## RADIATION HEAT TRANSFER IN A SUPERSONIC NOZZLE OF A SOLID-PROPELLANT ROCKET ENGINE

* Following from: *
Radiation heat transfer in a solid-propellant rocket engine

A principal difficulty of radiation heat transfer calculations in a supersonic nozzle is a significant variation of the medium temperature and density along the nozzle axis. In contrast to combustion chamber conditions, the temperature of condensed phase particles in the nozzle is several tens or even hundreds of degrees greater than the temperature of gas, and it is different for particles of various sizes. For this reason, a proper calculation of the radiation heat transfer in a supersonic nozzle is impossible without two-phase flow calculations.

Calculation of concentration and temperature fields of two-phase combustion products in a supersonic nozzle is a complicated physical and mathematical problem due to the substantial velocity and temperature nonequilibrium of the condensed phase. Strictly speaking, one should take into account the influence of particles on the gas flow field, as well as break up and coagulation of particles. Mathematically, the problem is additionally complicated by the 2D flow pattern, particularly in the transonic region, and also by complex particle size distribution of the condensed phase. Both break up and agglomeration of alumina melt particles appear to be important at separate regions of the nozzle flow. The latter makes the problem especially complicated.

The study of gas flow with particles in a supersonic nozzle has a long history. One can remember the early papers by Carlson and Hoglund (1964), Hoffman and Lorenc (1965), Jenkins and Hoglund (1969), and Regan et al. (1971), and the books by Sternin (1974), Pirumov and Roslyakov (1978), Yanenko et al. (1980), and Sternin and Shreiber (1994), which are well known in Russia. Some additional methodological results can be found in other papers (Chang, 1980, 1983; Crowe, 1982; Hwang and Chang, 1988; Carrier et al., 1991; Vasenin et al., 1995). The general methods of two-phase flow calculations have been already developed and one can use the present-day codes for nozzle flow calculations. In this article, taking into account the known difficulties with the use of general algorithms, the flow model considered is as simple as possible. We consider a conical nozzle and suppose that the gas flow field is 1D, and it has spherical symmetry with the center in the cone vertex. Strictly speaking, this assumption is not correct and results in a wrong flow pattern near the critical cross section of the nozzle (Pirumov and Roslyakov, 1978). Nevertheless, the 1D flow analysis seems to be applicable for investigation of the main regularities of heat transfer by radiation in supersonic nozzles of high expansion ratio. It was shown by Dombrovsky (1996a) that the variation of gas flow parameters in a nozzle cross section makes just no difference for the radiation heat transfer. At the same time, it is worth remembering that Klabukov et al. (1988) have used a more detailed flow model in the radiation problem.

The fields of velocity, concentration, and temperature of the condensed phase are determined by calculations of the velocity and temperature delay of single particles. The condition of dynamic and thermal equilibrium between the condensed phase particles and ambient gas is assumed in the initial cross section of the nozzle. The mathematical problem statement is as follows:

where u_{g}(x) and T_{g}(x) are the known profiles of gas velocity and temperature along
the nozzle axis, C_{D} is the drag coefficient, and Nu is the Nusselt number. The
values of C_{D} and Nu are determined by using the known semi-empirical
relations. Following the paper of Dombrovsky (1996a), consider two high-altitude
nozzles with the diameter of critical cross section d_{*} = 0.2 m and 0.02 m, the
angle between the nozzle wall generatrix and the axis θ_{0} = 21 deg, and the
expansion ratio ξ_{a} = d_{a}/d_{*} = 8.7. For brevity, the larger nozzle is hereafter
referred to as nozzle 1 and the smaller nozzle as nozzle 2. A comparison of the
results of calculations for these nozzles enables us to determine the effect of
the nozzle dimension on the heat transfer by radiation. We do not provide
the data for the composition and temperature of the combustion products;
however, for assessing the level of parameters, note that the temperature in the
combustion chamber is about 3600 K, the pressure is about 6 MPa, and the mass
fraction of condensed phase particles is about 0.34. These values are typical of
solid-propellant rocket engines (Shishkov et al., 1988). In the nozzles considered, the
gas temperature near the exit cross section is much lower than the melting
temperature of aluminum oxide, T_{m} = 2320 K. According to Reizer et al.
(1973), the crystallization of particles must be included in the case of nozzle
1. It was shown experimentally by Vazhinsky (1980) that the maximum
possible relative supercooling of alumina is about 20%, i.e., spontaneous
crystallization occurs at the temperature T_{s} ≈ 0.8T_{m}. The following expression was
given by Henderson (1977) for the velocity of a crystallization wave in a bulk
sample:

