A long roll of 2-m-wide and 0.5-cm-thick 1-Mn manganese steel plate coming off a furnace at 820^

step 1 In this question it is told that a monoatomic gas is compressed adiabatically to one -fourth of

A long roll of 2-m-wide and 0.5-cm-thick 1-Mn manganese...

A long roll of $2-\mathrm{m}$-wide and $0.5-\mathrm{cm}$-thick $1-\mathrm{Mn}$ manganese steel plate coming off a furnace at $820^{\circ} \mathrm{C}$ is to be quenched in an oil bath $\left(c_p=2.0 \mathrm{~kJ} / \mathrm{kg} \cdot{ }^{\circ} \mathrm{C}\right)$ at $45^{\circ} \mathrm{C}$. The metal sheet is moving at a steady velocity of $15 \mathrm{~m} / \mathrm{min}$, and the oil bath is 9 m long. Taking the convection heat transfer coefficient on both sides of the plate to be $860 \mathrm{~W} / \mathrm{m}^2 \cdot{ }^{\circ} \mathrm{C}$, determine the temperature of the sheet metal when it leaves the oil bath. Also, determine the required rate of heat removal from the oil to keep its temperature constant at $45^{\circ} \mathrm{C}$.

Key Concepts

Newton's Law of Cooling

Newton’s law of cooling states that the rate of heat transfer from a surface to the surrounding fluid is proportional to the temperature difference between the surface and the fluid. This principle is key in convection processes, where the convective heat transfer coefficient quantifies the proportionality, allowing for the calculation of cooling rates as a function of temperature difference and surface area.

Convection Heat Transfer Coefficient

The convection heat transfer coefficient is an empirical parameter that describes how effectively heat is transferred between a surface and a moving fluid. It is essential in determining the rate of convective heat transfer and plays a central role in calculations involving cooling or heating by immersion in a fluid.

Energy Balance in Convection

An energy balance involves accounting for the gain or loss of thermal energy in a system due to convective heat transfer. In cooling processes, particularly with moving materials, an energy balance helps establish how much heat is removed from the object over time or distance, resulting in a decrease in temperature.

Specific Heat Capacity

Specific heat capacity is a material property that indicates the amount of heat required to raise the temperature of a unit mass of a substance by one degree. This concept is crucial in thermal analyses because it relates the energy removed through convection to the resulting change in the temperature of the material.

Quenching Process

Quenching refers to the rapid cooling of a heated material by immersion in a fluid, often used to modify material properties. The process involves significant convective heat transfer and requires an understanding of the interplay between the heat transfer coefficient, the material’s thermal properties, and the cooling medium to accurately predict final temperatures and cooling rates.

Continuous Process Heat Transfer Analysis

When a material moves through a cooling medium, it is subject to a finite residence time during which heat can be removed. This continuous process analysis relates the length of the cooling zone, the speed of the material, and the convective heat transfer properties to determine the extent of cooling, linking spatial and temporal aspects of heat transfer.

Please give Ace some feedback