# Polymer Combustion as a Basis for Hybrid Propulsion: A Comprehensive Review and New Numerical Approaches

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

- (1)
- Safety. This is a major attraction. The solid fuel is inert, therefore it can be manufactured, transported and handled safely. In addition, because an intimate mixture of oxidizer and fuel is not possible, it is non-explosive.
- (2)
- Operating issues. Engine throttling and shutdown are significantly simplified by this technology. Throttling is achieved by liquid flow rate modulation, which is considerably simpler in this case compared to a liquid rocket engine where two liquid streams have to be synchronised. Termination is accomplished by cutting of the liquid flow rate. This opens possibility of quick and robust abort procedure.
- (3)
- Choice of fuel. A wide range of easily available solid fuels can be used, giving wider design flexibility compared to liquid or solid motors. Combustion performance of solid fuel is also more reliable since in a hybrid mode it is not sensitive to fuel-grain cracks.
- (4)
- Cost. Operational costs are obviously of great importance. In this regard hybrid systems benefit from simplified manufacturing procedures, due to their inherent safety. Consequently, fabrication (and therefore operation) costs are reduced.

- (1)
- Low regression rate. This is a major obstacle for a wide use of Hybrid Rocket Engines. Essentially non-energetic nature of fuels gives rise to requirement of very high regression rates, in order to achieve required thrust. In practice it leads to necessity to use multiple ports, and other modifications that complicates design.

- (2)
- Combustion efficiency is lower compared to liquid or solid engines, due to non-premixed nature of combustion.
- (3)
- Finally, ignition transient and thrust response to throttling is slower than in solid or liquid motors.

## 2. Fundamentals of Polymer Combustion

#### 2.1. Flammability Characteristics of Polymeric Materials

- Random chain cleavage followed by chain unzipping is characterized by high monomer yields and a slow decrease in the molecular weight of the polymer, e.g., poly(methyl methacrylate), poly(α-methylstyrene), polystyrene, polytetrafluoroethylene.
- Random chain cleavage followed by further chain scission is characterized by very low monomer yields amongst the degradation products and a rapid drop in molecular weight, e.g., polyethylene, polypropylene, poly(methylacrylate), polychlorotrifluoroethylene.
- An intra-chain chemical reaction followed by cross-linking reaction and formation of a carbonaceous residue, or random chain cleavage. This generates a relatively high yield of volatiles from the intra-chain reaction, but produces little monomer, and produces, no, or only a very slight, reduction in molecular weight during the initial stages of degradation, e.g., poly(vinyl chloride), poly(vinyl alcohol), polyacrylonitrile.

Polymer | LOI |
---|---|

Polypropylene | 18 |

Poly(butylene terephthalate) | 20 |

Poly(ethylene terephthalate) | 21 |

Nylon-6,6 | 24 |

Nylon-6 | 21 |

Cotton | 16 |

Polyester fabric | 21 |

Wool | 24 |

Polyacrylonitrile | 18 |

Polyaramid | 38 |

Polymer | Peak Heat Release Rate ^{a} (kW m^{−2}) |
---|---|

Polypropylene | 1095 |

Poly(butylene terephthalate) | 1313 |

Isophthalic polyester | 985 |

Nylon-6,6 | 1313 |

Nylon-6 | 863 |

Wool | 307 |

Acrylic fibres | 346 |

^{a}Measured under an irradiance of 40 kW m

^{−2}.

^{−1}, in a stream of nitrogen flowing at a rate of 80 cm

^{3}min

^{−1}. The thermal degradation products, thus obtained, were then mixed with a 20 cm

^{3}min

^{−1}stream of oxygen prior to entering a combustion chamber maintained at 900 °C. Each sample is run in triplicate and the data obtained are averaged over the three measurements [12]. The instrument also generates plots of the Heat Release Rates (HRR) against the temperature and gives values for the maximum amount of heat released per unit mass per degree of temperature (i.e., heat release capacity measured in Jg

^{−1}K

^{−1}), the latter being a reliable indicator regarding flammability of a material (Table 3).

**Table 3.**Heat release capacity, total heat released, and char yield of selected polymers [11].

Polymer Char | Heat Release Capacity (J g^{−1} K^{−1}) | Total Heat Released (kJ g^{−1}) | Residue (wt.%) |
---|---|---|---|

Polypropylene | 1571 | 41.1 | 0 |

Polyethylene (LDPE) | 1676 | 41.6 | 0 |

Polystyrene | 927 | 38.8 | 0 |

Poly(butyleneterephthalate) | 474 | 20.3 | 1.5 |

Poly(ethyleneterephthalate) | 332 | 15.3 | 5.1 |

Polymethylmethacrylate | 376–514 | 23.2 | 0 |

Polyoxomethylene | 169 | 10 | 0 |

Polyvinylchloride | 138 | 11.3 | 15.3 |

**Table 4.**Net heats of combustion of selected polymers by PCFC and oxygen bomb calorimetry [11].

Polymer | Net Heat of Combustion (kJ g^{−1}) | |
---|---|---|

PCFC | Oxygen Bomb | |

Polyethylene | 44.1 | 43.3 |

Polystyrene | 40.1 | 39.8 |

Polycarbonate | 29.1 | 29.8 |

Poly(butyleneterephthalate) | 26.3 | 26.7 |

Poly(ethyleneterephthalate) | 23.2 | 21.8 |

Polymethylmethacrylate | 25.0 | 25.4 |

Polyoxomethylene | 15.0 | 15.9 |

#### 2.2. Combustion of Some Representative Polymeric Solid Fuels

#### 2.2.1. Polyolefins

#### 2.2.2. Acrylics

#### 2.2.3. Elastomers

#### 2.3. Enhancement in Degradation and Regression Rates in Hybrid Fuels—Some Suggestions for Future Work

- (a)
- Spectroscopic (NMR and FT-IR) and elemental analyses—high field (500 MHz) solution state
^{1}H- and^{13}C-NMR for deciphering the microstructures of the polymers (this includes tacticity, composition, monomer sequencing, minor structures including structural defects, etc.). Limited, but complementary, information regarding the structural features of the polymer could also be obtained from the FT-IR spectra and heteroatom elemental analyses. - (b)
- Chromatographic and related techniques—these are primarily aimed at obtaining the molecular weights and their distributions. For polyolefin based-polymers, optionally melt-flow index measurements could be carried out.
- (c)
- Thermo-gravimetric analyses (TGA)—TGA runs need to be carried out on ca. 10–15 mg of the resin in nitrogen, air and in oxygen atmospheres, say, at 10 °C min
^{−1}, and from 30 to 1000 °C. The idea behind these runs is to get the general thermal- and thermo-oxidative degradation profiles of the material (i.e., under different oxidative atmospheres). This could be followed by repeating the runs, in a chosen atmosphere(s), with a view to estimating the Arrhenius parameters, if necessary. - (d)
- Differential Scanning Calorimetry (DSC)—here, milligrams of samples are heated in sealed aluminium pans, under a nitrogen atmosphere and usually at a heating rate of 10 °C min
^{−1}, up to a point where substantial thermal degradation starts. This is a very useful technique that often yields information regarding melting behaviours, glass transition temperatures, etc. that the material might undergo under the heating conditions imposed. - (e)
- Parallel Plate Rheometry—here again the sample, ideally in the shape of thin films, is heated whilst sandwiched between two heated parallel plates, at the same time a sinusoidal mechanical stress is applied. Generally, this constitute a good method for determining the moduli of elasticity (store and loss), the glass transition temperatures, and more importantly the melt flow behaviour of the resin
- (f)
- Combustion Bomb Calorimeter—this instrumentation is used to determine the heats of combustion (ΔH
_{comb}) of the resin. This parameter is a good indicator of the maximum heat out put on complete oxidation of the polymeric material in question. - (g)
- Pyrolysis Combustion Flow Calorimetry (PCFC)—this piece of instrumentation, often dubbed as the micro cone calorimeter, produces plots of Heat Release Rates against time, as well as generates parameters like the heat release capacity on milligrams of a material (i.e., the maximum amount of heat released per unit mass per degree of temperature (Jg
^{−1}K^{−1}, is a material property that appears to be a good predictor of flammability). - (h)
- Hyphenated techniques—attempts to identify the volatiles formed from thermal degradation of the materials could be made by hyphenating the TGA to an FT-IR or to a GC/MS. Such hyphenated technique is also available in a larger scale that, primarily, involves two consecutive tube furnaces in connected in series. Optionally, some of the gaseous-products formed upon degradation, in ambient atmosphere, collected through by using proprietary containers, will be subjected to GC/MS.

