The Physics of Energy, page 23
Conduction, convection, and radiation all contribute to heat loss from buildings. Conduction can be minimized by integrating thermal insulation into a building’s exterior surfaces. Thermal radiation can be obstructed by a very thin layer of metal foil, often included as a facing for insulation. Infiltration, a form of convection, can be reduced by sealing gaps. Interior convection loops near walls and windows can be obstructed with drapes, wall hangings, and other furnishings.
The first step in keeping a building at a higher temperature than its environment is to prevent air from escaping into the environment. Energy lost to in-/exfiltration accounts for roughly one third of thermal energy losses for US buildings. In many cases, older and/or poorly fitted windows and doors are principal culprits for excess infiltration. Some air exchange, however, is necessary for ventilation; the most ambitious ultra-efficient designs employ heat exchangers that use forced convection to transfer energy from exiting to entering air. Even with such devices, however, some thermal energy is lost as air leaves the building. In the following section (§6.4.1), we examine the problem of heat conduction through a building’s walls, and in §6.4.2, we look briefly at ways to reduce radiative and convective heat loss.
6.4.1 Insulation
Traditional structural materials such as concrete, brick, steel, glass, and to a lesser extent wood, have relatively high thermal conductivities (see Table 6.1). With one of these materials on the outside and only drywall on the inside, the thermal losses through building walls would be enormous (see Example 6.2 for a quantitative estimate). Other materials should therefore be sandwiched between the interior and exterior walls to suppress heat conduction. A look at Table 6.1 suggests that air is a wonderful thermal insulator. (Noble gases such as argon are better, but much more expensive.) Filling a wall cavity with air, however, would result in unacceptable thermal loss through convection loops that would arise when the interior and exterior surface temperatures differ significantly. In fact, the fundamental paradigm of insulation is to trap air (or another low-conductivity gas) in small cavities to avoid convection, using as low a density of material as possible to minimize heat conduction in the material itself. These goals are realized effectively by fiberglass, plastic foams, and shredded cellulose, the most common forms of insulation. Fiberglass, for example, is a fibrous material spun from glass. The thermal conductivity of glass is not too different from that of concrete (Table 6.1), but fiberglass is largely air, and its thermal conductivity is about 25 times lower than that of glass.
Example 6.2 Insulating a Simple House
To get a qualitative sense of the magnitude of heat transfer through walls and the importance of insulation, consider aseries of progressively more ambitious insulation efforts. First, consider an uninsulated 20 cm thick concrete wall with the temperature on the inside at 20∘C and on the outside at 0∘C. Taking W/m K from Table 6.1, we find m2 K/W, and the rate of heat flow is given by
This is a lot of energy loss. Consider a relatively small single-story dwelling with a floor area of 120 m2, horizontal dimensions 10 m m, and a height of 3 m. We can estimate the combined surface area of walls and ceiling (ignoring losses through the floor) to be 250 m2. So the power needed to maintain the building at a temperature 20 warmer than the environment is 250 m2 × 100 W/m2 ≈ 25 kW. The rate of heat loss works out to about 2 GJ per day, about 10 times the world average daily per capita energy use, or roughly twice the per capita use of the average American. We must conclude that conductive energy losses from a concrete (or stone or brick) building are unacceptably large.
We consider a series of steps to improve the situation. First, we consider walls half as thick made of (hard) wood. Then, using W/m K from Table 6.1, we find m2 K/W, and the rate of conductive energy loss drops by a factor of 2.8. Finally, if the hardwood were replaced by a sandwich of 3 cm of wood on the inside and outside with 20 cm of fiberglass insulation in between, the R-value increases to 4.6 m K/W (where 22.7 m K/W is the thermal resistance of one meter of fiberglass, which can be read off of Table 6.3) and the conductive energy loss drops by another factor of about nine to about one kW.
The moral of this example is, as might have been expected, that insulation is a great way to save energy. Insulation standards have been increasing steadily in recent years. The US Energy Efficiency and Renewable Energy Office recommends standards for home insulation. In 2010 those standards for the climate of Cambridge, Massachusetts were R38–R60 for attic space, R15.5–R21 for cavity walls plus insulated sheathing, and R25–R30 for floors (all in US units) [37]. The wall shown at right has been upgraded to R36 with several layers of insulation (Problem 6.7) (Image credit: Mark Bartosik, www.netzeroenergy.org)
Applying effective insulation thus gives a simple way of dramatically reducing thermal energy losses from buildings. Insulation is one of many ways of saving energy that may help reduce human energy needs if applied broadly; we explore this and other approaches to saving energy in §36. Note that gaps in insulation are particularly wasteful. According to the electric circuit analogy, the thermal resistance of parallel heat paths add inversely (see eq. (6.13)), so a path with zero resistance acts like a thermal short circuit, allowing a great deal of heat to escape. Problem 6.8 explores this phenomenon, finding that a gap of 1/60th the area of a wall reduces the effective thermal resistance of the wall by ~22%.
