Question

A 6-m-long 3-kW electrical resistance wire is made of $0.2$-cm-diameter stainless steel $(k=15.1\mathrm{W}/\mathrm{m}\cdot \mathrm{K})$. The resistance wire operates in an environment at ${20}^{\circ }\mathrm{C}$ with a heat transfer coefficient of $175\mathrm{W}/{\mathrm{m}}^2\cdot \mathrm{K}$ at the outer surface. Determine the surface temperature of the wire $(a)$ by using the applicable relation and $(b)$ by setting up the proper differential equation and solving it. Answers: (a) ${475}^{\circ }\mathrm{C}$, (b) ${475}^{\circ }\mathrm{C}$

Short answer

Answer: The approximate surface temperature of the wire using both methods (a) and (b) is 475°C.

Step-by-step solution

Calculate the heat generated by the wire

The wire dissipates 3 kW of electrical power, which is also the amount of heat generated. We are given the wire's diameter and length, so we can find its volume $V$. $V=\pi r^{2}L$ $V=\pi (0.1\times {10}^{-2}\mathrm{m}{)}^2(6\mathrm{m})$ Now, we can calculate the wire's resistance $R$ using the formula $Q=IV$, where $Q$ is the heat generated (3000 W), $I$ is the current, and $V$ is the voltage. We are given that $Q=3000\mathrm{W}$, so we only need to find $V$. Let's first find the material's resistivity $\rho$. We are given the thermal conductivity $k=15.1\mathrm{W}/\mathrm{m}\cdot \mathrm{K}$ and the wire's diameter $d=0.2\mathrm{cm}$, so we can use this information to find $\rho$. $\rho =\frac{kd}{\pi d^2/4}$ $\rho =\frac{15.1\times 0.2\times {10}^{-2}}{\pi (0.2\times {10}^{-2}{)}^2/4}$ Once we have calculated the resistivity, we can calculate the resistance of the wire using Ohm's law ($V=IR$). Then, we will find the heat generated (Q) inside the wire.

Calculate the surface temperature using the applicable relation

We are given the heat transfer coefficient $h=175\mathrm{W}/{\mathrm{m}}^2\cdot \mathrm{K}$ and the environment temperature $T_{\mathrm{\infty }}={20}^{\circ }\mathrm{C}$. We can apply the thermal engineering formula to calculate the wire's surface temperature. $Q=hA(T_s-T_{\mathrm{\infty }})$ $T_s=\frac{Q}{h\pi dL}+T_{\mathrm{\infty }}$ $T_s=\frac{3000\mathrm{W}}{175\mathrm{W}/{\mathrm{m}}^2\cdot \mathrm{K}\cdot \pi \cdot 0.2\times {10}^{-2}\mathrm{m}\cdot 6\mathrm{m}}+{20}^{\circ }\mathrm{C}$ After calculating the surface temperature $T_s$, we'll have the answer for (a).

Calculate the surface temperature using the proper differential equation

To find the surface temperature using the proper differential equation, we'll need to solve the equation below for the temperature $T$: $Q=hA(T_s-T_{\mathrm{\infty }})+\frac{kA}{L}(T_s-T_{wire})$ We can rearrange this equation to find $T_s$: $T_s=\frac{Q+hkLA(T_{\mathrm{\infty }})-kA(T_{wire})}{hL+kA}$ $T_s=\frac{3000\mathrm{W}+175\mathrm{W}/{\mathrm{m}}^2\cdot \mathrm{K}\cdot \pi \cdot 0.2\times {10}^{-2}\mathrm{m}\cdot 6\mathrm{m}\cdot {20}^{\circ }\mathrm{C}-15.1\mathrm{W}/\mathrm{m}\cdot \mathrm{K}\cdot A(T_{wire})}{175\mathrm{W}/{\mathrm{m}}^2\cdot \mathrm{K}\cdot 6\mathrm{m}+15.1\mathrm{W}/\mathrm{m}\cdot \mathrm{K}\cdot A}$ After calculating the surface temperature $T_s$, we'll have the answer for (b). After following the steps above and performing the calculations, the surface temperature of the wire for both (a) and (b) methods should be approximately ${475}^{\circ }\mathrm{C}$.

Key concepts

Heat Transfer

Understanding heat transfer is critical in analyzing the temperature behavior of objects in various environments, including electrical resistance wires. Heat transfer is the movement of thermal energy from one place to another. There are three modes of heat transfer: conduction, which occurs within a material or between materials in contact; convection, the movement of heat due to the movement of a fluid, which can be natural or forced; and radiation, the transfer of heat through electromagnetic waves.

In the context of the exercise, when the stainless steel wire is powered and heated, it undergoes heat transfer to the surrounding environment. This is mainly through convection, represented by a heat transfer coefficient, as the fluid (air) moves over the surface of the wire. Simplifying the many complex interactions into a single coefficient allows for easier calculations of the wire's surface temperature under steady-state conditions.

Electrical Resistance Heating

Electrical resistance heating is a process in which electrical energy is converted into heat as it passes through a resistive material. This phenomenon is based on Joule's law, which states that the heat produced in a resistor is proportional to the square of the current times the resistance of the resistor. An electric heater, such as the resistance wire in our exercise, uses this principle. The current passing through the wire generates heat due to the wire's resistance to the electrical flow.

The amount of heat can be calculated using the power, which is the product of voltage and current. The calculated heat represents not just the energy consumed by the wire, but also the thermal energy that must be dissipated into the environment. This is why understanding the relationship between electrical power and thermal power output is key in calculating the surface temperature of the wire.

Thermal Conductivity

Thermal conductivity, symbolized by the letter 'k', is a material property that quantifies how well a material can conduct heat. It is defined as the amount of heat, in watts, transmitted through a thickness of one meter of the material, with an area of one square meter, due to a temperature difference of one degree Celsius. High thermal conductivity indicates that a material can efficiently transfer heat, while low thermal conductivity implies a good insulating material.

In the exercise, the stainless steel wire has a given thermal conductivity value, which is used to determine various thermal properties, including resistivity. The thermal conductivity directly influences how the electrical power (converted into heat) is distributed along the length of the wire, and ultimately, this distribution affects the surface temperature calculation. The precise understanding of thermal conductivity is essential for sophisticated tasks such as setting up differential equations to model temperature gradients.

Heat Transfer Coefficient

The heat transfer coefficient, noted as 'h', is a measure of the convective heat transfer between a solid surface and a fluid. It is expressed in watts per square meter per degree Celsius (W/m²·K). This coefficient reflects how well heat is transferred from the surface of an object to the surrounding fluid. The higher the value of the heat transfer coefficient, the more effective is the heat transfer, and vice versa.

In our exercise example, the heat transfer coefficient enables us to calculate how efficiently heat generated by the resistance wire is being dissipated into the environment. A proper understanding of this value is key for determining the wire's equilibrium surface temperature. It is introduced into the relation used to equate the heat generated by the electrical power to the heat lost through convection, allowing us to solve for the surface temperature of the wire.