/*! 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 68 An uninsulated steam pipe is use... [FREE SOLUTION] | 91影视

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Chapter 7: Problem 68

An uninsulated steam pipe is used to transport hightemperature steam from one building to another. The pipe is of \(0.5-\mathrm{m}\) diameter, has a surface temperature of \(150^{\circ} \mathrm{C}\), and is exposed to ambient air at \(-10^{\circ} \mathrm{C}\). The air moves in cross flow over the pipe with a velocity of \(5 \mathrm{~m} / \mathrm{s}\). (a) What is the heat loss per unit length of pipe? (b) Consider the effect of insulating the pipe with a rigid urethane foam \((k=0.026 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\). Evaluate and plot the heat loss as a function of the thickness \(\delta\) of the insulation layer for \(0 \leq \delta \leq 50 \mathrm{~mm}\).

Short Answer

Expert verified
In this problem, we first calculate the heat loss per unit length for an uninsulated pipe by finding the convective heat transfer coefficient (h) and the heat loss (q). After that, we calculate the conductive heat transfer through the insulation and plot the heat loss as a function of insulation thickness \(\delta\). The results show that the heat loss per unit length decreases as the insulation thickness increases.

Step by step solution

01

Calculate Convective Heat Transfer Coefficient (h)

To find the heat loss per unit length of the pipe due to convection, we need to first calculate the convective heat transfer coefficient (h). We can use the following formula to calculate this: \(h = Nu \cdot \frac{k_a}{D}\) Where: - Nu is the average Nusselt number for the flow around the cylinder (dimensionless), - \(k_a\) is the thermal conductivity of air (W/m路K), - D is the diameter of the pipe (m). We can look up the thermal conductivity of air (\(k_a\)) at the film temperature (\((T_s + T_\infty) /2\)), where \(T_s = 150^{\circ} \mathrm{C}\) is the surface temperature of the pipe and \(T_\infty = -10^{\circ} \mathrm{C}\) is the ambient air temperature. We also need to find the Nusselt number (Nu) for this flow configuration. To do this, we need the Reynolds number (Re) and the Prandtl number (Pr) for the flow. We can look up these values in a fluid properties table at the film temperature.
02

Calculate Heat Loss per Unit Length for Uninsulated Pipe

Now that we have the convective heat transfer coefficient (h), we can calculate the heat loss per unit length of the pipe (q) due to convection: \(q = h \cdot \pi D \cdot (T_s - T_\infty)\) Where: - h is the convective heat transfer coefficient (W/m虏路K), - D is the diameter of the pipe (m), - \(T_s\) is the surface temperature of the pipe (掳C), - \(T_\infty\) is the ambient air temperature (掳C).
03

Calculate Conductive Heat Transfer Through Insulation

To calculate the heat loss per unit length of the insulated pipe, we need to analyze the conduction heat transfer through the insulation layer. The heat loss per unit length due to conduction can be calculated using the following formula: \(q = \frac{2 \pi k_i (T_s - T_\infty)}{ln(\frac{r_i + \delta}{r_i})}\) Where: - \(k_i\) is the thermal conductivity of the insulation material (W/m路K), - \(T_s\) is the surface temperature of the pipe (掳C), - \(T_\infty\) is the ambient air temperature (掳C), - \(r_i\) is the inner radius of the insulation (m), - \(\delta\) is the thickness of the insulation layer (m).
04

Plot Heat Loss as a Function of Insulation Thickness

Using the formula derived in Step 3, we can create a plot of the heat loss per unit length of the pipe as a function of the insulation thickness \(\delta\). The plot should cover the range specified in the problem, \(0 \leq \delta \leq 50 \mathrm{~mm}\). To create the plot, simply evaluate the heat loss formula for various values of \(\delta\), and then graph the results. The resulting plot will provide a visual representation of how the heat loss per unit length varies as the insulation thickness changes.

