D. Meyer, J. Friedland, T. Kohn and R. Güttel Institute of Chemical Engineering, Ulm University, D-89081 Ulm, Germany Modeling an unsteady-state isothermal CSTR with typical first-order reaction networks:
Investigation of unsteady-state chemical processes with transfer functions (TF) utilization of TF commonly used in electrical engineering deduce kinetic parameters with reduced mathematical effort
Evaluation of poles and zeros of TF
Objective: Investigation of the influence of kinetic constants on the frequency response.
analysis of cut-off frequency s,I investigation of residence time and kinetic constants ki
Reaction Unsteady-state isothermal CSTR model Analysis with two methods
Time Domain: Laplace Domain:
Method 1: Time domain analysis
Transfer Function
Figure 2: Reactant A (left) and product P (right) with different reaction rate ratios for a consecutive reaction for a two-step reaction system.
Each reaction step adds an additional pole to the TF of P additional -20 dB/decade magnitude -90° phase shift for each step increased frequency damping after s,P,1 Initial condition:
Poles Reactant A
Product P
A’s frequency response is unaffected by the number of reaction steps identical to a reaction methods match
Reaction
Method 2: Laplace domain analysis
Transfer Function
Figure 3: Reactant A (left) and product B (right) with different reaction rate ratios for a parallel reaction.
Poles Reactant A
Reaction
typical first-order - (left) and second-order (right) low-pass behavior s,A is a summation of -1 and the kinetic constants boundary case ( ):
Product B / C
s,B,1 s,B,2 s,B,2 s,B,2
s,A,1 s,A,1 s,A,1 s,A,1
Transfer Function
Reaction
Figure 1: Reactant A (left) and product B (right) with different reaction rate constants for a irreversible reaction. Left: Three typical regions for a frequency response are indicated for kB = 0 s-1 .
Transfer Function
Three distinct regions for frequency response:
Poles Reactant A
1. quasi-steady-state region (qss):
Product B
phase and magnitude unaffected by input frequency[1] steady-state kinetics can be applied 2. full transient region (ft): first-order/ second-order low-pass behavior for A / B s,A,1 of A increases with kB (Tab. 1) s,B,1 of B depends only on -1 (Tab. 1)
Figure 4: Reactant A (left) and product B (right) with different reaction rate ratios for a reversible reaction.
qss region: magnitude of A is constant, but decreases with decreasing backward reaction rate ft region: small kB: boundary case: large kB : complex behavior for increasing frequencies reactant A
product B
phase shift increases towards 0° magnitude stays constant
constant phase shift of – 90°
Table 1: Influence of ki on s kB /
s-1
Reactant A
Product B
s,A,1 / rad s-1
s,B,1 / rad s-1
s,B,2 / rad s-1
0
0.2
-
-
1
1.2
0.2
1.2
10
10.2
0.2
10.2
100
100.2
0.2
100.2
kB derived by analysis of s,A,1
in all investigated cases: three regions exist: qss ft rss
resonance-like state
Product B
Zeros Reactant A
system acts as an internal feedback loop with strong similarities to a chemical buffer
s,B,1 depends only on -1
derived by analysis of s,B,1 3. relaxed steady-state (rss): constant phase shift and constant decrease of the magnitude[1]: reactant A: - 90 ° / -20 dB/decade product B: -180° / -40 dB/decade Literature: [1] Periodic operation of reactors, 1st ed. (Eds: P.L. Silveston, R.R. Hudgins), Elsevier, Amsterdam, Boston, Kidlington, 2013.
simulation data in time domain matches simulation data in frequency domain typical regions for the frequency response (qss, ft and rss) are defined for first-order reactions kinetic data for linear systems can be derived by evaluation of the frequency response use of Bode Plot for analysis of frequency response for non-linear systems possible Perspective: Implementation chemical processes in electrical grid for integration of renewable resources into chemical value chains
further details see: D. Meyer, J. Friedland, T. Kohn, R. Güttel, Transfer Functions for Periodic Reactor Operation: Fundamental Methodology for Simple Reaction Networks, submitted.
Poles Reactant A
Contact:
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