Dispersed flow reactor response to spike input

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5 Dispersed flow reactor response to spike input Pe = c c t/t R

7 Fraction remaining Dispersed-flow reactor performance for k =.5/day Pe = (FMT) Pe = 1 Pe = 2 Pe = 1 Pe = (PFR) Residence time (days)

8 Dispersed-flow reactor performance for k =.5/day 1. Fraction remaining.1.1 Pe = (FMT) Pe = 1 Pe = 2 Pe = 1 Pe = (PFR) Residence time (days) 1

10 n= n= Dimensionless time, t/t R Dimensionless concentration, c/c

12 Tanks-in-series compared to dispersed flow reactor Dispersed flow Tanks-in-series

13 Tanks-in-series with exchange flow c c t/t R

15 Residence Time Distributions We have seen two extreme ideals: Plug Flow fluid particles pass through and leave reactor in same sequence in which they enter Stirred Tank Reactor fluid particles that enter the reactor are instantaneously mixed throughout the reactor Residence time distribution - RTD(t) represents the time different fractions of fluid actually spend in the reactor, i.e. the probability density function for residence time Q Inject slug of dye at inlet at t= V Q Measure dye concentration at outlet C(t) RTD (t) = (for steady flow) C(t)dt Note units - RTD is in inverse time by definition: RTD (t)dt = 1 (i.e., total probability = 1) t = t RTD(t) dt = first moment of RTD = tracer detention time D t R RTD 1 Plug flow RTD = δ(t-t R ) CSTR RTD = 1/t R exp(-t/t R ) t R 2t R t RTD math: Dirac delta function (or unit impulse function) 15

probability density function for residence time Q Inject slug of dye at inlet at t= V Q Measure dye concentration at outlet C(t) RTD (t) = (for steady flow) C(t)dt Note units - RTD is in inverse time

16 Represents a unit mass concentrated into infinitely small space resulting in an infinitely large concentration δ(t) = at t =, at t δ( t )dt = 1 Can think of Dirac delta function as extreme form of Gaussian M δ(t-τ) is spike of mass M at time τ Plug Flow RTD(t) = δ(t-t R ) with implied units of t -1 RTD (t)dt = δ(t tr )dt = 1 zeroth moment Note lower limit is and not - since you can t have negative residence time (i.e., fluid leaving before it entered) t = t RTD(t)dt = t δ(t t )dt = t first moment (mean) = D tracer detention time CFSTR RTD(t) = exp(-t/t R ) / t R units of t -1 RTD(t)dt = exp( t / t t R R R ) dt = t R R exp( t / t tr R ) = t R exp() = 1 tr t D = t exp( t / t = t RTD(t)dt = t t R[ exp() ( 1) ] = tr R R ) dt = 1 t R exp( t / t 2 1 tr R ) ( t / t 1) R Note: from CRC Tables: xe ax e dx = ( ax ) 2 a ax 1 Control Volume Models and Time Scales for Natural Systems What are actual systems like? Plug flow or Stirred reactor It depends upon the time scales: Mixing time for plug flow reactor is infinite: it never mixes Mixing time for stirred reactor is zero: it mixes instantaneously When are these assumptions realistic? We need to estimate the time of the real system to mix - t MIX compared to time to react If t MIX << t R stirred reactor If t MIX >> t R plug flow reactor 16

negative residence time (i.e., fluid leaving before it entered) t = t RTD(t)dt = t δ(t t )dt = t first moment (mean) = D tracer detention time CFSTR RTD(t) = exp(-t/t R ) / t R units of t -1 RTD(t)dt

17 Residence Time and Reactions RTD provides a means to estimate pollutant removal Consider a 1 st -order reaction: C(t) = C exp(-kt) This reaction applies to any water mass entering and exiting the system view from Lagrangian perspective (i.e., following the parcel of water) t 1 t 2 t 3 t 4 Exit concentration: C e = C exp(-kt 4 ) Consider a different parcel, taking a longer route: Exit concentration C e =C exp(-kt 6 ) where t 6 > t 4 t 1 t 2 t 3 t 4 t2 t 3 t 4 t 5 t 6 If a plug flow model applies, the exit concentration is simple: all parcels exit at exactly T R In a natural system, it is not perfect plug flow, therefore look at RTD RTD gives the probability that the fluid parcel requires a given amount of time to pass system On average: C = RTD(t) C exp( kt)dt e RTD At t 1 C e = C exp(-kt 1 ) At t 2 C e = C exp(-kt 2 ) t 1 t 2 t 17

view from Lagrangian perspective (i.e., following the parcel of water) t 1 t 2 t 3 t 4 Exit concentration: C e = C exp(-kt 4 ) Consider a different parcel, taking a longer route: Exit concentration C

18 Residence Time Distribution for Real Systems Q R C I At inlet Recirculation Q R C e At outlet Real circulation has: Short circuiting Dead zones (exclusion zones) RTD from tracer study plug flow or stirred tank reactor RTD t D t R t Detention time, T D t D = trtd (t)dt Note distinction with hydraulic residence time, t R = V/Q t D = t R if and only if there are no exclusion zones Variance of RTD is a measure of mixing σ 2 = ( t t D ) 2 RTD(t)dt As a dimensionless number, RTD σ d t = D 2 σ t As σ, no mixing, plug flow As σ, complete mixing, CFSTR 18

distinction with hydraulic residence time, t R = V/Q t D = t R if and only if there are no exclusion zones Variance of RTD is a measure

19 Residence Time Distribution for Real Systems Review some concepts: Two models for mixing Plug flow Stirred reactor Time scales: t R = V/Q mean hydraulic residence time (nominal residence time) t REACTION =1/k (or for 95% complete reaction or removal 3/k) t ADV = L/u Limitations of t R in describing residence times of true systems because of dead zones, recirculation, short circuiting Consider alteration of the real system: Add berms to control circulation! Lecture 4.doc 19

or removal 3/k) t ADV = L/u Limitations of t R in describing residence times of true systems because of dead zones,

23 Figure by MIT OCW. Adapted from: Camp, T. R. "Sedimentation and the design of settling tanks." Transactions ASCE 111 (1946):

3. Diffusion of an Instantaneous Point Source

3. Diffusion of an Instantaneous Point Source The equation of conservation of mass is also known as the transport equation, because it describes the transport of scalar species in a fluid systems. In this