The Physics of Energy, page 22
(6.14)
The units of U are [U] W/m2 K or BTU/ft2∘ F hr.
U-factors are measured under standardized conditions. The U-factors of windows quoted in Table 6.3, for example, were measured at the center of the glass surface between an “indoor” environment containing still air at 25∘C and an “outdoor” environment at 0∘C with a wind velocity of 7 m/s; these U-factors include the thermal resistance of the films of still air that abut the glass surface [28]. When used under other conditions, the actual U-factor of a fenestration product will depart from these standard values.
Table 6.3 Effective R-values for common building materials and U-factors for fenestration products [28]. The U-factorsquoted here for windows are measured at the center of the glass surface and include the thermal resistance of air-films.
Material Thickness R-value (SI) R-value (US)
(m2 K/W) (ft2∘F hr/BTU)
Gypsum drywall 1/2" 0.079 0.45
Plywood 3/4" 0.19 1.08
Hollow-backed vinyl siding 0.11 0.62
Wood shingles 0.15 0.87
Fiberglass batt 3.5"/8" 2.0/4.6 12/26
Blown cellulose 0.21–0.24/cm 2.9–3.4/in
Rigid (polyisocyanurate) foam 0.37–0.49/cm 5.3–7.1/in
Mineral fiber /cm /in
Common brick 4" 0.14 0.80
Concrete masonry unit (CMU) 4"/8" 0.14/0.19 0.80/1.11
Window type U-factor (SI) U-factor (US)
(W/m2 K) (BTU/ftF hr)
Single pane glazing 1/8" 5.9 1.04
Double pane glazing with " air space 3.1/2.7/2.3 0.55/0.48/0.40
Double glazing with " air space and emissivity coating 2.0 0.35
Double glazing, " space, argon filled, coating 1.7 0.30
Quadruple glazing, " space, argon filled, coating 0.97 0.17
Quadruple glazing, " space, krypton filled, coating 0.68 0.12
For a pure material placed between two surfaces at fixed temperatures, and in the absence of convection or radiation, there is an inverse relationship between the U-factor and R-value, . In most realistic situations, however, an empirically measured value of R is used to characterize conductive heat flow for homogeneous materials such as insulation, while U incorporates the full set of factors involved in thermal energy transport through a given building component. In the US, U-factors are generally quoted for windows, while R-values are generally used for insulation, walls, etc. [32]. Table 6.3 gives measured R-values for some common building materials and the U-factors for some representative windows of specific designs.
6.3Heat Transfer by Convection and Radiation
The ways that convection and radiation transfer heat were described qualitatively at the beginning of this chapter. Here we explore them at an elementary quantitative level. As mentioned earlier, both convective and radiative heat transfer can be quite complex, though we avoid most of the complexity in the situations treated in this book.
6.3.1 Forced Convection
Forced convection is a common way to transfer considerable heat quickly. We use it when we cool an overheated frying pan under a stream of water or when we blow on a burned finger. A cold fluid forced to flow over a hot surface carries away heat efficiently. Isaac Newton is credited with the empirical observation, known as Newton’s law of cooling, that the rate of heat transfer in situations involving forced convection is proportional to the temperature difference between the object (T) and the fluid (),
(6.15)
where is the net rate of heat transfer from the object to the fluid (integrated over the surface of the object), A is the object’s surface area, and is a constant characterizing the object and the fluid flow. is known as the heat transfer coefficient, and has units W/m2 K. can be thought of as the average over the object’s surface of a local heat transfer coefficient h.
R-values and U-factors
An R-value is an empirically measured thermal resistance for a given material. A U-factor is an empirically measured thermal conductivity for a building component that includes radiation and convection both within and in the vicinity of the structure in question. In the US, U-factors are used to rate windows and other fenestration products, while R-values are typically used for walls, roofs, and other construction materials.
Equation 6.15 can be understood as combining the effects of conduction and mass transport. When a fluid flows over an object, the fluid closest to the object actually adheres to its surface. The fluid only maintains its bulk flow rate and bulk temperature some distance from the surface. The object, therefore, is surrounded by a thin boundary layer, through which the temperature changes from T at the object’s surface to in the bulk of the fluid. Heat is conducted from the object through the layer of fluid that is at rest at its surface. The thinner the boundary layer, the larger the temperature gradient at the object’s surface and the faster the heat is conducted from the surface into the fluid. The role of convection is to transport the heated fluid away from the top of the boundary layer, thereby keeping the temperature gradient large. Since the conductive heat flux is proportional to the temperature gradient at the surface – which in turn is proportional to – the heat transferred from the surface to the fluid is proportional to , as Newton suggested. Figure 6.7 shows data that support Newton’s law of cooling. In reality, Newton’s law is only approximately valid and works best when is small. When grows large, the heat flow itself affects the dynamics of the boundary layer, introducing additional temperature dependence into eq. (6.15).
Figure 6.7 Measurements of heat flux () from an isothermal sphere to a steady (laminar) flow of water around the sphere (see inset) as a function of , for three values of the asymptotic flow velocity. The approximately linear slope illustrates Newton’s law of cooling. Data from [33].
