Last time (The Fundamentals of Vacuum Theory – Part 1) we talked about the Kinetic Theory of Gases and how it can be used to calculate gas properties. We also considered the relationship between molecular density, mean free path, molecular velocity and pressure. Now we turn our attention to a discussion about temperature and kinetic energy, pressure and kinetic energy, and types of flow in vacuum systems. Again, we will focus on the basics, using fundamental comparisons to explain the concepts significant to industrial vacuum systems.
Relevance of Temperature to the Kinetic Theory of Gases
Based on an atomic understanding of the world we live in, the Kinetic Theory reveals that gas properties are highly dependent on the speed of their molecules, which determines their kinetic energy, and therefore the gas pressure. When considering the effects of the Kinetic Theory, it is also important to understand the influence of temperature. Specifically, the speed of the molecules in a gas is dependent on its temperature (the higher the temperature the faster the gas molecules move). Another way to think of it is that the temperature of a gas is a measure of the average kinetic energy of that gas.
According to Kinetic Theory, a gas consists of a large number of tiny molecules, all in constant random motion, elastically colliding with each other and the vessel that contains them. Pressure is the net result of the impact force of those collisions against the vessel wall. The speed of the molecules during this motion is not random at all, but follows a bell curve, with a predictable distribution about the average. The molecular speed is dependent on the weight of
the molecules and their temperature, or inherent heat energy. So for a given gas, the heat energy contained in its molecules determines their energy level and therefore their speed.
In order to fully appreciate the influence of temperature on a gas, it is useful to understand degrees Kelvin (more correctly defined simply as Kelvin), which is an absolute temperature scale (degrees above so-called absolute zero where all motion stops). This is a true measurement of kinetic energy. As there is no temperature less than zero, and no volume less than zero, absolute zero is the lowest temperature possible, where all kinetic motion stops, and the gas’s volume is reduced to zero.
This is borne out when the volume of a gas at several different temperatures is measured and plotted (Fig. 1). When the graph is then extended to 0 K, the volume goes to zero. Keep in mind the gas molecules do not individually have zero volume, but the space between molecules approaches zero. Although absolute zero has never been attained, temperatures as low as a billionth of a kelvin have been achieved on small samples. For comparison between units, bear in mind the relationship between K and °C whereby absolute zero equals -273.16°C (−460.67°F). With an understanding of absolute temperature, we can discuss the effects of temperature on gas volume and pressure.
|Figure 1 | Volume vs. absolute temperature of a gas1|
It can be seen that for a given gas, as its temperature is increased, the average velocity of its molecules increases in proportion to the square root of its absolute temperature. Take nitrogen for example (Fig. 2). At 300 K (27° C or 81° F) its molecules are traveling an average of 400 meters per second (1,312 feet per second or 894 MPH). When the absolute temperature is increased by a factor of 4, to 1,200 K (927° C or 1,700° F), its average molecular speed
increases by a factor of 2 (which is the square root of 4), to 800 meters per second (2,624 feet per second or 1,789 MPH).
|Figure 2 | Molecular velocity distribution of nitrogen as a function of temperature2
Also note on figure 2 that at lower temperatures, the curves are narrower and taller. At higher temperatures there is a wider distribution of energy levels (and corresponding molecular velocities) among the population of molecules. At lower temperatures, there is less variability between the velocities of different molecules and so the curve is narrower at its base. When the (absolute) temperature of a gas is understood to be a measure of its kinetic energy, the implications of the Kinetic Theory of Gases in regard to temperature become clear. Recall that under the Kinetic Theory of Gases, the pressure exerted by a gas is the sum total of the force exerted by all the physical impacts between the gas molecules and the vessel (or vacuum chamber, or vacuum piping, etc.) containing it. Since the velocity of the molecules is directly related to (the square of) temperature, the pressure of the gas is therefore also directly related to temperature, by the square root of its absolute temperature.
