Non-Ideal Reactors, Residence Time Distribution & Dispersion Model Flashcards by Ellie J Bibby

Define space time and write the equation

The time needed to fill one reactor volume (macromixing)

tao = V/v0

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Define residence time

The amount of time molecules spend in the reactor (micromixing)

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State the assumptions of the residence time distribution

Steady state
Without reaction
The fluid is incompressible so no density change

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Why do we use the residence time distribution?

It allows the detection of non-ideal flows such as bypassing and short-circuiting by understanding the routes the fluid takes

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What is the pulse experiment?

An experiment where a pulse input where N0 moles of an inert tracer is injected at t=0 and the concentration at the outlet is analysed. The vessel must be closed.

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What is E(t)? What are the units?

The exit age distribution (1/time).

This shows the fraction of all the material leaving between a given time interval

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Write the general equation for the exit age distribution

E (t) = C (t)/integral [C (t) dt]

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Write the exit age distribution formulae in its discrete form

E (t) = C / sum [C delta (t)] = C / A
OR if the amount of tracer inputted is known:
E (t) = C v / N0

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What is F (t)?

The cumulative distribution function.

This shows the fraction of fluid that leaves the reactor with an age less than a given value of t.

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Write the general equation for the cumulative distribution function

F (t) = integral [E (t) dt]

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How do you calculate the fraction of fluid that leaves the reactor with an age greater than a given value of t?

1 - F (t) = 1 - integral [E (t) dt]

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Write the cumulative distribution function formulae in its discrete form

To find the F value for 1 specific time use:
F (t2) = F (t1) + (E (t1) + E (t2)/2) x delta (t2 - t1)

To find the fraction of fluid between 2 times (eg. 4 and 8 given in 2 min intervals):
F (t) = [ (E (4) + E (6))/2 ] x (6 - 4) + [(E (6) + E (8))/2 ] x (8 - 6)

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Write the equation for the area under the C-curve

A = integral [ C(t) dt ] (infinity, 0)
A = sum [ Ci delta ti ] 
A = N0/v

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Write the equation for the mean residence time of the C-curve

t = integral [ t E (t) dt ] (infinity, 0)

substituting in for E (t)

t = integral [ t C (t) dt ] (infinity, 0) / integral [ C (t) dt ] (infinity, 0)

t = sum [ ti Ci] / sum [ Ci ]

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What is the difference between the C-curve and the E-curve?

The E-curve is the normalised version of the C-curve which neglects the amount of tracer and the volumetric flow rate. This allows two sets of experimental data to be more accurately compared

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Write the equation for the area under the E-curve

A = N0/v
A = M/v

Write the equation for the mean residence time of the E-curve

t = integral [ t E (t) dt ] (infinity, 0)

Why would we use the E (theta) - curve? Define the units and necessary equations

To remove the effect of reactor volume

theta = ti/mean res time (t)

E (theta) = mean res time (t) x E

The units for E (theta) are dimensionless.

Write the equation for the variance of the C-curve or E-curve?

variance = integral [ ti^2 E (t) dt ] (infinity, 0) - (mean res time (t))^2

How do you diagnose reactor problems using the compartment model?

If reactor volume and volumetric flow rate is known we can use the compartment model.

Calculate V/v and compare this to the mean residence time observed from the data, if:

mean res time (t) = V/v then the reactor has no dead spaces
mean res time (t) ≠ V/v the reactor has dead spaces and the volume of dead space (Vd) can be calculated by:
Vp = mean res time (t) x v
Vd = V - Vp

If there is a bypass, the active volumetric flow rate (va) will be LESS than the total volumetric flow rate. To calculate the bypass volumetric flow (vb) use:
v = va + vb

Draw the curves for the following diagnoses in a PFR:

Stagnant backwaters
Parallel paths/channelling
Good flow
Internal recirculation
Late curve

See W9 notes

What are the 2 potential reasons for the late curve diagnosis in a PFR?

v or V has been measured incorrectly
The tracer used is not inert
The vessel is not closed

Draw the curves for the following diagnoses in a CSTR:

Stagnant fluud
Time lag
Good flow
Internal recirculation
Late curve
Shortcircuiting

See W9 notes

Outline the method for calculating a new conversion of reactant once a reactor diagnosis is improved

Write the design equation
Replace volume, V with Va (active volume) so that:
k Va/v = ( CA0 / (1 - XA) CA0 ) - 1
Calculate the RHS of the above equation (this is the ration for V/Va)
Equate k V/v = n k Va/v so that it becomes:
= n x (V/v) = number
The number calculated in 4 = (CA0/CA2) - 1 so rearrange to find CA2/CA0
The new conversion = 1 - the number in 5.

What does the dispersion coefficient represent? What are the units?

The measure of the degree of backmixing/spreading. (m^2 / s) - Large D = rapid spreading - Small D = Slow spreading

What does the dispersion coefficient represent? What are the units?

The measure of the degree of backmixing/spreading. (m^2 / s) - Large D = rapid spreading = CSTR = Broad RTD - Small D = Slow spreading = PFR = Narrow RTD

Write the equation for the vessel dispersion number

VDN = D/uL = variance^2 (theta) /2 Where: variance^2 (theta) = variance^2 / (mean res time)^2

Write the equation for the vessel dispersion number when the vessel dispersion number is small?

VDN = D/uL = variance^2 (theta) /2 Where: variance^2 (theta) the normalised variance = variance^2 / (mean res time)^2

How do you calculate the normalised E-curve for the dispersion model and find the maximum when the vessel dispersion number is small?

E (theta) = mean res time (t) x E The maximum occurs when theta = 1, therefore: E (theta) max = 1/sqrt [4 pi (D/u L)]

How can the vessel dispersion number be calculated from the normal distribution for small extents of dispersion (VDN < 0.01)?

1. At the point of inflection (0.61 E (theta) max) where the width = 2 sqrt [2 (D/u L)] 2. The width which includes 68% of the area where the shaded area = 0.68

What is the additivity of variances when the vessel dispersion number is small? When is it used?

The increase in variance will always be the same therefore, the difference in variance can be used to calculate the vessel dispersion number in systems that have a small extent of dispersion. delta variance ^2 = variance^2 (2) - variance^2 (1)

How can the mean residence time be found for a packed bed when only bed voidage (BV), length and velocity of fluid is given when the vessel dispersion number is small?

t = V/v V = L pi r^2 BV v = velocity pi r^2 t = (L BV)/(velocity)

What boundary condition gives a tracer curve that is identical to the E function?

close-close | laminar-laminar

When the vessel dispersion number is large and the boundary condition is closed-closed, how do you calculate the normalised variance?

variance^2 (theta) = variance^2 / (mean res time)^2 = 2 (D/u L) = 2 (D/u L)^2 (1 - e^(u L/D))

When the vessel dispersion number is large and the boundary condition is open-open, how do you calculate the normalised variance?

variance^2 (theta) = variance^2 / (mean res time)^2 = 2 (D/u L) + 8 (D/u L)^2

True or false: Tracer response curves for 'closed' vessels have larger deviations from plug flow

False 'open' vessels have a larger deviation

How do you know whether a dispersion is large or small by looking at the E curve?

If the curve is unsymmetrical and broad then it is likely to be large