Accelerating tube effective length

Separating variables and integrating them in the integral of [0, np0], the distance that the particle has to travel in the accelerating tube, or, in other words, the effective length of the accelerating tube needed, L.K, can be obtained as... 

A helium gas tube is shown schematically in Fig. 3.10a. It consists of an earthed steel tube filled with He the pressure is usually around 10 bar. A high voltage, 1800 V, anode runs down the length of the tube. The charged ionisation products caused by the proton and triton ( H", Eq. (3.5), are accelerated towards the anode, causing further ionisation and an avalanche effect. This results in a gain of up to a factor of 10 and single neutron detection is readily achievable. 

Figure 4-2. A schematic of the flow domain for pulsatile flow in a circular tube at very large values of Rm. In the core region, the velocity field is characterized by a length scale lc = R and the velocity field is dominated by inertia (acceleration) effects that are due to the time-dependent pressure gradient. In the near-wall region, on the other hand, the characteristic length scale for changes in velocity is much shorter, 0(RRZ1 2), and viscous effects remain important even for very large values of R, .

The conventional theory states that if the average value of the Mach number (Mam, defined as the ratio between the fluid velocity and the speed of sound) along a channel is less than 0.3 the flow can be modeled locally as incompressible. In addition, when the pressure drop along the tube length is lower than 5 % of the inlet pressure, the effect of the acceleration of the gas flow in the axial direction can be neglected (in other words, the variation of the gas density along the channel can be neglected). 

Since the pressure drop along the tube length is much larger in microchannels than in conventional-sized tubes, the effects of the gas acceleration caimot in general be neglected, even for very low values of the Mach numbers as observed theoretically and experimentally by several authors. 

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