### Problem 5.72: Cold Air Chamber for Quenching Steel Ball BearingsA cold air chamber is proposed for quenching steel ball bearings with a diameter of \( D = 0.2 \, \text{m} \) and an initial temperature of \( T_i = 400^\circ \text{C} \). The air in the chamber is maintained at \(-15^\circ \text{C}\) by a refrigeration system. The steel balls pass through the chamber on a conveyor belt. Optimum production of the bearings requires that 70% of the initial thermal energy content of the ball above \(-15^\circ \text{C}\) be removed. Radiation effects can be neglected, and the convection heat transfer coefficient within the chamber is \( 1000 \, \text{W/m}^2 \cdot \text{K} \).**Objectives:**- Estimate the residence time of the balls within the chamber.- Recommend a drive velocity for the conveyor.**Properties for Steel:**- Thermal conductivity (\( k \)): \( 50 \, \text{W/m} \cdot \text{K} \)- Thermal diffusivity (\( \alpha \)): \( 2 \times 10^{-5} \, \text{m}^2/\text{s} \)- Specific heat capacity (\( c \)): \( 450 \, \text{J/kg} \cdot \text{K} \)**Diagram Explanation:**The diagram illustrates the setup for cooling the steel ball bearings. The key elements are:- **Conveyor Belt**: Transports the ball bearings through the chamber.- **Ball Bearings**: Move from left to right on the conveyor.- **Cold Air**: Circulates within the chamber housing to cool the bearings.- **Chamber Length**: The chamber extends 5 meters.- **Velocity (\( v \))**: Represents the speed of the conveyor belt.This setup is designed to efficiently transfer heat from the bearings to the cold air, ensuring optimal cooling performance as required for production.

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5.72 A cold air chamber is proposed for quenching steel ball bearings of diameter D = 0.2 m and initial temperature T₁ = 400°C. Air in the chamber is maintained at -15°C by a refrigeration system, and the steel balls pass through the chamber on a conveyor belt. Optimum bearing production requires that 70% of the initial ther- mal energy content of the ball above -15°C be removed. Radiation effects may be neglected, and the convection heat transfer coefficient within the chamber is 1000 W/m².K. Estimate the residence time of the balls within the chamber, and recommend a drive veloc- ity of the conveyor. The following properties may be used for the steel: k = 50 W/m K, a = 2 × 105 m²/s, and c = 450 J/kg. K. Ball bearing -5m- Cold air Belt V Chamber housing

5.72 A cold air chamber is proposed for quenching steel ball bearings of diameter D = 0.2 m and initial temperature T₁ = 400°C. Air in the chamber is maintained at -15°C by a refrigeration system, and the steel balls pass through the chamber on a conveyor belt. Optimum bearing production requires that 70% of the initial ther- mal energy content of the ball above -15°C be removed. Radiation effects may be neglected, and the convection heat transfer coefficient within the chamber is 1000 W/m².K. Estimate the residence time of the balls within the chamber, and recommend a drive veloc- ity of the conveyor. The following properties may be used for the steel: k = 50 W/m K, a = 2 × 105 m²/s, and c = 450 J/kg. K. Ball bearing -5m- Cold air Belt V Chamber housing

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

ChapterMA: Math Assessment

Section: Chapter Questions

Problem 1.1MA

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### Problem 5.72: Cold Air Chamber for Quenching Steel Ball Bearings

A cold air chamber is proposed for quenching steel ball bearings with a diameter of \( D = 0.2 \, \text{m} \) and an initial temperature of \( T_i = 400^\circ \text{C} \). The air in the chamber is maintained at \(-15^\circ \text{C}\) by a refrigeration system. The steel balls pass through the chamber on a conveyor belt. Optimum production of the bearings requires that 70% of the initial thermal energy content of the ball above \(-15^\circ \text{C}\) be removed. Radiation effects can be neglected, and the convection heat transfer coefficient within the chamber is \( 1000 \, \text{W/m}^2 \cdot \text{K} \).

**Objectives:**
- Estimate the residence time of the balls within the chamber.
- Recommend a drive velocity for the conveyor.

**Properties for Steel:**
- Thermal conductivity (\( k \)): \( 50 \, \text{W/m} \cdot \text{K} \)
- Thermal diffusivity (\( \alpha \)): \( 2 \times 10^{-5} \, \text{m}^2/\text{s} \)
- Specific heat capacity (\( c \)): \( 450 \, \text{J/kg} \cdot \text{K} \)

**Diagram Explanation:**

The diagram illustrates the setup for cooling the steel ball bearings. The key elements are:
- **Conveyor Belt**: Transports the ball bearings through the chamber.
- **Ball Bearings**: Move from left to right on the conveyor.
- **Cold Air**: Circulates within the chamber housing to cool the bearings.
- **Chamber Length**: The chamber extends 5 meters.
- **Velocity (\( v \))**: Represents the speed of the conveyor belt.

This setup is designed to efficiently transfer heat from the bearings to the cold air, ensuring optimal cooling performance as required for production.

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