Assignments Involving PFR Analysis
Examples from REB, The Book
Example 9.6.1 The heat of reaction (1) is 44.8 kJ mol-1, and the reaction is irreversible. The rate expression is equation (2) where the pre-exponential factor is 7.22 x 106 mol atm-2 cm-3 s-1 and the activation energy is 84.1 kJ mol-1. A 10 foot long tubular reactor with a diameter of 1 inch is heated by a vapor condensing at 200 °C on the outside of the tube wall. The overall heat transfer coefficient is 7.48 x 104 J h-1 ft-2 K-1. Pressure drop through the reactor is negligible. If a gas phase mixture of 60% A and 40% B enters the reactor at 282 L min-1, 2.5 atm and 175 °C and if the heat capacities of A, B and Z are equal to 18.0, 12.25 and 21.2 cal mol-1 K-1, what steady state outlet temperature and conversion of B will result?
\[ A + B \rightarrow Z \tag{1} \]
\[ r_1 = k_1 P_A P_B \tag{2} \]
Example 9.6.2 A perfectly insulated, 40 L, tubular reactor is fed an aqueous solution containing A and B at concentrations of 1.0 and 1.2 M, respectively. This feed stream flows at 75 L min-1. Reagents A and B react according to reaction (1) with a rate of reaction that is accurately described by equation (2). The rate coefficient displays Arrhenius temperature dependence with a pre-exponential factor equal to 8.72 x 105 L mol-1 min-1 and an activation energy of 7200 cal mol-1. The heat of reaction (1) is –10,700 cal mol-1 and may be assumed to be constant. The heat capacity of the solution and the density of the solution may be taken to be constant and equal to those of water (1.0 cal g-1 K-1 and 1.0 g cm-3). The pressure drop in the reactor is negligible. What feed temperature is needed in order to convert 95% of the A, and what will the outlet temperature be?
\[ A + B \rightarrow Y + Z \tag{1} \]
\[ r_1 = k_1 C_A C_B \tag{2} \]
Example 9.6.3 The gas-phase decomposition of A according to reaction (1) takes place at steady-state in a PFR. The feed to the reactor is 75% A and 25% inert gas flowing at 100 cm3 s-1 (0.44 g s-1), 3 atm and 400 °C. The reactor diameter is 2.5 cm and its length is 8 m. It is surrounded by a shell containing a molten salt at a constant temperature of 375 °C. The heat transfer coefficient between the reactor and shell is 187 kJ h-1 m-2 K-1. The reaction is catalytic, and the reactor is packed with catalyst particles with a 0.25 cm diameter, a sphericity of 0.7, and a bed porosity of 0.60. The rate expression for reaction (1) is given in equation (2) where \(k_{0,f}\) = 9.0 x 1017 mol cm3 s-1 atm-1, \(𝐸_𝑓\) = 285 kJ mol-1, \(𝑘_{0,𝑟}\) = 4.09 x 10-4 mol cm-3 s-1 atm-4, and \(𝐸_𝑟\) = 85 kJ mol-1. The heat of reaction is constant and equal to 200 kJ mol-1. The heat capacities of the reagents are also constant: \(\hat{C}_{p,A}\) = 11.7 cal mol-1 K-1, \(\hat{C}_{p,Y}\) = 8.3 cal mol-1 K-1, \(\hat{C}_{p,Z}\) = 4.3 cal mol-1 K-1, and \(\hat{C}_{p,I}\) = 5.8 cal mol-1 K-1. The viscosity may be assumed to be constant and equal to 0.027 cP. What are the outlet temperature, pressure and conversion?
