The original purpose of a building is to provide shelter and to maintain a comfortable or at least liveable internal temperature. Other purposes include security, privacy and protection from wind and weather. To feel comfortable in a thermal sense, a human has to be able to release a well-defined amount of **Heat**. If this gets difficult, a person will either feel cold or hot. The human body operates as a chemical reactor that converts chemical energy of food and respiratory oxygen into mechanical work and heat. Heat output can vary from about 100 W for a sedentary person to 1000 W for an exercising person [ASHRAE Fundamentals (1993)]. (See also Physiology and Heat Transfer.)

To maintain body temperature within a narrow band, the heat produced by an occupant must be released to the indoor environment. If too much heat is lost, room temperature should be increased or warmer clothes be worn. The heat transfer on the human skin, the indoor temperature and the heat transfer through the building envelope are factors that influence thermal comfort [Mayer (1991)].

Figure 1 shows schematically the ranges of temperature variations of the human body, of the room air and outdoor air. The adjustment of heat transfer around the human alone (by variation of clothing or sweating) is not normally sufficient to control body heat release at large outdoor temperature variations without the thermal protection of the building envelope and heating or cooling. The dynamic storage of heat in building components is important to control indoor temperature variations.

Heat is one form of energy. It is a “lower” form of energy because no system can convert heat fed to it by heat transfer into mechanical work completely and continuously. Heat is the ultimate form of energy because systems tend toward a state where all energy is transformed into heat. The burning of fuel in a boiler turns chemical energy into heat. Heat is generated by the process of combustion, but the total amount of energy is unchanged.

In a closed system, energy is conserved. The energy contained in a building is increased, for instance, by sunshine, by the supply of electricity and fuel for heating and other purposes, and also by food brought in by occupants for preparing meals. The total amount of enclosed energy is reduced by heat losses and other transport mechanisms listed below. In the long-term, the average energy content of a building is almost constant. Energy flows “in transit” through the control volume of a building. The balance of heat transfer and energy flow determines the temperature level at which the interior settles.

According to the *Second Law of Thermodynamics*, heat transfer is only possible in the direction from a higher temperatures to a lower one. It becomes zero if temperatures are equal. The heat loss through an envelope should therefore be proportional to the difference T_{inside} – T_{outside}, or to a positive power of it for small differences. For a simple formula, a linear dependence on temperature difference is sufficient. Accepting further that heat loss grows linearly with surface area A, one finds:

The constant of proportionality, U, is the Overall Heat Transfer Coefficient in W/(m^{2}K). In the example of Figure 2, a building is represented by a cube of 5m × 5m × 5m. If no heat is lost into the solid and with U = 0.4 W/(m^{2}K), the total heat loss is:

**Figure 2. Estimation of the heat loss of a two-storey building. At 20 K temperature difference, this cube loses 1000 W to the atmosphere at an assumed overall heat transfer coefficient of 0.4 W/(m ^{2}K), and if heat loss to the ground is neglected.**

Equation (1) suggests three ways to reduce heat loss: 1) As the heat loss is proportional to the inside-outside temperature difference, the set-point for the indoor temperature can be reduced during the heating season; 2) The insulation of the envelope can be improved to reduce the overall heat transfer coefficient U; and 3) If possible, the surface area should be reduced without changing the enclosed volume. A spherical igloo would be optimal, but a cubical shape is still better than an elongated building with many wings. The opposite is true for the design of heat exchanger surfaces or *fin-tubes*, where the effective surface should be maximized.

The cumulated amount of lost heat is the time integral of the instantaneous heat flow,

The quantity of heat, Q, is measured in J (Joule). In the construction sector, it is often converted into kWh (kW-hours). The fuel consumed for heating is roughly proportional to the difference between Q and the sum of internal heat gains from sun, occupants, lights, equipment, and so forth. Therefore, the time-average of the temperature difference ΔT_{bal} = T_{balance} – T_{outside} during the heating season is of importance for estimating heating cost. Here, T_{balance} is that outside temperature at which no heating is required to maintain a prescribed inside temperature at given internal heat gains [Chapter 28, ASHRAE (1993)].

