Why Can't I Get the Correct Constants in My Chemical Reaction Calculations?

[saltine]

Homework Statement

This is an example from a book. I am having trouble in substituting the numbers to get the given result.

Homework Equations

The bar notation is to indicate the operating point. The subscript A is for the reactant A (which reacts and becomes product B in a A->B reaction). The subscript i stands for the input. So $$c_{A_i}$$ means the concentration of A entering the reactor.

f_c is the flow rate of the coolant
T_ci is the temperature of the coolant going in
T_c is the temperature of the coolant going out

f is the flow rate of the reactant going in
T_i is the temperature of the reactant going in
c_Ai is the concentration of the reactant going in
T is the temperature of the reactant going out
c_A is the concentration of the reactant going out

The Attempt at a Solution

My problem is that when I plug in the values, I don't get 2.07 for $$\tau_1$$. I first assumed that $$\bar{f} = f= 1.3364$$ and $$\bar{T}_{\bar{c}_A} = 678.9$$. But I got $$\tau_1 = 0.5148$$.

I assumed that my $$\bar{f}$$ was correct, so I set $$\tau_1=2.07$$ and rearranged the equation to solve for $$\bar{T}_{\bar{c}_A}$$.

$$\bar{T}_{\bar{c}_A} = -\frac{E}{R}(ln(\frac{1-\frac{\tau_1 \bar{f}}{V}}{2\tau_1 k_o}))^{-1}$$ . . . (2)

When I do that I got $$\bar{T}_{\bar{c}_A} = 631$$, which doesn't match any of the given temperature. So I thought $$\bar{f}$$ might be 0.8771. And I got $$\bar{T}_{\bar{c}_A} = 633$$, which still doesn't match any number.

I have tried some other equations but I couldn't get any result to match. Since I don't have a background in studying chemical reactions, I was suspecting that there is something fundamental that I am missing here.

Do you know how the author got 2.07 for $$\tau_1$$?

- Thank you.

 

[tiny-tim]

Hi Saltine!

Look at the dimensions …

it isn't e^(-E/RTc_A⁾

it's (e^(-E/RT))c_A

 

[saltine]

That is right!

A second problem:

Now I am trying to verify T2. Here, Cv is the heat capacity at constant volume of the reaction mixture (Unit of Cv: Btu/lb-°R). However, the book did not give its value. This is the same for other examples where it seems that every other value is listed, except the Cv. Is it directly derivable from the other values?

U is the overall heat transfer coefficient, assumed to be constant, Btu/ft²-°R-min
A is heat transfer area, ft²
∆Hr is the heat of reaction, Btu/lbmole of A reacted
r_A is the rate of reaction


I first computed $$\bar{r}_A = 0.03938$$. Since Cv is just alone in the numerator, I first evaluated T2 without using Cv and got 0.2833. Therefore,

Cv = -7.96 / 0.2833 = -28.10 Btu/lb-°R

There is nowhere else in this particular example where Cv is used, so I couldn't check Cv in another way. Does it make sense that Cv is negative? This is an exothermic reaction. The problem didn't state this, but I suppose it is since coolant is used.