\[ A \leftrightarrows Y + 3 Z \tag{1} \]
\[ r_1 = k_fP_A - k_rP_YP_Z^3 \tag{2} \]
Example 9.6.4 The water-gas shift, equation (1), is used to reduce the amount of CO in gas mixtures containing CO, CO2, H2, H2O, and other gases. It is exothermic, \(\Delta H\) = –9120 cal mol-1, and reversible with the equilibrium constant given by equation (3) with \(K_{0,1}\) = 0.132. The heat capacities of CO, H2O, CO2, H2, and inert, I, can be take to be constant and equal to 29.3, 34.3, 41.3, 29.2, and 40.5 J mol-1 K-1, respectively. Two reactors are often used, one operating at higher temperatures and the other at lower temperatures. The steam to CO feed ratio is often in the range from 3 to 6. Suppose the rate in the first reactor when using a certain catalyst is given by equation (2) with \(k_{0.1}\) = 0.0354 mol cm-3 min-1 atm-2 and \(E_1\) = 9740 cal mol-1. The feed to the adiabatic reactor is at 26 atm and 320 °C, and pressure drop is negligible. Using a feed with 1 mol CO h-1, 0.359 mol CO2 h-1, 4.44 mol H2 h-1, and 0.180 mol I h-1 as a basis, what inlet molar flow of H2O will minimize the reactor volume if 55% of the CO must be converted? What will the outlet temperature from a reactor of that volume equal?
\[ CO + H_2O \rightleftarrows CO_2 + H_2 \tag{1} \]
\[ r_1 = k_1 \left( P_{CO} P_{H_2O} - \frac{P_{CO_2}P_{H_2}}{K_1} \right) \tag{2} \]
\[ K_1 = K_{0,1} \exp{ \left( \frac{-\Delta H}{RT} \right)} \tag{3} \]
Example 9.6.5 An adiabatic PFR is used to produce Z via the catalyzed, liquid-phase reaction (1). The rate expression for the reaction is given in equation (2) where the pre-exponential factor is 6.632 x 108 cm6 g–1 mol–1 s–1 and the activation energy is 15 kcal mol–1. The reactor is 16 m long with a diameter of 2 cm. The catalyst bed density is 2.5 g cm–3. The reacting fluid density is 0.65 g cm–3, and its heat capacity is 0.5 cal g–1 K–1. The 0.228 L min–1 feed to the reactor contains 7.84 kg-mol m–3 of A and 2.32 kg-mol m–3 of B at 20 °C. The heat of reaction is –1.75 x 107 cal kg-mol–1. What are the conversion of A and the outlet temperature?
\[ A + B \rightarrow Z \tag{1} \]
\[ r_1 = k_1C_AC_B \tag{2} \]
Example 9.6.6 The gas-phase conversion of A to Z, equation (1), takes place at steady-state in a packed bed PFR operating isothermally at 400 °C. The reaction rate per unit catalyst particle volume is first order in A, equation (2), and at the operating temperature, the rate coefficient is equal to 0.15 cm3fluid cm–3particle s-1. The void fraction within the packed bed is equal to 0.4. Pure A, flowing at 1.0 cm3 s-1 (0.054 g s-1), 400 °C, and 30 atm is fed to the reactor. There are no temperature gradients anywhere in the reactor or in the catalyst particles, and there are no concentration gradients external to the spherical catalyst particles which have a diameter, \(D_p\) of 4 mm. The effective diffusion coefficient of A within the porous catalyst particles is equal to 0.0029 cm2 s-1 and the gas viscosity may be assumed to be constant and equal to 0.022 cP. If the reactor diameter is 1.5 cm and the conversion of A in the reactor is to equal 85%, what reactor length is required?
\[ A \rightarrow Z \tag{1} \]
\[ r_1 = k_1 C_A \tag{2} \]
Learning Activities from REB, The Course
Class 21 Learning Activity A gas mixture at 140 ºC and 1.0 atm and containing 30% A and 70% B is fed to a steady-state PFR at a flow rate of 0.45 L min−1. The PFR diameter is 4 cm and there is no pressure drop along its length. An external fluid, also at a temperature of 140 ºC, is fed at a rate of 20 g min−1 to a perfectly mixed shell surrounding the reactor tube. The heat capacity of that fluid is 1.4 J g–1 K–1, and the heat transfer coefficient is 1.3 x 10–6 kJ min−1 cm−2 K−1.