The thermal performance of buildings can be compared on the basis of *degree-days* [ASHRAE (1993)]. Heating degree-days, DD_{h}, for a geographical site between given dates correspond to a time integral of ΔT_{bal} in which only positive values of the difference are counted.

In building Heat Transfer, many different types of energy transport are effective. Often, heat is transported by different modes to or from the same place. Energy that reaches a point via different paths and modes may be added up for the heat balance. For instance, the heat loss of a human body is the sum of convection, radiation, and latent heat released by sweating and so forth.

Primary heat transport modes are:

**Conduction**(heat flow on a molecular scale. Medium at rest or moving);**Convection**(heat conveyed as internal thermal energy of mass that is displaced by mean or turbulent motion);**Radiation**(heat transfer by electromagnetic waves such as infrared or visible light).

(See entries on Conduction, Convective Heat Transfer and Radiative Heat Transfer.)

In buildings, heat is also transported by the following mechanisms, which basically belong to the convective mode:

Transfer of latent heat by transport of water or water vapor.

Thermal energy associated with the air replaced in a building by ventilation or by air leakage (infiltration).

Thermal energy associated with fresh and used domestic water and combustion air (including flue gases), and fluids feeding Heat Pumps.

The transport of energy in the above list is limited to energy in the form of *sensible* or *latent* heat. A change of sensible heat is characterized by a change of temperature while a change of *latent* heat is associated with some mass altering its phase. Phases are *gaseous*, *liquid*, *solid*. Transport of energy in forms other than heat are not considered.

Heat transfer in buildings may involve the listed types of transport. For an energy balance, other forms of energy — often referred to as energy sources or heat loads — and dynamic (time-dependent) storage of heat in solid, liquid, or gaseous media have to be taken into account.

Equation (1) above may be useful for a rough overall energy balance, but not for a detailed description of energy flows in a real building. Energy transport in a building is often analyzed by a network model, Figure 3. To each flow path, a mode of energy transport may be assigned.

**Figure 3. Flow paths of energy transport in a building are represented as networks: a) Single path, heat flows from a warmer place (black symbol) to a colder place (shaded circle) along a flow path (rectangle); b) Heat flow across a roof may pass through several layers in series; c) Parallel paths as through several windows in the same wall; d) Example with a human (B), surface of clothes (s), a heater (H), the room air (A), and the inside (S) and outside (O) wall surfaces. The hatched flow links in d) represent radiation.**

In reality, the air in a building is a *continuum* and the building structure can be divided into regions that may be considered continuous in themselves. So the network model is far too simple for a true description of heat transfer in a building. However, the network approach is quite successful in simulation of thermal building dynamics. For an introduction, see, e.g., Clarke (1985).

In a *network*, as sketched in Figure 3, the energy balance must be satisfied at each node. This leads to a system of algebraic equations. Their solution is a set of node temperatures, which may vary with time or be constant in a state of equilibrium. In a continuum, the temperature field and the associated energy transport are defined by partial differential equations. These equations, which contain derivatives with respect to all spatial coordinates and the time, must be satisfied at each point within the continuum.

To calculate the temperature distribution in a continuum and velocity field in air, the domain is subdivided into small cells. Discretization produces large systems of difference equations that are solved by computers (finite-difference, finite-volume and finite-element methods).

Heat transfer is best described by the local Heat Flux, , which amounts to a heat flow density. It is the heat flow per unit cross section. In Figure 4 the heat flow rate, , that passes through an infinitesimal surface element dA is illustrated in two different situations: in (Figure 4a) the heat is transferred across an interface between different materials of the same or different phase as, e.g., from air to a wall; in (Figure 4b) heat flows within one medium, as within a solid body.

