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The irreversible zero order reaction \( \mathbf{A} \rightarrow \mathbf{B} \) takes place in a porous cylindrical catalyst that is sealed at both ends as shown in figure. Assume dilute concentration and neglect any variations in the axis direction. (4)The steady state concentration profile is \( \frac{C_{A}}{C_{A_{S}}}=1+\frac{\Phi_{O}^{2}}{8}\left[\left(\frac{r}{R}\right)^{2}-1\right] \)Where \( \Phi_{0} \) is the Thiele modulus. For \( \Phi_{0}=4 \). Calculate the followingA. The range of \( r \) where \( \mathrm{C}_{\mathrm{A}}=0 \).B. Percentage of total radius of catalyst remain unused.?
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The irreversible zero order reaction \( \mathbf{A} \rightarrow \mathbf...
The Steady State Concentration Profile
To analyze the irreversible zero-order reaction \( \mathbf{A} \rightarrow \mathbf{B} \) in a porous cylindrical catalyst, we can use the given concentration profile:
\[
\frac{C_{A}}{C_{A_{S}}} = 1 + \frac{\Phi_{O}^{2}}{8}\left[\left(\frac{r}{R}\right)^{2} - 1\right]
\]
Where \( \Phi_{O} \) is the Thiele modulus.
A. Range of \( r \) where \( C_{A} = 0 \)
To find the range of \( r \) where \( C_{A} = 0 \), set the concentration equation to zero:
\[
0 = 1 + \frac{\Phi_{O}^{2}}{8}\left[\left(\frac{r}{R}\right)^{2} - 1\right]
\]
Solving for \( r \):
\[
\frac{\Phi_{O}^{2}}{8}\left[\left(\frac{r}{R}\right)^{2} - 1\right] = -1
\]
This leads to:
\[
\left(\frac{r}{R}\right)^{2} = 1 - \frac{8}{\Phi_{O}^{2}}
\]
Substituting \( \Phi_{O} = 4 \):
\[
\left(\frac{r}{R}\right)^{2} = 1 - \frac{8}{16} = 0.5
\]
Thus:
\[
\frac{r}{R} = \sqrt{0.5} \Rightarrow r = R \cdot \frac{1}{\sqrt{2}} \approx 0.707R
\]
B. Percentage of Total Radius of Catalyst Remaining Unused
The unused catalyst radius corresponds to the area where \( C_{A} = C_{A_{S}} \).
1. The effective reaction zone is from \( r = 0 \) to \( r = 0.707R \).
2. The total radius of the catalyst is \( R \).
Calculating the percentage of the unused area:
- Radius where \( C_{A} = C_{A_{S}} \): \( R - 0.707R = 0.293R \)
- Percentage of unused catalyst:
\[
\text{Percentage} = \left( \frac{0.293R}{R} \right) \times 100 \approx 29.3\%
\]
Thus, approximately 29.3% of the total radius of the catalyst remains unused, indicating that a significant portion of the catalyst is not participating in the reaction.
To analyze the irreversible zero-order reaction \( \mathbf{A} \rightarrow \mathbf{B} \) in a porous cylindrical catalyst, we can use the given concentration profile:
\[
\frac{C_{A}}{C_{A_{S}}} = 1 + \frac{\Phi_{O}^{2}}{8}\left[\left(\frac{r}{R}\right)^{2} - 1\right]
\]
Where \( \Phi_{O} \) is the Thiele modulus.
A. Range of \( r \) where \( C_{A} = 0 \)
To find the range of \( r \) where \( C_{A} = 0 \), set the concentration equation to zero:
\[
0 = 1 + \frac{\Phi_{O}^{2}}{8}\left[\left(\frac{r}{R}\right)^{2} - 1\right]
\]
Solving for \( r \):
\[
\frac{\Phi_{O}^{2}}{8}\left[\left(\frac{r}{R}\right)^{2} - 1\right] = -1
\]
This leads to:
\[
\left(\frac{r}{R}\right)^{2} = 1 - \frac{8}{\Phi_{O}^{2}}
\]
Substituting \( \Phi_{O} = 4 \):
\[
\left(\frac{r}{R}\right)^{2} = 1 - \frac{8}{16} = 0.5
\]
Thus:
\[
\frac{r}{R} = \sqrt{0.5} \Rightarrow r = R \cdot \frac{1}{\sqrt{2}} \approx 0.707R
\]
B. Percentage of Total Radius of Catalyst Remaining Unused
The unused catalyst radius corresponds to the area where \( C_{A} = C_{A_{S}} \).
1. The effective reaction zone is from \( r = 0 \) to \( r = 0.707R \).
2. The total radius of the catalyst is \( R \).
Calculating the percentage of the unused area:
- Radius where \( C_{A} = C_{A_{S}} \): \( R - 0.707R = 0.293R \)
- Percentage of unused catalyst:
\[
\text{Percentage} = \left( \frac{0.293R}{R} \right) \times 100 \approx 29.3\%
\]
Thus, approximately 29.3% of the total radius of the catalyst remains unused, indicating that a significant portion of the catalyst is not participating in the reaction.