Reactions (1) and (2) occur within the reactor at a space time of 20 min. The rate expressions for the reactions are given in equations (3) and (4). The pre-exponential factors for reactions (1) and (2) are equal to 1.65 × 104 mol min−1 cm−3 atm−2 and 3.24 × 104 mol min−1 cm−3 atm−2, respectively. The activation energies for reactions (1) and (2) are equal to 78 kJ mol−1 and 86 kJ mol−1, respectively. The standard heats of reactions (1) and (2) are −35.1 kJ mol−1 and −32.6 kJ mol−1, respectively. The heat capacities of the reagents, in J mol−1 K−1, are A: 78.3, B: 81.1, D: 75.4, Z: 68.3 and U: 76.3.
\[ A + B \rightarrow D + Z \tag{1} \]
\[ D + B \rightarrow U + Z \tag{2} \]
\[ r_1 = k_1 P_A P_B \tag{3} \]
\[ r_2 = k_2 P_D P_B \tag{4} \]
What are the conversion of A, the selectivity for D over U, the outlet temperature, and the outlet exchange fluid temperature?
Learning Activity 21 Calculations
Class 22 Learning Activity Hill and Root (“Introduction to Chemical Engineering Kinetics and Reactor Design,” 2nd Ed. Wiley, Hoboken, NJ, 2014.) describe the adiabatic oxidation of sulfur dioxide, reaction (1), in a steady-state, packed-bed PFR. A feed containing 8% SO2, 13% O2, and 79% N2 at 1 atm and 370 °C was fed to the first of two reactors at 0.149 lbmol s–1. It was assumed to have an effective viscosity of 0.09 lb h–1 ft–1. Spherical catalyst particles with a diameter of 0.25 in formed a packed bed with a porosity of 0.4 and a density of 0.6 g cm–3. The reactor diameter was 6 ft, and it operated at a conversion of 0.81. The rate expression shown in equation (2) was used with = 1.745 x 105 mol s–1 gcat–1 atm–1.5, = 31 kcal mol–1, = 7.59 x 109 mol s–1 gcat–1 atm, and = 53.6 kcal mol–1. Their analysis assumed no pressure drop, but after completing the analysis they estimated the pressure drop and found it to be significant. They suggested that the pressure drop could be decreased by using a larger reactor diameter.
Repeat their analysis, accounting for pressure drop, for reactors with diameters of 3, 6, and 9 ft and comment on the results. Specifically calculate the required catalyst volume and the pressure drop, and plot the temperature as a function of axial position in the reactor. Selected thermodynamic data are given in Table 1 where the heats of formation are in cal mol–1, and the heat capacities of the reagents in cal mol–1 K–1 can be calculated using the coefficients, \(\alpha_i\), \(\beta_i\), \(\gamma_i\), and \(\delta_i\), with temperatures in K.
\[ SO_2 + \frac{1}{2} O_2 \rightleftarrows SO_3 \tag{1} \]
\[ r = \frac{k_fP_{SO_2}P_{O_2} - k_rP_{SO_3}\sqrt{P_{O_2}}}{\sqrt{P_{SO_2}}} \tag{2} \]
\[ \hat C_{p,i} = \alpha _i + \beta _i T + \gamma _i T^2 + \delta _i T^3 \tag{3} \]
| i | \(\Delta H_{f,i}\big\vert_{298\,K}\) | \(\alpha_i\) | \(\beta_i\) | \(\gamma_i\) | \(\delta_i\) |
|---|---|---|---|---|---|
| SO2 | –70950 | 5.697 | 0.016 | –1.185 x 10–5 | 3.172 x 10–9 |
| O2 | 0 | 6.713 | –8.79 x 10–7 | 4.175 x 10–6 | –2.544x 10–9 |
| SO3 | –94470 | 12.13 | 0.00812 | ||
| N2 | 0 | 7.44 | –0.00324 | 6.4 x 10–6 | 2.79 X 10–9 |
Learning Activity 22 Calculations
Class 23 Learning Activity On a molar basis, a gas phase mixture contains 20% reagent A, 45% reagent B, 25% reagent C, and 10% reagent I at 220 °C and 3 atm. The mixture is fed at a rate of 1000 L min–1 to a PFR with a diameter of 5 cm and a length of 1 m. A fluid at a constant temperature of 185 °C surrounds the reactor tube which has a heat transfer coefficient of 850 cal m–2 min–1 K–1. The heat capacities of A, B, C, X, Y, ,Z and I are constant and equal to 12.7, 8.6, 11.3, 6.3, 14.4, 10.8, and 15.6 cal mol–1 K–1, respectively.
