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A sphere \(30 \mathrm{~mm}\) in diameter initially at \(800 \mathrm{~K}\) is quenched in a large bath having a constant temperature of \(320 \mathrm{~K}\) with a convection heat transfer coefficient of \(75 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). The thermophysical properties of the sphere material are: \(\rho=400 \mathrm{~kg} / \mathrm{m}^{3}, c=1600 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\), and \(k=1.7 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\). (a) Show, in a qualitative manner on \(T \dashv\) coordinates, the temperatures at the center and at the surface of the sphere as a function of time. (b) Calculate the time required for the surface of the sphere to reach \(415 \mathrm{~K}\). (c) Determine the heat flux \(\left(\mathrm{W} / \mathrm{m}^{2}\right)\) at the outer surface of the sphere at the time determined in part (b). (d) Determine the energy (J) that has been lost by the sphere during the process of cooling to the surface temperature of \(415 \mathrm{~K}\). (e) At the time determined by part (b), the sphere is quickly removed from the bath and covered with perfect insulation, such that there is no heat loss from the surface of the sphere. What will be the temperature of the sphere after a long period of time has elapsed? (f) Compute and plot the center and surface temperature histories over the period \(0 \leq t \leq 150 \mathrm{~s}\). What effect does an increase in the convection coefficient to \(h=200 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) have on the foregoing temperature histories? For \(h=75\) and \(200 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\), compute and plot the surface heat flux as a function of time for \(0 \leq t \leq 150 \mathrm{~s}\).

Step by step solution

01

Draw the temperature profile qualitatively

This is a qualitative kind of question. You should sketch the temperatures at the center and surface of the sphere as a function of time on a T vs. coordinates graph. You should notice that the temperature decreases gradually from the initial temperature, and the temperature difference between the center and surface increases initially and then decreases. The surface will reach a temperature of 415 K before the center.

02

Calculate the time required for the surface to reach 415 K

A common method to calculate the time required for the temperature to reach a certain point is by using the transient conduction heat transfer equation for spherical coordinates. It can also be solved by using dimensionless variables and the Biot number. The time required for the surface temperature to reach 415 K can be found using the equation: \[t = \frac{(\rho c V) (T_s - T_\infty)}{hA(T_i - T_\infty)}\] Where: \(t\) = time (s) \(\rho\) = density (kg/m³) \(c\) = specific heat (J/kg·K) \(V\) = volume of the sphere (m³) \(T_s\) = surface temperature (K) \(T_\infty\) = bath temperature (K) \(h\) = convection heat transfer coefficient (W/m²·K) \(A\) = surface area of the sphere (m²) \(T_i\) = initial temperature (K) Calculate \(V\) and \(A\), then substitute the given values and solve for \(t\).

03

Determine the heat flux at the surface at the calculated time

The heat flux at the surface can be calculated using the convection heat transfer equation: \(q'' = h(T_s - T_\infty)\) Substitute the values of \(h\), \(T_s\), and \(T_\infty\) to get the heat flux at the surface.

04

Calculate the energy lost during the cooling process

The energy lost by the sphere during the cooling process can be calculated as follows: \(Q = mc(T_i - T_s)\) Where: \(Q\) = energy lost (J) \(m\) = mass of the sphere (kg) \(c\) = specific heat (J/kg·K) Calculate the mass and then calculate the energy lost.

05

Determine the temperature of the sphere after insulating it for a long time

Assuming no heat loss after insulating the sphere, the temperature of the sphere will remain constant at the surface temperature of 415 K.

06

Plot the center and surface temperature histories, and the surface heat flux as a function of time

To plot the center and surface temperature histories and the surface heat flux as functions of time, we can use numerical methods such as finite difference or finite element methods. The resulting plots will show how the center and surface temperatures evolve over time and how the heat flux at the surface changes with time. Also, calculate and plot the temperature histories and surface heat flux with the increased convection coefficient \(h = 200 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). The increased convection coefficient will result in faster cooling and lower heat flux at the surface.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Transient Conduction Heat Transfer

Transient conduction heat transfer occurs when heat flows through a material whose temperature varies with time. This is a common scenario when an object, such as a hot sphere, is suddenly placed in a cooler environment and begins to cool down. The governing principle for transient heat conduction is the heat equation, which, in the case of a sphere, can be expressed in spherical coordinates. However, the complexity of the solution often requires simplifications or numerical methods.

For example, if we take a sphere initially at a uniform temperature and subject it to a sudden change in the surrounding temperature, the heat transfer within the sphere will not be instantaneous. Instead, there will be a time-dependent temperature distribution, resulting in a thermal gradient from the sphere's surface to its center. This effect can be visualized by plotting the temperature against time, showing how the surface cools down faster than the center due to the direct interaction with the cooler environment.

It's essential to understand that the exact behavior of temperature changes within the sphere involves mathematical methods that are challenging for beginners. Numerical approximations, such as the finite difference method, offer a practical approach to predict transient temperature distributions. Simplified models like the lumped capacitance method may also be used when certain conditions, like a small Biot number, are met.

Convection Heat Transfer Coefficient

The convection heat transfer coefficient, commonly denoted by 'h', is a measure of the heat transfer rate per unit area and temperature difference between a surface and a fluid moving past it. This coefficient plays a key role in quantifying the cooling or heating of objects through convection. It's crucial to realize that 'h' is affected by numerous factors, including the fluid's properties, surface roughness, fluid velocity, and the temperature difference itself.

In the context of the sphere quenching problem, 'h' determines how rapidly heat is transferred from the sphere's surface to the surrounding fluid. In the provided exercise, changing the value of 'h' alters the rate at which the sphere's temperature approaches that of the surrounding bath. A higher convection heat transfer coefficient, such as increasing 'h' from 75 to 200 W/m²·K, accelerates the heat transfer process, resulting in a quicker temperature decrease at the sphere's surface. This behavior can be monitored by calculating the surface heat flux, which is directly proportional to 'h' and the temperature difference between the sphere's surface and the quenching medium.

Temperature Distribution in Solids

The temperature distribution in solids refers to how temperature varies within a solid object at a given time. It is a spatial representation of temperature, and in the case of the sphere quenching problem, it illustrates the temperature variation from the sphere's surface to its center over time. Solids, like a metal sphere, exhibit thermal conductivity that allows heat to transfer through the material. However, this transfer is not instantaneous, leading to temperature gradients within the object.

In the quenching example, where the sphere is composed of a material with known thermophysical properties such as density, specific heat, and thermal conductivity, we can predict how the temperature will evolve over time. These parameters influence how quickly heat is distributed within the sphere. At the onset of quenching, a steep temperature gradient exists at the surface, with the center remaining at a higher temperature. Over time, this gradient lessens as the entire sphere approaches thermal equilibrium with the surrounding medium.

To fully grasp the concept of temperature distribution, one must be familiar with the principles of heat conduction and understand that conducting materials facilitate a certain distribution pattern driven by the difference in temperature. The use of graphical methods, like the plot of center and surface temperatures as functions of time, provides insights into the dynamic changes that occur during the cooling process.