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Chapter 9: Problem 101

A solar collector design consists of an inner tube enclosed concentrically in an outer tube that is transparent to solar radiation. The tubes are thin walled with inner and outer diameters of \(0.10\) and \(0.15 \mathrm{~m}\), respectively. The annular space between the tubes is completely enclosed and filled with air at atmospheric pressure. Under operating conditions for which the inner and outer tube surface temperatures are 70 and \(30^{\circ} \mathrm{C}\), respectively, what is the convective heat loss per meter of tube length across the air space?

Short Answer

Expert verified
The convective heat loss per meter of tube length across the air space between the two concentric tubes can be calculated as follows: 1. First, determine the Grashof number, Prandtl number, and Nusselt number for the given conditions. 2. Using the obtained Nusselt number, find the convective heat transfer coefficient. 3. Finally, calculate the convective heat loss per meter of tube length using the convective heat transfer coefficient, area of the tube, and the temperature difference between the inner and outer tubes. The resulting convective heat loss per meter of tube length can be determined using the formula: \(Q = h \cdot 2 \cdot \pi \cdot (0.15/2 + 0.10/2)/2 \cdot 1 \cdot (343.15 - 303.15)\)

Step by step solution

01

Calculate the Grashof number

(Calculate the Grashof number using the temperature difference and the dimensions of the air gap.) First, we need to determine the Grashof number, which is a dimensionless number that characterizes natural convection. The Grashof number can be calculated using the following formula: \(Gr = \frac{g \cdot \beta \cdot (T_i - T_o) \cdot D^3}{\nu^2}\) Where: - \(g\) is the gravitational acceleration (\(9.81 \mathrm{m/s^2}\)) - \(\beta\) is the volumetric thermal expansion coefficient, which is approximately \(1/T_{avg}\) for an ideal gas - \(T_i\) is the inner tube temperature (70掳C) - \(T_o\) is the outer tube temperature (30掳C) - \(D\) is the characteristic length, which is the difference between the outer and inner tube radii - \(\nu\) is the kinematic viscosity of air at the average temperature For an ideal gas, \(\beta = \frac{1}{T_{avg}}\), where \(T_{avg}\) is the average temperature in Kelvin. To calculate \(T_{avg}\), first convert the temperatures from Celsius to Kelvin: \(T_i = 70 + 273.15 = 343.15 \mathrm{K}\) \(T_o = 30 + 273.15 = 303.15 \mathrm{K}\) Then calculate the average temperature: \(T_{avg} = \frac{T_i + T_o}{2} = \frac{343.15 + 303.15}{2} = 323.15 \mathrm{K}\) Now, we need to find the kinematic viscosity (\(\nu\)) for air at this temperature. Consulting a table of air properties, we find that \(\nu = 1.6 \times 10^{-5} \mathrm{m^2/s}\) at 323.15 K. We can now calculate the Grashof number: \(Gr = \frac{9.81 \cdot \frac{1}{323.15} \cdot (343.15 - 303.15) \cdot (0.15/2 - 0.10/2)^3}{(1.6 \times 10^{-5})^2}\)
02

Calculate the Prandtl number

(Determine the Prandtl number for air at the average temperature.) Next, we need to find the Prandtl number, another dimensionless number representing the ratio of momentum diffusivity to thermal diffusivity. For air, we can consult a table of air properties to find the Prandtl number at the average temperature. At 323.15 K, the Prandtl number for air is: \(Pr = 0.7\)
03

Determine the Nusselt number

(Calculate the Nusselt number using the Grashof and Prandtl numbers, and a suitable correlation for natural convection between concentric cylinders.) Using the Grashof and Prandtl numbers, we can now find the Nusselt number. For natural convection between concentric cylinders, we can use the following correlation: \(Nu = C\cdot (Gr \cdot Pr)^{n}\) The constants C and n depend on the geometry and flow regime. For concentric cylinders and laminar flow, the constants can be approximated as \(C = 0.53\) and \(n = 0.25\). We can now calculate the Nusselt number: \(Nu = 0.53 \cdot (Gr \cdot Pr)^{0.25}\)
04

