## The Effect of Back Pressure on a Vortex Tube Part 2, Calculating Btu/Hr.

My previous blog post was about how Vortex Tubes react when there is back pressure due to a restriction on either the hot or cold discharge of the Vortex Tube.  In it I mentioned that there is a formula to calculate what the cooling capacity (Btu/Hr) will be if there is no way to avoid operating the Vortex Tube without back pressure on the discharge. That is the calculation focus of this blog – calculating Btu/hr of a Vortex Tube with back pressure.

To continue with the same example, the calculations from the previous blog are shown below.  Last time the example Vortex Tube was operating at 100 psig inlet pressure, 50% cold fraction, and 10 psi of back pressure. We will need some additional information to determine the Btu/Hr capacity. The additional information needed is the temperature of the supplied compressed air as well as the ambient air temperature desired to maintain.  For the example the inlet compressed air will be 70°F and desired ambient air temperature to maintain will be 90°F.

(100 psig + 14.7 psia) / (10 psig + 14.7 psia) = X / 14.7 psia
4.6437 = X / 14.7
X= 14.7 * 4.6437
X = 68.2628
(Values have been rounded for display purposes)

The calculation above gives the compensated operating pressure (X = 68.2628) which will be needed for the BTU/hr calculation. The rated air consumption value of the Vortex Tube will also need to be known.  A 30 SCFM rated generator will be used for this example, the normal BTU capacity of a Vortex Tube with a 30 SCFM generator is 2,000 BTU/hr.

First, determine the new consumption rate by establishing a ratio of the compensated pressure (68.2628 psi) against the rated pressure (100 psi) at absolute conditions (14.7 psia).

(68.2628 PSIG + 14.7 (atmospheric pressure)) / (100 PSIG (rated pressure) + 14.7) = .7233
.7233 x 30 SCFM  = 21.7 SCFM Input

Second, the volumetric flow of cold air at the previously mentioned cold fraction (50%) will be calculated.  To do this multiply the cold fraction setting (50%) of the Vortex Tube by the compensated input consumption (21.7 SCFM) of the Vortex Tube.

50% cold fraction x 21.7 SCFM input = 10.85 SCFM of cold air flow

Third, the temperature of air that will be produced by the Vortex Tube will need to be calculated.  For this consult the Vortex Tube performance chart which is shown below. To simplify the example the compensated operating pressure (68.2628 psi) will be rounded to 70 psig and to obtain the 70 psig value the mean between 80 psig and 60 psig performance from the chart will be used.

For the example: A 70 psig inlet pressure at 50% cold fraction will produce approximately an 88°F drop.
Fourth, subtract the temperature drop (88°F) from the temperature of the supplied compressed air temperature (70°F).

70°F Supply air – 88°F drop = -18°F Output Air Temperature

Fifth,  determine the difference between the temperature of the air being produced by the Vortex Tube (-18°F) and the ambient air temperature that is desired (90°F).

90°F ambient – -18°F air generated = 108°F difference.

The sixth and final step in the calculation is to apply the answers obtained above into a refrigeration formula to calculate BTU/hr.

1.0746 (BTU/hr. constant for air) x 10.85 SCFM of cold air flow x 108°F ΔT = 1,259 BTU/hr.

In summary, if a 2,000 BTU/hr. Vortex tube is operated at 100 psig inlet pressure, 50% cold fraction, 70°F inlet air to maintain a 90°F ambient condition with 10 psi of back pressure on the outlets of the Vortex Tube the cooling capacity will be de-rated to 1,259 BTU/hr.  That is a 37% reduction in performance.  If a back pressure cannot be avoided and the cooling capacity needed is known then it is possible to compensate and ensure the cooling capacity can still be achieved.  The ideal scenario for a Vortex Tube to remain at optimal performance is to operate with no back pressure on the cold or hot outlet.

Brian Farno
Application Engineer Manager
BrianFarno@EXAIR.com
@EXAIR_BF

## Which Vortex Tube Do I Need?

Last week, I wrote a brief introduction to vortex tubes, titled One Item Generates ¼ Ton of Refrigeration and Fits in the Palm of your Hand.” In it I introduced the Vortex Tube and the other products made from Vortex Tubes: Cabinet Coolers, Cold Guns, Adjustable Spot Cooler and Mini Coolers. I also introduced the idea of a cold fraction.  Today, I want to talk about specific Vortex Tube models.

The flow from the cold side of the Vortex Tubeis characterized in two different ways. First, we characterize the air by ΔT (temperature drop) from the starting compressed air temperature. With a supply pressure of 100 PSIG, the drop in temperature can range from 54° to 123° Fahrenheit. Second, we characterize the flow of air in Standard Cubic Feet per Minute. The different models of vortex tube are design to provide a range of flows and temperature.

When facing this list you have numerous choices that can be daunting. My priorities for selecting a Vortex Tube for a customer are twofold. First, you need the Vortex Tube that will work in your application. Second, I want to choose the model with the least amount of compressed air in order to solve their problem with the least amount of air possible. The smallest Vortex Tube is a model 3202. It also utilizes the least amount of compressed air, 2 SCFM. At 100 PSIG and an 80 percent cold fraction, it will produce a cold flow of 1.6 SCFM at 54° F  below your compressed air temperature. If your compressed air temperature is starting at 70° F, your cold temperature will 16° F. All of the Vortex Tubes will be able produce this same temperature drop, but depending on which Vortex Tube you use will determine the volume of flow produced at that temperature.

1.6 SCFM of flow 54° F below compressed air temperature will take 135 BTU/HR away from a small 100°F box, which is enough energy to cool a needle, a small sensor, or a tiny camera, but what if you have a bigger area you need to cool. Then you need to use a Vortex Tube that will produce more flow. The 3202, 3204, and 3208 will all produce air at the same temperature, but the 3204 and 3208 will produce more volume of cold air.  With the same parameters as above (100 PSIG of inlet pressure and 80 percent cold fraction) the 3204 will produce 3.2 SCFM of cold air and cool 275 BTU/Hr. out of a 100° F environment. The 3208 will produce 6.4 SCFM of cold air and cool 550 BTU/Hr. These larger Vortex Tubes could be used to cool a closed circuit camera in a hot environment or a small drill bit where coolant is prohibited or undesired. From here our product continue to produce more volume of flow and we can go up to our largest Vortex Tube, 3299 which will use 150 SCFM of compressed and cool up to 10,200 BTU/HR.

What if you have an application where you don’t need more air but 16°F  isn’t cold enough? Then you can adjust your cold fraction. Adjusting the cold fraction will allow you to increase the temperature drop. Opening the brass hot valve, will lower the cold fraction. As more air is allowed to escape out of the hot end of the Vortex Tube, the temperature and the flow rate of the cold flow decrease.  If you need to cool below a 50% cold fraction we recommend the 3400 series Vortex Tubes. At 100 PSIG this would occur when you need more than 100° F temperature drop.

Vortex Tubes can be used in a variety of cooling application. If you have any question about the topic discussed above please contact me or another application engineer.

Dave Woerner
Application Engineer
DaveWoerner@EXAIR.com
@EXAIR_DW