Mass Transport: Non-Ideal Flow Reactors Advanced Transport Phenomena Module 6 - Lecture 28 Dr. R. Nagarajan Professor Dept of Chemical Engineering IIT Madras 1
Simplest approach: apply overall material/ energy/ momentum balances to the reactor black box approach, insufficient Most rigorous: Divide into small subregions, approximate each region with PDEs Impractical Intermediate solution: model as discrete network of small number of interconnected ideal reactor types (SS PFR & WSR) 2
RTDF residence time distribution function (exit-age DF), E(t) E(t) dt fraction of material at vessel outlet stream that has been in vessel for times between t and t ± dt PFR: E(t) is a Dirac function, centered at residence time V/ m& / ρ ( ) 3
V vessel volume m& feed mass flow rate e.g., straight tube through which incompressible fluid flows with a uniform plug-flow velocity profile Partial recycle can alter RTDF 4
Tracer residence-time distribution functions for ideal and real vessels (for e.g., reactors) (adapted from Levenspiel (1972)) 5
Ideal plug-flow reactor (PFR) with partial recycle (recycle introduces a distribution of residence times, and reduces the residence time per pass within the PFR) 6
WSR: ( ) ( ) 1 E WSR = df/ dt= t exp flow t/ t flow Most likely residence time in a WSR is zero! Mean residence time = V/ m& / ρ ( ) Not all fluid parcels have same residence time, unlike PFR 7
WSR: Dimensionless variance s 2 about mean residence time indicator of spread of residence times Mean residence time related to first moment of E(t), i.e.: flow s 2 is related to 2 nd moment of E(t): σ t = 0 tet. () dt 2 1 ( ) 2 1 2 ( ) 2 t t. () 2 flow Et dt= t E t dt t 2 flow tflow 0 tflow 0 = 1 for a WSR, 0 for a PFR 8 PFR with infinite recycle behaves like WSR
RTDF for Composite Systems: If RTDF for vessel 1 is E 1 (t) and for vessel 2 is E 2 (t), RTDF for a series combination of the two is: (convolution formula) t 1 2 1 2 0 ( ) ( ) E + () t = E t. E t t dt ' ' ' 9
If vessel 1 is characterized by t flow,1, and s 12, and vessel 2 by t flow,2, and s 22, then for the series combination, mean residence times and variances are simply additive: t = t + t σ flow,1+ 2 flow,1 flow,2 = σ + σ 2 2 2 1+ 2 1 2 10
RTDF for Composite Systems: For a network of n-wsrs of equal volume, for which: (t flow 1 n 1 t t t flow E( n WSRs) =..exp ( n 1! ) tflow tflow V/( m& / ρ) ) for each vessel in series) 11
For vessels 1, 2, 3,., n in parallel, receiving fractions f 1, f 2, f 3,., f n of total flow: E= fe() t + f E() t +... + f E () t 1 1 2 2 n n Where i f = 1 i, and for each vessel: i 0 E t dt = 1 ( i= 1,2,..., n) ( ) 12
Real reactors as a network of ideal reactors: Modular modeling Network of ideal reactors can be constructed to approximate any experimental reactor RTDF: ωtracer() t Eexp() t = ωtracer ( t) dt 0 reactor exit (where tracer is input as a Dirac impulse function) 13
Real reactors as a network of ideal reactors: Modular modeling GT combustor; proposed interconnection of reactors comprising modular model (adapted from Swithenbank, et al.(1973)) 14
Real reactors as a network of ideal reactors: Modular modeling Info obtained from tracer diagnostics & from combustor geometry, cold-flow data, etc. Important since RTD-data alone cannot discriminate between alternative networks with identical RTDmoments t flow = te t dt, t E t dt,..., etc.) ( ) 2 ( ) 0 0 15
Equivalent vessel network is nonunique Each alternative may capture one aspect (e.g., combustor efficiency) but not another (e.g., domain of stable operation) 16
Real reactors as a network of ideal reactors: Modular modeling Tracer methods can: Guide development of modular models Diagnose operating problems with existing chemical reactors or physical contactors RTD data can show up dead-volumes, flowchanneling, bypassing (all cause inefficient operation) Geometric or fluid-dynamic changes in design can correct these flaws Perturbation in feed can be used as tracer 17
Real reactors as a network of ideal reactors: Modular modeling RTD function, E(t), does not capture role of concentration fluctuations due to turbulence, incomplete mixing (at molecular level micromixing ) When tracer concentration fluctuates at reactor exit, we only collect data on <E(t)> arithmetic average of N tracer shots, each yielding RTD E j (t) (j = 1, 2,., N) 18