## 3. CFD Modelling Framework for Hybrid Propulsion

#### 3.1. Fuel Regression

- Thermal decomposition
- Thermo-oxidative decomposition
- Decomposition of monomer MMA
- Combustion of pyrolysis products

_{x}is local Reynolds number and G is local mass flux due to oxidizer injection and upstream fuel addition. St and St

_{0}are Stanton numbers for turbulent boundary layer flow in the presence and in the absence of blowing, respectively. Further, u

_{e}is the velocity at the edge of the boundary layer, u

_{fl}is the velocity at the flame location, h

_{B}is stagnation enthalpy at the flame temperature, h

_{w}is gas enthalpy at the wall, and $\Delta {H}_{sg}^{tot}$ is the total heat of gasification. The latter is the total energy required to heat a unit mass of solid fuel up from its initial temperature and vaporise it.

#### 3.2. Polymer Decomposition Modelling

A_{p} | pyrolysis pre-exponential factor, kg/(m^{2} s) | ~8 × 10^{17} |

E_{p} | pyrolysis activation energy, J/mol | ~2.8 × 10^{5} |

^{−1}, 0.17 K s

^{−1}, and 0.5 K s

^{−1}to bring polymer sample from 373 K to 1003 K. Experiments were conducted in a nitrogen atmosphere.

_{D}are calculated by numerical differentiation of mass loss data, then the rate constants may be plotted against temperature. Assuming:

_{D}and h

_{C}are the heats of decomposition reactions and the heats of combustion of volatile decomposition products, respectively.

^{5}J/mol for the activation energy.

Polymer | A (s^{−1}) | E (J mol^{−1}) | h_{D} (J kg^{−1}) | h_{C} (J kg^{−1}) |
---|---|---|---|---|

PMMA | 8.5 × 10^{12} | 1.88 × 10^{5} | 8.7 × 10^{5} | 2.41 × 10^{7} |

HIPS | 1.2 × 10^{16} | 2.47 × 10^{5} | 1.0 × 10^{6} | 3.81 × 10^{7} |

HDPE | 4.8 × 10^{22} | 3.49 × 10^{5} | 9.2 × 10^{5} | 4.35 × 10^{7} |

^{5}J/mol) in the absence of oxygen at the PMMA surface than that (6.3 × 10

^{4}J/mol) for oxidative degradation.

E_{a} (kcal/mol) | ln A (s^{−1}) | T (°C) | Ref. | Comments |
---|---|---|---|---|

31 ± 3 ^{a} | – | cis-trans Isomerization 200–300 | [3] ^{a} | IR, vacuum |

15 ^{a} | – | Cross-linking and Cyclization 200–300 | [9] | IR, vacuum, first order, vinyl groups |

18.8 ^{b} | 2.5 | 328–420 | [4] | TGA |

27.6 ± 1.6 ^{b} | 16.2 | 350–400 | [13] ^{b} | TGA |

39 | – | 250 | [11] | Hardening data |

37.6 ^{c} | 25.6 | Chain Scission 450–532 | [15] | pyrolysis-GC, BD formation, first order |

40.7 ± 3 | – | N/A | [10] | estimate of 4-vinyl-1 cyclohexene formation |

42.1 ^{a} | – | 362–434 | [10] | DSC |

42.8 | – | – | [16] | Estimate |

46 ± 0.6 ^{b} | 22.8 | 436–470 | [4] | TGA |

51.9 ^{a} | 350–425 | [10] | TGA | |

60.0 ± 3.5 ^{a} | – | N/A | [10] | Estimate of BD formation |

60.1 ^{b} | 17.1 | 367–407 | [13] | TGA |

62 | – | 380–395 | [14] | weight change |

62 ± 4 ^{d} | – | 410–500 Unspecified Processes | [6] | TGA |

24.5–38.5 ^{e} | 12.2-20.8 | 350–550 | [7] | TGA |

28 ^{e} | 12.8 | 400–500 | [18] | TGA, order = 0.6–1 |

21.5–31.1 ^{b} | 8.9–13.8 | N/A | [19] | TGA |

^{a}1,4-polybutadiene;

^{b}HTPB;

^{c}cis-1,4-polybutadiene; dT/dt ≈ 5000 °C/s;

^{d}Polybutadiene-acrylonitrile-acrylic acid (PBAN);

^{e}Carbonyl-terminated polybutadiene (CTPB).

#### 3.3. Polymer Combustion Modelling

#### 3.3.1. Global Kinetics

B | combustion pre-exponential factor, m^{3} /(s mol K) | ~6.6 × 10^{6} |

E_{c} | combustion activation energy, J/mol | ~1.44 × 10^{5} |

_{i}is a stoichiometric requirement of the i-th species with respect to fuel. Again, as for decomposition, additional independent tests are desirable, and temperature-dependent treatment of parameters may be appropriate.Global combustion reaction is quite a crude approach, so priority should be given to developing more accurate models incorporating detailed chemistry.

