Tanks in Series Model
Levenspiel (Hfst. 14) p. 321 - 338
Tanks-inTanks-in-series model
Another model that is frequently used to simulate the behavior of actual reactor networks is a cascade of ideal CSTRs operating in series. series
Tanks-inTanks-in-series model
14
The actual reactor is replaced by n identical stirred tank reactors whose total volume is the same as that of the actual reactor.
1 2
Tanks-inTanks-in-series model
Model definition:
Tanks-inTanks-in-series model
Material balance – tank N:
t t⋅E = t
N −1
NN t e − tN t ; t = N t i ; σ 2 = (N − 1)! N
2 1 t e −t t i ; t i = ; σ 2 = N t i ( N − 1)! N
2
t ti ⋅ E = ti θi = t = dimensionl ess time based on mean residence time per tank, t i ti t θ = = dimensionl ess time based on mean residence time in all N tanks, t t θ i = 1; θ = 1; θ i = Nθ
3
N −1
Eθ = t ⋅ E = N Eθ i = t i ⋅ E =
( N θ )N −1 e − Nθ ; σ 2 = θ ( N − 1)! θ iN −1 e −θ i ; σ θ2i = N
1 N
(N − 1)!
4
1
Tanks-inTanks-in-series model
Chemical conversion Tanks-inTanks-in-series model
1st order reaction in one tank:
CA 1 = C A0 1 + k t
1st order reaction in N tanks-in-series:
N=
t2
σ
2
CA 1 = C A0 k t N 1 + N
5
Number of tanks-in-series N → ∞, plug flow
6
Class example
The concentration reading in the following table represents a continuous response to a pulse input. How many tanks in series is the reactor equivalent to?
Time t (min) 0 5 10 15 20 25 30 35 Tracer output concentration C(t) (g/L) 0 3 5 5 4 2 1 0
7
From previous class example:
N=
t2
σ2
= 4.737 ≈ 5
8
2
Tanks-inTanks-in-series model
Another model that is frequently used to simulate the behavior of actual reactor networks is a cascade of ideal CSTRs operating in series. series
Tanks-inTanks-in-series model
14
The actual reactor is replaced by n identical stirred tank reactors whose total volume is the same as that of the actual reactor.
1 2
Tanks-inTanks-in-series model
Model definition:
Tanks-inTanks-in-series model
Material balance – tank N:
t t⋅E = t
N −1
NN t e − tN t ; t = N t i ; σ 2 = (N − 1)! N
2 1 t e −t t i ; t i = ; σ 2 = N t i ( N − 1)! N
2
t ti ⋅ E = ti θi = t = dimensionl ess time based on mean residence time per tank, t i ti t θ = = dimensionl ess time based on mean residence time in all N tanks, t t θ i = 1; θ = 1; θ i = Nθ
3
N −1
Eθ = t ⋅ E = N Eθ i = t i ⋅ E =
( N θ )N −1 e − Nθ ; σ 2 = θ ( N − 1)! θ iN −1 e −θ i ; σ θ2i = N
1 N
(N − 1)!
4
1
Tanks-inTanks-in-series model
Chemical conversion Tanks-inTanks-in-series model
1st order reaction in one tank:
CA 1 = C A0 1 + k t
1st order reaction in N tanks-in-series:
N=
t2
σ
2
CA 1 = C A0 k t N 1 + N
5
Number of tanks-in-series N → ∞, plug flow
6
Class example
The concentration reading in the following table represents a continuous response to a pulse input. How many tanks in series is the reactor equivalent to?
Time t (min) 0 5 10 15 20 25 30 35 Tracer output concentration C(t) (g/L) 0 3 5 5 4 2 1 0
7
From previous class example:
N=
t2
σ2
= 4.737 ≈ 5
8
2