Tanks in series model

We have already seen that multiple MFRs in series approach PFR behavior as the number of MFRs increases. (Fig.6.3 & 6.5)

Conversely, we can think of a non-ideal PFR as a series of MFRs and develop quantitative analysis of the non-ideality as characterized by E curves (Fig.14.1)

EtENt

t

t

ttNt i

i

ii

Tracer balance on first tank

Recall, generally for MFR:

input – output = accumulation (no reaction term for tracer)

Assuming instantaneous addition of tracer pulse, no more input after time 0.

Tracer balance on subsequent tanks

input – output = accumulation (no reaction term for tracer)

The second tank receives time varying input from tank 1….>

The third tank receives time varying input from tank 2

. dt

dCVvCvC 3

332 etc.

The solutions to this set of equations are summarized in Box 3 and Fig.14.2

0−vC=VdCdt

which lead to:

E=vVe−vVt

vC1−vC2=V 2

dC2dt

n

x

2

2 Where is the

mean and n is the number of observations

Observations on Fig.14.2

• The E curve for the entire assembly (left figure) starts resembling a PFR E curve as N increases.

I.e overall spread decreases.

• The E curves for the individual reactors (right figure, Ei) get flatter (spread increases) as we move away from the feed end.

• Note however, that the spread for the individual tanks are measured relative to the individual mean residence times whereas the spread for the system as a whole is measured relative to the system mean residence time.

RTD for the tanks in series model (Fig.14.3)

• The spread or flatness of a distribution can be quantified by the variance:

• Fig 14.3 shows the relation between N and 2,

as well as E

One-shot tracer input

• In tracer studies, the input does not have to be an instantaneous spike.The input can be characterized by in

2

• And the output by: out2 (Fig. 14.4)

• The tanks in series model then says:

N

tinout

2222 )(

Where t is the time difference between the two peaks

Example 14.2 (Fig. E14.2)

Estimating the location of a spill in a river from the difference of spread at two downstream observation points.

• Over 119 miles the spread increases from 10.5 hr to 14 hr

• By considering that 2 is proportional to distance we can deduce that an instantaneous spill (pulse input) could have occurred 272 miles upstream, or, a sloppy input could have occurred closer.

Using the fact that the peak at Cincinnati occurred 26 hours after the peak at Portsmouth, and the 2 expression for the tanks-in-series model, we can find, for this stretch of river

Δσ 2=σOUT2 −σ IN

2 =(Δ t̄ )2

N

=(14 )2−(10 .5)2=(26 )2

NN=8

Example 14.3 (Fig. E14.3a)

From compartment models we know that multiple decaying peaks is a sign of recirculation (Fig.12.1, p.285)

Analyzing Fig E14.3a, we arrive at a tanks in series model depicted in Fig. E14.3b, 14.3c, 14.3d.

Example 14.4 (Fig E14.4a and 14.4b)

Vessel E curve from in2 and out

2

Equations used for tanks in series model:……..>

By: Devender Arora

Biotech 3rd year

Roll No.: 1229