10.1 Basic Concepts and Formulae
411
Rate of heat flow is directly proportional to the temperature gradient dT /dx and
the cross-sectional area and inversely proportional to the thickness. The constant of
proportionality k is known as thermal conductivity.
Heat conduction is through a composite slab made of two slabs of thickness d 1
and d 2 and thermal conductivity k 1 and k 2 , respectively, in series, the end temperatures being T 1 and T 2 . The equivalent conductivity is given by
k eq =
d 1 + d 2
d 1
k 1
+
d 2
k 2
(10.16)
The temperature of the interface is given by
T =
k 1 T 1
d 1
+
k 2 T 2
d 2
k 1
d 1
+
k 2
d 2
(10.17)
The rate of flow of heat in a composite slab made of n slabs in parallel:
−
dQ
dt
=
(T 1 − T 2 )
d
n
i=1
k i A i
(10.18)
k eq =
k i A i
A i
(10.19)
Convection
Rate of cooling:
−
dθ
dt
= C(θ − θ 0 )
(Newton’s law of cooling)
(10.20)
where θ is the mean temperature, θ 0 the room temperature and C a constant.
Radiation
Rate of energy loss:
−
dE
dt
= σ A
T
4
1 − T
4
2
(Stefan–Boltzmann formula)
(10.21)
where A is the area of the radiator, T 1 is its absolute temperature, T 2 is the absolute
temperature of the surroundings and σ is known as Stefan–Boltzmann constant.
Wien’s Law
λ m · T = b = 3 × 10
−3 m − K
(10.22)