 2.0K with CCVK 3

reactortransfered3

vIIITvtransfer ⋅=−=φ

63

In the reactor a reactant flow (φv = 2.5 m3/min, C0 = 3 kmol/m3) is converted into products by a first order irreversible reaction (− r = 0.25⋅C). In order to be able to predict the conversion in the reactor, the model depicted in the figure, which is based on the above considerations, is used. Both compartments are assumed to be ideally mixed. Calculate the conversion for the given situation. Problems 100 to 103 form the exam �Chemical Reactor Engineering 1 of January 27 1988. 100 In a certain biochemical process R is produced from A by the reaction: A + R ! 3.8 R, with

( )A2

RA1Ck1CCk

r⋅+⋅⋅

=− with k1 = 1.0 m3/(kmol⋅h) and k2 = 1.1 m3/(kmol⋅h)

The feed contains 2 kmol/m3 A and 0.2 kmol/m3 R and has a volume flow rate of 0.9 m3/h. The density of the liquid does not change by the reaction. For the continuous production of R a system consisting of an ideal CSTR followed by and ideal PFR is used. The liquid volume in each of these reactors is 0.8 m3. a Determine CA and CR both at the exit of the CSTR and at the exit of the PFR. For the

determination of the concentrations at the exit of the PFR use a numerical approach according to Runge-Kutta and an accuracy of 0.0001 kmol/m3. (Note: If you have not been able to determine the concentrations at the outlet of the CSTR, you may assume the (wrong) value CA,tank = 0.7000 kmol/m3).

b What amount of R is produced in both reactors together?

64

101 In an ideal PFR a stream of pure A (CA = 10 mol/m3) at 1 bar and 400 oC is converted for only 1% by a first order heterogeneously catalyzed reaction over a porous catalyst. Under the given circumstances, the pore diffusion resistance is negligible. The rate of conversion is:

!rA = 1.1 mol/(m3catalystAs)

Determine the maximum size of the catalyst particles if the effective diffusion coefficient is 1A10-7 m2/s. Indicate the assumptions you made concerning the shape of the catalyst particles. 102 An emulsion polymerization is performed in a continuous reactor. The feed to the reactor is a suspension of small monomer droplets in oil with a volume flow rate of 0.5 m3/min. The conversion rate of the monomer is second order in the monomer concentration. The reaction rate constant is 0.1 m3/(kmolAmin). The initial monomer concentration in the droplets is 1 kmol/m3. The system is completely segregated. From residence time measurements it is known that at a volume flow rate of 0.5 m3/min no volume element remains in the system more than 8 minutes, while for each t # 8 min E(t) is constant. a Calculate the conversion at the outlet of the reactor. b Calculate the reactor volume. 103 The feed to an ideal CSTR consists of three equimolar streams: one of pure A, one of pure B and one of pure D. In the reactor the following reactions take place:

A + D! R with rR = k1ACAACD B + D ! S with rS = k2ACBACD while k2/k1 = 0.2

Determine the fraction CR / (CR+CS) if: a The conversion of A is 50%. b The conversion of D is 50%. Problems 104 to 107 form the exam �Chemical Reactor Engineering 1 of May 26 1988. 104 Students examine the conversion of study acid (abbreviated: stac) to examinol in the liquid phase. This reaction is catalyzed by germanium(I) hydroxide on mole sieve 5A.

GeOH study acid ! examinol

The reaction is first order in study acid. An unknown type of isothermal reactor (V = 2 liter) is filled with spherical catalyst particles (see catalyst specifications below). When the feed has a volume flow rate φv of 0.1 liter/s (Cstac,in = πA√37 mol/liter) a steady- state conversion of study acid of 0.8 is measured. Check which of the following models can describe the conversion mentioned:

1 model �homogeneous ideal CSTR� with k = 0.072 s-1

2 model �homogeneous average� with k = 0.072 s-1

In this model it is assumed that the concentration everywhere in the reactor equals the algebraic average of the inlet and outlet concentrations: Cstac = (Cstac,in + Cstac,out) / 2.