#### 3.3.2. Detailed Kinetics

**Table 9.**Methanol combustion mechanisms proposed by Bell and Tipper [35].

CH_{3}OH + O_{2} → CH_{2}OH + HO_{2} | CHO + O_{2} → CO + HO_{2} |

CH_{3}OH + HO_{2} → CH_{2}OH + H_{2}O_{2} | CH_{2}O + HO_{2} → CHO + H_{2}O_{2} |

CH_{2}OH + O_{2} → CH_{2}O + HO_{2} | HO_{2} + O_{2} → inner products |

CH_{2}O + O_{2} → CHO + HO_{2} |

HCHO + O_{2}→HO_{2} + CO | HCO_{3}H → OH + Products |

HCHO + O_{2}→OH + Products | HO_{2} + HCHO → H_{2}O_{2} + HCO |

(HCOOH or H_{2} + OO + H_{2}O) | |

HCHO + OH → H_{2}O + HOO | HO_{2} + HO_{2} → H_{2}O_{2} |

HOO + M → H + OO + M | H_{2}O_{2} → HO_{2} + 1/2 O_{2} |

HOO + O_{2} → HO_{2} + OO | H_{2}O_{2} + M → 2OH + M |

HOO + O_{2} → HCO_{3} | HO_{2} → 1/2 H_{2}O + 3/4 O_{2} |

HCO_{3} + HCHO → HCO_{3}H + HCO |

C_{2}H_{2} + M C_{2}H + H + M | CH_{2} + CH_{2} C_{2}H_{4} + H |

C_{2}H_{2} + C_{2}H_{2} C_{4}H_{2} + H | CH_{2} + CH_{2} C_{2}H_{4} + H_{2} |

C_{2}H_{2} + O_{2} CHO + CHO | CH_{2} + CH_{2} C_{2}H_{5} + H |

C_{2}H_{2} + H C_{2}H + H_{2} | CH_{2} + H CH + H_{2} |

C_{2}H_{2} + O CH_{2} + CO | CH_{2} + O_{2} CO_{2} + H + H |

C_{2}H_{2} + O CHCO + H | CH_{2} + CH_{2} C_{2}H_{2} + H + H |

C_{2}H_{2} + CH C_{2}H + H_{2}O | C_{2}H_{6} CH_{2} + CH_{2} |

C_{2}H_{2} + CH CH_{2}CO + H | C_{2}H_{6} + H C_{2}H_{5} + H_{2} |

C_{2}H_{2} + CH C_{2}H_{2} | C_{2}H_{6} + CH C_{2}H_{5} + H_{2} |

C_{2}H_{2} + CH_{2} C_{2}H_{4} | C_{2}H_{6} + CH_{2} CH_{4} + C_{2}H_{5} |

C_{2}H_{2} + CH_{2} C_{2}H + CH_{4} | C_{2}H_{5} C_{2}H_{4} + H |

C_{2}H_{2} + C_{2}H C_{4}H_{2} + H | C_{2}H_{4} + M C_{2}H_{2} + H_{2} + M |

C_{2}H_{2} + C_{2}H_{2} C_{4}H_{4} + H | C_{2}H_{4} + H C_{2}H_{2} + H_{2} |

C_{2}H + O_{2} CO + CO + H | C_{2}H_{4} + O CH_{2} + CHO |

CH_{2}CO + M CO + CH_{2} + M | C_{2}H_{4}+ CH C_{2}H_{2} + H_{2}O |

CH_{2}CO + H CHCO + H_{2} | C_{2}H_{2} + M C_{2}H_{2} + H + M |

CH_{2}CO + H CH_{2} + CO | C_{2}H_{2} + H C_{2}H_{2} + H_{2} |

CH_{2}CO + O CH_{2} + CO_{2} | C_{2}H_{2} + O CH_{2}CO + H |

CH_{2}CO + O CH_{2}O + CO | C_{2}H_{2} + O_{2} C_{2}H_{2} + HO_{2} |

CH_{2}CO + CH CH_{2}O + CHO | C_{2}H_{2} + O_{2} CH_{2}O + CO + H |

CH_{2}CO + CH CH_{2}CH + CO | C_{2}H_{2} + CH_{2} C_{2}H_{2} + CH_{4} |

CH_{2}CO + CH_{2} C_{2}H_{4} + CO | C_{2}H_{4} + M C_{2}H_{2} + H + M |

CH_{2}CO+ CH_{2} CHCO + CH_{3} | C_{2}H_{4} + H C_{2}H_{2} + H_{2} |

CH_{2}CO+ CH_{2} C_{2}H_{5} + CO | C_{2}H_{4} + H C_{2}H_{2} + CH_{2} |

CH_{2}CO+ CH_{2} CHCO + CH_{4} | C_{2}H_{4} + CH_{2} C_{2}H_{2} + CH_{4} |

CHCO + O CO + CO + H | C_{4}H_{4} CH_{2}CO + CH_{2} |

CHCO + OH CO + CO + H_{2} | C_{4}H_{4} C_{4}H_{2}+ H |

CHCO + O_{2} CO + CO + CH | C_{4}H_{4} C_{2}H_{2} + C_{2}H_{2} |

CHCO + H CH_{2}+ CO | C_{4}H_{4} C_{4}H_{2} + H_{2} |

CHCO + CH_{2} C_{2}H_{2} + CO | C_{4}H_{4} + H C_{4}H_{2} + H_{2} |

CHCO + CH_{2} C_{2}H_{4} + CO | C_{4}H_{4} + H C_{4}H_{2} + H_{2} |

CHCO + CHCO CO + CO + C_{2}H_{2} | C_{4}H_{2} + M C_{4}H_{2} + H + M |

CH_{2}O + H CHO + H_{2} | C_{4}H_{2} + H C_{4}H_{2} + H_{2} |

CH_{2}O + CH CHO + H_{2}O | C_{4}H_{2} + C_{2}H C_{2}H_{2} + H |

CHO + M CO + H+ M | O_{2}+ H CH + O |

CHO + H H_{2}+ CO | H_{2}+ O CH + H |

CHO + O_{2} HO_{2}+ CO | H_{2}O + H CH + H_{2} |

CH_{2}O + M CH_{2}O + H+ M | H + O_{2}+ M HO_{2} + M |

CH_{2}OH + M CH_{2}O + H+ M | HO_{2} + H H_{2} + O_{2} |

CO + CH CO_{2} + H | HO_{2} + H CH + OH |

CH_{4}+ M CH_{2} + H + M | N_{2}O + M N_{2} + O + M |

CH_{4}+ H CH_{2} + H_{2} | N_{2}O + O N_{2} + O_{2} |

CH_{4}+ O CH_{2} + OH | N_{2}O + O NO + NO |

CH_{4}+ CH CH_{2}+ H_{2}O | N_{2}O + H N_{2}+ OH |

CH_{4}+ CH_{2} CH_{2} + CH_{2} | N_{2}O + CH_{2} CH_{2}O + N_{2} |

CH_{2}+ H CH_{2} + H_{2} | N_{2}O + CH_{2} CH_{2}O + N_{2} |

CH_{2}+ O CH_{2}O + H | N_{2}O + C_{2}H_{2} CH_{2}CHO + N_{2} |

CH_{2}+ CH CH_{2}O + H_{2} | N_{2}O + CO CO_{2} + N_{2} |

CH_{2}+ CH CH_{2}CH + H | N_{2}O + CHO CO_{2} + H + N_{2} |

CH_{2}+ O_{2} CH_{2}O + O | N_{2}O + CHCO CO + CHO + N_{2} |

CH_{2}+ HO_{2} CH_{2}O + CH | N_{2}O + C_{2}H_{2} CHCO + H+ N_{2} |

**Table 12.**The content of combustion products of PMMA and MMA in air and Ar at different temperatures [33].

Product | Content v/v % | |||||||
---|---|---|---|---|---|---|---|---|

PMMA (air) | MMA (Ar) | MMA (air) | ||||||

300 °C | 500 °C | 300 °C | 400 °C | 500 °C | 300 °C | 400 °C | 500 °C | |

MMA | 95.5 | 78.9 | 92.7 | 83.8 | 74.2 | 91.6 | 79.8 | 68.2 |

CH_{4} | 0.8 | 1.3 | 0.6 | 1.5 | 2.6 | 0.5 | 1.2 | 2.2 |

CH_{2}-CHCH_{3} | 1.7 | 1.1 | 2.3 | 3.4 | 1.0 | 1.4 | 2.9 | |

CH_{2}-C(CH_{3})_{2} | 1.9 | 1.0 | 2.4 | 3.8 | 1.2 | 1.8 | 3.0 | |

CH_{3}OH | 1.8 | 3.2 | 1.6 | 3.6 | 5.8 | 1.5 | 2.9 | 4.9 |

HCHO | 0.3 | 0.5 | 0.8 | |||||

CH_{3}COCH_{3} | 0.6 | 0.2 | 0.5 | 1.0 | 0.8 | 1.3 | 1.6 | |

CH_{3}COCOOCH_{3} | 0.8 | 1.2 | 2.0 | |||||

CO_{2} | 0.8 | 6.0 | 0.9 | 2.0 | 3.3 | 1.4 | 5.2 | 8.0 |

CO | 0.2 | 0.3 | 1.4 | 3.3 | 5.4 | 1.1 | 0.4 | 0.3 |

H_{2}O | 0.4 | 4.5 | 0.6 | 3.8 | 5.6 |

#### 3.3.3. Fourier Transform Infrared Spectroscopy (FTIR) and Gas Chromatography/Mass Spectrometry (GC/MS)

#### 3.4. Crucial Submodels and Implementation

#### 3.4.1. Flow Model

_{in}and the shear-stress tensor are given by:

#### 3.4.2. Combustion Modeling

#### 3.4.2.1. Fast Chemistry Approach

_{i}(ζ) replaced by ${\overline{Y}}_{i}(\overrightarrow{x},t)$. In practical computations, the form of sgpdf is normally prescribed based on experimental insights. (Most comprehensive but extremely expensive Probability Density Function (PDF) approach solves transport equations for relevant pdfs). The most conventional form (β-function) is set as:

#### 3.4.2.2. Flamelet Approach

_{st}is its value at stoichiometry. (Conventional summation over repeating indices is assumed in (19), (20), and further on).

_{st}⋅f(ζ), rather than χ = χ

_{st}= const in the above equations. Particular form of f(ζ) can be found in the literature, e.g., [45,46,47,48].

_{st}variables must be known.

^{2}≈ 2 ([45,49])

**Figure 8.**Schematic diagram of the temporally evolving shear layer with counter-directed streams ${u}_{1}=-\frac{1}{2}\Delta u$, ${\rho}_{1}={\rho}_{0}$, ${Y}_{O}={Y}_{O.0}$; ${u}_{2}=-\frac{1}{2}\Delta u$, ${\rho}_{2}={\rho}_{0}$, ${Y}_{F}={Y}_{F.0}$ (Δu is prescribed velocity difference).

