(371u) Optimal Heating Profiles in Tubular Reactors with Solid-Phase Axial Wall Conduction for Isothermal Operation

Sunjeev Venkateswaran, Benjamin Wilhite, Costas Kravaris

Department of Chemical Engineering, Texas A&M
University,

Optimal heating
profiles in tubular reactors with solid-phase axial wall conduction for
isothermal operation

In heat transfer limited
processes like endothermic methane steam reforming, the challenge is to
maintain a sufficiently high temperature throughout the reactor. In such highly
endothermic reactions a large decrease in temperature is observed at the inlet
which may decrease the overall conversion of the reactor. However excess
heating can lead to catalyst sintering and material stability issues which may
compromise the reactor lifetime. Thus, the reactor temperature should be chosen
carefully to maintain required conversion while ensuring catalyst/material
stability. Following which, appropriate heat inputs should be given to maximize
temperature uniformity along the reactor length at this temperature. The
primary goal of this study is to obtain heat input profiles to minimize
deviation from isothermal operation.

In
this regard a one-dimensional model is formulated capturing the main transport
effects in microchannels and optimal control theory is applied to obtain optimal
heat inputs. Though the model is
developed with micro-reactors as the main application, it is applicable to even
conventional tubular reactors with significant solid phase axial conduction. The
resulting model is rendered dimensionless and the magnitude of solid phase
axial heat conduction is captured by the conduction parameter term (CP) which gives
the relative importance of conductive heat transfer compared to the energy
carried by the fluid-phase.

For the optimal control problem, an appropriate objective function is framed to minimize the temperature deviation from the inlet temperature/desired
temperature. The optimal control analysis is divided into three cases (i) negligible solid-phase axial heat conduction (Very Low
CP), (ii) significant solid-phase axial heat conduction (Moderate CP) and (iii)
very high solid-phase axial heat conduction.

(i)            
Very Low CP-
Order of magnitude analysis is employed to reduce the original model to a classical
non-isothermal tubular reactor model. It is shown that for negligible solid-phase
conduction, perfectly isothermal operation is possible provided the maximum
heat input available is high enough.

(ii)           
Moderate CP-
Perfectly isothermal operation requires the presence of infinite impulses at
the boundaries of the reactor to satisfy the boundary conditions of the solid-phase.
Singular optimal control theory is used to obtain the optimal inputs in the case
of bounded inputs. The resulting optimal inputs are a finite input approximation
of the inputs observed in the case when inputs are unbounded (impulsive), and consist of bang-bang (maximum-minimum) and singular
parts.

(iii)         
Very High CP- As
CP keeps increasing deviation from isothermality is
shown to increase and in the limit of very high CP, order of magnitude analysis
is used to show that the fluid phase temperature is uncontrollable by the heat
input.

References

1.     Moreno, A., Murphy, K.,
& Wilhite, B. A. (2008). Parametric study of solid-phase axial heat
conduction in thermally integrated microchannel networks. Industrial
& Engineering Chemistry Research, 47(23), 9040-9054

2.     Logist F, Van Erdeghem P, Smets I, Van Impe J. Optimal
design of dispersive tubular reactors at steady-state using optimal control
theory. Journal of Process Control. 2009;19(7):1191-1198.

3.     Smets IY, Dochain D, Van Impe JF. Optimal temperature
control of a steady‐state exothermic plug‐flow reactor. AIChE Journal. 2002;48(2):279-286