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Chapter 1: Problem 24

Under conditions for which the same room temperature is maintained by a heating or cooling system, it is not uncommon for a person to feel chilled in the winter but comfortable in the summer. Provide a plausible explanation for this situation (with supporting calculations) by considering a room whose air temperature is maintained at \(20^{\circ} \mathrm{C}\) throughout the year, while the walls of the room are nominally at \(27^{\circ} \mathrm{C}\) and \(14^{\circ} \mathrm{C}\) in the summer and winter, respectively. The exposed surface of a person in the room may be assumed to be at a temperature of \(32^{\circ} \mathrm{C}\) throughout the year and to have an emissivity of \(0.90\). The coefficient associated with heat transfer by natural convection between the person and the room air is approximately \(2 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\).

Short Answer

Expert verified
More heat loss occurs in winter due to lower wall temperature, making the person feel colder.

Step by step solution

01

Understand the Problem

The room temperature is constant at \(20^{\circ} \text{C}\) both in summer and winter, but the wall temperatures differ, being \(27^{\circ} \text{C}\) in summer and \(14^{\circ} \text{C}\) in winter. Our task is to explain why a person feels colder in winter and provide supporting calculations.
02

Determine Radiation Heat Transfer

The rate of heat loss from the body due to radiation can be calculated using the Stefan-Boltzmann law. The heat transfer due to radiation is given by:\[ q_{\text{rad}} = \varepsilon \sigma A (T_s^4 - T_w^4) \]where \(\varepsilon\) is the emissivity (0.90), \(\sigma\) is the Stefan-Boltzmann constant \(5.67 \times 10^{-8} \text{W/m}^2\cdot\text{K}^4\), \(A\) is the surface area of the person, \(T_s\) is the surface temperature of the person (32掳C or 305 K), and \(T_w\) is the wall temperature (300 K in summer and 287 K in winter).
03

Calculate Radiation Heat Loss for Summer

Substitute the summer wall temperature into the equation:\[ q_{\text{rad, summer}} = 0.90 \times 5.67 \times 10^{-8} \times A (305^4 - 300^4) \]Calculate the value to find the rate of heat loss by radiation in summer.
04

Calculate Radiation Heat Loss for Winter

Substitute the winter wall temperature into the equation:\[ q_{\text{rad, winter}} = 0.90 \times 5.67 \times 10^{-8} \times A (305^4 - 287^4) \]Calculate the value to find the rate of heat loss by radiation in winter.
05

Determine Convection Heat Transfer

The convection heat transfer coefficient is given as \(2 \text{ W/m}^2\cdot\text{K}\). The heat transfer due to convection can be determined by:\[ q_{\text{conv}} = h_c A (T_p - T_a) \]where \(h_c\) is the convection coefficient, \(T_p = 32^{\circ}C\), and air temperature \(T_a = 20^{\circ}C\) which is constant throughout the year.
06

Calculate Convection Heat Loss

Compute the heat loss due to convection:\[ q_{\text{conv}} = 2 \times A (32 - 20) \]Both in summer and winter since air temperature is constant.
07

Summarize and Compare

Summarize total heat loss as the sum of radiation and convection for both seasons:\[ q_{\text{total, summer}} = q_{\text{rad, summer}} + q_{\text{conv}} \]\[ q_{\text{total, winter}} = q_{\text{rad, winter}} + q_{\text{conv}} \]Compare the total heat loss for summer and winter. Greater total heat loss in winter explains why a person feels colder.

