To begin our journey, we start with the energy balance equation (11-9) and then proceed to finally arrive at the equations given in Table 11-1 by first dissecting two terms:
An animated version of what follows for the derivation of the energy balance can be found in the reaction engineering games “Heat Effects 1” and “Heat Effects 2” on the DVD-ROM. Here, equations move around the screen, making substitutions and approximations to arrive at the equations shown in Table 11-1. Visual learners find these two ICGs a very useful resource.
We will now consider flow systems that are operated at steady state. The steady-state energy balance is obtained by setting (dÊsys/dt) equal to zero in Equation (11-9) in order to yield
To carry out the manipulations to write Equation (11-10) in terms of the heat of reaction, we shall use the generalized reaction
The inlet and outlet summation terms in Equation (11-10) are expanded, respectively, to
and
where the subscript I represents inert species.
We next express the molar flow rates in terms of conversion.
In general, the molar flow rate of species i for the case of no accumulation and a stoichiometric coefficient υi is
Specifically, for Reaction (2-2), , we have
We can substitute these symbols for the molar flow rates into Equations (11-11) and (11-12), then subtract Equation (11-12) from (11-11) to give
The term in parentheses that is multiplied by FA0X is called the heat of reaction at temperature T and is designated ΔHRx(T).
All enthalpies (e.g., HA, HB) are evaluated at the temperature at the outlet of the system volume, and, consequently, [ΔHRx(T)] is the heat of reaction at a specific temperature T. The heat of reaction is always given per mole of the species that is the basis of calculation [i.e., species A (joules per mole of A reacted)].
Substituting Equation (11-14) into (11-13) and reverting to summation notation for the species, Equation (11-13) becomes
Combining Equations (11-10) and (11-15), we can now write the steady-state [i.e., ] energy balance in a more usable form:
If a phase change takes place during the course of a reaction, this form of the energy balance [i.e., Equation (11-16)] that must be used.
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