Heterogeneous catalytic reactions (1) and (2) take place within the PFR with no pressure drop. The bed density is 2.3 g cm–3. The rate expressions are given in equations (3) and (4). The pre-exponential factors for reactions (3) and (4) are 4.35 x 105 mol g–1 atm–2 min–1 and 2.17 x 106 mol g–1 atm–2 min–1, respectively, and the activation energies are 19.7 and 21.3 kcal mol–1, respectively. The heat of reaction 1 is 28.3 kcal mol–1 and the heat of reaction (2) is 29.8 kcal mol–1.
What are the conversion of B, the selectivity in moles of Y per mole of Z, and the temperature at the reactor outlet?
\[ A + B \rightarrow X+Y \tag{1} \]
\[ B + C \rightarrow X + Z \tag{2} \]
\[ r_1 = k_1P_AP_B \tag{3} \]
\[ r_2 = k_2P_BP_C \tag{4} \]
Practice Assignments from REB, The Course
Class 21 Practice Assignment A perfectly insulated PFR is being considered for producing Y and Z according to reaction (1). It would be fed a total of 100 mol s–1 of a mixture containing 20% inerts (I), 60% A and 20% B at a temperature of 150 °C and 2 atm. The gas phase reaction, equation (1), is first order in A and one–half order in B. The pre–exponential factor is equal to 2 x 1015 L0.5 mol–0.5 s–1 and the activation energy is 25,100 cal mol–1. The heat capacities in cal mol–1 K–1 are given in Table 1. The reaction can be assumed to be irreversible with a constant heat of –10 kcal mol–1. If 95% of the limiting reagent must be converted, how large must the reactor be, assuming there is no pressure drop in the reactor, and what would the final temperature equal?
\[ A + B \rightarrow Y + Z \tag{1} \]
| Reagent | Heat Capacity (cal mol⁻¹ K⁻¹) |
|---|---|
| \(A\) | \(5\) |
| \(B\) | \(7\) |
| \(Y\) | \(6.5\) |
| \(Z\) | \(5.5\) |
| \(I\) | \(4.2\) |
Class 21 Practice Assignment Solution
Class 21 Practice Assignment Calculations
Class 22 Practice Assignment A PFR surrounded by a perfectly mixed jacket is going to be used to produce Z according to liquid-phase reaction (1). The feed to the reactor is at 45 psi and flows at 120 ft3 h–1. It contains only A at a concentration of 0.025 lbmol ft–3 and a temperature of 120 °F. Water at 75 °F flows into the shell at 2000 lb h–1. The reactor tubes have a diameter of 1 in, a length of 125 ft, a heat transfer coefficient of 150 BTU ft–2 h–1 °F–1, and a Darcy friction factor of 0.018.
\[ A \rightarrow Z \tag{1} \]
The reacting fluid heat capacity is 8 BTU lbmol–1 °F–1. Its viscosity is 1 lb ft–1 h–1, and its density is 57 lb ft–3. The cooling water heat capacity is 1 BTU lb–1 °F–1. The reaction is exothermic with a heat of reaction of –30,500 BTU lbmol–1. The rate coefficient equals 0.059 s–1 at 120 °F, and the activation energy is 14,000 BTU lbmol–1.
Plot the conversion, reacting fluid temperature and pressure along the length of the PFR.
Class 22 Practice Assignment Solution
Class 22 Practice Assignment Calculations
Class 23 Practice Assignment Suppose reaction (1) is taking place in an isothermal, steady-state PFR. The reaction is catalyzed by porous spherical catalyst particles wherein the effective diffusion coefficient is 2.7 x 10–7 cm2 s–1. The rate, normalized with respect to the bed volume, is first order in A, and the rate coefficient is 0.019 s–1. The flow rate of the feed is 1 L min–1, and there are no external concentration gradients and no pressure drop in the reactor. Calculate the catalyst bed volume necessary to achieve 90% conversion of a 1 M feed of A for pellet diameters of 0.04, 0.27, 0.55, and 0.77 mm.
\[ A \rightarrow Z \tag{1} \]
Additional Assignments for Extra Practice
Additional assignments will be added as they become available.
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