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Key Concepts

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

Radiation Heat Transfer
Radiation heat transfer is a mode where heat is transferred in the form of electromagnetic waves, primarily in the infrared spectrum. This occurs without the need for a medium, meaning heat can be transferred through a vacuum, and is largely influenced by the temperatures and emissivities of the surfaces involved. In our situation, when a person is indoors, their body continuously exchanges thermal radiation with the surrounding walls.
The rate of radiation heat transfer between the person's body and the room walls can be determined using the Stefan-Boltzmann law, which is expressed as:
  • Formula: \( q_{\text{rad}} = \varepsilon \sigma A (T_s^4 - T_w^4) \)
  • \( \varepsilon \): Emissivity of the human skin (0.90 in this case)
  • \( \sigma \): Stefan-Boltzmann constant, \( 5.67 \times 10^{-8} \text{ W/m}^2\cdot\text{K}^4 \)
  • \( A \): Surface area of the person
  • \( T_s \): Surface temperature of the person (305 K)
  • \( T_w \): Temperature of the walls, varying between 300 K in summer and 287 K in winter
In winter, since the wall temperature is lower, the rate of heat loss from the body due to radiation is higher compared to summer, leading to a sensation of being chilled.
Convection Heat Transfer
Convection is another mode of heat transfer that depends on the movement of fluid or air. In the scenario where a person is in a room, natural convection occurs as the air around the person heats up, becomes less dense, and moves away, allowing cooler air to take its place.
The heat transfer due to convection can be calculated using the formula:
  • Formula: \( q_{\text{conv}} = h_c A (T_p - T_a) \)
  • \( h_c \): Convective heat transfer coefficient (given as 2 W/m虏路K)
  • \( A \): Surface area of the person
  • \( T_p \): Temperature of the person (305 K or 32掳C)
  • \( T_a \): Ambient air temperature (293 K or 20掳C)
As the air temperature remains constant at 20掳C throughout the year, the convection heat loss does not change between summer and winter. However, this mode contributes to the total heat loss felt by the person along with radiation.
Thermal Comfort
Thermal comfort is a subjective experience and is primarily about how warm or cold a person feels in a given environment. It's affected by a delicate balance between heat generation in the body and heat loss to the environment.
Factors influencing thermal comfort include:
  • Temperature differences between the body and surrounding surfaces
  • Humidity and air velocity
  • Clothing insulation
In the winter, even if the air temperature is the same as in summer, lower wall temperatures result in greater radiation heat loss from the body, disrupting thermal comfort. Meanwhile, in summer, the wall temperature is higher, reducing the radiation heat loss and making the environment feel warmer and more comfortable. Hence, understanding both radiation and convection heat transfers is key in explaining the seasonal differences in thermal comfort.

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Most popular questions from this chapter

In the thermal processing of semiconductor materials. annealing is accomplished by heating a silicon wafer according to a temiperature-titue recipe and then maintaining a fixed elevated temperature for a prescribed period of time. For the process tool arrangement shown as follow5, the wafer is in an evacuated chamber whose walls are maintained at \(27 \mathrm{C}\) and wiahin which heating lamps maintain a radiant flux \(q\), at its upper surface. The wafer is \(0.78 \mathrm{~mm}\) thick, has a thermul conductivity of 30 W/m \(\cdot \mathrm{K}\), and an emiskivity that cquals its absorptivity to the radiant flux \(\left(\varepsilon=\alpha_{i}=0.65\right)\). For \(q_{t}^{n}=3.0 \times 10^{5}\) \(\mathrm{W} / \mathrm{m}^{2}\), the temperalure on its lower surface is measured by a ridiation themometer and found to have a value of \(T_{\alpha l}=997^{\circ} \mathrm{C}\). To avoid warping the wafer and inducing slip planes in the crystal structure, the temperature difference across the thickness of the wafer must be less than \(2^{\circ} \mathrm{C}\). Is this cundition being met?

A vacuum system, as used in sputiering electrically conducting thin films on microcircuits, is comprised of a buseplate maintained by an electrical heater at \(300 \mathrm{~K}\) and a shroud within the enclosure maintained at \(77 \mathrm{~K}\) by a liquid-nitrogen coolant loop. The circular baseplate, insulated on the lower side, is \(0.3 \mathrm{~m}\) in diameter and has an emissivity of \(0.25\). (a) How much clectrical power must be provided to the baseplate heater? (b) At what rate must liquid nitrogen be supplied to the shiroud if its heat of vaporization is \(125 \mathrm{~kJ} / \mathrm{kg}\) ? (c) To reduce the liquid-nitrogen consumption, it is proposed to bond a thin sheet of aluminura foil \((\varepsilon=0.09)\) to the baseplate. Will this have the desfred effect?