This formula was recommended for condensed phase particles at A_{f} = 0.72 × 10^{-6}
m/(s·K^{1.8}), n_{f} = 1.8. The value of L_{m} = 1070 kJ/kg was taken for the latent heat of
alumina melting. The results of calculations by taking into account the crystallization
kinetics are given in Fig. 1 from books by Dombrovsky (1996b) and Dombrovsky and
Baillis (2010). In the graphs, ξ = d/d_{*} is the current expansion ratio, and
ρ_{p}(ξ) = ρ_{p}(ξ)/ρ_{p}(1) is the relative mass concentration of the condensed phase
particles. A temperature increase during the crystallization of particles is explained
by the release of the latent heat of melting. The mass concentration and temperature
of particles decrease significantly with ξ. This leads to the corresponding variation
in radiative characteristics of combustion products. Note that there is a
considerable difference in the temperature of large particles between nozzles 1 and
2.

**Figure 1. Variation of particle temperature (a) and relative mass concentration of
condensed phase (b) along the nozzle axis: solid lines-nozzle 1; dashed lines--nozzle
2; 1-a= 1 μm, 2-a= 3 μm, 3-a= 5 μm**

The spectral absorption coefficient and transport extinction coefficient of two-phase combustion products are calculated as follows:

where α_{λ}^{g} is the absorption coefficient of the molecular gas mixture calculated by
using the box model (Modest, 2003). Note that this approximation for the absorption
coefficient of gases is justified by the predominance of the effects due to the
presence of numerous particles. The results of calculations of α_{λ} and β_{λ}^{tr}
for nozzle 1 are shown in Fig. 2 from the book by Dombrovsky and Baillis
(2010). The calculations were performed for the polydisperse condensed phase
with gamma distribution of particles by size [see Eq. (10) from the article,
“Radiative properties of polydisperse systems of independent particles”] with
parameters A = 3 μm^{-1}, B = 2. One can see that α_{λ} drops by about three
orders of magnitude when the expansion ratio ξ varies from 1 to ξ_{a}. As a
result, the relative contribution of the gas phase increases appreciably. The
transport extinction coefficient β_{λ}^{tr} decreases not as fast as α_{λ} since the value of
β_{λ}^{tr} is insensitive to the particle temperature. As a result, the transport
albedo ω_{λ}^{tr} = 1 - α_{λ}/β_{λ}^{tr} increases along the nozzle and the scattering
considerably dominates over the absorption, especially in the large-scale
nozzle.

**Figure 2. Variation of the spectral absorption coefficient α _{λ} and transport
extinction coefficient β_{λ}^{tr} along the axis of nozzle 1 (I) and nozzle 2 (II):
1--λ = 1 μm, 2--λ = 2 μm**

In calculations of the thermal radiation transfer in a nozzle, one should take into account the differences between the temperatures of particles of various sizes and the gas temperature. For this reason, the term corresponding to the emission of thermal radiation in the radiative transfer equation is written in the form

where the first term is the condensed phase contribution and the second term corresponds to the gas phase contribution.

The numerical solution for the 2D radiative transfer problem in a nozzle becomes complicated by the following circumstances: a complex shape of the computational region, a sharp variation of parameters along the nozzle in the acceleration region and in the region of c-phase crystallization, a variation of the wall surface temperature, and an indeterminate radiation from the exhaust jet toward the nozzle.

When a lot of condensed phase particles are in the boundary layer on the nozzle wall, the thermal boundary layer may be not quite transparent for thermal radiation. In this case, the problem of the wall radiation flux determination becomes considerably complicated, the more so because the particle temperature in the boundary layer may be much greater than that in the main flow. In this section, we consider only the case of low particle concentration in the boundary layer. Note that this assumption is usually satisfied for the nozzles with a large spread angle, which is used to avoid intense mechanical erosion of the nozzle by impact of condensed phase particles. Under the favorable condition of a transparent thermal boundary layer, the convective and radiative heat transfer do not influence each other, and the total heat flux toward the nozzle wall can be determined as a sum of convective flux and radiation flux from the central region of the flow. It is known that particle concentration in the boundary layer on the nozzle wall may be considerable (Soo, 1989). In this case, one should solve a combined heat transfer problem in the boundary layer.

