1
Heat flow in composite rods – an old problem reconsidered
T. Kranjca)
Department of Physics and Technology, Faculty of Education,
University of Ljubljana, Kardeljeva ploscad 16, 1000 Ljubljana, Slovenia
and
J. Peternelj
Faculty of Civil and Geodetic Engineering,
University of Ljubljana, Jamova 2, 1001 Ljubljana, Slovenia
The interface temperature of two rods with equal cross section joined at one end and with
different initial temperatures, initially always acquires the value characteristic for two semiinfinite rods. This value, which is shown to be a consequence of energy conservation is, in
general, different from the thermal equilibrium temperature in finite rods. To illustrate this
result, two particular cases are discussed.
I. INTRODUCTION
In this brief paper we wish to examine certain aspects of the well known problem of linear
flow of heat in composite rods. The usual treatment of the problem gives a mathematical
solution but lacks a clear physical picture. It is our goal to clarify a certain point concerning
the time dependence of interface temperature and the approach to thermal equilibrium which,
at least to our knowledge, has not been addressed and clarified in the past. We give a clear
explanation together with a clean mathematical solution that confirms it.
Consider two semi-infinite rods of equal and uniform cross section joined at one end and
insulated on the sides. If the initial temperatures are T01 and T02 > T01, the corresponding
temperature distribution in the rods for t > 0 is well known to be1, 2 (for the benefit of the
reader a brief derivation is presented in the Appendix)
T1 ( x, t ) =
 −x 
κT01 + T02 T02 − T01
 , x < 0,
−
erf 
2 χ t 
κ +1
κ +1
1


(1a)
T2 ( x, t ) =
κT01 + T02 κ (T02 − T01 )  x 
+
erf
, x > 0,
2 χ t 
κ +1
κ +1
2 

(1b)
and
χ2
where k1 and k2 are thermal conductivities of the left (x < 0) and the right (x > 0)
χ1
rod, and χ1 and χ2 are the corresponding thermal diffusivities. From the above equations it
κ=
k1
k2
follows
lim T1 ( x, t ) = lim T2 ( x, t ) = T1 (0, t > 0) = T2 (0, t > 0) =
t →∞
t →∞
κT01 + T02
κ +1
(2)
2
which is saying that the interface temperature is constant and equal to the final equilibrium
temperature of the rods.
On the other hand, if the rods are of finite and equal length L, the equilibrium temperature,
obtained by an elementary calculation using the conservation of energy is
Teq =
ρ1c1T01 + ρ 2 c 2T02
=
ρ1c1 + ρ 2 c 2
(
)
χ 2 / χ 1 κT01 + T02
(
)
χ 2 / χ1 κ + 1
(3)
where ρi and ci are the density and the specific heat of the respective rod. This result which
does not depend on the length of the rods seems to be at odds with the corresponding result
for semi-infinite rods given by eq. (2).
II. TEMPERATURE DISTRIBUTION IN COMPOSITE RODS OF FINITE LENGTH
Consider two rods of lengths L1 and L2, and of equal cross-section aligned with the x-axis
and joined at x = 0. The left rod with the initial temperature T01 covers the interval −L1 < x <
0, and the right rod with the initial temperature T02 > T01 covers the interval 0 < x < L2.
Assuming that the rods are insulated on the sides implies that T = T(x, t). Consequently, the
temperature distribution is determined by the heat diffusion equation of the form
∂Ti
∂ 2T
= χ i 2i , i = 1, 2,
∂t
∂x
(4)
together with the initial conditions
T1(x, 0) = T01, T2(x, 0) = T02,
(5a)
the boundary conditions at the ends,
∂T1
∂x
=
x = − L1
∂T2
∂x
=0,
(5b)
x = L2
and the boundary conditions at the interface,
T1(0, t > 0) = T2(0, t > 0),
− k1
∂T1
∂x
= −k2
x =0
∂T2
∂x
.
(5c)
(5d)
x=0
Using the Laplace transformation3,
∞
~
T ( x, s) = ∫ e − st T ( x, t )dt ,
0
~
the heat diffusion equation in terms of the Laplace transform T ( x, s) is the following ordinary
inhomogeneous second order differential equation.
3
− χi
~
d 2Ti ( x, s)
~
~
− sT ( x, s ) i = −Ti ( x,0) , i = 1, 2.
2
dx
(6)
The corresponding initial and boundary conditions are
~
~
T1 ( x,0) = T01 , T2 ( x,0) = T02 ,
~
~
T1 (0, s ) = T2 (0, s) ,
~
~
dT1
dT
= 2
= 0,
dx x=− L
dx x= L
1
2
~
~
dT1
dT2
.
− k1
= −k2
dx x=0
dx x=0
(7a)
(7b)
(7c)
(7d)
The solution of (6) satisfying the initial and boundary conditions (7) is