3 model �heterogeneous ideal PFR�, ks = 10-8 m3/(m2As) Catalyst specifications:

mean pore diameter dpo 5 Å internal surface area Sg 200 m2/g particle diameter dp 8 mm particle density ρp 1200 kg/m3 bed porosity εbed 0.6 - effective diffusion coeff. study acid & examinol in particles Deff 2.4A10-8 m2/s

65

105 In a biochemical process A is converted to R by the following reaction: A + R ! 3.8R with !rA = kACAACR and k = 2.0 m3/(kmolAh) The process stream that is to be converted contains 2 kmol/m3 A and 0.2 kmol/m3 R and has a volume flow rate of 0.6 m3/h. The density of the liquid does not change by the reaction. For the continuous production of R a system consisting of an ideal CSTR followed by and ideal PFR is used. The liquid volume in each of these reactors is 0.12 m3. a Determine CA and CR at the outlet of the CSTR. b Derive the concentrations CA and CR at the outlet of the PFR. Which of the following four

answers is correct? b1 0.1102 kmol/m3 b2 0.1206 kmol/m3 b3 0.1304 kmol/m3 b4 0.1403 kmol/m3

c How much R is produced in both reactors together? d Sketch 1/(!rA ) versus CA. Indicate the calculated concentrations and other relevant

information. At a certain time the volume flow rate is increased to 0.7 m3/h. e Using sketch d, estimate whether the steady state production of R will

- decrease - increase - remain constant compared to the production obtained in c. Explain your answer!

If you have not been able to determine the concentrations at the outlet of the CSTR, you may assume the (wrong) value CA,tank = 0.8 kmol/m3.

Note: ( ) xbxaln

a1

bxaxdx +−=+∫

106 Three ideal PFRs, volumes: 2, 4 and 6 m3 are interconnected as shown in the figure below. The water feed (volume flow rate φv = 3 m3/min) is divided over both parallel PFRs as shown:

At t = 0, a pulse of 2 moles of salt is injected in the feed at point A. a Sketch the salt concentration versus time curves at the points B, C, D and E. Indicate the

scale divisions on the axes. After this experiment the split ratio of the feed over the two PFRs is changed, as shown in the next figure.

66

Again, at t =0 a pulse of 2 moles of salt is injected in the feed at point A. b Again sketch the salt concentration versus time curves at the points B�, C�, D� and E�.

Indicate the scale divisions on the axes. c Sketch the F-curve for the overall system of figure a and the overall system of figure b. Note: Neglect the volume of the tubing connecting the PFRs! 107 Besides the �porous-sphere model� and the �porous-slab model� another model is available to describe a catalyst: the �straight-pore model�. In this model all of the catalytically active surface is located on the walls of straight cylindrical pores. These pores: − are all parallel to each other − are perpendicular to the external catalyst surface − all have the same diameter D − all have the same length L The �bottom� and the �round� surface of the straight pores constitute the active catalyst surface. This model is used to describe the conversion A ! B (gas phase; first order kinetics). a Derive the differential equation describing the change in concentration of product B as a

function of the penetration depth x. Start with the formulation (in words) of a molar balance of B over a slice of catalyst perpendicular to the axes of the pores. Sketch the direction of the molar flows.

b Determine the boundary condition(s) of this differential equation.

67

ANSWERS 1 Ideal reactors: basics 2 Ideal reactors: in a little more depth

pfr cstr batch

A,pfr A,cstr A,batch

s,pfr s,cstr

A A B

s

s

1a 30 s ; 18.3 sb 1.715 mmol/l ; 0.726 mmol/l

2 230 s ; 624 s3a r 4b 0.0485 min

c cstr4a R/(R+S)=0.636 b R/(R+S)=0.6675a R=0.31 molR/molA b 330 s

C C C

C C

τ τ τ

τ τ

τ

τ

= = =

= = =

= =

− = •=

=c T=330 K

-1

-1

1A

1 A,1 11A,1

v v

3 1v

v 0

v 0 R1

R v

A

6a 0.06 min

0.06 min( )1 b ( )( ) ln with 0.4512 10 min( )

V 2= min

c 0.146 m mind( )7a

dt

b (1 exp( ))

c will decrease a

k

kk k

k t t tk k

CV C kCV

C VC kkV

C

τ

τ

ξ ξτ ξ

τφ φ

φ

φ

φφ

−

=

=− + − + − = = =− + =

=

= −

= − −

1

2A

A2

AA A,0

-1v,cond

2

RA A,0

R

fter 8 198 m

d9a 0dx

d b 0: ; : 0dx

c

10 100(1 exp( )) ml min

11 3 2exp( )

