ABSTRACT
Combining this with the mass balance expressed in (9.169), the energy balance becomes (assuming that only one reaction is taking place and using = dh1j dT)
s s dT s dc·dT SQ* """'har + '\'vc·C ·-- """""DdC ·-1 -= --L....~ .1 1 ~ 1 p,; dz L-"" p,J ,l-cl7 j= 1 J~ 1 .i= I " " ~
(9.184)
Equation (9.184) can be reduced to dT d2T SQ*
(t:,.H) r + vCr dz-D 1hCp dz2 - (9.185)
or
dz dz~ CP CpAc (9.186)
where the energy flux by axial mass dispersion in Eq. 184) is approximated as s dc·dT d2T
f;DdCp,J d: dz = DrhCp dz2 (9.187)
v d~- Dth d27; = t ( -t:.H;) r;- SQ* cL d~ i=l CPA, (9.188)
The energy balance for axially dispersed tubular reactors is also a second-order differential equation; therefore two boundary conditions are needed for its solution. As discussed in the formulation of the mass balance, temperature shows a discontinuity at the reactor inlet. As a result, the energy flux balance \vill be the inlet boundary condition,
by which the set of boundary conditions can be expressed as [similar to Eqs. (9.173) and 74)]
dT =O dz
(9.190)
Using the dimensionless distance as defined in Eq. the energy balance equation (9.186) and corresponding boundary conditions can be expressed as
dT 1 d2 T L'fr_(-t:,.H;)r; LSQ* dZ - Pe1h dZ2 = --v-Cc:-P-- - -vC_r_A_c
T(O-) = T(O+) - -~ - dT (0+) Pe1hdZ
at
(9.19 I)
(9.193)
where Pe1h is the dimensionless Peclet number, which is defined as the ratio of the convection and heat fluxes:
Mass and energy dispersions are dynamic properties of the reactive system. In Peclet numbers corresponding to mass and energy dispersions can be expressed in tenns of the Reynolds number, Re, and mass transfer and heat transfer Prandtl numbers, Prm and Prth:
where pvL vL
and Pe111 = Re Pr th
. - y Pl m-Dd, y
(9.195)
(9.196)
where y is the kinematic viscosity (y = 1~1 p, whre f.L is the viscosity and pis the density). For reactor design purposes, published experimental data as well as new measurements of Pcclet numbers for the geometry, and operating conditions of a reactive system would always be needed.