s 
 L2 s 
cosh ( x + L1 )
sinh



χ 1 
χ 2 
T01  T02 − T01  
~



T1 ( x, s) =
+
⋅
,
s 
s








 
s
s
s
s








 cosh L1 χ  sinh  L2 χ  + κ sinh  L1 χ  cosh L2 χ  
1 
2 
1 
2 





–L1 < x < 0, (8a)





s 
 L1 s 
cosh ( x − L2 )
sinh



χ 2 
χ 1 
T02
~
 T02 − T01  



− κ
T2 ( x, s ) =
⋅
,
s
s









 
s
s
s
s








 cosh L1 χ  sinh  L2 χ  + κ sinh  L1 χ  cosh L2 χ  
1
2
1
2









0 < x < L2. (8b)
In particular, taking the limit L1, L2 → ∞ we get
T
T −T
~
~
lim T1 ( x, s ) ≡ T1 (∞ ) ( x, s) = 01 + 02 01 ⋅ e
L1 , L2 →∞
s
s(1 + κ )
s
χ1
x
−
T
T −T
~
~
lim T2 ( x, s ) ≡ T2( ∞ ) ( x, s) = 02 − κ 02 01 ⋅ e
L1 , L2 →∞
s
s (1 + κ )
,
(9a)
s
χ2
x
.
These limits represent the Laplace transforms of (1a) and (1b) respectively.
Applying the final value theorem for Laplace transforms3,
(9b)
4
~
lim T ( x, t ) = lim sT ( x, s) ,
t →∞
s →0
to the eqs. (9) we recover the result (2). When we apply this theorem to eqs. (8) we obtain
'
lim T1 ( x, t ) = lim T2 ( x, t ) ≡ T eq =
t →∞
t →∞
((L / L ) χ / χ )κT + T
((L / L ) χ / χ )κ + 1
1
2
1
2
1
2
2
01
02
,
(10)
1
χ1 / χ 2 .
which reduces to (3) for L1 = L2 and to (2) for L1/L2 =
To find the inverse Laplace transforms of (8a) and (8b) we use the residue theorem3, i.e.,
~
T ( x, t ) = ∑ residues of e stT ( x, s ) ,
~
where the sum runs over all poles of T ( x, s) .
L2 χ1
s
and iβ = L1
. From the expressions (8a) and (8b) we deduce that
L1 χ 2
χ1
~
~
T1 ( x, s) and T2 ( x, s ) are single-valued functions of s with simple poles at s = 0 and
Let σ =
s=−
χ1
L12
β m2 where ±βm, m = 1, 2, 3,..., are the roots of the equation
cos β sin σβ + κ sin β cos σβ = 0 ,
(11)
which is obtained by equating denominator of (8a) or (8b) to zero.
In what follows we will consider two cases only. Case 1 where we set L1 = L2 = L and
Case 2 where we choose L1 / L2 = χ1 / χ 2 .
Case 1:
Applying the residue theorem in this case (noting that in this case σ = χ1 / χ 2 ), it follows
∞
T1 ( x, t ) = Teq + 2(T02 − T01 )∑
m=1
e − χ1β mt / L sin σβ m cos β m (1 + x / L )
, (12a)
[(σ + κ ) cos β m cos σβ m − (1 + σκ ) sin β m sin σβ m ]
2
1
βm
⋅
2
and
e − χ1β mt / L sin β m cos σβ m ( x / L − 1)
T2 ( x, t ) = Teq − 2κ (T02 − T01 )∑
⋅
(12b)