Generally speaking, thermal radiation affects the temperature of particles in a supersonic nozzle, i.e., the radiation in the axial direction is partially absorbed by comparably cold particles, and they are heated. This effect depends on the particle size. The calculations by Dombrovsky (1996a,b) showed that particles of radius of about 1 μm are heated by radiation in the ξ < 3 zone and cooled by own thermal radiation at ξ > 3. The particles of radius a = 5 μm are cooled by thermal radiation during their motion in the nozzle, even at ξ > 1.7. A variation of the particle temperature under the effect of radiation is small (on a level of 10 K), and this effect can be neglected in practical calculations. It should be noted that the effect of thermal radiation on the temperature of particles may be considerable under off-design conditions of the nozzle flow, i.e., the particles approaching the shock wave are heated by radiation propagating upstream. According to the data by Klabukov et al. (1988), the related variation of the particle temperature may reach 200 K.

To the best of our knowledge, only a few papers are concerned with the radiation
heat transfer in rocket nozzles. This may be explained by two reasons. In the first
place, the early estimates based on 1D models indicated the insignificant role of the
radiation as compared with the convective heat transfer in the nozzle (Pearce, 1978).
In the second place, as was mentioned above, a correct calculation of the radiative
transfer in a supersonic nozzle is very complicated, i.e., one should calculate the flow
field and optical properties of two-phase combustion products and solve
the 2D radiative transfer problem in an essentially inhomogeneous medium
taking into account a variable temperature of the nozzle surface. Vafin et al.
(1981) and Dombrovsky and Barkova (1986) have shown that the previously
existing view of the minor importance of thermal radiation for heat transfer
in the nozzle of a solid-propellant rocket engine (SPRE) is incorrect, and
radiative heat transfer should be taken into account in thermal calculations.
Vafin et al. (1981) were probably the first to solve the 2D radiative transfer
problem for an SPRE nozzle. The radiative transfer model was based on the
P_{1} approximation. The corresponding boundary-value problem was solved
by a finite difference method. Absorption and scattering of radiation by
condensed phase particles were taken into account. A series of calculations led
to the conclusion about a significant error of a 1D approximation in the
supersonic part of the nozzle. At the same time, some of the basic assumptions by
Vafin et al. (1981) restrict the possibility of using the obtained results. In
particular, the condensed phase was considered as monodisperse, and the thermal
and dynamic nonequilibrium of the condensed phase, as well as particle
crystallization, were not taken into account. The main, more important,
assumption was in ignoring the thermal radiation of the internal surface of the
nozzle. Thus, the radiative transfer calculated in the paper by Vafin et al.
(1981) may be referred to only at the beginning of the engine running. As
shown by Dombrovsky (1996a,b,c), thermal radiation of the nozzle walls is
an important factor forming the radiation field in the nozzle of an SPRE.
The above-mentioned limitations in the problem statement were removed in
papers by Dombrovsky (1996a,c), which are used as a basis of the present
section.

An analysis of the radiative heat transfer in a liquid-propellant rocket engine (LPRE) by taking into account the opacity of combustion products due to the presence of soot has been performed by Hammad and Naraghi (1989). The combustion products were considered as a gray medium with isotropic scattering. An additional difference from a similar problem for an SPRE is the cooling of the walls, which makes their radiation insignificant. It was found by Liu and Tiwari (1996) and Nelson (1997) that radiative effects on the wall heat transfer may be significant in chemically reacting nozzles and scramjet combustors. In a recent paper by Byun and Baek (2007), a detailed numerical investigation of radiation heat transfer in an LPRE was presented. The authors used a spectral radiation model with a present-day description of radiative properties of gases and soot particles. It was shown that the effect of radiation (mainly from soot particles) on the heat transfer to the nozzle wall may be significant.

