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Chapter 9: Problem 86

As is evident from the property data of Tables A.3 and A.4, the thermal conductivity of glass at room temperature is more than 50 times larger than that of air. It is therefore desirable to use windows of double-pane construction, for which the two panes of glass enclose an air space. If heat transfer across the air space is by conduction, the corresponding thermal resistance may be increased by increasing the thickness \(L\) of the space. However, there are limits to the efficacy of such a measure, since convection currents are induced if \(L\) exceeds a critical value, beyond which the thermal resistance decreases. Consider atmospheric air enclosed by vertical panes at temperatures of \(T_{1}=22^{\circ} \mathrm{C}\) and \(T_{2}=-20^{\circ} \mathrm{C}\). If the critical Rayleigh number for the onset of convection is \(R a_{L} \approx 2000\), what is the maximum allowable spacing for conduction across the air? How is this spacing affected by the temperatures of the panes? How is it affected by the pressure of the air, as, for example, by partial evacuation of the space?

Short Answer

Expert verified
The maximum allowable spacing for conduction across the air space enclosed by vertical panes of double-pane windows is approximately 0.025 meters or 2.5 centimeters. This spacing is influenced by the temperatures of the panes, as a larger temperature difference will affect the Rayleigh number formula. Additionally, the pressure of the air in the space affects the properties, such as kinematic viscosity and thermal diffusivity, which are used in the Rayleigh number formula. If the pressure is reduced, these properties will change, thus affecting the maximum allowable spacing.

Step by step solution

01

Define the critical Rayleigh number and Thermal expansivity formula

The Rayleigh number (Ra) can be defined as (assuming the air behaves like an ideal gas): \[Ra_L = \frac{g \beta \Delta T L^3}{\nu \alpha}\] Where: - \(g\) is the acceleration due to gravity, \(9.81 m/s^2\) - \(\beta\) is the thermal expansivity of air, given by \(\beta = 1/T\) - \(\Delta T\) is the temperature difference between the panes - \(L\) is the thickness of the air space - \(\nu\) is the kinematic viscosity - \(\alpha\) is the thermal diffusivity We need to find the maximum allowable spacing (\(L\)).
02

Calculate the temperature difference

Given the temperatures of the panes, \(T_1 = 22^{\circ} \mathrm{C}\) and \(T_2 = -20^{\circ} \mathrm{C}\), we can find the temperature difference \(\Delta T\): \[\Delta T = T_1 - T_2 = 22 - (-20) = 42^{\circ} C\]
03

Convert temperatures to Kelvin

To work with thermal expansivity (\(\beta\)), we need to convert the temperature difference to Kelvin: \[\Delta T = 42^{\circ} C + 273.15 K = 315.15 K\]
04

Calculate the average temperature and thermal expansivity

The average temperature of the air can be found as: \[T_{avg} = \frac{T_1 + T_2}{2} = \frac{22 + (-20)}{2} = 1^{\circ} C = 274.15 K\] Now we can calculate the thermal expansivity (\(\beta\)) of the air: \[\beta = \frac{1}{T_{avg}} = \frac{1}{274.15 K} = 3.65 \times 10^{-3} K^{-1}\]
05

Identify given Rayleigh number and rearrange the formula for L

We are given that the critical Rayleigh number for the onset of convection is \(Ra_L \approx 2000\). Now, we will rearrange the Rayleigh number formula to solve for the maximum allowable spacing (\(L\)): \[L = \sqrt[3]{\frac{Ra_L \nu \alpha}{g \beta \Delta T}}\]
06

Find the values of kinematic viscosity and thermal diffusivity

Using Tables A.3 and A.4 (or other reference sources), we can find the values of kinematic viscosity (\(\nu\)) and thermal diffusivity (\(\alpha\)) for air at the calculated average temperature: - For air at \(1^{\circ} C\), \(\nu \approx 1.56 \times 10^{-5} m^2/s\) - For air at \(1^{\circ} C\), \(\alpha \approx 2.19 \times 10^{-5} m^2/s\)
07

Calculate the maximum allowable spacing

Now we can substitute the values into the rearranged Rayleigh number formula and calculate the maximum allowable spacing (\(L\)): \[L = \sqrt[3]{\frac{2000 \times 1.56 \times 10^{-5} \times 2.19 \times 10^{-5}}{9.81 \times 3.65 \times 10^{-3} \times 315.15}} \approx 0.025 m\] Thus, the maximum allowable spacing for conduction across the air space is approximately 0.025 meters or 2.5 centimeters. To answer the additional questions: 1. The spacing is affected by the temperatures of the panes since a larger temperature difference will increase \(\Delta T\) in the Rayleigh number formula, which influences the maximum allowable spacing. 2. The pressure of the air in the space will affect the properties, such as kinematic viscosity and thermal diffusivity, that are used in the Rayleigh number formula. If the pressure is reduced, these properties will change, thus affecting the maximum allowable spacing.

