CEE255A-L4-Non-Ideal Flow Models

CEE255A-L4-Non-Ideal - CEE 255A Lecture 4 Non-Ideal Flow& Reactor Models Dr Eric M.V Hoek Civil& Environmental Engineering General Reactor

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Unformatted text preview: CEE 255A Lecture 4 Non-Ideal Flow & Reactor Models Dr. Eric M.V. Hoek Civil & Environmental Engineering General Reactor Design Algorithm 1. Mole Balances (constant volume) PFR FA0dX/dV = -rA rA = kACA CBCC/K) MFR V = FA0(Xout Xin)/(-rA)out rA = kACA/(1 + KCA) Batch NA0dX/dt = -rAV rA = kA(CA 2. Rate Laws (first order, enzymatic, reversible,...) 3. Stoichiometry Flow (constant Q) CA= FA/Q0; FA = FA0(1 X) X) Batch (constant V) CA= NFA/V; NA = NA0(1 4. Combine and Integrate FA0dX/dV = rA = kACA = kAFA/Q0 = kAFA0(1 X)/Q0 Scale-Up Design of a MFR Mole Balance Rate Law Stoichiometry Combine (alternate) (n-CSTRs) V = FA0X/(-rA)exit -rA = kCA (1st order) CA= FA/Q0; CA = CA0(1 X) = (CA0 - CA)/kCA ; = V/Q0 X = 1 - 1 /(1 + k) X = 1 - 1 /(1 + k)n st Scale-Up Design of a PFR Mole Balance Rate Law Stoichiometry Combine (alternate) (alternate) (alternate) V = FA00 dX/(-rA)exit -rA = kCA (1st order) X CA= FA/Q0; CA = CA0(1 X) = -ln(1 - X)/k ; = V/Q0 = -ln(CA/CA0)/k X = 1 - exp(-k) CA/CA0 = exp(-k) MFR Performance Equations Benjamin & Lawler, Physicochemical Processes, Chapter 4, draft (2006). PFR Performance Equations Benjamin & Lawler, Physicochemical Processes, Chapter 4, draft (2006). Residence Time Distribution (RTD) The residence time is the "age" of a fluid element, or the time elapsed since it entered the reactor. Since fluid elements may take different "routes" through a reactor, they have different residence times. The residence time distribution (RTD) is a statistical distribution of residence times of all fluid elements leaving the reactor. The RTD is determined experimentally by injecting an inert chemical, molecule, or atom, called a tracer in the reactor at some time, t = 0, and measuring the changing concentration in the effluent stream with time. Tracer Curves for Ideal Reactors Accumulation = Inflow Outflow Degradation VdC/dt = 0 QC 0 C = C0e-t/ PFR MFR Residence Time Distribution (RTD) Conservative Tracer (dyes, salts, radioisotopes) soluble, non-reactive, non-adsorbing, detectable Reactor Pulse Input C-curves Step Input C-curves Residence Time Distribution Pulse Input Consider injection of pulse tracer for a single-input/output system in which only flow (i.e., no dispersion) carries the tracer material across system boundaries. As the time increment approaches zero, t 0, N = C(t)Q t ; N = amount of material leaving reactor over t N/N0 = C(t)Q t/N0 = fraction of material with RT b/w t & t+ t Normalizing by the amount injected at t = 0, N0 gives E(t): The "E-function" is the residence time distribution function (a.k.a., "age distribution function" or "exit age") For a pulse injection of tracer, let E(t) = C(t)Q/N0 (= N/N0 t) E(t) = fraction of exiting material with "age" b/w t and t + t Residence Time Distribution Pulse Input As 0 , E(t) = dN/N0dt, and since the volumetric flow rate, Q, is maintained constant and N0 = 0 QC(t)dt, we can define E(t) as, dN C(t)Qdt C(t)Q E(t) = = = = N 0 dt QC(t)dt dt Q 0 C(t)dt 0 C(t) @ 0 C(t) 0 C(t)dt C(t)Dt An alternative way of interpreting the RTD function is in its integral form: fraction of material leaving the reactor that has resided in the reactor for times between t1 and t2 = E (t )dt t1 t2 Also, the fraction of material that has left the reactor at time t = is given by E (t )dt = 1 0 Residence Time Distribution Pulse Input Example: Constructing the C(t) and E(t) curves. 