m =1 β m [(σ + κ ) cos β m cos σβ m − (1 + σκ ) sin β m sin σβ m ]
∞
1
2
2
where Teq is defined by (3). Moreover, the eq. (11) can be rewritten as
sin β sin σβ (cot β + κ cot σβ ) = 0 .
The roots of (11) are thus seen to be the roots of equation
(13)
5
cot β + κ cot σβ = 0
(14a)
which are all real and simple4, together with the common roots of
sin β = 0, sin σβ = 0 .
(14b)
However, these common roots, if they exist, do not contribute to the sums (12a) and (12b)
because of the presence of sinβm and sinσβm factors in the numerators of these expressions.
Therefore the sums in (12) run only over the roots of eq. (14a).
Moreover, we can always approximate σ as a ratio of two integers namely, C = r/p. In this
case the roots of (14a) can be written as
βm → βν,n = βν + n(pπ), n = 1, 2, ...,
(15)
with βν representing the roots of (14a) on the interval from 0 to pπ. Taking into account (15)
and using (14a) we can write the temperature distribution in the rods in the final form as
T1(x, t) = Teq –
sin β cos βν sin 2 σβν
– 2(T02 – T01) ∑ 2 ν
2
ν sin σβν + κσ sin βν
2
2
sin βν sin σβν
+ 2(T02 – T01) ∑ 2
2
ν sin σβν + κσ sin βν
∞
∑
−
χ1t
−
χ1t
e
L2
( βν + npπ ) 2
n =0
∞
∑
e
L2
( βν + npπ ) 2
n =0
cos[(βν + npπ )( x / L)]
βν + npπ
sin[(βν + npπ )( x / L)]
βν + npπ
(16a)
and
T2(x, t) = Teq +
+ 2κσ (T02 – T01) ∑
ν
2
sin σβν cos σβν sin βν
sin 2 σβν + κσ sin 2 βν
sin 2 σβν sin 2 βν
+ 2κσ (T02 – T01) ∑ 2
2
ν sin σβν + κσ sin βν
−
∞
∑
χ 2t
e
L2
(σβν + nrπ ) 2
n =0
∞
∑
n =0
−
e
χ 2t
L2
(σβν + nrπ ) 2
cos[(σβν + nrπ )( x / L)]
σβν + nrπ
sin[(σβν + nrπ )( x / L)]
.
σβν + nrπ
(16b)
Case 2: Since L1 / χ1 = L2 / χ 2 , the results (8) simplify to the following expressions,

s
cosh  (x + L1 )
χ
T
T − T01
~
1

T1 ( x, s ) = 01 + 02
⋅
s
(κ + 1) s

s 
cosh  L1
χ1 




,
(17a)
6

s
cosh  ( x − L2 )
χ
κ (T02 − T01 )
T
~
2

T2 ( x, s ) = 02 −
⋅
s
(κ + 1) s


s 
cosh  L2

χ
2 




.
(17b)
The corresponding inverse Laplace transforms can be found in Ref. 3, p. 213, for example,
and are
(
)2
− n − 12 π 2 χ1t / L12
2(T02 − T01 ) 1 ∞ e
κT + T
T1 ( x, t ) = 01 02 +
⋅ ∑
κ +1
κ +1
π n=1 (n − 12 )
(

πx 
sin  (n − 12 )  ,
L1 

)2
− n− 12 π 2 χ 2t / L22
2κ (T02 − T01 ) 1 ∞ e
κT + T
T2 ( x, t ) = 01 02 +
⋅ ∑
κ +1
κ +1
π n=1 (n − 12 )

