Lecture 4 Mole balance: calculation of membrane reactors and unsteady state in tank reactors. nalysis of rate data
Mole alance in terms of Concentration and Molar Flow Rates Working in terms of number of moles (N, N,..) or molar flow rates (F, F etc) rather than conversion could be more convenient at some instances The difference in calculation: we will write mole balance for each and every species in the reactor
Isothermal reaction design algorithm
Membrane reactors used to increase conversion when the reaction is thermodynamically limited (e.g. with small K) or to increase selectivity in when multiple reactions are occurring CH 3H + CH 6 12 2 6 6 inert membrane reactor with catalyst pellet on the feed side (IMRCF) catalyst membrane reactor (CMR)
Membrane reactors CH 3H + CH 6 12 2 6 6 3+ C df Mole balances: dv = r df C C dv = r Δ + Δ = 0 V V+ΔV generation F F R V r V IN by flow OUT by diffusion df dv = r R OUT by flow no accumulation
Membrane reactors df dv = r R R = W a= k a( C C ) c S Diffusion flux area per volume concentration in the sweep gas channel mass transfer coefficient a π DL = = 2 π LD /4 4 D ssuming C S =0 and introducing k c R = k C = kca c
Example: Dehydrogenation reaction Typical reactions: dehydrogenation of ethylbenzene to styrene; dehydrogenation of butane to butene dehydrogenation of propane to propene Problem: for a reaction of type + C where an equilibrium constant Kc=0.05 mol/dm 3 ; temperature 227ºC, pure enters chamber at 8.2 atm and 227ºC at a rate of 10 mol/min write differential mole balance for,, C plot the molar flow rate as a function of space and time calculate conversion at V=400 dm 3. ssume that the membrane is permeable for only, catalyst density is ρ b =1.5 g/cm 3, tube inside diameter 2cm, reaction rate k=0.7 min -1 and transport coefficient k c =0.2 min -1.
Mole balance: df r dv = Rate law r = k C Example df r R dv = C CC Transport out of the reactor R Stoichiometry = k C c F K C C df dv = r C = CT0 C = CT0 C T0 F T F T C C 0 0 = T0 = = RT0 r = rc = r P F 0.2 mol/dm 3 C F C = C F C T
POLYMTH solution Example
Use of Membrane reactors to enhance selectivity + C+ D is fed uniformly through the membrane df r dv = + R
Unsteady state operation of stirred reactors + C+ D reactor start-up semibatch w. cooling reactive distillation during the start up of a reactor: slow addition of component to a large quantity of e.g. when reaction is highly exothermic or unwanted side reaction can occur at high concentration of one of the products (C) is vaporized and withdrawn continuously.
Startup of CSTR Conversion doesn t have any meaning in startup so we have to use concentrations dn F0 F + rv = dc V0 For liquid phase with C0 C + rτ = τ, τ = constant overflow v0 v= v ; V = V 0 0 For the 1 st order reactions e.g. to reach 99% steady state concentration dc 1+ τ k C r = kc + C = τ τ C 0, C C0 t = 1 exp ( 1+ τ k) 1+ τ k τ S C0 τ =, ts = 4.6 1+ τ k 1+ τ k for small k: t s ts = 4.6τ for large k: = 4.6 k
Semibatch reactors semibatch reactors could be used e.g. to improve selectivity k + D D k + U U r r = kc C 2 D = kc C 2 U S selectivity: / DU r k C = = r k C D D U U Selectivity can be increased by keeping concentration high and concentration low
Semibatch equations For component : no flow in/out rv 0 ( ) dn d C V VdC CdV = = = + V = V0 + v0t vc + rv= VdC dc v = r V 0 C dn VdC For component : = rv + F0 C dv + = rv + vc 0 0 dc v = r + C C V ( ) 0 0
Example 4.9 Production of methyl bromide is carried as an irreversible liquid phase reaction in isothermal semibatch reactor. Consider reaction to be elementary. initial volume of fluid initial concentration CNr flow of CH 3 NH 2 solution concentration CH 3 NH 2 rate constant CNr + CH3NH2 CH3r + NCNH2 V 0 0 0 = 5dm 3 C ( CNr) = 0.05 mol / dm 3 0.05 / v = dm s C ( CH NH ) = 0.025 mol / dm 0 3 2 3 k = 2.2 dm / s mol 3 3
Solution mole balance: rate law: combining dc v0 dc v0( C0 C ) = r C; = r + V V r = kc C dc v0 dc v0( C0 C ) = kcc C; = kcc + V V dcc v0 dcd v0 = kcc CC; = kcc CD; V V V = V + v t 0 0 N0 N C0V0 CV X = = N C V 0 0 0
Polymath calculation
nalysis of rate data Main question: How to collect rate data and deduce the reaction rate law? The lgorithm: 1. Postulate a rate law 2. Select appropriate reactor type and mole balance 3. Determine the variables and process the experimental data. pply simplification and create a model 4. Fit model to the data and find the coefficients. 5. nalyze your rate model for goodness of fit.