6.4.2 Preventing Radiative and Convective Heat Transfer
A thin metal film provides a very effective barrier to radiative energy transport between regions or materials at different temperatures. With high reflectivity and low emissivity, such a film works in two ways. First, the radiation incident from the hotter region is reflected back, and second the film radiates only weakly into the colder region because of its low emissivity. Such films have many uses (Figure 6.10).
Figure 6.10 Marathoners avoid hypothermia by wrapping in radiation reflecting films after the 2005 Boston Marathon. (Credit: Stephen Mazurkiewicz, RunnersGoal.com)
Reflective films are used in building construction where radiative transfer (in either direction) would otherwise be large and other structural considerations, such as the need to allow the flow of water vapor, allow it. It is common, for example, to have reflective films installed below roof insulation in hot climates (Figure 6.11). Convection does not transmit heat downward from a hot roof, but radiation does. A metallic film reflects the heat back and does not radiate downward into the living space. In colder climates, a reflective film on the outside of insulating layers similarly helps to trap heat within the living space.
Figure 6.11 Foil-faced 120 mm fiberglass batts insulate the ceiling of this home in Italy where heat load on the roof is an important consideration. (Credit: Alec Shalinsky, Vetta Building Technologies)
Windows cannot be obstructed by metal films. Instead, glass coatings have been developed that are transparent to visible light, but reflect the infrared radiation (see §4.3) that dominates the spectrum of thermal radiation at room temperature. Windows with , known as low-e windows, are commonly available.
Finally, a few words about convection. Convective loops form wherever air is free to circulate along vertical surfaces at a temperature different from that of the ambient air. Conventional bulk insulation blocks convection within (and infiltration and conduction through) walls. More problematic are the convection loops that can form both inside and outside a structure near windows and outside by the cool outside walls of buildings. In these situations, the motion of air increases the rate of heat transfer in a similar fashion to that discussed earlier for forced convection. Some of the paths for convection near and within a multi-pane window are shown in Figure 6.12. Convection within the window can be reduced by decreasing the thickness of the air space (which unfortunately also diminishes its thermal resistance), by evacuating the air from the space between the glass panes, or by using a gas such as argon that has higher viscosity than air.3 The convection loops on the inside can be suppressed by window treatments such as shades or drapes that can also have insulating value.
Figure 6.12 A sketch of a section of a triple-glazed window in an exterior wall and its immediate environs, showing the convection loops that can form near and within the window.
In most real situations, convection and radiative heat transfer combine with conduction to give a total rate of heat transfer through given building components that is given by the U-factor described in §6.2.5. For example, the U-factors for different types of windows given in Table 6.3 can be understood as coming from a composition of all three types of heat transfer (Problems 6.10, 6.11).
6.5The Heat Equation
In our analysis so far, we have assumed that the temperature in the materials we study does not change over time. Often this is not the case. A hot object cools in a cold environment; heat introduced at one end of an object flows throughout its bulk. To study such systems we need to understand the dynamics of heat flow in time-dependent situations. We explore this phenomenon in the context of conductive heat transfer in a homogeneous medium that is fixed in place (as in an incompressible solid).
6.5.1 Derivation of the Heat Equation
The key ingredient in the dynamics of conductive heat transfer is the conservation law (6.5) that relates heat flow to thermal energy density,
(6.18)
When other parameters are held fixed, u is a function of the temperature T, with
(6.19)
where we have replaced the volumetric heat capacity by the product of the specific heat capacity and the density ρ (§5.4.3). With this replacement, eq. (6.18) becomes
(6.20)
When the heat flux is due only to conduction, Fourier’s law, , allows us to eliminate q from this equation and obtain a single equation relating the space and time derivatives of the temperature,
(6.21)
In general ρ, , and k could depend on the temperature and thus on x and t. If the temperature does not depart too much from some mean value , however, we can evaluate ρ, , and k at and treat them as constants. In that case, k may be moved out in front of the space derivatives, and combined with ρ and into a new constant , known as the thermal diffusivity. The equation then simplifies to
(6.22)
where is the Laplacian operator (6.22) to incorporate the expansion or compression of the material involved. In solids, where the heat equation is most frequently applied, and the distinction is irrelevant.
The Heat Equation
Time-dependent conductive heat flow in a rigid homogeneous medium is described by the heat equation
where is the thermal diffusivity of the material.
6.5.2 Solving the Heat Equation
Equation (6.1. Like the thermal conductivity, the values of a range over many orders of magnitude. Since the time rate of change of T is proportional to a, materials with large thermal diffusivities, such as metals, conduct heat rapidly.
Like the wave equation studied in §4, the heat equation is linear and homogeneous; i.e. each term is linear in T. Therefore, like waves, solutions to the heat equation can be superposed: if and are two solutions of eq. (6.22), then so is for any constants and . Unlike waves, which propagate over long distances, local temperature variations die away in characteristic solutions to the heat equation. The magnitude of initial temperature inhomogeneities in a material with a fixed spatially averaged temperature decreases exponentially in time as the temperature through the material equilibrates. Similarly, if a material initially contains localized high-temperature regions, the thermal energy dissipates away through the material over time. From the perspective of energy conservation, thermal energy moves from regions of high temperature to regions of low temperature as the system approaches thermal equilibrium.
