DIRECTION OF HEAT TRANSFER
Statement
Given two bodies initially at temperatures T1 and T2, deduce from Suniv>0 that heat must flow from T1
to T2, if put in contact.
Dados dos cuerpos inicialmente a temperatura T1 a T2, demostrar, a partir de que Suniv>0 que el calor
ha de fluir de T1 a T2, si entran en contacto
Solution
The universe (i.e. all participating systems) is here comprised of the two bodies, and their variation of
entropy, assuming a small and slow energy exchange (without internal degradation), will be dS1=dQ1/T1
and dS2=dQ2/T2, according to Eq. (2.11). But the global energy balance (First Thermodynamic Principle)
forces dQ1+dQ2=0, and the global entropy increase (Second Thermodynamic Principle) forces
dSuniv=dS1+dS2>0, and their combination, dSuniv=dQ1(1/T1-1/T2)=dQ1(T2-T1)/(T1T2)>0, shows that the
sign of dQ1 must be opposite to the sign of the difference T1-T2, i.e. energy must be lost by the hotter and
gained by the colder, i.e. heat can only flow from hot to cold bodies. In summary:
0  dEuniv  dE1  dE2   dQ1  dW1    dQ2  dW2 
0  dSuniv

1 1
Q1  0 if T1  T2

 dQ1
  dQ2
  dQ1     0  
 dS1  dS2  
 dSgen,1   
 dSgen,2  
Q1  0 if T1  T2
 T1 T2 
 T1
  T2

Comments
It was in 1822, well before any idea of entropy was advanced, that J. Fourier first stated explicitly in
mathematical language that the flow of heat, being proportional to the temperature gradient and a material
constant, was in the direction opposite to that gradient, i.e. the minus sign in the so called Fourier's
equation q  k T . The aim here in this exercise is to check that this fact may be deduced from the more
general idea that the entropy of the universe must increase in any real process.
It is important to keep in mind the statement of the problem and not only its result, because it might be
erroneously concluded from this exercise that it is impossible to heat up a hot object with a colder one, for
instance, or that a temperature gradient always implies a heat flow. In all kinds of refrigerators a system is
cooled (the working fluid) without others being cooler, and heat is extracted from a cool ambient (the
fridge interior) and (after some additional processing) heat is cast to a hotter system (the fridge
environment). A steady state with a temperature gradient and no heat flow can be maintained in nonequilibrium mixtures, for instance (explained in Chap. 10).
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Direction of heat transfer
1