Week 09: The concept of transmission probability and molecular flow, Dynamical analysis of molecular flow using transmission probability concepts, Exercise problems

4.7 The Concept of Transmission Probability and Molecular Flow

Consider, as in Figure 4.7, a pipe of length L, diameter D, and cross-sectional area A connecting two regions of low pressure p1 and p2, such that l >> L, D, and conditions are molecular. The total number of molecules per second crossing the plane EN to enter the pipe is J1A. They approach it from all directions within a solid angle 2π in the left-hand volume.                       

                                      Fig. 4.7  Molecular flow through a pipe

Relatively few molecules, like molecule (1), will be traveling in such a direction as to pass right through the pipe without touching the sides, but most will not. The majority, like molecule (2), will collide with the wall at a place such as X and return to the vacuum in a random direction as discussed earlier.

There are now three possible outcomes (a), (b), or (c) as shown by the dashed trajectories.

The molecule may (a) return to the left-hand volume, (b) go across the pipe to Y, and then another “three-outcome” event, or (c) leave the pipe through the exit plane EX into the right-hand volume.

These three outcomes occur with different probabilities. Furthermore, for a molecule which goes to Y at a different distance along the pipe, the balance of probabilities for where it next goes will have changed accordingly. A little reflection shows that this three-dimensional problem is quite complex to analyze.

Dynamical analysis of Molecular Flow through Long Pipe using concept of Transmission Probability

Considering the total number of molecules per second, J1A, which cross plane EN and enter the pipe, and the diverse possibilities for their subsequent trajectories, it is clear that some molecules will eventually be transmitted through the exit plane EX. The remainder will return through the plane EN as shown in Fig.4.7.

The fraction that does pass through EX into the right-hand region may be defined as the transmission probability α of the pipe so that

                                                Transmitted flux = α(J1A)

We expect that will be large for short pipes, and that for L << D it will approach unity, corresponding to the flow through an aperture in a thin wall. Increase of L, because of the increased number of randomizing collisions with the pipe wall, must cause α to decrease. For flow in the right-to-left direction, similar considerations must apply.

The transmission probability of the pipe must be the same in both directions, but the flux J2 corresponds to the lower pressure p2. The right-to-left flow (backward flux) is, therefore,

                                              Backward flux = α(J2A)

 and the net observable flow is the difference of the flows in each direction.

                                  Net forward flow (flux) = α(J1 - J2)A

Thus, multiplying this net flow rate α(J1 - J2)A by kT to get a throughput gives

                                                                                                                    

Substituting for J gives                              

 and therefore comparing with fundamental equation of throughput                                                                                                       

We obtain conductance of long pipe for molecular flow                                                                                                        

For an aperture, clearly,α = 1

For long circular pipes                                                     

Modern methods (Computer simulations) show, however, that even for L/D = 20, which would normally be regarded as a safe approximation to being long pipe, this formula gives values that are 10% too high.

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