Determine the convective heat transfer coefficient

(Calculate the convective heat transfer coefficient using the Nusselt number and the thermal conductivity of air at the average temperature) Now that we have the Nusselt number, we can find the convective heat transfer coefficient using the formula: \(h = \frac{Nu \cdot k}{D}\) Where \(k\) is the thermal conductivity of air, which can be found in a table of air properties at 323.15 K. At this temperature, the thermal conductivity of air is \(k = 0.0262 \mathrm{W/(m\cdot K)}\). We can now calculate the convective heat transfer coefficient: \(h = \frac{Nu \cdot 0.0262}{(0.15/2 - 0.10/2)}\)
05

Calculate the convective heat loss per meter of tube length

(Determine the convective heat loss per meter of tube length using the convective heat transfer coefficient, the area of the tube, and the temperature difference between the inner and outer tubes.) Finally, we can determine the convective heat loss per meter of tube length using the convective heat transfer coefficient, the area of the tube, and the temperature difference: \(Q = h \cdot A \cdot \Delta T\) Where: - \(Q\) is the convective heat loss per meter of tube length - \(A\) is the area of the tube, which can be calculated as \(A = 2 \cdot \pi \cdot r_{avg} \cdot L\), with \(r_{avg}\) as the average radius between the inner and outer tubes, and \(L = 1 \mathrm{~m}\) as the length of the tube - \(\Delta T = T_i - T_o\) We can now calculate the convective heat loss per meter of tube length: \(Q = h \cdot 2 \cdot \pi \cdot (0.15/2 + 0.10/2)/2 \cdot 1 \cdot (343.15 - 303.15)\)

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Key Concepts

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

Grashof Number
The Grashof Number is crucial in understanding natural convection phenomena. It is a dimensionless number that helps to describe the flow regimes (laminar or turbulent) within a fluid when there is a temperature difference. Essentially, the Grashof Number relates the buoyancy forces to the viscous forces in the fluid. The formula for calculating the Grashof Number is \[ Gr = \frac{g \cdot \beta \cdot (T_i - T_o) \cdot D^3}{u^2} \]Where:
  • \( g \) is the gravitational acceleration.
  • \( \beta \) is the coefficient of volumetric thermal expansion for the fluid. For gases under consideration, it can be approximated by \( 1/T_{avg} \).
  • \( T_i \) and \( T_o \) are the temperatures of the inner and outer tubes in Kelvin, respectively.
  • \( D \) is the characteristic length.
  • \( u \) is the kinematic viscosity of the fluid.
Understanding the Grashof Number is essential in optimizing designs for systems involving natural convection, such as passive cooling devices and heat exchangers.
Prandtl Number
The Prandtl Number provides insights into the fluid flow behavior during heat transfer processes. This dimensionless number indicates the ratio of momentum diffusivity (kinematic viscosity) to thermal diffusivity. It essentially tells us how adept the fluid is at conducting thermal energy compared to its ability to transport momentum due to viscous forces. The Prandtl number in general terms is defined as:\[ Pr = \frac{u}{\alpha} \]Where:
  • \( u \) is the kinematic viscosity of the fluid.
  • \( \alpha \) is the thermal diffusivity.
For most gases like air, the Prandtl Number is usually around 0.7. A lower Prandtl Number indicates that thermal diffusion dominates over momentum diffusion, which is observed in gases. Understanding the Prandtl Number is key in analyzing heat transfer efficiencies in fluid systems.
Nusselt Number
The Nusselt Number stands as a measure of convective heat transfer relative to conductive heat transfer across a boundary. This dimensionless number correlates the heat transfer enhancement brought about by convection as opposed to conduction alone. The Nusselt Number is defined using correlations commonly dependent on the Grashof and Prandtl numbers, like:\[ Nu = C \cdot (Gr \cdot Pr)^{n} \]Here, \( C \) and \( n \) are constants determined by the specific system configuration and the flow regime, such as laminar or turbulent. In systems with concentric cylinders under natural convection, these constants help in calculating the Nusselt Number which is pivotal for computing the heat transfer rates in the system.The higher the Nusselt Number, the greater the convective heat transfer relative to conduction. This understanding facilitates enhancements in thermal system performance through optimized designs.
Natural Convection
Natural Convection is a heat transfer mechanism where fluid motion is induced by buoyancy forces, which arise due to density differences caused by temperature variations within the fluid. Unlike forced convection, where an external source drives the fluid flow, natural convection relies on natural causes. In the given solar collector problem, natural convection occurs within the annular space between the tubes filled with air. Here, the warmer air tends to rise while the cooler air descends, creating a convective flow path which aids in the heat transfer between the inner and outer tubes. It's crucial to consider natural convection for scenarios involving passive heat dissipation or where no mechanical sources are viable. Designers often leverage the principles of natural convection to augment thermal management without the need for powered components, leading to more efficient and maintenance-free systems.
Thermal Conductivity
Thermal Conductivity is a material-specific property that measures the ability of a substance to conduct heat. This property is vital for analyzing and predicting heat transfer across materials.In engineering and thermal management, knowing the thermal conductivity of a fluid, like air in the case of the solar collector design, is integral to calculating the heat transfer coefficient and ultimately understanding the system's efficiency. For air around the average temperature in the example, its thermal conductivity is approximately \( k = 0.0262 \text{ W/m} \cdot \text{K} \).The level of thermal conductivity informs how effectively heat will flow through a material when subjected to a temperature gradient, thus influencing design decisions in both natural and forced convection applications. A higher thermal conductivity means better heat transfer capacity of the material or fluid, essential for optimizing thermal systems.

