Heat Transfer
HEAT TRANSFER
Heat transfer is the energy flow that occurs between bodies as a result of a temperature difference. There are three commonly accepted modes of heat transfer: conduction, convection, and radiation. Although it is common to have two or even all three modes of heat transfer present in a given process, we will initiate the discussion as though each mode of heat transfer is distinct.
CONDUCTION
When a temperature difference exists in or across a body, an energy transfer occurs from the hightemperature region to the lowtemperature region. This heat transfer, q, which can occur in gases, liquids, and solids, depends on a change in temperature, ΔT, over a distance, Δx (i.e., ΔT/Δx) and a positive constant, k, which is called the thermal conductivity of the material. In equation form, the rate of conductive heat transfer per unit area is written as where q is heat transfer, A is normal (or perpendicular to flow of heat), k is thermal conductivity, ΔT is the change in temperature, and Δx is the change in the distance in the direction of the flow. The minus sign is needed to ensure that the heat transfer is positive when heat is transferred from the hightemperature to the lowtemperature regions of the body.
The thermal conductivity, k, varies considerably for different kinds of matter. On an orderofmagnitude basis, gases will typically have a conductivity range from 0.01 to 0.1 W/(m–K) (0.006 to 0.06 Btu/hft°F), liquids from 0.1 to 10 W/(m–K) (0.06 to 6 Btu/hft°F), nonmetallic solids from 0.1 to 50 W/(m–K) (0.06 to 30 Btu/hft°F), and metallic solids from 10 to 500 W/(m–K) (6 to 300 Btu/hft°F). Obviously, gases are among the lowest conductors of thermal energy, but nonmetallic materials such as foamed plastics and glass wool also have low values of thermal conductivity and are used as insulating materials. Metals are the best conductors of thermal energy. There is also a direct correlation between thermal conductivity and electrical conductivity—that is, the materials that have a high thermal conductivity also have a high electrical conductivity. Conductive heat transfer is an important factor to consider in the design of buildings and in the calculation of building energy loads.
CONVECTION
Convective heat transfer occurs when a fluid (gas or liquid) is in contact with a body at a different temperature. As a simple example, consider that you are swimming in water at 21°C (70°F), you observe that your body feels cooler than it would if you were in still air at 21°C (70°F). Also, you have observed that you feel cooler in your automobile when the airconditioner vent is blowing directly at you than when the air stream is directed away from you. Both of these observations are directly related to convective heat transfer, and we might hypothesize that the rate of energy loss from our body due to this mode of heat transfer is dependent on not only the temperature difference but also the type of surrounding fluid and the velocity of the fluid. We can thus define the unit heat transfer for convection, q/A, as follows: where q is the heat transfer per unit surface area, A is the surface area, h is the convective heat transfer coefficient W/(m^{2} – K), T_{i} is the temperature of the body, and T_{∞} is the temperature of the fluid.
Thus convective heat transfer is a function of the temperature difference between the surface and the fluid; the value of the coefficient, h, depends on the type of fluid surrounding the object, and the velocity of the fluid flowing over the surface. Convective heat transfer is often broken down into two distinct modes: free convection and forced convection. Free convection is normally defined as the heat transfer that occurs in the absence of any external source of velocity—for example, in still air or still water. Forced convection has some external source, such as a pump or a fan, which increases the velocity of the fluid flowing over the surface. Convective heat transfer coefficients range widely in magnitude, from 6W/(m^{2} – K) (1 Btu/hft^{2}°F) for free convection in air, to more than 200,000 W/(m^{2} – K) (35,000 Btu/hft^{2}°F) for pumped liquid sodium. Convective heat transfer plays a very important role in such energy applications as power boilers, where water is boiled to produce highpressure steam for power generation.
RADIATION
Radiative heat transfer is perhaps the most difficult of the heat transfer mechanisms to understand because so many factors influence this heat transfer mode. Radiative heat transfer does not require a medium through which the heat is transferred, unlike both conduction and convection. The most apparent example of radiative heat transfer is the solar energy we receive from the Sun. The sunlight comes to Earth across 150,000,000 km (93,000,000 miles) through the vacuum of space. Heat transfer by radiation is also not a linear function of temperature, as are both conduction and convection. Radiative energy emission is proportional to the fourth power of the absolute temperature of a body, and radiative heat transfer occurs in proportion to the difference between the fourth power of the absolute temperatures of the two surfaces. In equation form, q/A is defined as: where q is the heat transfer per unit area, A is the surface area, σ is the StefanBoltzmann constant 5.67×10^{8} W/(m^{2} – K^{4}), T_{1} is the temperature of surface one, and T_{2} is the temperature of surface two.
