Modern Engineering Thermodynamics

19.5 Thermomechanical Coupling 779
Pressure p 1 Pressure p 2
Temperature T 2
Temperature T 1
Capillary tube, porous plug,
or permeable membrane
FIGURE 19.9
The thermomechanical effect.
In Example 19.4, the pressure difference and the temperature difference have opposite signs. However, the
osmotic heat conductivity (k o ) can be negative for certain substances, and this produces pressure and temperature
differences with the same algebraic sign.
Figure 19.9 shows two rigid insulated vessels connected by a capillary tube (or a tube filled with a porous substance,
or one having a permeable membrane). If we construct this system so that there is either (1) zero heat
transfer between the vessels, (2) zero temperature difference between the vessels, or (3) zero pressure difference
between the vessels, then the following phenomena occur.
19.5.1 Zero Heat Transfer
When the vessels are at different pressures and temperatures but no heat transfer occurs between the vessels
(J Q = 0), Eq. (19.49) requires that a temperature gradient exists along the connecting tube of the form
dT
dx = − k o dp
k t + μk 2 o /ðTk pÞ dx
Then, from Eq. (19.50), the following mass flux occurs through the connecting tube:
J JQ M =0 = ρk o
Tk t + μk 2 o /k − ρk
p dp
p μ dx
which vanishes only under the improbable circumstance of Tk t = 0.
(19.54)
19.5.2 Zero Temperature Gradient
When both vessels are at different pressures but equal temperatures, T = constant and dT/dx = 0. Then, Eqs.
(19.49) and (19.50) give
and
So that, using Eq. (19.50), we can write
J Q
J M
J T
dp
Q =constant
= −k o
dx
J T M =constant
= − ρk
p dp
μ dx
T
=constant
= μk o
ρk p
= − T dp
JM
ρ dx
=0
At this point, we define the isothermal energy transport rate _Q i that is “induced” by the thermomechanical mass
flow rate _m with 14
J Q T Q
= _ i
=constant
_m = μk o
(19.55)
ρk p
J M
14 Notice that we appear to have a heat transfer here that is not due to a temperature difference. It is because such thermal energy
transports exist that we chose earlier to define heat transfer in general terms as a nonwork, nonmass flow energy transport.