Pe = ∞

0cc

t/tR

Dispersed flow reactorresponse to spike input

Dispersed-flow reactor performance for k = 0.5/day

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Frac

tion

rem

aini

ng

Pe = 0 (FMT) Pe = 1 Pe = 2 Pe = 10 Pe = ∞ (PFR)

0 2 4 6 8 10

Residence time (days)

Dispersed-flow reactor performance for k = 0.5/day

0.001

0.010

0.100

1.000

0 2 4 6 8 Residence time (days)

Frac

tion

rem

aini

ng

Pe = 0 (FMT) Pe = 1 Pe = 2 Pe = 10 Pe = ∞ (PFR)

10

0

0.5

1

1.5

2

2.5

3

0 0.5 1 1.5 2Dimensionless time, t/tR

Dim

ensi

onle

ss c

once

ntra

tion,

c/c

0

n=1

2

5

10

20

40n=∞

Tanks-in-series compared to dispersed flow reactor

Dispersed flow Tanks-in-series

Tanks-in-series with exchange flow

t/tR

0cc

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

∫∞

=

0

dt)t(C

)t(C)t(RTD (for steady flow)

Note units - RTD is in inverse time

by definition: ∫∞

=0

1dt)t(RTD (i.e., total probability = 1)

∫∞

=0

D dt)t(RTDtt = first moment of RTD = tracer detention time

RTD math: Dirac delta function (or unit impulse function)

V Q Q

Measure dye concentration at outlet

Inject slug of dye at inlet at t=0

tR•RTD

tR t

CSTR RTD = 1/tR exp(-t/tR)

Plug flow RTD = δ(t-tR)

1

0.14

0.38

2tR

16

Represents a unit mass concentrated into infinitely small space resulting in an infinitely large concentration

δ(t) = ∞ at t = 0, 0 at t ≠ 0

∫∞

∞−

=δ 1dt)t(

Can think of Dirac delta function as extreme form of Gaussian M0δ(t-τ) is spike of mass M0 at time τ

Plug Flow RTD(t) = δ(t-tR) with implied units of t-1

∫ ∫∞ ∞

=−δ=0 0

R 1dt)tt(dt)t(RTD zeroth moment

Note lower limit is 0 and not -∞ since you can’t have negative residence time (i.e., fluid leaving before it entered)

∫ ∫∞ ∞

=−δ==0

R0

RD tdt)tt(tdt)t(RTDtt first moment (mean) =

tracer detention time CFSTR

RTD(t) = exp(-t/tR) / tR units of t-1

1t

)0exp(0tt

)t/texp(tdtt

)t/texp(dt)t(RTDR

R0R

RR

0 R

R

0

=⎥⎦

⎤⎢⎣

⎡−−=⎥

⎦

⎤⎢⎣

⎡ −−=

−=

∞∞∞

∫∫

( )

( )[ ] RR

0

R2R

R

R0 R

R

0D

t10)0exp(0t

1t/tt1

)t/texp(t1dt

t)t/texp(tdt)t(RTDtt

=−−−=

⎥⎦

⎤⎢⎣

⎡−−

−=

−==

∞∞∞

∫∫

Note: from CRC Tables: ∫ −= )ax(aedxxe

axax 1

2

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 - tMIX

compared to time to react If tMIX << tR → stirred reactor If tMIX >> tR → plug flow reactor

17

Residence Time and Reactions

RTD provides a means to estimate pollutant removal Consider a 1st-order reaction: C(t) = C0 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)

Exit concentration: Ce = C0 exp(-kt4)

Consider a different parcel, taking a longer route: Exit concentration Ce=C0 exp(-kt6) where t6 > t4

If a plug flow model applies, the exit concentration is simple: all parcels exit at exactly TR

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:

dt)ktexp(C)t(RTDC0

0e −= ∫∞

At t1 Ce = C0 exp(-kt1) At t2 Ce = C0 exp(-kt2)

t1

t2

t3t4

RTD

t t1 t2

t1

t2t3

t4t2t3 t4

t5t6

18

Residence Time Distribution for Real Systems

Real circulation has: Short circuiting Dead zones (exclusion zones)

RTD from tracer study ≠ plug flow or stirred tank reactor

Detention time, TD

tD = ∫∞

0

dt)t(tRTD

Note distinction with hydraulic residence time, tR = V/Q tD = tR if and only if there are no exclusion zones

Variance of RTD is a measure of mixing

∫∞

−=σ0

2D

2 dt)t(RTD)tt(

As a dimensionless number, 2

Dtd ⎟⎟

⎠

⎞⎜⎜⎝

⎛ σ=

As σ → 0, no mixing, plug flow As σ → ∞, complete mixing, CFSTR

QR CI At inlet

QR Ce At outlet

Recirculation

RTD

ttD tR

RTD

t σ

19

Residence Time Distribution for Real Systems

Review some concepts: Two models for mixing

Plug flow Stirred reactor

Time scales: tR = V/Q mean hydraulic residence time (nominal residence time) tREACTION =1/k (or for 95% complete reaction or removal 3/k) tADV = L/u

Limitations of tR 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

Figure by MIT OCW.

Adapted from: Camp, T. R. "Sedimentation and the design of settling tanks."Transactions ASCE 111 (1946): 895-936.