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Chapter 2: Problem 162

The conduction equation boundary condition for an adiabatic surface with direction \(n\) being normal to the surface is (a) \(T=0\) (b) \(d T / d n=0\) (c) \(d^{2} T / d n^{2}=0\) (d) \(d^{3} T / d n^{3}=0\) (e) \(-k d T / d n=1\)

Short Answer

Expert verified
Answer: \(\frac{dT}{dn}=0\)

Step by step solution

01

Understand the concept of adiabatic surface

An adiabatic surface is defined as a surface with no heat transfer across it. In other words, the rate of heat transfer through the surface is zero. To find the correct boundary condition for an adiabatic surface, we need to analyze each option in terms of heat transfer.
02

Analyze each option

Using the concept of no heat transfer across an adiabatic surface, we can determine which option corresponds to this condition. (a) \(T=0\): This states that the temperature of the surface is zero. This does not necessarily imply that there is no heat transfer across the surface, as the temperature could be non-zero inside the material. (b) \(\frac{dT}{dn}=0\): This states that the temperature gradient in the direction normal to the surface (n) is zero. As Fourier's Law states that the rate of heat transfer through a material is proportional to the temperature gradient, a zero gradient means no heat transfer across the adiabatic surface. This is the correct choice. (c) \(\frac{d^2T}{dn^2}=0\): This refers to the curvature of the temperature profile in the direction normal to the surface. This has no direct relation to the heat transfer across the surface. (d) \(\frac{d^3T}{dn^3}=0\): This refers to the rate of change of curvature of the temperature profile in the direction normal to the surface. This has no direct relation to the heat transfer across the surface. (e) \(-k\frac{dT}{dn}=1\): This states that the heat flux (rate of heat transfer per unit area) across the surface is equal to 1. This contradicts the definition of an adiabatic surface, which has no heat transfer.
03

Identify the correct boundary condition

Based on the analysis in Step 2, we can conclude that the correct conduction equation boundary condition for an adiabatic surface with direction \(n\) being normal to the surface is: (b) \(\frac{dT}{dn}=0\)

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Key Concepts

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

Heat Transfer
Heat transfer is a process in which thermal energy is exchanged between physical systems. There are three primary modes of heat transfer: conduction, convection, and radiation. In this discussion, we'll focus on conduction.

Conduction occurs when heat is transferred through a material without the movement of the material itself. It relies on the interaction between particles in a substance, such as atoms or molecules. These particles collide, transferring energy from the hotter region to the cooler one.

In an adiabatic surface, which is a surface that does not permit any heat to pass through, the heat transfer rate is effectively zero. This means the system is perfectly insulated, preventing any loss or gain of heat. Understanding this concept is crucial in fields such as chemistry and engineering, as it allows for the design of efficient insulation systems.
Temperature Gradient
The temperature gradient is a physical quantity that describes the direction and rate of temperature change in a particular region. It is defined mathematically as the difference in temperature per unit distance. In simpler terms, it tells us how the temperature varies from one point to another.

A high temperature gradient indicates a sharp change in temperature, which could cause faster heat transfer, while a low temperature gradient shows a more gradual change. Typically, in the presence of heat transfer through a material, you'll find that the temperature gradient guides the flow of heat, moving from hot to cold regions. This gradient is crucial because it represents the driving force behind heat conduction between materials.
  • In an adiabatic surface, the temperature gradient normal to the surface is zero, ensuring no heat transfer.
  • Understanding gradients helps in analyzing thermal systems, where controlling temperature is critical for efficiency.
Fourier's Law
Fourier's Law is a fundamental principle that describes the conduction of heat within solid materials. Formulated by Jean-Baptiste Joseph Fourier, this law relates the rate of heat transfer through a material to its temperature gradient and its thermal conductivity, denoted as \( k \).

The mathematical expression of Fourier's Law is given by:\[ q = -k \frac{dT}{dx} \]where:
  • \( q \) is the heat flux, the heat transfer per unit area over time.
  • \( k \) represents the thermal conductivity of the material, indicating its ability to conduct heat.
  • \( \frac{dT}{dx} \) denotes the temperature gradient in the direction of heat flow.
In the context of adiabatic surfaces, applying Fourier's Law helps confirm the absence of heat transfer since the temperature gradient, \( \frac{dT}{dn} \), is zero, leading to zero flux. This theory is pivotal in engineering and physics, as it helps predict how heat diffuses through different materials under thermal gradients.

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Most popular questions from this chapter

Consider steady one-dimensional heat conduction in a plane wall in which the thermal conductivity varies linearly. The error involved in heat transfer calculations by assuming constant thermal conductivity at the average temperature is \((a)\) none, \((b)\) small, or \((c)\) significant.

Consider a short cylinder of radius \(r_{o}\) and height \(H\) in which heat is generated at a constant rate of \(\dot{e}_{\text {gen. }}\). Heat is lost from the cylindrical surface at \(r=r_{o}\) by convection to the surrounding medium at temperature \(T_{\infty}\) with a heat transfer coefficient of \(h\). The bottom surface of the cylinder at \(z=0\) is insulated, while the top surface at \(z=H\) is subjected to uniform heat flux \(\dot{q}_{H}\). Assuming constant thermal conductivity and steady two-dimensional heat transfer, express the mathematical formulation (the differential equation and the boundary conditions) of this heat conduction problem. Do not solve.

Consider a large plane wall of thickness \(L=0.8 \mathrm{ft}\) and thermal conductivity \(k=1.2 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft} \cdot{ }^{\circ} \mathrm{F}\). The wall is covered with a material that has an emissivity of \(\varepsilon=0.80\) and a solar absorptivity of \(\alpha=0.60\). The inner surface of the wall is maintained at \(T_{1}=520 \mathrm{R}\) at all times, while the outer surface is exposed to solar radiation that is incident at a rate of \(\dot{q}_{\text {solar }}=300 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft}^{2}\). The outer surface is also losing heat by radiation to deep space at \(0 \mathrm{~K}\). Determine the temperature of the outer surface of the wall and the rate of heat transfer through the wall when steady operating conditions are reached.

A spherical communication satellite with a diameter of \(2.5 \mathrm{~m}\) is orbiting around the earth. The outer surface of the satellite in space has an emissivity of \(0.75\) and a solar absorptivity of \(0.10\), while solar radiation is incident on the spacecraft at a rate of \(1000 \mathrm{~W} / \mathrm{m}^{2}\). If the satellite is made of material with an average thermal conductivity of \(5 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) and the midpoint temperature is \(0^{\circ} \mathrm{C}\), determine the heat generation rate and the surface temperature of the satellite.

Consider a function \(f(x)\) and its derivative \(d f l d x\). Does this derivative have to be a function of \(x\) ?
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