Advanced Building Simulation

64 Degelman
3.4 Model for temperatures
3.4.1 Deterministic model
The modeling of temperatures uses both deterministic methods and stochastic methods.
The deterministic portion is the shape of the diurnal pattern. This shape is fairly
consistent from day to day as shown in Figure 3.1, even though the values of the
peaks and valleys will vary.
After the morning low temperature (Tmin) and the afternoon high temperature
(Tmax) are known, hourly values along the curve can be closely estimated by fitting a
sinusoidal curve between the two end-points. Likewise, after the next morning’s low
temperature (Tmin1) is known, a second curve can be fit between those two endpoints.
Derivation of the hourly values then can be done by the following equations.
From sunrise to 3:00 p.m.
T t�T ave0�(�T/2) cos[�(t�t R)/(15�t R)] (3.1)
where, T t is the temperature at time t; T ave0, the average morning temperature, (T min�
T max)/2; �T, the diurnal temperature range, (T max�T min); �, the universal value of
PI �3.1415927; t R, the time of sunrise; and 15, the hour of maximum temperature
occurrence (used as 3:00 p.m.).
From 3:00 p.m. to midnight
T t�T ave1�(�T�/2) cos[�(t�15)/(t R��9)] (3.2)
where T t, is the temperature at time t; T ave1, the average evening/night temperature,
(T max�T min1)/2; �T�, the evening temperature drop, (T max�T min1); and t R�, the time
of sunrise on next day.
From midnight to sunrise the next day
T t�T ave1�(�T�/2) cos[�(t�9)/(t R��9)] (3.3)
The time step can be chosen to be any value. Most energy simulation software uses
1-h time steps, but this can be refined to 1-min steps if high precision is required. The
thermal time lag of the building mass usually exceeds 1h, so it is not necessary to use
a finer time step than 1h; however, a finer time step may be desirable when simulating
T ave = T min + T max
2
T max
Tmin Tmin1 Sunrise 3:00 p.m. Sunset Sunrise
Figure 3.1 Characteristic shape of the daily temperature profile.
This symbol indicates
statistically-derived
max–min data points.