91Ó°ÊÓ

In a material processing experiment conducted aboard the space shuttle, a coated niobium sphere of \(10-\mathrm{mm}\) diameter is removed from a furnace at \(900^{\circ} \mathrm{C}\) and cooled to a temperature of \(300^{\circ} \mathrm{C}\). Although properties of the niobium vary over this temperature range, constant values may be assumed to a reasonable approximation, with \(\rho=8600 \mathrm{~kg} / \mathrm{m}^{3}, c=290 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\), and \(k=\) \(63 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\). (a) If cooling is implemented in a large evacuated chamber whose walls are at \(25^{\circ} \mathrm{C}\), determine the time required to reach the final temperature if the coating is polished and has an emissivity of \(\varepsilon=0.1\). How long would it take if the coating is oxidized and \(\varepsilon=0.6\) ? (b) To reduce the time required for cooling, consideration is given to immersion of the sphere in an inert gas stream for which \(T_{\infty}=25^{\circ} \mathrm{C}\) and \(h=\) \(200 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). Neglecting radiation, what is the time required for cooling? (c) Considering the effect of both radiation and convection, what is the time required for cooling if \(h=200 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) and \(\varepsilon=0.6\) ? Explore the effect on the cooling time of independently varying \(h\) and \(\varepsilon\).

Step by step solution

01

Consider the Stefan-Boltzmann law

The heat transfer rate by radiation is given by: \(q_{\text{rad}}=A \varepsilon \sigma(T_{1}^{4}-T_{2}^{4})\) where \(q_{\text{rad}}\): Heat transfer rate due to radiation \(A\): Surface area of the sphere \(\varepsilon\): Emissivity of the sphere \(\sigma\): Stefan-Boltzmann constant, \(5.67\cdot10^{-8} \, W/m^2K^4\) \(T_{1}\): Temperature of the sphere \(T_{2}\): Temperature of the surrounding

02

Solve for cooling time using the energy balance

To calculate the time required to reach a specific temperature, we'll use the energy balance equation: \(q_{\text{rad}}t = mc(T_{i}-T_{f})\) Where: \(t\): Time to reach final temperature \(m\): Mass of the sphere \(c\): Specific heat capacity of the material \(T_{i}\): Initial temperature of the sphere \(T_{f}\): Final temperature of the sphere Solving for the time \(t\), we have: \(t=\frac{mc(T_{i}-T_{f})}{A \varepsilon \sigma(T_{1}^{4}-T_{2}^{4})}\)

03

Calculate the time for polished and oxidized coating cases

For the polished coating, where \(\varepsilon = 0.1\), and for the oxidized coating, where \(\varepsilon = 0.6\), insert the given values, and calculate the cooling time for each case. (b) Cooling by convection:

04

Consider Newton's law of cooling

The heat transfer rate by convection is given by: \(q_{\text{conv}}=hA(T_{1}-T_{2})\) Where \(q_{\text{conv}}\): Heat transfer rate due to convection \(h\): Heat transfer coefficient

05

Solve for cooling time using the energy balance

Similar to the case with radiation, we can use the energy balance equation: \(q_{\text{conv}}t = mc(T_{i}-T_{f})\) And solve for the time \(t\): \(t=\frac{mc(T_{i}-T_{f})}{hA(T_{1}-T_{2})}\)

06

Calculate the time for convection cooling

Insert the given values, and calculate the cooling time for the case with convection. (c) Cooling by both radiation and convection:

07

Consider the combined heat transfer rate

The total heat transfer rate is given by: \(q_{\text{total}}=q_{\text{rad}}+q_{\text{conv}}\) And the energy balance equation becomes: \((q_{\text{rad}}+q_{\text{conv}})t = mc(T_{i}-T_{f})\)

08

Solve for cooling time using the energy balance

Solving for the time \(t\): \(t=\frac{mc(T_{i}-T_{f})}{q_{\text{rad}}+q_{\text{conv}}}\)

09

Calculate the time for combined cooling

Insert the given values, calculate the time for cooling by both radiation and convection.

10

Varying \(h\) and \(\varepsilon\) independently

Explore the cooling time's dependency on the heat transfer coefficient \(h\) and emissivity \(\varepsilon\) by calculating the time for a range of \(h\) and \(\varepsilon\) values and observing how the cooling time changes.

