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Chapter 2: Problem 32

Starting with an energy balance on a volume element, derive the two- dimensional transient heat conduction equation in rectangular coordinates for \(T(x, y, t)\) for the case of constant thermal conductivity and no heat generation.

Short Answer

Expert verified
Question: Derive the two-dimensional transient heat conduction equation in rectangular coordinates with constant thermal conductivity and no heat generation. Answer: The two-dimensional transient heat conduction equation in rectangular coordinates with constant thermal conductivity and no heat generation is given by: $$\frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} = \frac{\rho c_p}{k} \frac{\partial T}{\partial t}$$

Step by step solution

01

Define the Problem Variables

First, let us define the variables that we will use to describe the heat conduction in two-dimensional rectangular coordinates. - \(T(x, y, t)\): Temperature (K) at position \((x, y)\) and time \(t\) - \(k\): Thermal conductivity (W/m K) – constant - \(q\): Heat transfer rate per unit volume (W/m³) – no heat generation, so \(q=0\)
02

Write the energy balance equation for a volume element

We will now write the energy balance equation for a volume element \(\delta x \delta y \delta z\) located at position \((x, y)\). The energy balance equation states that the rate of heat entering the volume element must be equal to the rate of change of energy stored in the element. $$\frac{\partial}{\partial x} \left( k \frac{\partial T}{\partial x} \right) \delta x \delta y \delta z + \frac{\partial}{\partial y} \left( k \frac{\partial T}{\partial y} \right) \delta x \delta y \delta z = \rho c_p \frac{\partial T}{\partial t} \delta x \delta y \delta z$$ Here, \(\rho\) is the density (kg/m³) and \(c_p\) is the specific heat capacity at constant pressure (J/kg K).
03

Simplify the energy balance equation

As we know that \(k\) (thermal conductivity) is constant, we can take it out of the partial derivatives, and then we divide the entire equation by \(\delta x\delta y\delta z\) to simplify. $$k \frac{\partial^2 T}{\partial x^2} + k \frac{\partial^2 T}{\partial y^2} = \rho c_p \frac{\partial T}{\partial t}$$
04

The final two-dimensional transient heat conduction equation

Finally, we divide both sides of the equation by \(k\) and obtain the desired two-dimensional transient heat conduction equation in rectangular coordinates for \(T(x, y, t)\), with constant thermal conductivity and no heat generation: $$\frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} = \frac{\rho c_p}{k} \frac{\partial T}{\partial t}$$

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Most popular questions from this chapter

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 steady one-dimensional heat conduction through a plane wall, a cylindrical shell, and a spherical shell of uniform thickness with constant thermophysical properties and no thermal energy generation. The geometry in which the variation of temperature in the direction of heat transfer will be linear is (a) plane wall (b) cylindrical shell (c) spherical shell (d) all of them (e) none of them

Consider a 20-cm-thick large concrete plane wall \((k=0.77 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\) subjected to convection on both sides with \(T_{\infty 1}=27^{\circ} \mathrm{C}\) and \(h_{1}=5 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) on the inside, and \(T_{\infty 2}=8^{\circ} \mathrm{C}\) and \(h_{2}=12 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) on the outside. Assuming constant thermal conductivity with no heat generation and negligible radiation, (a) express the differential equation and the boundary conditions for steady one-dimensional heat conduction through the wall, (b) obtain a relation for the variation of temperature in the wall by solving the differential equation, and \((c)\) evaluate the temperatures at the inner and outer surfaces of the wall.

What is the difference between the degree and the order of a derivative?

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 allowed to cool in ambient air at \(T_{\infty}\) by convection and radiation. The emissivity of the outer surface of the cylinder is \(\varepsilon\), and the temperature of the surrounding surfaces is \(T_{\text {surr }}\). The average convection heat transfer coefficient is estimated to be \(h\). Assuming variable 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.
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