4. MODELS FOR THE THERMAL BALANCE OF A ROOM
4.6 Models based on the thermal response of the room
4.6.3 Method of the periodic transfer functions
This method permits to in the equation (4.31) to calculate all values of Σqc,i, without recurring to the detailed thermal balance of the room, as described in chapter 4.3.1. The method here described is valid if the room is subject to a periodic thermal behaviour 24 h long. The method also starts from considering the superposition effect, i.e. the different convective heat gains can be considered separately. For this purpose the concept of transfer function can be applied.
It is possible to define periodic transfer functions, where the unitary solicitation is repeated according to the period of the solicitation. In building simulations the time step is usually ∆τ=1h and the period is one day, hence the amount of response factors Dj is 24. Based on the these assumptions response of a system to a periodic solicitation in a certain instant τ can be defined as:
The transfer functions depend on The U-value of the walls, on their specific mass mf and on two parameters defined effective primary (µp) and secondary (µs) mass.
Defined n the overall number of walls and d the amount of walls facing outwards, the parameters mean thermal transmittance Um, average primary specific effective mass Mp
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The coefficient fc is defined as:
019 2 with respect to the overall internal heat exchange coefficient.
It has to be underlined that bz are calculated for a factor Fs = 1, i.e. for µp = µs. For
where bz’ are calculated for Fs=1 as described above.
The radiant heat gains and due to solar radiation (qs) and internal gains (ql) are partly
113
where k = τ – z +1 if τ – z +1 ≥ 0 and k = 24 + τ – z +1 if τ – z +1 < 0.
The coefficient fr takes into account that part of absorbed solar radiation by the internal surface can be transferred towards outside; this coefficient can be calculated as:
025 2
. 0 248
. 0 000 .
1 m m
r U U
f = − ⋅ + ⋅ (4.116)
Also in this case the transfer functions uz are calculated under the condition Fs=1. In case Fs ≠ 1 the modified transfer functions can be calculated as:
23 ' )1 1 ( ' '
1 1
1
F u u
F u
u u
s z
s z
− − +
=
=
z =2,...,24 (4.117)
At the end, superposing the effects, the following equation can be written:
b d n
i
c q q
q = +
∑
=1(4.118)
where qd is described by equation (4.112) and qb by equation (4.115).
At the end equation (4.31) can be written in steady state conditions as:
=0 + + +
+ b C g p
d q q q q
q (4.119)
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