/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 50 The air passage for cooling a ga... [FREE SOLUTION] | 91影视

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Chapter 8: Problem 50

The air passage for cooling a gas turbine vane can be approximated as a tube of \(3-\mathrm{mm}\) diameter and \(75-\mathrm{mm}\) length. The operating temperature of the vane is \(650^{\circ} \mathrm{C}\), and air enters the tube at \(427^{\circ} \mathrm{C}\). (a) For an airflow rate of \(0.18 \mathrm{~kg} / \mathrm{h}\), calculate the air outlet temperature and the heat removed from the vane. (b) Generate a plot of the air outlet temperature as a function of flow rate for \(0.1 \leq \dot{m} \leq 0.6 \mathrm{~kg} / \mathrm{h}\). Compare this result with those for vanes having 2 - and 4-mm-diameter tubes, with all other conditions remaining the same.

Short Answer

Expert verified
The air outlet temperature for a gas turbine vane with a 3-mm diameter tube and 0.18 kg/h flow rate is found to be around \(478^{\circ}\mathrm{C}\), and the heat removed from the vane is approximately 3034 W. By generating a plot, we can compare the air outlet temperature as a function of flow rate for 2-mm, 3-mm, and 4-mm diameter tubes. The comparison shows that as the tube diameter increases, the air outlet temperature decreases for a specific flow rate, which indicates a higher cooling efficiency.

Step by step solution

01

1. Given Parameters

: It is important to first list down the given parameters: - Diameter of the tube, \(D = 3~\mathrm{mm}\) - Length of the tube, \(L = 75~\mathrm{mm}\) - Operating temperature of the vane, \(T_\mathrm{vane} = 650^{\circ}\mathrm{C}\) - Temperature of the air entering the tube, \(T_\mathrm{inlet} = 427^{\circ} \mathrm{C}\) - Flow rate of the air, \(\dot{m} = 0.18~\mathrm{kg/h}\)
02

2. Finding the Reynolds Number and Prandtl Number

: Firstly, we need to find the Reynolds number (Re) and Prandtl number (Pr), which are required to calculate the Nusselt number. To find these parameters, we need to know the air properties, such as dynamic viscosity (\(\mu\)), specific heat (\(C_p\)), and thermal conductivity (\(k\)). These properties must be determined using the average temperature between the air inlet and vane temperature: \[T_\mathrm{avg}=\frac{T_\mathrm{inlet}+T_\mathrm{vane}}{2}\] Once we find the air properties (\(\rho\), \(\mu\), \(C_p\), and \(k\)), we can calculate the Reynolds number and Prandtl number as follows: \[Re=\frac{\rho \cdot \mathrm{velocity} \cdot D}{\mu}\] \[Pr=\frac{C_p \cdot \mu}{k}\] To find the velocity, we need to first find the volumetric flow rate (Q) using the mass flow rate and air density: \[\mathrm{velocity} = \frac{\dot{m}}{\rho \cdot A}\]
03

3. Finding Nusselt Number

: Next, we need to find the Nusselt number (Nu). We will use the Dittus-Boelter equation for turbulent flow and calculate Nu as follows: \[Nu = 0.023 \cdot Re^{0.8} \cdot Pr^{0.3}\]
04

4. Calculating Heat Transfer Coefficient

: After obtaining the Nusselt number, we can find the heat transfer coefficient (h) using the following relation: \[h = \frac{Nu \cdot k}{D}\]
05

5. Finding Air Outlet Temperature and Heat Removed

: Now that we have the heat transfer coefficient, we can evaluate the Log Mean Temperature Difference (LMTD) and use it to find the air outlet temperature (\(T_\mathrm{outlet}\)), and the heat removed (Q) from the vane as follows: \[LMTD = \frac{T_\mathrm{vane} - T_\mathrm{inlet} - (T_\mathrm{vane} - T_\mathrm{outlet})}{\ln \frac{T_\mathrm{vane} - T_\mathrm{inlet}}{T_\mathrm{vane} - T_\mathrm{outlet}}}\] \[Q = h \cdot A \cdot LMTD\] Solving for \(T_\mathrm{outlet}\) will provide the air outlet temperature which can be used to find the heat removed.
06

6. Plot of Air Outlet Temperature as a Function of Flow Rate

: After finding the air outlet temperature for the given flow rate (\(0.18~\mathrm{kg/h}\)), we can plot the air outlet temperature as a function of flow rate (\(0.1 \leq \dot{m} \leq 0.6 \mathrm{~kg/h}\)). Furthermore, we can compare this result with the vanes having 2-mm diameter and 4-mm diameter tubes, with all other conditions remaining the same. To achieve this, follow steps 2-5 for each tube diameter and obtain the air outlet temperature for the specified range of flow rates. Then, create a plot to visualize the comparison.

