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Chapter 12: Problem 113

It is not uncommon for the night sky temperature in desert regions to drop to \(-40^{\circ} \mathrm{C}\). If the ambient air temperature is \(20^{\circ} \mathrm{C}\) and the convection coefficient for still air conditions is approximately \(5 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). can a shallow pan of water freeze?

Short Answer

Expert verified
Yes, given the conditions, the water can potentially freeze.

Step by step solution

01

Understand the Concept

To determine if the water can freeze, we need to calculate the rate of heat transfer from the water to the night sky by convection. If the heat transfer is sufficient to lower the water temperature to 0掳C or below, the water can freeze.
02

Identify Given Data

The problem gives us the following data: ambient air temperature, \( T_{air} = 20^{\circ} C \); night sky temperature, \( T_{sky} = -40^{\circ} C \); and convection coefficient, \( h = 5 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K} \).
03

Use the Heat Transfer Formula

We use the convective heat transfer equation for the rate of heat transfer per unit area, \( q = h (T_{air} - T_{water}) \). Since we're checking for freezing, we'll use \( T_{water} = 0^{\circ} C \) for our calculation.
04

Calculate the Heat Transfer Rate

Plug in the known values into the formula: \( q = 5 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K} \times (20^{\circ} C - 0^{\circ} C) \). Therefore, \( q = 5 \times 20 = 100 \mathrm{~W} / \mathrm{m}^{2} \).
05

Assess Freezing Condition

The calculated heat transfer rate of 100 W/m虏 is the maximum rate at which heat can be removed from the water. If this rate is sufficient to lower the water temperature to 0掳C and continue to extract heat, the water can freeze.

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Key Concepts

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

Convective Heat Transfer
Convective heat transfer is a process by which heat is transported through the movement of fluids, such as liquids or gases. This process can happen naturally or be forced. In the context of the original problem, the movement of air molecules around the shallow pan of water is an example of natural convection.
Natural convection occurs due to the temperature difference between the water and the surrounding air. This temperature difference causes air molecules to move and transfer heat energy away from the water, effectively cooling it.
Understanding the principles of convective heat transfer is crucial for various applications, including weather predictions, HVAC systems, and even understanding the heat loss properties of buildings and vehicles.
  • Heat is transferred by the mass motion of fluids (such as air).
  • It is influenced by the temperature difference between the object and the surrounding medium.
  • Convective cooling can lower the temperature of an object, such as water in a pan.
Convection Coefficient
The convection coefficient, often denoted as \( h \), is a measure that quantifies the ability of a fluid to conduct heat away from a surface it flows over. In simple terms, it represents the effectiveness of convective heat transfer under specific conditions.
A higher convection coefficient means more efficient heat transfer from the object's surface to the fluid, while a lower coefficient suggests slower heat transfer. In the exercise, the specified convection coefficient is \( 5 \, \mathrm{W/m^2 \cdot K} \). This value is typical for still air, indicating that the air is not moving much and therefore transfers heat at a modest rate.
To apply the convection coefficient in practical scenarios, the equation used is: \[ q = h (T_{surface} - T_{fluid}) \] This equation helps estimate the rate at which heat is leaving or being absorbed by the surface, crucial in determining whether the pan of water will reach a freezing point.
  • Specifies the effectiveness of heat transfer between a surface and a fluid.
  • Depends on factors like fluid type, flow characteristics, and temperature.
  • Used to calculate the rate of heat transfer in engineering applications.
Freezing Point
The freezing point is the temperature at which a liquid turns into a solid. For water, this happens at \( 0^{\circ} C \), known as the freezing point of water. To determine if the water in the pan will freeze, it's necessary for the pan to reach this specific temperature.
In the problem scenario, for water to freeze, the rate at which heat is removed from the water by the surrounding air and night sky must be enough to reduce its temperature to \( 0^{\circ} C \). The calculated heat transfer rate tells us if the cold temperatures and convective processes are capable of meeting or exceeding the heat loss required for freezing.
  • The freezing point is constant for pure substances but can change with impurities or pressure changes.
  • Monitoring the ambient and surface temperature differences helps in predicting freezing.
  • A balance between environmental conditions and heat transfer rates determines the freezing outcome.

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Most popular questions from this chapter

A spherical satellite of diameter \(D\) is in orbit about the carth and is coated with a diffuse material for which the spectral absorptivity is \(\alpha_{\lambda}=0.6\) for \(\lambda \leq 3 \mu \mathrm{m}\) and \(\alpha_{2}=0.3\) for \(\lambda>3 \mu \mathrm{m}\). When it is on the "dark" side of the earth, the satellite sees irradiation from the carth's surface only. The irradiation may be assumed to be incident as parallel rays, and its magnitude is \(G_{E}=340 \mathrm{~W} / \mathrm{m}^{2}\). On the "bright" side of the earth the satellite sees the earth irradiation \(G_{E}\) plus the sola irradiation \(G_{5}=1353 \mathrm{~W} / \mathrm{m}^{2}\). The spectrul distribetion of radiation from the earth may be approximuted as that of a blackbody at \(280 \mathrm{~K}\), and the temperature of the satellite may be ussumed to remain below \(500 \mathrm{~K}\). What is the steady-state temperature of the satellite when it is on the dark side of the earth and when it is on the bright side?

One scheme for extending the operation of gas turbine blades to higher temperatures involves applying a ceramic coating to the surfaces of blades fabricated from a superalloy such as inconel. To assess the reliability of such coatings, an apparatus has been developed for testing samples under laboratory conditions. The sample is placed at the bottom of a large vacuum chamber whose walls are cryogenically cooled and which is equipped with a radiation detector at the top surface. The detector has a surface area of \(A_{d}=\) \(10^{-5} \mathrm{~m}^{2}\), is located at a distance of \(L_{2-d}=1 \mathrm{~m}\) from the sample, and views radiation originating from a portion of the ceramic surface having an area of \(\Delta A_{c}=10^{-4}\) \(m^{2}\). An electric heater attached to the bottom of the sample dissipates a uniform heat flux, \(q_{h}^{n}\), which is transferred upward through the sample. The botiom of the heater and sides of the sample are well insulated. Consider conditions for which a ceramic coating of thickness \(L_{\gamma}=0.5 \mathrm{~mm}\) and thermal conductivity \(k_{\mathrm{c}}=6 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) has been sprayed on a metal substrute of thickness \(L,=8 \mathrm{~mm}\) and thermal conductivity \(k_{3}=25 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\). The opaque surface of the ceramic may be approximated as diffuse and gray. with a total, hemispherical emissivity of \(\varepsilon_{c}=0.8\). (a) Consider steady-state conditions for which the bottom surface of the substrate is maintained at \(T_{1}=1500 \mathrm{~K}\), while the chamber walls (including the surface of the radistion detector) are maintained at \(T_{\mathrm{w}}=90 \mathrm{~K}\). Assuming negligible thermal contact resistance at the ceramic-substrate interface, determine the ceramic top surface temperature \(T_{2}\) and the hest flux \(q_{\hbar-}^{\prime \prime}\) (b) For the prescribed conditions, what is the rate at which radiation emitted by the ceramic is intercepted by the detector? (c) After repeated experiments, numerous cracks develop at the ceramic- substrate interface, creating an interfacial thermal contact resistance. If \(T_{w}\) and \(q_{h}^{\prime \prime}\) are maintained at the conditions associated with part (a), will \(T_{1}\) increase, decrease, or remain the same? Similarly, will \(T_{2}\) increase, decrease, or remain the same? In cach case, justify your answer.

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