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Chapter 2: Problem 33

Starting with an energy balance on a ring-shaped volume element, derive the two-dimensional steady heat conduction equation in cylindrical coordinates for \(T(r, z)\) for the case of constant thermal conductivity and no heat generation.

Short Answer

Expert verified
Question: Derive the 2D steady heat conduction equation in cylindrical coordinates for a temperature function, T(r, z), with constant thermal conductivity and no heat generation. Answer: The 2D steady heat conduction equation in cylindrical coordinates is given by: $$\frac{\partial^2 T}{\partial r^2}+\frac{1}{r}\frac{\partial T}{\partial r}+\frac{\partial^2 T}{\partial z^2}=0.$$

Step by step solution

01

Visualize the ring-shaped volume element in cylindrical coordinates

Consider a ring-shaped volume element in cylindrical coordinates \((r, \theta, z)\) with inner radius \(r\), outer radius \(r + \Delta r\), height \(\Delta z\), and thickness \(\Delta \theta\). The volume of the element is given by \(V = \Delta V = (r\Delta\theta)(\Delta r)(\Delta z)\).
02

Set up the energy balance equation

For steady-state heat conduction with no heat generation, the energy balance equation states that the amount of heat entering the volume element is equal to the amount of heat leaving it. We will consider heat transfer in the radial and axial directions only, since the problem is two-dimensional. Thus, we can write the energy balance as: $$q_r(r) - q_r(r+\Delta r) + q_z(z) - q_z(z+\Delta z) = 0.$$
03

Express heat flux in terms of temperature and thermal conductivity

The heat flux in the radial and axial directions can be expressed using Fourier's law of heat conduction: $$q_r = -k\frac{\partial T}{\partial r}\text{ and }q_z = -k\frac{\partial T}{\partial z},$$ where \(k\) is the constant thermal conductivity.
04

Substitute the expressions for heat flux and evaluate the limits

Substitute the expressions for \(q_r\) and \(q_z\) into the energy balance equation and evaluate the result in the limit as \(\Delta r\) and \(\Delta z\) approach zero: $$(-k\frac{\partial T}{\partial r})_{r} + (-k\frac{\partial T}{\partial r})_{r+\Delta r}+(-k\frac{\partial T}{\partial z})_{z} + (-k\frac{\partial T}{\partial z})_{z+\Delta z}=0.$$ Divide the entire equation by \(k\Delta r\Delta z\) and simplify, $$\frac{1}{\Delta r} \left(\frac{\partial T}{\partial r}|_{r+\Delta r} - \frac{\partial T}{\partial r}|_{r}\right) + \frac{1}{\Delta z} \left(\frac{\partial T}{\partial z}|_{z+\Delta z} - \frac{\partial T}{\partial z}|_{z}\right) =0.$$
05

Recognize the limits as partial derivatives and obtain the 2D steady-state heat conduction equation in cylindrical coordinates

As \(\Delta r \to 0\) and \(\Delta z \to 0\), the expression on the left-hand side of the equation represents the sum of two second-order partial derivatives: $$\frac{\partial^2 T}{\partial r^2}+\frac{1}{r}\frac{\partial T}{\partial r}+\frac{\partial^2 T}{\partial z^2}=0.$$

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Most popular questions from this chapter

Does heat generation in a solid violate the first law of thermodynamics, which states that energy cannot be created or destroyed? Explain.

A spherical vessel is filled with chemicals undergoing an exothermic reaction. The reaction provides a uniform heat flux on the inner surface of the vessel. The inner diameter of the vessel is \(5 \mathrm{~m}\) and its inner surface temperature is at \(120^{\circ} \mathrm{C}\). The wall of the vessel has a variable thermal conductivity given as \(k(T)=k_{0}(1+\beta T)\), where \(k_{0}=1.01 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\), \(\beta=0.0018 \mathrm{~K}^{-1}\), and \(T\) is in \(\mathrm{K}\). The vessel is situated in a surrounding with an ambient temperature of \(15^{\circ} \mathrm{C}\), the vessel's outer surface experiences convection heat transfer with a coefficient of \(80 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). To prevent thermal burn on skin tissues, the outer surface temperature of the vessel should be kept below \(50^{\circ} \mathrm{C}\). Determine the minimum wall thickness of the vessel so that the outer surface temperature is \(50^{\circ} \mathrm{C}\) or lower.

A spherical metal ball of radius \(r_{o}\) is heated in an oven to a temperature of \(T_{i}\) throughout and is then taken out of the oven and dropped into a large body of water at \(T_{\infty}\) where it is cooled by convection with an average convection heat transfer coefficient of \(h\). Assuming constant thermal conductivity and transient one-dimensional heat transfer, express the mathematical formulation (the differential equation and the boundary and initial conditions) of this heat conduction problem. Do not solve.

Is the thermal conductivity of a medium, in general, constant or does it vary with temperature?

Consider a large 5-cm-thick brass plate \((k=\) \(111 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) ) in which heat is generated uniformly at a rate of \(2 \times 10^{5} \mathrm{~W} / \mathrm{m}^{3}\). One side of the plate is insulated while the other side is exposed to an environment at \(25^{\circ} \mathrm{C}\) with a heat transfer coefficient of \(44 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). Explain where in the plate the highest and the lowest temperatures will occur, and determine their values.
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