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Key Concepts

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

Convection Heat Transfer

Convection heat transfer is a mode of heat transfer that occurs between a solid surface and a fluid (like air or a liquid) which comes into contact with it. In convection, the heat is carried away from or toward a surface by a fluid motion. It is one of the primary ways heat is transferred in many practical situations, such as a fan blowing air over a hot surface to cool it.

In our exercise, when a hot niobium sphere is cooled with an inert gas stream, convection is the main method of heat transfer. The formula for calculating the heat transfer rate in this setup is given by Newton's law of cooling:

• \( q_{\text{conv}} = hA(T_{1} - T_{2}) \)

where:

• \( q_{\text{conv}} \): The heat transfer rate due to convection
• \( h \): Heat transfer coefficient
• \( A \): Surface area of the sphere
• \( T_{1} \): Initial temperature of the sphere
• \( T_{2} \): Temperature of the surrounding gas

The convection heat transfer coefficient \( h \) reflects how effectively the fluid can carry heat away from the surface. Increasing \( h \) often leads to faster cooling. Convection is generally more effective when there is a greater temperature difference or when the fluid movement, like in forced convection, is enhanced.

Radiation Heat Transfer

Radiation heat transfer is the process of energy transfer in the form of electromagnetic waves or photons. This process does not require a medium, meaning it can occur in vacuums, such as space. All objects emit radiation energy, and the amount of energy emitted increases with temperature.

The Stefan-Boltzmann law helps us calculate radiation heat transfer. The formula is:

• \( q_{\text{rad}} = A \varepsilon \sigma (T_{1}^{4} - T_{2}^{4}) \)

where:

• \( q_{\text{rad}} \): Heat transfer rate due to radiation
• \( \varepsilon \): Emissivity, a measure of how effective a surface is at emitting thermal radiation
• \( \sigma \): Stefan-Boltzmann constant, \( 5.67 \times 10^{-8} \ W/m^2K^4 \)
• \( A \): Surface area of the sphere
• \( T_{1} \) and \( T_{2} \): Temperatures of the sphere and the surroundings respectively

For the niobium sphere in a large evacuated chamber, radiation is the primary mode of cooling since there isn't a medium to support conduction or convection heat transfers. Depending on the emissivity, which varies based on the surface condition (polished or oxidized), the cooling time significantly changes. High emissivity means more effective radiation and faster cooling.

Emissivity

Emissivity is a key concept in radiation heat transfer, representing the efficiency of a surface in emitting thermal radiation compared to a perfect blackbody. Emissivity is a dimensionless quantity that ranges from 0 to 1.

A perfect blackbody has an emissivity of 1, meaning it is the most efficient at radiating energy. Conversely, a highly polished surface often has a low emissivity value, reflecting more energy than it emits. In the context of the exercise, the emissivity of the niobium sphere’s coating affects how quickly it cools when placed in a vacuum.

• For a polished surface with emissivity \( \varepsilon = 0.1 \), the cooling will be slower as compared to a higher emissivity surface because less heat is radiated away.
• An oxidized coating with \( \varepsilon = 0.6 \) will emit more radiation and therefore cool faster.

Emissivity plays a crucial role in designing and optimizing thermal control systems across a range of applications, from industrial processing to aerospace engineering. It determines how well objects, like the niobium sphere, manage thermal radiation as they absorb and emit heat.

Heat Transfer Coefficient

The heat transfer coefficient \( h \) is a measure that indicates the convective heat transfer capability of a fluid surrounding a body. It quantifies the rate of heat exchange between the fluid and the surface it contacts. The higher the value of \( h \), the more efficient the process is at transferring heat.

The heat transfer coefficient is important in calculating convection heat transfer and is part of the Newton's law of cooling. It's used to determine how effectively a fluid like air or an inert gas can absorb heat from a surface. Here is how it appears in the formula for the convective heat transfer rate:

• \( q_{\text{conv}} = hA(T_{1} - T_{2}) \)

where \( q_{\text{conv}} \) is the heat transfer rate through convection, \( A \) is the surface area, and \( T_{1} \) and \( T_{2} \) are the temperatures of the surface and the fluid, respectively.
Increasing \( h \) (by for instance enhancing the flow rate of the surrounding fluid) will speed up the cooling process. This makes \( h \) an essential parameter in heat exchanger design and thermal regulation strategies across many engineering fields.