_{1}, y = x

_{2}and z = x

_{3}):

_{ω}is the initial vorticity thickness of the shear layer, defined as:

_{1}direction. The values of the mean μ

_{x}and the standard deviation σ

_{x}can be found in Pantano et al. [50]. Results of flamelet modelling should be compared to experiments. It is expected that SLFM should provide fair results in the view of fast kinetics of major heat-generating reactions. If, however, this approach is found to be deficient, extension to unsteady flamelet model is possible, although its implementation in LES is not as straightforward [51].

#### 3.4.3. Fuel Regression Model

_{1}= const fuel surface S

_{f}is described by the time-dependent function F

_{x1}(x

_{2},x

_{3}) so that ${S}_{f}=\{\overrightarrow{x},t:{F}_{{x}_{1},t}({x}_{2},{x}_{3})=0\}$.

_{sg}is heat of pyrolysis (positive if thermal decomposition is exothermal, and negative if decomposition is endothermal). Here $({S}_{f}\cap \{\alpha \cdot \overrightarrow{n}\})$ denotes a point of intersection between fuel surface and the line $\{\alpha \cdot \overrightarrow{n}\}$ $\alpha \in \Re $, normal to the surface (Figure 9).

_{f}is fuel density, ${\dot{m}}_{f}$ is fuel thermal decomposition rate. The approaches to model the rate ${\dot{m}}_{p}$ have been discussed above (sub-Section 3.2). Evolution equation for the surface-defining function F

_{x1,t}(x

_{2},x

_{3}) is:

_{f}and material properties. Experiments are needed to approximate the form of the function f.

#### 3.4.4. Radiation Heat Transfer

#### 3.4.4.1. Radiation Correction

_{R}is added to the energy Equation (7).

_{m}is local absorption coefficient.

#### 3.4.4.2. Comprehensive Radiation Modeling

_{n}) and leaves (I

_{n+1}) control volumes.

#### 3.4.5. Injector Spray Model

_{x}) injector flow. It is assumed that liquid oxygen stream is atomized into droplets that evaporate further downstream and thus provide gaseous oxidant supply for diffusion combustion.

^{1/2}), this distribution can be sampled randomly to obtain the instantaneous velocity U

_{´}.

_{p}is the particle relaxation time:

_{e}. Time of interaction between the particle and the eddy is taken as the shortest of the two times, t

_{e}and t

_{p}.

_{m}is defined as:

^{3}[61].

_{SGS}). Heat-up and evaporation rates can be calculated by conventional procedures described above.

_{x}) properties of injected droplets can be specified from table data (liquid oxygen has a density of approximately 1.14 g/cm

^{3}and boiling point of 90.188 K (−182.96 °C) at 101.325 kPa).

## 4. Preliminary Simulations

^{3}] denotes the density, u

_{i}(i = 1,2,3) [m/s] are the Cartesian velocity components corresponding to (u, v, w), p [Pa] the pressure, e [J/m

^{3}] the total energy per unit volume, respectively. In order to close the above equations, the equation of state for perfect gas is employed which is defined as:

_{ij}the components of rate-of-strain tensor which are defined as follows:

_{j}[J/(m

^{2}s)] is given by:

_{p}[J/(kg K)] denotes the specific heat at constant pressure and Pr the Pandtl number, respectively.

_{j}are the component of normal vector to ∂Ω.

#### 4.1. Implicit Large Eddy Simulation

^{F}denotes the total energy per unit mass, and we introduce the computable shear-strain tensor which depends on the computable rate-of-strain tensor , and the computable heat flux .

_{j}appears in the momentum equation. On the other hand, as can be seen from Equation (79), there are a lot of SGS terms for compressible LES because of the interaction of GS components with SGS components in the filtered energy conservation equation, which are the SGS temperature flux:

#### 4.2. Combustion Model

_{x}H

_{y}and the inflow is an air as oxidant. A non-premixed flame in the boundary layer is obtained by the combustion between fuel and oxidant. The chemical reaction is governed by one step reaction as follows:

_{x}H

_{y}, O

_{2}, CO

_{2}, H

_{2}O and N

_{2}, are considered, while N

_{2}does not contribute to the above reaction. For flow includes multi species, the definition of the specific heat c

_{p}[J/(kg K)] and the enthalpy h [J/kg] for mixture gas are given respectively by:

_{p}(T) [J/(kg K)] and enthalpy h

_{i}(T) [J/kg] for -th chemical species are given by following equations [89]:

_{j}[J/(kg K)] denotes the gas constant for j -th chemical species and gas constant for mixture gas is defined by R = ∑

_{j}Y

_{j}R

_{j}. Using the definition of internal energy ε = e / ρ − (u

^{2}+ v

^{2}+ w

^{2})/2 = ∫ c

_{v}(T)dT = h(T) − RT, where c

_{v}(T) [J/(kg K)] denotes the specific heat at constant volume, the temperature can be calculated by iterative procedure which is, for example, Newton-Raphson method:

^{0}T

^{0}= 8314.51 [J/(kg mol K)] denotes the universal gas constant n [kgmol/kg] the mole number per unit mass of mixture gas defined by using the mole number per unit mass of species n

_{j}[kgmol/kg], as follows:

_{j}as follows:

_{2}, the Equation (88) is given as follows:

_{i}[m

^{−3}], density of mixture gas ρ [kg/m

^{3}] and the Avogadro constant N

_{A}=6.02214179 [mol

^{−1}] as follows:

_{j}and m

_{i}[kg] denote the mass of j-th molecule and the mass of i-th element, respectively. For the elements of carbon, hydrogen, oxygen and nitrogen, the masses of element are given by:

_{C}: ν

_{H}= x : y

_{O}(0 ≤ A

_{O}≤ 1), the ratio of number density between oxygen and nitrogen becomes constant except for in no air stream:

_{C}, are detected, mole density of ν

_{H}, ν

_{O}and ν

_{N}are also determined by Equations (91), (93) and (95). Then, the mole number of element b

_{i}

^{0}can be determined by Equation (90).

_{j}satisfy the following relations obtained by Equations (89) and (106):

_{st}and add some assumption using ξ

_{st}. The stoichiometric mixture fraction ξ

_{st}is given by following procedure. According to the chemical reaction Equation (80), the stoichiometric mole ratio of O

_{2}to C

_{x}H

_{y}is x + y / 4. Therefore, using Equations (94) and (95), the ratio among the mole numbers per unit mass for each element at the stoichiometric condition is obtained as follows:

_{st}can be calculated by solving Equations (108) and (109):

_{st}, it is assumed that there is no fuel after chemical reaction because the amount of fuel is smaller than that of oxidant. This assumption is equivalent to assuming n

_{CxHy}= 0. Then, the Equation (107) can be solved as follows:

_{j}denotes the molecular weight. Then, the mass fractions of all species are obtained as follows:

_{st}, it is similarly assumed that there is no oxidant after chemical reaction because the amount of oxidant is smaller than that of fuel. This assumption is equivalent to assuming n

_{O}

_{2}= 0. Then, the Equation (107) can be solved as follows:

_{st}, it is assumed that there is no oxidant and fuel after chemical reaction. This assumption is equivalent assuming n

_{CxHy}= n

_{O}

_{2}= 0. Then, the Equation (107) can be solved as follows:

_{O}= 0.256.

**Figure 13.**Mass fraction profiles of each chemical species against mixture fraction ξ for methane and air combustion.

#### 4.3. Numerical Methodology

#### 4.3.1. Time Integration

#### 4.3.2. Numerical Flux Function

_{x}is Cartesian components of a normal vector from the left to the right and $\dot{m}$ represents mass flux, respectively. For one dimensional case, n

_{x}= 1, while, for three dimensional case, $\dot{N}={[0,{n}_{x},{n}_{y},{n}_{z},0,0]}^{T}$ and that components satisfy ${n}_{x}^{2}+{n}_{y}^{2}+{n}_{z}^{2}=1$. In Equation (127), $\tilde{p}$ denotes the corrected pressure term.

