3. HEAT CONDUCTION IN BUILDING ELEMENTS
3.2 Thermal conduction in steady state conditions in a linear element
∂ =
∂ ρ τ
1 (3.13)
where x is the axe parallel to the direction of the propagating heat flux.
In the following paragraphs the bodies are assumed without internal heat generation.
First the case of one dimensional problem is shown and then the two dimensional problem is presented. In fact the one dimensional problem can be used for usual building structures. On the other hand the two dimensional approach has to be studied when looking at thermal bridges or if the building element contains pipes.
3.2 Thermal conduction in steady state conditions in a linear element
Let us consider a piece of material with parallel surfaces as usual happens in a wall section. As shown in Figure 3.2a, considering steady state conditions, integrating equation 3.13 leads to the following equation:
(
tsi tse)
x
q k −
= ∆
* (3.14)
As can be seen, it is evident an analogy between the heat transfer problem and the electrical Ohm equation, as will be discussed later. In equation (3.14) the ratio k /∆x = Γ, named thermal conductance; the same problem can be solved by using the thermal resistance R = ∆x/k. When a wall is made of different material layers, equation (3.14) can be written as:
∑
= −
j j
se si
R t
q* t (3.15)
a b
Figure 3.2: Heat conduction through a building element: from inner surface to outer surface (a) and from indoor temperature and outdoor temperature (b)
Sometimes constructions may not be built with uniform and homogeneous materials; in these cases the thermal resistance has to be evaluated by means of laboratory tests. In
Q = 0 W Q = 0 W
t si
t se
Q = 0 W Q = 0 W
t i
t se t e
t si
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any case, once determined the thermal resistance or conductance of the building element and the thickness ∆x, the equivalent conductivity k* of the building element can be used.
3.2.1 Thermal transmittance of a building component
As described in Chapters 2 and 4, in the problems of heating/cooling buildings usually the indoor temperature ti and the outdoor temperature tamb are used. This means that the overall balance of the inner surface has to deal with both convection with the air and infrared radiation with the other surfaces. Details on these two problems are reported in Chapter 4. Looking at the heat flow through a building element in steady state conditions means that the indoor temperature is fixed and the outdoor temperature is constant (see 2.3.1.1) and neither solar radiation nor internal gains are present. In this case the overall surface balance can be estimated by an overall surface heat exchange coefficient hsi (or via the reciprocal internal resistance Rsi):
( )
The same problem has to be solved on the external surface of the wall. In this case there are two possible ways as described in Chapter 4.5. In the simplest model the infrared
In this way a wall can be schematized as a series of thermal resistances and the heat flux can be estimated as:
tot
The reciprocal of the overall thermal resistance is named transmittance of the wall, known also as U-value: U = 1/Rtot, therefore the heat flux through a linear wall can be expressed as: transfer which cannot be properly solved via the simplifications adopted in 3.2.1. In this case the temperature and the heat transfer inside the building element are difficult to analyse since it is characterized by multi-dimensional heat transfer (Figures 3.3 and 3.4), and therefore they cannot be adequately approximated by the one-dimensional models (U-values).
Usually thermal bridges are associated to an extra flux compared to the heat flow calculated through U-values method and, as a consequence, the internal surface of the thermal bridge presents a lowering in temperature, which is called minimum surface temperatre tsi.
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Both the problems can be solved by means of the use of 2-D FDM (3.3.2.1) or FEM (3.3.2.2) methods, as described afterwards. In particular, the local minimum temperature may lead to moisture problems, which are known as surface condensation.
3.2.2.1 The problem of the heat flux through a thermal bridge
Let us consider a corner as shown in Figure 3.5. The overall heat flowing through the considered element can be calculated by means of a 2-D numerical model imposing the proper boundary condition, i.e.:
• Internal surface: internal surface resistance Rsi and ti = 1°C
• External surface: external surface resistance Rse and tamb = 0°C.