πx 
sin  (n − 12 )  .
L2 

(18a)
(18b)
III. NUMERICAL RESULTS AND DISCUSSION
From eqs. (1) and (18) it follows that the interface temperature is constant in the case of
semi-infinite rods and rods with their length ratio L1/L2 = χ1 / χ 2 , and is equal to the value
given by eq. (2). On the other hand, the interface temperature for the rods of equal length is,
for example,
∑
ν
T1 (0, t ) = Teq − 2(T02 − T01 )
2
sin βν cos βν sin σβν
sin 2 σβν + σκ sin 2 βν
∞
∑
n =0
−
e
χ1t
L2
(βν + npπ )2
βν + npπ
.
(19)
It is plotted in Fig. 1 as a function of χ1t/L2. We observe, however, that even in this case the
interface temperature whose initial value is again given by eq. (2) remains approximately
constant for a relatively large fraction of time L2/χ, which is the time characteristic for the
rods to reach thermal equilibrium. This initial and apparently general behavior of interface
temperature is understandable because for times such that χt << L2 the heat flow is
independent of the length of the rods. For semi-infinite rods this condition is clearly satisfied
at all times and, consequently, the interface temperature is thus rigorously constant. For finite
rods with arbitrary ratio of their lengths the interface temperature, of course, starts to change
with increasing time and eventually approaches the equilibrium value given by eq. (10). This
is illustrated in Fig. 2 showing the temperature distribution along the rods of equal lengths for
various values of time. For comparison, the temperature distribution in semi-infinite rods is
also shown in Fig. 3 and, for finite rods with the length ratio L1/L2 = χ1 / χ 2 , in Fig. 4.
7
Fig. 1. Temperature at the contact for the case of a copper (initial temperature T01 = 10 ºC) and
aluminum rod (initial temperature T02 = 100 ºC), both of length L = 1 m, as a function of χ1t/L2. The
upper horizontal line is the initial contact temperature T1(0, t > 0) = T2(0, t > 0) = 65.12 ºC given by
Eq. (2), the lower horizontal line is the final equilibrium temperature Teq = 44.85 ºC given by Eq. (3).
Fig. 2. Spatial distribution of temperature (T = T(x, t)) in a copper and an aluminum rod, both of length
L = 1 m, in contact at x = 0, for various times: t = 1 s, 100 s, 1000 s, 5000 s, 10000 s, 20000 s, 50000 s
and 100000 s. The initial temperature at the contact is T(0, t = 0+) = 65.12 ºC, the final (equilibrium)
temperature Teq = T(0, t → ∞) = 44.85 ºC.
8
Fig. 3. Spatial distribution of temperature (T = T(x, t)) in a copper and aluminum rod of semi-infinite
length, in contact at x = 0, for various times: t = 1 s, 100 s, 1000 s, 10000 s, 20000 s, 50000 s and
100000 s. The temperature at the contact (T(0, t > 0) is 65.12 ºC at all times.
Fig. 4. Spatial distribution of temperature (T = T(x, t)) in a copper and aluminum rod of length ratio
L1/L2 = χ1 / χ 2 , in contact at x = 0, for various times: t = 1 s, 100 s, 1000 s, 10000 s, 20000 s, 50000
s and 100000 s. The temperature at the contact does not change and is at all times 65.12 ºC which is
also the final equilibrium temperature of the rods.
On the basis of these results we conclude that the interface temperature is given initially,
in all cases, by the value obtained from eq. (2). Moreover, we have shown that this value
9
corresponds to the thermal equilibrium value Teq′ of two finite rods whose length ratio is L1/L2
= χ1 / χ 2 . Thai is,
Teq′ =
(
)
( L / L ) χ 2 / χ1 κT01 + T02 κT01 + T02
L1 ρ1c1T01 + L2 ρ 2 c2T02
,
= 1 2
=
L1 ρ1c1 + L2 ρ 2 c2
κ +1
(L1 / L2 ) χ 2 / χ1 κ + 1
(
)
(20)
in agreement with eq. (10). The condition that the interface temperature remains
approximately constant is χ1t << L12, χ2t << L22. When this condition ceases to be satisfied,
the interface temperature starts to change, except in the case of semi-infinite rods, where this
condition is never violated, and in the particular case of finite rods with their length ratio
adjusted in such a way that the equilibrium temperature Teq′ coincides with the initial interface
temperature.
The equilibrium temperature for semi-infinite rods must therefore be compared to the
equilibrium temperature of two finite rods of lengths L1 and L2 with their ratio equal to L1/L2
= χ1 / χ 2 . The transition to the limit L1, L2 → ∞, determined by the heat conduction process
itself, is such that this ratio is kept fixed.
Thus, when we join two rods or two bodies with different temperatures, small regions on
each side of the interface whose sizes are roughly of the order χ1t and χ 2t , respectively,
almost immediately reach the equilibrium temperature (20). The respective portion of the
body with initial temperature T02 > T01 cools by the amount
T02 − Teq′ =
κ
1+ κ
(T02 − T01 ) ,
(21a)
while the respective portion of the colder body (with initial temperature T01) heats up by the
amount
Teq′ − T01 =
T02 − T01
,
1+ κ
(21b)
k1 χ 2
k1 ρ1c1
. A practical example illustrating this result is the following
=
k 2 χ1
k 2 ρ 2 c2
well-known situation. When we step with bare feet (T02) on a cold surface (T01) our feet will
cool off by an amount given by (21a) which is the larger the larger is κ. Consequently, those
surfaces for which κ, as defined above, is large will be perceived as colder than surfaces with
small κ even though their temperatures are the same.
where κ =
a)
1
Electronic mail: tomaz.kranjc@pef.uni-lj.si
H. S. Carslaw and J.C. Jaeger, Conduction of Heat in Solids (2nd Edition, Oxford U. P.,
Oxford, 1959), p. 87.
2
A. V. Luikov, Analytical Heat Diffusion Theory (Academic Press, New York, 1968), p. 401.
3
M. R. Spiegel, Laplace Transforms (Schaum’s Outline Series, McGraw-Hill, New York,
1965).
4
Reference 1, p. 324.
5
Reference 1, p. 89.
10
APPENDIX
In the case of temperature diffusion in semi-infinite rods aligned with the x-axis the only
quantity with dimension of length in addition, of course, to the coordinate x is χt .
Consequently the only dimensionless quantity on which the temperature distribution may
depend is x / χt . Since the diffusion equation is linear and homogeneous with respect to
temperature its unit can be assigned arbitrarily. If T0 is some temperature characteristic to the
problem at hand, we may write
 x 
T ( x, t )
 ≡ f (ξ ) ,
(A1)
= f
 2 χt 
T0