Liquid oxygen, which has a boiling point of \(90 \mathrm{~K}\) and a latent heat of vaporization of \(214 \mathrm{~kJ} / \mathrm{kg}\), is stored in a spherical container whose outer surface is of \(500-\mathrm{mm}\) diameter and at a temperature of \(-107 \mathrm{C}\). The container is housed in a laboratery whose air and walls are at \(25^{\circ} \mathrm{C}\) (a) If the surface cmissivity is \(0.20\) and the heat transfer coefficient associated with free consection at the outer surface of the container is 10 \(\mathrm{W} / \mathrm{m}^{2}+\mathrm{K}\), what is the rate, in \(\mathrm{kg} / \mathrm{s}\), at which oxygen vapor must be vented from the system? (b) Moisture in the ambient air will result in frost formation on the container, causing the surface emissivity to increase. Assuming the surface temperature and convection coefficient to remain at \(-10^{\circ} \mathrm{C}\) and \(10 \mathrm{~W} / \mathrm{m}^{2}-\mathrm{K}\), respectively, compute the oxygen evaporation rate \((\mathrm{kg} / \mathrm{s})\) as a function of surface cmissivity over the range \(0.2 \leq \varepsilon \leq 0.94\).

A hair dryer may be idealized as a circular doct through which a mmall fan draws ambient air and withie which the air is heated as it flows over a coiled electric resistance wire. (a) If a dryer is designed to operate with an electric power consumption of \(P_{\text {tesc }}=500 \mathrm{~W}\) and to heat air from an ambient temperature of \(T_{i}=20^{\circ} \mathrm{C}\) to a discharge temperature of \(T_{n}=45^{\circ} \mathrm{C}\), at what volumetric flow rate \(\forall\) should the fan operate? Heat loss from the casing to the ambient air and the surroundings nuy be neglected. If the duct has a diameler of \(D=70 \mathrm{~mm}\), what is the discharge velocity \(V_{0}\) of the uir? The density and specific heat of the air maly be approximated as \(\rho=1.10 \mathrm{~kg} / \mathrm{m}^{3}\) and \(c_{p}=1007\) \(\mathrm{J} / \mathrm{Kg}+\mathrm{K}\), respectively. (b) Consider a dryer duct length of \(L=150 \mathrm{~mm}\) and a surface cmisuivity of \(e=0.8\). If the coeflicient associaled with heat transfer by natural convection from the casing to the ambient air is \(h=4 \mathrm{~W} / \mathrm{m}^{2}-\mathrm{K}\) and the temperature of the air and the surroundings is \(T_{\mathrm{w}}=\) \(T_{w e}=20^{\circ} \mathrm{C}\), confirm that the heat loss from the casing is, in fact, negligitle. The casing may be assumed to have an average surface temperature of \(T_{3}=40^{\circ} \mathrm{C}\).

A computer consists of an array of five printed circuit boards (PCBs), each dissipating \(P_{h}=20 \mathrm{~W}\) of power. Cooling of the electronic components on a board is provided by the forced flow of air, equally distributed in passages formed by adjoining boards, and the convection coefficient associated with heat transfer from the components to the air is approximately \(h=200 \mathrm{~W} / \mathrm{m}^{2}\). \(\mathrm{K}\). Air enters the computer console at a temperiture of \(T_{6}=20^{\circ} \mathrm{C}\), and flow is driven by a fin whese power consumption is \(P_{f}=25 \mathrm{~W}\). Iniet ain \(\forall_{1} T_{i}\) (a) If the tempensture rise of the air flow. \(\left(T_{e}-T_{i}\right)_{\text {is }}\) is ot to exceed \(15^{\circ} \mathrm{C}\), what is the minimum allowable volumetric flow rate \(\forall\) of the air? 'The density and specific heat of the air may be approximated is \(\rho=1.161\) \(\mathrm{kg} / \mathrm{m}^{3}\) and \(c_{p}=1007 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\), respectively. (b) The component that is most suncejtible to thermal failure dissipates I W/cm \({ }^{2}\) of surface area. To minimive the potential for thermal failure, where should the component be installed on a PCB? What is its surface lemperature at this location?
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