Knetcs and Reactor Desgn Ph.D.Qualfyng Examnaton January 2006 Instructons Ph.D. Qualfyng Examnaton n Knetcs and Reactor Desgn January 2006 Unversty of Texas at Austn Department of Chemcal Engneerng 1. The exam s closed book. There are fve equally weghted problems n ths exam and you must answer them all to receve full credt. 2. Wrte your Code Number clearly on each page of your answer sheets. 3. Please use the paper that we have provded to you for your solutons and number each page of your soluton to each problem sequentally. Keep the soluton to each problem on separate sheets of paper and when you are fnshed wth the exam, staple each problem soluton separately and place n the stack for that problem number. Ths wll allow gradng of the problems smultaneously so that the results can be returned to you rapdly. Please wrte only on one sde of each of your soluton sheets. Clearly ndcate your answer. 4. Useful formulas and nformaton are attached mmedately followng ths cover page. 5. Wrte your name on a slp of paper (wth the Code Number) and seal n the envelope. 6. Turn n your exam, solutons (agan, place each problem n the stack for that problem), and the envelope to the proctor when the exam perod s over. 1
Desgn Equatons Batch Reactor d(v R c j ) dt = n rxns ν j r V R nrxns 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 = n rxns ν j r 2 dt dv = R U(T a T ) n rxns r H R Q n components j c j C pj Strred Tank Reactor d(v R c j ) dt n rxns = Q f c jf Qc j + ν j r V R dt dt = UA(T a T ) nrxns 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) n rxns 0 = Q f c jf Qc j + ν j r V R n rxns n components Tf 0 = UA(T a T ) r H R V R + Q f c jf C pj dt j T 2
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 (n 1)b (a + b n 1 mb x) (a + bx) m 1 (a + b x) n 1 dx ] x m (a + bx) n dx = xm+1 (a + bx) n m + n + 1 + 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 4 +... + 4f (N 1) + f N where: X 0 f(x)dx = h 3 h = X N X 0 N 3
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 Ap = 1 Φ ( ) R c A = c As r snh Φ( 3r R ) snh(3φ) [ 1 tanh 3Φ 1 ] ( k 1 c As ) 3Φ Φ = R 3 k 1 D e For some heterogeneous cases t s approprate to use where η = R j R js η 1 [ 1 Φ tanh 3Φ 1 ] 3Φ Φ = V [ p n + 1 S p 2 knc s n 1 ] 0.5 D e tanh x = ex e x e x + e x 4
Knetcs and Reactor Desgn Ph.D. Qualfyng Examnaton Questons Problem 1 A ntraton reacton can be modeled as A + 2B 2C + D and s carred out n an adabatc, steady-state CSTR. The reacton rate s frst order n the concentraton of A and second order n the concentraton of B. The rate constant s gven by [ ( 40 (kj/mol) 1 k = 0.090 exp R 303 1 )] T (K) The nlet condtons and the thermodynamc propertes are Varable Value Unts T f 303 K N Af 10 mol/mn N Bf 30 mol/mn Q f 1000 l/mn c Af 0.01 mol/l H r -370.1 kj/mol C pa 84.5 J/(mol-K) C pb 137 J/(mol-K) C pc 170 J/(mol-K) C pd 75 J/(mol-K) (L/mol) 2 (mn) 1 Ths reacton occurs n the lqud phase and the concentratons are dlute so that mole change wth reacton does not change the overall densty of the reactng flud. Assume the C p s are constant wth respect to temperature. (A) Calculate the reactor volume to acheve a 35% converson of A. (B) Wll the reactor volume ncrease or decrease f reactor coolng s allowed? Why? 5
Knetcs and Reactor Desgn Ph.D. Qualfyng Examnaton Questons Problem 2 Consder a porous catalyst partcle n the shape of a thn dsk (of thckness 2d), such that the surface area of the edge of the dsk s small n comparson to that of the two crcular faces. Partcles such as these are planned for use n a reactor to catalyze the reacton of A P r = kc A The effectve dffusvty of A n the catalyst s D e, and the concentraton of A on the surface of the partcle s c As. (A) What s the concentraton of A n the partcle? (B) Derve an expresson for the effectveness factor for ths partcle. (C) Derve approxmate expressons for the effectveness factor n the lmt of large and small kd2 D e. Problem 3 The mass acton statement for the formaton of A 2 B s 2A + B A 2 B Two elementary reactons are proposed to descrbe ths formaton n whch AB* s not observed at sgnfcant concentratons and s consdered a reacton ntermedate. A + B k 1 AB k 1 AB k 2 A 2 B k 2 (A) What s the rate of formaton of A 2 B n terms of stable (and measurable) speces? (B) Expermental data reveal that R A2 B = 0.72c2 A c B 1 + 2c A What does ths expermental result reveal about the mechansm and the relatve rates of the reactons? How are the coeffcents, 0.72 and 2, related to the rate constants? 6
Knetcs and Reactor Desgn Ph.D. Qualfyng Examnaton Questons + + + kp (I) k (II) + Fgure 1: The polymerzaton somerzaton reactons durng catonc polymerzaton of 3-methyl-1-butene. Problem 4 Isomerzaton polymerzaton s a term appled to polymerzatons where the repeat unt somerzes after addng to the chan, but before the next unt s added. Ths phenomenon s partcularly common n catonc polymerzatons, where secondary carbocatons wll rearrange va a hydrde shft to form much more stable tertary carbocatons. The polymerzaton somerzaton reactons n the catonc somerzaton polymerzaton of 3-methyl-1-butene are gven n Fgure 1 where denotes the rest of the polymer chan, (I) represents the normal non-somerzed repeat unt, and (II) represents the somerzed repeat unt. Kennedy and coworkers [J. P. Kennedy, R. M. Thomas, L. S. Mnckler, G. G. Wanless, J. Polm. Sc. A 2, 1441 (1964).] found that when the polymerzaton was conducted at -130 C, the fracton of somerzed unts (II) n the polymer chan, f, was approxmately 0.95. When the polymerzaton was carred out at -100 C, however, f 0.70. In both cases, the polymerzatons were taken to low conversons, such that the monomer concentraton may be consdered as constant. Both somerzaton and polymerzaton are very favorable energetcally such that both reactons may be consdered rreversble. Fnally the reactons can be treated as elementary for the purposes of wrtng the knetc statements. Gven the above and the value of the gas constant R = 8.314 J/mol-K, estmate the followng: (A) The fracton of somerzed unts, f, when the polymerzaton s carred out at -70 C. (B) The monomer concentraton that would yeld f = 0.95 when the polymerzaton s carred out at -100 C. 7
Knetcs and Reactor Desgn Ph.D. Qualfyng Examnaton Questons Problem 5 Consder the gas-phase reacton 2A B r = kc A k = 3.4 sec 1 The feed to an sothermal PFR s pure A at 8,120 g/sec. The molecular weght of A s 58 g/gmole. The temperature s 650 K and the pressure 1.5 atm. (A) Determne the reactor volume requred for 85% converson of A. (B) Pure A s stll fed at 8,120 g/sec at the same temperature and pressure, only now the reactor dameter s doubled. Provded the flow remans turbulent, by what factor wll the nlet velocty change and by what factor wll the reactor volume change for a converson of 85%? Does the answer you gve for the volume make sense? 8