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Chapter 4: Problem 37

Does the temperature distribution during steady heat transfer in an object with b.c.'s of only the first kind depend on \(k\) ? Explain.

Short Answer

Expert verified
The temperature distribution does not depend on \(k\); it is independent of it during steady heat transfer with boundary conditions of the first kind.

Step by step solution

01

Understand the Problem Statement

The problem is asking whether the temperature distribution depends on the thermal conductivity, denoted by \(k\), in a situation involving steady heat transfer with boundary conditions of the first kind. Boundary conditions of the first kind imply that the temperatures at the boundaries are held constant.
02

Analyze Steady Heat Transfer

Steady heat transfer implies that the temperature distribution does not change with time within the object. This means the heat entering any part of the object is equal to the heat leaving that part.
03

Examine the Role of Thermal Conductivity \(k\)

Thermal conductivity \(k\) is a property of the material that affects the rate of heat transfer. It appears in the heat equation, \( abla^2 T = 0 \), which describes steady-state heat conduction, but \(k\) only affects the rate at which heat is transferred, not the resulting distribution of temperature under steady conditions with constant boundary temperatures.
04

Derive the Heat Equation for Steady State

For steady-state heat transfer, the governing equation is \( abla^2 T = 0 \), known as Laplace's equation. This equation states that the temperature distribution is determined by the shape of the domain and the boundary conditions, not by \(k\).
05

Conclusion on the Dependence of Temperature Distribution on \(k\)

Since the boundary conditions are of the first kind (constant boundary temperatures), and the solution to the Laplace's equation does not include \(k\), the temperature distribution within the object is not dependent on \(k\). \(k\) affects the rate of heat transfer, not the steady-state distribution.

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Key Concepts

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

Thermal Conductivity
Thermal conductivity, often represented by the symbol \(k\), is an intrinsic property of materials that measures their ability to conduct heat. It quantifies how easily heat can pass through a material.
Think of it as a factor that determines how much resistance the material poses to heat flow.
  • A material with a high thermal conductivity will allow heat to pass through it more quickly.
  • Conversely, a material with low thermal conductivity will slow down the transfer of heat.
However, in the context of steady-state heat transfer with constant boundary conditions of the first kind, thermal conductivity does not influence the temperature distribution within the material. While it is integral to computing how fast the heat moves, the final steady-state temperature profile remains unaffected by \(k\) when boundary temperatures are fixed.
Laplace's Equation
Laplace's Equation, \(abla^2 T = 0\), is a cornerstone in the analysis of steady-state heat conduction. It defines the condition where the temperature distribution does not change over time, indicating that heat entering and leaving any section of the domain are equal.
This equation implies that the system has reached thermal equilibrium in terms of temperature distribution. To solve such problems, the spatial domain, boundary conditions, and initial temperature setup must be carefully considered.
When Laplace's Equation is applied to a domain with fixed boundary temperatures (first-kind boundary conditions), the materials' thermal conductivity \(k\) does not alter the resulting temperature pattern. Instead, the distribution depends solely on the boundary conditions and the physical geometry of the object involved.
  • Laplace's Equation describes a harmonic function, which is crucial in determining the temperature profile when there are no internal sources or sinks.
  • Its solutions rely heavily on constants set on the boundaries rather than material properties like thermal conductivity.
Boundary Conditions
Boundary conditions are essential settings that help determine how a system behaves in steady heat transfer analysis. They dictate the conditions applied at the surface or boundary of a domain.
There are several types of boundary conditions, but in the context of this problem, we focus on first-kind boundary conditions, also known as Dirichlet boundary conditions, where the temperature at the boundaries is constant.
This type of boundary condition provides a stable reference temperature for the system, allowing us to solve the heat conduction problems without considering the effects of thermal conductivity on the temperature distribution.
  • When employing first-kind boundary conditions, the known values (boundary temperatures) are used as inputs to derive the temperature distribution across the entire domain.
  • These stable values ensure that the solution of Laplace's Equation results in a unique and well-defined temperature field within the domain, irrespective of the thermal properties of the materials, as long as the boundary values themselves do not change.

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Most popular questions from this chapter

Steam condenses on the inside of a small pipe, keeping it at a specified temperature, \(T_{i}\). The pipe is heated by electrical resistance at a rate \(\dot{q} \mathrm{~W} / \mathrm{m}^{3}\). The outside temperature is \(T_{\infty}\) and there is a natural convection heat transfer coefficient, \(\bar{h}\) around the outside. (a) Derive an expression for the dimensionless expression temperature distribution, \(\Theta=\left(T-T_{\infty}\right) /\left(T_{i}-T_{\infty}\right)\), as a function of the radius ratios, \(\rho=r / r_{o}\) and \(\rho_{i}=r_{i} / r_{o}\); a heat generation number, \(\Gamma=\dot{q} r_{o}^{2} / k\left(T_{i}-T_{\infty}\right)\); and the Biot number. (b) Plot this result for the case \(\rho_{i}=2 / 3, \mathrm{Bi}=1\), and for several values of \(\Gamma\). (c) Discuss any interesting aspects of your result.

The highest heat flux that can be achieved in nucleate boiling (called \(q_{\max }-\) see the qualitative discussion in Section 9.1) depends upon \(\rho_{g}\), the saturated vapor density; \(h_{f g}\), the latent heat vaporization; \(\sigma\), the surface tension; a characteristic length, \(l\); and the gravity force per unit volume, \(g\left(\rho_{f}-\rho_{g}\right)\), where \(\rho_{f}\) is the saturated liquid density. Develop the dimensionless functional equation for \(q_{\max }\) in terms of dimensionless length.

The heat tansfer coefficient, \(\bar{h}\), resulting from a forced flow over a flat plate depends on the fluid velocity, viscosity, density, specific heat, and thermal conductivity, as well as on the length of the plate. Develop the dimensionless functional equation for the heat transfer coefficient (cf. Section 6.5).

The left side of a slab of thickness \(L\) is kept at \(0^{\circ} \mathrm{C}\). The right side is cooled by air at \(T_{\infty}{ }^{\circ} \mathrm{C}\) blowing on it. \(\bar{h}_{\mathrm{RHS}}\) is known. An exothermic reaction takes place in the slab such that heat is generated at \(A\left(T-T_{\infty}\right) \mathrm{W} / \mathrm{m}^{3}\), where \(A\) is a constant. Find a fully dimensionless expression for the temperature distribution in the wall.

Thin fins with a \(0.002 \mathrm{~m}\) by \(0.02 \mathrm{~m}\) rectangular cross section and a thermal conductivity of \(50 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) protrude from a wall and have \(\bar{h} \simeq 600 \mathrm{~W} / \mathrm{m}^{2} \mathrm{~K}\) and \(T_{0}=170^{\circ} \mathrm{C}\). What is the heat flow rate into each fin and what is the effectiveness? \(T_{\infty}=\) \(20^{\circ} \mathrm{C}\).
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