Modern Engineering Thermodynamics

776 CHAPTER 19: Introduction to Coupled Phenomena
EXAMPLE 19.3 (Continued )
Thus, if the copper lead wires have the same end point junction temperatures, then they do not contribute any net
Seebeck voltage to the circuit. Adding up all the potential differences around the circuit and using T a = T f and T b = T e
gives
Now,
Δϕ af =
Z Tb
Ta
Z Ta Z Tc
α cu dT + α cu dT + α al dT +
T b T b
Z Td
Tc
Z Tb
α ch dT + α al dT
T d
Z Tc Z Tb Z Tc Z Td
α al dT + α al dT = α E dT = − α al dT
Tb
Td
Td
Tc
so
Z T d
Z T d
ϕ af = ðα ch − α E ÞdT = ðα ch-alÞdT = α ch-alðT d − T c Þ
Tc
Tc
Consequently, the potentiometer measures the chromal-alumel thermoelectric effects only so long as all the lead wires
have equal junction temperatures.
19.5 THERMOMECHANICAL COUPLING
In this section, we investigate the open system coupling phenomena resulting from the coupling of simultaneous
heat transfer and mass flow. This type of coupling is commonly called the thermomechanical effect.
In 1873, W. Feddersen reported having observed a flow of air through a porous plug brought about by only a
temperature difference on the opposite sides of the plug. There was no pressure drop across the plug, yet there
was a flow of air. We call the phenomenon of fluid flow induced by the presence of a temperature difference
rather than a pressure difference thermal osmosis. 12 This effect has been observed in both gases and liquids.
In 1939, J. G. Daunt and K. Mendelssohn reported observing isothermal heat transfer caused by only a pressure
difference in a fluid. This is the reciprocal of the thermal osmosis effect and is often called the mechanocaloric
effect.
The thermal osmosis and mechanocaloric effects are secondary or coupled effects induced by the primary effects
of Fourier conduction heat transfer and Darcy-Weisbach pressure-driven mass flow. In thermomechanical systems,
the fluxes are
1. Heat flux: J Q = _Q /A = _q
2. Mass flux: J M = _m /A = ρV, where ρ is the local density and V is the local velocity of the moving fluid. The
generalized forces are
Temperature gradient: X Q = − 1
dT
T 2 dx
Pressure gradient: X M = − v
dp
T dx
where v is the local specific volume of the moving fluid. Then, for near equilibrium conditions, Eq. (19.9)
gives the coupled heat and mass flux equations as
J Q = − L
QQ dT
T 2 − vL
QM dp
(19.34)
dx T dx
and
J M = − L
QQ dT
T 2 − vL
MM dp
(19.35)
dx T dx
12 This phenomenon is also known as Knudsen effect and the fountain effect in the literature.