Usually the adiabatic surfaces have to be imposed at least 0.8 m far away from the zone where the thermal bridge is present (usually 1 m is considered), in order to avoid to affect simulation results.
a b
Figure 3.3: Thermal bridge due to a corner: materials (a) and temperature distribution (b)
a b
Figure 3.4: Thermal bridge due to a pillar: materials (a) and temperature distribution (b) Once the simulation has been run, L2D represents the overall specific heat flow exiting the thermal bridge [W/(m K)]. This includes the heat flows calculated via the use of the one-dimensional models (U-values) and the extra-flux ψy. The following equation can be therefore written:
y y y z yz y x xy
D U l l U l l l
L2 = + +ψ (3.20)
Considering ly = 1 m, as usual can be assumed when dealing with 2-D simulations (Figure 3.5b):
y z yz x D Uxyl U l
L2 = + +ψ (3.21)
leading hence to:
53 z
yz x D xy
y =L2 −U l −U l
ψ (3.22)
As can be seen, depending on the dimensions chosen as reference for the calculations (lx and lz), the following definitions apply:
• ψe: external dimensions
• ψi: internal net dimensions
• ψoi: internal gross dimensions
a b
Figure 3.5: Example of thermal bridge: a section of the corner (a) and boundary conditions for the simulation (b)
3.2.2.2 The problem of surface condensation on a thermal bridge
The problem of surface condensation is related on both the minimum surface temperature and the indoor conditions in terms of air temperature and relative humidity [25].
As for the minimum temperature the following parameter is defined (temperature factor):
amb i
amb Rsi tsi t
t f t
−
= − (3.23)
As can be observed, the temperature factor may vary between 0 (if the minimum internal temperature is equal to the outdoor temperature, i.e. no insulation is present) and 1 (minimum internal surface temperature equal to indoor temperature, i.e. perfect insulation which leads to no heat flux on the outer wall).
Once carried out the simulation of the thermal bridge in the 2-D numerical model, the minimum temperature on the internal surface (again Figure 3.5b), by imposing 1°C difference between the outdoor temperature and the internal temperature, the minimum surface temperature of the simulation is equal to fRsi.
For checking if surface condensation occurs, the indoor conditions have to be evaluated according to equation (4.34). Based on specific humidity conditions, the surface condensation has to be evaluated for the critical RHcr = 1 in the case of light structures and for the critical RHcr = 0.8 in the case of massive structures (Figure 3.6). In the first case the average minimum outdoor temperature has to be set as boundaqry condition; in
l x
X Y
l z
l y
t i
R si, t si
R se, t se
t amb
adiabatic
adiabatic
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the second case the calculation has to be set month by month and the average month conditions have to be considered.
Once calculated ξi, pi can be calculated by means of the equation (2.36). The saturation pressure psat,cr corresponding to the critical relative humidity RHcr has to be calculated as:
cr cr i
sat RH
p , = p (3.24)
Once calculated the saturation pressure psat, the correspondent temperature can be calculated by means of the equations (2.37) and (2.38).
The temperature calculated in this way is the minimum internal surface temperature tsi,min which leads to the minimum surface temperature factor:
e i
e Rsi si
t t
t f t
−
= ,min−
min
, (3.25)
The temperature factor calculated with the 2-D simulation has to be greater than the minimum surface temperature factor.
Figure 3.6: Surface moisture problem in a thermal bridge 3.2.3 Building elements containing pipes
In the case of building elements containing pipes the simulation has to be carried out by means of a detailed 2-D calculation method. Both FDM and FEM methods can be used [26], ensuring accurate results.
For defining the boundary conditions, as can be seen in Figure 3.7, it can be assumed that the temperature difference of the water between two adjacent pipes is negligible (t1 = t2 = t3), hence the section between two pipes can be considered adiabatic.
As for the other conditions, usually the convective heat exchange coefficient of the water inside the pipes can be considered negligible.
As for the surface heat exchange coefficients above and below the radiant system, the values reported in Table 3.1.
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Figure 3.7: Assumptions for determining the thermal field inside building elements containing pipes
Table 3.1: Overall heat exchange coefficients on upward and downward surfaces Heating Cooling
Floor 8.92 ∙ (ts-ti)0.1 7
Wall 8 8
Ceiling 6 8.92 ∙ (ti-ts)0.1