12 exp

tx

CDiff kC

Cx C C x L

kDiff

kt

V kt

VC C ktV V

φ

=

• − =

= = = =

= + − ⋅

= − −

= − +

68

3 Ideal reactors: multiple reactors

-1

-3 -3A mol,A,0 A B

-1 -1

a . min and satisfy the data( exp( )) b ; ( ) . kmol m ; ( ) . kmol m

a ln . min ; ln . min

first assumption needed; second not0 0 0

t 5 t 5

13 k 0 5 n 11 ktC C 1 84 C 6 84

kV1 c 1 cV14 a a 0 05 b b 0 0612t c t c v

= =

= =− −= φ = ⋅ = ⋅

− −= → = = → =

( )

A A,0A,0

-3 -3A,1 A,2 A,cstr A,0 A,s A,p

A,0 sA

s s

%

15 exp

( ) ( ) . kmol m ; . kmol m( ) . min

18 exp

K20 model 4=axial dispers

12

4

2ktC C 3 1C

16 C C C C 1 2 929 C 1 71617 t 1 01

C 1 kC 1 t

1 k

19 T 450

= − = = = − ξ = ⋅ = ⋅

=

+ τ = − − + τ τ

=ion model

model 5=ideal pfrmodel 6=ideal cstr

3,1 3,2 3,3 3

A,1 A,2 P,1 P,2

B T TB BT3 1 3 -1

v,2 v,8

-3 -3A,7 A,13

3

2122 en 23 All proposals decrease the overall conversion2425a 6 m s en 8 m s

b 3.68 kmol m en 0.92 kmol m

26 10000 m per

C C C C

C C

V

ξ ξ ξ ξ

ξ ξ ξ ξφ φ−

< = =< >

= < <

= ⋅ = ⋅

= ⋅ = ⋅

=-1

mol,B

-1 -1A

3

pfr

1 2

A,0A,2

1 2

1

2

basin27 26 kmol min

28a 1 ; 8.3 MJ kmol ; 10 sb 500 K ; 33.2 m

29a,b c 22.3 liter d30a

b independant of sequence(1 )(1 )

c 1

n E k

T V

V V VC

Ck k

VV

φ

τ

τ τ

∞

= ⋅

= = ⋅ =

= =−

=

−= +

=+ +

=

69

4 Ideal reactors: a financial question 5 Ideal reactors: no reaction without heat

31 2 7 3 8

32 1 5 6

1 3

2 5 4 8

1 12

33 6 00

≤ ≤ ≤ ≤

= ⋅ = ⋅

= ⋅ = ⋅

= ⋅ = ⋅

= ⋅ = ⋅

= ⋅

φ φ

φ

φ

φ

φ

v,1 v,2

v,13 -1

A,1-3

v,23 -1

A,2-3

v,33 -1

A,3-3

v,43 -1

A,4-3

out,model-3

en

m min kmol m

m min kmol m

m min kmol m

m min kmol m

kmol m

.

. .

.

C

C

C

C

C

34 135 80

0 82

37 2 0 5 1

38 6

2 61 2 1 3

1 1 1 4

8

40

40

2

nf

V

V

C C

C C C

=

=

≈ − ≈

=

=+ +FHG

IKJ

= = ⋅ = ⋅ = ⋅

= ⋅

= ⋅

per year36

minimal total costs if m

m independant of the price of B

39a

b h ; kmol m kmol m kmol m

c m h

a 12 cycles / day b 4 cycles / day c 12 cycles / day

41 kmol h

A,optimal

R3

optimal3

C A,0

A-3

B-3

C-3

v,optimal3 -1

mol,B-1

ξ

ττ τ

τφ

φ

.

ln( . ) .4

( )( )

; ;

42 2000

43 200

44 0 9 120 4 1045 450

5

246 0 39 513

201

16000

1 2 1048 250 175

5

5

0

40%

U A

T HT

C

Ct TT

U A

T T

⋅ = ⋅ ⋅

= ⋅

= = ° = − ⋅=

= ⋅

= ⋅

= === °

⋅ = ⋅ ⋅

= ⋅= ° = °

kcal K s

Mcal h

C ; kJ kmol Aa K

b kmol m

c kmol m h ; K

47a C

b kJ h K

c kJ ha C ; C

b stable c no change

-1 -1

heat-1

A k-1

A,0,max-3

A,0,max-3

-1 -1

heat-1

f

φξ

φ

. ;

( ) .

.

