Dynamics of a Coupled Reactor-Separator System with Time Delay

This Demonstration analyzes the effect of time delay on the behavior of a coupled non-isothermal continuous-flow stirred tank reactor (CSTR) with a separator.
The effluent of the reactor is fed to an isothermal separator and the liquid stream of the separator is recycled to the reactor. A first-order exothermic irreversible reaction takes place in the reactor and there is a time delay in the transport from the reactor to the separator. The dimensionless delay-differential equations that describe the system (equations 8 and 9 in [1]) are
,
,
,
.
In these equations, represents the mole fraction of species in the reactor, is the reactor dimensionless temperature, and is the dimensionless time. , , and are the Damköhler number, the dimensionless heat transfer coefficient, and the dimensionless adiabatic temperature rise; these dimensionless numbers are defined in terms of system variables in the reference.
The mole fractions of species in the reactor fresh feed, the distillate stream, and the recycle stream are , , . The equations are solved with and . In the absence of delay, the coupled system exhibits damped oscillations leading to a steady state for low and high values of the Damköhler number and oscillations without a steady state for intermediate values of the Damköhler number. Delay induces new regions of dynamic instability: increasing the delay beyond a lower threshold value can either destabilize the system or lead to isolated states of stability.

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References
[1] P. Balasubramanian, S. Pushpavanam, and A. Kienle, "Delay Effect in a Coupled Reactor-Separator System Sustaining Non Isothermal Reaction," Proceedings of the National Conference on Nonlinear Systems & Dynamics, Indian Institute of Technology, Kharagpur, India, Dec. 28–30, 2003 pp. 189–192. ncnsd.org/proceedings/proceeding03/html/pdf/189-192.pdf.
[2] Wikipedia. "Damköhler Numbers." (Feb 26, 2013) en.wikipedia.org/wiki/Damk% C3 % B6hler_numbers.
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