A non-ideal flow model is used when
Answer: A
Real vessels always lie somewhere between the two ideals, and the model quantifies where. It is fitted to a measured residence time distribution rather than derived from geometry. Every real vessel lies between the two ideals.
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Match each non-ideal flow model with its adjustable parameter.
List I
A. Tanks in series
B. Axial dispersion
C. Compartment model
D. Segregation model
List II
1. the fraction of dead and active volume
2. the number of equivalent tanks
3. no parameter beyond the distribution itself
4. the dispersion or Peclet number
Answer: B
Each is a one parameter description fitted to the measured curve, and the choice between them is largely one of convenience. The dispersion model suits tubes while tanks in series suits vessels. Each carries a single adjustable number.
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The axial dispersion model represents deviation from plug flow as
Answer: C
The dispersion coefficient plays the role of a diffusivity and spreads the tracer pulse. It lumps together molecular diffusion, velocity profile effects and turbulent mixing into one parameter. The coefficient is fitted rather than looked up.
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Assertion (A): The dispersion coefficient in the axial dispersion model is not a molecular diffusivity.
Reason (R): It lumps together velocity profile effects, turbulence and molecular diffusion into a single empirical quantity.
Answer: D
Both are true and the reason gives its character: it is fitted to the tracer response rather than looked up, and its value can exceed the molecular diffusivity by many orders of magnitude. It is an effective quantity, not a physical one.
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The vessel dispersion number is defined as
Answer: A
It is dimensionless and is the reciprocal of the Peclet number for the vessel. A value near zero means plug flow and a large value means intense back mixing. It is dimensionless by construction. Zero and infinity are the two ideal limits.
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Consider the following statements about the dispersion number.
1. A value approaching zero corresponds to plug flow.
2. A large value corresponds to a well mixed vessel.
3. It is the same quantity as the Reynolds number.
Answer: B
The first two are correct and set the two limits. The third is quite wrong, since the Reynolds number describes the flow regime while the dispersion number describes the mixing that results from it. The two limits bracket every real vessel.
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Assertion (A): The dispersion number and the Peclet number carry the same information.
Reason (R): One is the reciprocal of the other.
Answer: C
Both are true and the reason gives the relation, though care is needed because the Peclet number may be defined on the vessel length or on the particle diameter and the two differ by a large factor. Which definition is meant should always be stated.
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All of the following increase axial dispersion in a tube EXCEPT
Answer: D
A long tube in turbulent flow is the closest practical approach to plug flow. The other three all spread the residence times and move the vessel away from the ideal. Turbulence and length both help. Laminar flow is the worst case of the four.
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A tracer test gives a variance of 4 and a mean residence time of 10 minutes. What is the equivalent number of tanks in series?
Answer: A
Correct answer: 25
For the tanks in series model the number of tanks is the square of the mean divided by the variance: \[ N = \frac{\tau^{2}}{\sigma^{2}} = \frac{10^{2}}{4} = 25. \] A large \( N \) means a narrow distribution and behaviour close to plug flow, so this vessel deviates only slightly from it. Dividing the mean by the variance rather than its square gives the 2.5 offered.
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Assertion (A): The tanks in series number is obtained from the mean and variance of the tracer response.
Reason (R): For a cascade of equal tanks the variance is the square of the mean divided by the number.
Answer: B
Both are true and the reason gives the formula, and inverting it is exactly how a measured curve is turned into a model parameter. A fractional value is perfectly acceptable, since the number is a fitted description rather than a count of vessels.
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