ISOTHERMAL PISTON FLOW REACTORS
3.3 TRANSPIRED-WALL REACTORS
Tubular reactors sometimes have side entrance points for downstream injection. Like the case of fed-batch reactors, this raises the question of how quickly the new ingredients are mixed. Mixing in the radial direction is the dominant con- cern. If radial mixing is fast, the assumption of piston flow may be reasonable and the addition of new ingredients merely reinitializes the problem. The equiva- lent phenomenon was discussed in Section 2.6.2 for fed-batch reactors.
This section considers the case where the tube has a porous wall so that reac- tants or inerts can be fed gradually. Transpiration is used to cool the walls in high-temperature combustions. In this application, there is usually a change of phase, from liquid to gas, so that the cooling benefits from the heat of vapor- ization. However, we use the term transpiration to include transfer through a porous wall without a phase change. It can provide chemical protection of the wall in extremely reactive systems such as direct fluorinations. There may be selectivity advantages for complex reactions. This possibility is suggested by Example 3.9.
Assume that the entering material is rapidly mixed so that the composition is always uniform in the radial direction. The transpiration rate per unit length of tube is q¼ qðzÞ with units of m2/s. Component A has concentration atrans¼ atransðzÞ in the transpired stream. The component balance, Equation (3.4), now becomes 1 Ac dð _NNAÞ dz ¼ 1 Ac dðQaÞ dz ¼ 1 Ac dðAcuuaÞ dz ¼ atransq Ac þ R A ð3:46Þ
We also need a total mass balance. The general form is
Q ¼ Qininþ Zz 0
qtransdz ð3:47Þ
Analytical solutions are possible in special cases. It is apparent that transpira- tion will lower the conversion of the injected component. It is less apparent, but true, that transpired wall reactors can be made to approach the performance of a CSTR with respect to a transpired component while providing an environ- ment similar to piston flow for components that are present only in the initial feed.
Example 3.9: Solve Equation (3.46) for the case of a first-order reaction where , q and atrans are constant. Then take limits as Qin! 0 and see what happens. Also take the limit as q! 0.
Solution: With constant density, Equation (3.47) becomes Q¼ Qinþ qz
Substitute this into the Qa version of Equation (3.46) to obtain a variable- separable ODE. Integrate it subject to the initial condition that a¼ ain at z¼ 0: The result is aðzÞ ¼ qatrans Ackþ q qatrans Ackþ q ain 1þ qz Qin ðAckþqÞ=q ð3:48Þ
Taking the limit as Qin! 0 gives a¼ qatrans
Ackþ q¼ atrans AcLk
Qout þ 1
The z dependence has disappeared! The reactor is well mixed and behaves like a CSTR with respect to component A. Noting that Qout¼ qL gives
aout¼ atrans 1þ Vk Qout ¼ atrans 1þ ktt
which is exactly the behavior of a CSTR. When a transpired-wall reactor has no initial feed, it behaves like a stirred tank. When Qin> 0 but ain¼ 0, it will still have a fairly uniform concentration of A inside the reactor while behaving much like a piston flow reactor for component B, which has bin> 0 but btrans¼ 0. For this component B,
bðzÞ ¼ bin 1þ qz
Qin
ðAckþqÞ=q
Physical insight should tell you what this becomes in the limit as q! 0. Problem 2.7 shows the mathematics of the limit.
This example shows an interesting possibility of achieving otherwise unob- tainable products through the use of transpired-wall reactors. They have been proposed for the manufacture of a catalyst used in ammonia synthesis.1 Transpiration might be useful in maintaining a required stoichiometry in copolymerizations where the two monomers polymerize at different rates, but a uniform product is desired. For the specific case of an anionic polymerization, transpiration of the more reactive monomer could give a chemically uniform copolymer while maintaining a narrow molecular weight distribution. See Section 13.4 for the background to this statement.
Membrane reactors, whether batch or continuous, offer the possibility of selective transpiration. They can be operated in the reverse mode so that some
products are selectively removed from the reaction mix in order to avoid an equilibrium limitation. Membrane reactors can be used to separate cell mass from fermentation products. See Section 12.2.2.
