CONSERVATION: MASS, MOMENTUM, AND ENERGY BALANCES - Biomass as a Sustainable Energy Source for the Future - page 75

A balance for a chemical species i in an arbitrarily chosen system, in terms of the
generic Equation (3.1), is easily derived as

Accumulation rate of species i = rate of supply of species i rate of release of species i

+ rate of production of species i

−

ð Eq : 3 : 5 Þ

This equation can be mathematically reformulated in macroscopic form as
(Westerterp et al., 1988)

dm i
dt

= φ m , i , in − φ m , i , out +MW i R i h i V= − Δ φ m , i +MW i R i h i V ð Eq : 3 : 6 Þ

s −1 ) of species i caused by the chemical reaction
with R i , expressed as the molar conversion in time per reactor volume (mol

MW i R i is the rate of production (kg

m −3 s −1 )

and representing the average over the space in which the reaction takes place. R i

usually depends on the concentrations of the species present and on temperature;
accompanying reaction rate expressions are dealt with in Chapter 5.
The aforementioned macroscopic mass balance equation can also be rewritten so as
to give the macroscopic mole balance for species i:

h…i

dn i
dt =

φ n , i , in − φ n , i , out + R hi V= − Δ φ n , i + R hi V ð Eq : 3 : 7 Þ

When chemical conversions are considered, it is important to define towhich extent such
reactions take place in a reactor. For this purpose, the degree of conversion ( )isintro-

duced, which is based on the mass fractions of the reacting species (Y i ). It is defined as

ξ

ξ ≡ j Y i 0 − Y i j ð Eq : 3 : 8 Þ

Now, consider a reaction |v A |A + |v B |B v X X+v Y Y.

If no species are added to or removed from the reacting system, we can write

!

Y A +Y B +Y X +Y Y = constant :

The following relation holds with respect to the moles of the reacting species:

ξ A
ν jj

ξ B
ν jj

ξ X
ν X MW X =

ξ Y
ν Y MW Y MW A = MW B =

ð Eq : 3 : 9 Þ

For more convenience in calculations, the degree of conversion is related to the con-
centration of the species originally present. Thus, the relative degree of conversion of
a reactant (

ζ

) is defined as

ζ i = ξ i

Y i
Y i 0 Y i 0 =1

− ð Eq : 3 : 10 Þ