Modern Engineering Thermodynamics

19.5 Thermomechanical Coupling 777
Under isothermal conditions (dT = 0), Eq. (19.34) gives
J T¼constant Q = − vL
QM dp
T dx
This equation describes the mechanocaloric effect, which is usually modeled after Fourier’s law as
J T¼constant
dp
Q = − k o
dx
(19.36)
(19.37)
where k o is an empirical material constant called the osmotic heat conductivity coefficient. Comparing Eqs. (19.36)
and (19.37), using Onsager’s reciprocity relation and ρ = 1/v, we see that
L QM = Tk o /v = ρTk o = L MQ (19.38)
Similarly, the mass flux under isothermal conditions (dT = 0) is given by Eq. (19.35) as
J T¼constant M = vL
MM dp
T dx
(19.39)
Now, two flow models can be used to interpret this equation. The first was formulated empirically in 1856 by
the French hydraulic engineer Henri Philibert Gaspard Darcy (1803–1858) for flow through porous media.
Appropriately called Darcy’s law, it states that the bulk fluid velocity in a porous material is given by
V = − k p
μ
dp
dx
(19.40)
where μ is the fluid viscosity and k p is called the permeability of the porous material. 13 Comparing Eqs. (19.39)
and (19.40) and introducing the mass flux definition along with the relation ρ = 1/v produces
J T¼constant M = ρV = − ρk
p dp
= − vL
MM dp
(19.41)
μ dx T dx
from which it is clear that the mass flow primary coefficient L MM is
L MM = ρ 2 Tk p /μ (19.42)
The other flow model that can be used in Eq. (19.39) was developed for flow through a circular tube and
is called the Darcy-Weisbach equation after Henri Darcy and the German engineer Julius Ludwig Weisbach
(1806–1871). In this model, the isothermal mass flux is given by
J T M =constant
= ρV = − 2Dg
c dp
(19.43)
fV dx
where D is the tube diameter and f is an empirically determined “friction factor.” The value of f can be found
in most fluid mechanics textbooks from a generalized curve of f versus the dimensionless Reynolds number,
ρVD/μ. It can be shown that, if the flow is laminar (ρVD/μ < 2000), then
J M
T¼constant = ρV = − ρD2
32μ
dp
dx
(19.44)
By comparing Eqs. (19.41) and (19.44), we see that these equations are similar and they become identical when
we define the effective permeability of a circular tube as
Thus, for very slow flow through a circular tube, we can write
k p = D 2 /32 (19.45)
L MM = TðρDÞ2
(19.46)
32μ
Finally, combining Eqs. (19.34) and (19.35) to eliminate the term (v/T)(dp/dx) gives
J Q = − L 2
QQL MM − L QM dp
L MM T 2
− L QM
dx T J M (19.47)
13 The common unit of measure of permeability is the darcy, which has the rather awkward definition of 1 darcy being the
permeability that allows the flow of 1 cm 3 /s of fluid with a pressure gradient of 1 atm/cm. The darcy is not an SI unit. The SI unit for
permeability is m 2 , where 1 darcy = 10 −12 m 2 .