CONSERVATION: MASS, MOMENTUM, AND ENERGY BALANCES - Biomass as a Sustainable Energy Source for the Future - page 76

Both conversion definitions use mass fractions; however, usually, it is more conven-
ient to use molar concentrations, which are measured with analytical instruments, or
partial pressures, which are related to volume fractions in gas-phase reaction systems.
For a gas-phase reaction system, one can write the molar concentration as follows
(assuming that the ideal gas law holds):

c i = X i p
R u T

ð Eq : 3 : 11 Þ

The molar concentration and mass fraction are interrelated via

c i = ρ

Y i
MW i

ð Eq : 3 : 12 Þ

So for reactant A, one can write

−

= ρ
ρ 0

c A0
ρ 0

ξ A
MW A

c A = ρ c A0 1 ð − ζ A Þ ð Eq : 3 : 13 Þ

Exercise: Derive Equation (3.13).

In industrial practice, two other concepts, namely, selectivity and yield , are of
importance as often (undesired) by-products are formed in a reaction system. The
selectivity, , toward reaction product X is defined as the ratio between the amount

of X formed and the amount of key reactant A converted. Thus,

σ

σ X = ξ X ν jj

MW A
ξ A ν X MW X

= n X , formed

n A , reacted ν jj

ð Eq : 3 : 14 Þ

ν X

At constant density of the reaction mixture, this is equal to

ð c X − c X0 Þ ν jj

ν X

σ X = ð Eq : 3 : 15 Þ

ð c A0 − c A Þ

, of product X from reactant A, which is the
amount of X formed relative to the amount of A fed as reactant:

The main aim is to realize a high yield, η

η X = σ X ζ X ð Eq : 3 : 16 Þ

Using the alternative formulation of Equation (3.2), for f , we can substitute ρ i ,theden-

!

! i . Species i can be formed or consumed
via a chemical reaction, so that a chemical source term exists: s f =

sity of species i, and for ϕ

f , we can substitute

ρ

: i . This results in

ω

+ =

∂ ρ i
∂

, ∂ ρ i
∂

v !

:

v !

:

−r ρ

ω i

r ρ

ω i : :

t = t +

for i = 1, … ,N ð Eq 3 17 Þ