Fluid Mechanics Applications/B47 Vacuum Cleaner

Vacuum cleaner is device that is used to remove dust or dirt by sucking mechanism.
The dust is either collected in a dust bag or a cyclone for later disposal. Vacuum
cleaners come in variety of sizes and models suited to achieve the purpose
differently. The most efficient and environment friendly are diaphragm pumps.
A diaphragm pump can save upto 90% of the operating costs compared to a
water jet pump.^[1]

Basic Terms

Working pressure, P_w

Vacuum cleaners use pumps to create vacuum. During the evacuation process the gasses and vapours are also removed from the vessel. Vapours may get condensed on increasing pressure unlike gasses at the prevailing temperatures.^[2] Because of this the analytically calculated pressure is not achieved. Thus the pressure required is achieved by additional pumping. This pressure obtained is known as working pressure.

Particle number density, n

The pressure exerted by a gas depends on both temperature and number of gas molecules as explained by the Kinetic Gas Theory.
It's unit is cm⁻³

P=\,nkT

n is particle number density

k is The Boltzmann’s constant

We may also say that at a particular working condition(considering it to be isothermal) the pressure exerted by a gas is dependent on the number of gas molecules. Concentrating on the Kinetic Gas Theory formulae we can conclude that the pressure does not depend on gas particle characteristics i.e. mass of the particle.

Pump throughput, q

Pump throughput is the pumping capacity of the pump. It is equal to the mass flow rate at pump inlet or to the pV flow through the pump's intake port.^[3]

q={\frac {m}{t}}

q={\frac {pV}{t}}

Conductance, C

The pumping capacity of a pump is proportional to the pressure difference between the inlet and outlet of a pipe, valve etc.^[4] The constant of proportionality here is known as conductance, C.

q=\,C(P_{1}-P_{2})

Here P₁ - P₂ is the pressure difference between inlet and outlet ends of the piping system. The value of C is affected by the geometry of piping element and can also be calculated for a system of piping elements connected in series or parallel configurations.
We may convert our capacitance of each element into a term R, resistance to flow.

R={\frac {1}{C}}

Now this value of R can be used as follows

Pipes in series

R_{T}=R_{1}+R_{2}+R_{3}+...+R_{n}

Pipes in Parallel

\sum _{i=1}^{n}{\frac {1}{R_{i}}}={\frac {1}{R_{T}}}

where R_T is the total resistance to flow of a piping system

Conductance for a straight pipe, which is not too short, of length l, with a circular cross section of diameter d for the laminar, Knudsen and molecular flow ranges, valid for air at 20 °C can be calculated by this formulae^[5]

C={\frac {\,135Pd^{4}}{l}}+{\frac {\,(12.1d^{3})(1+192Pd)}{\,(l)(1+237Pd)}}

where

P={\frac {P_{1}+P_{2}}{2}}

P₁ is the pressure at the start of the pipe in mbar

P₂ is the pressure at the end of the pipe in mbar

Interesting Fact

We all know that Helium is only about 5 ppm in the atmosphere. "In ultra high vacuum systems sealed with Vitron or incorporate glass or quartz, this helium gas is able to permeate these substances to a considerable extent."^[6]

Vacuum Ranges and Their Characterization^[7]

Range Name	Abbreviation	Pressure Range	Unit
Rough Vacuum	RV	1000 - 1	mbar
Medium Vacuum	MV	1 - 10⁻³	mbar
High Vacuum	HV	10⁻³ - 10⁻⁷	mbar
Ultrahigh Vacuum	UHV	10⁻⁷ - 10⁻¹⁴	mbar

Vacuum Generation

Vacuum pumps are used to reduce the gas pressure in certain volume and thus reducing gas density. There are variety of vacuum pumps and varieties in each type too. Sticking to my topic vacuum cleaner and obviously those which are used in homes. I stick to only two vacuum cleaners.
1. Roots vacuum pump
2. Turbomolecular

Roots Vacuum Pump

The designing of this vacuum pump was complete in 1848 and twenty years later it was invented. Since 1954 roots pump have been used in vacuum engineering. The Root pumps are used in combination with backing pumps and can be used to work in a medium vacuum range. Two stage Roots pump extend this to high vacuum range.