All of the complex physics of the fluid in the boundary layer, as well as heat and mass transport through it, enter into the heat transfer coefficient . Like the drag coefficient and thermal conductivity k, is a phenomenological parameter that is quite difficult to calculate from first principles and is instead usually taken from experiment. Any physical effect that influences the thickness of the boundary layer modifies the rate of heat transfer and thus changes . The value of is much larger, for example, when the fluid flow is turbulent as opposed to laminar,1 because the boundary layer is thinner for turbulent flow. Another case where is large is when T is so high that it causes the fluid in contact with the object to boil. When the fluid begins to boil, the buoyancy of the vapor carries it rapidly away from the surface, enhancing heat transfer. This is one of several reasons why phase change is used in large-scale heat transfer devices such as refrigerators and steam boilers (see §12).
Forced convection can be a much more effective means of heat transfer than conduction. To quantify this, we can compare the local (convective) heat transfer coefficient h at a surface with the thermal conductivity of the fluid k. Since it only makes sense to compare quantities with the same dimensions (otherwise the comparison depends on the choice of units), it is customary to multiply h by a length scale characteristic of the surface over which heat is being transferred. Thus the ratio
(6.16)
known as the Nusselt number, is a measure of (forced) convective relative to conductive heat transport at a surface. The subscript L on Nu indicates which length scale has been used in the definition. The Nusselt number for water flowing at 2 m/s over a 60 mm long flat plate, for example, is found to be approximately 60 – an indication of the greater effectiveness of heat transfer by forced convection in this situation compared with conduction [27].
Forced Convection
Forced convection transfers heat effectively through motion of a fluid forced to flow past an object at a different temperature. Newton’s law of cooling,
parameterizes the rate of energy transfer in terms of a heat transfer coefficient , which depends on the properties of the flowing fluid and the object. Under a wide range of conditions, forced convection is a more efficient way to transfer heat than conduction.
6.3.2 Free Convection
The most familiar form of free convection is driven by the force of gravity. When a packet of fluid is heated, it expands. The resulting reduction in gravitational force per unit volume (buoyancy) then pushes the expanded fluid upwards relative to the surrounding cooler and denser fluid. This drives a circulation pattern that transports heat away from the source. Such free convection would not take place in the absence of a gravitational field. Other density-dependent forces, real or apparent, however, can also drive free convection. For example, the centrifugal force felt by material whirling in a centrifuge, and the Coriolis force that figures in the large-scale circulation of the atmosphere and ocean currents (§27.2) can drive free convection patterns that transport heat in different environments.
The structure of convection patterns depends strongly on the geometry of the system. In this book we encounter two quite different idealized situations: one in which a fluid is bounded by horizontal planes, and the other in which the boundaries are vertical planes. We encounter the first case, for example, when studying convection in Earth’s atmosphere. This situation can be modeled in the laboratory by placing a fluid between two horizontal plates held at temperatures below and above. If is higher than , no convection occurs since the warmer, less dense air is above the cooler denser air already in equilibrium. Even when is lower than , convection does not start until a critical value of is reached. This is described in more detail in the context of Earth’s atmosphere in §34.2. Above , convection cells known as Bénard cells appear. When is just above , these cells are few in number and orderly in form, but as increases they become more numerous and the flow in the cells becomes turbulent. Figure 6.8 shows a cartoon of two Bénard cells between plates.
Figure 6.8 Two Bénard cells convecting heat between horizontal plates at temperatures and . The horizontal length scale for the convective cells is set by the separation L between the plates.
The second case, convection in a region bounded by one or more vertical surfaces, is relevant to convective heat transfer near and within walls and windows. A basic model is given by two vertical plates held at different temperatures and respectively with a fluid in between. In contrast to the horizontal case, there is always convection in this geometry, with warm fluid flowing upwards in the vicinity of the hotter surface and cool fluid flowing downward along the colder surface. The fluid velocity goes to 0 at the boundary, as discussed above in the context of forced convection. The velocity profile is shown in Figure 6.9. Like forced convection, this flow transports heat very efficiently from the hotter to the colder surface. This kind of convection is a serious source of heat loss in and near the walls and windows of buildings, where convection is driven at a single boundary fronting the large heat reservoir provided by interior or exterior spaces.
Figure 6.9 Velocity of vertical convection as a function of distance between vertical plates separated by . The scale of the vertical velocity depends upon the properties of the fluid, the temperature difference, and the distance between the plates [34].
