Burning rate mass flux

The above experimenters have used the technique described to obtain flow rate measurements of the liquid wall-film at various mass velocities, tube dimensions, etc., and some typical results from Staniforth and Stevens (S7) are shown in Fig. 7. Also shown are the values of burn-out heat flux obtained at the four different mass velocities indicated. It can be seen that the liquid-film flow rate decreases steadily with increasing heat flux until at burn-out the flow rate becomes zero or very close to zero. We thus have confirmation of a burn-out mechanism in the annular flow regime which postulates a liquid film on the heated wall diminishing under the combined effects of evaporation, entrainment, and deposition until at burn-out, the film has become so thin that it breaks up into rivulets which cause dry spots and consequent overheating. 

Since at y = 0, the mass flux pv is the mass loss rate of burning rate evolved from the condensed phase. From Equation (9.18) we also realize that we have a constant pressure process... 

With T) = 1300 °C, Too = 25 °C and 7Fj0 = 1, we have two equations in two unknowns m" and nip for the extinction conditions. However, under only suppression of the burning rate, Equation (9.80) applies, giving a nearly linear result in terms of q" and m" as well as 02,oo. The nonlinearity of the blocking effect can be ignored as a first approximation, and the experimental results can be matched to this theory. By subtracting Equation (9.81a) from Equation (9.80), we express the critical mass loss flux as... 

Equation (9.106) can be modified by eliminating all of the mass terms in favor of the burning rate (flux) term. This gives... 

If the fuel responds fast to the compartment changes, such a quasi-steady burning rate model will suffice to explain the expenditure of fuel mass in the compartment. The fuel heat flux is composed of flame and external (compartment) heating. The flame temperature depends on the oxygen mass fraction ( Yq2 ), and external radiant heating depends on compartment temperatures. 

In Zone III combustion, the burning rate is determined by the diffusive flux of oxygen through the particle boundary layer. The particle density remains constant throughout burnout and the particle size continually decreases as mass is removed solely from the particle surface (constant-density combustion). 

Fig. 11.13 shows the temperature profiles in the combustion waves of AP py-rolants with and without B parhcles at 1 MPa. It is evident that the temperature gradient above the burning surface is increased by the addition of B particles. Thus, the heat flux transferred back from the gas phase to the propellant is increased, and hence the burning rate is increased. The heat of reaction is also increased as the mass frachon of B particles is increased, as shown in Fig. 11.14. The results indicate that the B particles act as a fuel component in the gas phase, undergoing oxidation just above the burning surface.

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. 

With empirical pyrolysis models, a material s burning rate is zero until its surface is heated to its ignition temperature (Tig), at which time ignition occurs. After ignition, the mass loss rate of a fuel element is estimated from the net heat flux to the fuel s surface (q"a) divided by the effective heat of gasification (A7/g) ... 

Fuel loading or burning rate group (Time average mass burn rate to convective mass flux). 

Another key issue is the sensitivity of the mass flux or burning rate and surface temperature to the independent variables, P, To, qr. The sensitivity of burning rate to pressure and initial temperature is obviously important for internal ballistics and rocket motor performance prediction for quasi-static operation. The sensitivity of surface temperature is not quite as obvious but is related to the unsteady combustion behavior through Zeldovich-Novozhilov (ZN) theory. The sensitivity parameters are derivatives of the steady equations as defined in the nomenclature. Equation (15) or (16) can be differentiated with respect to initial temperature (To), pressure (Dg), and radiative flux (qr) to give... 

The physical processes in the gas-phase and subsurface regions must be matched at the interface by requiring continuities of mass and energy fluxes. This procedure eventually determines propellant surface conditions and burning rate as the eigenvalues of the problem. The interfacial boundary conditions are expressed as follows ... 

The two phases are coupled explicitly with mass flux and energy flux preserved. Thermal radiation from gas phase reaction is neglected in this model only laser radiation and conductive heat feedback from adjacent gas phase grids are taken into account as energy source terms for the condensed phase, which is critical to realize the pressure dependence of the mass burning rate. 

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