Solar energy use in buildings - Solar Energy Sciences and Engineering Applications - page 605

Thus, the energy balance equation leads to:

x o = x o −

dx 2 x o

λA d 2 T

λA dT
dx

ρdVc dT
dt

λA dT
dx

− − dx

(18.8.10)

d 2 T
dx 2

λ
ρc

dT
dt

=

ρc [ m 2 s ] is termed the thermal diffusivity, which lies between 10 − 7 m 2 s − 1 for

wood and 10 − 4 m 2 s − 1 for metals.

λ

where a =

18.9 COMPONENTTEMPERATURES FOR SUDDEN
TEMPERATURE INCREASES

The differential equation can be solved by a product approach for periodic bound-
ary conditions or with temperature-equalizing processes, wherein one function is
dependent only on time and the other only on place.
The heat flow i s proportional to the so-called heat penetration coefficient b = √ λρc

and falls with 1 / √ t as shown in Figure 18.9.1.
The integration of the heat flow over time results in the total amount of heat
penetrating into the component when there is a surface temperature jump.

dt

λρc
π T c

t 0

√ π λ ρ c

√ t 0 T c

Q
A =

dQ
A

1
√ t

2

= − =− (18.9.1)

b

0

Figure 18.9.2 shows that the amount of energy stored within a component is, like
the heat flow, directly proportional to the heat penetration coefficient b and to the
temperature jump T at the surface, but it also rises with the root of time.

Figure 18.9.1 Heat flux as a function of time for a concrete and a wooden floor with a 10 K temperature
jump on the surface.