#### 4.3.3. High Order Interpolation Method

_{i}= q

_{i}

_{+1}− q

_{i}. The parameter k decides the spatial accuracy of the scheme shown in Table 13.

k | Scheme |
---|---|

−1 | second-order upwind |

0 | second-order |

1/3 | third-order upwind |

1/2 | QUICK (second-order upwind) |

1 | second-order central |

_{i}) is introduced as follows:

_{i}which is given by Anderson et al. [95] can be used:

^{-6}is introduced to avoid the denominator becoming zero.

#### 4.4. Non-Premixed Flame in Three-Dimensional Flowfield

#### 4.4.1. Brief Description for Computation

_{∞}= 1.0 atm and T

_{∞}= 300 K, respectively. The Riemann boundary condition is employed along the inflow and upper boundary. Along the outlet boundary, a zeroth extrapolation is used to give the dependent variables. At both side walls of the computational domain, periodic boundary conditions are enforced. The total number of grid points in the computational domain is 253 × 182 × 151 in the streamwise direction, normal direction to the wall surface and spanwise direction, respectively.

#### 4.4.2. Results and Discussions

_{st}≅ 0.06 indicating flame location, and the yellow one denotes ξ = 0.75 which is one of iso-surface in the fuel stream. In Figure 16b, the three-dimensionality of iso-surfaces appears obviously by comparing those in Figure 16a, however, this cannot indicate that the turbulent flow has developed because the computation time is insufficient.

_{CO2 + H2O}and the projected velocity vectors, respectively, on the cross section at x = 0.9 m where the mixing is promoted. These figures are obtained at the same instance with Figure 16b. In these figures, one can find that the high temperature region corresponds to the region where the large amount of products exists. This high temperature gas of combustion products is convected toward fuel stream side due to the vortices induced by the Kelvin-Helmholtz instability in the shear layer. Again, the turbulent flow has not yet developed in this instance while it can be found that the longitudinal vortices appear in the fuel stream.

_{4}, oxidizer O

_{2}+ N

_{2}and products CO

_{2}+ H

_{2}O, respectively. In this figure, one can find that the fuel is consumed and the products increase about twice as much as initial value as time advances. This is because a large number of products near the diffusion flame are convected and chemical reaction is promoted.

**Figure 17.**Temperature contours, contours of mass fraction of products (Y

_{CO2 + H2O}) and projected velocity vectors on the cross section at x = 0.9 m. (

**a**) Temperature; (

**b**) Mass fraction of products; (

**c**) Projected velocity field.

Resolution | Grid Points | CPU Numbers | Total CPU Times per 10^{4} Time Steps |
---|---|---|---|

Fine | 253 × 182 × 151 | 16 | 10.64 h |

Coarse | 201 × 151 × 101 | 8 | 5.34 h |

**Figure 19.**The instantaneous temperature contours in the cross section at z = 0 m, t = 0.536 s obtained by using coarser grid.

## 5. Conclusions

## Acknowledgements

## Nomenclature

flux Jacobian matrix, | |

A_{O} | mass fraction of element O in the air (0 ≤ A _{O} ≤ 1) |

mole number for i-th element per unit mass of mixture gas [kgmol/kg] | |

c_{p}, c_{p}(T) | specific heat at constant pressure [J/(kg K)] |

c_{pj}(T) | specific heat at constant pressure for j-th chemical species [J/(kg K)] |

c_{v}(T) | specific heat at constant volume [J/(kg K)] |

C_{D} | drag coefficient |

c_{f} | friction coefficients |

C_{μ} | k-ε model constant |

d | diameter [m] |

D | diffusion coefficient [m ^{2}/s] |

D_{j} | subgrid scale (SGS) viscous diffusion |

e | total energy per unit volume [J/m ^{3}] |

e_{in} | internal energy [J/kg] |

E | total energy per unit mass [J/kg] |

inviscid flux vector | |

numerical flux vector at the cell interface | |

obtained flux by flux vector splitting methods | |

G | kernel of filter |

h | enthalpy per unit mass [J/kg] |

h_{C} | convective heat transfer coefficient [W/ m ^{2} K] |

h_{j}(T) | internal enthalpy per unit mass j-th chemical species [J/kg] |

H | total enthalpy per unit mass [J/kg] |

I | total radiation intensity [W/ m ^{2}/sr] |

J_{j} | subgrid scale (SGS) turbulent diffusion |

k | parameter deciding the spatial accuracy of MUSCL;turbulent kinetic energy [m ^{2}/s^{2}] |

L_{fg} | latent heat of liquid evaporation [kJ/kg] |

m_{i} | mass of i-th element [kg] |

m_{j} | mass of j-th molecule [kg] |

mass flux [kg/(m ^{2} s)] | |

pyrolysis rate [kg/(m ^{2} s)] | |

Nu | Nusselt number |

n | mole number for mixture gas per unit mass of mixture gas [kgmol/kg] |

inner normal to fuel surface | |

n_{j} | mole number for j-th species per unit mass of mixture gas [kgmol/kg] |

n_{x}, n_{y}, n_{z} | Cartesian components of a normal vector from the left to the right at cell interface |

n_{j} | component of normal vector to ∂Ω |

p, P | pressure [Pa] |

corrected pressure term [Pa] | |

Pr | Pandtl number |

q_{j} | heat flux [J/(m ^{2} s)] |

(q_{j})^{C} | computed heat flux [J/(m ^{2} s)] |

q_{i} | primitive variables |

radiative flux [W/m ^{2}] | |

droplet heating rate [W] | |

Δq_{i} | = q _{i+1} − q_{i} |

Q_{j} | subgrid scale (SGS) temperature flux |

conservation vector | |

R | gas constant for mixture gas [J/(kg K)] |

R_{j} | gas constant for j-th chemical species [J/(kg K)] |

Re | Reynolds number |

r | stoichiometric requirement |

r_{i} | = ∆q _{i−}_{1}/q_{i} stoichiometric requirement for species i with respect to fuel |

R_{f} | fuel surface |

Sc | Schmidt number |

S_{R} | radiation source term [W/m ^{3}] |

St, St_{0} | Stanton numbers |

S_{ij} | rate-of-strain tensor |

S_{i} | slope limiter function by Van Leer |

T | temperature [K] |

T_{f} | solid fuel temperature [K] |

T^{0} | reference temperature |

u_{i} | Cartesian velocity components corresponding to (u, v, w) [m/s] |

cartesian coordinate [m] | |

Y | mass fraction |

Δ_{x}, Δ_{y}, Δ_{z} | cut-off scale in each direction of the Cartesian coordinate |