**Figure 4. Heat flux across an interface, a), and within a homogeneous medium, b), dA is the surface element;
, the heat flow rate.**

The local heat flux, measured in W/m^{2}, is:

The local heat flux, , is the basic quantity for the analysis of heat transfer. It is associated with a point on a surface, as in Figure 4a, or with a point and a direction, as the arrow dn in 4b. In mathematical terms, heat flux in a continuum is a vector with components in x, y, z directions:

The at an interface is the vector component normal to the surface. Heat flow rate, , across a finite area may be obtained from by integration or by multiplying the area with the mean value of heat flux over that area.

For the Conduction transfer mode, the heat flux in a continuum (Figure 4b) is:

with the thermal conductivity, λ, in W/(mK) and the gradient of the temperature field, T.

The heat transfer from room air to the wall surface is an example for the interface heat flux illustrated in Figure 4a. In this particular case, the interface is thought to comprise the surface and the boundary layer. This is a thin layer of air flow retarded by wall friction. Depending on room size, the boundary layers may have a thickness of several centimeters.

The local heat flux,
, in this example is a function of temperature T_{1} and T_{2} on either side of the interface and of properties of air and of the boundary layer. The air temperature is measured outside of the boundary layer, for instance, 0.1 m from the surface point where is determined.

The Heat Transfer Coefficient, α, is itself a function of T_{1}, T_{2} and of flow parameters. The relation at right defines α such that
is positive when heat flows from 1 to 2, and vanishes if the two temperatures are equal (Figure 4a). On the inside surface of a wall, α is of the order of 3 to 5 W/(m^{2}K) if it does not include radiation. Heat transfer coefficients for different situations have been measured and correlated by nondimensional parameters, as in Chapter 3 of ASHRAE (1993).

The transmission of heat through a building wall, Eq. (1), may now be considered as a network with three resistances in series (Figure 3b). The overall heat transfer coefficient, U, becomes:

where

These equations are obtained by combining Eqs. (6) and (7) and eliminating intermediate temperatures.

To reduce heat loss of a building, different modes of transport have to be considered. The example of Figure 2 does not account for thermal energy associated with the air replaced by ventilation and air leakage. This additional heat loss is now estimated for the example of Figure 2. At an air change rate of n = 1 h^{−1} for ventilation and leakage together (i.e., all the air in the building is, on the average, renewed once every hour), the ventilation heat loss is:

At a volume, V = 125 m^{3}; air density, ρ = 1.2 kg/m^{3}; specific heat capacity of air at constant pressure, c_{p} = 1000 J/(kgK); and a temperature difference of 20 K, the ventilation heat loss becomes:

This is of the same order as the transmission heat loss estimated at 1000 W. If only a small portion of the air change is caused by leakage, exhaust air heat recovery may reduce this loss.

The transport of latent heat by humid air can amount to a substantial heat loss [Hens (1991, 1995)]. The heat required to evaporate 1 kg water is 2500 kJ. Saturated air of 20°C holds about 0.016 kg water vapor per kg air. Condensation of this water releases about 0.016 × 2500 = 40 kJ latent heat (per kg air).

The ventilation heat loss was estimated above for *dry air*. For the same example, latent heat transport is computed for a relative humidity of 50% indoors and outdoors. The warm indoor air holds more water vapor than cold air. To maintain indoor relative humidity, water must be evaporated. This evaporation energy has to balance the loss of latent heat in the extract air. This additional heat loss is
= 550 W. In this example, latent heat loss accounts for about 40% of the ventilation loss.

Figure 5 shows an example of a faulty insulation layer between warm and cold surfaces. A small-scale but steady circulation of humid air may develop in cavities of building components, or between heated and unheated spaces of a house. These streams transport latent heat, that is released when vapor condenses on cool surfaces. (See also Humidity.)

In the heat transfer mode of Convection, moving packets of air transport energy as their internal thermal energy. The *“vector”* or carrier of energy is the air. In turbulent convection, the individual packets are small and their size is of the order of the length scale of the Turbulence. These are the turbulent eddies. Heat is also convected by the mean motion of turbulent or laminar flow. Depending on the driving forces for the motion of the packets, two types of convection are defined:

Heat transfer by forced convection (air motion is independent of heat transfer);

Heat transfer by natural or Free Convection (driven by buoyancy forces acting on heated or cooled air).