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The irreversible zero order reaction \( \mathbf{A} \rightarrow \mathbf{B} \) takes place in a porous cylindrical catalyst that is sealed at both ends as shown in figure. Assume dilute concentration and neglect any variations in the axis direction. (4)The steady state concentration profile is \( \frac{C_{A}}{C_{A_{S}}}=1+\frac{\Phi_{O}^{2}}{8}\left[\left(\frac{r}{R}\right)^{2}-1\right] \)Where \( \Phi_{0} \) is the Thiele modulus. For \( \Phi_{0}=4 \). Calculate the followingA. The range of \( r \) where \( \mathrm{C}_{\mathrm{A}}=0 \).B. Percentage of total radius of catalyst remain unused.?
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The irreversible zero order reaction \( \mathbf{A} \rightarrow \mathbf{B} \) takes place in a porous cylindrical catalyst that is sealed at both ends as shown in figure. Assume dilute concentration and neglect any variations in the axis direction. (4)The steady state concentration profile is \( \frac{C_{A}}{C_{A_{S}}}=1+\frac{\Phi_{O}^{2}}{8}\left[\left(\frac{r}{R}\right)^{2}-1\right] \)Where \( \Phi_{0} \) is the Thiele modulus. For \( \Phi_{0}=4 \). Calculate the followingA. The range of \( r \) where \( \mathrm{C}_{\mathrm{A}}=0 \).B. Percentage of total radius of catalyst remain unused.? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about The irreversible zero order reaction \( \mathbf{A} \rightarrow \mathbf{B} \) takes place in a porous cylindrical catalyst that is sealed at both ends as shown in figure. Assume dilute concentration and neglect any variations in the axis direction. (4)The steady state concentration profile is \( \frac{C_{A}}{C_{A_{S}}}=1+\frac{\Phi_{O}^{2}}{8}\left[\left(\frac{r}{R}\right)^{2}-1\right] \)Where \( \Phi_{0} \) is the Thiele modulus. For \( \Phi_{0}=4 \). Calculate the followingA. The range of \( r \) where \( \mathrm{C}_{\mathrm{A}}=0 \).B. Percentage of total radius of catalyst remain unused.? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The irreversible zero order reaction \( \mathbf{A} \rightarrow \mathbf{B} \) takes place in a porous cylindrical catalyst that is sealed at both ends as shown in figure. Assume dilute concentration and neglect any variations in the axis direction. (4)The steady state concentration profile is \( \frac{C_{A}}{C_{A_{S}}}=1+\frac{\Phi_{O}^{2}}{8}\left[\left(\frac{r}{R}\right)^{2}-1\right] \)Where \( \Phi_{0} \) is the Thiele modulus. For \( \Phi_{0}=4 \). Calculate the followingA. The range of \( r \) where \( \mathrm{C}_{\mathrm{A}}=0 \).B. Percentage of total radius of catalyst remain unused.?.
The irreversible zero order reaction \( \mathbf{A} \rightarrow \mathbf{B} \) takes place in a porous cylindrical catalyst that is sealed at both ends as shown in figure. Assume dilute concentration and neglect any variations in the axis direction. (4)The steady state concentration profile is \( \frac{C_{A}}{C_{A_{S}}}=1+\frac{\Phi_{O}^{2}}{8}\left[\left(\frac{r}{R}\right)^{2}-1\right] \)Where \( \Phi_{0} \) is the Thiele modulus. For \( \Phi_{0}=4 \). Calculate the followingA. The range of \( r \) where \( \mathrm{C}_{\mathrm{A}}=0 \).B. Percentage of total radius of catalyst remain unused.? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about The irreversible zero order reaction \( \mathbf{A} \rightarrow \mathbf{B} \) takes place in a porous cylindrical catalyst that is sealed at both ends as shown in figure. Assume dilute concentration and neglect any variations in the axis direction. (4)The steady state concentration profile is \( \frac{C_{A}}{C_{A_{S}}}=1+\frac{\Phi_{O}^{2}}{8}\left[\left(\frac{r}{R}\right)^{2}-1\right] \)Where \( \Phi_{0} \) is the Thiele modulus. For \( \Phi_{0}=4 \). Calculate the followingA. The range of \( r \) where \( \mathrm{C}_{\mathrm{A}}=0 \).B. Percentage of total radius of catalyst remain unused.? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The irreversible zero order reaction \( \mathbf{A} \rightarrow \mathbf{B} \) takes place in a porous cylindrical catalyst that is sealed at both ends as shown in figure. Assume dilute concentration and neglect any variations in the axis direction. (4)The steady state concentration profile is \( \frac{C_{A}}{C_{A_{S}}}=1+\frac{\Phi_{O}^{2}}{8}\left[\left(\frac{r}{R}\right)^{2}-1\right] \)Where \( \Phi_{0} \) is the Thiele modulus. For \( \Phi_{0}=4 \). Calculate the followingA. The range of \( r \) where \( \mathrm{C}_{\mathrm{A}}=0 \).B. Percentage of total radius of catalyst remain unused.?.
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