In papers by Dombrovsky and Barkova (1986) and Dombrovsky (1996a), the
calculations of 2D radiative transfer in supersonic nozzles were performed by use of
the P_{1} approximation and finite element method. The boundary value problem for
the spectral radiation energy density I_{λ}^{0} is as follows:

where D_{λ} = 1/(3β_{λ}^{tr}) is the radiation diffusion coefficient, γ_{w} is the boundary
condition parameter equal to ε_{w}/(2 - ε_{w}) on the nozzle wall (ε_{w} is the wall
emissivity), S_{w} = B_{λ}(T_{w}) on the nozzle wall, T_{w} is the temperature of the nozzle
internal surface. The condition of zero radiation flux (γ_{w} = 0) was assumed in the
critical cross section of the nozzle (at ξ = 1). To determine a correct boundary
condition in the exit cross section of the nozzle ξ = ξ_{a}, taking into account the small
self-emission of a high-altitude jet and radiation backscattering, the computational
region was enlarged by conventional initial zone of the jet ξ_{a} < ξ < ξ_{0} with
extrapolated optical properties of combustion products. A variation of parameters of
the jet initial zone showed that a boundary condition of zero external flux (γ_{w} = 1,
S_{w} =0) at ξ = ξ_{a} would be crude. This condition leads to additional radiation
flux to the nozzle wall due to an error of assumed linear approximation
of the angular dependence of radiation intensity in the P_{1} approximation.
As a result, the calculated integral radiation flux from the wall near the
nozzle exit may be ~ 30% less than the exact value. As pointed out above,
this problem was overcome by insignificant enlarging of the computational
region.

The spectral and integral radiation fluxes to the nozzle wall are determined by the following relations:

Following the method described in the article, “Finite element method for radiation diffusion in nonisothermal and nonhomogeneous media,” the boundary-value problem (6) was formulated as a variational problem when the minimum of the functional

should be found. This problem was solved numerically by the finite element method.
The solution was obtained in spherical coordinates for the computational region
0<θ<θ_{0}, 1<ξ <ξ_{0}. In analyzing the main results of calculations of heat transfer by
radiation in a nozzle, we will restrict ourselves to the computational data for the
following two approximations of temperature profiles of the nozzle internal
surface:

which correspond to different time moments of the engine running. Some results of calculations for nozzle 1 are shown in Fig. 3 from the books by Dombrovsky (1996) and Dombrovsky and Baillis (2010). One can see that the integral radiation flux is comparable with the convective heat flux in the region of high expansion ratios. The relative role of the radiation may vary severalfold during the nozzle wall heating. The gaseous phase emission and absorption are insignificant, whereas the crystallization of the condensed phase increases appreciably the radiation flux to the wall at ξ > 3. It can be shown that the monodisperse approximation discussed above as applied to the combustion chamber may lead to essential errors in the region of ξ > 3 due to the incorrect determination of the condensed phase temperature.

**Figure 3. Convective heat flux q _{w}^{c} and integral radiation flux q_{w} on the wall of
nozzle 1: a--k= 1, b--k = 2; 1--calculation ignoring the gas radiation (α_{λ}^{g} = 0),
2--calculation ignoring the crystallization of particles (L_{m} = 0)**

To determine the radiation flux to the nozzle wall in a 1D approximation, the
assumption of constant gas flow parameters in every cross section of the nozzle
was used. When the flow field is considered as central-symmetric, the flow
parameters are constant only along the spherical surfaces. With transfer
to the axisymmetrical 1D problem, we use the flow parameters near the
wall. An error of this approximation is not significant. The radiation flux to
the nozzle wall was calculated by using the analytical solution in the P_{1}
approximation. A comparison of the 2D numerical solution with the 1D
approximation (see Fig. 4) demonstrates that 2D effects must be included everywhere
except for the region adjoining the critical cross section. For nozzle 1, a 1D
calculation proves acceptable in the case of ξ < 3, and for nozzle 2, only at
ξ < 1.5. The radiative transfer along the nozzle axis leads to a substantial
variation of the radiation flux to the wall in the region of high expansion ratios.
The examples discussed above are characterized by the predominance of
the heat removal due to self-radiation of the wall toward the nozzle exit
rather than by heat supply from the high-temperature zone located upstream.
Figure 4, reproduced from the book by Dombrovsky and Baillis (2010),
further illustrates the important role of the scattering of the radiation by
particles. The presence of scattering leads to partial shielding of thermal
radiation from the high-temperature region, and to reduction of the radiation
flux from the walls toward the nozzle exit cross section. A failure to include
radiation results both in incorrect determination of radiation flux and in
considerable distortion of the entire pattern of thermal radiation transfer in
the nozzle. The hot walls of the nozzle in the region from ξ ≈ 1.5 to 3.5
radiate freely toward the nozzle exit, and the walls in the region ξ > 6-7 are
additionally heated by this radiation. The importance of heat transfer by
radiation is especially great in the case of small nozzles (see the data for nozzle
2).

