Tanks in series model

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Tanks in series model

• We have already seen that multiple MFRs in series
  approach PFR behaviour 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)

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Fig. 6.3
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Fig. 6.5
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Fig. 14.1
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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.

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

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Box 3
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Figure 14.2
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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, E?i) 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.

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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?

Where ? is the mean and n is the number of
observations
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Fig.14.3
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One-shot tracer input

• In tracer studies, the input does not have to be
  an instantaneous spike. The input can be
  characterized by ??in2
• And the output by ??out2 (Fig. 14.4)
• The tanks in series model then says
• Where ?t is the time difference between the two
  peaks

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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.

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(Fig. E14.2)
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miles
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• 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

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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.

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Fig E14.3a
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Fig. E14.3b, 14.3c, 14.3d
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Example 14.4 (Fig E14.4a and 14.4b)

• Vessel E curve from ??in2 and ??out2
• Equations used for tanks in series model

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(Fig E14.4a and 14.4b)
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