Design Equations. ν ij r i V R. ν ij r i. Q n components. = Q f c jf Qc j + Continuous Stirred Tank Reactor (steady-state and constant phase)

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1 Desgn Equatons Batch Reactor d(v R c j ) dt = ν j r V R n dt dt = UA(T a T) r H R V R ncomponents V R c j C pj j Plug Flow Reactor d(qc j ) dv = ν j r 2 dt dv = R U(T a T) n r H R Q n components j c j C pj Strred Tank Reactor d(v R c j ) dt = Q f c jf Qc j + ν j r V R dt dt = UA(T a T) n ncomponents r H R V R + Q f j c jf (H jf H j ) ncomponents V R j c j C pj Contnuous Strred Tank Reactor (steady-state and constant phase) 0 = Q f c jf Qc j + ν j r V R n n components Tf 0 = UA(T a T) r H R V R + Q f c jf C pj dt j T 2

2 General Informaton on Mass Transfer wth Reacton For heterogeneous reactons one must determne the rate per unt volume of pellet. R jp = 1 R j dv = S p dc j D j V p V p V p dr r=rp The dmensonless steady-state concentraton wthn a symmetrcal pellet s found wth [ 2 a 2 ] R js c + R = 0 R = R j c js D j R js For a frst-order reacton (A B) wth a sphercal pellet surface concentraton of c As ( ) R c A = c As r snh Φ( 3r R ) snh(3φ) R Ap = 1 Φ [ 1 tanh3φ 1 ] ( k 1 c As ) 3Φ Φ = R 3 k 1 For some heterogeneous cases (A B) t s approprate to use D e η = R j R js η 1 [ 1 Φ tanh3φ 1 ] 3Φ Φ = V [ p n + 1 S p 2 knc ] s n D e where n s the reacton order n component A and the overall reacton order. 3

3 Useful Integrals 1 a + bx dx = 1 ln(a + bx) b (a + bx) n dx = (a + bx)n+1 (n + 1)b ;n 1 ( dx (a + bx)(a + b x) = 1 a ab a b ln + b ) x a + bx a + bx bx a + b dx = x b + ab a b b 2 ln ( a + b x ) (a + bx) m (a + b x) n dx = [ 1 (a + bx) m ] (a + bx) m 1 (n 1)b (a + b n 1 mb x) (a + b x) n 1dx x m (a + bx) n dx = xm+1 (a + bx) n m + n an m + n + 1 x m (a + bx) n 1 dx where: Smpson s three-eghth s rule for numercal ntegraton: X3 X 0 f(x)dx = 3h 8 [f(x 0) + 3f(X 1 ) + 3f(X 2 ) + f(x 3 )] h = X 3 X 0 3 X 1 = X 0 + h X 2 = X 0 + 2h Integraton of N + 1 ponts, where N s even: XN ] [f 0 + 4f 1 + 2f 2 + 4f 3 + 2f f (N 1) + f N where: X 0 f(x)dx = h 3 h = X N X 0 N 4

4 Problem 1 The elementary, lqud-phase, rreversble reacton A + B C s carred out n a CSTR operated at 300 K. Reactant B also somerzes to form undesred product D. The rates of reacton are gven by: A + B C B D r 1 = k 1 c A c B r 2 = k 2 c B There are two reactant streams that are mxed before enterng the reactor. The frst reactant stream contans A (but no B) at a concentraton of 1 mol/lter and a flow rate of 1 lter/mn. The second reactant stream contans B (but no A) at a concentraton of 1 mol/lter and a flow rate of 1 lter/mn. The two reactant streams mx to form a sngle feed stream that s equal molar n A and B, wth a total volumetrc flow rate of 2 lters/mn. Addtonal Informaton: At 300 K, k 1 = 1 lters/mol-mn, k 2 = 0.1 mn -1 Reactor volume V R = 100 lters a) What s the resdence tme n the CSTR? b) What fracton of ncomng A s converted n the CSTR? c) What fracton of ncomng B s converted n the CSTR? d) What are the concentratons of products C (desred) and D (undesred) n the output stream of the CSTR? Problem 2 Sulfurc acd s produced by the exothermc, reversble reacton of SO 2 and ar over a vanadum pentoxde catalyst at o F. The reacton s exothermc and has no sde-reactons. The feed gas to the catalytc reactors s produced by burnng sulfur n ar to produce SO 2 then blendng n more ar. a) One early reactor desgn uses 3 or 4 stages of catalyst wth ntercoolng between the stages to obtan an overall converson of SO 2 to SO 3 of 95 to 98%. Ths s followed by a scrubber at lower T wth addton of water to condense H 2 SO 4 from the gas. The effluent gas contanng ppm SO 2 was vented. 1) What tradeoffs determne the T of the reactor? 2) Why are multple reactor stages useful to ncrease converson? b) A later system desgn addresses the emssons of SO 2 by usng two scrubbers. Gas from the frst scrubber s reheated to 800 o F then passes over addtonal catalyst stages wth ntercoolng to get % converson to SO 3. The gas s then cooled and passed through another scrubber to remove the SO 3 as sulfurc acd. 1) Why does ths work? c) What would be requred to desgn the process wth a classc recycle of unreacted SO 2 so that 99.99% converson would be acheved? 5

