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Chapter 1: Problem 8

A thermodynamic analysis of a proposed Brayton cycle gas turbine yields \(P=5 \mathrm{MW}\) of net power production. The compressor, at an average temperature of \(T_{c}=400^{\circ} \mathrm{C}\), is driven by the turbine at an average temperature of \(T_{h}=1000^{\circ} \mathrm{C}\) by way of an \(L=1\)-m-long, \(d=70-\mathrm{mm}-\) diameter shaft of thermal conductivity \(k=40 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\). (a) Compare the steady-state conduction rate through the shaft connecting the hot turbine to the warm compressor to the net power predicted by the thermodynamics-based analysis. (b) A research team proposes to scale down the gas turbine of part (a), keeping all dimensions in the same proportions. The team assumes that the same hot and cold temperatures exist as in part (a) and that the net power output of the gas turbine is proportional to the overall volume of the device. Plot the ratio of the conduction through the shaft to the net power output of the turbine over the range \(0.005 \mathrm{~m} \leq L \leq 1 \mathrm{~m}\). Is a scaled-down device with \(L=0.005 \mathrm{~m}\) feasible?

Short Answer

Expert verified
In summary, the steady-state conduction rate through the shaft is very small compared to the net power output of the gas turbine at its original size. For a scaled-down device with L=0.005 m to be feasible, the ratio of the conduction rate to net power output must remain similar when all the dimensions are in the same proportions. Therefore, it is essential to plot the ratio over the given range and compare the ratios to evaluate the feasibility of the smaller device.

Step by step solution

01

Calculate the temperature difference between the hot turbine and warm compressor

To calculate the steady-state conduction rate through the shaft, we first need to find the temperature difference between the hot turbine and the warm compressor: 螖T = T_h - T_c Where 螖T is the temperature difference, T_h is the average temperature of the hot turbine, and T_c is the average temperature of the warm compressor. Given that T_h = 1000掳C and T_c = 400掳C, we can calculate 螖T: 螖T = 1000 - 400 = 600 K
02

Calculate the steady-state conduction rate

Now that we have the temperature difference, we can calculate the steady-state conduction rate (Q_dot) using Fourier's law of heat conduction: \( Q_{dot} = \frac{k \cdot A \cdot \Delta T}{L} \) Where k is the thermal conductivity, A is the cross-sectional area of the shaft, 螖T is the temperature difference, and L is the length of the shaft. Given that k = 40 W/m路K, L = 1 m, and d = 70 mm, we can calculate A and then Q_dot: A = 蟺(d/2)^2 = 蟺(0.035 m)^2 鈮� 0.0038 m^2 \( Q_{dot} = \frac{40\ \mathrm{W/m} \cdot \mathrm{K} \cdot 0.0038\ \mathrm{m^2} \cdot 600\ \mathrm{K}}{1\ \mathrm{m}} \approx 91.6\ \mathrm{W} \)
03

Compare the conduction rate to the net power output

Given that the net power output (P) is 5 MW, we can calculate the ratio of the conduction rate to the net power output: \( \frac{Q_{dot}}{P} = \frac{91.6\ \mathrm{W}}{5\ \mathrm{MW}} \approx 1.83 \times 10^{-5} \) (a) The steady-state conduction rate through the shaft is very small compared to the net power output of the gas turbine.
04

Calculate the ratio of conduction rate to net power over the range of lengths

In this step, we will assume that the net power output of the gas turbine (P) is proportional to the overall volume of the device, which means the ratio between Q_dot and P remains constant. We will vary the length L between 0.005 m and 1 m and plot the ratio. Note: Due to editing limitations, a plot like this cannot be shown here, but the main steps regarding the equation and how to determine feasibility will be provided. Since the ratio of conduction rate to net power remains constant as the device is scaled down, we can simplify the equation by setting up the following proportion: \( \frac{Q_{dot}}{P}=\frac{k \cdot A \cdot \Delta T}{L \cdot P_{scaled}} \) Where P_scaled is the net power output of the scaled-down device, proportional to the overall volume of the device.
05

Determine feasibility of the scaled-down device

To determine if the scaled-down device with L = 0.005 m is feasible, we need to calculate the ratio of the conduction rate to the net power output for this length and compare it to the original ratio calculated in Step 3. If the ratio is not drastically different from the original ratio, it is likely that the scaled-down device is feasible. If the ratio remains similar to the ratio obtained in the original configuration when L=0.005 m, then a scaled-down device that keeps all the dimensions in the same proportions would be feasible.

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Key Concepts

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

Thermodynamic Analysis
Thermodynamic analysis refers to the assessment of energy transformations within a thermodynamic system. In the context of a Brayton cycle gas turbine, such analysis helps evaluate the turbine's efficiency and power output. The Brayton cycle involves processes such as compression, heat addition, expansion, and heat rejection. For the problem given, thermodynamic analysis shows a net power production of 5 MW. This power is generated by utilizing the temperature difference between the hot turbine at 1000掳C and the warm compressor at 400掳C.

By determining these key temperatures and the associated pressure changes, calculations of work done and efficiency can be made. In practical terms, this helps designers optimize systems for maximum power output and minimize losses. Understanding thermodynamics is crucial for engineering efficient turbines that supply necessary power with minimum waste.
Steady-State Conduction Rate
The steady-state conduction rate describes the heat transfer rate through a material under constant state conditions. It is the point where heat entering one side of the material equals the heat exiting the other; thus, temperatures remain constant over time.

In this specific example, the problem asks us to compare the steady-state conduction rate through a shaft connecting hot and warm parts of a gas turbine. This rate was computed using Fourier's law of heat conduction, indicating a relatively small amount of heat transfer (approximately 91.6 W) compared to the turbine's power output of 5 MW.

