Critical mass velocity

Green (G3) has proposed an alternate approach based on the concept of a critical mass-velocity required to produce a Mach number of 1 in a constant-area channel. Green showed this approach was able to correlate the erosive-burning data he obtained for both a double-base propellant and a composite propellant. 

Studies on fluidization. 1. The critical mass velocity" by C van Heerden, A P P Nobel and D W van Krevelen. 

Subjected to steady acceleration, a droplet is flattened gradually. When a critical relative velocity is reached, the flattened droplet is blown out into a hollow bag anchored to a nearly circular rim which contains at least 70% of the mass of the original droplet. Surface tension force is sufficient to allow the bag shape to develop. The bag, with a concave surface to the gas flow, is stretched and swept off in the downstream direction. The rupture of the bag produces a cloud of very fine droplets presumably via a perforation mode, and the rim breaks up into relatively larger droplets, although all droplets are at least an order of magnitude smaller than the initial droplet size. This is referred to as bag breakup (Fig. 3.10)T2861... 

Yarin and Weiss[357] also determined the number and size of secondary droplets, as well as the total ejected mass during splashing. Their experimental observations by means of a computer-aided charge-coupled-device camera and video printer showed that the dependence of the critical impact velocity, at which splashing initiates, on the physical properties (density, viscosity, and surface tension) and the frequency of the droplet train is universal, and the threshold velocity may be estimated by ... 

Perhaps the simplest classification of flow regimes is on the basis of the superficial Reynolds number of each phase. Such a Reynolds number is expressed on the basis of the tube diameter (or an apparent hydraulic radius for noncircular channels), the gas or liquid superficial mass-velocity, and the gas or liquid viscosity. At least four types of flow are then possible, namely liquid in apparent viscous or turbulent flow combined with gas in apparent viscous or turbulent flow. The critical Reynolds number would seem to be a rather uncertain quantity with this definition. In usage, a value of 2000 has been suggested (L6) and widely adopted for this purpose. Other workers (N4, S5) have found that superficial liquid Reynolds numbers of 8000 are required to give turbulent behavior in horizontal or vertical bubble, plug, slug or froth flow. Therefore, although this classification based on superficial Reynolds number is widely used... 

The decrease in film burn-out heat flux with increasing mass velocity of flow at constant quality has been explained by Lacey et al. in the following way. At constant quality, increasing total mass flow rate means increasing mass flow of vapor as well as liquid. It has been shown that above certain vapor rates increased liquid rates do not mean thicker liquid layers, because the increased flow is carried as entrained spray in the vapor. In fact, the higher vapor velocity, combined with a heat flux, might be expected to lead to easy disruption of the film with consequent burn-out, which seems to be what actually occurs at a constant steam mass velocity over very wide ranges of conditions—that is, the critical burn-out steam quality is inversely proportional to the total mass flow rate. 

In film breakdown at burn-out, nucleation may be a factor together with loss by entrainment and evaporation (in excess of spray deposition), and instabilities associated with surface tension. There is evidence for the existence of a critical vapor mass velocity, independent of pressure, above which the film is easily disrupted by heat fiux it is also clear that upstream conditions, including the inlet arrangements, must strongly influence the film breakdown at the exit. 

When it is desired to determine the discharge rate through a nozzle from upstream pressure p0 to external pressure p2, Equations (6-115) through (6-122) are best used as follows. The critical pressure is first determined from Eq. (6-119). If p2 > p , then the flow is subsonic (subcritical, unchoked). Then p, = p2 and M, may be obtained from Eq. (6-115). Substitution of Mx into Eq. (6-118) then gives the desired mass velocity G. Equations (6-116) and (6-117) may be used to find the exit temperature and density. On the other hand, if p2< p , then the flow is choked and M = 1. Then j> = p , and the mass velocity is G obtained from Eq. (6-122). The exit temperature and density may be obtained from Eqs. (6-120) and (6-121). 

Although the value of critical liquid mass velocity for kapp/kv 1 is above 0.3 g cm-2 s it is not a universal constant. It is a function of reactor geometry and catalyst-bed packing and, in some instances, no effect of mass velocity on the contacting effectiveness can be observed. 

The effect of confinement on the heat transfer coefficient before dry-out was found to be an increase of 74% when the hydraulic diameter decreased from 2 to 0.77 mm. The effect of confinement on dry-out was found to be a decrease in the critical quality from 0.3-0.4 to 0.1-0.2 for the same reduction of the hydraulic diameter. Heat flux dependent boiling prevailed in the 2 mm hydraulic diameter tube while quality dependent boiling prevailed in the 0.77 hydraulic diameter tube because of the difference in boiling and confinement numbers. The transition from one regime to another occurred for Bo - (1 - x) si 2.2-10 regardless of the heat and mass velocity. Moreover it was found that dry-out could even be the dominant boiling mechanism at low qualities. The results obtained with the 2 mm hydraulic diameter tube were in total agreement with Huo et al. (2004) s work. Finally frictional pressure losses seem to dominate up to mass velocities of 469 kg/m s. 

It can be seen from Fig. 20 that Mq 10 5 M0 yr 1 for the considered 60 M0 sequences. It should be stressed that, although the model is simplified and the analysis only qualitative, the derived mass loss rate at the H-limit is expected to be even quantitatively correct, since it is established by the rate of angular momentum loss which is associated with the mass loss. Once the critical rotational velocity is reached, the angular momentum loss rate is set by the expansion time scale of the star (i.e. its speed in the HR diagram) but does not depend on the value of the critical velocity. 

A further striking feature shown in Fig. 20 is the convergence of the rotational velocities to the value of v( nt,iinm (i.e., lOOkrns ), which is produced by the fact that the time dependence of the critical rotational velocity is almost independent of the mass loss history, in particular its minimum value. Note, however, that although the rotation velocities of the four sequences displayed in Fig. 20 at the end of the main sequence evolution are almost identical, their masses at that time are greatly different, and thus their ensuing post-main sequence evolution is expected to be very different as well. 

Figure 20. Upper panel Evolution of the equatorial rotation velocity with time during the core hydrogen burning phase of four 60 M sequences with different initial rotation rates (see at t = 0). The evolution of the critical rotation velocity (Eq. 5.31) is displayed for the sequence with Vrot.init. = 100 kms-1 by the triangles. It is very similar for the other sequences. For Vcrit — ( rot, the stars evolve at the Q-limit. Lower panel Evolution of the stellar mass with time for the same 60M sequences. The initial equatorial rotation velocities are given as labels. For comparison, the evolution of a non-rotating star is shown in addition.

The heart of the energy generation process is the same in all reactors. A critical mass of uranium is assembled in a tank with a moderator. One atom fissions into two lighter atoms and several high energy (high velocity) neutrons. A moderator slows down the neutrons without reacting with them. 

FIGURE 15.134 Fraction of liquid evaporated before onset of critical heat flux as a function of mass velocity for a plain tube and for a tube with helical tape inserts q" is the flux corresponding to the evaporation of all the injected water (from Moeck et al. [326], with permission). 

Fines are migrated in the medium if the salinity of the aqueous phase, CSjW, is below a critical value or if a critical flow velocity is surpassed (u > iqc). The critical salinity concept does not apply to fines migration in the oleic phase, therefore ki0dl = 0. It is assumed that the rate at which fines are trapped at pore throats is proportional to the mass flux of the particles, Furthermore, for fines that... 

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