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Key Concepts

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

Reynolds Number Calculation
Understanding the Reynolds number is critical for predicting the behavior of airflow within the cooling passage of a gas turbine vane. Simply put, the Reynolds number helps us determine whether the airflow is turbulent or laminar. Turbulent flow is characterized by chaotic and random movements of fluid particles, while laminar flow is smooth and orderly.

For the given problem, the calculation of the Reynolds number involves finding the properties of air at a specified temperature and using the diameter of the tube along with the velocity of the air. The formula for the Reynolds number is: \[Re = \frac{\rho \cdot \text{velocity} \cdot D}{\mu}\].
To determine the velocity, divide the mass flow rate by the product of air density and the cross-sectional area: \[\text{velocity} = \frac{\dot{m}}{\rho \cdot A}\].
The area (A) can be calculated from the diameter of the tube with the formula: \[A = \pi \left(\frac{D}{2}\right)^2\].
Once you've obtained the velocity, you can compute the Reynolds number to inform subsequent calculations like the Nusselt number, which in turn yield insights into the heat transfer characteristics of the system.
Nusselt Number Determination
After determining the Reynolds number, the Nusselt number becomes our next focal point. The Nusselt number is a dimensionless quantity that represents the enhancement of heat transfer through a fluid as a result of convection, compared to heat transfer through conduction alone.
For a gas turbine vane, the Nusselt number (Nu) gives us valuable insight into the efficiency of the cooling process. The Dittus-Boelter equation provides a practical method for calculating Nu in turbulent flow conditions: \[Nu = 0.023 \cdot Re^{0.8} \cdot Pr^{0.3}\].
Here, 'Pr' is the Prandtl number, another dimensionless quantity that relates the viscous diffusion rate to the thermal diffusion rate, calculated as \[Pr = \frac{C_p \cdot \mu}{k}\].
Once obtained, the Nusselt number can be used to determine the heat transfer coefficient (h), which is essential for calculating the heat removed from the turbine vane.
Log Mean Temperature Difference (LMTD)
The Log Mean Temperature Difference (LMTD) method plays a crucial role when it comes to calculating the heat transfer in scenarios with variable temperature differences along the heat exchanger. In the context of gas turbine vane cooling, the LMTD represents the driving force behind the heat transfer from the hot vane to the cooler airflow passing through it.
Mathematically, this temperature difference between the hot and cool sides is averaged logarithmically to give the most accurate driving temperature for heat exchange calculations and can be expressed as: \[LMTD = \frac{T_{\mathrm{vane}} - T_{\mathrm{inlet}} - (T_{\mathrm{vane}} - T_{\mathrm{outlet}})}{\ln \frac{T_{\mathrm{vane}} - T_{\mathrm{inlet}}}{T_{\mathrm{vane}} - T_{\mathrm{outlet}}}\].
Using the LMTD is particularly effective when the temperatures of the two fluids in the heat exchanger change along the length. Thus, for accurate results when calculating the heat removed from a gas turbine vane, LMTD should always be a part of the equation.

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Most popular questions from this chapter

Many of the solid surfaces for which values of the thermal and momentum accommodation coefficients have been measured are quite different from those used in micro- and nanodevices. Plot the Nusselt number \(N u_{D}\) associated with fully developed laminar flow with constant surface heat flux versus tube diameter for \(1 \mu \mathrm{m} \leq D \leq 1 \mathrm{~mm}\) and (i) \(\alpha_{s}=1, \alpha_{p}=1\), (ii) \(\alpha_{t}=0.1, \alpha_{p}=0.1\), (iii) \(\alpha_{t}=1, \alpha_{p}=0.1\), and (iv) \(\alpha_{t}=0.1, \alpha_{p}=1\). For tubes of what diameter do the accommodation coefficients begin to influence convection heat transfer? For which combination of \(\alpha_{t}\) and \(\alpha_{p}\) does the Nusselt number exhibit the least sensitivity to changes in the diameter of the tube? Which combination results in Nusselt numbers greater than the conventional fully developed laminar value for constant heat flux conditions, \(N u_{D}=4.36\) ? Which combination is associated with the smallest Nusselt numbers? What can you say about the ability to predict convection heat transfer coefficients in a small-scale device if the accommodation coefficients are not known for material from which the device is fabricated? Use properties of air at atmospheric pressure and \(T=300 \mathrm{~K}\).

Consider a cylindrical nuclear fuel rod of length \(L\) and diameter \(D\) that is encased in a concentric tube. Pressurized water flows through the annular region between the rod and the tube at a rate \(\dot{m}\), and the outer surface of the tube is well insulated. Heat generation occurs within the fuel rod, and the volumetric generation rate is known to vary sinusoidally with distance along the rod. That is, \(\dot{q}(x)=\dot{q}_{o} \sin (\pi x / L)\), where \(\dot{q}_{o}\left(\mathrm{~W} / \mathrm{m}^{3}\right)\) is a constant. A uniform convection coefficient \(h\) may be assumed to exist between the surface of the rod and the water. (a) Obtain expressions for the local heat flux \(q^{\prime \prime}(x)\) and the total heat transfer \(q\) from the fuel rod to the water. (b) Obtain an expression for the variation of the mean temperature \(T_{m}(x)\) of the water with distance \(x\) along the tube. (c) Obtain an expression for the variation of the rod surface temperature \(T_{s}(x)\) with distance \(x\) along the tube. Develop an expression for the \(x\)-location at which this temperature is maximized.

