Chapter 5, Problem 5.7E

Appelpolscher has just left a meeting with Stella J. Smarly, IGC’s vice-president for process operations anddevelopment. Smarly is concerned about an upcoming extended plant test of a method intended to improvethe yields of a large packed-bed reactor. The basic idea, which came from IGC’s university consultant and wasrecently tested for feasibility in a brief run, involves operating the reactor cyclically so that nonlinearities in thesystem cause the time-average yield at the exit to exceed the steady-state value. Smarly is worried about thepossibility of sintering the catalyst during an extended run, particularly in the region of the “hotspot” (axiallyabout one-third of the way down the bed and at the centerline) where temperatures invariably peak.Appelpolscher, who plans to leave the next day on a two-week big game photo safari, doesn’t want to cancelhis vacation. On the other hand, Smarly has told him he faces early, unexpected retirement in Botswana if themeasurement device (located near the hot spot) fails to alert operating people and the reactor catalyst sinters.Appelpolscher likes Botswana but doesn’twant to retire there. He manages to pull together the following dataand assumptions before heading for the airport and leaves them with you for analysis with the offer of the useof his swimming pool while he is gone. What do you report to Smarly?

Data:
$\begin{array}{l}\text{Frequency of cyclic operation}=0.\text{1 cycles}/\text{mm}\\ \text{Amplitude of thermal wave }\left(\text{temperature}\right)\text{ at the measurement point obtained experimentally in the recent brief run}=\text{15 }°\text{C}\\ \text{Average operating temperature at the measurement point},T_{\text{meas}}=\text{35}0°\text{C}\\ \text{Time constant of temperature sensor and thermowell}=\text{1}.\text{5 min}\\ \text{Temperature at the reactor wall}=\text{2}00°\text{C}\\ \text{Temperature at which the catalyst sinters if operated for several hours}=\text{7}00°\text{C}\\ \text{Temperature at which the catalyst sinters instantaneously}=\text{715 }°\text{C}\end{array}$

Assumptions:
The reactor operational cycle is approximately sinusoidal at the measurement point.
The thermowell is located near the reactor wall so as to measure a radial average’ temperature rather thanthe centerline temperature.
The approximate relation is
   $T=\frac{T_{\text{center}}+2T_{\text{wall}}}{3}$
which also holds during transient operation.