0 0.0 0.000 I[C(t)dt] = I[C(t)dt] = I[C(t)dt] = 1 1.0 0.020 47.4 2.6 50.0 2 5.0 0.100 3 8.0 0.160 4 10.0 0.200 5 8.0 0.160 I[E(t)dt] = I[E(t)dt] = I[E(t)dt] = 6 6.0 0.120 0.95 0.05 1.00 7 4.0 0.080 8 3.0 0.060 9 2.2 0.044 10 1.5 0.030 12 0.6 0.012 14 0.0 0.000 t = 0 to t = 10 t = 10 to t = 14 t = 0 to t = 14 t = 0 to t = 10 t = 10 to t = 14 t = 0 to t = 14 t (min) C (g/m3) E(t) (min-1) 12 10 C (g/m 3) 8 6 4 2 0 0 1 2 3 4 5 6 7 t (min) 8 9 10 11 12 13 14 Integrals determined via trapezoid rule, Simpson's rule, or graphically... see text... Residence Time Distribution Pulse Input Example: Constructing the C(t) and E(t) curves. 0 0.0 0.000 I[C(t)dt] = I[C(t)dt] = I[C(t)dt] = 1 1.0 0.020 47.4 2.6 50.0 2 5.0 0.100 3 8.0 0.160 4 10.0 0.200 5 8.0 0.160 I[E(t)dt] = I[E(t)dt] = I[E(t)dt] = 6 6.0 0.120 0.95 0.05 1.00 7 4.0 0.080 8 3.0 0.060 9 2.2 0.044 10 1.5 0.030 12 0.6 0.012 14 0.0 0.000 t = 0 to t = 10 t = 10 to t = 14 t = 0 to t = 14 t = 0 to t = 10 t = 10 to t = 14 t = 0 to t = 14 t (min) C (g/m3) E(t) (min-1) 0.3 0.2 E (min -1 ) 0.2 0.1 0.1 0.0 0 1 2 3 4 5 6 7 t (min) 8 9 10 11 12 13 14 Integrals determined via trapezoid rule, Simpson's rule, or graphically... see text... Residence Time Distribution Step Input Consider a constant rate of tracer addition to a feed that is initiated at time t = 0. C0(t) = 0 ...t<0 C0 (constant) ... t 0 The concentration of tracer in the feed to the reactor is kept at this level until the concentration in the effluent is indistinguishable from that in the feed. Based on the known relationship between a "pdf" and "cumulative density function, cdf" we can say, C = C0 E (t )dt = C0 F (t ) 0 t Where F(t) is the fraction of effluent which has been in the reactor for less time than t. Residence Time Distribution Step Input Dividing by C0 yields: C = 0 E (t )dt = F (t ) C0 step d C (t ) E (t ) = dt C0 step t Differentiating this expression yields the RTD function: In general: fraction of material leaving the reactor that has resided in the reactor for less than time t = F (t ) = E (t )dt 0 t t m = tE (t )dt = 0 Giving as the definition for the "mean residence time," tm: is the first moment or mean of the RTD function, E(t) Residence Time Distribution Moments First Moment = "mean residence time" Average time fluid elements spend in the reactor... = tE (t )dt = 1 0 Second Moment = "variance or square of std. dev." An indication of the spread of the distribution... = ( t - ) E (t )dt 2 2 0 Third Moment = "skewness" An indication of how skewed a distribution is w.r.t. the mean... s = 3 1 3/ 2 ( t - ) 0 3 E (t )dt Residence Time Distribution Summary In real reactors, each fluid element (and the solutes contained therein) can spend a unique amount of time in the reactor due to non-idealities. Short-circuiting, dead zones (poor mixing), channeling, etc. Key parameters describing the distribution of residence times include: Mean, =f[E(t)]; Variance, 2=f[E(t)]; and Skewness, s3=f[E(t)] E(t) is the "normalized tracer concentration as a function of time in response to a pulse input." E(t)dt = 1 F(t) is the "normalized tracer concentration as a function of time in response to a step input." 0 F(t) 1 F(t) = E(t)dt = Ei t 0 0 Residence Time Distribution Summary The objective of a tracer study is to determine the mean residence time and the shape of the distribution. Mean, = tE(t)dt E(t) = C(t)/ C(t)dt = tC(t)dt/ C(t)dt = Ci / Ci t 0 0 0 0 = tiCi t / Ci t Variance, = (t )2E(t)dt = [(t )2C(t)/ C(t)dt] dt = [(ti )2Ci / Ci t] dt 2 0 0 0 5-Point Numerical Integration, A = h/3 (f1 + 4f2 + 2f3 + 4f4 + ... + 4fn-1 + fn) 2 Residence Time Distribution Example t (min) 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 12.0 14.0 C dt = C dt = C dt = a C (g/m3) 0.0 1.0 5.0 8.0 10.0 8.0 6.0 4.0 3.0 2.2 1.5 0.6 0.0 47.4 2.60 50.0 t = 0 to 10 t = 10 to 14 t = 0 to 14 E (t ) (min-1) 0.000 0.020 0.100 0.160 0.200 0.160 0.120 0.080 0.060 0.044 0.030 0.012 0.000 =tEdt = =tEdt = tE (t ) (-) 0.00 0.02 0.20 0.48 0.80 0.80 0.72 0.56 0.48 0.40 0.30 0.14 