Page 107 - The engineering of chemical reactions

Conversion in a Constant-Density CSTR 91

but this equation becomes an &h-order polynomial,

tkC; + CA - CA0 = 0
and to solve for CA we must find the proper root among the II roots of an nth-order
polynomial.
We took the + sign on the square root term for second-order kinetics because the
other root would give a negative concentration, which is physically unreasonable. This is
true for any reaction with nth-order kinetics in an isothermal reactor: There is only one real
root of the isothermal CSTR mass-balance polynomial in the physically reasonable range
of compositions. We will later find solutions of similar equations where multiple roots are
found in physically possible compositions. These are true multiple steady states that have
important consequences, especially for stirred reactors. However, for the &h-order reaction
in an isothermal CSTR there is only one physically significant root (0 < CA < CA,) to the
CSTR equation for a given t .

Fractional conversion

As shown in the previous chapter, we can use another variable, the fractional conversion
X. which we defined as
CA = cA,(l - x)
as long as the system is at constant density. In order for this variable to go from 0 to 1, it is
necessary that we base it in terms of a reactant A whose concentration is CA (the limiting
reactant with VA = -1). We can now write the mass-balance equation for the CSTR in
terms of X as

CA0 - CA = cA,x = tkCA(X)
and solve for X(t) rather than CA(~). For the first-order irreversible reaction this equation
becomes

cA,x = tkcAo(l - x)
which can be solved for t
1 x
*=--
k l - X
or for X,
kt
X(t) = -
1 +kt
From the definitions of CA and Cg in terms of X, we can use this X(r) to find the same
expressions as above for CA(t),

C’A(T)=CA,(~-X)=c~,,(l-&)=&

and for CB (t), and
CA&
cc(t) = C.&x = __
l+kt
which are the same answers we obtained by solving for CA directly.