CAL CHEM CORPORATION

To: CHE Juniors Date: Winter Quarter

File: CHE 322L

From: CHE faculty

Laboratory Managers

Subject: Cooling by Hilsch Tube

The Diamond Bar plant has a nitrogen stream at 3.0 atm and 300.0oK that is presently vented to the atmosphere. The management would like to use this stream to satisfy some of the heating or cooling requirements of other processes. Their engineer, James Hank, a Cal Poly graduate, has devised a process that will produce equal amounts of a hot stream at 450oK and a cold stream at 150oK and thus satisfy simultaneously some heating and cooling requirements. Furthermore, James claims that his device will be self-sustaining because no additional heat or work need be supplied to the device. The management would like an explanation of this device.

The principles behind James’ device are believed to be the same as the Hilsch-Ranque Vortex Tube, which is available in our laboratory. Please take the necessary data to verify the claims by James Hank.

Operate the Hilsch tube over the widest possible range of the operating variables. The inlet air pressure to the Hilsch tube can be set by the air regulator. The ratio of the hot air flow rate to the cold air flow rate (hot/cold) can be changed by turning the knob at the hot outlet of the Hilsch tube.

You will be measuring temperatures and flow rates. Use thermocouples for the temperature measurement. Ask the technician for the vane anemometer to measure air flow rates. The vane anemometer is a revolution counter with jeweled bearing, actuated by a small windmill. Air velocity is displayed in ft/s or m/s by a selector switch.


CHE 322 TRANSPORT LABORATORY

Experiment No. 3

COOLING BY HILSCH TUBE

The actual flow rate will be cancelled out in the analysis, so it is not important to have an accurate flow rate. However, the ratio of the hot to cold air flow rates is required in the calculation, the measurements of the hot and cold air velocity must be consistent to yield good results. Air leaving the Hilsch tube flows through a pipe where the anemometer is used to measure air flow rate. The mass flow rate leaving a pipe can be estimated by

= Avr

where A is the inside area of the pipe, v is the air velocity, and r is the air density. The temperature of the air leaving the anemometer Th,pipe and Tc,pipe must be measured to determine the air density by ideal gas law. This temperature is not the hot temperature Th or the cold temperature Tc used in Eq. (1). Th and Tc must be measured just inside the Hilsch tube.

The laboratory work is relatively easy to complete. Be certain that your data should have at least four different inlet pressures at eight different (hot/total) for a total of thirty-two different conditions. Note the time required to reach steady state operation. Compute the standard error in your experimental readings and in your calculated results.

Analysis

This is mostly a “think” experiment that tests your knowledge of thermodynamics. When you perform a thermodynamic analysis of a process remember:

- both the first and second laws must be followed for the process to be theoretically possible.

- always do the first law analysis before the second law analysis.

- if you are analyzing a process and the first law is violated you must stop at that point. Go over the first law calculations and, if possible, discover the error. Examine carefully any assumption that you made. Only after the first law analysis is correct can you perform the second law analysis.

Apply the first law to the Hilsch tube

Ti = hTh + (1 - h)Tc (1)

where h = (hot/total), total = hot + cold, Ti = inlet air temperature, Tc = outlet cold air stream, and Th = outlet hot air stream.

The maximum temperature spread from the Hilsch vortex tube will be obtained for a reversible operation where the total entropy change is zero (Ref. 1, pg. 123).

DS = hCpln(Th/Ti) + (1 - h)Cpln(Tc/Ti) + Rln(Pi/Po) = 0 (2)

where R = gas constant, Pi = inlet air pressure, Po = outlet air pressure, and Cp = heat capacity of air.

Substitute Th/Ti from Eq. (1) into Eq. (2) to obtain

hCpln{[1- (1- h)(Tc/Ti)]/h} + (1 - h)Cpln(Tc/Ti) + Rln(Pi/Po) = 0 (3)

Eq. (3) can be solved for Tc/Ti if h and (Pi/Po) are known.

- Solve Eqs. (1) and (3) for Th/Tc, Tc/Ti , and Th/Ti at (Pi/Po) = 2 over the range 0.15 h 0.85. Plot the results.

- Solve Eqs. (1) and (3) for Th/Tc, Tc/Ti , and Th/Ti at h = 0.5 over the range 1.5 (Pi/Po) 8.5 obtainable from the experiment. Plot the results.