PROBLEMS
3.1. The first-order sequence A!kI B!kII C is occurring in a constant- density piston flow reactor. The residence time is tt.
(a) Determine bout and cout given that bin¼ cin¼ 0 and that kI¼ kII. (b) Find a real chemical example, not radioactive decay, where the
assumption that kI¼ kII is plausible. As a last resort, you may consider reactions that are only pseudo-first-order.
3.2. Suppose l ¼ 1 0 1 1 1 1 0 0 2 6 6 4 3 7 7 5
gives the stoichiometric coefficients for a set of elementary reactions.
(a) Determine the elementary reactions and the vector of reaction rates that corresponds tol.
(b) Write the component balances applicable to these reactions in a PFR with an exponentially increasing reactor cross section, Ac¼ AinletexpðBzÞ:
3.3. Equation (3.10) can be applied to an incompressible fluid just by setting tt ¼ V=Q. Show that you get the same result by integrating Equation (3.8) for a first-order reaction with arbitrary Ac¼ AcðzÞ.
3.4. Consider the reaction B!k 2A in the gas phase. Use a numerical solu- tion to determine the length of an isothermal, piston flow reactor that achieves 50% conversion of B. The pressure drop in the reactor is negli- gible. The reactor cross section is constant. There are no inerts. The feed is pure B and the gases are ideal. Assume bin¼ 1, and ain¼ 0, uuin¼ 1, and k¼ 1 in some system of units.
3.5. Solve Problem 3.4 analytically rather than numerically.
3.6. Repeat the numerical solution in Example 3.2 for a reactor with variable cross section, Ac¼ AinletexpðBzÞ. Using the numerical values in that example, plot the length needed to obtain 50% conversion versus B for 1 < B < 1 (e.g. z ¼ 0:3608 for B ¼ 0). Also plot the reactor volume V versus B assuming Ainlet¼ 1.
Assume
Kequil¼
PSTYPH2 PEB
¼ 0:61 atm at 700C
3.8. Annular flow reactors, such as that illustrated in Figure 3.2, are some- times used for reversible, adiabatic, solid-catalyzed reactions where pres- sure near the end of the reactor must be minimized to achieve a favorable equilibrium. Ethylbenzene dehydrogenation fits this situation. Repeat Problem 3.7 but substitute an annular reactor for the tube. The inside (inlet) radius of the annulus is 0.1 m and the outside (outlet) radius is 1.1 m.
3.9. Consider the gas-phase decomposition A ! B þ C in an isothermal tubular reactor. The tube i.d. is 1 in. There is no packing. The pressure drop is 1 psi with the outlet at atmospheric pressure. The gas flow rate is 0.05 SCF/s. The molecular weights of B and C are 48 and 52, respec- tively. The entering gas contains 50% A and 50% inerts by volume. The operating temperature is 700C. The cracking reaction is first order with a rate constant of 0.93 s1. How long is the tube and what is the conversion? Use ¼ 5 105Pas. Answers: 57 m and 98%. 3.10. Suppose B !k 2A in the liquid phase and that the density changes from
1000 kg/m3 to 900 kg/m3 upon complete conversion. Find a solution to the batch design equation and compare the results with a hypothetical batch reactor in which the density is constant.
3.11. A pilot-scale, liquid-phase esterification with near-zero heat of reaction is being conduced in a small tubular reactor. The chemist thinks the reac- tion should be reversible, but the by-product water is sparingly soluble in the reaction mixture and you are not removing it. The conversion is 85%. Your job is to design a 100 scaleup. The pilot reactor is a 31.8 mm i.d. tube, 4 m long, constructed from 12 BWG (2.769 mm) 316 stainless steel. The feed is preheated to 80C and the reactor is jacketed with tempered water at 80C. The material begins to discolor if higher temperatures are used. The flow rate is 50 kg/h and the upstream gauge pressure is 1.2 psi. The density of the mixture is around 860 kg/m3. The viscosity of the material has not been measured under reaction conditions but is believed to be substantially independent of conversion. The pilot plant discharges at atmospheric pressure.
