Batch reactors operated adiabatically are often used to determine the reaction orders, activation energies, and specific reaction rates of exothermic reactions by monitoring the temperature–time trajectories for different initial conditions. In the steps that follow, we will derive the temperature-conversion relationship for adiabatic operation.
For adiabatic operation () of a batch reactor (Fi0 = 0) and when the work done by the stirrer can be neglected (), Equation (13-10) can be written as
It is shown in the Summary Notes on the Web and DVD-ROM that if we combine Equation 13-12 with Equation 2-6, we can do a lot of rearranging and integrating to arrive at
We note that for adiabatic conditions, the relationship between temperature and conversion is the same for batch reactors, CSTRs, PBRs, and PFRs. Once we have T as a function of X for a batch reactor, we can construct a table similar to Table E11-3.1 and use techniques analogous to those discussed in Section 11.3.2 to evaluate the following design equation to determine the time necessary to achieve a specified conversion.
However, if you do not have that much time or your hands to form a table and use Chapter 2 integration techniques, then use Polymath to solve the differential form of mole balance equation (2-6) and the energy balance equation (13-14) simultaneously.
Example 13-1. Adiabatic Batch Reactor
It is still winter, and although you were hoping for a transfer to the plant in the Bahamas, you are still the engineer of the CSTR of Example 12-3, in charge of the production of propylene glycol.
You are considering the installation of a new glass-lined 175-gal CSTR, and you decide to make a quick check of the reaction kinetics. You have a nice insulated instrumented 10-gal stirred batch reactor available. You charge this reactor with 1 gal of methanol and 5 gal of water containing 0.1 wt % H2SO4. For safety reasons, the reactor is located in a storage shed on the banks of Lake Wobegon (you don’t want the entire plant to be destroyed if the reactor explodes). At this time of year, the initial temperature of all materials is 38°F. We have to be careful here! If the reactor temperature increases above 580°R, a secondary, more exothermic reaction will take over, causing runaway and subsequent explosion.
a. How many minutes should it take the mixture inside the reactor to reach a conversion of 51.5% for adiabatic operation? Use the data and the reaction rate law given in Example 12-3.
b. What would be the temperature at 51.5% conversion?
Recall that for liquid batch V = V0
From the data in Example 12-3,
From Example 12-3, ΔCP = –7 Btu/lb mol · °F, and consequently, the second term on the right-hand side of the expression for the heat of reaction,
is very small compared with the first term (–36,400 Btu/mol) [less than 2% at 51.5% conversion (from Example 12-3)]. Taking the heat of reaction at the initial temperature of 515°R, we obtain
Because terms containing ΔCp are very small, we assume that
ΔCP ≃ 0
In calculating the inlet temperature after mixing T0, we must include the temperature rise (17°C) from the heat of mixing the two solutions initially at 38°C.
A summary of the heat and mole balance equations is given in Table E13-1.1.
Table E13-1.1. Summary for First Order Adiabatic Batch Reaction
A software package (e.g., Polymath) was also used to combine Equations (E13-1.3), (E13-1.5), and (E13-1.7) to determine conversion and temperature as a function of time. Table E13-1.2 shows the program, and Figures E13-1.1 and E13-1.2 show the solution results.
Table E13-1.2. Polymath Program
Analysis: The temperature-time and conversion-time trajectories show the reaction going to completion. For 51.5% conversion we would need to quench the reactor at 2560s (40 minutes) from a temperature of 561°F by rapidly lowering the temperature well below 515°R. We note that if the quench system fails, the temperature will continue to increase above 580°R and the rate of the second reaction would become significant and we could have a runaway reaction similar to that in Example 13-6.
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