MODELING
OF A SIMPLE BUILDING EXPOSED TO PERIODIC TEMPERATURE CHANGES
Consider the
simple model of a building in the form of a rectangular box, with identical
walls, exposed to the influence of the external temperature Te. One-dimensional heat
transfer through the walls is assumed. The building is ventilated; the air
exchange velocity is constant in time. All other effects are neglected.
Let the
external temperature Te be
a harmonic function of time, with angular frequency w and amplitude ATe. The steady-state periodic temperature Ti is also a harmonic
function of time with angular frequency w but amplitude ATi and some time shift tTi
of the maximum with respect to the maximum of
Te:
. (19)
The lower the value of the ATi/ ATe
ratio, the better the thermal stability of the system.
The equation of
the heat balance for this system has the following form:
,
(20)
where qi is
the heat flux across the internal surfaces of walls, Sw is the total surface area of the walls, CV = r cpV
is the air volume thermal capacity, n
(h−1) is the air exchange frequency.
Solving Eq.
(20) with respect to Ti,
with qi given by Eq. (17),
the following expression is obtained:
.
(21)
The response function 1/B is in the numerator and D/B is in the denominator, both multiplied
by the area of the walls. Therefore, in general, the amplitude of the
temperature Ti increases with the amplitude of 1/B and decreases with the amplitude of D/B.
A simple recipe for good thermal stability of this system is thus a low
response to external temperature variations and high response to internal
temperature variations.
Values of the
amplitude ratio ATi/ATe and the time shift tTi calculated for walls 1–6, with room
dimensions of 4.5 ´ 4.5 ´ 2.7 m and an air exchange
frequency n = 1 (h-1), are presented in Table 3.
Table 3. Amplitude ratio and time shift of the
internal and external temperature
oscillations for the one-room building with walls 1–6
(shown in Figure 1)
|
Wall no. |
Layers (in.)a |
ATi /ATe |
tTi |
|
Gypsum—concrete—insulation—concrete—stucco |
|||
|
1 |
½–3–4–3–¾ |
0.040 |
−2.878 |
|
2 |
½–4–4–2–¾ |
0.041 |
−2.490 |
|
3 |
½–6–4–0–¾ |
0.047 |
-−1.996 |
|
Gypsum–insulation–concrete–insulation–stucco |
|||
|
4 |
½–4–6–0–¾ |
0.222 |
−5.330 |
|
5 |
½–1–6–3–¾ |
0.142 |
−2.087 |
|
6 |
½–2–6–2–¾ |
0.184 |
−2.880 |
|
Homogeneous core |
½–10–¾ |
0.094 |
−2.101 |
a 1 in. = 25.4 mm
Results of the analysis shown in Table 3 for the
simple building model indicate that buildings having walls with massive
concrete inside layers are stable. The amplitude ratio ATi/ATe
for walls 1–3, with high values of the structure factor jii and admittance response amplitude, is
about 5.5 times lower than for wall 4 and about 4 times lower than for wall 6,
with low values of jii
and D/B amplitude. Wall 5 is again an exception; a comparatively high
value of jii,
which does not guarantee a high value of the admittance response amplitude
(Table 2), also does not guarantee a low value of the internal and
external temperature amplitude ratio.
High thermal
stability of the building may reduce heating and cooling loads, especially when
outdoor conditions are not far from thermal comfort conditions. Analysis of the
simple model indicates that a high value of the internal admittance amplitude
for exterior walls is more important for stability than a low value of the
transmittance amplitude. Modifying the model by adding massive interior walls
and changing the air exchange velocity has no effect on this general
conclusion.
Figure 5
depicts the dependence of the internal and external temperature amplitude ratio
on the thermal mass factor Cjii, calculated using Eq. (20), for the representative
set of walls from the ASHRAE
Handbook—Fundamentals [18, 19]. Walls 1–6 are also included. ATi/ATe decreases with Cjii down to its value of about
200 kJ/m2K. This is the level of Cjii
at which the admittance response amplitude stops increasing (see
Figure 4). Like the D/B amplitude, ATi/ATe is
approximately constant for Cjii > 200 kJ/m2K.
