# Solution Explanation Let's break down the problem step-by-step: ## **Given Data** - Applied Pressure = 1200 kPa - \( E = 24 \) MPa - \( R_1 = R_2 = 300 \) mm - \( L = 2000 \) mm - Web speed = 200 m/min - \( T_{in} = 200 \) mm - \( T_{out} = 65 \) mm We are asked to find: - **Contact time** - **Gap between rollers** --- ## **1. Contact Time Calculation** **Contact time** is the time for which a point on the web stays in contact with the rollers. It is given by: \[ t = \frac{2d}{V} \] Where: - \( 2d \): Contact width (distance the web travels while in contact with the rollers) - \( V \): Web speed ### **a) Find Contact Width \( 2d \)** For two rollers of equal radii (\( R_1 = R_2 = R \)), the formula for contact width is: \[ 2d = \sqrt{ \frac{P R}{E L} } \] Where: - \( P \): Applied pressure (in N/m²) - \( R \): Radius of rollers (in m) - \( E \): Young's modulus (in N/m²) - \( L \): Length of roller (in m) **Let's use the units:** - \( P = 1200 \) kPa = \( 1,200,000 \) N/m² - \( R = 300 \) mm = \( 0.3 \) m - \( E = 24 \) MPa = \( 24,000,000 \) N/m² - \( L = 2000 \) mm = \( 2 \) m Plug the values in: \[ 2d = \sqrt{ \frac{1,200,000 \times 0.3}{24,000,000 \times 2} } \] \[ 2d = \sqrt{ \frac{360,000}{48,000,000} } \] \[ 2d = \sqrt{0.0075} \] \[ 2d \approx 0.0866 \text{ m} = 8.66 \text{ cm} \] --- ### **b) Calculate Contact Time** Web speed (\( V \)) = 200 m/min = \( \frac{200}{60} \) m/s ≈ 3.33 m/s \[ t = \frac{2d}{V} = \frac{0.0866}{3.33} \approx 0.026 \text{ s} \] So, **contact time ≈ 0.026 s (26 ms)** --- ## **2. Gap between Rollers** The **gap** is the distance between the rollers at the point of minimum separation, i.e., the thickness of the material at the exit (\( T_{out} \)). Given: - \( T_{out} = 65 \) mm So, the **gap between rollers = 65 mm**. --- ## **Summary Table** | Parameter | Value | |---------------------------|--------------| | Contact width (\(2d\)) | 8.66 cm | | Contact time (\(t\)) | 0.026 s | | Gap between rollers | 65 mm | --- ### **Final Answers** - **Contact time:** \( 0.026 \) seconds (or 26 ms) - **Gap between rollers:** \( 65 \) mm --- **Let me know if you need any further explanation on the derivations or formulas!**