The leak rate is composed of two elements: Q = Ql + Qo, where Ql is the true leak rate (i.e. due to a hole in the wall) and Qo is a virtual leak rate. A virtual leak is one which originates inside the system volume; it can be caused by degasssing from the walls, or from trapped volumes, which are to be strongly avoided.
The solution of the pump-down equation has:
For example, if the system volume V = 50 liter, roughly 50x20x50 cm3, then A is roughly 1 m2. Qo = qA, with a typical (good) value for q around 10-8 mbar.liter.m-2.s-1, pu = 2x10-10 mbar. This is a pressure to aim for after bakeout. The bakeout is required to desorb gases, particularly H2O, from the walls.
Note: In doing problems on the pump-down equation, some students used it too uncritically, or deduced solutions which went against their experience in the laboratory. For example, to deduce, via point i) above, that you can get down to 10-6 mbar, say, in a minute or so, is not correct. It is however correct to deduce that in that time the term -Vdp/dt becomes less than +Q; but Q itself varies (decreases) with time, as the walls outgas. This means that for almost all UHV situations we are interested in the long time limit of the equation, but with variable Q, depending on the bakeout and other treatments of the vacuum system.
where Ci are measured in liters/s. In this case, inverse conductances and pumping speeds therefore add as add as resistances in series.
Thus we need to choose Ci large enough so that S is not much less than S0; or equivalently, if S is sufficient, we can economise on the size (S0) of the pump. As with all design problems, we need to have enough in hand so that our solution works routinely and is reliable. On the other hand, over-provision is (very) expensive. We will consider actual values of C in the next section (handout).
Sometimes, if high pumping speed is essential, or if the geometrical aspect ratio is unfavorable (as in the accelerator examples given in sect 2.1), we would use multiple pumps distributed along the length of the apparatus. In this case the conductances are distributed in parallel, and
Whether this is a good solution should be clear from geometry. Obviously, a solution involving one UHV pump is simpler, if possible. Sometimes we use more than one pump because different pumps have different characteristics, as described in the next section.
By continuity, we have p0S0 = pS = Q, the flow rate. But the flow rate is also equal to C(p-p0), as this is the definition of the conductance. So, rearranging, we can deduce that
Thus there is a big error in the measurement of p at position p0, if S0 is large and/or C small. One can also use the above relations to prove that S-1 = ΣCi-1 + S0-1.
Note that in general both S and C can be functions of pressure. In the molecular flow regime, at low p where the gas molecules only collide with the walls, and where we are not near the ultimate pressure of the pump, then they are, in fact, both constant.
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