Non-Ideal Reactors, Residence Time Distribution & Dispersion Model

Question 1

Define space time and write the equation

Answer 1

The time needed to fill one reactor volume (macromixing)

tao = V/v0

Question 2

Define residence time

Answer 2

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

Question 3

State the assumptions of the residence time distribution

Answer 3

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

Question 4

Why do we use the residence time distribution?

Answer 4

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

Question 5

What is the pulse experiment?

Answer 5

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.

Question 6

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

Answer 6

The exit age distribution (1/time).

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

Question 7

Write the general equation for the exit age distribution

Answer 7

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

Question 8

Write the exit age distribution formulae in its discrete form

Answer 8

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

Question 9

What is F (t)?

Answer 9

The cumulative distribution function.

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

Question 10

Write the general equation for the cumulative distribution function

Answer 10

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

Question 11

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

Answer 11

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

Question 12

Write the cumulative distribution function formulae in its discrete form

Answer 12

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)

Question 13

Write the equation for the area under the C-curve

Answer 13

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

Question 14

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

Answer 14

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 ]

Question 15

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

Answer 15

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

Question 16

Write the equation for the area under the E-curve

Answer 16

A = N0/v
A = M/v

Question 17

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

Answer 17

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

Question 18

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

Answer 18

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.

Question 19

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

Answer 19

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

Question 20

How do you diagnose reactor problems using the compartment model?

Answer 20

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:

1. mean res time (t) = V/v then the reactor has no dead spaces
2. 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

Question 21

Draw the curves for the following diagnoses in a PFR:

1. Stagnant backwaters
2. Parallel paths/channelling
3. Good flow
4. Internal recirculation
5. Late curve

Answer 21

See W9 notes

Question 22

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

Answer 22

1. v or V has been measured incorrectly
2. The tracer used is not inert
3. The vessel is not closed

Question 23

Draw the curves for the following diagnoses in a CSTR:

1. Stagnant fluud
2. Time lag
3. Good flow
4. Internal recirculation
5. Late curve
6. Shortcircuiting

Answer 23

See W9 notes

Question 24

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

Answer 24

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

Question 25

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

Answer 25

The measure of the degree of backmixing/spreading. (m^2 / s)

• Large D = rapid spreading
• Small D = Slow spreading

Question 26

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

Answer 26

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

Question 27

Write the equation for the vessel dispersion number

Answer 27

VDN = D/uL = variance^2 (theta) /2

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

Question 28

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

Answer 28

VDN = D/uL = variance^2 (theta) /2

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

Question 29

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

Answer 29

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)]

Question 30

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

Answer 30

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

Question 31

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

Answer 31

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)

Question 32

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?

Answer 32

t = V/v

V = L pi r^2 BV

v = velocity pi r^2

t = (L BV)/(velocity)

Question 33

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

Answer 33

close-close

laminar-laminar

Question 34

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

Answer 34

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

Question 35

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

Answer 35

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

Question 36

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

Answer 36

False

‘open’ vessels have a larger deviation

Question 37

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

Answer 37

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