# Residence Time Distribution for Continuous Stirred-Tank Reactors in Series Using the First Four Moments

Residence Time Distribution for Continuous Stirred-Tank Reactors in Series Using the First Four Moments

Consider CSTRs (continuous strirred-tank reactors) subject to a tracer impulse experiment. The residence time distribution (RTD) can be found exactly by solving a system of ODEs obtained from mass balances in all reactors or by Laplace inversion of the system's transfer function. It turns out that the first four moments can be easily obtained from the transfer function expression and by means of a Gram–Charlier series for the approximatie RTD. This Demonstration presents a comparison of these three methods for user-set values of , the number of CSTRs in series. It is clear that, as increases, the exact result and the approximate result (derived from the moment expressions in Gram–Charlier theory) show close agreement.

N

N

N

The transfer function of the system is given by , where is the residence time, taken to be five hours for all tanks.

G(s)=

1

1+s

τ

N

N

τ

N

The system of ODEs is,

for the first CSTR, =δ(t)-C(t);

dC

1

dt

1

for subsequent CSTRs, =C(t)-C(t), for .

dC

j

dt

j-1

j

j=2,…,N

The expression for the tracer's concentration based on the four moments is

C(t)=exp1-m(3Z-Z)/6+(m-3)(Z-6Z+3)/24

N

1

σ

2π

(t-τ)

2

2σ

2

3

2

4

4

2

where =, =, , =, and .

σ

2

τ

2

1

N

γ

3

τ

3

2

N

2

m=

3

γ

σ

3

δ

4

τ

4

3(N+2)

N

3

m=

4

δ

σ

4

The first, second, third, and fourth moments are measures of the mean, variance, skewness, and kurtosis, respectively.