Question

For identical feed comp., flow rate, conversion and for all positive reaction orders the ratio of the volume of mixed reactor to the volume of plug flow reactor

• A. Is independent of the order of reaction
• B. Increases with increase in the order of reaction
• C. Decreases with increase in the order of reaction
• D. Increases with increase in the percentage of conversion

Hint:

For positive reaction orders, a CSTR requires a larger volume than a PFR, and the volume ratio \( V_{\text{CSTR}} / V_{\text{PFR}} \) increases with reaction order due to the lower reaction rate in a CSTR at the outlet concentration.

Answer 1

The Correct Option is B

Step 1: Define the reactors and the problem. 
Mixed reactor (CSTR): A continuous stirred-tank reactor where the concentration is uniform throughout, equal to the outlet concentration.
Plug flow reactor (PFR): A reactor where the concentration decreases along the length, with no mixing in the axial direction.
The question asks for the ratio of the volume of a CSTR (\( V_{\text{CSTR}} \)) to the volume of a PFR (\( V_{\text{PFR}} \)) for the same feed composition, flow rate, conversion, and for all positive reaction orders (\( n>0 \)). 
Step 2: Set up the design equations. 
Assume a reaction \( A \to \text{products} \), with a rate law \( -r_A = k C_A^n \), where \( n \) is the reaction order (\( n>0 \)), \( k \) is the rate constant, and \( C_A \) is the concentration of A. Let:
\( F_{A0} \): Molar flow rate of A at the inlet,
\( X \): Fractional conversion of A,
\( C_{A0} \): Inlet concentration of A,
\( v_0 \): Volumetric flow rate (assumed constant for simplicity).
The outlet concentration is: \[ C_A = C_{A0} (1 - X). \] - CSTR design equation: For a CSTR, the concentration is uniform and equal to the outlet concentration \( C_A \). The design equation is: \[ V_{\text{CSTR}} = \frac{F_{A0} X}{-r_A} = \frac{F_{A0} X}{k C_A^n} = \frac{F_{A0} X}{k [C_{A0} (1 - X)]^n}. \] Since \( F_{A0} = C_{A0} v_0 \): \[ V_{\text{CSTR}} = \frac{C_{A0} v_0 X}{k [C_{A0} (1 - X)]^n} = \frac{v_0 X}{k C_{A0}^{n-1} (1 - X)^n}. \] PFR design equation: For a PFR, the concentration varies along the reactor. The design equation is: \[ V_{\text{PFR}} = \int_0^X \frac{F_{A0} dX}{-r_A}. \] Substitute \( -r_A = k C_A^n \), and \( C_A = C_{A0} (1 - X) \): \[ -r_A = k [C_{A0} (1 - X)]^n, \] \[ V_{\text{PFR}} = \int_0^X \frac{F_{A0} dX}{k [C_{A0} (1 - X)]^n} = \frac{F_{A0}}{k C_{A0}^n} \int_0^X \frac{dX}{(1 - X)^n}. \] Evaluate the integral: \[ \int_0^X \frac{dX}{(1 - X)^n} = \int_0^X (1 - X)^{-n} dX. \] Let \( u = 1 - X \), so \( dX = -du \), and the limits change: when \( X = 0 \), \( u = 1 \); when \( X = X \), \( u = 1 - X \): \[ \int_0^X (1 - X)^{-n} dX = \int_1^{1-X} u^{-n} (-du) = \int_{1-X}^1 u^{-n} du. \] \[ = \left[ \frac{u^{-n+1}}{-n+1} \right]_{1-X}^1 = \frac{1^{-n+1}}{-n+1} - \frac{(1-X)^{-n+1}}{-n+1} = \frac{1}{-n+1} \left[ 1 - (1-X)^{-n+1} \right] \quad (\text{for } n \neq 1). \] For \( n = 1 \), the integral is: \[ \int_0^X \frac{dX}{1 - X} = -\ln(1 - X) \Big|_0^X = -\ln(1 - X) + \ln(1) = -\ln(1 - X). \] So: For \( n \neq 1 \): \[ V_{\text{PFR}} = \frac{F_{A0}}{k C_{A0}^n} \cdot \frac{1 - (1-X)^{-n+1}}{-n+1} = \frac{v_0 C_{A0}}{k C_{A0}^n} \cdot \frac{(1-X)^{-n+1} - 1}{n-1}. \] For \( n = 1 \): \[ V_{\text{PFR}} = \frac{v_0 C_{A0}}{k C_{A0}} [-\ln(1 - X)] = \frac{v_0}{k} [-\ln(1 - X)]. \] 