As an example, consider a one-dimensional world where at some initial time, , the temperature varies sinusoidally in space about a constant value , so . Then it is straightforward to check that the time-dependent function
(6.23)
agrees with the initial condition at and satisfies eq. (6.22). The shape of the sinusoidal temperature variation thus remains fixed in place as its amplitude decays away to zero. Such a solution is sketched in Figure 6.13. The larger the thermal diffusivity, the faster the heat dissipates.
Figure 6.13 A sinusoidal temperature distribution at dies away with t according to .
From Fourier’s theorem (B.3) any initial condition for the heat equation can be written as a linear combination of sine wave modes. Each of these modes evolves in time according to eq. (6.23). Since the heat equation is linear, this means that the solution for any initial condition can be computed as a superposition of the solutions for the sine wave components. In fact, it was in this context that Fourier developed the methods that now bear his name.
Another example of a solution to eq. (6.22) that illustrates the dissipation of thermal energy is shown in Figure 6.14. In this example, the temperature is initially elevated above the surroundings by in the interval . As time goes on, the heat diffuses outward. This kind of problem can be solved using Fourier’s methods.
Figure 6.14 At time zero the temperature in the region between and L is elevated by above the surroundings. The three curves, computed numerically, show the time-dependent temperature distribution that solves the heat equation (6.22) at times , , and .
6.5.3 Seasonal Variation of Ground Temperature
An application of the heat equation with relevance to energy systems is the computation of ground temperature at various depths in response to daily and seasonal fluctuations on the surface. A contour map of annual mean ground surface temperature in the continental United States is shown in Figure 6.15. The temperature of the soil at depths greater than a few meters is roughly constant throughout the year, and matches very closely with the average annual surface temperature. The seasonal variation in soil temperature within the first few meters of the surface is an important consideration in building foundation design and in the specification of ground source heat pumps (see §32 (Geothermal energy)).
Figure 6.15 A map of annual mean ground surface temperatures in the contiguous US; the small isolated region in the western US indicates an area with high geothermal heat flux (§32). (Credit: Tester, J.W. et al., The Future of Geothermal Energy)
As a simple model of annual fluctuations in soil temperature, we assume that the temperature at the soil/air interface varies sinusoidally over the year with an average of and a maximum variation of ,
(6.24)
where /y. (In §36 (Systems) we fit empirical data to show that this is a good approximation to seasonal variations in surface temperature.) We neglect diurnal (daily) fluctuations for now, but return to this issue at the end of this analysis. We model the soil as a semi-infinite mass of material with thermal diffusivity a, beginning at the surface, where , and running to arbitrarily large (positive) depth z. We assume that there is no net heat flux into or out of the deep earth; the depth at which this assumption breaks down depends upon the local geothermal heat flux, which is discussed further in §32 (see also Problem 6.15). We also assume that the temperature distribution does not depend on the coordinates x and y that run parallel to Earth’s surface. So we need to solve the one-dimensional heat equation
(6.25)
subject to the condition that is given by eq. (6.24). The general theory of differential equations is beyond the level of this book; we proceed simply by trying a form for the solution that captures some features of the solutions discussed in the previous section: we look for a solution that oscillates in time with frequency ω and decays exponentially with z. Because we expect that the temperature at depth lags behind the surface temperature that is driving it, we anticipate that the phase of the time oscillation will vary with depth in the soil. All of this leads to a guess of the form,
(6.26)
where α and c are parameters adjusted to solve eq. (6.25). Indeed, eq. (6.26) satisfies eq. (6.25) if
(6.27)
Box 6.2 Solving the Heat Equation Numerically
Generally, we wish to solve for the temperature T in a medium of fixed extent. In this case, in addition to the initial temperature distribution, we need to know the behavior of the temperature or the heat flux atthe boundaries of the medium – information known as boundary conditions. When boundary conditions are complicated, or when a is spatially dependent, it is often desirable to solve eq. (6.22) numerically. Tremendous amounts of computer time (and therefore energy) are consumed worldwide in solving such partial differential equations. The fact that the heat equation involves only one time derivative simplifies this task. Knowledge of at any instant in time is sufficient to determine the temperature function at all future times. In contrast, an equation that involves a second time derivative, such as Newton’s Law, , requires knowledge of both x and at an initial time.
Suppose is known for all x. Then the derivatives that appear in can be computed (in Cartesian coordinates, for example, ), and eq. (6.22) can be used to compute the temperature at a slightly later time, ,
If is sufficiently small, then the higher-order terms can be neglected in a numerical solution. In practice, can be computed approximately on a mesh of lattice points, , where are integers and δ is a small lattice spacing. In this case, a simple discrete update rule can be used to compute the leading approximation to eq. (6.22) in the limit of small and δ. Writing as , we have
