Question 10.7: Quasi-Autothermal Operation of a Semibatch Reactor Using Add...

The material balance equations have been solved earlier with the result that

m_{ A }=m_{ A i} e^{-k\left(t-t_i\right)}+\frac{\phi_{ A 0}}{k}\left[1-e^{-k\left(t-t_i\right)}\right]

where m_{A} is the mass of A remaining in the reactor at time t, m_{Ai} is the mass of A in the reactor at time t_{i}, and \phi_{A0} is the mass feed rate of A appropriate to the time interval between t and t_{i}.

An energy balance on the reactor can be derived from equation (10.1.1) by omission of appropriate terms:

\frac{d E_{ sys }}{d t}=\dot{Q}+h_{ in } \dot{m}_{ in }

Multiplying by dt and integrating between time zero and time t gives

E_{sys} − E_{sys , 0} = Q + h_{in}m_{in}

If the temperature of the inlet stream is taken as the datum temperature for enthalpy and internal energy calculations,

h_{in} = 0

and

E_{ sys }-E_{ sys , 0}=Q

However, for condensed phases, the difference between internal energy and enthalpy is usually negligible:

Q \approx H_{ sys }-H_{ sys , 0}            (A)

The enthalpy difference is given by

where Ĉ is the average heat capacity per unit mass and where we have assumed that the initial material sump (m_{B0}) is at 163°C.

Equations (A) and (B) can be combined with appropriate numerical values to obtain

where Q is expressed in calories and m is expressed in grams. If m is expressed in pounds and Q in Btu, then

Q=128.7 m_{ A }-20.7\left(m_{ B }-m_{ B 0}\right)

The numerical values characterizing m_{A} \text{  and  } m_{B} at various times are obtained as indicated in Illustration 8.11. They may be used to determine the net heat required between time zero and time t. The pertinent results are presented in Table I10.7-1. It is evident that heat must be supplied to the reactor during the first several hours, but the reactor then reaches a point beyond which cooling must be supplied.

Instantaneous heat transfer requirements may be determined by recognizing that the thermal energy supplied must be used to effect either a sensible heat change or the reaction.
If \dot{Q} is the heat transfer rate,

\dot{Q}=\phi_{ A 0} \int_{T_0}^T \hat{C} d T+r V_R \Delta H_R

where V^{\prime}_{R} is the liquid-phase volume at time t. Because
C_{A} = m_{A}/V^{\prime}_{R} and the reaction obeys first-order kinetics,

\dot{Q}=\phi_{ A 0} \int_{T_0}^T \hat{C} d T+k m_{ A } \Delta H_R

Substitution of numerical values gives

\begin{aligned}\dot{Q} & =\phi_{ A 0}(0.5)(163-20)+0.8 m_{ A }(-83) \\& =71.5 \phi_{ A 0}-66.4 m_{ A }\end{aligned}

where \dot{Q} is expressed in cal/h, ϕ_{A0} is expressed in g/h, and m_{A} is expressed in grams.

In more conventional engineering units,

\dot{Q}=128.7 \phi_{ A 0}-119.5  m_{ A }

where \dot{Q} is now expressed in Btu /h, ϕ_{A0} is expressed in lb/h, and m_{A} is expressed in pounds. Calculated values of \dot{Q} are also presented in Table I10.7-1.

If one were to operate this semibatch reactor under a filling schedule, which for the first 11 h is identical to that considered previously, and then proceed to feed A at the maximum rate of 400 lb/h for an additional 2.875 h, the same total amount of A would have been introduced to the reactor. However, in this case, the heat transfer
requirements would change drastically. There would be a strong exotherm beginning at the moment the cold feed is stopped. The results for this case are presented in Table I10.7-2.

Consequently, if one desires to minimize the possibility that this reaction will run away (be out of control) as a result of exceeding the capacity of the cooling network, it would be advisable to operate in a mode such that the feed rate is gradually diminished instead of being abruptly terminated at a high feed flow srate.