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Most popular questions from this chapter

Certain wood stove designs rely exclusively on heat transfer by radiation and natural convection to the surroundings. Consider a stove that forms a cubical enclosure, \(L_{s}=1 \mathrm{~m}\) on a side, in a large room. The exterior walls of the stove have an emissivity of \(\varepsilon=0.8\) and are at an operating temperature of \(T_{s s s}=500 \mathrm{~K}\). The stove pipe, which may be assumed to be isothermal at an operating temperature of \(T_{s, p}=400 \mathrm{~K}\), has a diameter of \(D_{p}=0.25 \mathrm{~m}\) and a height of \(L_{p}=2 \mathrm{~m}\), extending from stove to ceiling. The stove is in a large room whose air and walls are at \(T_{\infty}=T_{\text {sur }}=300 \mathrm{~K}\). Neglecting heat transfer from the small horizontal section of the pipe and radiation exchange between the pipe and stove, estimate the rate at which heat is transferred from the stove and pipe to the surroundings.

Consider window blinds that are installed in the air space between the two panes of a vertical double-pane window. The window is \(H=0.5 \mathrm{~m}\) high and \(w=0.5 \mathrm{~m}\) wide, and includes \(N=19\) individual blinds that are each \(L=25 \mathrm{~mm}\) wide. When the blinds are open, 20 smaller, square enclosures are formed along the height of the window. In the closed position, the blinds form a nearly continuous sheet with two \(t=12.5 \mathrm{~mm}\) open gaps at the top and bottom of the enclosure. Determine the convection heat transfer rate between the inner pane, which is held at \(T_{s, i}=20^{\circ} \mathrm{C}\), and the outer pane, which is at \(T_{s, o}=-20^{\circ} \mathrm{C}\), when the blinds are in the open and closed positions, respectively. Explain why the closed blinds have little effect on the convection heat transfer rate across the cavity.