The following Matlab progam solves Eqs. (1) and (3) for Th/Tc, Tc/Ti , and Th/Ti at (Pi/Po) = 2 over the range 0.15 h 0.85 and at h = 0.5 over the range 1.5 (Pi/Po) 8.5.

------Vortex ------

% Estimate Cold and Hot Temperature of a Vortex tube

% Use First and Second Law, Assume reversible process

%

h=.1;Cp=29.3;R=8.314;

Pr=input('Pi/Po = '); % Pr = Pi/Po

f='h*log((1-(1-h)*x)/h)+(1-h)*log(x)+(R/Cp)*log(Pr)';

fp='h*(h-1)/(1-(1-h)*x)+(1-h)/x';

fprintf('At Pi/Po = %g\n',Pr)

fprintf(' h Th/Tc Tc/Ti Th/Ti\n')

x=.5;

for i=1:15

h=h+.05;ex=.1;

while abs(ex)>.0001

ex=eval(f)/eval(fp);

x=x-ex;

end

ThTi=(1-(1-h)*x)/h;ThTc=ThTi/x;

fprintf('%3.2f %4.3f %4.3f %4.3f \n',h,ThTc,x)

fprintf('%f \n',ThTi)

end

h=.5;Pr=1;

fprintf('At h = .5\n')

fprintf('Pi/Po Th/Tc Tc/Ti Th/Ti\n')

x=.5;

for i=1:15

Pr=Pr+.5;ex=.1;

while abs(ex)>.0001

ex=eval(f)/eval(fp);

x=x-ex;

end

ThTi=(1-(1-h)*x)/h;ThTc=ThTi/x;

fprintf('%3.2f %4.3f %4.3f %4.3f',Pr,ThTc,x)

fprintf(' %f \n',ThTi)

end

------

The Hilsch vortex is not a reversible device, hence the maximum temperature spread will be lower than the value predicted by Eqs. (1) and (3). An analytical solution by B. Ahlborn et al focuses on the influence of the bulk kinetic energy on the energetics of each of the two fluid streams from a vortex tube (Ref. 2). The heating in a vortex tube is attributed to conversion of kinetic energy into heat and the cooling is attributed to the reverse process. This model yields the upper limit for the temperature increase on the hot side:

Th Ti[1 + X(g - 1)/g] (4)

where g = Cp/Cv and X = (Pi - Po)/Pi

and a lower limit for the temperature reduction on the cold side:

Tc ³ Ti(1 - X)(g-1)/g (5)

Experimental Work

1)  Read the barometric pressure. This is the outlet pressure Po.

2)  Measure room temperature. This is the inlet temperature Ti.

3)  Set the pressure ratio Pi/Po = 2. The barometric pressure must be added to the reading from the pressure gage to obtain Pi in absolute value.

4)  Set a value for h by turning the knob at the hot outlet of the Hilsch tube. Measure Th, Tc, Th,pipe, and Tc,pipe. Measure the hot and air velocity using the anemometer.

5)  Repeat step 4 for at least a total of eight different values of h. Be sure to cover the whole range of h.

6)  Repeat step 3 at pressure ratios of 4, 6, and 8 (or the higest possible ratio).

Data Analysis

1)  For each data point, evaluate

DQc = (cold)Cp(Tc - Ti), where Cp = 1.013 kJ/kg×oK (heat capacity of air)

DQh = (hot)Cp(Th - Ti)

DQ = DQc + DQh

DSc = (cold){Cpln(Tc/Ti) + (R/29)ln(Pi/Po)}, where R = 8.314 kJ/kmol×oK

DSh = (hot){Cpln(Th/Ti) + (R/29)ln(Pi/Po)}, where R = 8.314 kJ/kmol×oK

DS = DSc + DSh

Discuss the results.

2)  At each value of Pi/Po, plot Th/Tc as a function of h. On the same graph, plot Th/Tc calculated Eqs. (1), (3) and from Eqs. (4), (5). Could you determine whether Th/Tc is independent of hot/total as predicted by Eqs. (4), (5).

,

References

1. Sandler S. I., Chemical and Engineering Thermodynamics, Wiley, (1998)

2. B Ahlborn, J U Keller, R Staudt, G Treitz and E Rebhan, J. Phys. D: Appl. Phys. (1994) 480-488.

3. Tester, J. W., and Modell, M., Thermodynamics and Its Applications, Prentice-Hall, (1997)

4. Kyle B. G., Chemical and Process Thermodynamics, Prentice-Hall, (1999)

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