∆

70

6 Ideal reactors: with a solid catalyst

A

B B B-1

49a 0.69 b positive lower ; zero equal ; negative higher

c 33 kJ (mol K)50a 3 cubes b after 14 minutes c add cubes as late as possible (or crush and feed them continuousl

E E E

E

ξξ ξ ξ

=

≈ ⋅ ⋅

0

y at 560 K) d pyrex can be used up to approx. 770 K, so no cubes needed51 slice [K] [-]

1 0.0333 0.0112 15.0 0.01122 0.0115 0.0227 20.1

t T Tξ ξ∆ ∆ −

0.0339

2 3

1 6 7 8

A,I

A,II

p,I

p,II

v,I

v,II

-1 -1eff,MZ eff,Al O

7uit

52 ... 1 ; ... 0.1102

53 varies from 2 (at low temperature) to 2.355 (at high temperature)

a) 0.6

b) 2.0

54a 0.072 s ; 0.0036 s

b 2.67 10 0.3

E E E E

rr

ddSS

k k

C −

= =

=

=

= =

= =

A1 2

A

B-1

v

-1 -1homog s

-1B homog

ppm(vol)

55a 1 1386 K

3 b 2773 K

56a 3336 ml s b =0.381 c =0.99257 from data sponge 4s and 0.02 m s

using spheres 353.6 88.8 s new situation better

EE ER

EE

R

k k

k

φφξξ

φ

= = → =

= → =

= ⋅

= = ⋅

= → =

71

7 Non-ideal reactors: residence time distribution

58 0

0 31

0a s b = 0.65 c probably conversion found less than calculated59

-1

mon

E =

=

.447

.

ξ

ξ

out

-1v

-1r

A

A

49out

v,2A,out

v,0

60a91 b = s6

c 0.4865

61a 4 l min ; =2.5 min volume=10 liter

b 0.231 min ; 0.4162 0.87563 0.961

64a b,c d equal, although order is 2 ! explain!65

66b

k B

C

τ

ξφ τ

ξξξ

φφ

−

=

= ⋅ →

= ===

=−

−

= v,1A,0 A,0

v,0 T

11 1

2 2

AA

A,0

2ss

11

6768a

1 b 0 : 0 ; : exp

c 1

d the sequence of the reactors (NOT: data on degree of mixing!)69a no

b exp

c

C Ck

tt E t E

CC

t tE

φφ τ

ττ ττ τ

ξ

ττ

++

−−

−≤ ≤ = ≥ = −

= −

−= −

A d 0.9970a,b

ξ =−

72

8 Non-ideal reactors: the alternative of dispersion 9 Various

71 0 98572

0 96

4 5

ξ

ξ

A

A

2 -1

length approx. 300 m73a b

c m s

=

−=

= ⋅

.

.

.D

74

76

ab 696 K

75a 0.1468 10 kmol m

b 0.289 10 kmol m

-3 -3

-3 -3

−

⋅

⋅−

-3

-1

7880 3.28kmol m81a 154.2 & 266.8 min (593 & 713 K)

bc 29.49 & 8.94 kJ min ( 0.491 and 0.149 kW)

82a. 0.4022bc 0.3900

83a C,NC,NC b NC,C,NC c NC,NC,C d C,NC,NC e NC,C,NCf C,NC,NC g NC,C,NC h NC,NC,C i NC,

−•

−

⋅ =

−

-3

-3

-3

NC,C84a 2.5, 1.0, 2.5, 1.0 mol m

b 1.0 mol m c 1.09375 mol md justified

8586a87a

⋅

⋅

⋅

−−−

73

10 Exams

3

-3

3

-3

3

93 see 9094a 2.75 m

bc 0.641d,e

95a 1.25 kmol m b a= c a =

96a 0.6 m b m/s c E = 1 d 1140 s, 431.6 K e 6.23 kmol m

97a,bc 0.318

98ab 0.986 (cat 1) c 600 K

99a 0.4442b,c

100a cstr: 0.6573 kmol/m ,S RC C

−

−

⋅∞∞

⋅−

−

−

= 3

3 3

3.9595 kmol/m ,

pfr: 0.0130 kmol/m , 5.7636 kmol/m5.0072 kmol/h

101 radius < 0.953 mm

S RC Cb

=

= =

8889a 1min

0.9190a 0.5432 must be 0.8641

b 0.02311 (largest), 0.6321 (smallest) c

91 see 4892a,b

c 1/3 min d 0.58

bc

−

−

−

−

74

102a 0.265 b 2 ma 75% b 77.5%

104 cstr = 0.590, homogeneous mean = 0.837, pfr = 0.7543 a 0.763105a = 0.841 mol m , = 3.4452 mol m

b 0.1304 kmol m c 3.141 kmol h de increase expected

106107

3

A-3

R-3

-3

-1

103

`C C⋅ ⋅

⋅

⋅−

−−