# The Heat Balance Method

The heat balance method is able to provide dynamic simulations of the building load. It is the foundation for all calculation methods that can be used to estimate the heating and cooling loads. Since all energy flows in each zone must be balanced, a set of energy balance equations for the zone air and the interior and exterior surfaces of each wall, roof, and floor must be solved simultaneously. The energy balance method combines various equations, such as equations for transient conduction heat transfer through walls and roofs, algorithms or data for weather conditions, and internal heat gains.

The method can be illustrated by considering a zone consisting of six surfaces, four walls, a roof, and a floor. The zone receives energy from solar radiation coming through windows, heat conducted through exterior walls and the roof, and internal heat gains due to lighting, equipment, and occupants. The heat balance on each of the six surfaces is generally represented by

4¡,e = hci(t a,e t¡,e) + £ gj (tj,e - t¡,e) A + 4si,e + 4u,e + 4ra,e j=1,j .

where qi,9 = rate of heat conducted into surface i at the inside surface at time 9 (W). i = surface number (1 to 6). ns = number of surfaces in the room. Ai = area of surface i (m2).

hci = convective heat transfer coefficient at interior of surface i (W/m2-K). gj = linearized radiation heat transfer factor between interior surface i and interior surface j (W/m2-K). tafi = inside air temperature at time 9 (°C). tjfi = average temperature of interior surface i at time 9 (°C). tj,9 = average temperature of interior surface j at time 9 (°C). qsi,e = rate of solar heat coming through the windows and absorbed by surface i at time 9 (W).

qli e = rate of heat from the lighting absorbed by surface i at time 9 (W). qei,0 = rate of heat from equipment and occupants absorbed by surface i at time 9 (W).

The equations governing conduction within the six surfaces cannot be solved independent of Eq. (6.1), since the energy exchanges occurring within the room affect the inside surface conditions, which in turn affect the internal conduction. Consequently, the aforementioned six formulations of Eq. (6.1) must be solved simultaneously with the equations governing conduction within the six surfaces to calculate the space thermal load. Among the possible ways to model this process are numerical finite element and time series methods. Most commonly, due to the greater computational speed and little loss of generality, conduction within the structural elements is formulated using conduction transfer functions (CTFs) in the general form

qifi = ^ Yk,m tofi-m+1 ~ ^ Zk,m tofi-m+1 + ^ Fm qifi-m (6.2)

where i = inside surface subscript. k = order of CTF. m = time index variable. M = the number of nonzero CTF values. o = outside surface subscript. t = temperature (°C). 9 = time.

Y = cross CTF values. Z = interior CTF values. Fm = flux history coefficients.

Conduction transfer function coefficients generally are referred to as response factors and depend on the physical properties of the wall or roof materials and the scheme used for calculating them. These coefficients relate an output function at a given time to the value of one or more driving functions at a given time and at a set period immediately preceding (ASHRAE, 2005). The Y (cross CTF) values refer to the current and previous flow of energy through the wall due to the outside conditions, the Z (interior CTF) values refer to the internal space conditions, and the Fm (flux history) coefficients refer to the current and previous heat flux to zone.

Equation (6.2), which utilizes the transfer function concept, is a simplification of the strict heat balance calculation procedure, which could be used in this case for calculating conduction heat transfer.

It must be noted that the interior surface temperature i;,e is present in both Eqs. (6.1) and (6.2), and therefore a simultaneous solution is required. In addition, the equation representing the energy balance on the zone air must also be solved simultaneously. This can be calculated from the cooling load equation:

^hci(ti,e ta,e) A + pCpQi,e(to,e ta,e) + pCpQv,e(tv,e ta,e) i=i

where tafi = inside air temperature at time 9 (°C). tofi = outdoor air temperature at time 9 (°C). tvfi = ventilation air temperature at time 9 (°C). p = air density (kg/m3). cp = specific heat of air (J/kg-K).

Qifi = volume flow rate of outdoor air infiltrating into the room at time 9 (m3/s).

Qv0 = volume rate of flow of ventilation air at time 9 (m3/s).

Qs0 = rate of solar heat coming through the windows and convected into the room air at time 9 (W). ql9 = rate of heat from the lights convected into the room air at time 9 (W). qe9 = rate of heat from equipment and occupants convected into the room air at time 9 (W).