Ideal Flow Reactors - Bioprocess Engineering: Kinetics, Biosystems, Sustainability and Reactor Design - page 236

Normally, one needs to integrate numerically as k f and k b are functions of temperature,
which varies along the reactor or with mixture composition. Numerical techniques such
as the Simpson's rule, integrator such as OdexLims can be applied for this purpose. In
this particular case, integration can be carried out analytically. Substituting the tempera-
ture Eqn (E5-10.10) , the rate constants, initial concentration, and the pressure into Eqn
(E5-10.13) , we obtain

s ¼ 965:089 R

0:20636

d f A

1 þ f A exp

1 þ f A 2 exp

f A

21:75 1 f A

20000
8:314 623:15

60000
8:314 623:15

0

367500

þ 965:089 0:8
0:20636

d f A

1 f A
f A

! 1 f A

f A

!

1
2

3
2

f A
1 þ f A 2

21:75 1 f A
1 þ f A

4:4416 10 3 8:7623 10 8

367500

which is integrated to yield

s ¼ 248:03

0:20636

f A þ 3:25762 ln 0 : 3431108 þ f A

0:3431108 f A

0

þ 14985:09 h 1 þ f A Þð1 f A Þ 1=2 arc sin f A i 0:8

0:20636

s

¼ 16771:86 s ¼ 4:659 h

Plotted in Fig. E5-10 is also the variation of conversion with reactor space time. This is
achieved by integrating for the space time as a function of conversion.
3. Since the heat capacity is negligible, temperature change can be achieved with negligible
thermal energy. Therefore, the heat generated during reaction must be removed. Energy
balance gives

X

N S

d Q d W s F j 0 dH j þ rDH R d V ¼

j¼1

There is no work input or output. When heat capacity is negligible, the energy balance
equation reduces to

d Q
d V ¼ rDH R ¼ rðE f E b Þ

which gives the heat required for the reactor. It is the negative of what needs to be
removed.

Figure E5-10 d shows the heat generated (or removal requirement) along the PFR. One can
observe that more heat is being generated at the front end of the reactor.