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A procedure for measuring the thermal conductivity of solids at elevated temperatures involves placement of a sample at the bottom of a large furnace. The sample is of thickness \(L\) and is placed in a square container of width \(W\) on a side. The sides are well insulated. The walls of the cavity are maintained at \(T_{w}\), while the bottom surface of the sample is maintained at a much lower temperature \(T_{e}\) by circulating coolant through the sample container. The sample surface is diffuse and gray with an emissivity \(\varepsilon_{s}\). Its temperature \(T_{s}\) is measured optically. (a) Neglecting convection effects, obtain an expression from which the sample thermal conductivity may be evaluated in terms of measured and known quantities \(\left(T_{w}, T_{s}, T_{c}, \varepsilon_{s}, L\right)\). The measurements are made under steady-state conditions. If \(T_{w}=1400 \mathrm{~K}, T_{s}=1000 \mathrm{~K}, \varepsilon_{s}=0.85, L=\) \(0.015 \mathrm{~m}\), and \(T_{c}=300 \mathrm{~K}\), what is the sample thermal conductivity? (b) If \(W=0.10 \mathrm{~m}\) and the coolant is water with a flow rate of \(\dot{m}_{c}=0.1 \mathrm{~kg} / \mathrm{s}\), is it reasonable to assume a uniform bottom surface temperature \(T_{c}\) ?

Step by step solution

01

Define Fourier's Law for heat conduction

Fourier's Law states that the heat flux, \(q''\), is proportional to the temperature gradient, \(\frac{dT}{dx}\), and is given by: \(q'' = -k \frac{dT}{dx}\) Here, \(k\) is the thermal conductivity of the sample that we need to evaluate.

02

Set up a steady-state energy balance

Since we neglect convection, the net heat flux that enters the sample through the walls (\(q''_\text{rad}\)) must be equal to the net heat flux that leaves through the bottom surface by conduction (\(q''_\text{cond}\)), as the sample's temperature is maintained at a much lower value on the bottom surface. Therefore, we have: \(q''_\text{rad} = q''_\text{cond}\)

03

Express radiant heat flux

The radiant heat flux can be obtained using the Stefan-Boltzmann Law, considering the emissivity of the sample: \(q''_\text{rad} = \varepsilon_{s} \sigma (T_{w}^4 - T_{s}^4)\) Here, \(\varepsilon_{s}\) is the emissivity, \(\sigma\) is the Stefan-Boltzmann constant, and \(T_{w}\) and \(T_{s}\) are the temperatures of the cavity walls and sample surface, respectively.

04

Express conductive heat flux

We can express the conductive heat flux using Fourier's Law. We know that: \(q''_\text{cond} = k \frac{T_{s} - T_{c}}{L}\) Where \(k\) is the thermal conductivity, \(T_{c}\) is the temperature at the bottom surface, and \(L\) is the thickness of the sample.

05

Solve the energy balance equation

Now, we will equate the radiant and conductive heat flux expressions: \(\varepsilon_{s} \sigma (T_{w}^4 - T_{s}^4) = k \frac{T_{s} - T_{c}}{L}\) Then, we can solve for the thermal conductivity \(k\): \(k = \frac{\varepsilon_{s} \sigma L (T_{w}^4 - T_{s}^4)}{T_{s} - T_{c}}\)

06

Calculate the sample thermal conductivity

Now plug in the given values for \(T_{w} = 1400 \, K\), \(T_{s} = 1000 \, K\), \(\varepsilon_{s} = 0.85\), \(L = 0.015 \, m\), and \(T_{c} = 300 \, K\): \(k = \frac{0.85 \cdot 5.67 \times 10^{-8} \cdot 0.015 (1400^4 - 1000^4)}{1000 - 300}\) On solving, we get: \(k \approx 13.62 \, W / (m \cdot K)\)

07

Assess the reasonability of the assumption

The width of the container, \(W = 0.10 \, m\), and coolant flow rate \(\dot{m}_{c} = 0.1 \, kg / s\). Since water has high thermal conductivity and high specific heat capacity, and due to the large width compared to the thickness, the assumption that there is a uniform bottom surface temperature of \(T_{c}\) is reasonable. The flow rate of the coolant is fast enough to maintain a constant surface temperature, consequently minimizing temperature fluctuations along the bottom surface.

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Key Concepts

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

Fourier's Law

Fourier's Law is a foundational concept in thermal physics, connecting temperature gradients with thermal conduction. This law tells us how heat transfer occurs through materials due to a difference in temperature. The formula for Fourier's Law is written as:\[ q'' = -k \frac{dT}{dx} \]where:

• \( q'' \) is the heat flux, representing the rate of heat transfer per unit area.
• \( k \) is the thermal conductivity of the material, which describes how well the material can conduct heat.
• \( \frac{dT}{dx} \) is the temperature gradient, showing how temperature changes with distance within the material.

Fourier's Law is widely applied when designing thermal systems, ensuring that materials can handle the expected thermal loads. In our context, it helps in determining the thermal conductivity of a sample by connecting it to known temperatures at the sample's interfaces.

Stefan-Boltzmann Law

The Stefan-Boltzmann Law is crucial for understanding how objects radiate heat. This law describes the radiant heat flux emitted by a surface, indicating how much energy is radiated as electromagnetic waves due to an object's temperature. The law is expressed as follows:\[ q''_\text{rad} = \varepsilon \sigma (T^4_\text{object} - T^4_\text{surrounding}) \]where:

• \( q''_\text{rad} \) is the radiant heat flux, the power radiated per unit area.
• \( \varepsilon \) is the emissivity of the surface, showing how effectively it radiates energy compared to a perfect black body.
• \( \sigma \) is the Stefan-Boltzmann constant, approximately \( 5.67 \times 10^{-8} \, W/m^2K^4 \).
• \( T_\text{object} \) and \( T_\text{surrounding} \) are the temperatures of the radiating object and its surroundings, respectively.

This law is especially important in the exercise context as it helps calculate the radiant heat exchange between the sample's surface and its surroundings, contributing to the total heat flux across the sample.

Radiant Heat Flux

Radiant heat flux refers to the energy transferred as heat through radiation, often influenced greatly by the temperature and properties of the surfaces involved. Unlike conduction, which requires direct contact, radiation can transfer heat across empty space. In the exercise, the radiant heat flux is a critical factor in determining the sample's thermal conductivity. This flux is defined by the Stefan-Boltzmann Law and depends upon:

• Emissivity of the surface, \( \varepsilon_s \).
• Surface temperature difference, emphasized by the objects' temperatures to the fourth power.

The effectiveness of heat transfer via radiation is highly dependent on surface characteristics like emissivity and temperature. High emissivity means better ability to emit energy as radiation, thus higher radiant heat flux.

Steady-State Energy Balance

In thermal analyses, a steady-state energy balance implies that all incoming and outgoing energy flows are equal, leading to no net energy change over time. This assumption allows simplification in calculations and is crucial in measuring thermal conductivity as seen in the exercise. Under steady-state conditions, the system reaches thermal equilibrium, meaning:

• Heat entering a system equals the heat exiting the system.
• Temperature distributions stabilize and do not change with time.

This concept is used to equate the radiant heat flux entering the sample with the conductive heat flux leaving it, ensuring an accurate calculation of thermal conductivity. By focusing on the steady-state energy balance, the calculation relies on maintained temperature differences across the sample at equilibrium, simplifying the computational process and ensuring consistent measurements.