In this equation we are assuming that all of the energy leaving surface one is received by surface two, and that both surfaces are ideal emitters of radiant energy. The radiative exchange equation has to be modified to account for real situations—that is, where the surfaces are not ideal and for geometrical arrangements in which surfaces do not exchange their energies only with each other. Two factors are usually added to the above radiative exchange equation to account for deviations from ideal conditions. First, if the surfaces are not perfect emitters, a factor called the emissivity is added. The emissivity is a number less than 1, which accounts for the deviation from nonideal emission conditions. Second, a geometrical factor called a shape factor or view factor is needed to account for the fraction of radiation leaving a body that is intercepted by the other body. If we include both of these factors into the radiative exchange equation, it is modified as follows: where Fε is a factor based on the emissivity of the surface and F_{G} is a factor based on geometry.
The geometric factor can be illustrated by considering the amount of sunlight (or radiative heat) received by Earth from the Sun. If you draw a huge sphere with a radius of 150 million km (93 million miles) around the sun that passes through Earth, the geometric factor for the Sun to Earth would be the ratio of the area on that sphere's surface blocked by Earth to the surface area of the sphere. Obviously, Earth receives only a tiny fraction of the total energy emitted from the Sun.
Other factors that complicate the radiative heat transfer process involve the characteristics of the surface that is receiving the radiant energy. The surface may reflect, absorb, or transmit the impinging radiant energy. These characteristics are referred to as the reflectivity, absorbtivity, and transmissivity, respectively, and usually are denoted as ρ, α, and τ. Opaque surfaces will not transmit any incoming radiation (they absorb or reflect all of it), but translucent and clear surfaces will transmit some of the incoming radiation. A further complicating factor is that thermal radiation is wavelengthdependent, and one has to know the wavelength (spectral) characteristics of the material to determine how it will behave when thermal radiation is incident on the surface. Glass is typical of a material with wavelengthdependent properties. You have observed that an automobile sitting in hot sunlight reaches a temperature much higher than ambient temperature conditions. The reason is that radiant energy from the Sun strikes the car windows, and the veryshortwavelength radiation readily passes through the glass. The glass, however, has spectrally dependent properties and absorbs almost all the radiant energy emitted by the heated surfaces within the car (at longer wavelengths), effectively trapping it inside the car. Thus the car interior can achieve temperatures exceeding 65°C (150°F). This is the same principle on which a greenhouse works or by which a passively heated house is warmed by solar energy. The glass selectively transmits the radiation from the Sun and traps the longerwavelength radiation emitted from surfaces inside the home or the greenhouse. Specially made solar glasses and plastics are used to take advantage of the spectral nature of thermal radiation.
R VALUES
Many everyday heat flows, such as those through windows and walls, involve all three heat transfer mechanisms—conduction, convection, and radiation. In these situations, engineers often approximate the calculation of these heat flows using the concept of R values, or resistance to heat flow. The R value combines the effects of all three mechanisms into a single coefficient.
R Value Example: Wall
Consider the simple wall consisting of a single layer of Sheetrock, insulation, and a layer of siding as shown in Figure 1. We assume it is a cool autumn evening when the room air is 21°C (70°F) and the outside air is 0°C (32°F), so heat will flow from inside the wall to the outside. Convective and radiative heat transfer occurs on both the inside and outside surfaces, and conduction occurs through the Sheetrock, insulation, and siding. On the outside, the convective heat transfer is largely due to wind, and on the inside of the wall, the convective heat transfer is a combination of natural and forced convection due to internal fans and blowers from a heating system. The room surfaces are so close to the outside wall temperature that it is natural to expect the radiative heat transfer to be very small, but it actually accounts for more than half of the heat flow at the inside surface of the wall in this situation. When treated using the R value concept, the total R value of the wall is the sum of individual R values due to the inside surface, the Sheetrock, the insulation, the siding, and the outer surface. Hence the heat flow through this wall may be written as
The resistances of the Sheetrock, siding, and insulation may be viewed as conductive resistances, while the resistance at the two surfaces combine the effects of convection and radiation between the surface and its surroundings. Typical values for these resistances in units of (W/m^{2} – W/m^{2}°C)^{1} ((Btu/hft^{2}°F)^{1}) are as follows:
R_{inside surf}  = 0.12  ( 0.68) 
R_{Sheetrock}  = 0.08  ( 0.45) 
R_{insulation}  = 1.94  (11.0) 
R_{siding}  = 0.14  (0.80) 
R_{outside surf}  = 0.04  (0.25) 
R_{total}  = 2.32  (13.18) 
For this case, the heat flow through the wall will be (21–0)°C/2.32 (W/m^{2}°C)^{1} = 9.05 W/m^{2} (2.87 Btu/hft^{2}). While the surfaces, Sheetrock, and siding each impede heat flow, 80 percent of the resistance to heat flow in this wall comes from the insulation. If the insulation is removed, and the cavity is filled with air, the resistance of the gap will be 0.16 (W/m^{2}°C)^{1} (0.9 (Btu/hft^{2}°F)^{1}) and the total resistance of the wall will drop to 0.54 (W/m^{2}°C)^{1} (3.08 (Btu/hft^{2}°F)^{1}) resulting in a heat flow of 38.89 W/m^{2}(12.99 Btu/hft^{2}). The actual heat flow would probably be somewhat different, because the Rvalue approach assumes that the specified conditions have persisted long enough that the heat flow is "steadystate," so it is not changing as time goes on. In this example the surface resistance at the outer wall is less than half that at the inner wall, since the resistance value at the outer wall corresponds to a wall exposed to a wind velocity of about 3.6 m/s (8 mph), which substantially lowers the resistance of this surface to heat flow.