enthalpy of formation [J/kg] | |

ΔH_{sg} | Heat of pyrolysis [kJ/kg] |

ε | internal energy [J/kg] energy dissipation rate [m ^{2}/s^{3}] |

ϕ | primitive variables |

ϕ(r_{i}) | flux limiter |

γ | ratio of specific heats |

κ | thermal conductivity (gas) [J/(m s K)] |

κ_{s} | solid thermal conductivity (gas) [J/(m s K)] |

μ | coefficient of molecular viscosity [kg/(m s)] |

μ_{i} | molecular weight [kg/mol] |

ρ | density [kg/m ^{3}] |

∜ | set of real numbers |

υ_{i} | number density for i-th element [m ^{−3}] |

χ | scalar dissipation rate [s ^{−1}] |

σ_{ij} | shear-stress tensor |

(σ_{ij})^{C} | computed shear-stress tensor |

τ_{ij} | subgrid scale (SGS) scale stress |

Ω | a control volume |

∂Ω | a surface of the control volume |

ω_{i} | species production rate [kg/m ^{3}/s] |

ξ | mixture fraction of fuel |

ξ_{st} | mixture fraction of fuel at the stoichiometric condition |

## Constant values

N_{A} | Avogadro constant = 6.02214179 [mol ^{−1}] |

g | gravity acceleration =9.81 [m ^{2}/s] |

R^{0} | the universal gas constant = 8314.51 [J/(kgmol K)] |

σ | Stefan-Boltzmann constant = 5.670 10 ^{−8} [W/m^{2}/K^{4}] |

## Superscripts

F | fuel |

g | gas |

l | liquid |

n | the time step |

O | oxygen |

pr | product |

s | surface |

SGS | sub-grid scale |

→ | vector |

- | filtered value |

()^{F} | Favre averaged value |

()^{C} | value computed by the Favre averaged values |

∞ | Free stream |

⌒ | point of curves intersection |

|| || | norm |

(·) | dot product |

## Subscripts

i, j | direction in the Cartesian coordinate system (i, j = 1, 2, 3) |

i | elements C, H, O and N |

j | chemical species C _{x}H_{y}, O_{2}, CO_{2}, H_{2}O and N_{2} |

## References

- Chiaverini, M.J.; Kuo, K.K. Fundamentals of Hybrid Rocket Combustion and Propulsion (Progress in Astronautics and Aeronautics); AIAA: Reston, VA, USA, 2007. [Google Scholar]
- Altman, D.; Holzman, A. Overview and history of hybrid rocket propulsion. In Fundamentals of Hybrid Rocket Combustion and Propulsion (Progress in Astronautics and Aeronautics); Chiaverini, M.J., Kuo, K.K., Eds.; AIAA: Reston, VA, USA, 2007; Volume 218, Chapter 1; pp. 1–36. [Google Scholar]
- Cullis, C.F.; Hirschler, M.M. The Combustion of Organic Polymers; Clarendon Press: Oxford, UK, 1981. [Google Scholar]
- Joseph, P.; Ebdon, J.R. Recent developments in flame-retarding thermoplastics and thermosets. In Fire Retardant Materials; Horrocks, A.R., Price, D., Eds.; Woodhead Publishing Limited: Cambridge, UK, 2000; pp. 220–263. [Google Scholar]
- Arnold, C., Jr. Stability of high temperature polymers. J. Polym. Sci. Macromol. Rev.
**1979**, 14, 265–378. [Google Scholar] [CrossRef] - Martel, B. Charring process in thermoplastic polymers: Effect of condensed phase oxidation on the formation of chars in pure polymers. J. Appl. Polym. Sci.
**1988**, 35, 1213–1226. [Google Scholar] [CrossRef] - Fenimore, C.P.; Martin, F.J. Flammability of polymers. Combust. Flame
**1966**, 10, 135–139. [Google Scholar] [CrossRef] - Babrauskas, V. Development of a cone calorimeter—a bench-scale heat release apparatus cased on oxygen consumption. Fire Mater.
**1984**, 8, 81–95. [Google Scholar] [CrossRef] - Schartel, B.; Bartholmai, M.; Knoll, U. Some comments on the use of cone calorimetric data. Polym. Degrad. Stab.
**2005**, 88, 540–547. [Google Scholar] [CrossRef] - De Ris, J.L.; Khan, M.M. A sample holder for determining material properties. Fire Mater.
**2000**, 24, 219–226. [Google Scholar] [CrossRef] - Lyon, R.E.; Walters, R.N. Pyrolysis combustion flow calorimetry. J. Anal. Appl. Pyrolysis
**2004**, 71, 27–46. [Google Scholar] [CrossRef] - Cogen, J.M.; Lin, T.S.; Lyon, R.E. Correlations between pyrolysis flow combustion calorimetry and conventional flammability tests with halogen-free flame retardant polyolefin compounds. Fire Mater.
**2009**, 33, 33–50. [Google Scholar] [CrossRef] - Ebdon, J.R.; Hunt, B.J.; Jones, M.S.; Thorpe, F.G. Chemical modification of polymers to improve flame retardance—II: The influence of silicon-containing groups. Polym. Degrad. Stab.
**1996**, 69, 395–400. [Google Scholar] [CrossRef] - Ebdon, J.R.; Guisti, L.; Hunt, B.J.; Jones, M. The effects of some transition-metal compounds on the flame retardance of poly(styrene-co-4-vinyl pyridine) and poly(methyl methacrylate-co-4-vinyl pyridine). Polym. Degrad. Stab.
**1998**, 60, 401–407. [Google Scholar] [CrossRef] - Armitage, P.; Ebdon, J.R.; Hunt, B.J.; Jones, M.S.; Thorpe, F.G. Chemical modification of polymers to improve flame retardance—I. The influence of boron-containing groups. Polym. Degrad. Stab.
**1996**, 54, 387–393. [Google Scholar] [CrossRef] - Ebdon, J.R.; Price, D.B.; Hunt, B.J.; Joseph, P.; Gao, F.; Milnes, G.J.; Cunliffe, L.K. Flame retardance in some polystyrenes and poly(methyl methacrylate)s with covalently bound phosphorus-containing groups: initial screening experiments and some laser pyrolysis mechanistic studies. Polym. Degrad. Stab.
**2000**, 69, 267–277. [Google Scholar] [CrossRef] - Ebdon, J.R.; Hunt, B.J.; Joseph, P. Thermal degradation and flammability characteristics of some polystyrenes and poly(methyl methacrylate)s chemically modified with silicon-containing groups. Polym. Degrad. Stab.
**2004**, 83, 181–185. [Google Scholar] [CrossRef] - Zhang, S.; Hull, T.R.; Horrocks, A.R.; Smart, G.; Kandola, B.K.; Ebdon, J.; Hunt, B.; Joseph, P. Thermal degradation analysis and XRD characterisation of fibre-forming synthetic polypropylene containing nanoclay. Polym. Degrad. Stab.
**2007**, 92, 727–732. [Google Scholar] [CrossRef] - Marxman, G.A.; Gilbert, M. Turbulent boundary layer combustion in the hybrid rocket. In Ninth International Symposium on Combustion; Academic Press: New York, NY, USA, 1963; pp. 371–383. [Google Scholar]
- Marxman, G.A.; Wooldridge, C.E.; Muzzy, R.J. Fundamentals of hybrid boundary layer combustion. In Heterogeneous Combustion (AIAA Progress in Astronautics and Aeronautics); Wolfhard, H.G., Glassman, I., Green, L., Jr., Eds.; Academic Press: New York, NY, USA, 1964; Volume 15, pp. 485–521. [Google Scholar]
- Chiaverini, M.J. Review of solid-fuel regression rate behavior in classical and nonclassical hybrid rocket motors. In Fundamentals of Hybrid Rocket Combustion and Propulsion (Progress in Astronautics and Aeronautics); Chiaverini, M.J., Kuo, K.K., Eds.; AIAA: Reston, VA, USA, 2007; Volume 218, pp. 37–125. [Google Scholar]
- Lengelle, G. Solid-fuel pyrolysis phenomena and regression rate. In Fundamentals of Hybrid Rocket Combustion and Propulsion (Progress in Astronautics and Aeronautics); Chiaverini, M.J., Kuo, K.K., Eds.; AIAA: Reston, VA, USA, 2007; Volume 218, pp. 127–165. [Google Scholar]
- Zeng, W.R.; Li, S.F.; Chow, W.K. Review on chemical reactions of burning poly(methylmethacrylate) PMMA. J. Fire Sci.
