Dispersion model the dispersed plug flow model

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Dispersion model (the dispersed plug flow model)

• Fig 13.1 and 13.3 show the basic features of the
  scenario.
• The spreading of the tracer curve is related to
  the dimensionless vessel dispersion number
• D dispersion coefficient, (m2/s)
• u average fluid velocity, m/s
• L length of system, m

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• We characterize the Cpulse curve by the mean
  residence time and the variance

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Fig. 13.1 and 13.3
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Dispersion model

• You will study the advection-dispersion equation
  (5 and 6) in detail in the contaminant transport
  course next term.

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Dispersion model, D/uLlt0.01

• The solution in non-dimensional form gives the E?
  vs ? curve (Fig. 13.4) similar to the E? vs ?
  curve for the tanks in series model (Fig. 14.3)
• Box 8 equations give the relevant relationships.
  Note the limitation of D/uLlt0.01 for the
  solution.

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Fig. 13.4
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Box 8
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Determination of the vessel dispersion number

• Compare experimental E? vs ? curve with the
  solution in Fig 13.4 to get the D/uL parameter by
  one of the following methods
• by calculating its variance
• by measuring its maximum height
• by measuring its width at the point of inflection
• by finding the width which includes 68 of the
  area

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• Additivity for vessels in series.

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One-shot tracer input (arbitrary shape)

• The additivity of variances enables us to use
  tracer inputs of arbitrary shape and use the
  difference in variance to quantify the
  dispersion Fig. 13.6
• For small extents of dispersion (D/uL lt 0.01) we
  can also use
• However, if the tracer curve shapes are very
  different from symmetrical and D/uLgt0.01 the
  models basic assumptions begin to fail. Using
  an alternate model (tanks in series?) may be a
  better idea.

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Figure 13.6
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Dispersion model, D/uLgt0.01

• For large D/uL values we have two sets of
  solutions, for open and closed boundary
  conditions. (Fig. 13.7)
• Fig. 13.8 and Box 13 for closed vessel boundary
  conditions.
• Fig. 13.10 and Box 14-15 for open vessel boundary
  conditions.

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Figure 13.7
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Fig. 13.8
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Box 13
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Fig. 13.10
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Box 14-15
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Example 13.1

• Based on previous example 11.1 data

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Correlations for Dispersion number
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• The Schmidt number captures the fluid properties

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Correlations for Dispersion number

• These would help predict the dispersion number,
  as opposed to running an experiment on a full
  scale system.

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Effect of dispersion on conversion

• Recall Examples 11.1 and 11.4
• Example 11.1 Residence time distribution in PFR
• Example 11.4 Impact of the RTD on conversion
• We will now quantify the effect of dispersion on
  conversion using the dispersed plug flow model

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RTD of Example 11.1
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Example 11.4 First order reaction in non-ideal PFR

• Cpulse vs time data from Example 11.1
• -rA(0.307 min-1)CA
• Ideal PFR gives CA/CA0 0.01, i.e. XA0.99
• The reactor with E from Example 11.1 gives
• CA/CA0 0.0469, i.e. XA0.9531

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RTD and dispersion number

• In examples 11.1 and 11.4 the nonideality of flow
  was quantified by the E vs t curve
• The dispersed plug flow model uses D/uL
  (dispersion number) as the parameter to quantify
  non-ideality.
• Figures 13.19 and 13.20 relate the volume of a
  reactor with given dispersion number to the
  volume of an ideal PFR

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Figures 13.9 and 13.20

• These are similar in function to Figures 6.16 and
  6.17 which quantified the effect of recycle on
  the performance of a PFR.
• The performance of the PFR deteriorates with
  increasing
• Recycle ratio
• or
• D/uL

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Example 11.4 with dispersion model

• Refer to Excel worksheet nonideal_flow.xls

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