Page 318 - Introduction to chemical reaction engineering and kinetics

12.3 Design Equations for a Batch Reactor 299

where xi is the mole fraction of species i , and CPi is its molar heat capacity as a pure
species; m, is the total (specific) mass of the system:

N
m, = 1 mi (12.3-14)
i=l

and cP is the specific heat capacity of the system, approximated by

(12.3-15)
‘P k 5 wicPi

where wi is the weight (mass) fraction of species i , and cpi is its specific heat capacity
as a pure species.
Equations similar to 12.3-10 to -15 may be written in terms of internal energy,
U, with C,, the heat capacity at constant volume, replacing C,. For liquid-phase
reactions, the difference between the two treatments is small. Since most single-phase
reactions carried out in a BR involve liquids, we continue to write the energy balance
in terms of H, but, if required, it can be written in terms of U . In the latter case, it is
usually necessary to calculate AU from AH and C, from C,, since AH and C, are the
quantities listed in a database. Furthermore, regardless of which treatment is used, it
may be necessary to take into account the dependence of AH (or AU) and C, (or C,)
on Ta2
From equations 12.3-8, -9, and -10, the energy balance for a BR, equation 12.3-7,
becomes

I 1
UA,(T, - T), + (-AH&(-rA)V = n,C,$ (12.3-16)
/ 1

Equation 12.3-16 is valid whether heat is transferred to or from the system, and whether
the reaction is exothermic or endothermic. Note that each term on the left side of equa-
tion 12.3-16 may be an input or an output. Furthermore, C, is the molar heat capacity
of the system, and is given by equation 12.3-13; as such, it may depend on both T and
composition (through fA). The right side of equation 12.3-16 may also be expressed on
a specific-mass basis (12.3-11). This does not affect the consistency of the units of the
terms in the energy balance, which are usually J s-i.

2 AH = AU + A(PV) (12.3-17)
(12.3-18)
cp - cv = CY’VT/K~
where (Y is the coefficient of cubical expansion, and KT is the isothermal compressibility;

dAHldT = A& (12.3-19)
where ACp is the heat capacity change corresponding to that of AH. The dependence of Cp on T is usually
given by an empirical expression such as

Cp = a + bT + CT’ (12.3-20)

in which the coefficients a, b, and c depend on the species, and must be given.