5 Problem 3 A gas phase thermal decomposton reacton takes place n a plug-flow reactor. The reactant A s decomposed nto products B and C. The reacton has the followng rate constant: log ] = 2 T 1 10( k[ s ) 15.1 The system operates sothermally at temperature T = 1000Kand pressure P = 1 atm. The feed stream conssts of pure reactant A, and has molar flow rate N Af = 5 kmol/hr and volumetrc flow rate Q f. The feed stream s at the same temperature and pressure as the reactor. The gases follow deal gas behavor. The converson of A at the effluent s x A = Please compute: a) Q f and Q, whch are the volumetrc flow rates at the reactor nlet and outlet, respectvely. b) The reactor volume (V R ) requred to attan a converson of A of x A = R = lter atm mol -1 K -1 Problem 4 The followng frst-order catalytc reacton s conducted n a fxed-bed reactor. The reactant and product are gases. A 2B r = kc A The ntrnsc rate constant s k = exp( 19,124/T) s -1, where T s n Kelvn. The feed conssts of pure A at 2 atm and a total molar flow rate of N Af = 12 mol/s. At ths feed rate there are no external mass transfer lmtatons. The reactor s sothermal at a temperature of 578 K. The catalyst pellets are sphercal wth a radus of 0.3 cm. The effectve dffusvty s D eff = cm 2 /s. The catalyst bed has a vod fracton of Under these condtons, the fxed bed reactor was desgned to yeld a converson of A correspondng to x A = 0.9. There was a problem wth the purty of the feed for several days, the catalyst was posoned by an mpurty and now the catalyst s ten tmes less actve. For all practcal purposes the ntrnsc rate constant s now k posoned = exp( 19,124/T) s -1. Assumng the dffusvty wll not change sgnfcantly wth temperature, what new reactor temperature wll you need to ensure the exstng reactor contnues to provde x A = 0.9? You should assume you can adjust the feed and reactor to ths new temperature so the system s stll sothermal. R = 82 cm 3 atm K 1 mol 1 6

6 Problem 5 The lqud-phase parallel reactons A B A C r 1 = k 1 c A r 2 = k 2 c A are to be carred out n a CSTR. The feed enters at 65 C and conssts of pure A at a concentraton of c Af = 5 mol/lter. Varable Value Unts ρq 93,000 g/mn H R1-3,200 cal/mn H R2 2,000 cal/mn V R 100 lter Q f 100 lter/mn T a 50 C C pf 0.22 cal/g-k U 0.65 cal/mn-cm 2 -K A 15,400 cm 2 k exp(-12,500/t) mn -1 ; T n K k exp(-8,500/t) mn -1 ; T n K Under these condtons the reactor has three possble operatng condtons (two stable and one unstable) as llustrated n Fgure 1 on Page 8. What process parameters would you change to ensure operaton at the upper branch of stable solutons? You do not need to have the same operatng temperature as llustrated n Fgure 1. You do need to realze hgh converson of reactant A. Explan how changng the parameter(s) you select wll work to make the system have a stable soluton. 7

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