Transient Modeling of Hybrid Rocket Low Frequency Instabilities

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The reference-state indicated by subscript “ref‘ coirqonds to the nomina! 0pe:ztjng point around which the system i s perturbed.

Equation 2 relates the wall heat flux to the regression rate in the Laplace space and establishes one block in our overall transient modcI that is shown schematically in Figure 2. Var io~s important characteristics of a lincar system including stability can be inferred from rhe dcnominator of its transfer finction commonly referred as the characteristic equation. This transfer function for the therninl systcm, which contains a square root term, produces a phase lead between the heat flux and thc rcgrcssion rate in the low end of the frequency dornaj~i’~. AS it will be discussed in the following sections, this phase lcad capabiIity will play a critical rolc in the production of low frequency hstabilitics.

5) Gas Phase Combustion hlodel:

In the thermal lag model we have treated thc surface hear flux as a parameter that can directly be controlled. However, in a hybrid motor, the oxidizer mass flow rate is the primary input variable that can be contsolleti direal). The actual responsc of the motor to a chan~~e in the oxidizer mass flow i s rather complicatcd during a transient. As the oxidizer mass flow rate of the motor changes, thc mass flux at a characteristic point in the port reacts to the change as does thc turbulent boundary layer developed ovcr tbc fuel surface. In this section we will summarize rhe modeling ofthe boundary laytr combustion dynamics and investigate its interactions wjth the thermal lags in the solid. For the sake ofsiiqdicity, wc will ignore the radiative heat transfer to thc fuel surface, which is typically a rclaiively small fraction of the total heat flux. Unless the radiation dominates the convection component of rhe surface heat transfer, the transient model dcvclopcd in this paper is expected to be valid.

h our preliminary inodel we assume that rhe boundary-layer response is quasi-steady, namely the boundary-layer diffusion lag times are smdl cornpared to the thernial lag times in t h t solid. Under the quasi-steady assumption one can use thc cl~sictd approach15 to calculate the response of the wall heat flux to rile ChZUIgeS in mass flux. However, rhe results of classicd hybrid theory m o t be used dircctly during u transient in the solid, due to the fact that thc convective heat transfer to the wall. depends cxpliciUy on the instantaneous regression ratc through The blocking

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generated by the blowing of the gaseous fuel iiom the d a c e . In the presence of the therm21 !ags, the blocking generates a coupling mechanism in thc gas phase between the regression rate and the heat flow to the surface. Based on this understanding, the classical theory can be modified EO obtain a functional relation between wall heat flux and oxidizer mass flux-regression rate combination in terms of nondimensional parameters:

(3)

is rhe oxidizer flux exponent k is the blowing correction cxponcnt first defined by Marxman (i.e. C, /cH0 = B~ 1- Here the local total rnass fiw &at appears in the original M m m formulation is replaced by the oxidizer mass flux for convenience. The justificalion for that transformation is given in Ref. 14.

Thus far in the development of thc transient hybrid combustion theory we assumed rhat the boundary layer responds rapidly to the chmges in the m a s flux compared to thc other transient time s c a l a in the rocket motor such as the gas dynamic lags or the t h e m i lags. This assumption fails to be valid especially for largc hybrid motors. In ordu to develop n realistic model for the dynamics ofthe hybrid motor, the boundary layer lags must be considered. Since this complex dynamic phenomenon is extremely difficult to investigate both theoretically and experimentally, we consider the simpler cases reported in the literature of an incompressible turbulent boundary Iaycr with no blowing or chemical reactionsIG. The most important conclusion for this simplified case is rhat the time required for transition from the initial equilibrium profile to the final equilibrium profile at any a..ial Iocation,~, is proportional to the time of fight of a fluid particle from the leading edge of the boundary layer to the specific axial location at the spoed of the free stream flow,#,. Thhis very important result can be hrmuiated for hybrid boundary layers as

Here c‘ is a wnstant that needs to be determincd empirically. We will caIl this time required for c q d i b d w , the characteristic response time of the boundary laycr, rb,. it is

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importar to note that the physical na.re of the boundary layer transient time is nor related to the propagation of the disturbarms with thc spced of the port velocity as suggested by Equation 4. The delay rathcr dcpcnds on the di€fusion time scale across the boundary layer which is proportional to the ratio of the local boundmy layer thickness to the diffusion speed, 561 = J/U' I The diffision speed is defined in terms of thc shcar strcss and mean gas density as U* = m. The bounday layer delay time, after the substitution of the standard (incompressible) turbuknt boundary layer expressions17 for rhe sliear stress, ro = 0.0288~ u, Re:', and the thickness,

6=0.37zRcJo2, bccorncs z,,, = 2 . 1 8 R ~ ~ ~ ' z / ~ i , . Here the local Reynolds nuiuber is defined as Rez =u,zp/ /r . Note that the coefficient c' is found to be a weak function of the local Reynolds number. Thus, for simplicity, we assume that c' is constant. For Reynolds numbers corresponding to typical hybrid operation, c' is astirnatcd to ba approximately 0.55. In a real hybrid boundary layer with combustion and blowing, c' can be different from this estimation and for that reason, it is determined empirically.

For the purpose of this paper, it is convenient to consider an avcragc boundary laycr delay and replace he locaI distance z, with the half of the grain lengrh, L/2. Note that the empirical constant C' accounts for tlic correction to the inaccurate selection of the length scale, L. However, we recognize that in reality there is a range of boundary layer delay times that should be considered. Thus the significant observation here is that a relatively broad band of oscillation frequencies is expected as opposed to a very shvp peak at the center frequciicy coi-rcsponding to L / 2 .

In our studies the response of the boundary layer to the changes jn the mass ffux is

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accounted for by simply inserting time delays in the hear-flux expressions derived under the assumption of quasi-steady response. The implementation o f th is idea in the linearized version of Eq. 3 yiclds

whcre

Hcre T~,, and T~~~ arc the time delays

experienced by the wall heat flux (&I to thc changes in the oxidizer mass flux and the regression rate, respectively. The scaling of the time delays rb1, and zbl2 obey thc general scaling law given by Eq. 4. However the c' cwfficients for zd,, and zb,? are expected to be different since each of these delays represents a different adjustment mechanism for the boundary layer. The model presented by Equation 5 is central to the Dnalysis in that it relates the heat conduction in the solid to thc boundary layer combwXion.

6) Thermnl-Combustion (TC) CoupIed System

Now, with the use of Equation 5 the thermal lags in the solid can be coupled to the combustion transients in thc boundary layer. This coupling yields the following transfer function between the oxidizer mass flux (?(s)) and the

rcgrmsion rate (R,(s)) and represent the Hybrid Combustion block in the overall transient model (Figure 2).

This transfer function (TC coupled ~ I - O C S S ~ in the solid and approximates the system) that rcprcscnts tlic cornbustion combustion dynamics in the turbulent boundary phenomenon includes the dynamics of thc thermal layer of the rocket motor. Equation 6 can be used

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combastiaa luslabii m Liqvid RopclhoC Rockel Motors", Agardograph No. 8,1956.

Diffirence Methods for Fluid Dynmics", 1993. 25. K h m u d T. P o h q Ti

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Fg.= 4: Tbe uns?abk p o k of the TC coupled system m the s plane fir, = 0 , ry2 = 38 msez This plot is for a HTPB system with E, = I5 kcaUmole.

22 ArncTican lmtitute of Aeronautics and Astronautics

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