0.00 4.57 0.58 5.16 a t - (t - )2E (t ) (min) 0.00 0.35 0.99 0.74 0.27 0.00 0.09 0.27 0.49 0.65 0.70 0.56 0.00 4.14 1.97 6.11 (min) -5.16 -4.16 -3.16 -2.16 -1.16 -0.16 0.84 1.84 2.84 3.84 4.84 6.84 8.84 2 2 =(t - ) E (t )= 2 2 =(t - ) E (t )= 2 2 =(t - ) E (t )= =tEdt = This column is completed only after the mean residence time is found. Note: and 2 have units of "min" and "min2", respectively. Sources of Non-Ideal Flow Non-Ideal Reactors Mixing and/or resident time distribution in the reactor does not meet the ideal assumptions due to: Short Circuiting Channeling Stagnant Zones Dispersion Examples Residence Time Distribution Examples Reactor Modeling with the RTD In most cases, the fluid in a reactor is neither well-mixed, nor does it approximate plug flow. Conversion for a non-ideal reactor can not be calculated from the performance equations for CSTR and PFR. For 1st order reactions, performance of non-ideal reactors can be calculated directly from the measured E-function. How can we use the RTD to predict conversion in a real reactor? 1. Zero Parameter Reactor Models primary use = CSTR Complete Segregation and Maximum Mixedness Tanks in Series and Dispersion 2. One Parameter Reactor Models primary use = PFR/PBR 3. Two Parameter Reactor Models more complex models Reactor Modeling with the RTD RTD = how long the fluid elements are in the reactor, but it tells us nothing about the exchange of matter between the elements, i.e., the "mixing". 1st Order knowledge of the length of time each molecule spends in the reactor is all that is needed to predict conversion. Recall, dX/dt = k (1 X) Conversion is independent of concentration; hence, mixing between surrounding molecules is not important. Therefore, once the RTD is known, we can predict the conversion that will be achieved in a real reactor provided we know the rate constant. For reactions other than 1st Order, knowledge of the RTD is not sufficient to predict conversion and the degree of mixing of molecules must be known in addition to how long each molecule spends in the reactor. We must develop models that account for the mixing of molecules inside the reactor ... macromixing and micromixing Reactor Modeling with the RTD Macromixing Produces a distribution of residence times without specifying how molecules of different ages encounter one another. Describes how molecules of different ages encounter one another in the reactor. Micromixing Reactor Modeling with the RTD Macromixing Produces a distribution of residence times without specifying how molecules of different ages encounter one another. Describes how molecules of different ages encounter one another in the reactor. Two extremes exist: Micromixing Complete Segregation: All molecules of an age group remain together and are not mixed until they exit the reactor. Maximum Mixedness: Molecules of different age groups are completely mixed at the molecular level as soon as they enter the reactor. The two extremes of micromixing provide upper and lower limits on the conversion in a non-ideal reactor. r = kCn, n < 1: maximum mixedness highest conversion r = kCn, n = 1: conversion is independent of micromixing *** r = kCn, n > 1: complete segregation highest conversion Reactor Modeling with the RTD Complete Segregation Assumption: each fluid element passes through the reactor without intermixing with adjacent fluid elements. = concentration of reactant remaining + in an element of age between t and t + dt fraction of exit stream of age between t and t + dt mean concentration of reactant in the exit stream C = Celement Edt 0 C = C0 exp(- kt ) E (t )dt C0 exp(- kt ) Ei t 0 i Each fluid element can be treated as a batch reactor because there is no intermixing or flow in/flow out of the element. 