The effect of temperature on pressure as described by kinetic theory can be illustrated by a simple experiment using a party balloon and liquid nitrogen (Fig. 3). When the inflated balloon is temporarily submerged in a container of liquid nitrogen, which is at a temperature of 77 kelvin (-196 °C or -320° F), the air molecules inside it immediately drop in temperature and lose kinetic energy (i.e. their speed decreases). As a result of the decrease in molecular speed, two things happen; (a) the collisions between molecules become less frequent, allowing the space between them to decrease, and (b) the force with which the molecules collide with their container (the balloon) decreases, reducing the pressure the air exerts on the balloon. This causes the balloon to shrink. The reduced volume of the air is proportional to its absolute temperature. Since the temperature of the air inside the balloon was decreased from 293 K (20° C, room temperature) down to 77 K, the resulting volume is 77/293 = 26% of the balloon’s original volume.
|Figure 3 | Pressure in a balloon decreases when submerged in liquid nitrogen2
Types of Flow in Vacuum Systems; Continuum, Molecular and Knudsen
The manner in which a gas flows in a vacuum system is dependent on the gas pressure. At rough vacuum pressures above approximately 1 mbar (.015 PSI), continuum (or viscous) flow prevails. At these pressures, the molecules are relatively close together, and their collisions more frequent. Therefore flow is governed by interaction between molecules. As a result, the entire volume of gas, or group of molecules can be made to move in an ordered motion, that is flow (Fig .4). This ordered motion is superimposed, or added to, the normal random motion of the individual molecules. The gas at these pressures can be thought of as having a viscosity, or stickiness, that permits their ordered motion due to the internal friction between molecules. Therefore the preferred speed and direction of molecule flow will be the same as for the macroscopic gas flow.
|Figure 4 | Continuum flow of gas molecules through a pipe
Molecular flow, on the other hand, is seen at pressures below .001 mbar (.000015 PSI), which is in the high and ultra-high vacuum range. At these pressures intermolecular collisions are much less frequent due to the fact that there is so much space between gas molecules (Fig. 5). Molecules move freely without any mutual interference, and therefore there is no ordered group flow possible. The molecules individually move in a straight line without colliding, until they strike the wall of the vessel or pipe containing them. Molecular flow is present where the mean free path length (average distance a molecule must travel before colliding with another molecule) is much larger than the diameter of the pipe the gas is flowing through and therefore the molecules are free to travel until they collide with the pipe walls. As a consequence, a gas particle can move in any arbitrary direction in a high vacuum and macroscopic, continuum flow is no longer possible.
|Figure 5 | Molecular flow of gas molecules through a pipe
In the transitional range between continuum flow and molecular flow, Knudsen flow prevails. In this range both wall collisions and intermolecular collisions are influential in determining flow characteristics.
Pump Technology In The Continuum and Molecular Flow Range
Vacuum pumps that operate in the continuum (viscous) flow range such as roots blowers, screw pumps, claw pumps and rotary vane pumps function by moving the molecules as a group. They have the advantage of using the interactions between molecules (the viscous quality of the gas) to their advantage. They create a suction to draw the volume of gas to the pump inlet, then push it through the pump mechanism, and expel it at atmospheric pressure. As a result they can generate a high throughput and provide a quick drawdown in the roughing and low vacuum phases.
Pumps used to create high and ultra-high vacuum include diffusion pumps, cryogenic pumps, and ion pumps and must operate in the molecular flow range. They therefore use different technology than roughing and low-vacuum pumps. Since there is no true macroscopic flow in the molecular range, the pumps used at high and ultra-high vacuum cannot “pull” the gas out of the vacuum chamber. Rather their mode of operation is to simply capture the molecules that randomly enter the pump inlet. As a result, when the chamber pressure reaches the high and ultra-high vacuum ranges, the drawdown becomes much slower and the pump’s job becomes more difficult.
In order to understand the operation of a pump in the molecular flow range it is useful to imagine the molecules in a vacuum chamber as billiard balls on a pool table (Fig. 6) that has only one open pocket, which simulates the pump inlet. If the balls are all set in motion in random directions, such as after a break, but are allowed to keep bouncing off the bumpers without slowing down (as gas molecules do after colliding with the chamber walls), some of them would begin to fall into the pocket. Since the pocket does not have the ability to “pull” the balls toward itself, it relies on the chance occurrence that a ball will fall into it. At first, while more balls are present, there is a high probability that a ball will fall into the pocket, and the balls are removed rather quickly. As more of the balls fall into the pocket, there are fewer and fewer of them remaining, reducing the likelihood of a ball falling into the pocket. Eventually there are only a few balls left, and the probability of a ball falling into the pocket becomes very low. At this point, further ball removal is very unlikely and for practical purposes, no more balls will be removed in a reasonable amount of time. The equivalent of ultra-high vacuum has been attained.
|Figure 6 | Ultra-High vacuum pump operation simulated by balls on a pool table3 (modified by the author)
Next Time: We will discuss flow speed, conductance, diffusion, and effusion, again focusing on the fundamental concepts.
1. The Ideal Gas Law