Frozen flow

Regime 3. If the punc ture is above the initial liquid level but becomes covered by the swell, there will be noncondensables mixed with the liquid (ot, > 0). If also T, < T no flashing occurs. This is called a. frozen flow situation, since the mass fraction of compressible component X, is constant during discharge. 

To model the temporal behaviour of the turbulence induced aberrations we assume that a single layer of turbulence can be considered as frozen , but translated across the aperture by the wind. This is known as Taylor s frozen-flow hypothesis. The temporal behaviour can then be characterized by a time constant,... 

Subcooled liquid mixed with noncondensable gases (a0 > 0) (frozen flow)... 

The omega method HEM solution for orifice flow is plotted in Fig. 23-36. The solution for flashing liquids without noncondensables is to the right of = 1, and the solution for frozen flow with subcooled liquids plus noncondensables is to the left. The omega method HEM solution for horizontal pipe flow is plotted in Fig. 23-37 as the ratio ot pipe mass flux to orifice mass flux. 

Note that for most cases of interest, T is close to unity since the flowing gas mass fraction x 1. For the case of frozen flow (i.e., no heat transfer between the two phases), T would be replaced by k (or CpJC . ) in Eq. (23-106). However, the difference between the two limiting cases is small (< 10 percent) in terms of the flow capacity [Leung and Epstein, A Generalized Correlation for Two-Phase Nonflashing Homogeneous Choked Flow, Trans. ASME J. Heat Transfer 112 (May), pp. 528-530, 1990],... 

For gassy systems, G should be calculated assuming non-flashing two-phase flow, sometimes called "frozen flow V Possible methods for the calculation of G for gassy systems (using the.homogeneous frozen flow model (HFM) which is a version of the HEM) are ii ... 

Sizing formulae for flashing two-phase flow through relief devices were obtained through DIERS. It is based on Fauske s equilibrium rate model (ERM) and assumes frozen flow (non-flashing) form a stagnant vessel to the relief device throat. This is followed by flashing to equilibrium in the throat. The orifice area is expressed by... 

IV. E. 2. Frozen flow efficiencies of some potential heat... 

Approximate procedures have been evolved which permit one to determine the state of the expansion process for a given system. In fact these procedures permit the performance to be calculated when the chemical rates are finite and thus do not correspond to frozen, essentially zero chemical rate or equilibrium, essentially infinite chemical rate,flow. As one would expect intuitively, the results of these finite rate determinations show that the flow remains nearly in chemical equilibrium at the beginning of the expansion process, and at a given temperature or point in the nozzle the composition becomes frozen and remains so throughout the expansion process. Finite rate performance calculations are very complex and are presently limited to only a few systems due to lack of kinetic data at the temperatures of concern. Thus most performance calculations are made for either or both equilibrium and frozen flow and it is kept in mind that the actual results must lie somewhere between the two. For most systems equilibrium calculations are very satisfactory. 

Fig. II. C. 2 Variation of composition in a nozzle to show transition to frozen flow...

Performance values determination for frozen flow are much simpler. Since the flow is frozen the composition at the exhaust is the same as that in the chamber and therefore known. For the given exhaust pressure and composition, the entropy equation (Eq., n. C. 4.) is solved to determine the exhaust temperature. Since one does not have to make a product composition determination, the entropy equation is solved quite easily for the temperature and thus the total enthalpy at the exhaust determined just as readily. The exhaust temperature for the frozen composition case is always less than the exhaust temperature for equilibrium flow. 

In frozen flow the concentration of (J) is much greater than its -equilibrium value (Jeq) since none have recombined. Thus near frozen flow yeq y and ... 

Figure n. C. 4. also taken from (27) shows the effect of the chamber pressure on the results. Again excellent agreement is noted between the complete kinetic and Bray methods. However, notice further that as the chamber pressure increases the kinetic solutions approach the equilibrium solution. This result would be expected because at higher chamber pressures there is less dissociation and higher possible kinetic recombination ratesdue to the high mixture densities. Similarly at very low chamber pressures, sav about 1 nsia. the results would approach that of frozen flow. 

As discussed previously, in many propulsion systems the recovery of a large fraction of the dissociation energy in the nozzle expansion through recombination is difficult to achieve. While the assumption of frozen flow with respect to recombination reactions appears necessary for many heat transfer rocket nozzle expansions, it is possible that condensation phenomena are sufficiently rapid to provide near equilibrium flow with respect to phase changes. For this special possibility, phase equilibrium in the presence of frozen dissociation, it has been shown theoretically (48) that the performance in terms of specific impulse of propellants containing light metallic elements can exceed the performance of hydrogen. 

If a large fraction of the energy absorbed by the propellant is trapped by frozen flow expansion in dissociated or vaporized species, then it is convenient to consider the efficiency of energy utilization. A frozen flow efficiency is defined as ... 

An even more pronounced effect of chamber pressure would be expected and is observed in the frozen flow expansion of these same propellants, as is shown in figure V.A.3. Unlike the equilibrium expansion case, no recovery of the energy of dissociation is possible so that the positive effect of the inhibition of dissociation in the combustion chamber brought about by higher pressures is evidenced strongly in the predicted specific impulse. The shift of optimum mixture ratio toward the stoichiometric value with increasing chamber pressure again is observed. 

A chamber pressure effect of probable significant importance but as yet ill-defined is related to the acceleration of reaction kinetics at elevated pressure. Increased pressures in the combustion chamber should speed reaction kinetics and favor production of equilibrium combustion products which in turn, generally yields increased performance. Similarly, the gases in the nozzle will be at higher pressures and, thus, the exothermic three body recombination reactions will be accelerated. The transition from equilibrium to frozen flow in the nozzle, as discussed in Chapter m, should thereby be delayed and specific impulse increased. In comparing the effects of operation at very high and very low chamber pressures, the changes in reaction kinetics are likely to play an important role. 

Similarly at very low chamber pressures, say about 1 psia, the results would approach that of frozen flow. 

An equilibrium calculation is performed from the chamber to the throat and then a frozen flow calculation from the throat to nozzle exit. 

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