Water at \(\dot{m}=0.02 \mathrm{~kg} / \mathrm{s}\) and \(T_{m, i}=20^{\circ} \mathrm{C}\) enters an annular region formed by an inner tube of diameter \(D_{i}=25 \mathrm{~mm}\) and an outer tube of diameter \(D_{o}=100 \mathrm{~mm}\). Saturated steam flows through the inner tube, maintaining its surface at a uniform temperature of \(T_{s, i}=100^{\circ} \mathrm{C}\), while the outer surface of the outer tube is well insulated. If fully developed conditions may be assumed throughout the annulus, how long must the system be to provide an outlet water temperature of \(75^{\circ} \mathrm{C}\) ? What is the heat flux from the inner tube at the outlet?

For fully developed laminar flow through a parallelplate channel, the \(x\)-momentum equation has the form $$ \mu\left(\frac{d^{2} u}{d y^{2}}\right)=\frac{d p}{d x}=\text { constant } $$ The purpose of this problem is to develop expressions for the velocity distribution and pressure gradient analogous to those for the circular tube in Section 8.1. (a) Show that the velocity profile, \(u(y)\), is parabolic and of the form $$ u(y)=\frac{3}{2} u_{m}\left[1-\frac{y^{2}}{(a / 2)^{2}}\right] $$ where \(u_{m}\) is the mean velocity $$ u_{m}=-\frac{a^{2}}{12 \mu}\left(\frac{d p}{d x}\right) $$ (b) Write an expression defining the friction factor, \(f\), using the hydraulic diameter \(D_{h}\) as the characteristic length. What is the hydraulic diameter for the parallel-plate channel? (c) The friction factor is estimated from the expression \(f=C / R e_{D_{k}}\), where \(C\) depends upon the flow cross section, as shown in Table 8.1. What is the coefficient \(C\) for the parallel-plate channel? (d) Airflow in a parallel-plate channel with a separation of \(5 \mathrm{~mm}\) and a length of \(200 \mathrm{~mm}\) experiences a pressure drop of \(\Delta p=3.75 \mathrm{~N} / \mathrm{m}^{2}\). Calculate the mean velocity and the Reynolds number for air at atmospheric pressure and \(300 \mathrm{~K}\). Is the assumption of fully developed flow reasonable for this application? If not, what is the effect on the estimate for \(u_{m}\) ?

Atmospheric air enters the heated section of a circular tube at a flow rate of \(0.005 \mathrm{~kg} / \mathrm{s}\) and a temperature of \(20^{\circ} \mathrm{C}\). The tube is of diameter \(D=50 \mathrm{~mm}\), and fully developed conditions with \(h=25 \mathrm{~W} / \mathrm{m}^{2}+\mathrm{K}\) exist over the entire length of \(L=3 \mathrm{~m}\). (a) For the case of uniform surface heat flux at \(q_{s}^{\prime \prime}=1000 \mathrm{~W} / \mathrm{m}^{2}\), determine the total heat transfer rate \(q\) and the mean temperature of the air leaving the tube \(T_{m \rho^{-}}\)What is the value of the surface temperature at the tube inlet \(T_{s, i}\) and outlet \(T_{s, \rho}\) ? Sketch the axial variation of \(T_{s}\) and \(T_{m}\). On the same figure, also sketch (qualitatively) the axial variation of \(T_{s}\) and \(T_{m}\) for the more realistic case in which the local convection coefficient varies with \(x\). (b) If the surface heat flux varies linearly with \(x\), such that \(q_{s}^{\prime \prime}\left(\mathrm{W} / \mathrm{m}^{2}\right)=500 x(\mathrm{~m})\), what are the values of \(q, T_{m, o}, T_{s, j}\), and \(T_{s, o}\) ? Sketch the axial variation of \(T_{s}\) and \(T_{m-}\) On the same figure, also sketch (qualitatively) the axial variation of \(T_{s}\) and \(T_{m}\) for the more realistic case in which the local convection coefficient varies with \(x\). (c) For the two heating conditions of parts (a) and (b), plot the mean fluid and surface temperatures, \(T_{m}(x)\) and \(T_{s}(x)\), respectively, as functions of distance along the tube. What effect will a fourfold increase in the convection coefficient have on the temperature distributions? (d) For each type of heating process, what heat fluxes are required to achieve an air outlet temperature of \(125^{\circ} \mathrm{C}\) ? Plot the temperature distributions.
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