Page 477 - Advanced thermodynamics for engineers

20.3 THERMODYNAMIC FORCES AND THERMODYNAMIC VELOCITIES 469

Thus, the total change of entropy for the whole system is

dS dS 1 dS 2 dQ 1 1 dQ T 2 T 1
¼ þ ¼ ¼ (20.3)
dt dt dt dt T 1 T 2 dt T 2 T 1
Now T 2 > T 1 and therefore the rate of change of entropy, dS > 0:
dt
To understand the meaning of this result, it is necessary to consider a point in the bar. At the point ‘
from the left-hand end, the thermometer reading is T. This reading is independent of time and is the
reading obtained on the thermometer in equilibrium with the particular volume of the rod in contact
with it. Hence, the thermometer indicates the ‘temperature’ of that volume of the rod. Since the
temperature is constant, the system is in a ‘steady state’ and at each point in the rod, the entropy is
invariant with time. However, there is a net transfer of entropy from the left-hand reservoir to the right-
hand reservoir, i.e. entropy is ‘flowing’ along the rod. The total entropy of the composite system is
increasing with time and this phenomenon is known as ‘entropy production’.

20.3 THERMODYNAMIC FORCES AND THERMODYNAMIC VELOCITIES
It has been suggested that for systems not far removed from equilibrium, the development of the
relations used in the thermodynamics of the steady state should proceed along analogous lines to the
study of the dynamics of particles, i.e. the laws should be of the form
J ¼ LX (20.4)
where
J is the thermodynamic velocity or flow;
X is the thermodynamic force; and
L is a coefficient independent of X and J and is scalar in form, while both J and X are vector
quantities.
The following simple relationships illustrate how this law may be applied.
Fourier’s equation for one-dimensional conduction of heat along a bar is
dQ dT
¼ kA ; (20.5)
dt d‘
where Q ¼ quantity of energy (heat); T ¼ temperature; A ¼ area of cross-section; ‘ ¼ length;
k ¼ thermal conductivity.
Ohm’s law for flow of electricity along a wire, which is also one-dimensional, is

dq ‘ de
I ¼ ¼ lA (20.6)
dt d‘
where I ¼ current; q [ ¼ charge (coulomb); e ¼ potential difference (voltage); A ¼ area of
cross-section of wire; [ ¼ length; l ¼ electrical conductivity.
Fick’s law for the diffusion due to a concentration gradient is, in one dimension,

dn i dC i
¼ k (20.7)
dt d‘