1.
1 Derivation of Concentration Profile within a Porous
Catalyst with a 1st Order Reaction
The concentration through a porous catalyst for a first order reaction can be described by
the differential equation,
d2 CA 2 d CA k1′′ Sa ρc
+ − CA = 0 (1.1.1)
d r2 r dr De
This equation can be turned into a dimensionless version by defining 2 dimensionless
variables,
CA
φ= (1.1.2)
CAs
r
λ= (1.1.3)
R
This means that we can write,
r = λR (1.1.4)
d r = Rd λ (1.1.5)
and,
CA = CAs φ (1.1.6)
d CA dφ
= CAs (1.1.7)
dr dr
d2 CA d2 φ
= CAs 2 (1.1.8)
d r2 dr
Combining with our definition of d r we then produce,
d CA CAs d φ
= (1.1.9)
dr R dλ
2
d CA CAs d2 φ
= (1.1.10)
d r2 R 2 d λ2
Substituting this into equation 1.1.1 produces,
CAs d2 φ 2CAs d φ k1′′ Sa ρc
+ − CAs φ = 0
R 2 d λ2 λR2 d λ De
d2 φ 2 d φ k1′′ Sa ρc R2
+ − φ=0 (1.1.11)
d λ2 λ d λ De
We can now define a final dimensionless parameter,
k1′′ Sa ρc R2
ϕ2 = (1.1.12)
De
Thus producing a dimensionless form of the differential equation for the concentration
through a porous catalyst,
d2 φ 2 d φ
2 + − ϕ2 φ = 0 (1.1.13)
dλ λ dλ
©T.L. Rodgers 2021
The boundary conditions can also be defined in terms of the dimensionless parame-
ters1 ,
CA = CAs at r = R ⇒ φ = 1 at λ = 1
d CA dφ
= 0 at r = 0 ⇒ = 0 at λ = 0 (1.1.14)
dr dλ
Equation 1.1.13 can be solved by defining a new variable y = φλ, which means that,
y
φ= (1.1.15)
λ
dφ 1 dy 1
= −y 2 (1.1.16)
dλ λdλ λ
d2 φ 1 d2 y 1 dy 1 dy 2
2 = 2 − 2 − 2 +y 3
dλ λdλ λ dλ λ dλ λ
2
1d y 2 dy 2
= 2 − 2 + 3y (1.1.17)
λdλ λ dλ λ
Substituting these into equation 1.1.13 produces,
1 d2 y 2 dy 2 2 dy 2 1
2 − 2 + 3y + 2 − 3 y − ϕ2 y = 0
λdλ λ dλ λ λ dλ λ λ
1 d2 y 21
2 −ϕ y=0
λdλ λ
d2 y
− ϕ2 y = 0 (1.1.18)
d λ2
This differential equation can now be solved by substituting an exponential form for y
as,
y = eβλ (1.1.19)
where β is a value to find. Substituting this expression into equation 1.1.18 allows β to
be calculated as,
β 2 eβλ − ϕ2 eβλ = 0
β 2 − ϕ2 = 0
(β − ϕ) (β + ϕ) = 0
β = ±ϕ (1.1.20)
This means that from the linear combination of all solutions, then the full integrated ex-
pression can be given as,
y = Aeϕλ + Be−ϕλ (1.1.21)
or using the definition of y, equation 1.1.15,
A ϕλ B −ϕλ
φ= e + e (1.1.22)
λ λ
1
It should be noted here that if the reaction order is less than 1 it is possible to have what is called a
dead zone, i.e. the concentration of the reactant drops to 0 before the centre of the catalyst, this means that
the second boundary condition is not true in this case and needs to be replaced with φ = 0 at λ = λc where
λc has to be calculated from the balance equations. For example, see R. L. York, K. M. Bratlie, L. R. Hile,
and L. K. Jang, (2011) Catalysis Today, 160:204-212.
©T.L. Rodgers 2021
We can now use the boundary conditions, equation 1.1.14, to find the constants A and B.
The differential of equation 1.1.22 is,
dφ A B
= 2 eϕλ (ϕλ − 1) − 2 e−ϕλ (ϕλ + 1) (1.1.23)
dλ λ λ
Thus with our boundary condition at the centre of the particle we get,
0 = A(1)(−1) − B(1)(1)
A= −B (1.1.24)
With this and our boundary condition at the surface of the particle we get,
1 = Aeϕ + Be−ϕ
1 = A eϕ − e−ϕ
1 = 2A sinh ϕ
1
A= = −B (1.1.25)
2 sinh ϕ
eax − e−ax
where we have used the mathematical definition of sinh ax = . Substituting
2
these constant values into equation 1.1.22 gives,
e − e−ϕλ
ϕλ
1
φ=
λ sinh ϕ 2
1 sinh ϕλ
φ= (1.1.26)
λ sinh ϕ
©T.L. Rodgers 2021
