Tanks In Series Model

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)

t = Nti

θi =

t ti

θ =

t t

θi = Nθ

Eθ = tE

dC dt which lead to :

0 − vC = V v

v − t E= e V V

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)

vC1 1….> −vC2 The second tank receives time varying input from tank

= V2

dC2 dt

The third tank receives time varying input from tank 2

vC 2 − vC 3 = V3

dC 3 dt

.

etc.

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

Observations on Fig.14.2 •

The Eθcurve for the entire assembly (left figure) starts resembling a PFR Eθ curve as N increases. I.e overall spread decreases.

•

The Eθ curves for the individual reactors (right figure, Eθi) 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 σ2, as well as Eθ

One-shot tracer input •

In tracer studies, the input does not have to be an instantaneous spike. The input can be characterized by σin2

•

And the output by: σout2 (Fig. 14.4)

•

The tanks in series model then says:

2 (∆ t ) 2 ∆σ 2 = σ out − σ in2 = N

Where ∆t is the time t M + N difference = t M + t N between the two peaks

σ M2 + N = σ M2 + σ N2 2 2 ∆σ 2 = σ OUT − σ IN =

(∆t ) 2 N

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 σ2is 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 ∆σ2 expression for the tanks-in-series model, we can find, for this stretch of river

Example 14.3 (Fig. E14.3a) From compartment models we know that multiple decaying peaks is a sign of (∆t ) 2 2 2 recirculation (Fig.12.1, p.285) ∆σ 2 = σ − σ = OUT IN

N

Analyzing Fig E14.3a, we arrive at a tanks (26) 2 in series model depicted in Fig. E14.3b, = (14) 2 − (10.5) 2 = 14.3c, N 14.3d.

N =8

2 ∆σ 2 = σout −σin2 =

E=

t N −1 t

Example 14.4 (Fig E14.4a and 14.4b) Vessel E curve from σin2 and σout2 Equations used for tanks in series model:……..>

N

( ∆t ) 2 N

NN e −t / t i ( N −1)!

By: Devender Arora

Biotech 3rd year Roll No.: 1229