(a) Propose alternative designs based on scaling in parallel, in series, by geometric similarity, and by constant pressure drop. Estimate the Reynolds number and pressure drop for each case.
(b) Estimate the total weight of metal needed for the reactor in each of the designs. Do not include the metal needed for the water jacket in your weight estimates. Is the 12 BWG tube strong enough for all the designs?
(c) Suppose the full-scale reactor is to discharge directly into a finishing reactor that operates at 100 torr. Could this affect your design? What precautions might you take?
(d) Suppose you learn that the viscosity of the fluid in the pilot reactor is far from constant. The starting raw material has a viscosity of 0.0009 PaEs at 80C. You still have no measurements of the viscos- ity after reaction, but the fluid is obviously quite viscous. What influence will this have on the various forms of scaleup?
3.12. A pilot-scale, turbulent, gas-phase reactor performs well when operated with a inlet pressure of 1.02 bar and an outlet pressure of 0.99 bar. Is it possible to do a geometrically similar scaleup by a factor of 10 in throughput while maintaining the same mean residence time? Assume ideal gas behavior and ignore any change in the number of moles due to reaction. If necessary, the discharge pressure on the large reactor can be something other than 0.99 bar.
3.13. Refer to the results in Example 3.7 for a scaling factor of 100. Suppose that the pilot and large reactors are suddenly capped and the vessels come to equilibrium. Determine the equilibrium pressure and the ratio of equilibrium pressures in these vessels assuming
(a) Pin=Pout1¼ 100 (b) Pin=Pout1¼ 10 (c) Pin=Pout1¼ 2 (d) Pin=Pout1¼ 1:1
3.14. An alternative to Equation (3.16) is Fa ¼ 0:04Re0:16. It is more conser- vative in the sense that it predicts higher pressure drops at the same Reynolds number. Use it to recalculate the scaling exponents in Section 3.2 for pressure drop. Specifically, determine the exponents for P when scaling in series and with geometric similarity for an incompressible fluid in turbulent flow. Also, use it to calculate the scaling factors for SR and SLwhen scaling at constant pressure.
3.15. An integral form of Equation (3.15) was used to derive the pressure ratio for scaleup in series of a turbulent liquid-phase reactor, Equation (3.34). The integration apparently requires to be constant. Consider the case where varies down the length of the reactor. Define an average viscosity as ^ ¼1 L ZL 0 ðzÞ dz
Show that the Equation (3.34) is valid if the large and small reactors have the same value for and that this will be true for an isothermal or adia-^ batic PFR being scaled up in series.
3.16. Suppose an inert material is transpired into a tubular reactor in an attempt to achieve isothermal operation. Suppose the transpiration rate q is independent of z and that qL¼ Qtrans.Assume all fluid densities to be constant and equal. Find the fraction unreacted for a first-order reac- tion. Express your final answer as a function of the two dimensionless parameters, Qtrans=Qin and kV=Qin where k is the rate constant and
Qin is the volumetric flow rate at z¼ 0 (i.e., Qout¼ Qinþ Qtrans). Hint: the correct formula gives aout=ain¼ 0:25 when Qtrans=Qin¼ 1 and kV=Qin¼ 1:
3.17. Repeat Problem (3.16) for a second-order reaction of the 2A !k=2 B type. The dimensionless parameters are now Qtrans=Qinand kainV=Qin.
REFERENCE
1. Gens, T. A., ‘‘Ammonia synthesis catalysts and process of making and using them,’’ U.S. Patent 4,235,749, 11/25/1980.
SUGGESTIONS FOR FURTHER READING
Realistic examples of variable-property piston flow models, usually nonisother- mal, are given in
Froment, G. F. and Bischoff, K. B., Chemical Reactor Analysis and Design, 2nd Ed., Wiley, New York, 1990.
Scaleup techniques are discussed in