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Chapter 1: Problem 84

Bus bars proposed for use in a power transmission station have a rectangular cross section of height \(H=600 \mathrm{~mm}\) and width \(W=200 \mathrm{~mm}\). The electrical resistivity, \(\rho_{e}(\mu \Omega \cdot \mathrm{m})\), of the bar material is a function of temperature, \(\rho_{e}=\rho_{e, o}\left[1+\alpha\left(T-T_{o}\right)\right]\), where \(\rho_{e, a}=\) \(0.0828 \mu \Omega \cdot \mathrm{m}, T_{o}=25^{\circ} \mathrm{C}\), and \(\alpha=0.0040 \mathrm{~K}^{-1}\). The emissivity of the bar's painted surface is \(0.8\), and the temperature of the surroundings is \(30^{\circ} \mathrm{C}\). The convection coefficient between the bar and the ambient air at \(30^{\circ} \mathrm{C}\) is \(10 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). (a) Assuming the bar has a uniform temperature \(T\), calculate the steady-state temperature when a current of \(60,000 \mathrm{~A}\) passes through the bar. (b) Compute and plot the steady-state temperature of the bar as a function of the convection coefficient for \(10 \leq h \leq 100 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). What minimum convection coefficient is required to maintain a safe-operating temperature below \(120^{\circ} \mathrm{C}\) ? Will increasing the emissivity significantly affect this result?

Short Answer

Expert verified
In summary, to find the steady-state temperature of a busbar with a given electrical resistivity, emissivity, and convection coefficient, we need to balance the Joule heating power with the heat removed by convection and radiation. Solving this heat balance equation allows us to determine the steady-state temperature and analyze its dependence on the convection coefficient and emissivity. To maintain a safe-operating temperature below 120掳C, we can find the minimum convection coefficient required and analyze the impact of increasing the emissivity on this result.

Step by step solution

01

Find the resistance of the busbar

To find the resistance of the busbar, we can use the formula: \[R = \frac{\rho L}{A}\] Where R is the resistance, 蟻 is the resistivity, L is the length, and A is the cross-sectional area. Since we know the dimensions of the busbar, we can find the cross-sectional area as: \[A = W \times H\]
02

Calculate the Joule heating power

Now, we can find the Joule heating power using the formula: \[P_{joule} = I^2 R\] Where P_joule is the power dissipation due to Joule heating, I is the current (60,000 A) and R is the resistance of the busbar we found earlier.
03

Calculate heat removed by convection and radiation

The heat removed by convection can be found using the following formula: \[Q_{conv} = h A_s (T - T_s)\] Where Q_conv is the heat removed by convection, h is the convection coefficient, A_s is the surface area of the busbar, T is the temperature of the busbar, and T_s is the temperature of the surroundings. The heat removed by radiation can be found using the following formula: \[Q_{rad} = \sigma \varepsilon A_s (T^4 - T_s^4)\] Where Q_rad is the heat removed by radiation, 蟽 is the Stefan-Boltzmann constant (5.67 脳 10^-8 W/m^2K^4), 蔚 is the emissivity of the busbar (0.8), and A_s is the surface area of the busbar.
04

Balance the heat equation

In steady-state, the heat generated by the Joule effect must equal the heat removed by convection and radiation, so we have: \[P_{joule} = Q_{conv} + Q_{rad}\] We can now solve this equation for the steady-state temperature T.
05

Find the minimum convection coefficient

To find the minimum convection coefficient required to maintain a safe-operating temperature below 120掳C, we will need to analyze how the steady-state temperature changes as a function of the convection coefficient and find the value of the convection coefficient at which the temperature equals 120掳C.
06

Analyze the impact of increasing emissivity

After finding the minimum convection coefficient, we can analyze how increasing the emissivity of the busbar will affect the steady-state temperature and if it will significantly change the minimum convection coefficient required for safe operation. Using the above steps, the problem can be solved, and the various relationships can be analyzed to fully understand the impacts of the convection coefficient and increased emissivity.

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Key Concepts

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

Heat Transfer
Understanding heat transfer is essential in a wide range of engineering applications, from power stations to cooling systems. It refers to the movement of thermal energy from one place to another due to a temperature difference. There are three modes of heat transfer: conduction, which involves heat moving through a static material; convection, which is the transfer of heat by the movement of fluids such as air or water; and radiation, which involves emitting heat energy in the form of electromagnetic waves.

For example, in the case of the bus bars in a power transmission station, heat transfer occurs through both convection and radiation. The bars emit heat to the surrounding air (convection) and also radiate heat to the environment. The steady-state temperature of the bar will be reached when the heat generated within the bar equals the heat removed by these two processes. This balance is crucial to prevent overheating and ensure the system's safety and efficiency.
Joule Heating
Joule heating, also known as resistive or Ohmic heating, is the process by which the passage of an electric current through a conductor produces heat. This effect is a result of the interaction between the moving electrons and the atoms within the conductor that resist the flow of electrical current.

In our exercise, the bus bars work as a conductor for the electric current. When a current of 60,000 A flows through it, Joule heating occurs due to the bar鈥檚 electrical resistance. The formula \(P_{joule} = I^2 R\) allows us to calculate the power dissipated as heat. Ensuring we can calculate this value is critical because it must match the heat removed by convection and radiation for the system to maintain a steady temperature. If not properly managed, excessive Joule heating can lead to failure or damage within electrical systems.
Convection Coefficient
The convection coefficient, denoted as 'h', represents the efficiency of heat transfer between a solid surface and a fluid moving over it. It's a measure of how well the fluid can remove heat from the surface, and its unit is \(W/m^2K\). High convection coefficients indicate efficient cooling, while low coefficients suggest poor heat transfer.