**Figure 4. Integral radiation flux to the nozzle wall: a--k= 1, b--k =
2; 1--complete two-dimensional calculation, 2--one-dimensional calculation,
3--two-dimensional calculation ignoring scattering**

The foregoing inferences about the importance of the 2D effects under conditions
of heat transfer by radiation in supersonic nozzles and about the role of scattering
are corroborated by the spectral dependences given in Fig. 5 from the book by
Dombrovsky and Baillis (2010). In particular, the positions of the extrema of the
curves q_{wλ}(λ), calculated in disregard of scattering, is indicative of the radiation
source, i.e., in the cross section ξ = 5.8 (Fig. 5a), the small maximum at λ < 1 μm
corresponds to radiation from the hot region, while the pronounced maximum at λ =
1.8 μm corresponds to intensive thermal self-radiation of the comparably cold wall of
the nozzle. In the exit cross section of the nozzle (Fig. 5b), the predominant
effect is that of the radiation received by the wall with a maximum at the
wavelength λ = 1.2 μm (that is, from the high-temperature region) and
minor uncompensated thermal self-radiation of the wall shows up at λ > 2.6
μm.

**Figure 5. Spectral radiation flux to the wall of nozzle 1 for k = 1: 1--complete 2D
calculation, 2--1D calculation, 3--2D calculation ignoring scattering**

The spectral dependences of the radiation flux shown in Fig. 5 indicate that correct calculations of radiative heat transfer in supersonic nozzles, generally speaking, cannot be performed in a gray approximation. Of course, this does not rule out the possibility of using “gray” engineering models that are oriented to a rough estimate of the integral radiation flux and do not pretend to a high accuracy. The simplest theoretical estimate of the integral radiation flux from the internal surface of a high-altitude nozzle can be obtained by calculating the radiation of the given ring element of the surface to the exit cross section of the nozzle. In so doing, the problem reduces to determining the view factor ψ, and the radiation flux is calculated by the formulas

In the vicinity of the nozzle exit cross section, the expression for the view factor is considerably simplified,

Equations (10) include no dependence on any linear dimensions. Therefore, this
approximate solution can correspond to the exact calculations for nozzles of different
sizes only on an average. A comparison shown in Fig. 6 indicates that Eq. (10) can be
used for estimating the maximum heat removal by radiation from the internal surface
of a nozzle with d_{*} > 0.05 m in the range of ξ > 3.5-4, where a 1D approximation
would be obviously invalid. Note that the error of such an estimation in
calculating the thermal state of the nozzle walls proves rather insignificant
because of the nonlinear dependence of the thermal radiation on the surface
temperature. The examples of temperature calculations for the nozzle with a
titanium wall protected from internal heating by a layer of phenolic carbon and
for the carbon-carbon nozzle cooled by radiation from the external surface
were considered in the book by Dombrovsky (1996b). It was shown that
approximation (10) has an acceptable accuracy, and can be employed for estimates of
radiative heat removal from the internal surface of the high-altitude rocket
nozzles.

**Figure 6. Spectral radiation flux to the wall of nozzle 1 for k = 1: 1--complete 2D
calculation, 2--1D calculation, 3--2D calculation ignoring scattering**

The results obtained in this study of heat transfer by radiation in supersonic nozzles may seem to be trivial. Therefore, it should be remembered that the calculation of thermal radiation fluxes in the supersonic nozzle by a 1D approximation or disregarding radiation scattering by combustion products, as well as disregarding thermal radiation of the nonisothermal surface of the nozzle, yields physically incorrect results and may lead to considerable errors in determining the temperature of high-expansion-ratio nozzles.

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