Modern Engineering Thermodynamics

784 CHAPTER 19: Introduction to Coupled Phenomena
coupling deals mainly with thermal osmosis and mechanocaloric effects. Equations are developed to describe
the entropy production rates of both the thermoelectric and the thermomechanical coupling processes. A
modern case study of electrohydrodynamic coupling is discussed and the concept of the viscoelectric effect is
introduced to explain a number of known electrostatic phenomena in moving dielectric fluids and solids.
Modeling lightning as a viscoelectric effect may lead to its use as an important new renewable energy source.
Problems (* indicates problems in SI units)
1. Equation (19.11) gives the EPRD for a general binary system
as σ = L 11 X 12 +2L 12 X 1 X 2 + L 22 X 22 > 0. Show that, if X 1 is held
constant, a minimum value in σ occurs when J 2 = L 12 X 1 + L 22 X 2 =0.
Assume all the L ij are constant here.
2. Equation (19.11) gives the EPRD for a general binary system as
σ = L 11 X 12 +2L 12 X 1 X 2 + L 22 X 22 > 0. This quadratic form has to
be positive for all positive or negative values of X 1 and X 2 . Show
that this requires that
a. L 11 > 0.
b. L 22 > 0.
c. L 11 L 22 > 2L 12 .
3. Show that the difference in Kelvin coefficients for a
thermocouple made of conductors A and B can be written in
terms of the open circuit voltage as τ AB = Td ð 2 ϕ AB /dT 2 Þ:
4. Show that, if the Kelvin thermoelectric effect did not exist (i.e.,
if τ AB were zero), then the voltage produced by a thermocouple
would always depend linearly on temperature.
5.* Table 19.6 gives the temperature-voltage conversion for an ironconstantan
thermocouple. Estimate its Seebeck coefficient at
2.00°C and at 200.°C, and compute the percent difference based
on the 20.0°C value.
Table 19.6 Problem 5
Temp. (°C)
18 0.916
19 0.967
20 1.019
21 1.070
22 1.122
– –
198 10.666
199 10.721
200 10.777
201 10.832
202 10.888
Seebeck voltage (mV)
6.* The absolute Seebeck coefficient for a material is given by α =
(300. + 150. T − T 2 ) × 10 −6 V/K, where T is in °C. Determine
the formula for the (a) Peltier and (b) Kelvin coefficients.
7.* Using the temperature to Seebeck voltage relationship for the
iron-copper thermocouple given in Example 19.2, determine
formulae for the Seebeck, Peltier, and difference in Kelvin
coefficients and evaluate them at 0.00°C and 200.°C.
8. Seebeck voltage (ϕ) to temperature (T) conversion for
thermocouples is often written in a power series of the form
T = a 0 + a 1 ϕ + a 2 ϕ 2 +a 3 ϕ 3 + … + a n ϕ n . Using this representation,
determine the formula for the (a) Seebeck, (b) Peltier, and
(c) Kelvin coefficients.
9.* Determine the EPRD in a semiconductor thermoelectric junction
that has the following physical properties:
Electrical resistivity = 1.10 × 10 −5 ohm meters
thermal conductivity = 1.30 W(m · K)
Seebeck coefficient = (200. + 10.0 T − 0.0100 T 2 ) × 10 −6 V/K
Junction temperature = 400. K
The temperature gradient at the junction is 1000. K/m and the
voltage gradient is 0.0300 V/m.
10.* A thermocouple is connected to a battery. The cold junction is
maintained at 0.00°C while the temperature of the hot junction
varies. When the hot junction is at 100.°C, the relative Peltier
coefficient is measured to be 2.75 × 10 −3 W/A, and when it is at
200.°C, it is 4.51 × 10 −3 W/A. If the thermocouple potential is
given by ϕ = aT + b 2 , where T is in K, determine
a. The constants a and b.
b. The value of ϕ when the hot junction is at 100.°C and 200.°C.
11. Show that the entropy production rate per unit volume (σ) is
always positive for a thermocouple regardless of the values of
α, dT/dx, and dϕ/dx.
12. Show that the entropy production rate per unit volume (σ) for a
thermocouple must always be greater than (k t /T)(dT/dx) 2 .
13.* Suppose the vessels used in Example 19.5 and 19.6 are arranged
so that no heat transfer occurs between them along the
interconnecting tube. If the pressure difference between the
vessels is 10.0 kPa, determine the required temperature
difference and mass flow rate in the interconnecting tube.
14.* Determine the numerical values of the thermomechanical
coupling coefficients (L 11 , L 12 , L 21 , and L 22 ) for a liquid
undergoing very slow flow in a 1.00 mm diameter circular tube
at 0.00°C. The fluid data are
Viscosity = 9.20 × 10 −5 kg/(m·s)
Thermal conductivity = 0.105 W/(m · K)
Density = 927 kg/m 3
Osmotic heat conductivity = 3.72 × 10 −3 m 2 /s.
15.* Determine the numerical values of the thermomechanical coupling
coefficients (L 11 , L 12 , L 21 ,andL 22 ) for a liquid that has the same
physical properties as saturated liquid water. The isothermal heat
flux is 2.76 W/m 2 and the isothermal mass flux is 0.0100 kg/
(m 2 · s) both at 20.0°C and a pressure gradient of −14.0 MPa/m.
16. Starting with Eq. (19.2), use Eqs. (19.38), (19.41), (19.42), and
(19.43) to derive the formula for σ for
a. Viscous flow through porous media.
b. Turbulent flow through circular pipes.
c. Laminar flow through circular pipes.
17. a. Show that, for an ideal gas in a thermomechanical system,
Eq. (19.54) can be written as
T dp
p dT
JM =0
= dðlnðpÞ
dðlnðTÞ
where R is the specific gas constant.
JM =0
= − _ S i
_mR