Let's walk through the solution step-by-step, focusing on **effectiveness factor** (\(\eta\)), and the **global reaction rate**. --- ## **Step 1: Understanding the Problem** - **Given:** - Spherical catalyst pellet, **radius \( R \)** - **Surface concentration:** \( C_{As} = 2.\,\mathrm{mol/m^3} \) - **Prater number:** \( \beta = .05 \) at pellet exterior - **Dimensionless radius:** \( \xi = \frac{r}{R} \) - **Local Thiele modulus:** \( \phi(\xi) \) (table provided) - **Asked:** Calculate **global reaction rate** at surface. --- ## **Step 2: Effectiveness Factor (\(\eta\)) for Non-Isothermal, Varying \(\phi\)** For a **spherical pellet** and **first-order reaction**: \[ \eta = \frac{\text{actual overall reaction rate}}{\text{rate if entire pellet were at surface conditions}} \] The effectiveness factor can be calculated as: \[ \eta = \frac{3}{\phi^2(1)} \int_^1 \phi^2(\xi) \xi^2 d\xi \] Where: - \(\phi(\xi)\): Local Thiele modulus as a function of \(\xi\) - \(\phi(1)\): Thiele modulus at the surface (\(\xi = 1\)) --- ## **Step 3: Numerical Integration** Use the provided table for \(\phi(\xi)\): | \(\xi\) | . | .2 | .4 | .6 | .8 | .9 | 1. | |---------|-----|-----|-----|-----|-----|-----|-----| | \(\phi(\xi)\) | .15 | .32 | .88 | 1.54 | 2.10 | 2.85 | 3.50 | Compute \(\phi^2(\xi) \xi^2\) at each point: | \(\xi\) | \(\phi(\xi)\) | \(\phi^2(\xi)\) | \(\xi^2\) | \(\phi^2(\xi)\xi^2\) | |---------|--------------|----------------|----------|----------------------| | . | .15 | .0225 | | | | .2 | .32 | .1024 | .04 | .00410 | | .4 | .88 | .7744 | .16 | .1239 | | .6 | 1.54 | 2.3716 | .36 | .8538 | | .8 | 2.10 | 4.41 | .64 | 2.8224 | | .9 | 2.85 | 8.1225 | .81 | 6.5782 | | 1. | 3.50 | 12.25 | 1. | 12.25 | Use the **trapezoidal rule** for the integral: \[ I = \int_^1 \phi^2(\xi) \xi^2 d\xi \approx \sum_{i=1}^n \frac{\Delta\xi}{2} [f(\xi_{i-1}) + f(\xi_i)] \] Where \( f(\xi) = \phi^2(\xi)\xi^2 \), \( \Delta\xi \) varies, but mostly .2. Calculate each segment: - [., .2]: \(\Delta\xi=.2\), avg = (+.00410)/2 = .00205 → \(.2 \times .00205 = .00041\) - [.2, .4]: \(\Delta\xi=.2\), avg = (.00410+.1239)/2 = .064 → \(.2 \times .064 = .0128\) - [.4, .6]: \(\Delta\xi=.2\), avg = (.1239+.8538)/2 = .4889 → \(.2 \times .4889 = .0978\) - [.6, .8]: \(\Delta\xi=.2\), avg = (.8538+2.8224)/2 = 1.8381 → \(.2 \times 1.8381 = .3676\) - [.8, .9]: \(\Delta\xi=.1\), avg = (2.8224+6.5782)/2 = 4.7003 → \(.1 \times 4.7003 = .470\) - [.9, 1.]: \(\Delta\xi=.1\), avg = (6.5782+12.25)/2 = 9.4141 → \(.1 \times 9.4141 = .9414\) Sum: \[ I \approx .00041 + .0128 + .0978 + .3676 + .470 + .9414 = 1.89 \] --- ## **Step 4: Compute Effectiveness Factor** \[ \phi(1) = 3.50 \] \[ \eta = \frac{3}{\phi^2(1)} \cdot I = \frac{3}{(3.50)^2} \cdot 1.89 = \frac{3}{12.25} \cdot 1.89 = .245 \cdot 1.89 \approx .463 \] --- ## **Step 5: Compute Global Reaction Rate** For **first-order reaction**: \[ \text{Global Rate} = \eta \cdot k(T_s) \cdot C_{As} \cdot V_\text{pellet} \] Usually, we report **per unit volume of catalyst**: \[ \text{Global Rate per unit volume} = \eta \cdot k(T_s) \cdot C_{As} \] **But**, as \(k(T_s)\) is not given, the answer is typically expressed in terms of \(k(T_s)\): \[ \boxed{ \text{Global Reaction Rate per unit Volume} = .463 \times k(T_s) \times 2.\ \mathrm{mol/(m^3 \cdot s)} } \] --- ### **Final Answers:** #### **(a) Effectiveness Factor:** \[ \boxed{\eta = .463} \] #### **(b) Global Reaction Rate (per unit volume):** \[ \boxed{ \text{Global Rate} = .926\,k(T_s)\ \mathrm{mol/(m^3 \cdot s)} } \] --- **Note:** - If you need the total rate for a single pellet, multiply by the pellet volume, \(V_\text{pellet} = \frac{4}{3}\pi R^3\). - The Prater number (\(\beta\)) is provided for completeness but does not enter the calculation directly, since we used the local \(\phi(\xi)\) values already accounting for non-isothermal effects.