Step 3: Compute the volume ratio. 
For \( n \neq 1 \): \[ \frac{V_{\text{CSTR}}}{V_{\text{PFR}}} = \frac{\frac{v_0 X}{k C_{A0}^{n-1} (1 - X)^n}}{\frac{v_0 C_{A0}}{k C_{A0}^n} \cdot \frac{(1-X)^{-n+1} - 1}{n-1}} = \frac{X}{(1 - X)^n} \cdot \frac{n-1}{(1-X)^{-n+1} - 1} \cdot C_{A0}^{1-n} \cdot C_{A0}^{n-1} = \frac{X (n-1)}{(1 - X)^n [(1-X)^{-n+1} - 1]}. \] Simplify: \[ \frac{V_{\text{CSTR}}}{V_{\text{PFR}}} = \frac{X (n-1)}{(1 - X)^n \left[ \frac{1}{(1-X)^{n-1}} - 1 \right]} = \frac{X (n-1)}{(1 - X)^n \cdot \frac{1 - (1-X)^{n-1}}{(1-X)^{n-1}}} = \frac{X (n-1)}{(1 - X) [1 - (1-X)^{n-1}]}. \] For \( n = 1 \): \[ V_{\text{CSTR}} = \frac{v_0 X}{k (1 - X)}, \] \[ V_{\text{PFR}} = \frac{v_0}{k} [-\ln(1 - X)], \] \[ \frac{V_{\text{CSTR}}}{V_{\text{PFR}}} = \frac{\frac{v_0 X}{k (1 - X)}}{\frac{v_0}{k} [-\ln(1 - X)]} = \frac{X}{(1 - X) [-\ln(1 - X)]}. \] 
Step 4: Analyze the effect of reaction order \( n \). 
For \( n = 1 \), the ratio is \( \frac{X}{(1 - X) [-\ln(1 - X)]} \).
For \( n = 2 \): \[ \frac{V_{\text{CSTR}}}{V_{\text{PFR}}} = \frac{X (2-1)}{(1 - X) [1 - (1-X)^{2-1}]} = \frac{X}{(1 - X) [1 - (1-X)]} = \frac{X}{(1 - X) X} = \frac{1}{1 - X}. \] For \( n = 3 \): \[ \frac{V_{\text{CSTR}}}{V_{\text{PFR}}} = \frac{X (3-1)}{(1 - X) [1 - (1-X)^{3-1}]} = \frac{2X}{(1 - X) [1 - (1-X)^2]}. \] Notice that as \( n \) increases:
The denominator \( 1 - (1-X)^{n-1} \) becomes smaller because \( (1-X)^{n-1} \) decreases more rapidly for higher \( n \).
The numerator increases with \( n-1 \).
Thus, the ratio \( \frac{V_{\text{CSTR}}}{V_{\text{PFR}}} \) increases as the reaction order \( n \) increases. For positive reaction orders (\( n>0 \)), a CSTR requires a larger volume than a PFR to achieve the same conversion, and this difference grows with higher reaction orders because the reaction rate in a CSTR is based on the outlet (lowest) concentration, while a PFR benefits from higher concentrations along its length. 
Step 5: Evaluate the options. 
(1) Is independent of the order of reaction: Incorrect, as the ratio depends on \( n \). Incorrect.
(2) Increases with increase in the order of reaction: Correct, as shown by the increasing trend with \( n \). Correct.
(3) Decreases with increase in the order of reaction: Incorrect, as the ratio increases, not decreases. Incorrect.
(4) Increases with increase in the percentage of conversion: Incorrect, as the trend with reaction order is the focus, and the effect of conversion varies depending on \( X \). Incorrect.
Step 6: Select the correct answer. 
For identical feed composition, flow rate, conversion, and for all positive reaction orders, the ratio of the volume of a mixed reactor to the volume of a plug flow reactor increases with an increase in the order of reaction, matching option (2).