atch reactor data For irreversible reactions, it is possible to determine reaction order α and ratee constant k by either nonlinear regression or differentiating concentration vs time data Differential method is most applicable when: rate is function of one concentration only: method of excess is used + Products r = k C α Products r = k C C α β b
atch reactor: differential method Combing mole balance and rate law: dc α = kc dc ln = ln k + α ln C k ( dc / ) = ( C ) p α
atch reactor: differential method Techniques to obtain dc/ from the data: Graphical Numerical Differentiation of polinomial fit. Graphical: ΔC Δt plotted vs. t, smooth curve plotted to appoximate the area under the histogramm
atch reactor: differential method Numerical method: three-point differentiation formulas first point interior point last point dc dc t0 = dc ti tf = = 3C + 4C C C ( + 1) i 0 1 2 2Δt 2Δt C i ( 1) C 4C + 3C ( f 2) ( f 1) f 2Δt
atch reactor: differential method Polynomial fit: fit C with a polynomial and differentiate 2 n C = a0 + at 1 + a2t +... + ant dc = a1 + 2a2t + 3 a3t +... + na t 2 n 1 n significant error in derivative 3rd order polynomial 5th order polynomial
atch reactor: integral method Used when reaction order is known to evaluate the rate constant. The procedure: assume the reaction order integrate the model equation plot the data in appropriate coordinates and fit the data Example Products 0th order 1st order dc dc = r; = r; dc dc = k = kc C = C0 kt C0 ln = kt C 2nd order dc = r; dc 2 = kc 1 1 = kt C C 0
atch reactor: integral method
atch reactor: non-linear regression parameters in the non-linear equation (e.g. reaction order and rate constant) are varied to minimize the sum of squares: measured value s 1 σ = = 2 N 2 2 ( rim ric) N K N K i experimental value number of parameters to be determined number of runs steepest gradient descent trajectory to find α and k
Example 5-3 ( ) ( ) C H CCl + CH OH C H COCH + HCl 6 5 3 3 6 5 3 3 + C + D The reaction of trityl and methanol is carried out in a solution of benzene and pyridine. Pyridine reacts with HCl and precipitate so the reaction is irreversible. Initial concentration of methanol 0.5 mol/dm 3. and trityl 0.05 mol/dm 3. Determine reaction order with trityl. dc α = kc 1 C C t = k 1 α 1 α 1 α 0
Method of initial rates Differential method might be problematic if backward reaction rate is significant. In this case method of initial rates should be used instead Series of experiments are carried out at different initial concentrations C 0 and initial rate r 0 is determined rate dependence is plotted to determine the reaction order, e.g. r = kc α 0 0
Method of half-lives Half-life time of the reaction (time for the concentration of the reactant to fall by half) is determined as function of initial concentration 1 t = t at C = C 2 1/2 0 Products dc t t = r = kc α 1 1 1 = k( α 1) C C α 1 α 1 0 α 1 2 1 1 1/2 = α 1 k( α 1) C0
Differential reactor consist of a tube with a thin catalyst disk (wafer) conversion of the reactants, concentration and temperature should be small most commonly used catalytic reactor to obtain experimental data F0 Fe r = ΔW r = r ( C ) bed concentration is equal to the inlet 0
Problems P4-26: large component in the processing train for fuel cell technology is the water gas shift membrane reactor, where H 2 can diffuse out the sides of the membrane while the other gases cannot. CO + H O CO + H 2 2 2 ased on the following information plot the concentration and molar flow rates of each of the reacting species down the length of the membrane reactor. ssume: the volumetric feed is 10dm 3 /min at 10atm; equil molar feed of CO and water vapour with C T0 =0.4mol/dm 3, equilibrium constant K e =1.44, reaction rate k=1.37 dm 6 /mol kg cat min, mass transfer coefficient for H 2 kc=0.1dm 3 /mol kg cat min.compare with PFR. P5-13