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Key Concepts

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

Convective Heat Transfer Coefficient
Understanding the convective heat transfer coefficient is essential when evaluating how heat is lost from surfaces, like our steam pipe example. This coefficient, denoted by the symbol 'h', is a measure of the convective heat transfer rate per unit area and temperature difference between a surface and the surrounding fluid. It depends on factors such as the properties of the fluid, the velocity of the fluid flow, and the characteristic dimensions of the surface in question.

To measure 'h' in practical scenarios, we typically rely on empirical correlations which can involve dimensionless numbers such as the Nusselt number, the Reynolds number, and the Prandtl number. In simpler terms, think of 'h' as a number that tells you how good or bad the environment (in our case, air) is at removing heat from the pipe's surface. A high 'h' value means the environment is very effective at removing heat, leading to greater heat loss, which is why it is so crucial in our pipe鈥檚 heat loss calculation.
Nusselt Number
Moving on to the Nusselt number, this is another dimensionless value used extensively in heat transfer calculations, symbolized as 'Nu'. The Nusselt number relates the convective heat transfer to conductive heat transfer within a fluid. It's essentially telling us the enhancement of heat transfer through a fluid as a result of fluid motion versus when the fluid is still.

The value of 'Nu' is crucial as it is used directly to calculate the convective heat transfer coefficient 'h'. To determine 'Nu', a complex interplay of fluid dynamics and thermal properties is considered, including the motion of the fluid (via the Reynolds number) and its ability to transfer heat (via the Prandtl number). Understanding 'Nu' and how to calculate it, as seen in the solution where it's used to find 'h', enables students to make sense of how efficiently heat is being transferred in situations of fluid flow over solid surfaces.
Thermal Conductivity
The final piece of the puzzle is thermal conductivity, represented by the symbol 'k'. This property is a measure of a material鈥檚 ability to conduct heat. In basic terms, it tells us how well heat will pass through a material. Different materials have different 'k' values 鈥� metals, for instance, usually have high thermal conductivity, which means they conduct heat well.

In our exercise, there are two distinct 'k' values to consider: the thermal conductivity of air (ka), affecting convective heat loss, and the thermal conductivity of the insulation material (ki), affecting conductive heat loss. For insulation, we want a material with a low 'k' value to minimize heat conduction and thereby reduce the heat loss from our steam pipe, which is apparent when plotting the heat loss against the insulation thickness in the solution steps.

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Most popular questions from this chapter

Motile bacteria are equipped with flagella that are rotated by tiny, biological electrochemical engines which, in turn, propel the bacteria through a host liquid. Consider a nominally spherical Escherichia coli bacterium that is of diameter \(D=2 \mu \mathrm{m}\). The bacterium is in a water-based solution at \(37^{\circ} \mathrm{C}\) containing a nutrient which is characterized by a binary diffusion coefficient of \(D_{\mathrm{AB}}=0.7 \times 10^{-9} \mathrm{~m}^{2} / \mathrm{s}\) and a food energy value of \(\mathcal{N}=16,000 \mathrm{~kJ} / \mathrm{kg}\). There is a nutrient density difference between the fluid and the shell of the bacterium of \(\Delta \rho_{\mathrm{A}}=860 \times 10^{-12} \mathrm{~kg} / \mathrm{m}^{3}\). Assuming a propulsion efficiency of \(\eta=0.5\), determine the maximum speed of the E. coli. Report your answer in body diameters per second.

Consider atmospheric air at \(u_{\infty}=2 \mathrm{~m} / \mathrm{s}\) and \(T_{\infty}=300 \mathrm{~K}\) in parallel flow over an isothermal flat plate of length \(L=1 \mathrm{~m}\) and temperature \(T_{s}=350 \mathrm{~K}\). (a) Compute the local convection coefficient at the leading and trailing edges of the heated plate with and without an unheated starting length of \(\xi=1 \mathrm{~m}\). (b) Compute the average convection coefficient for the plate for the same conditions as part (a). (c) Plot the variation of the local convection coefficient over the plate with and without an unheated starting length.

