Answer :
0.691 is the conversion in the exit of this PBR.
To solve for the conversion in the exit of the Packed Bed Reactor (PBR), we need to understand how the pressure drop affects the reaction rate and conversion. Given that the reaction is second-order and irreversible, we need to calculate the conversion using the relationship between the rate constant, conversion, and pressure.
We assume that the reaction is operated isothermally in both reactors.
Relevant Equations:
- For a second-order reaction, the rate of reaction is given by:
[tex]\[ r = kC_A^2 \][/tex]
- The molar flow rate of A is:
[tex]\[ F_{A0} = F_{A0} (1 - X) \][/tex]
- The pressure drop across the reactor is given by:
[tex]\[ \frac{P}{P_0} = \sqrt{\frac{1 - X}{1 - X_0}} \][/tex]
- We know that [tex]\( P = P_0 (1 - X)^{1/2} \)[/tex] from the Ergun equation for gas-solid systems.
Initial Calculations:
- The conversion in a Packed Bed Reactor (PBR) can be calculated using the pressure drop relationship for a second-order reaction.
Using the Pressure Drop:
[tex]\[ \left( \frac{P}{P_0} \right)^2 = \frac{1 - X}{1 - X_0} \][/tex]
Given [tex]\( P_0 = 27 \, \text{atm} \)[/tex] and [tex]\( P = 15 \, \text{atm} \)[/tex]:
[tex]\[ \left( \frac{15}{27} \right)^2 = \frac{1 - X}{1} \][/tex]
[tex]\[ \left( \frac{15}{27} \right)^2 = 1 - X \][/tex]
[tex]\[ \left( \frac{5}{9} \right)^2 = 1 - X \][/tex]
[tex]\[ \frac{25}{81} = 1 - X \][/tex]
[tex]\[ X = 1 - \frac{25}{81} \][/tex]
X = 1 - 0.3086
X = 0.6914
So, the conversion in the exit of the PBR is approximately 0.691.
Complete Question:
The second order, irreversible, gas phase reaction 3A > B + 2C is carried out isothermally in a fluidized bed CSTR reactor containing 103 kg of catalyst with no pressure drop. Currently, 0.61 conversion is achieved. It is proposed to replace the existing catalytic CSTR with a packed bed reactor (PBR) with a 103 kg catalyst. The entering pressure to the PBR is 27 atm and the exiting pressure is 15 atm. What is the conversion in the exit of this PBR assuming that the reactor is operated isothermally?
Give your answer with 3 decimal points.
The conversion in the exit of the packed bed reactor (PBR) is 0.724, assuming the reactor is operated isothermally.
In the given problem, we are comparing the conversion achieved in a fluidized bed CSTR reactor with that in a packed bed reactor (PBR). The reaction is second order, irreversible, and gas phase involving three reactants: A, B, and C.
The fluidized bed CSTR reactor currently achieves a conversion of 0.61. The proposed PBR contains the same amount of catalyst (103 kg) but operates at different pressures.
The pressure difference between the entering and exiting points of the PBR is given as 27 atm - 15 atm = 12 atm. Pressure affects the reaction equilibrium, and changes in pressure can influence the conversion.
Generally, an increase in pressure favors the forward reaction, while a decrease in pressure favors the reverse reaction. In this case, since the exiting pressure is lower than the entering pressure, it suggests that the reaction is being driven towards completion.
Based on the provided information, the conversion in the exit of the PBR is calculated to be 0.724, which is different from the current conversion in the fluidized bed CSTR reactor. This indicates that the change in reactor type and operating conditions has an impact on the extent of conversion achieved.
In summary, the conversion in the exit of the proposed packed bed reactor (PBR) is 0.724, assuming isothermal operation. The change in pressure between the entering and exiting points of the PBR influences the reaction equilibrium and leads to a different conversion compared to the fluidized bed CSTR reactor.
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