Type 4 Home Problem
Problems that are under-specified and require the student to consult other information sources.

PART 2 - Type 4 Solution

In Part 2, we must derive the equilibrium constant (K_C) and the diffusion rate constant (k_B). Often the exact value that is required is either too cumbersome to calculate or unavailable in the literature. Because we are just looking for estimates of K_C and k_B, several assumptions will be made. We will start with our derivation of K_C.

We begin with a thermodynamic equation for K_C:

From Perry's Chemical Engineering Handbook ¹, we find the following free energies of formation at 298 K for our components:

We then calculate the change in free energy for our reaction:

and insert the value into our K_C equation:

This value for K_C favors our products.

Our next task is to calculate our diffusion rate constant (k_B). To calculate it, we will need to introduce three new dimensionless constants--the Reynolds Number (Re), the Schmidt Number (Sc), and the Sherwood Number(Sh):

A correlation from Fundamentals of Momentum, Heat and Mass Transfer ² by Welty, Wicks, and Wilson (or simply WWW), we can relate k_B to Re and Sc through the Sherwood number (Sh):

This equation is valid for 2000 < Re < 70000 and 1 < Sc < 2260.

First, we will approximate the bulk properties of our fluid by assuming that throughout the reactor, our fluid is half A and half C. From WWW, we found m_C=1.785*10^-5 kg /(m*s) (1 cP=0.1 Pa*s or kg/(m*s)), and from Perry's (Table 3-311), we found the viscosity of acetone. For the sake of these calculations we will assume that the viscosities of acetone and formaldehyde (A) are similar, m_A=0.75*10^-5 kg/(m*s).

The density of the bulk fluid will also be calculated using the 50-50% assumption:

We need to calculate the superficial velocity of our fluid (u_bulk). To do so we will assume the volumetric flow rate (v_o) stays constant throughout the reactor:

Now, we calculate Re:

In order to determine Sc, we need to find the kinematic viscosity of the hydrogen (B). In WWW, we find that u_B = 1.096*10^-4 sec/m². We then calculate the Schmidt number (Sh):

Using the Sherwood correlation, we solve for k_B':

Unfortunately, k_B' is the rate constant per membrane surface area. We need the diffusion rate constant (k_B) per volume of the reactor. This conversion can be performed quite easily. We multiply by the perimeter and divide by the cross-sectional area:

The diffusion rate constant per minute:

k_B = 4.2/min

Go to Part 3 of the Solution - Polymath

Back to PART 1 of the Solution

CRE Thoughts Ten Types Homogeneous Example 4  

REFERENCES:

1. Perry, R.H. and Green, D.W. (Editors). Perry's Chemical Engineering Handbook, New York: MacGraw-Hill Inc., 1984.
   

2. Welty, J.R., Wicks, C.E., & Wilson, R.E. Fundamentals of Momentum, Heat and Mass Transfer, New York: Wiley & Sons, 1984.