2nd order differential equation for Catalyst

[FaNgS]

Homework Statement

There's a catalyst pellet in a reactor and I'm supposed to prove an equation for the maximum temperature which is:

Tmax=Ts +[(-H)*(D*Cas)]/k

Tmax= max temp,
Ts = temp at catalyst surface,
T=temperature,
H=enthalpy of rxn,
D=diffusivity,
Cas= concentration at catalyst surface,
Ca=concenration
k= heat transfer coefficient,
r= radius,
R=radius of catalyst sphere,
Rxn= rate of reaction

2. The attempt at a solution
First thing I did was develop an Energy Balance across the spherical catalyst and I got the following equation:

(1/r^2)*d/dr(r^2*k*dT/dr) + (-H)*(-Rxn) = 0

expanding i get:

d^2T/dr^2 + (2/r)*dT/dr + (-H)*(-Rxn)/k = 0

Boundary Conditions: at r=0, dT/dr =0 AND at r=R, T=Ts (at the catalyst surface i.e. r=R the temperature T = Ts (catalyst surface temperature))

Using the boundary conditions and integrating factor I got
T=Ts + [(-H)(-Rxn)*(R^2-r^2)]/(6*k) ...(eqn 1)

Now for the mole balance across the catalyst I got (where Ca is the concentration):

d^2Ca/dr^2 + (2/r)*dCa/dr - (k/D)*Ca =0 ... (eqn 2)

Boundary Condition: at r=R, Ca=Cas

Can someone confirm the equation I got for temperature (eqn 1) and also for the concentration (eqn 2) how do I go about to solve it??

I tried another way to solve it, as suggested by my instructor, which is by using the un-expanded forms of the energy and mole balance equations which are:

(1/r^2)*d/dr(r^2*k*dT/dr) + (-H)*(-Rxn) = 0

(1/r^2)*d/dr(r^2*D*dCa/dr) + (Rxn) = 0

So here I have the "Rxn" term common in both equations and I combined and got the following after some simplifications:

d^2Ca/dr^2 + 2*dCa/dr = [k/(H*D)]* { r*d^2T/dr^2 + 2*dT/dr}

But in this case it looks way more complicated and I'm not sure how to deal with this type of an equation, since I have 2 derivates on both sides of the equation one with respect to the Concentration Ca and one with respect to temperature T.

not sure which way to proceed and how to proceed

 

[HallsofIvy]

Homework Statement

There's a catalyst pellet in a reactor and I'm supposed to prove an equation for the maximum temperature which is:

Tmax=Ts +[(-H)*(D*Cas)]/k

Tmax= max temp,
Ts = temp at catalyst surface,
T=temperature,
H=enthalpy of rxn,
D=diffusivity,
Cas= concentration at catalyst surface,
Ca=concenration
k= heat transfer coefficient,
r= radius,
R=radius of catalyst sphere,
Rxn= rate of reaction

2. The attempt at a solution
First thing I did was develop an Energy Balance across the spherical catalyst and I got the following equation:

(1/r^2)*d/dr(r^2*k*dT/dr) + (-H)*(-Rxn) = 0

expanding i get:

d^2T/dr^2 + (2/r)*dT/dr + (-H)*(-Rxn)/k = 0

Since it was already partially solved for you don't expand!
$$d(r^2k(dT/dr)= -HRxn r^2 dr$$
gives a first integral of
$$r^2k (dT/dr)= -(HRxn/3)r^3+ C$$
and then
$$k dT= (-(HRxn/3)r+ Cr^{-2}) dr$$
integrates to
$$kT= -(HRxn/6)r^2- C/r+ D$$

Boundary Conditions: at r=0, dT/dr =0

The fact that dT/dr exists at r= 0 means that C= 0

AND at r=R, T=Ts (at the catalyst surface i.e. r=R the temperature T = Ts (catalyst surface temperature))

so $kTs= -(HRxn/6)R^2+ D$ and $D= kTs+ (HRxn/6)R^2$

Using the boundary conditions and integrating factor I got
T=Ts + [(-H)(-Rxn)*(R^2-r^2)]/(6*k) ...(eqn 1)

Now for the mole balance across the catalyst I got (where Ca is the concentration):

d^2Ca/dr^2 + (2/r)*dCa/dr - (k/D)*Ca =0 ... (eqn 2)

Boundary Condition: at r=R, Ca=Cas

Can someone confirm the equation I got for temperature (eqn 1) and also for the concentration (eqn 2) how do I go about to solve it??

I tried another way to solve it, as suggested by my instructor, which is by using the un-expanded forms of the energy and mole balance equations which are:

(1/r^2)*d/dr(r^2*k*dT/dr) + (-H)*(-Rxn) = 0

(1/r^2)*d/dr(r^2*D*dCa/dr) + (Rxn) = 0

So here I have the "Rxn" term common in both equations and I combined and got the following after some simplifications:

d^2Ca/dr^2 + 2*dCa/dr = [k/(H*D)]* { r*d^2T/dr^2 + 2*dT/dr}

But in this case it looks way more complicated and I'm not sure how to deal with this type of an equation, since I have 2 derivates on both sides of the equation one with respect to the Concentration Ca and one with respect to temperature T.

not sure which way to proceed and how to proceed

 

[FaNgS]

so here you used D as the integration constant after the second integration and got D=kTs+ (H*Rxn/6)*R^2, if i substitute that back into the equation won't everything cancel out??

Edit 1: nevermind, what i said earlier...ok so now this confirms the first equation.

can you give me any hints on how to proceed to solve the equation for the concentration?


Edit 2: sorry for being jumpy, i figured it out and got the same equation as required (Tmax)
thanks a lot for your help i really appreciate it, I've been stuck on this for some time

thank you sir!