In parts 1 and 2 we talked about the Kinetic Theory of Gases and in particular, we showed how the movement of atoms affects principles such as molecular density, mean free path, and molecular velocity and how they are used to analyze the macroscopic properties of gases such as pressure, temperature, and flow in a vacuum environment. Here we will discuss the related topic of gas flow, gas speed, conductance, diffusion, and effusion once again focusing on the fundamental concepts.
Speed of Gas Flow
When designing and using vacuum systems, it is critical to be able to predict the time required to draw the pressure down to the desired level. Since this time is directly related to the speed of the gas flowing through the system, it is therefore important to understand what influences the speed of gas flow. Of practical importance is the influence of pipe diameter, pipe bends, and devices such as filters and condensers. Flow phenomena are not easily understood unless the differing properties of the gas in the so-called continuum and molecular flow ranges are taken into account.
Choked (Blocked) Flow
At low flow rates in the continuum, or viscous range of gas flow (occurring at pressures above roughly 1 mbar), the flow rate through a pipe or orifice is directly proportional to the pressure differential across the pipe or orifice. This changes when the flow speed of the gas reaches the speed of sound, referred to as the sonic velocity (approximately 1,223 km/h or 760 miles per hour), which is the speed of sound in that gas.
Increasing the pressure further will not increase the flow rate (Fig. 1). This condition is referred to as choked flow. Choked flow occurs when the pressure differential across the orifice, or through the pipe, is such that the pressure on the ratio between the high-pressure side and the low-pressure side of the orifice reaches a certain value specific to that gas. For air this value is 52.8% (PIN / POUT = 0.528).
This is of critical importance when venting a vacuum chamber. When the vent is opened, air at atmospheric pressure flows into the chamber at no greater than sonic velocity, no matter how low the pressure is inside the chamber. As venting continues and the chamber fills with gas, the pressure inside the chamber rises until it reaches PATM x 0.528, or 528 mbar (7.76 psi). After the pressure ratio rises above this point, the flow rate becomes proportional to the pressure difference across the vent. The takeaway is this – since the velocity of gas flow cannot be increased above sonic speed, the only way to increase the venting speed of a chamber is by using a larger vent.
|Figure 1 | Airflow velocity is limited to subsonic speed for a pressure ratio of 0.528.|
Conductance is the characteristic of a vacuum component or piping to “conduct” the gas through the system. Analogous to electrical systems, where the conductivity of a wire permits the flow of electrons through a circuit caused by the electrical potential, conductance in a vacuum system permits the flow of gas molecules through a vacuum system caused by the pressure differential generated by the pump. Conductance must be closely considered when selecting the vacuum pump and other components, to prevent reduced pumping rates and extended drawdown times. The capacity of the pump must be increased to accommodate the resistance or the inverse of the conductance.
Conductance has units of volumetric flow rate divided by pressure drop, expressed as liters per second or cubic feet per minute. The conductance between two points is defined as the gas flow rate flowing through a device divided by the resulting pressure drop.
Conductance is greatest in the viscous flow region (Cv), smallest in the molecular flow region (Cm), and in between in the transitional flow region (Ct). In other words, Cm < Ct < Cv. The flow resistance (the reciprocal of conductance) is greatest in the molecular flow range, at higher vacuum, and lowest in the viscous flow range, at lower vacuum. This may seem counterintuitive since gas is denser at higher pressures and less dense at higher vacuum. Why would a gas be easier to pump when in a denser state, and harder to pump while in a less dense state? The answer lies in the interaction of the gas molecules in the viscous flow vs molecular flow ranges.
At higher pressure, which is where viscous flow occurs, the gas molecules are relatively close together and move collectively as a group. Note that in Figure 2 the chamber pressure is greater than the vacuum pump inlet pressure. In the viscous flow range, collisions between molecules are frequent, since they are relatively close together, and when a pressure differential is exerted, the molecules move as a group. In the molecular range, on the other hand (Fig. 3), the molecules are so spread out that collisions are very infrequent. Therefore when a pressure is exerted on the chamber end of the pipe, the molecules can’t “push” each other through the pipe because they almost never collide, but rather move independently. The pump must rely on the random motion of the molecules to enter the pump inlet, at which time they are simply captured. The pump does not have the ability to draw the molecules toward it.
|Figure 2 | Flow through a pipe in the continuum range (adapted from multiple sources by the author).
|Figure 3 | Flow through a pipe in the molecular range (adapted from multiple sources by the author).
A few essential facts about conductance worth reviewing:
1. Conductance can be calculated for a system. Pipe conductance is commonly taken from charted values and is dependent on pipe diameter, pipe length, flow rate, and pressure. Conductance values for components such as valves, filters, and traps are published by their manufacturers and are based on empirical values.