If the wall in the example had sunlight shining on it, the heat absorbed on the outer surface of the wall would reduce the flow of heat from inside to outside (or could reverse it, in bright sunshine), even if the temperatures were the same.
R Value Example: Window
A window consisting of a single piece of clear glass can also be treated with Rvalue analysis. As with the wall, there is convective and radiative heat transfer at the two surfaces and conductive heat transfer through the glass. The resistance of the window is due to the two surface resistances and to the conductive resistance of the glass, R_{glass}. For typical window glass, R_{glass} = 0.003 (W/m^{2}°C)^{1} (0.02 (Btu/hft^{2}°F)^{1}) so the total resistance of the window is R_{window} = (0.12 + 0.003 + 0.04) (W/m^{2}°C)^{1 }= 0.163 (W/m^{2}°C)^{1} (0.95 (Btu/hft^{2}°F)^{1}). Thus the heat flow will be q = (21 – 0)°C/0.163(W/m^{2}°C)^{1} = 128.8 W/m^{2} (40.8 Btu/hft^{2}), or fourteen times as much as that through the insulated wall. It is interesting to note that the heat flow through an ancient window made from a piece of oilskin, or even a "window" made from a piece of computer paper, would not increase by more than 2 percent from that of the glass window, because the resistance of the glass is so small.
When sunlight is shining through a window, the heat transfer becomes more complicated. Consider Figure 2.
Suppose the outdoor and indoor temperatures are still 0°C and 21°C, but the window now has 600 W/m^{2} (190 Btu/ft^{2}h) of sunlight striking it (q_{r}), typical of a fairly sunny window on an autumn day. About 8 percent (ρq_{r}) will be reflected back into the atmosphere, about 5 percent (αq_{r}), or 30 W/m^{2}, will be absorbed in the glass, and the remaining 87 percent (τq_{r}), or 522W/m^{2}, will be transmitted into the room, providing light before it strikes the walls, floor, ceiling, and furnishings in the room. Again, some will be reflected, providing indirect lighting, and the remainder will be absorbed and converted to heat. Eventually, all—except any that may be reflected back out the window—will be absorbed and converted to heat in the room. Thus the amount of sunlight coming through the window is about four times as great as the amount of heat flowing outward. The 30 W/m^{2 }that is absorbed in the glass heats the glass slightly, reducing the conductive heat flow through the glass to 121 W/m^{2}, so the net effect of the sunlight is to result in the window providing 401 W/m^{2} of heating (522–121) to the room instead of losing 128 W/m^{2}.
In this example, we have assumed that the outside temperature is lower than inside; therefore the heat flow due to the temperature difference is from inside to outside. In the summer it will be hotter outside, and heat will flow from outside to inside, adding to the heat gained from the sunlight. Because windows represent a large source of heat gain or the heat loss in a building, a number of schemes are used to reduce the heat gains in summer and heat losses in winter. These include the use of double (two glass panes) or triple glazing for windows. The glass is separated by an air space, which serves as an added insulating layer to reduce heat transfer. Reflective films are used to reduce heat gain from sunlight. Some of these films also serve to reduce the heat transfer by longwave (nonsolar) radiation as well.
There are numerous other examples where heat transfer plays an important role in energyusing systems. One is the production of steam in large boilers for power production, where steam is boiled from water. Hot combustion gases transfer heat to the water by radiation, conduction through the pipe walls, and convection. The boiling of the water to produce steam is a special case of convective heat transfer where very high heat transfer coefficients are obtained during the boiling process. Another very practical example of convective cooling is in automobiles. The "radiator" is, in fact, a "convector" where pumped water is forced through cooling tubes and is cooled by air forced over the tubes and fins of the radiator. Forced convective heat transfer occurs at both the air side and the water side of the "radiator." Cooling towers and airconditioning coils are other practical examples of combined heat transfer, largely combined conduction and convection.
While heat transfer processes are very useful in the energy field, there are many other industries that rely heavily on heat transfer. The production of chemicals, the cooling of electronic equipment, and food preparation (both freezing and cooking) rely heavily on a thorough knowledge of heat transfer.
W. Dan Turner David E. Claridge
See also: Air Conditioning; Furnaces and Boilers; Heat and Heating; Heat Pumps; Insulation; Refrigerators and Freezers; Solar Energy; Solar Energy, Historical Evolution of the Use of; Thermodynamics; Windows.
BIBLIOGRAPHY
ASHRAE. (1997). 1997 ASHRAE Handbook: Fundamentals. Atlanta: Author.
Incropera, F. P., and DeWitt, D. P. (1996). Fundamentals of Heat and Mass Transfer, 4th ed. New York: John Wiley & Sons.
Turner, W., ed. (1982). Energy Management Handbook. New York: John Wiley & Sons.
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