Modern Engineering Thermodynamics

Problems 785
b. For a Knudson gas, we know that _S i / _mR= −1/2: Integrate
the result of part a to show that, for a Knudson gas in a
thermomechanical system, p 2 /p 1 = (T 2 /T 1 ) 1/2 .
18. Figure 19.12 illustrates the basic operation of an
electrohydrodynamic (EHD) generator. This device exploits the
coupling of mass flow and electric current such that an electric
current is induced (i.e., generated) simply by the mass flow of
the liquid as shown. The system contains no moving parts
except for the liquid and easily produces a 20.0 × 10 3 V
potential difference. This device was used by William Thomson
(Lord Kelvin) to test the continuity of the transatlantic cable as
it was being laid at sea in 1858. The mass and electrical current
fluxes are related by
and
J mass = ρV = −ðL ME /TÞðdϕ/dxÞ − ðνL MM /TÞðdp/dxÞ
J current = I/A = −ðL EE /TÞðdϕ/dxÞ − ðνL EM /TÞðdp/dxÞ
subject to the following special “effects”:
Ohm’s law,
Darcy-Weisbach law,
ðJ current Þ p =constant = − k e ðdϕ/dxÞ
ðJ mass Þ ϕ= constant = − ρk p =μ ðdp=dxÞ
¼ −ðρD 2 =32μÞðdp=dxÞ
The mechanoelectric effect, (dϕ/dp) I=0 = k m /p 1/2 , where k m is the
mechanoelectric coefficient.
Nozzle
Water
drops
FIGURE 19.12
Problem 18.
Water reservoir
+
−
Flow
control
valves
Stream breaks
into droplets
Metal rings
+
−
Electrical
power
output
Metal recievers
(act as storage
capacitors)
Insulators
a. Find formulae for L EE , L MM , L EM , and L ME in terms of
measurable quantities (i.e., T, k e , D, μ, etc.).
b. Predict the existence of and find a formula for the mass flow
induced by the flow of an electric current when there is no
pressure drop. This is the reverse of the EHD generator
shown previously and is called EHD pumping.
c. Find a formula for the short-circuit current induced by the
mass flow of the EHD generator. This is called the streaming
current of the device.
Computer Problems
19.* The Seebeck voltage (ϕ) to relative temperature (T) conversion
for an iron-constantan thermocouple with an ice point (0.00°C)
reference is given over the range 0.00°C to 760.°C byT = a 0 +
a 1 ϕ + a 2 ϕ 2 + a 3 ϕ 3 + a 4 ϕ 4 + a 5 ϕ 5 , where T is in °C and ϕ is the
Seebeck voltage in microvolts (i.e., 10 −6 V). The polynomial
coefficients given by the National Bureau of Standards are
a 0 = −0.048868252
a 1 = 19873.14503
a 2 = −218614.5353
a 3 = 11569199.78
a 4 = −264917531.4
a 5 = 2018441314.
a. Write an interactive computer program that asks the user for
the thermocouple voltage (in the proper units) and returns
the temperature, Seebeck, Peltier, and difference in Kelvin
coefficients (in proper units) to the screen.
b. Plot the Seebeck voltage and the Seebeck and Peltier
coefficients vs. temperature over the temperature range of
0.00°C to 700.°C using at least 100 points per curve.
20.* The Seebeck voltage (ϕ) to relative temperature (T) conversion for
a copper-constantan thermocouple with an ice point (0.00°C)
reference is given over the range −160.°C to 400.°C byT = a 0 a 1 ϕ +
a 2 ϕ 2 + a 3 ϕ 3 + … + a 7 ϕ 7 ,whereT is in °C andϕ is the Seebeck
voltage in microvolts (i.e., 10 −6 V). The polynomial coefficients
given by the National Bureau of Standards are
a 0 = 0.100860910
a 1 = 25727.94369
a 2 = −767345.8295
a 3 = 78025595.81
a 4 = −9247486589
a 5 = 6.97688 × 10 11
a 6 = −2.66192 × 10 13
a 7 = 3.94078 × 10 14
a. Write an interactive computer program that asks the user for
the thermocouple voltage (in the proper units) and returns
the temperature, Seebeck, Peltier, and difference in Kelvin
coefficients (in proper units) to the screen.
b. Plot the Seebeck voltage and the Seebeck and Peltier
coefficients vs. temperature over the temperature range of
100.°C to 400.°C using at least 100 points per curve.