Question

1. A CSTR and a PFR of equal volume (VCSTR = VPFR = V) are used to degrade a compound in a first-order manner under steady-state conditions.

(a) If the CSTR gives a 80% degradation what is the efficiency of the PFR?

(b) If instead of one CSTR of volume V, you use 5 CSTRs in series, each having a volume of 0.2V, what would be the overall degradation efficiency?

(c) Show that the overall efficiency will be equal to that of the PFR (volume V) if you use an infinite number (n→∞) of CSTRs in series each having a volume of V/n.

Answer 1

a)For a CSTR, the Characteristic equation for a first order is

KT= XA/(-1XA)

Where T = Volume of reactor/ Volumetric flow rate

K= Rate constant

XA= percent conversion

KT= 0.8/0.2= 4

For a PFR KT= -ln(1-XA) , 4= -ln(1-XA), ln(1-XA)= -4 , 1-XA= exp(-4) , XA =0.9817

Let CO= inlet concentration of the ist reactor , C1, C2, C3, C4,C5= outlet concentrations from each of the five reactors

For first reactor of volume 0.2 V,

T1= space time

KT=CO/C1- 1, CO/C1= _{1+ KT1}

C1= CO/(1+KT1) C2= CO/ (1+KT1)² , C₃= C_O/ (1+KT1)³ and C₅= C_O/ (1+KT1)⁵

But given KT= 4 and T= 5T1  

T1= T/5

And KT1= KT/5 =0.8

C5= CO/ (1+0.8)⁵ = CO*0.05292

1-C5/CO= conversion = 1-0.05292 =0.947

When there are n reactors in series

C_n/CO= 1/(1+KT1)ⁿ

T1= T/n

Where T= overall space time

When n tends to infinity

C_n/CO= 1/(1+KT/n)ⁿ =(1+KT/n)^-n= 1- n*KT/n + +    , 1- KT

For a PFR of first order

C/CO =e^-KT   expanding in Taylor series gives =1- KT