Mach disk

Behind the sampler a concentric shock wave structure is formed, which surrounds a zone of silence and ends in a shock wave front called the Mach disk. The interface should be such that the skimmer aperture still lies in the zone of silence for an optimal extraction. 

The distance from the orifice to the Mach disk may be crudely estimated from experimental work (36) as 0.67D(P /p )l/2, where D is the orifice diameter and Pf is the fluid pressure. Thus, if Pf = 400 bar, Py = 1 torr, and D 1 Jim, the distance to the Mach disk is 0.4 mm. The "droplets" formed during the expansion of a dense gas result primarily from solvent cluster formation during adiabatic cooling in the first stages of the expansion process. 

Several Type HI studies were performed at very high injectant-to-chamber pressure ratios to imitate SCRAMJET conditions. Wu and Chen [24, 25] and Lin and Cox-Stouffer [26] studied the location of shock structures resulting fi"om this type of injection. Far from the critical point, jet behavior resembled ideal-gas expansion. In contrast, homogeneous droplet nucleation was observed near the critical point. Locations of the observed shock structures, i.e. Mach disks, matched well with those from under-expanded ideal-gas jet predictions. However, the Mach disks disappeared as the injectant-to-chamber pressure ratio decreased. 

Two other quantities must be considered when designing this type of experiment the stream velocity of the carrier gas, and the location of the Mach disk or shock front. The stream velocity, v, defines the flight time for molecules in the jet and thus must be known in order to correctly time the firing of the production and probe lasers. This quantity can conveniently be calculated from the Mach number, which can be expressed as ... 

The location of the Mach disk must be considered when designing a continuous nozzle expansion. When the expanding gas meets this shock front, the turbulence destroys the desirable properties of the jet. 

At first sight this equation might be taken to imply that the Mach disk can be moved further from the nozzle simply by increasing the backing pressure. However, if we express the chamber pressure, P, in terms of... 

For He at 300K, v is equal to 1.26 x 10 cm s. Equation 14 implicitly assumes that the system pumping speed is independent of the chamber pressure. Within this relatively good approximation, we see that the Mach disk location is independent of the nozzle diameter and the backing pressure. Only the type of gas and the pumping speed affect the shock-front location. 

As a numerical example, consider a free jet pumped by a small Roots system (S = 70. s ). The Mach disk in a helium expansion from a 300K reservoir is located --11 mm from the nozzle tip - more than 70 nozzle diameters for a typical 150 /zm orifice. 

Figure 3 Schematic of the ICP-MS interface showing the supersonic expansion formed in the expansion chamber, the barrel shock, and position of the Mach disk.

Figure 5.14 Expansion of gas and ions from atmospheric pressure into a vacuum, (a) simple case showing shockwaves (barrel shock and Mach disk) and zone of silence, (b) generation of a beam of gas and ions by saiurling from the silent zone via a skimmer penetrating through the Mach disk, (c) sampling of gas and ions with a skimmer located downstream from the Mach disk, leading to the beam of gas and ions being scattered by passage through the Mach disk. Reproduced from Bruins (1991), Mass Spectrom. Revs. 10, 53, with permission of John Wiley Sons, Ltd.

Empirically, it has been found that the Mach disk location Xm, given in nozzle diameters, is... 

Thus, the ratio of the mass flow rate through the Mach disk to that of the entire free jet exiting the plain orifice can be estimated from the following equation ... 

Note that both the plain orifice and the convergent-divergent nozzle are assumed to undergo an isentropic expansion from the same initial conditions thus, the velocities and densities in each device are identical. The denominator a3xcd/of2cd in Eq. (49) can be calculated with Eq. (47) if the Mach number that the flow attains when entering the Mach disk is known. To calculate aM/ot2, one needs to know only the diameter of the Mach disk Dm and of the nozzle exit D2 ... 

Based on an extensive experimental study. Bier and Schmidt (36) developed a general correlation between the dimensionless diameter of the Mach disk and its dimensionless distance from the nozzle exit for ideal gases ... 

Here zu is the distance from the nozzle exit to the Mach disk, Ku is a constant that depends on y, and Pq is the stagnation pressure at the nozzle inlet conditions for a perfectly isentropic expansion. The dimensionless distance, zm/D2, was found by Ashkenas and Sherman (36) to be a function of the isentropic expansion pressure ratio ... 

Combining Eq. (52) with a log-linear interpolation between Eqs. (51a) and (51b), we obtain a generalized expression for the size of the Mach disk. The resulting expression indicates that the dimensionless diameter of the Mach disk generated by a freely expanding supersonic jet is a function of only the expansion pressure ratio and the thermodynamic properties of the gas ... 

Ashkenas and Sherman (36) were also able to obtain an expression for the Mach number of the flow at any position z between the nozzle exit and the Mach disk ... 

For Eq. (54) to be valid, the distance from the nozzle exit, z, must be at least 2.5 capillary diameters. Like /fM, the constants Am and zmo are functions of y. The variation of these constants with y is given in Table 1. Finally, the above equations can be used to obtain the desired expression for the fraction of the free-jet flow that passes through the Mach disk. Equation (53) is substituted into Eq. (50) then this result and Eq. (47) are substituted into Eq. (49) to obtain the desired result ... 

By combining Eqs. (52) and (54), an expression for the Mach number immediately before the Mach disk, Masx, is obtained ... 

The mass fraction of total free jet flow that passes through the Mach disk is an ideal gas as a function of the expansion pressure ratio. 

Zero-dimensional, compressible-fluid theory can also be used to analyze the pressure change that occurs across the Mach disk. Momentum, energy, and continuity equations are written for both the upstream (3x) and downstream (3y) sides of the Mach disk, with the ideal gas law being used as the equation of state. In addition, the second-law requirement that the entropy must either increase or remain constant across the disk is used. Simultaneous solution of these equations (the classical solution technique is graphical) followed by considerable algebraic manipulation yields the downstream Mach number Msy as a function of y and the upstream Mach number M3X (21) ... 

The static pressures of the fluids entering and leaving the Mach disk are derived from the application of the momentum and continuity equations, along with the ideal gas law ... 

Thus, the stagnation pressures on each side of the Mach disk are related by... 

A plot of P y/PA vs. the expansion pressure ratio, Po/Pa, is given as Figure 7 and shows that, for typical RESS conditions, the stagnation pressure on the downstream side of the Mach disk never exceeds 1 or 2. 

From the above analysis of the Mach disk, we can draw several conclusions. First, the normal shock that occurs at the Mach disk is an effective dissipator that takes the fluid from supersonic to subsonic Mach numbers. In fact, Eq. (57) shows that the higher the upstream Mach number, Masx, the lower the downstream Mach number, Masy. In RESS, the entropy of the fluid always increases, and the largest fraction of this entropy increase (particularly... 

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