91Ó°ÊÓ

Turbine blades mounted to a rotating disc in a gas turbine engine are exposed to a gas stream that is at \(T_{\infty}=1200^{\circ} \mathrm{C}\) and maintains a convection coefficient of \(h=250 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) over the blade. The blades, which are fabricated from Inconel, \(k \approx 20 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\), have a length of \(L=50 \mathrm{~mm}\). The blade profile has a uniform cross-sectional area of \(A_{c}=6 \times 10^{-4} \mathrm{~m}^{2}\) and a perimeter of \(P=110 \mathrm{~mm}\). A proposed blade- cooling scheme, which involves routing air through the supporting disc, is able to maintain the base of each blade at a temperature of \(T_{b}=300^{\circ} \mathrm{C}\). (a) If the maximum allowable blade temperature is \(1050^{\circ} \mathrm{C}\) and the blade tip may be assumed to be adiabatic, is the proposed cooling scheme satisfactory? (b) For the proposed cooling scheme, what is the rate at which heat is transferred from each blade to the coolant?

Step by step solution

01

Define the variables and determine the resistance

Since both convection and conduction are occurring alongside each other, we define their respective variables: - Convection: Gas stream temperature \((T_{\infty}=1200^{\circ}\mathrm{C})\) and convection coefficient \(h\) - Conduction: Conductivity of Inconel (\(k\)), length, cross-sectional area, and perimeter of the blade profile The resistance (\(R\)) to heat transfer for the blade can be found using the formula: \[R = \frac{L}{kA_c} + \frac{1}{hA_s}\] where \(A_s\) is the surface area of the blade.

02

Calculate the surface area of the blade

We need to find \(A_s\) to calculate resistance. The blade surface area (\(A_s\)) can be obtained using the blade's perimeter (\(P\)) and length (\(L\)) as follows: \[A_s = P \times L\] First, we convert the given perimeter to meters: \[P = 110~\mathrm{mm} \times \frac{1~\mathrm{m}}{1000~\mathrm{mm}} = 0.11~\mathrm{m}\] Now we can calculate the surface area: \[A_s = 0.11~\mathrm{m} \times 50 \times 10^{-3}~\mathrm{m} = 0.0055~\mathrm{m}^2\]

03

Calculate the resistance to heat transfer

Now we have all the necessary values to find the resistance (\(R\)): \[R = \frac{L}{kA_c} + \frac{1}{hA_s} = \frac{50 \times 10^{-3}~ \mathrm{m}}{20~ \mathrm{W/m\cdot K} \times 6\times10^{-4}~\mathrm{m}^2} + \frac{1}{250~\mathrm{W/m^2\cdot K} \times 0.0055~\mathrm{m}^2}\] \[R \approx 0.03130~\mathrm{K/W}\]

04

Calculate the temperature at the blade tip

Considering the adiabatic condition at the blade tip, the temperature difference across the blade can be calculated using the resistance and heat transfer rate. Since heat transfer rate \(q = \frac{\Delta T}{R}\), we have \[\Delta T = q \cdot R\] Now we know the temperature difference and the base temperature (\(T_b = 300^{\circ}\mathrm{C}\)), we can calculate the temperature at the blade tip (\(T_t\)) using the following formula: \[T_t = T_b + \Delta T\] First, we need the heat transfer rate (\(q\)) as follows: \[q = \frac{T_\infty - T_b}{R} = \frac{1200 - 300}{0.03130} \Rightarrow q \approx 28746.65~\mathrm{W}\] Now we can find the temperature at the blade tip: \[T_t = 300 + 28746.65 \times 0.03130~\mathrm{K/W}\] \[T_t \approx 902.3^{\circ} \mathrm{C}\]

05

Determine the cooling scheme's effectiveness

Since the calculated temperature at the blade tip (\(T_t = 902.3^{\circ} \mathrm{C}\)) is less than the maximum allowable temperature (\(1050^{\circ} \mathrm{C}\)), the proposed cooling scheme is satisfactory.

06

Calculate the rate of heat transfer to the coolant

We already calculated the heat transfer rate in step 4 (\(q = 28746.65~\mathrm{W}\)). Therefore, the rate at which heat is transferred from each blade to the coolant is approximately \(28746.65~\mathrm{W}\).

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Key Concepts

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

Convection and Conduction

Heat transfer is a crucial concept when evaluating objects like turbine blades exposed to high temperatures. Two significant modes of heat transfer are convection and conduction.

• Conduction: This occurs within a solid body. It transfers heat through a material due to temperature gradients.
• Convection: This occurs between a solid surface and a fluid, such as air or water, involving heat exchanged between the surface and the fluid.

For turbine blades in a gas turbine, convection takes place as the hot gas stream flows over the blades, promoting heat exchange. Convection is characterized by the convection coefficient, denoted as \( h \), which in this exercise is \( 250 \, \mathrm{W/m^2 \cdot K} \).
The material of the blade, Inconel, promotes conduction within the blade's structure. In this exercise, the thermal conductivity \( k \) of Inconel is \( 20 \, \mathrm{W/m \cdot K} \). Together, these heat transfer modes determine how efficiently the blade can dissipate heat, ensuring it doesn't exceed its maximum allowable temperature.
The synergy between convection and conduction ensures turbine blades remain effective under extreme conditions by regulating temperature within controllable limits.

Turbine Blade Cooling

Cooling of turbine blades is essential to maintain performance and structural integrity in high-temperature environments. In the given problem, the blades must be cooled to avoid exceeding \( 1050^{\circ} \mathrm{C} \). This cooling mechanism is achieved through air routed via the supporting disc.
The blade's base is maintained at a lower temperature, specifically \( 300^{\circ} \mathrm{C} \), which is essential for effective cooling. With the tip of the blade assumed adiabatic, meaning no heat is transferred from the tip to the surrounding, the cooling must evenly regulate temperature along the blade.
In our scenario, the proposed cooling scheme effectively reduces the maximum blade temperature. This is demonstrated through using thermal resistance and calculating the resulting temperature, showing it doesn't exceed the dangerous threshold, which validates the cooling strategy used.

Thermal Resistance Calculation

Thermal resistance is a pivotal concept in understanding how effectively heat is transferred from the turbine blades to the surrounding coolant. It represents the temperature difference driving the heat flow divided by the amount of heat transferred. The formula used in this analysis is: \[R = \frac{L}{kA_c} + \frac{1}{hA_s}\] where:

• \( R \) is the total thermal resistance, which combines both conduction and convection resistances.
• \( L \) is the blade's length of \( 50 \, \mathrm{mm} \) or \( 0.05 \, \mathrm{m} \).
• \( k \) is the thermal conductivity of Inconel, given as \( 20 \, \mathrm{W/m \cdot K} \).
• \( A_c \) is the cross-sectional area \( 6 \times 10^{-4} \, \mathrm{m^2} \).
• \( h \) is the convection coefficient \( 250 \, \mathrm{W/m^2 \cdot K} \).
• \( A_s \) is the surface area found using the blade’s perimeter and length \( 0.0055 \, \mathrm{m^2} \).

Using these values, the total thermal resistance came out to be approximately \( 0.03130 \, \mathrm{K/W} \). From this resistance, the rate of heat transfer can be calculated, ensuring that the proposal for turbine blade cooling remains efficient and prevents overheating, confirming the effectiveness of the design.