6.3.3 Radiative Heat Transfer
Heat transfer by radiation obeys quite different rules than conduction and convection. In particular, radiation is much less sensitive to the material characteristics of the emitting object and its environment. In an ideal (but still useful) limit the radiant power emitted by a hot object depends only on its temperature T and surface area A. This is summarized by the Stefan–Boltzmann law,
(6.17)
where σ is the Stefan–Boltzmann constant W/m K. The Stefan–Boltzmann law is derived and described in more detail in §22 (Solar energy), after we have introduced the basic notions of quantum mechanics (§7) and entropy (§8), and given a more precise definition of thermal equilibrium (§8). The Stefan–Boltzmann law seems quite uncanny at first sight, since it requires that hot objects radiate electromagnetic energy simply by virtue of being hot, but independent of the density or other material characteristics of the radiating body. This result follows, however, from the quantum nature of thermodynamic equilibrium between any material containing charged matter (such as electrons) and the electromagnetic field. The frequency spectrum of thermal radiation has a characteristic shape, which is derived in §22. The Sun radiates most of its energy as visible light, but thermal radiation from objects at temperatures below 1000 K peaks in the infrared part of the electromagnetic spectrum where it is invisible to the unaided eye; only the high-frequency tail of the spectrum is visible as a red or orange glow for sufficiently hot objects.
Actually, eq. (6.17) holds only for perfect emitters, which absorb and re-emit all radiation that impinges on them. Materials with this property are also known as black bodies, and their radiation is known as blackbody radiation. For less ideal materials, the Stefan–Boltzmann law is modified by a factor ε, the emissivity, which is less than or equal to one. The Sun, as an example, radiates approximately like a black body with temperature 5780 K and emissivity (Problem 6.4).
When electromagnetic radiation falls on a surface, it can be absorbed, reflected, or transmitted. As explained in detail in Box 22.1, conservation of energy requires that the fractions of the light absorbed (the absorptivity), reflected (reflectivity), and transmitted (transmittance) must sum to one at each wavelength of light. An opaque object is one that transmits no light, so for an opaque object the absorptivity and reflectivity sum to one. Furthermore, the laws of thermodynamics require that the absorptivity of an opaque object is equal to its emissivity for every wavelength of light (Kirchhoff’s law). This explains the peculiar term “black body”: in order to be a perfect emitter of radiation, a body must also be a perfect absorber, i.e. “black.” Since the transmittance of a sample varies with its thickness, the absorptivity ( emissivity) usually quoted for a specific material is measured for a sample thick enough that negligible light is transmitted.
Emissivities in general depend on radiation frequency, but in many situations can be approximated as frequency-independent constants. Most non-conducting materials have emissivities of order one over the range of frequencies that dominate thermal radiation by objects at room temperature. A list of common materials with emissivity ε greater than 0.9 includes ice (0.97), brick (0.93), concrete (0.94), marble (0.95), and most exterior paints. In contrast, polished metals with high electrical conductivity are highly reflective and have low emissivity. They do not emit like black bodies. Examples include aluminum alloys (0.1), brass (0.03), gold (0.02), and silver (0.01).
The optical properties of materials can vary dramatically with the wavelength of light. A 5 mm thick sheet of ordinary glass, for example, transmits more than 90% of the visible light that falls on it at perpendicular incidence; most of the rest is reflected, and very little is absorbed. In contrast, the same sheet of glass is virtually opaque to the infrared radiation emitted by objects at room temperature, with an emissivity of about 0.84. Films exist that reflect, rather than absorb, most infrared radiation yet transmit visible light. Such films, as well as metal foils with high reflectivity, play an important role in the control of thermal radiation.
If two objects are at different temperatures, radiation transfers energy between them. When , the objects are in thermal and radiative equilibrium (described in more detail in §8, §22, §34). A typical (male human) adult with surface area 1.9 m2 at a temperature of 37∘C radiates energy at a rate of W. If, however, he sits in an environment at 28∘C, energy radiates back to him from the environment at W, so the net radiative energy loss is ~100 W.
Radiative Heat Transfer
An object at temperature T radiates thermal energy according to the Stefan–Boltzmann law,
where W/m2 K4 is the Stefan–Boltzmann constant and A is the object’s surface area. ε is the object’s emissivity, which equals its absorptivity, the fraction of incident light that it absorbs. defines a black body.
Most non-conducting materials have emissivity close to one. Highly reflective materials such as polished metal surfaces have and can be used to block thermal radiation.
6.4Preventing Heat Loss from Buildings
Keeping buildings warm in a cold environment is a problem as old as human civilization. Enormous quantities of energy are used in the modern era for space heating. Roughly ~30 EJ, or about 6% of human energy consumption, was used for residential space heating in 2009 [35]. The fraction goes up to roughly 1/6 in temperate climates. This is one area, however, where understanding the basic physics, making use of modern materials, and adopting ambitious standards have already led to considerable energy savings – and much more is possible. In 1978, 7.3 EJ were used to provide space heating for 76.6 million housing units in the US compared with 4.5 EJ for 111.1 million units in 2005 – an almost 60% reduction in the energy use per housing unit over less than three decades [36].2
We can identify the critical steps to reducing heat transfer from the warm interior of a building to the colder environment: (1) eliminate infiltration, the most dramatic form of convection in which colder air simply flows into the building as warmer air flows out; (2) use materials for walls, windows, and roofs that have high thermal resistance to minimize heat conduction; (3) include a layer of highly reflective metal foil between the building’s insulating layers and outer walls to obstruct outgoing radiation (see §6.4.2); and (4) obstruct convection loops that would otherwise form within or near walls and windows.
Minimizing Heat Loss from Buildings