Two networks with identical <E(t)> but with different shot-to-shot variations, as measured by variance: lim 1 N N N j= 1 0 ( ) E t E() t dt j will perform differently as chemical reactors 2 19
Statistical micro flow (Random Eddy Surface-Renewal) models of interfacial mass transport in turbulent flow systems Mass/ energy transport visualized to occur during intervals of contact between turbulent eddies & surface stale eddies replaced by fresh ones Effective transport coefficient calculated by timeaveraging RTDF-weighted instantaneous St(t) 20
Statistical micro flow (Random Eddy Surface-Renewal) models of interfacial mass transport in turbulent flow systems If E(t) is defined such that: Relative portion of each unit interfacial area E() t dt covered by fluid eddies having"ages" between tandt+dt, 21
then: St = Stt (). Et () dt 0 St(t) calculated from transient micro fluid-dynamical analysis of individual eddy flow St time-averaged transfer coefficient Interfacial region being viewed as a thin vessel w.r.t eddy residence time 22
Statistical microflow (Random Eddy Surface-Renewal) models of interfacial mass transport in turbulent flow systems Earliest & simplest model: each eddy considered to behave like a translating solid body Large compared to transient diffusion BL (penetration) thickness Dimensional time-averaged mass-transfer coefficient given by: 23
4DA '' ja, m ( w πt = ρ ω ) Ab, ωaw, D A tm 1/2 1/2 [ ] fore( PFR) ( Higbie1935 [ ] for E( WSR) ( Danckwerts 1951) t m mean eddy contact time (1/(average renewal frequency)) Related to prevailing geometry & bulk-flow velocity Versatile alternative to Prandtl-Taylor eddy diffusivity approach 24
Extinction, ignition, parametric sensitivity of chemical reactors: Simplest modular model for steady-flow behavior of combustors: WSR + PFR 25
m& max upper limit to total mass flow rate, at each upstream condition (T u, p u, mixture ratio Φ) above which extinction of exoergic reaction (flame-out) abruptly occurs For m& < m& max m&,, two possible SS conditions exist: one corresponding to high fuel consumption & high temperature in WSR, the other to negligible fuel consumption & rise in T 26
Extinction, ignition, parametric sensitivity of chemical reactors: Simple, two-ideal reactor modular model of gas turbine, ramjet, or rocket engine combustor 27
Extinction, ignition, parametric sensitivity of chemical reactors: Parametric sensitivity: change in reactor performance for a small change in input or operating parameter (e.g., T u ) 28
Example: WSR module with following overall stoichiometric combustion reaction: 1gm O + f gmf 1+ f gram P+ fqcal(heat) ( ) 29
Extinction, ignition, parametric sensitivity of chemical reactors: Allow a 2 nd generation reactant (oxidant) & associated heat Governs WSR operating temperature, T 2 WSR species mass balance: (i = O, F, P) m&. r,, T. V ( ) ''' ω ( ) 2 ω 1 = & ω 2 ω 2 2 i i i O F WSR 30
Extinction, ignition, parametric sensitivity of chemical reactors: Overall energy balance: ( ) ''' = & ( ω ω ) mc &. T T r,, T QV. p 2 1 F O 2 F 2 WSR Source terms for oxidizer & fuel related by: r& = r& / f ''' O So, ω O 2 and ω F 2 can be expressed in terms of T 2 ''' F 31
Extinction, ignition, parametric sensitivity of chemical reactors: Overall kinetics represented by Arrhenius-type massaction rate law: ''' E 1 pm vo vf r& F = A.exp... ω vo vf 1 Oω F RT MOMF RT LHS straight line intersecting RHS at 3 distinct T 2 values, middle one unstable, upper ignited WSR SS, lower extinguished WSR SS (no chemical reaction) n 32
Extinction, ignition, parametric sensitivity of chemical reactors: Influence of feed mass flow rate on WSR operating temperature and space (volumetric heating rate(shr);(straight line is the LHS of the energy balance equation) 33
Extinction, ignition, parametric sensitivity of chemical reactors: Maximum volumetric rate of fuel consumption (hence, maximum chemical heating rate) occurs at WSR temperature: T ''' r& Tb max 1 + n( RT / E) Only slightly > extinction temperature (previous Figure) T b adiabatic, complete-combustion temperature Typical E, n values listed in following Table b 34