Heat transfer in a hair-dryer or in a room with mechanical ventilation is an example of forced convection; the upward flow that develops on a vertical radiator induces natural convection.

A third heat transfer mode active in a room is *radiation*. The heat exchanged by radiation between two black surfaces 1 and 2 is:

The geometric configuration factor F_{1–2} accounts for the distance between and for the relative orientation of the two surfaces and for the size of surface 2 [see Siegel and Howell (1992)]. The *Stefan–Boltzmann constant* is σ = 5.7 × 10^{−8} W/(m^{2}K^{4}). Each surface has a uniform surface temperature, T_{1} or T_{2}, respectively, that is measured in degree K. (See Radiative Heat Transfer.)

A simplified example shows the contributions to the heat transfer by radiation and convection between walls of a room (Figure 6). In the illustrated cavity, heat is exchanged between the opposing walls by radiation and natural convection. The other bounding surfaces of the cube are assumed to be perfect adiabatic mirrors. The radiative transfer may therefore be evaluated by Eq. (11) with a geometric configuration factor F1–2 = 1. According to a correlation by Henkes [Henkes (1990)], the turbulent free-convection heat flow rate in air is approximately proportional to the 4/3-power of the temperature difference. The heat flow rates in the example of Figure 6 are:

T_{1} |
T_{2} | (black body) | 1 → 2 | 2 → 1 | |

25°C | 20°C | 4.7 W | 29 W | 447 W | 418 W |

40°C | 20°C | 30 W | 126 W | 544 W | 418 W |

The radiation rate is given for perfectly black surfaces. For grey surfaces, the actual rate may be as low as 50% of the listed value. Even then, radiation accounts for two to three times the free-convection heat transfer. And the radiation in *one direction* is even larger and demonstrates that radiation — even at room temperatures — acts as a “short-cut” transfer mechanism between walls. The radiation from far surfaces to the human skin is of significance for thermal comfort.

**Figure 6. A cube with edges of 1 m has hot and cold opposing black walls. The other sides are reflective for thermal radiation and well insulated. The air in the cavity moves up near the warm surface and down at the cold surface. The heat transfer by natural convection and by radiation are compared.**

Longwave radiation may also be absorbed and emitted by a gas. A small portion of the radiation that crosses a room is absorbed by the room air and by water vapor, or CO_{2}. Depending on its temperature, the so-called participating gas emits radiation diffusely into all directions. Gas radiation may have an effect in air with high moisture content or in large rooms, and it may affect the transient reaction of air temperature to a sudden temperature change of a bounding surface. Moisture participates effectively at wave lengths of its absorption spectrum. For water vapor, each absorption line must be considered.

#### REFERENCES

ASHRAE (1993) ASHRAE Handbook, *1993 Fundamentals, SI Edition*; one of four volumes published periodically by the American Society of Heating, Refrigeration and Air-Conditioning Engineers, Inc., Atlanta, GA 30329, USA.

Clarke, J. A. (1985) *Energy Simulation in Building Design.* Adam Hilger Ltd., Bristol and Boston.

Henkes, R. A. W. M. (1990) *Natural-Convection Boundary Layer*, Ph.D. Thesis, Delft University of Technology, The Netherlands.

Hens, H. editor (1991) *Condensation and Energy, Volume 1: Sourcebook.* International Energy Agency, IEA, Energy Conservation in Buildings and Community Systems (BCS), Annex 14.

Hens, H. editor (1995) *Heat, Air and Moisture Transfer in Insulated Envelope Parts*, several volumes. International Energy Agency, IEA, Energy Conservation in Buildings and Community Systems (BCS), Annex 24.

Mayer, E. (1991) *Überprüfung einer neuen Bewertunsgrösse für Luftbewegungen in Räumen*, Bericht 3/1/34/91 der Forschungsvereingung für Luft- und Trocknungstechnik, Frankfurt.

Siegel, R. and Howell, J. R. (1992) *Thermal Radiation Heat Transfer*, 3rd edition, Hemisphere Publishing Corp., Washington DC 20005, USA.