Within our exercise context, the convection coefficient plays a vital role in determining the steady-state temperature of the bus bar. A higher coefficient means that the air can remove more heat from the bus bar, potentially reducing its steady-state temperature. As we compute the steady-state temperature for different convection coefficients from \(10 W/m^2K\) to \(100 W/m^2K\), we aim to find the value below which the bar's temperature remains safe, that is, under \(120^\circ C\). This value of 'h' ensures the bus bar doesn't overheat, thereby maintaining the reliability and safety of the power station鈥檚 operation.

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Most popular questions from this chapter

An instrumentation package has a spherical outer surface of diameter \(D=100 \mathrm{~mm}\) and emissivity \(\varepsilon=0.25\). The package is placed in a large space simulation chamber whose walls are maintained at \(77 \mathrm{~K}\). If operation of the electronic components is restricted to the temperature range \(40 \leq T \leq 85^{\circ} \mathrm{C}\), what is the range of acceptable power dissipation for the package? Display your results graphically, showing also the effect of variations in the emissivity by considering values of \(0.20\) and \(0.30\).

A cartridge electrical heater is shaped as a cylinder of length \(L=200 \mathrm{~mm}\) and outer diameter \(D=20 \mathrm{~mm}\). Under normal operating conditions, the heater dissipates \(2 \mathrm{~kW}\) while submerged in a water flow that is at \(20^{\circ} \mathrm{C}\) and provides a convection heat transfer coefficient of \(h=5000 \mathrm{~W} / \mathrm{m}^{2}+\mathrm{K}\). Neglecting heat transfer from the ends of the heater, determine its surface temperature \(T_{s}\). If the water flow is inadvertently terminated while the heater continues to operate, the heater surface is exposed to air that is also at \(20^{\circ} \mathrm{C}\) but for which \(h=50\) \(\mathrm{W} / \mathrm{m}^{2} \cdot \mathrm{K}\). What is the corresponding surface temperature? What are the consequences of such an event?

Most of the energy we consume as food is converted to thermal energy in the process of performing all our bodily functions and is ultimately lost as heat from our bodies. Consider a person who consumes \(2100 \mathrm{kcal}\) per day (note that what are commonly referred to as food calories are actually kilocalories), of which \(2000 \mathrm{kcal}\) is converted to thermal energy. (The remaining \(100 \mathrm{kcal}\) is used to do work on the environment.) The person has a surface area of \(1.8 \mathrm{~m}^{2}\) and is dressed in a bathing suit. (a) The person is in a room at \(20^{\circ} \mathrm{C}\), with a convection heat transfer coefficient of \(3 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). At this air temperature, the person is not perspiring much. Estimate the person's average skin temperature. (b) If the temperature of the environment were \(33^{\circ} \mathrm{C}\), what rate of perspiration would be needed to maintain a comfortable skin temperature of \(33^{\circ} \mathrm{C}\) ?

The heat flux through a wood slab \(50 \mathrm{~mm}\) thick, whose inner and outer surface temperatures are 40 and \(20^{\circ} \mathrm{C}\), respectively, has been determined to be \(40 \mathrm{~W} / \mathrm{m}^{2}\). What is the thermal conductivity of the wood?

In considering the following problems involving heat transfer in the natural environment (outdoors), recognize that solar radiation is comprised of long and short wavelength components. If this radiation is incident on a semitransparent medium, such as water or glass, two things will happen to the nonreflected portion of the radiation. The long wavelength component will be absorbed at the surface of the medium, whereas the short wavelength component will be transmitted by the surface. (a) The number of panes in a window can strongly influence the heat loss from a heated room to the outside ambient air. Compare the single- and double-paned units shown by identifying relevant heat transfer processes for each case. (b) In a typical flat-plate solar collector, energy is collected by a working fluid that is circulated through tubes that are in good contact with the back face of an absorber plate. The back face is insulated from the surroundings, and the absorber plate receives solar radiation on its front face, which is typically covered by one or more transparent plates. Identify the relevant heat transfer processes, first for the absorber plate with no cover plate and then for the absorber plate with a single cover plate. (c) The solar energy collector design shown in the schematic has been used for agricultural applications. Air is blown through a long duct whose cross section is in the form of an equilateral triangle. One side of the triangle is comprised of a double-paned, semitransparent cover; the other two sides are constructed from aluminum sheets painted flat black on the inside and covered on the outside with a layer of styrofoam insulation. During sunny periods, air entering the system is heated for delivery to either a greenhouse, grain drying unit, or storage system. Identify all heat transfer processes associated with the cover plates, the absorber plate(s), and the air. (d) Evacuated-tube solar collectors are capable of improved performance relative to flat-plate collectors. The design consists of an inner tube enclosed in an outer tube that is transparent to solar radiation. The annular space between the tubes is evacuated. The outer, opaque surface of the inner tube absorbs solar radiation, and a working fluid is passed through the tube to collect the solar energy. The collector design generally consists of a row of such tubes arranged in front of a reflecting panel. Identify all heat transfer processes relevant to the performance of this device.
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