Chapter 5: Isothermal Reactor Design: Conversion
Converting from Length (z) to Catalyst Weight (W):
Starting with this equation
\(\frac{dP}{dz} = -\beta_0 \frac{P_0}{P} \frac{T}{T_0} \frac{F_T}{F_{T0}} \tag{1}\) (1)
we begin by multiplying the top and bottom of the left-hand side of equation (1) by the same combination of constants (which is equivalent to multiplying equation (1) by one):
\(\frac{dP}{dz} = \frac{\frac{dP}{dz} A_c (1 - \phi) \rho_c}{A_c (1 - \phi) \rho_c}\)
Since the product of \(A_c (1 - \phi) \rho_c\) is constant, we can take it inside the derivative in the denominator to combine it with our length (z):
\(\frac{dP}{dz} = \frac{\frac{dP}{d\left[z A_c (1 - \phi) \rho_c\right]}}{A_c (1 - \phi) \rho_c}\)
We'll convert from reactor length (z) as our dimension to catalyst weight (W) by making use of the equation for catalyst weight:
\(W = z A_c \rho_b = z A_c (1 - \phi) \rho_c\)
Where:
\(\rho_b = \text{bulk density}\)
\(\rho_c = \text{solid catalyst density}\)
\(\phi = \text{porosity (a.k.a., void fraction)}\)
Then:
\(\frac{dP}{dz} = \frac{dP}{dW} A_c (1 - \phi) \rho_c\)
With a little rearranging:
\(\frac{dP}{dW} = \frac{dP}{dz} \frac{1}{A_c (1 - \phi) \rho_c}\)
which we substitute into equation (1) to get:
\(\frac{dP}{dW} = \frac{-\beta_0 P_0 T F_T}{A_c (1 - \phi) \rho_c P T_0 F_{T0}}\)