Question

A 2-kW resistance heater wire whose thermal conductivity is $k=10.4 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft} \cdot \mathrm{R}$ has a radius of \(r_{o}=0.06\) in and a length of \(L=15 \mathrm{in}\), and it is used for space heating. Assuming constant thermal conductivity and one-dimensional heat transfer, express the mathematical formulation (the differential equation and the boundary conditions) of this heat conduction problem during steady operation. Do not solve.

Short answer

Question: Write down the mathematical formulation of the heat conduction differential equation during steady operation, considering one-dimensional cylindrical coordinate system (radial direction only) and constant thermal conductivity for a resistance heater wire with constant thermal conductivity, radius, and length. Answer: 1. Differential equation: \(\frac{1}{r} \frac{d}{dr} \left( r \frac{dT}{dr} \right) = -\frac{q_g}{k}\) 2. Boundary conditions: a. At the center of the wire: \(\frac{dT}{dr}|_{r=0} = 0\) b. At the surface of the wire: \(q_r = -k \frac{dT}{dr}|_{r=r_o}\)

Step-by-step solution

Identify Fourier's Law in cylindrical coordinates (radial direction only)

For a one-dimensional heat conduction problem in a cylindrical coordinate system (radial direction only), the thermal conductivity is constant, and Fourier's Law is expressed as: \(q_r = -k \frac{dT}{dr}\) Where \(q_r\) is the radial heat flux, \(k\) is the thermal conductivity, and \(\frac{dT}{dr}\) is the temperature gradient.

Express the heat generation rate

As the wire is generating heat, the heat generation rate per unit volume (\(q_g\)) can be expressed as: \(q_g = \frac{P}{V}\) Where \(P\) is the total power generated (2 kW) and \(V\) is the volume of the wire.

Write the general heat conduction equation

The general heat conduction equation in cylindrical coordinates (radial direction only), with heat generation, is given by: \(\frac{1}{r} \frac{d}{dr} \left( r \frac{dT}{dr} \right) = -\frac{q_g}{k}\)

Express heat flux at the wire's surface

We know that the heat flux at the wire's surface (\(r = r_o\)) is given by: \(q_r = -k \frac{dT}{dr}|_{r=r_o}\)

Express boundary condition at the center of the wire

As the wire is generating heat uniformly, the temperature at the center of the wire will be maximum, and the temperature gradient will be zero: \(\frac{dT}{dr}|_{r=0} = 0\)

Write down the mathematical formulation

As requested, we do not need to solve the problem, but the mathematical formulation of the heat conduction differential equation during steady operation, considering one-dimensional cylindrical coordinate system (radial direction only) and constant thermal conductivity is: 1. The differential equation: \(\frac{1}{r} \frac{d}{dr} \left( r \frac{dT}{dr} \right) = -\frac{q_g}{k}\) 2. Boundary conditions: a. At the center of the wire: \(\frac{dT}{dr}|_{r=0} = 0\) b. At the surface of the wire: \(q_r = -k \frac{dT}{dr}|_{r=r_o}\)