1st Order Batch C(t) = C0e-kt 1st Order Batch X(t) = 1 e-kt X = 1 - X (t ) E (t )dt 1 - X i Ei t 0 i Reactor Modeling with the RTD Find the conversion of a contaminant in a real reactor, and compare with a PFR and MFR of the same size (i.e., ). Assume first order and kA = 0.307 min-1 Assume RTD and E-function from previous example apply Assume segregation model applies... 1st Order in PFR: XPFR = 1 exp(k) XREAL = 1 XiEi t Xi = 1 exp(-kti) Solution: 1st Order in Real Reactor: 1st Order in MFR: XMFR = k /(1+ k) Reactor Modeling with the RTD Given: t (min) 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 12.0 14.0 C (g/m3) 0.0 1.0 5.0 8.0 10.0 8.0 6.0 4.0 3.0 2.2 1.5 0.6 0.0 k = = E (t ) (min-1) 0.000 0.020 0.100 0.160 0.200 0.160 0.120 0.080 0.060 0.044 0.030 0.012 0.000 t = 0 to 10 t = 10 to 14 t = 0 to 14 min-1 min X (t )E (t ) (min-1) (-) 0.000 0.000 0.264 0.005 0.459 0.046 0.602 0.096 0.707 0.141 0.785 0.125 0.841 0.101 0.883 0.071 0.914 0.055 0.937 0.041 0.954 0.029 0.975 0.012 0.986 0.000 X real = 0.690 X real = 0.050 0.740 X real = 0.307 5.16 X (t ) X PFR = X REAL = X CSTR = 0.795 0.740 0.613 X = 1i X i E i Dt Reactor Modeling with the RTD Modeling Non-Ideal Flow in PF and PB Reactors Model characterizes non-ideal flow (mixing) within reactor Model parameters determined from experimental data by fitting Dispersed Flow Model model parameter is Per Tanks in Series Model model parameter is n Axial dispersion of reactants (described by analogy with Fick's law) is superimposed on the flow through the reactor. Most common models: Dispersed Flow Model Reactor Modeling with the RTD Dispersed Flow Model (cont'd) Molar Flow Rate of Tracer, FT by convection and dispersion: FT = JTAc = -DaAc CT/ x + UAcCT Da = effective dispersion coefficient (we can determine this value) U = superficial velocity (for PBR we use U/; = bed porosity) Mole Balance on the Tracer Ac CT/ t = - FT / x = - [-DaAc CT/ x + UAcCT]/ x Dividing through by Ac and re-arranging yields: CT/ t = Da 2 CT/ x2 U CT/ x Putting the equation in dimensionless form by assuming: C* = C/C0; = Ut/L; x* = x/L ...produces: C*/ = (Da/UL) 2 C*/ x2 - C*/ x* Reactor Modeling with the RTD Dispersed Flow Model (cont'd) By imposing certain boundary conditions, the dispersion equation can be solved. - (1 - ) 2 C* = exp 3 4 / Per 2 / Per 1 2 t m = 1 + Pe r 2 2 8 = + 2 t m Per Per 2 For Pe ...convection dominated ... PFR (a.k.a., Da/UL 0 ... no mixing) For Pe 0 ...dispersion dominated ... CSTR (a.k.a., Da/UL ... well-mixed) Conversion in a reactor with dispersion : 4 exp( Per / 2) X = 1- (1 + ) 2 exp( Per / 2) - (1 - ) 2 exp(- Per / 2) Reactor Modeling with the RTD Reactor Modeling with the RTD Tanks in Series Model Model considers n-equal sized MFRs in series. n and XREAL can be determined from a PULSE INPUT by fitting experimental data to the model equations: n -1 t t E (t ) = exp - ; setting = t/ and i = / n produces : n (n - 1)! i i n(n) n -1 E ( ) = exp( - n ) (n - 1)! 1 2 n= 2 = 2 X REAL = 1 - 1 ; i = V / nQ n (1 + k i ) If n 1, the reactor approaches MFR performance If n , the reactor approaches PFR performance Reactor Modeling with the RTD Tanks in Series Model Model considers n-equal sized MFRs in series. n and XREAL can be determined from a PULSE INPUT by fitting experimental data to the model equations: n -1 t t E (t ) = exp - ; setting = t/ and i = / n produces : n (n - 1)! i i n(n) n -1 E ( ) = exp( - n ) (n - 1)! 1 2 n= 2 = 2 X REAL = 1 - 1 ; i = V / nQ n (1 + k i ) If n 1, the reactor approaches MFR performance If n , the reactor approaches PFR performance Reactor Modeling with the RTD Reactor Modeling with the RTD MFR PFR ...
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This note was uploaded on 02/02/2012 for the course CEE 255A taught by Professor Erichoek during the Fall '11 term at UCLA.

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