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10 Heat and Matter
Fig. 10.7 Heat flow in the
composite slab made of two
slabs in series
dQ 1
dt
=
dQ 2
dt
=
dQ
dt
(3)
Using (3) in (1) and (2)
T 1 − T =
−d 1
k 1 A
dQ
dt
(4)
T − T 2 =
−d 2
k 2 A
dQ
dt
(5)
Adding (4) and (5)
T 1 − T 2 =
−1
A
d 1
k 1
+
d 2
k 2
dQ
dt
(6)
or
−
dQ
dt
=
A(T 1 − T 2 )
d 1
k 1
+
d 2
k 2
(7)
(b) Rewriting (7)
−
dQ
dt
=
A(T 1 − T 2 )
d
d
d 1
k 1
+
d 2
k 2
=
A(T 1 − T 2 )k
d
(8)
with d = d 1 + d 2 , and the equivalent conductivity
k eq =
d 1 + d 2
d 1
k 1
+
d 2
k 2
(9)
Formula (9) can be generalized to any number of slabs in series.
k =
d i
d i
k i
(10)
(c) Eliminating dQ/dt between (4) and (5)
T =
(k 1 T 1 /d 1 ) + (k 2 T 2 /d 2 )
(k 1 /d 1 ) + (k 2 /d 2 )
(11)