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Modern Engineering Thermodynamics

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Problems 785<br />

b. For a Knudson gas, we know that _S i / _mR= −1/2: Integrate<br />

the result of part a to show that, for a Knudson gas in a<br />

thermomechanical system, p 2 /p 1 = (T 2 /T 1 ) 1/2 .<br />

18. Figure 19.12 illustrates the basic operation of an<br />

electrohydrodynamic (EHD) generator. This device exploits the<br />

coupling of mass flow and electric current such that an electric<br />

current is induced (i.e., generated) simply by the mass flow of<br />

the liquid as shown. The system contains no moving parts<br />

except for the liquid and easily produces a 20.0 × 10 3 V<br />

potential difference. This device was used by William Thomson<br />

(Lord Kelvin) to test the continuity of the transatlantic cable as<br />

it was being laid at sea in 1858. The mass and electrical current<br />

fluxes are related by<br />

and<br />

J mass = ρV = −ðL ME /TÞðdϕ/dxÞ − ðνL MM /TÞðdp/dxÞ<br />

J current = I/A = −ðL EE /TÞðdϕ/dxÞ − ðνL EM /TÞðdp/dxÞ<br />

subject to the following special “effects”:<br />

Ohm’s law,<br />

Darcy-Weisbach law,<br />

ðJ current Þ p =constant = − k e ðdϕ/dxÞ<br />

ðJ mass Þ ϕ= constant = − ρk p =μ ðdp=dxÞ<br />

¼ −ðρD 2 =32μÞðdp=dxÞ<br />

The mechanoelectric effect, (dϕ/dp) I=0 = k m /p 1/2 , where k m is the<br />

mechanoelectric coefficient.<br />

Nozzle<br />

Water<br />

drops<br />

FIGURE 19.12<br />

Problem 18.<br />

Water reservoir<br />

+<br />

−<br />

Flow<br />

control<br />

valves<br />

Stream breaks<br />

into droplets<br />

Metal rings<br />

+<br />

−<br />

Electrical<br />

power<br />

output<br />

Metal recievers<br />

(act as storage<br />

capacitors)<br />

Insulators<br />

a. Find formulae for L EE , L MM , L EM , and L ME in terms of<br />

measurable quantities (i.e., T, k e , D, μ, etc.).<br />

b. Predict the existence of and find a formula for the mass flow<br />

induced by the flow of an electric current when there is no<br />

pressure drop. This is the reverse of the EHD generator<br />

shown previously and is called EHD pumping.<br />

c. Find a formula for the short-circuit current induced by the<br />

mass flow of the EHD generator. This is called the streaming<br />

current of the device.<br />

Computer Problems<br />

19.* The Seebeck voltage (ϕ) to relative temperature (T) conversion<br />

for an iron-constantan thermocouple with an ice point (0.00°C)<br />

reference is given over the range 0.00°C to 760.°C byT = a 0 +<br />

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<br />

Seebeck voltage in microvolts (i.e., 10 −6 V). The polynomial<br />

coefficients given by the National Bureau of Standards are<br />

a 0 = −0.048868252<br />

a 1 = 19873.14503<br />

a 2 = −218614.5353<br />

a 3 = 11569199.78<br />

a 4 = −264917531.4<br />

a 5 = 2018441314.<br />

a. Write an interactive computer program that asks the user for<br />

the thermocouple voltage (in the proper units) and returns<br />

the temperature, Seebeck, Peltier, and difference in Kelvin<br />

coefficients (in proper units) to the screen.<br />

b. Plot the Seebeck voltage and the Seebeck and Peltier<br />

coefficients vs. temperature over the temperature range of<br />

0.00°C to 700.°C using at least 100 points per curve.<br />

20.* The Seebeck voltage (ϕ) to relative temperature (T) conversion for<br />

a copper-constantan thermocouple with an ice point (0.00°C)<br />

reference is given over the range −160.°C to 400.°C byT = a 0 a 1 ϕ +<br />

a 2 ϕ 2 + a 3 ϕ 3 + … + a 7 ϕ 7 ,whereT is in °C andϕ is the Seebeck<br />

voltage in microvolts (i.e., 10 −6 V). The polynomial coefficients<br />

given by the National Bureau of Standards are<br />

a 0 = 0.100860910<br />

a 1 = 25727.94369<br />

a 2 = −767345.8295<br />

a 3 = 78025595.81<br />

a 4 = −9247486589<br />

a 5 = 6.97688 × 10 11<br />

a 6 = −2.66192 × 10 13<br />

a 7 = 3.94078 × 10 14<br />

a. Write an interactive computer program that asks the user for<br />

the thermocouple voltage (in the proper units) and returns<br />

the temperature, Seebeck, Peltier, and difference in Kelvin<br />

coefficients (in proper units) to the screen.<br />

b. Plot the Seebeck voltage and the Seebeck and Peltier<br />

coefficients vs. temperature over the temperature range of<br />

100.°C to 400.°C using at least 100 points per curve.

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