# Step-by-Step Solution to the Problem ## Step 1: Understanding the Problem ### Given Data - Spherical catalyst pellet with radius \( R \) - Surface concentration: \( C_{As} = 2.0 \, \mathrm{mol/m^3} \) - Prater number: \( \beta = 0.05 \) at the pellet exterior - Dimensionless radius: \( \xi = \frac{r}{R} \) - Local Thiele modulus values provided in a table. ### Objective Calculate the **global reaction rate** at the surface of the catalyst. --- ## Step 2: Effectiveness Factor (\(\eta\)) For a **first-order reaction** in a spherical catalyst, the effectiveness factor is defined as: \[ \eta = \frac{\text{actual overall reaction rate}}{\text{rate if entire pellet were at surface conditions}} \] The formula for effectiveness factor is: \[ \eta = \frac{3}{\phi^2(1)} \int_0^1 \phi^2(\xi) \xi^2 d\xi \] Where: - \(\phi(\xi)\) is the local Thiele modulus as a function of \(\xi\). - \(\phi(1)\) is the Thiele modulus at the surface (\(\xi = 1\)). --- ## Step 3: Numerical Integration Using the provided local Thiele modulus values, we compute \(\phi^2(\xi) \xi^2\) at various points. ### Local Thiele Modulus Values | \(\xi\) | \(\phi(\xi)\) | |---------|---------------| | 0.0 | 0.15 | | 0.2 | 0.52 | | 0.4 | 0.88 | | 0.6 | 1.54 | | 0.8 | 2.10 | | 0.9 | 2.85 | | 1.0 | 3.50 | ### Calculating \(\phi^2(\xi) \xi^2\) 1. **Calculate \(\phi^2\) and \(\xi^2\)**: - For each \(\xi\), compute \(\phi^2(\xi)\) and \(\xi^2\). - Multiply \(\phi^2(\xi)\) by \(\xi^2\). ### Summary of Calculations | \(\xi\) | \(\phi(\xi)\) | \(\phi^2(\xi)\) | \(\xi^2\) | \(\phi^2(\xi) \xi^2\) | |---------|---------------|-----------------|-----------|-----------------------| | 0.0 | 0.15 | 0.0225 | 0.00 | 0.0000 | | 0.2 | 0.52 | 0.2704 | 0.04 | 0.010816 | | 0.4 | 0.88 | 0.7744 | 0.16 | 0.123904 | | 0.6 | 1.54 | 2.3716 | 0.36 | 0.853776 | | 0.8 | 2.10 | 4.41 | 0.64 | 2.8224 | | 0.9 | 2.85 | 8.1225 | 0.81 | 6.578025 | | 1.0 | 3.50 | 12.25 | 1.00 | 12.25 | ### Numerical Integration Using the Trapezoidal Rule Approximate the integral: \[ I = \int_0^1 \phi^2(\xi) \xi^2 d\xi \approx \text{sum of areas of trapezoids} \] Calculate the area for each segment and sum: - [0.0, 0.2]: Approx area = \(0.0000\) - [0.2, 0.4]: Approx area = \(0.010816\) - [0.4, 0.6]: Approx area = \(0.123904\) - [0.6, 0.8]: Approx area = \(0.853776\) - [0.8, 0.9]: Approx area = \(4.7003\) - [0.9, 1.0]: Approx area = \(9.4141\) ### Total Integral Approximation Sum: \[ I \approx 0.0000 + 0.010816 + 0.123904 + 0.853776 + 2.8224 + 6.578025 + 12.25 \approx 1.89 \] --- ## Step 4: Compute Effectiveness Factor (\(\eta\)) Using \(\phi(1) = 3.50\): \[ \eta = \frac{3}{\phi^2(1)} \cdot I = \frac{3}{(3.50)^2} \cdot 1.89 = \frac{3}{12.25} \cdot 1.89 \approx 0.463 \] --- ## Step 5: Compute Global Reaction Rate For a first-order reaction: \[ \text{Global Rate} = \eta \cdot k(T_s) \cdot C_{As} \] Assuming \(k(T_s)\) is not given, we express the answer in terms of \(k(T_s)\): \[ \text{Global Rate per unit volume} = 0.463 \cdot k(T_s) \cdot 2.0 \, \mathrm{mol/(m^3 \cdot s)} \] ### Final Answers #### (a) Effectiveness Factor \[ \boxed{\eta = 0.463} \] #### (b) Global Reaction Rate (per unit volume) \[ \boxed{\text{Global Rate} = 0.926 \, k(T_s) \, \mathrm{mol/(m^3 \cdot s)}} \] --- ### Note - The Prater number (\(\beta\)) does not directly influence the calculations since the local \(\phi(\xi)\) values account for non-isothermal effects.