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An air heater may be fabricated by coiling Nichfome wire and passing air in cross flow over the wire. Consider a heater fabricated from wire of diameter \(D=\) \(1 \mathrm{~mm}\), electrical resistivity \(p_{e}=10^{-6} \mathrm{n} \cdot \mathrm{m}\), thernal conductivity \(k=25 \mathrm{~W} / \mathrm{m}=\mathrm{K}\), and emissivity \(\varepsilon=0.2\). The heater is designed to deliver air at a temperatiue of \(T_{m}=50^{\circ} \mathrm{C}\) under flow conditions that provide a convection coefficient of \(h=250 \mathrm{~W} / \mathrm{m}^{2}\). \(\mathrm{K}\) for the wire. The temperature of the housing that encloses the wire and through which the air flows is \(T_{\text {er }}=50^{\circ} \mathrm{C}\). If the maximum allowable temperature of the wire is \(T_{\max }=1200^{\circ} \mathrm{C}\), what is the maximum allowable clextric current \(n\) ? If the maximum available voltage is \(\Delta E=110 \mathrm{~V}\), what is the corresponding length \(L\) of wire that may be used in the heater and the power raing of the heater? Himt: In your solution, assume negligible temperature variations within the wire, but after obtaining the desired results, assess the validity of this arsumption.

Step by step solution

01

Calculate Maximum Allowable Current

Using the given maximum allowable temperature of the wire, apply the energy balance for the Nichrome wire. Start with the expression for ohmic heating: \[ q = I^2 R \]where \( q \) is the heat generated per unit length, \( I \) is the current, and \( R \) is the resistance. The wire experiences convection and radiation to the surroundings. The equation becomes:\[ I^2 R = \, h \pi D (T_{w} - T_{m}) + \varepsilon \sigma \pi D (T_{w}^4 - T_{ ext{sur}}^4) \].Substitute known values and solve for \( I \).

02

Determine Resistance per Unit Length

From the resistivity \( \rho_e \) of the wire, calculate the resistance per unit length using the formula:\[ R' = \frac{\rho_e}{A} \],where \( A = \frac{\pi D^2}{4} \) is the cross-sectional area of the wire. Substitute the values and calculate \( R' \).

03

Calculate Corresponding Length of Wire

The resistance of the wire is also given by \( R = R' L \). Using the available voltage \( \Delta E \), we relate to ohm's law \( V = IR \):\[ \Delta E = I \cdot R' \cdot L \].Solve for the length \( L \) by substituting \( I \) from Step 1 and calculating \( R' \) from Step 2.

04

Calculate Power Rating of Heater

The power rating of the heater is given by electrical power, \( P = I \cdot \Delta E \). Substitute \( I \) from Step 1 and the voltage \( \Delta E = 110 \, V \). Calculate the power \( P \).

05

Validate Assumption of Negligible Temperature Variation

Assess whether the assumption of negligible temperature variations within the wire holds by comparing the Biot number \( Bi = \frac{hD}{k} \) to a critical value, usually much less than 0.1. Calculate \( Bi \) to validate if the assumption is reasonable.

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Key Concepts

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

Ohmic Heating

Ohmic heating, also known as Joule heating, occurs when electricity flows through a conductor, like a wire, causing it to heat up. This happens because of the resistance the material offers to the flow of electric current. It's quantified using the formula \[ q = I^2R \]where- \( q \) is the heat generated,- \( I \) is the current,- \( R \) is the electrical resistance. This effect is key when designing devices such as air heaters, where managing heat output is critical. The amount of heat generated depends directly on how much current passes through the wire and the resistance of the wire material, ensuring that the resulting heat is sufficient for the intended application.

In the context of the exercise, Ohmic heating is used to determine how much heat can be safely produced by the wire without exceeding its maximum temperature limit of 1200°C. Proper calculation ensures efficiency and safety in applications where heat generation is controlled by electricity.

Convection Coefficient

The convection coefficient, denoted as \( h \), represents how efficiently heat is transferred from a surface to a fluid moving over it. It's a measure of convective heat transfer performance. An important parameter in thermal analysis, it varies based on fluid properties and flow conditions.

In the exercise, the convection coefficient is given as \( 250 \, \mathrm{W/m}^2\cdot \mathrm{K} \), which indicates a highly efficient heat transfer from the wire's surface to the air flowing over it. This parameter directly influences the heat dissipation rate from the wire, playing a critical role in preventing overheating.

In formula terms, the heat transfer via convection can be described by:\[ q = h \cdot A \cdot (T_{w} - T_{m}) \]where- \( q \) is the heat transfer,- \( A \) is the surface area,- \( T_{w} \) is the wire temperature,- \( T_{m} \) is the temperature of the moving fluid.Understanding and calculating the convection coefficient is crucial in system design where cooling efficiency is needed to manage heat generated by electrical resistance.

Thermal Conductivity

Thermal conductivity \( k \) is a property of a material that indicates its ability to conduct heat. High thermal conductivity means a material can transfer heat efficiently, minimizing temperature variations. In the context of the problem, the wire has a thermal conductivity of \( 25 \, \mathrm{W/m} \cdot \mathrm{K} \). This value plays a significant role in ensuring that the wire can distribute heat uniformly across its length, enhancing the performance and safety of the heater.

Thermal conductivity helps determine how much the wire can maintain temperature uniformity, influencing assumptions like negligible temperature variations. In thermal analysis, the Biot number \( Bi \) is used to assess whether temperature distribution within a body is uniform. It's given by:\[ Bi = \frac{hD}{k} \]where- \( h \) is the convection coefficient,- \( D \) is the characteristic length (such as the wire's diameter),- \( k \) is the thermal conductivity.In the scenario, checking the Biot number helps validate the assumption of uniform temperature in the wire, ensuring reliable calculations for maximum current.

Electrical Resistivity

Electrical resistivity \( \rho_e \) defines how much a material opposes the flow of electric current. It is given in \( \Omega \cdot \mathrm{m} \), meaning how much resistance is present per unit length for a specific cross-sectional area. In the exercise, the resistivity of the Nichrome wire is provided as \( 10^{-6} \, \Omega \cdot \mathrm{m} \). This low value indicates that Nichrome is a good conductor, though it still provides enough resistance for heating purposes when current flows through it.

• The resistance \( R \) of the wire can be calculated from its resistivity using:
\[ R' = \frac{\rho_e}{A} \]where- \( R' \) is the resistance per unit length,- \( A \) is the wire's cross-sectional area given by \( \frac{\pi D^2}{4} \).
• Resistance increases with the length of wire and decreases with greater cross-sectional area.

This principle is crucial for calculating how much wire can be used (length) to achieve desired resistance and heating levels, ensuring effective design of electric heating devices. Understanding electrical resistivity helps balance between enough resistance for heating and maintaining efficient electric current flow.