Extinction, ignition, parametric sensitivity of chemical reactors: a Supplemented, rounded (and selected) values based on Table 4.4 of Kanury (1975) b Units are: 10 14 s -1 (g-moles/cm 3 ) -(n-1 ), where n is the overall reaction order. c units are: 109 BTU/ft 3 /hr d Stoichiometric mixture, no diluent ( diluent is N 2 ) unless otherwise specified 35
Extinction, ignition, parametric sensitivity of chemical reactors: Black-box modular-models capture many important features of real reactors, useful for correlating performance data on full-scale & small-scale models Predictive ability limited compared to more-detailed pseudo-continuum mathematical models All have, as their basis, macroscopic conservation principles outlined earlier in this course. 36
PROBLEM1 The length requirement for a honeycomb-type automotive exhaust catalytic converter is set by the need to reduce the CO concentration in the exhaust to about 5% of the inlet concentration (i.e., 95% conversion). Consider the basic conditions: Inlet gas temperature 700K Inlet gas pressure 1 atm Inlet gas composition y(n 2 )=0.93, y(co)=0.02, (mole fraction) y(o 2 )=0.05 37
PROBLEM1 Inlet gas velocity 10 3 cm/s Channel cross-section dimensions 1.5mm by 1.5mm (each channel) Assumed channel wall temperature 500 K Assume that the Pt-based catalyst used on the walls of each channel is active enough to cause the surfacecatalyzed CO oxidation reaction to be diffusion-controlled, that is, the steady-state value of the CO-mass fraction established at (1 mean-free-path away from) 38
PROBLEM1 the wall, ω CO,w, is negligible compared to ω CO,b (z) within each channel. Also assume that the gas-phase kinetics of CO oxidation under these conditions preclude appreciable (uncatalyzed) homogeneous COassumption in the available residence times. Answer the following questions: a. By what mechanism is CO(g) mass transported to the channel wall, where chemical consumption (to produce CO 2 ) occurs? What is the relevant transport coefficient 39
PROBLEM1 and to what energy-transfer process and transport properly coefficient is this analogous? b. Are the mass-heat transfer analogy conditions (MAC, HAC) discussed in this module approximately met in this application? What is the inlet mass fraction of CO gas? Sc v D / CO mix c. Estimate the Schmidt number mix for CO Fick diffusion through the prevailing combustion gas mixture, using the experimental observation that 40
PROBLEM1 D CO N 2 1.73 2 0.216 T cm. p 300 s where p is the prevailing pressure (expressed in atmospheres) and T the mixture temperature (expressed in kelvins) d. Under the flow rate, temperature, and pressure conditions given above and using the mass-transfer analog, estimate the catalytic duct length 41
PROBLEM1 required to consume 95% of the inlet CO concentration, and the mixing cup (bulk) stream temperature at this length. e. List and defend the principal assumptions made in arriving at the length estimate (of Part (d)) f. If the catalyst were poisoned (e.g., by lead compounds), what could happen to the CO exit concentration? Which of the assumptions used in predicting the required converter length (Part (d)) would be violated? 42
PROBLEM1 g. If the heat of combustion of CO(g) is about 67.8 kcal/gmole CO consumed, calculate how much must be removed to maintain the channel-wall temperature constant at 500 K? h. Automatic operating conditions are never strictly steady, so that in practice the mass-flow rate, temperature, and gas composition entering the catalytic afterburner will be time-dependent. Under what circumstances (be 43
PROBLEM1 quantitative) can the design equations you used be defended if used to predict the conditions exiting the duct at each instant?(quasi-steady approximation) i. At the design condition, estimate the fractional pressure drop, p/ p 0, in the honeycomb-type catalytic afterburner. If, instead of the honeycomb type converter, a packed bed device were used to achieve the same reduction in CO-concentration, would you expect p/ p to be larger or smaller than the honeycomb device of your preliminary design? 0 44