4. MODELS FOR THE THERMAL BALANCE OF A ROOM
4.6 Models based on the thermal response of the room
4.6.2 Method for the transfer functions of the room
The storage factor depends on the duration of the presence of the internal gains and on the specific mass of the room MR. The storage factor begins (time step 0, as reported in Figure 4.35) when the internal gain starts.
Figure 4.35: Storage factor for 10 hours internal gains in a room as a function of the thermal capacity of the room
The method is easy to implement and has been widely used. Some lacks in accuracy derives by the fact that the mass involved in dumping the solar radiation and in storing the internal gains do not consider the position of insulation material on the outer walls. A more detailed method is the one presented hereafter.
4.6.2 Method for the transfer functions of the room
This method is a special case of the calculation of heat flow through building components. This approach uses sol-air temperature to represent outdoor conditions, and an assumed constant indoor air temperature ta.
Furthermore, both indoor and outdoor surface heat transfer coefficients are assumed constant. Thus, the heat gain through a wall or roof (HGd) at current time step is given by:
109
where Si is the indoor surface area of a wall or roof, bj,cj ,dj are conduction transfer function coefficients, n is the number of exterior walls and T the number of factors for the transfer function of the wall.
Stephenson and Mitalas [31], Mitalas and Stephenson [63], and Kimura and Stephenson [64] related Heat Gain (HG) to the corresponding Cooling Load (CL) by a Room Transfer Function (RTF), which depends on the nature of the heat gain and on the heat storage characteristics of the space (i.e., of the walls, floor, etc., that enclose the space, and of the contents of that space). Where the heat gain is given at equal time intervals, the corresponding cooling load at time τ can be related to the current value of heat gain and the preceding values of cooling load and heat gain by the following equation:
∑
where N is the number of heat gain components and M the number of relevant time steps to be considered. The terms vi and wi are the coefficients of the RTF:
which relates the transform of the corresponding parts of the cooling load and of the heat gain. These coefficients depend on the size of the time interval ∆τ between successive values of heat gain and cooling load, on the nature of the heat gain (how much is in the form of radiation and where it is absorbed), and on the heat storage capacity of the room and its contents. Therefore, different RTFs are used to convert each distinct heat gain component to cooling load.
The RTF procedure distributes all heat gained during a 24-h period throughout that period in the conversion to cooling load. Thus, individual heat gain components rarely appear at full value as part of the cooling load unless representing a constant 24-h input.
This concept is further complicated by the premise of “constant interior space temperature” (i.e., operation of an HVAC system 24 h a day, seven days a week with fixed control settings). The effect of intermittent system operation is seen primarily during the first hours of operation for a subsequent day, see again Figure 4.33, and can impact equipment size selection significantly.
Finally, the total cooling load for a space can be calculated by simple addition of the individual components, which is slightly incorrect in a theoretical sense. However, means for compensation for these limitations fall within the range of acceptable error that must be expected in any estimate of cooling load.
To obtain appropriate room transfer function data for use in equation (4.97), the value of w1 has to be selected for the approximate space envelope construction and range of air circulation, and the values of v0 and v1 have to be derived based on the appropriate heat gain component and range of space construction mass.
Sensible cooling load from strictly convective heat gain elements is instantaneous, hence it is added directly to the results via the following equation:
C g
c q q
CL = + (4.99)
110
where qC and qg are calculated by means of equations (4.32) and (4.33).
The method assumes the maintenance of a constant interior temperature and the hourly total removal of all cooling load entering the space.
For allowing TFM in predicting temperature swings in the space and the ability of cooling equipment to extract heat when operated in a building with extended off cycles (nights and weekends) as well as any loss back to the environment.
The cooling loads determined by the TFM serve as input data for estimating the resultant room air temperature and the heat extraction rate (ER) with a particular type and size of cooling unit, or set of operating conditions, or both. In addition, the characteristics of the cooling unit (i.e., heat extraction rate versus room air temperature), the schedule of operation, and a Space Air Transfer Function (SATF) for the room that relates room air temperature and heat extraction rate must also be included to run these calculations.
The heat extraction characteristics of the cooling unit can be approximated by a linear expression of the form:
ta
S W
ER = + ⋅ (4.100)
where W and S are parameters characterizing performance of specific types of cooling equipment.
This linear relationship only holds when ta is within the throttling range of the control system. When ta lies outside of this range, ER has the value of either ERmax or ERmin, depending on whether the temperature ta is above or below the throttling range. The value of S is the difference ERmax - ERmin divided by the width of the throttling range, and W is the value ER would have if the straightline relationship between it and ta held at ta1 = 0. This intercept depends on the set point temperature of the control system (ta*), which may be taken as the temperature at the middle of the throttling range:
2 *
min
max ER S ta
W ER + + ⋅
= (4.101)
The heat extraction rate and the room air temperature are related by the Space Air Transfer Function (SATF):
( ) ∑ ( )
∑
= − ∆= −∆ − − ∆ = 2 −
0
, 1
0 r
r a ac r r
r
r ER CL g t t
p τ τ τ (4.102)
where gr and pr are the SATF coefficients, and CL is the calculated cooling load for the room, based on an assumed constant room temperature of tac. Normalized values of g and p depend on the thermal inertia of the constructions.
In calculating the design cooling load components previously described, it was assumed that all energy transferred into the space eventually appears as space cooling load.
However, this is not quite true over an extended period, because a fraction of the input energy can instead be lost back to the surroundings. This fraction Fc [-] depends on the thermal conductance between the space air and the surroundings and can be estimated as a function of the unit length conductance between the space air and surroundings K [W/(m K)] by means of the following equations:
K
Fc =1−0.0116 ⋅ (4.103)
111
Where LF is the length of space exterior wall, subscripts R for roof, W for window, OW for outside wall, and P for partition.
Depending on heat loss to surroundings or not, and using the appropriate values of gr and pr, equations (4.100) and (4.101) can be solved simultaneously for ER:
0
The space air transfer function coefficients are calculated based on the appropriate space envelope construction as a function of the floor area Sf, on the total conductance K between space air and surroundings, on the ventilation rate, and on infiltration rate.
If the value of ER calculated by Equation (4.105) is greater than ERmax, it is made equal to ERmax; if it is less than ERmin, it is made equal to ERmin. Then air temperature is calculated from the expression:
g0
ER ta I−
= (4.107)