The human eye contains aqueous humor, which separates the external cornea and the internal iris-lens structure. It is hypothesized that, in some individuals, small flakes of pigment are intermittently liberated from the iris and migrate to, and subsequently damage, the cornea. Approximating the geometry of the enclosure formed by the cornea and iris-lens structure as a pair of concentric hemispheres of outer radius \(r_{o}=10 \mathrm{~mm}\) and inner radius \(r_{i}=7 \mathrm{~mm}\), respectively, investigate whether free convection can occur in the aqueous humor by evaluating the effective thermal conductivity ratio, \(k_{\mathrm{efI}} / k\). If free convection can occur, it is possible that the damaging particles are advected from the iris to the cornea. The iris-lens structure is at the core temperature, \(T_{i}=37^{\circ} \mathrm{C}\), while the cornea temperature is measured to be \(T_{o}=34^{\circ} \mathrm{C}\). The properties of the aqueous humor are \(\rho=990 \mathrm{~kg} / \mathrm{m}^{3}, k=0.58 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\), \(c_{p}=4.2 \times 10^{3} \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}, \mu=7.1 \times 10^{-4} \mathrm{~N} \cdot \mathrm{s} / \mathrm{m}^{2}\), and \(\beta=3.2 \times 10^{-4} \mathrm{~K}^{-1}\)

Convection heat transfer coefficients for a hot horizontal surface facing upward may be determined by a gage whose specific features depend on whether the temperature of the surroundings is known. For configuration A, a copper disk, which is electrically heated from below, is encased in an insulating material such that all of the heat is transferred by convection and radiation from the top surface. If the surface emissivity and the temperatures of the air and surroundings are known, the convection coefficient may be determined from measurement of the electrical power and the surface temperature of the disk. Configuration B is used in situations for which the temperature of the surroundings is not known. A thin, insulating strip separates semicircular disks with independent electrical heaters and different emissivities. If the emissivities and temperature of the air are known, the convection coefficient may be determined from measurement of the electrical power supplied to each of the disks in order to maintain them at a common temperature. (a) In an application of configuration A to a disk of diameter \(D=160 \mathrm{~mm}\) and emissivity \(\varepsilon=0.8\), values of \(P_{\text {elec }}=10.8 \mathrm{~W}\) and \(T=67^{\circ} \mathrm{C}\) are measured for \(T_{\infty}=T_{\text {sur }}=27^{\circ} \mathrm{C}\). What is the corresponding value of the average convection coefficient? How does it compare with predictions based on a standard correlation? (b) Now consider an application of configuration \(B\) for which \(T_{\infty}=17^{\circ} \mathrm{C}\) and \(T_{\text {sur }}\) is unknown. With \(D=160 \mathrm{~mm}, \varepsilon_{1}=0.8\), and \(\varepsilon_{2}=0.1\), values of \(P_{\text {elect, } 1}=9.70 \mathrm{~W}\) and \(P_{\text {elec, } 2}=5.67 \mathrm{~W}\) are measured when \(T_{1}=T_{2}=77^{\circ} \mathrm{C}\). Determine the corresponding values of the convection coefficient and the temperature of the surroundings. How does the convection coefficient compare with predictions by an appropriate correlation?

Saturated steam at 4 bars absolute pressure with a mean velocity of \(3 \mathrm{~m} / \mathrm{s}\) flows through a horizontal pipe whose inner and outer diameters are 55 and \(65 \mathrm{~mm}\), respectively. The heat transfer coefficient for the steam flow is known to be \(11,000 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). (a) If the pipe is covered with a \(25-\mathrm{mm}\)-thick layer of \(85 \%\) magnesia insulation and is exposed to atmospheric air at \(25^{\circ} \mathrm{C}\), determine the rate of heat transfer by free convection to the room per unit length of the pipe. If the steam is saturated at the inlet of the pipe, estimate its quality at the outlet of a pipe \(30 \mathrm{~m}\) long. (b) Net radiation to the surroundings also contributes to heat loss from the pipe. If the insulation has a surface emissivity of \(\varepsilon=0.8\) and the surroundings are at \(T_{\text {sur }}=T_{\infty}=25^{\circ} \mathrm{C}\), what is the rate of heat transfer to the room per unit length of pipe? What is the quality of the outlet flow? (c) The heat loss may be reduced by increasing the insulation thickness and/or reducing its emissivity. What is the effect of increasing the insulation thickness to \(50 \mathrm{~mm}\) if \(\varepsilon=0.8\) ? Of decreasing the emissivity to \(0.2\) if the insulation thickness is \(25 \mathrm{~mm}\) ? Of reducing the emissivity to \(0.2\) and increasing the insulation thickness to \(50 \mathrm{~mm}\) ?
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