**2002**, 20, 401–433. [Google Scholar] [CrossRef] - Ananth, R.; Ndubizu, C.C.; Tatem, P.A. Burning rate distributions for boundary layer flow combustion of a PMMA plate in forced flow. Combust. Flame
**2003**, 135, 35–55. [Google Scholar] [CrossRef] - Stoliarov, S.I.; Crowley, S.; Lyon, R.E.; Linteris, G.T. Prediction of the burning rates of non-charring polymers. Combust. Flame
**2009**, 156, 1068–1083. [Google Scholar] [CrossRef] - Arisawa, H.; Brill, T.B. Kinetics and mechanisms of flash pyrolysis of poly(methyl methacrylate) (PMMA). Combust. Flame
**1997**, 109, 415–426. [Google Scholar] [CrossRef] - Bedir, H.; T’ien, J.S. A Computational Study of Flame Radiation in PMMA Diffusion Flames Including Fuel Vapor Participation. In Proceedings of the Twenty-Seventh Symposium (International) on Combustion, Combustion Institute, Pittsburgh, PA, USA, 2–7 August 1998; Volume 27, pp. 2821–2828.
- Vovelle, C.; Delfau, J.L.; Reuillon, M.; Bransier, J.; Laraqui, N. Experimental and numerical study of the thermal degradation of PMMA. Combust. Sci. Technol.
**1987**, 53, 187–207. [Google Scholar] [CrossRef] - Krishnamurthy, L.; Williams, F.A. Fourteenth Symposium (International) on Combustion; The Combustion Institute: Pittsburgh, PA, USA, 1974. [Google Scholar]
- Kashiwagi, T.H.; Brown, J.E. Thermal and oxidative degradation of poly(methyl methacrylate) molecular weight. Macromolecules
**1985**, 18, 131–138. [Google Scholar] [CrossRef] - Kumar, R.N.; Stickler, D.B. Polymer-degradation theory of pressure-sensitive hybrid combustion. Proc. Symp. (Int.) Combust.
**1971**, 13, 1059–1072. [Google Scholar] [CrossRef] - Madorsky, S.L. Thermal Degradation of Organic Polymers; Interscience Publishers: New York, NY, USA, 1964. [Google Scholar]
- Zeng, W.R.; Li, S.F.; Chow, W.K. Preliminary studies on burning behavior of poly(methylmethacrylate) (PMMA). J. Fire Sci.
**2002**, 20, 297–317. [Google Scholar] [CrossRef] - GRI-MECH Database Homepage. Available online: http://www.me.berkeley.edu/gri-mech/ (accessed on 21 October 2011).
- Bell, K.M.; Tipper, C.F.H. The slow combustion of methylalcohol, a general investigation. Proc. R. Soc. Lond. Ser. A
**1956**, 238, 256–268. [Google Scholar] [CrossRef] - Vardanyan, I.A.; Sachyan, G.A.; Nalbandyan, A.B. Kinetics and mechanism of formaldehyde oxidation. Combust. Flame
**1971**, 17, 315–322. [Google Scholar] [CrossRef] - Hay, J.M.; Hessam, K. The oxidation of gaseous formaldehyde. Combust. Flame
**1971**, 16, 237–242. [Google Scholar] [CrossRef] - Hidaka, Y.; Hattori, K.; Okuno, T.; Inami, K.; ABE, T.; Koike, T. Shock-tube and modeling study of acetylene pyrolysis and oxidation. Combust. Flame
**1996**, 107, 401–417. [Google Scholar] [CrossRef] - Wilkie, C.A. TGA/FTIR: An extremely useful technique for studying polymer degradation. Polym. Degrad. Stab.
**1999**, 66, 301–306. [Google Scholar] [CrossRef] - Raemaekers, K.G.H.; Bart, J.C.J. Applications of simultaneous thermogravimetry-mass spectrometry in polymer analysis. Thermochim. Acta
**1997**, 295, 1–58. [Google Scholar] [CrossRef] - Kaisersberger, E.; Post, E. Practical aspects for the coupling of gas analytical methods with thermal-analysis instruments. Thermochim. Acta
**1997**, 295, 73–93. [Google Scholar] [CrossRef] - Maciejewski, M.; Baiker, A. Quantitative calibration of mass spectrometric signals measured in coupled TA-MS system. Thermochim. Acta
**1997**, 295, 95–105. [Google Scholar] [CrossRef] - Marsanich, K.; Barontini, F.; Cozzani, V.; Petarca, L. Advanced pulse calibration techniques for the quantitative analysis of TG-FTIR data. Thermochim. Acta
**2002**, 390, 153–168. [Google Scholar] [CrossRef] - Branley, N.; Jones, W.P. Large eddy simulation of a turbulent non- premixed flame. Combust. Flame
**2001**, 127, 1914–1934. [Google Scholar] [CrossRef] - Peters, N. Laminar diffusion flamelet models in non-premixed turbulent combustion. Prog. Energy Combust. Sci.
**1984**, 10, 319–339. [Google Scholar] [CrossRef] - Pitsch, H.; Chen, M.; Peters, N. Unsteady flamelet modeling of turbulent hydrogen-air diffusion flames. Proc. Symp. (Int.) Combust.
**1998**, 27, 1057–1064. [Google Scholar] [CrossRef] - Pitsch, H.; Cha, C.M.; Fedotov, S. Interacting Flamelet Model for Non-Premixed Turbulent Combustion with Local Extinction and Re-Ignition; Annual Research Briefs 2001; Center for Turbulence Research, Stanford University: Menlo Park, CA, USA, 2001. [Google Scholar]
- Pitsch, H.; Peters, N. A consistent flamelet formulation for non-premixed combustion considering differential diffusion effects. Combust. Flame
**1998**, 114, 26–40. [Google Scholar] [CrossRef] - Gran, I.R.; Melaaen, M.C.; Magnussen, B.F. Numerical simulation of local extinction effects in turbulent combustor flows of methane and air. Proc. Symp. (Int.) Combust.
**1994**, 25, 1283–1291. [Google Scholar] [CrossRef] - Pantano, C.; Sarkar, S.; Williams, F.A. Mixing of a conserved scalar in a turbulent reacting shear layer. J. Fluid Mech.
**2003**, 481, 291–328. [Google Scholar] [CrossRef] - Pitsch, H. Extended Flamelet Model for LES of Non-Premixed Combustion; Annual Research Briefs 2000; Center for Turbulence Research, Stanford University: Menlo Park, CA, USA, 2000. [Google Scholar]
- Chiaverini, M.J.; Serin, N.; Johnson, D.K.; Lu, Y.; Kuo, K.K.; Risha, G.A. Regression rate behavior of hybrid rocket solid fuels. J. Propul. Power
**2000**, 16, 125–132. [Google Scholar] [CrossRef] - Sankaran, V. Computational fluid dynamics modeling of hybrid rocket flowfields. In Fundamentals of Hybrid Rocket Combustion and Propulsion (Progress in Astronautics and Aeronautics); Chiaverini, M.J., Kuo, K.K., Eds.; AIAA: Reston, VA, USA, 2007; Volume 218, pp. 323–349. [Google Scholar]
- Hossain, M.; Jones, J.C.; Malalasekera, W. Modelling of a bluff-body nonpremixed flame using a coupled radiation/flamelet combustion model. Flow Turbul. Combust.
**2001**, 67, 217–234. [Google Scholar] [CrossRef] - Novozhilov, V. Computational fluid dynamics modelling of compartment fires. Prog. Energy Combust. Sci.
**2001**, 27, 611–666. [Google Scholar] [CrossRef] - Lockwood, F.C.; Shah, N.G. A new radiation solution method for incorporation in general combustion prediction procedures. Proc. Symp. (Int.) Combust.
**1981**, 18, 1405–1414. [Google Scholar] [CrossRef] - Novozhilov, V.; Harvie, D.J.E.; Kent, J.H.; Apte, V.B.; Pearson, D. A computational fluid dynamics study of wood fire extinguishment by water sprinkler. Fire Saf. J.
**1997**, 29, 259–282. [Google Scholar] [CrossRef] - Novozhilov, V.; Harvie, D.J.E.; Green, A.R.; Kent, J.H. A computational fluid dynamic model of fire burning rate and extinction by water sprinkler. Combust. Sci. Technol.
**1997**, 123, 227–245. [Google Scholar] [CrossRef] - Kuo, K.K. Principles of Combustion; Wiley: New York, NY, USA, 1986. [Google Scholar]
- Gosman, A.D.; Ioannides, E. Aspects of computer simulation of liquid-fuelled combustors. In Proceedings of the American Institute of Aeronautics and Astronautics, Aerospace Sciences Meeting, St. Louis, MO, USA, 12–15 January 1981.