2. Conductance changes during the three modes of flow through the system: continuum flow, molecular flow and Knudsen flow (the transition between the two). Recall that continuum flow occurs at higher pressure (low vacuum), and molecular occurs at lower pressure (high vacuum). Therefore the conductance of a given vacuum component is not a constant value but varies with the system pressure. When calculating conductance, therefore, it is necessary to realize that only the conductance values applicable to a certain pressure range may be applied in that range.
3. In order to select a vacuum pump, it is necessary to calculate the total conductance of the system being designed. This is usually done as an approximation, with a safety factor added. The pump can then be selected to provide the desired flow rate in each pressure range and the cumulative flow determined over time.
4. In the molecular flow region, the conductance value is independent of pressure (Fig. 4). The curves flatten out in the lower pressure range. In other words, at high and ultra-high vacuum the conductance remains constant at different pressures. This is not the case in the continuum and Knudsen ranges, where conductance is highly dependent on pressure.
5. System Geometry:
a. Shorter length and larger diameter piping are best
b. Manifold and piping diameter should be the same as, or greater than, the inlet to the vacuum pump
c. The pumping system should be physically located as close as possible to the chamber
d. The number of bends, elbows, and turns should be kept to a minimum
|Figure 4 – Conductance values for pipes of different diameters at different pressures (courtesy of Edwards Vacuum)
Effusion describes the manner in which gases to pass through a small hole from a region of higher pressure to lower pressure (Fig. 5). It is relevant to vacuum systems because it explains the action of vacuum leaks under high vacuum conditions when molecular flow prevails.
In order to understand effusion, consider a barrier between an area of low pressure (high vacuum) and of high pressure (Fig. 5). If a very small (diameter smaller than the root mean path) pinhole is created in the barrier, the kinetic motion of the gas molecules dictates that molecules will pass through the hole when their path randomly directs them through it. Recall that the root mean path (Fig. 6) is the distance a gas molecule travels before colliding with another molecule. The root mean path varies greatly over the pressure range that vacuum systems operate. Since at high vacuum any hole or leak in a vacuum system is smaller than this distance, the gas will not flow through the hole in ordered, group motion. Rather, over time this random motion causes some particles to eventually pass through the hole.
|Figure 5 – Gas Passes Through a Hole by Effusion (courtesy of prenhall.com)
Graham’s laws dictate that the effusion rate of a gas is greater for lighter gases than heavier gases. This is due to the fact that lighter molecules move faster. Since the only way for a molecule to escape from its vessel is for it to “hit” the hole, the faster the molecules are moving, the greater the likelihood that a molecule will hit the hole and effuse. Therefore hydrogen and helium, which have low molecular weight and high velocity, will pass through a leak much faster than air.
|Table 1 | Mean Free Path of Nitrogen Molecule at 0° C (adapted by the author from information supplied by Pfeiffer Vacuum).
Effusion can also be used to describe the manner in which molecules enter a high vacuum pump. Since the diameter of a vacuum pump will be greater than the mean free path for pressures below about 10-4 mbar, essentially all molecules which arrive at the inlet continue and pass into the pump because collisions between molecules in the region of the inlet are negligible.
The term effusion also helps clarify the manner in which high vacuum pumps collect the molecules that enter the pump, rather than suck the molecules toward the inlet. The molecules could be said to effuse through the pump inlet.
According to the Kinetic Theory of Gases, gas molecules are in a constant state of random motion, moving at varying speeds and in many different directions. Because of their kinetic energy at temperatures above absolute zero, all particles undergo diffusion or movement from an area of high concentration to one of low concentration. The rate of this movement is related to temperature, the viscosity of the medium, and the mass of the molecules. Diffusion causes the gradual mixing of different gases.
The principle of diffusion has relevance for the field of vacuum technology in a couple of ways. First, diffusion is the means by which gas molecules pass through a solid material to contaminate a vacuum process. Even if the components of a vacuum system could be designed with no physical leaks, as well as valves and other components that mechanically seal perfectly, small molecules such as helium can permeate polymer seals and even the metal walls of vacuum chambers. Various strategies are used to overcome this, including relying on the vacuum pump to simply remove the invading molecules, and using materials with better permeation constants.
The other relevance of diffusion in vacuum systems involves the diffusion pump (Fig. 6), which is commonly used to achieve a high vacuum. Its principle of operation relies on diffusion of the molecules of pumped gas into the pump media, such as oil or polymer. As discussed previously, all high vacuum pumps rely on the capture of the gas molecules, and diffusion pumps are able to capture the molecules by maintaining a low concentration of gas molecules diffused in the pumping media. The molecules of pumped gas are then removed from the media through condensation, and this cleaned media is constantly reintroduced, which induces further diffusion of the gas by preventing the media from becoming saturated with the gas.
|Figure 6 | Anatomy of a diffusion pump (courtesy of Edwards Vacuum)