This comparison shows that, even though heat conduction is ongoing, its impact on power output is minimal. Engineers use this insight to ensure there's enough insulation or other mitigation measures to maintain efficiency and prevent energy losses through heat conduction. Therefore, steady-state conduction rates are essential for understanding how effectively a turbine operates, especially in maintaining the desired temperature gradients.
Fourier's Law of Heat Conduction
Fourier's law of heat conduction is fundamental in understanding how heat is transferred through materials. It states that the rate of heat transfer through a material is directly proportional to the negative gradient of temperatures and the area through which heat is transferred.

Mathematically, it's expressed as:
  • \( Q_{dot} = -k abla T A \)
where:
  • \( Q_{dot} \) is the heat transfer rate,
  • \( k \) is the thermal conductivity of the material,
  • \( abla T \) is the temperature gradient,
  • \( A \) is the cross-sectional area.
For the shaft in the exercise, the application's focus is on calculating \( Q_{dot} \) given a set temperature gradient (螖T) and cross-sectional area (A). This calculation helps determine how much energy is lost via conduction.

Engineers rely heavily on Fourier's law to predict how materials will behave under thermal loads. This allows them to engineer materials and designs that can withstand specific conditions, enhancing the overall efficiency and performance of thermal systems like gas turbines.

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Most popular questions from this chapter

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.

A concrete wall, which has a surface area of \(20 \mathrm{~m}^{2}\) and is \(0.30 \mathrm{~m}\) thick, separates conditioned room air from ambient air. The temperature of the inner surface of the wall is maintained at \(25^{\circ} \mathrm{C}\), and the thermal conductivity of the concrete is \(1 \mathrm{~W} / \mathrm{m}=\mathrm{K}\). (a) Determine the heat loss through the wall for outer surface temperatures ranging from \(-15^{\circ} \mathrm{C}\) to \(38^{\circ} \mathrm{C}\), which correspond to winter and summer extremes, respectively. Display your results graphically. (b) On your graph, also plot the heat loss as a function of the outer surface temperature for wall materials having thermal conductivities of \(0.75\) and \(1.25 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\). Explain the family of curves you have obtained.

During its manufacture, plate glass at \(600^{\circ} \mathrm{C}\) is cooled by passing air over its surface such that the convection heat transfer coefficient is \(h=5 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). To prevent cracking, it is known that the temperature gradient must not exceed \(15^{\circ} \mathrm{C} / \mathrm{mm}\) at any point in the glass during the cooling process. If the thermal conductivity of the glass is \(1.4 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) and its surface emissivity is \(0.8\), what is the lowest temperature of the air that can initially be used for the cooling? Assume that the temperature of the air equals that of the surroundings.

If \(T_{s} \approx T_{\text {sur }}\) in Equation \(1.9\), the radiation heat transfer coefficient may be approximated as $$ h_{r, a}=4 \varepsilon \sigma \bar{T}^{3} $$ where \(\bar{T} \equiv\left(T_{s}+T_{\text {sur }}\right) / 2\). We wish to assess the validity of this approximation by comparing values of \(h_{r}\) and \(h_{r, a}\) for the following conditions. In each case, represent your results graphically and comment on the validity of the approximation. (a) Consider a surface of either polished aluminum ( \(\varepsilon=\) \(0.05)\) or black paint \((\varepsilon=0.9)\), whose temperature may exceed that of the surroundings \(\left(T_{\text {sur }}=25^{\circ} \mathrm{C}\right)\) by 10 to \(100^{\circ} \mathrm{C}\). Also compare your results with values of the coefficient associated with free convection in air \(\left(T_{\infty}=T_{\text {sur }}\right)\), where \(h\left(\mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\right)=0.98 \Delta T^{1 / 3}\). (b) Consider initial conditions associated with placing a workpiece at \(T_{s}=25^{\circ} \mathrm{C}\) in a large furnace whose wall temperature may be varied over the range \(100 \leq\) \(T_{\text {sur }} \leq 1000^{\circ} \mathrm{C}\). According to the surface finish or coating, its emissivity may assume values of \(0.05\), \(0.2\), and \(0.9\). For each emissivity, plot the relative error, \(\left(h_{r}-h_{r, a}\right) / h_{r}\), as a function of the furnace temperature.

An aluminum plate \(4 \mathrm{~mm}\) thick is mounted in a horizontal position, and its bottom surface is well insulated. A special, thin coating is applied to the top surface such that it absorbs \(80 \%\) of any incident solar radiation, while having an emissivity of \(0.25\). The density \(\rho\) and specific heat \(c\) of aluminum are known to be \(2700 \mathrm{~kg} / \mathrm{m}^{3}\) and \(900 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\), respectively. (a) Consider conditions for which the plate is at a temperature of \(25^{\circ} \mathrm{C}\) and its top surface is suddenly exposed to ambient air at \(T_{\infty}=20^{\circ} \mathrm{C}\) and to solar radiation that provides an incident flux of \(900 \mathrm{~W} / \mathrm{m}^{2}\). The convection heat transfer coefficient between the surface and the air is \(h=20 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). What is the initial rate of change of the plate temperature? (b) What will be the equilibrium temperature of the plate when steady-state conditions are reached? (c) The surface radiative properties depend on the specific nature of the applied coating. Compute and plot the steady-state temperature as a function of the emissivity for \(0.05 \leq \varepsilon \leq 1\), with all other conditions remaining as prescribed. Repeat your calculations for values of \(\alpha_{S}=0.5\) and \(1.0\), and plot the results with those obtained for \(\alpha_{S}=0.8\). If the intent is to maximize the plate temperature, what is the most desirable combination of the plate emissivity and its absorptivity to solar radiation?
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