- Faeth, G.M. Evaporation and combustion of sprays. Prog. Energy Combust. Sci.
**1983**, 9, 1–76. [Google Scholar] [CrossRef] - Crowe, C.T.; Sharma, M.P.; Stock, D.E. Theparticle-source-incell(PSI Cell) model for gas-droplet flows. J. Fluids Eng.
**1977**, 99, 325–332. [Google Scholar] [CrossRef] - Crowe, C.T. Heat transfer in dispersed-phase flow. In Two Phase Momentum, Heat and Mass Transfer in Chemical, Process and Energy Engineering Systems; Afgan, N.H., Tsiklauri, G.V., Eds.; Hemisphere: McGraw-Hill: New York, NY, USA, 1978; Volume 1, pp. 23–32. [Google Scholar]
- Shuen, J.S.; Chen, L.D.; Faeth, G.M. Evaluation of a stochastic model of particle dispersion in a turbulent round jet. AIChE J.
**1983**, 29, 167–170. [Google Scholar] [CrossRef] - Shuen, J.S.; Chen, L.D.; Faeth, G.M. Predictions of the structure of turbulent, particle. Laden, round jets. AIAA J.
**1983**, 21, 1483–1484. [Google Scholar] [CrossRef] - Novozhilov, V. On some integrable cases of particle motion in a fluid, mathematics in engineering. Sci. Aerosp.
**2010**, 1, 371–380. [Google Scholar] - Novozhilov, V. Flashover control under fire suppression conditions. Fire Saf. J.
**2001**, 36, 641–660. [Google Scholar] [CrossRef] - Putnam, A. Integrable form of droplet drag coefficient. Am. Rocket Soc. J.
**1961**, 31, 1467–1468. [Google Scholar] - Faeth, G.M. Evaporation and Combustion of Sprays. Prog. Energy Combust. Sci.
**1983**, 9, 1–76. [Google Scholar] [CrossRef] - Shirolkar, J.S.; Coimbra, C.F.M.; McQuay, M.Q. Fundamental aspects of modeling turbulent particle dispersion in dilute flows. Progr. Energy Combust. Sci.
**1993**, 22, 363–399. [Google Scholar] [CrossRef] - Ranz, W.E.; Marshall, W.R., Jr. Evaporation from drops: Part I. Chem. Eng. Progress.
**1952**, 48, 141–146. [Google Scholar] - Ranz, W.E.; Marshall, W.R., Jr. Evaporation from drops: Part II. Chem. Eng. Progress.
**1952**, 48, 173–180. [Google Scholar] - Faeth, G.M.; Lazar, R.S. Fuel droplet burning rates in a combustion gas environment. AIAA J.
**1971**, 9, 2165–2171. [Google Scholar] [CrossRef] - Migdal, D.; Agosta, V.D. A Source Flow Model for Continuum Gas-Particle Flow. J. Appl. Mech.
**1967**, 34, 860–865. [Google Scholar] [CrossRef] - Kumar, S.; Heywood, G.M.; Liew, S.K. Superdrop Modelling of a Sprinkler Spray in a Two-phase Cfdparticle Tracking Model. In Proceedings of the Fifth International Symposium on Fire Safety Science, Melbourne, Australia, 3–7 March 1997; pp. 889–900.
- Lin, C.L.; Chiu, H.H. Numerical Analysis of Spray Combustion in Hybrid Rocket, AIAA 95-2687. In Proceedings of the 31st AIANASMUSAUASEE Joint Propulsion Conference and Exhibition, San Diego, CA, USA, 1995.
- Kawamura, T.; Kuwahara, K. Computation of High Reynolds Number Flow Around A Circular Cylinder with Surface Roughness. In Proceedings of the 22nd American Institute of Aeronautics and Astronautics, Aerospace Sciences Meeting, Reno, NV, USA, 9–12 January 1984.
- Boris, J.P.; Grinstein, F.F.; Oran, E.S.; Kolbe, R.L. New insights into large eddy simulation. Fluid Dyn. Res.
**1992**, 10, 199–228. [Google Scholar] [CrossRef] - van Leer, B. Towards the ultimate conservative difference scheme V: A second-order sequel to Godunov’s method. J. Computat. Phys.
**1979**, 32, 101–136. [Google Scholar] [CrossRef] - Shu, C.W.; Osher, S. Efficient implementation of essentially non-oscillatory shock-capturing schemes, I. J. Comput. Phys.
**1988**, 77, 439–471. [Google Scholar] [CrossRef] - Shu, C.W.; Osher, S. Efficient implementation of essentially non-oscillatory shock-capturing schemes, II. J. Comput. Phys.
**1989**, 83, 32–78. [Google Scholar] [CrossRef] - Liu, X.D.; Osher, S.; Chan, T. Weighted essentially non-oscillatory schemes. J. Comput. Phys.
**1994**, 115, 200–212. [Google Scholar] [CrossRef] - Jiang, G.S.; Shu, C.W. Efficient implementation of weighted ENO schemes. J. Comput. Phys.
**1996**, 126, 202–228. [Google Scholar] [CrossRef] - Deng, X.G.; Zhang, H.X. Developing high-order weighted compact nonlinear schemes. J. Comput. Phys.
**2000**, 165, 22–44. [Google Scholar] [CrossRef] - Drikakis, D.; Hahn, M.; Mosedale, A.; Thornber, B. Large eddy simulation using high-resolution and high-order methods. Philos. Trans. R. Soc. A
**2009**, 367, 2985–2997. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Hahn, M.; Drikakis, D. Implicit large-eddy simulation of swept-wing flow using high-resolution methods. AIAA J.
**2009**, 47, 618–630. [Google Scholar] [CrossRef] - Panaras, A.G.; Drikakis, D. High-speed unsteady flows around spiked-blunt bodies. J. Fluid Mech.
**2009**, 632, 69–96. [Google Scholar] [CrossRef] - Shimada, Y.; Thornber, B.; Drikakis, D. High-order implicit large eddy simulation of gaseous fuel injection and mixing of a bluff body burner. Comput. Fluids
**2011**, 44, 229–237. [Google Scholar] [CrossRef] - Gordon, S.; McBride, B.J. Computer Program for Calculation of Complex Chemical Equilibrium Compositions and Applications, II. Users Manual and Program Description; Nasa Reference Publication 1311; Lewis Research Center: Cleveland, OH, USA, 1996. [Google Scholar]
- Liou, M.S.; Steffen, C.J., Jr. A new flux splitting scheme. J. Comput. Phys.
**1993**, 107, 23–39. [Google Scholar] [CrossRef] - Liou, M.S. A sequel to AUSM, Part II: AUSM+-up for all speeds. J. Comput. Phys.
**2006**, 214, 137–170. [Google Scholar] [CrossRef] - Shima, E.; Kitamura, K. On New Simple Low-Dissipation Scheme of AUSM-Family for All Speeds; AIAA Paper 2009-136; AIAA: Reston, VA, USA, 2009. [Google Scholar]
- Roe, P.L. Characteristic-based schemes for the Euler equations. Annu. Rev. Fluid Mech.
**1986**, 18, 337–365. [Google Scholar] [CrossRef] - van Leer, B. Towards the ultimate conservative difference scheme II: Monotonicity and conservation combined in a second-order scheme. J. Comput. Phys.
**1974**, 14, 361–370. [Google Scholar] [CrossRef] - Anderson, W.K.; Thomas, J.L.; van Leer, B. A Comparison of Finite Volume Flux Vector Splittings for the Euler Equations; AIAA Paper 85-122; AIAA: Reston, VA, USA, 1985. [Google Scholar]

© 2011 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

## Share and Cite

**MDPI and ACS Style**

Novozhilov, V.; Joseph, P.; Ishiko, K.; Shimada, T.; Wang, H.; Liu, J.
Polymer Combustion as a Basis for Hybrid Propulsion: A Comprehensive Review and New Numerical Approaches. *Energies* **2011**, *4*, 1779-1839.
https://doi.org/10.3390/en4101779

**AMA Style**

Novozhilov V, Joseph P, Ishiko K, Shimada T, Wang H, Liu J.
Polymer Combustion as a Basis for Hybrid Propulsion: A Comprehensive Review and New Numerical Approaches. *Energies*. 2011; 4(10):1779-1839.
https://doi.org/10.3390/en4101779

**Chicago/Turabian Style**

Novozhilov, Vasily, Paul Joseph, Keiichi Ishiko, Toru Shimada, Hui Wang, and Jun Liu.
2011. "Polymer Combustion as a Basis for Hybrid Propulsion: A Comprehensive Review and New Numerical Approaches" *Energies* 4, no. 10: 1779-1839.
https://doi.org/10.3390/en4101779