In the production of sheet metals or plastics, it is customary to cool the material before it leaves the production process for storage or shipment to the customer. Typically, the process is continuous, with a sheet of thickness \(\delta\) and width \(W\) cooled as it transits the distance \(L\) between two rollers at a velocity \(V\). In this problem, we consider cooling of an aluminum alloy (2024-T6) by an airstream moving at a velocity \(u_{\infty}\) in counter flow over the top surface of the sheet. A turbulence promoter is used to provide turbulent boundary layer development over the entire surface. (a) By applying conservation of energy to a differential control surface of length \(d x\), which either moves with the sheet or is stationary and through which the sheet passes, derive a differential equation that governs the temperature distribution along the sheet. Because of the low emissivity of the aluminum, radiation effects may be neglected. Express your result in terms of the velocity, thickness, and properties of the sheet \(\left(V, \delta, \rho, c_{p}\right)\), the local convection coefficient \(h_{x}\) associated with the counter flow, and the air temperature. For a known temperature of the sheet \(\left(T_{i}\right)\) at the onset of cooling and a negligible effect of the sheet velocity on boundary layer development, solve the equation to obtain an expression for the outlet temperature \(T_{a}\). (b) For \(\delta=2 \mathrm{~mm}, V=0.10 \mathrm{~m} / \mathrm{s}, L=5 \mathrm{~m}, W=1 \mathrm{~m}\), \(u_{\infty}=20 \mathrm{~m} / \mathrm{s}, T_{\infty}=20^{\circ} \mathrm{C}\), and \(T_{i}=300^{\circ} \mathrm{C}\), what is the outlet temperature \(T_{a}\) ?

In an extrusion process, copper wire emerges from the extruder at a velocity \(V_{e}\) and is cooled by convection heat transfer to air in cross flow over the wire, as well as by radiation to the surroundings. (a) By applying conservation of energy to a differential control surface of length \(d x\), which either moves with the wire or is stationary and through which the wire passes, derive a differential equation that governs the temperature distribution, \(T(x)\), along the wire. In your derivation, the effect of axial conduction along the wire may be neglected. Express your result in terms of the velocity, diameter, and properties of the wire \(\left(V_{e}, D, \rho, c_{p}, \varepsilon\right)\), the convection coefficient associated with the cross flow \((\bar{h})\), and the environmental temperatures \(\left(T_{w}, T_{\text {sur }}\right)\). (b) Neglecting radiation, obtain a closed form solution to the foregoing equation. For \(V_{c}=0.2 \mathrm{~m} / \mathrm{s}\), \(D=5 \mathrm{~mm}, V=5 \mathrm{~m} / \mathrm{s}, T_{\infty}=25^{\circ} \mathrm{C}\), and an initial wire temperature of \(T_{i}=600^{\circ} \mathrm{C}\), compute the temperature \(T_{o}\) of the wire at \(x=L=5 \mathrm{~m}\). The density and specific heat of the copper are \(\rho=8900 \mathrm{~kg} / \mathrm{m}^{3}\) and \(c_{p}=400 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\), while properties of the air may be taken to be \(k=0.037 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \nu=3 \times\) \(10^{-5} \mathrm{~m}^{2} / \mathrm{s}\), and \(P r=0.69\). (c) Accounting for the effects of radiation, with \(\varepsilon=0.55\) and \(T_{\text {sur }}=25^{\circ} \mathrm{C}\), numerically integrate the differential equation derived in part (a) to determine the temperature of the wire at \(L=5 \mathrm{~m}\). Explore the effects of \(V_{e}\) and \(\varepsilon\) on the temperature distribution along the wire.

A computer code is being developed to analyze a temperature sensor of \(12.5-\mathrm{mm}\) diameter experiencing cross flow of water with a free stream temperature of \(80^{\circ} \mathrm{C}\) and variable velocity. Derive an expression for the convection heat transfer coefficient as a function of the sensor surface temperature \(T_{s}\) for the range \(20
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