where we have included a factor of 2 for later convenience. This transformation is originally
due to Boltzmann5. Inserting this function into diffusion equation (4) we obtain
1 df 1 d 2 f
.
− ξ
=
2 dξ 4 dξ 2
(A2)
Assuming that two rods with initial temperatures T1 (left rod) and T2 (right rod) are joined at x
= 0 and integrating the above equation, we write the solutions in the form
T ( x < 0, t ) = A1 + B1
T ( x > 0, t ) = A2 + B2
 −x
−ξ 2

e
d
=
A
+
B
erf
ξ
1
1
2 χ t
π x / 2∫ χ1t
1

2
2
π
0
x / 2 χ 2t
∫
0

,


 x 
2
,
e −ξ dξ = A2 + B2 erf 
2 χ t 
2 

(A3)
(A4)
where erf(x) is the error function defined as
erf ( x) =
2
π
x
∫e
−ξ 2
dξ ,
0
erf (∞) = 1 ,
erf (− x) = − erf ( x) .
Imposing the initial and boundary conditions (5) enables us to determine the integration
constants with the results
A1 =
κT1 + T2
T − T2
, B1 = 1
, A2 = A1 , B2 = −κB1 .
1+ κ
1+ κ
Inserting these expressions into (A3) and (A4) we obtain the results quoted in the
introduction.