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Key Concepts

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

Thermal Conductivity
Thermal conductivity is a material's ability to conduct heat. Imagine it as how easily heat spreads through a substance.
For instance, metals usually have high thermal conductivity, meaning they warm up quickly.
Conversely, materials like air and wool have low thermal conductivity, making them good insulators.
In the context of double-pane windows:
  • Double-pane windows use the air between the panes as an insulating layer.
  • A high thermal conductivity means more heat escapes through the glass.
  • This outcome is not desirable if the goal is to keep heat in or out.
  • Thus, using a material with low thermal conductivity is crucial.
Understanding thermal conductivity can help in selecting materials for insulation in construction, especially for energy-efficient homes.
Critical Rayleigh Number
The critical Rayleigh number represents a threshold where convection starts.
To visualize, think about gently warming a pot of water. As it warms, heat initially transfers through simple conduction, but after a point, the water starts moving around鈥攕purring convection currents.
In the context of this problem, we use:
  • The critical Rayleigh number to determine when air between window panes starts moving due to temperature differences.
  • If the Rayleigh number exceeds the critical value (\(Ra_L = 2000\)), convection currents form, reducing the insulation effect.
  • The Rayleigh number also changes when pane temperatures or air pressure vary, altering how much space can be left between panes before convection occurs.
This helps in designing windows that maximize energy efficiency by managing how heat is transferred.
Thermal Resistance
Thermal resistance is how well a material can resist heat flow. Think of it as the opposite of thermal conductivity.
In simpler terms:
  • The higher the thermal resistance, the better it blocks heat passing through.
  • In window design, increasing the thermal resistance means the gaps between panes help keep indoor temperatures stable.
  • Adjusting the air space thickness can raise or lower a window's thermal resistance.
However, if the gap becomes too wide, air convection might begin, nullifying the benefit of increased resistance.
Proper balance is essential to enhance thermal resistance effectively, especially in varied climate conditions.
Thermal Expansivity
Thermal expansivity measures how much a material expands when heated. This property becomes crucial when dealing with gases like air between window panes.
The formula used:
\[\beta = \frac{1}{T_{avg}}\]
Here, \(\beta\) reflects changes in density with temperature.
  • For air, thermal expansivity affects how much it will expand as it warms up due to temperature differences between panes.
  • It is pivotal to assess how the air behaves inside the gap, impacting the onset of convection.
  • By understanding thermal expansivity, calculations ensure that panes are spaced appropriately, maintaining an effective insulating barrier.
Preventing unnecessary expansion or contraction helps in conserving energy by keeping the indoor climate stable and reducing heating or cooling demands.

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Most popular questions from this chapter

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?

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 \(\beta=3.2 \times 10^{-4} \mathrm{~K}^{-1}\)

A rectangular cavity consists of two parallel, \(0.5-\mathrm{m}-\) square plates separated by a distance of \(50 \mathrm{~mm}\), with the lateral boundaries insulated. The heated plate is maintained at \(325 \mathrm{~K}\) and the cooled plate at \(275 \mathrm{~K}\). Estimate the heat flux between the surfaces for three orientations of the cavity using the notation of Figure 9.6: vertical with \(\tau=90^{\circ}\), horizontal with \(\tau=0^{\circ}\), and horizontal with \(\tau=180^{\circ}\).

In the central receiver concept of a solar power plant, many heliostats at ground level are used to direct a concentrated solar flux \(q_{s}^{\prime \prime}\) to the receiver, which is positioned at the top of a tower. However, even with absorption of all the solar flux by the outer surface of the receiver, losses due to free convection and radiation reduce the collection efficiency below the maximum possible value of \(100 \%\). Consider a cylindrical receiver of diameter \(D=7 \mathrm{~m}\), length \(L=12 \mathrm{~m}\), and emissivity \(\varepsilon=0.20\). (a) If all of the solar flux is absorbed by the receiver and a surface temperature of \(T_{s}=800 \mathrm{~K}\) is maintained, what is the rate of heat loss from the receiver? The ambient air is quiescent at a temperature of \(T_{\infty}=300 \mathrm{~K}\), and irradiation from the surroundings may be neglected. If the corresponding value of the solar flux is \(q_{S}^{\prime \prime}=10^{5} \mathrm{~W} / \mathrm{m}^{2}\), what is the collector efficiency? (b) The surface temperature of the receiver is affected by design and operating conditions within the power plant. Over the range from 600 to \(1000 \mathrm{~K}\), plot the variation of the convection, radiation, and total heat rates as a function of \(T_{s}\). For a fixed value of \(q_{S}^{\prime \prime}=10^{5} \mathrm{~W} / \mathrm{m}^{2}\), plot the corresponding variation of the receiver efficiency.

A biological fluid moves at a flow rate of \(\dot{m}=0.02 \mathrm{~kg} / \mathrm{s}\) through a coiled, thin-walled, 5 -mm-diameter tube submerged in a large water bath maintained at \(50^{\circ} \mathrm{C}\). The fluid enters the tube at \(25^{\circ} \mathrm{C}\). (a) Estimate the length of the tube and the number of coil turns required to provide an exit temperature of \(T_{m, o}=38^{\circ} \mathrm{C}\) for the biological fluid. Assume that the water bath is an extensive, quiescent medium, that the coiled tube approximates a horizontal tube, and that the biological fluid has the thermophysical properties of water. (b) The flow rate through the tube is controlled by a pump that experiences throughput variations of approximately \(\pm 10 \%\) at any one setting. This condition is of concern to the project engineer because the corresponding variation of the exit temperature of the biological fluid could influence the downstream process. What variation would you expect in \(T_{m, o}\) for a \(\pm 10 \%\) change in \(\dot{m}\) ?
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