A Roots vacuum pump is a rotary positive-displacement type of pump where two symmetrically-shaped impellers rotate inside the pump casing past each other in close proximity and synchronized by a toothed gear. The clearance between the rotors and the casing wall as well as between the rotors themselves amounts only to a few tenths of a millimeter.
Roots pump are not oil sealed. Compression ratios of 10-100 can only be achieved.
During the compression phase these surface areas (rotors and casing) are loaded with gas (boundary layer); during the suction phase this gas is released. The thickness of the traveling gas layer depends on the clearance between the two rotors and between the rotors and the casing wall.^[8]

Turbomolecular Pump

The principle of the molecular pump is that the gas particles to be pumped are hit with the rapidly moving surfaces of a rotor, an impulse in a required flow direction. The spaces between the stator and the rotor disks are made in the order of millimeters for larger tolerances. However, a pumping effect of any significance is only attained when the circumferential velocity (at the outside rim) of the rotor blades reaches the order of magnitude of the average thermal velocity of the molecules which are to be pumped.^[9]

Air Density variation near Blades of Vacuum Pump

The flow profile followed in the Roots pump and Turbomolecular pump is quite much complex. It was far beyond my scope. The Roots pump follows Double Folium Cycloidal profile. This complex problem could not be solved by me. I apologize for my lack of knowledge.

Dust bag and Cyclone Working

Cyclone Working

Conventionally cyclone is installed as first separating device followed by bag filter.
The working of Cyclone is as follows.

1. The dust-laden air enters the cyclone chamber.
2. The dust /powder separates due to cyclonic action and then enters the bag filter chamber.
3. The air is uniformly distributed avoiding channelling while the powder separated by cyclone slides down from sides.
4. Initially a coat of material forms on the bags. Subsequently, the coat acts as the filtering medium.
5. The dust is accumulated on filter elements while the air passes through the filter bags from outside to inside.
6. The accumulated powder is dislodged from the bags by reverse pulsejet air intermittently.
7. The dislodged powder falls on bottom cone and is discharged through powder discharge valves.
8. The dust free air is sucked by induced draft fan and is exhausted to atmosphere.
Note : Knockers are provided on conical portion especially for sticky/hygroscopic powders/dust.

Dust bag Working

The bag filter system, often called bag house, has three sections
1. Clean air Chamber
2. Housing
3. Hopper
Dust is drawn into the bag through duct lines. The chamber where the dust bearing common duct enters is called dusty chamber and the dust free chamber is called as clean air chamber. The clean air is pulled out by draft generating fan (induced draft fan). The dust is retained in the dust bag.

References

1. Dr. Walter Umrath et al, Fundamentals of Vacuum Technology, (Cologne, June 2007), 20
2. Dr. Walter Umrath et al, Fundamentals of Vacuum Technology, (Cologne, June 2007), 9
3. Dr. Walter Umrath et al, Fundamentals of Vacuum Technology, (Cologne, June 2007), 11
4. Dr. Walter Umrath et al, Fundamentals of Vacuum Technology, (Cologne, June 2007), 11
5. Dr. Walter Umrath et al, Fundamentals of Vacuum Technology, (Cologne, June 2007), 16
6. Dr. Walter Umrath et al, Fundamentals of Vacuum Technology, (Cologne, June 2007), 13
7. Dr. Walter Umrath et al, Fundamentals of Vacuum Technology, (Cologne, June 2007), 14
8. Dr. Walter Umrath et al, Fundamentals of Vacuum Technology, (Cologne, June 2007), 27
9. Dr. Walter Umrath et al, Fundamentals of Vacuum Technology, (Cologne, June 2007), 46