In a manufacturing facility, 2 -in-diameter brass balls (k=64.1 Btu / h ·ft · ^∘ F, ρ=532 lbm /

Name: Problem 15, Chapter 11: Introduction To Thermodynamics and Heat Transfer
Uploaded: 2023-10-27T12:36:58-08:00
Duration: 2 min
Channel: Anand Jangid

step 1 So here in this question to determine the rate at which the heat needs to be removed from water

In a manufacturing facility, 2 -in-diameter brass balls...

In a manufacturing facility, 2 -in-diameter brass balls $\left(k=64.1 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft} \cdot{ }^{\circ} \mathrm{F}, \rho=532 \mathrm{lbm} / \mathrm{ft}^3\right.$, and $c_p=$ $0.092 \mathrm{Btu} / \mathrm{lbm} \cdot{ }^{\circ} \mathrm{F}$ ) initially at $250^{\circ} \mathrm{F}$ are quenched in a water bath at $120^{\circ} \mathrm{F}$ for a period of 2 min at a rate of 120 balls per minute. If the convection heat transfer coefficient is $42 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft}^2 \cdot{ }^{\circ} \mathrm{F}$, determine (a) the temperature of the balls after quenching and (b) the rate at which heat needs to be removed from the water in order to keep its temperature constant at $120^{\circ} \mathrm{F}$.

Key Concepts

Heat Transfer Rate Calculation

Calculating the heat transfer rate requires determining the amount of thermal energy removed or added per unit time, based on the object's mass, specific heat, and change in temperature. This calculation is crucial for establishing a steady operation in systems where the continuous removal of heat, such as maintaining a constant bath temperature, is necessary.

Transient Energy Balance

A transient energy balance involves setting up an equation that accounts for the accumulation or depletion of heat energy in a system over time. By equating the rate of change of energy within the object to the convective heat flux from its surface, one can predict the temperature evolution during processes like quenching.

Convection Heat Transfer

Convection is the transfer of heat between a solid surface and a moving fluid, governed by the temperature difference and the convection heat transfer coefficient. This concept explains how the fluid's motion enhances the rate of heat removal or addition at the surface, playing a significant role in processes like quenching.

Lumped Capacitance Method

This method simplifies transient heat transfer analysis by assuming that the entire object remains at a uniform temperature at any instant during the cooling or heating process. It is valid when the Biot number is small, meaning the internal thermal resistance is much lower than the boundary layer resistance, allowing the use of an ordinary differential equation to describe the temperature change over time.

Newton's Law of Cooling

Newton's law of cooling is a principle that establishes a relationship between the rate of heat loss of a body and the difference in temperature between the body and its environment. It serves as the foundation for formulating the energy balance in situations where convective heat transfer dominates, leading to an exponential decay model of the object's temperature over time.

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