## Abstract

In this paper, we formulate and analyse an elementary model for autoignition of cylindrical laminar jets of fuel injected into an oxidizing ambient at rest. This study is motivated by renewed interest in analysis of hydrothermal flames for which such configuration is common. As a result of our analysis, we obtain a sharp characterization of the autoignition position in terms of the principal physical and geometrical parameters of the problem.

## 1. Introduction

Autoignition is the process of an abrupt growth of the reaction rate in an explosive system being initially in a non-reactive state. Theoretical studies of autoignition can be traced back to the classical works of Semenov, Frank-Kamentskii and Zeldovich, see [1,2]. Analysis of autoignition of premixed and non-premixed flames has been performed by many authors in different situations and at different levels of complexity [3–7]. In particular, in the context of non-premixed flames, the ignition studies were predominantly undertaken for a counter-flow configuration [8–10]. In this paper, we propose an elementary model for autoignition of diffusion flames in laminar jets of fuel injected into an oxidizing ambient at rest. To the best of our knowledge, mathematical studies of this configuration have not previously been undertaken. Our analysis of autoignition in this particular configuration is motivated by recent interest in hydrothermal flames for which such a configuration is common.

Hydrothermal flames are diffusion flames produced in aqueous environments at conditions above the thermodynamic critical point of water with temperatures and pressures exceeding 374°C and 221 bar, respectively. These flames were first reported in the late 1980s [11] and since then have been studied by many researchers (see [12] for a recent review of experimental results). When conditions are suitable (i.e. temperatures above the ignition temperature with reactant concentrations at appropriate levels), hydrothermal flames can be observed. These conditions are often reached (sometimes unintentionally) during the oxidation reactions taking place in reactors designed to operate above the critical point of water. Due to the renewed interest in supercritical water oxidation (SCWO) as an advantageous technology for the complete and highly efficient destruction of aqueous organic waste streams, recent research has been directed towards understanding the physical processes that cause flames to evolve from the relatively low-temperature oxidation reactions for which most SCWO reactors are designed.

In a typical SCWO reactor, an aqueous waste stream comprising 10–30% of organic waste is introduced at conditions above the critical point of water. Organic compounds and gases (e.g. O_{2}) readily mix in supercritical water and when temperatures are sufficiently high oxidation reactions take place at very high reaction rates resulting in nearly perfect conversion efficiencies (often above 99.9%) in very brief reactor residence times [12–14]. The SCWO processing of organic waste shows many strong advantages over conventional technologies from both the perspective of efficiency and the perspective of being a ‘green technology’, owing to the fact that there is often sufficient thermal energy released from the oxidation of the organic waste to allow the conversion process to be self-sustaining [13]. These advantages have attracted considerable attention from scientists and engineers over the past decade making the analysis of SCWO processes in general, and hydrothermal flames in particular, of great interest. Apart from the technological relevance, hydrothermal flames provide a canonical system for studying combustion processes at very high pressures.

While a number of experimental studies of hydrothermal flames exist in the literature, there are only a few theoretical papers on the subject (e.g. [15,9] and references therein). In this paper, we formulate and analyse a simple model that provides a sharp characterization of autoignition in terms of the principal physical and geometric parameters. We hope that our results will be useful for guiding future studies of autoignition of laminar jets.

The paper is organized as follows. In §2, we derive a model of autoignition for laminar jets. In §3, we state the main results of the analysis of this model and provide heuristic arguments to give physical insight of these results as well as results of numerical simulations. Section 4 is dedicated to proof of the main results. In §5, we provide a summary of our results and a brief discussion of the direction of future research in this area.

## 2. Formulation of the model

A common experimental configuration for the studies of ignition and dynamics of hydrothermal flames consists of a combustion vessel and an injection inlet [16]. The vessel is filled with a fuel-rich mixture at supercritical conditions and a small amount of oxidizer in supercritical conditions is injected into the ambient at rest. Alternatively, a fuel-rich jet is injected into an oxidizing ambient, both being at a supercritical state. This process may lead to the ignition of an inverse diffusion or diffusion flame, respectively. Under certain hydrodynamic conditions, the jet of injected substance has approximately cylindrical shape with a circular cross section (figure 1). In such jets, the flow may be regarded as unidirectional with an axially uniform profile.

To model the process, we will make the following assumptions: the jet has a prescribed shape of a cylinder with circular cross section and fixed height; the velocity inside the jet is constant; an ignition, if successful, has to occur within the jet; the activation energy of the reaction is large; the chemical reaction occurs exclusively on the surface of the jet; prior to autoignition, both the fuel and the oxidizer are in excess on the surface of the jet; the heat exchange between the jet and the ambient is negligible; diffusion in a vertical direction is negligible in comparison with advection; the temperature assumes a steady-state profile.

Under these assumptions, the equation governing the temperature field *T* inside the jet of radius *r*_{j} and height *h*_{j} is as follows:
*κ* is the thermal conductivity, *u* is the flow velocity within the jet, *c*_{p} and *ρ* are specific heat and density, *r* and *z* are coordinates in radial and vertical directions.

This equation should be complemented by a condition prescribing the jet temperature at the entrance to the vessel, namely
*Q* is the reaction intensity, *C*_{o} and *C*_{f} are concentrations of the oxidizer and the fuel, *E* is the effective activation energy and *R* is the universal gas constant. Here *C*^{0}_{o} means zero-order reaction with respect to oxygen [17].

Using the assumptions given above, we set the concentration of fuel and oxidizer on the surface of the jet to be equal to the initial concentrations

Introducing scaled temperature *θ*, scaled radius λ and height of the jet, *μ*, as well as the scaled radial *ξ* and vertical *ζ* coordinates
*θ*(*ξ*,*ζ*) inside the jet:

A study of autoignition of the jet therefore reduces to the analysis of solutions of problem (2.7)–(2.9) that depend on two parameters: scaled radius of the jet λ and its scaled height *μ*. As we will show in §3, for fixed λ there exists *μ*<*ζ**(λ) and blows up (becomes unbounded) as *ζ*→*ζ** when *μ*≥*ζ**. In the framework of this theory, as in any theory based on the Frank–Kamenetskii approximation, autoignition is associated with thermal runaway, that is the ‘blow up’ of the solution of corresponding differential equations. Consequently, the autoignition condition reads

It is important to note that problem (2.7)–(2.9) and autoignition condition (2.11) are quite different from classical models of autoignition [1,2]. The main difference of the model considered here is that the nonlinear reaction term appears in boundary condition (2.9), whereas in classical models the reaction term is present in the equation describing temperature field in the bulk. Models involving boundary reaction, however, were previously studied in the literature [4–6]. It is also worth mentioning that as the activation energy *E* is usually large, parameters λ and *μ* in problem (2.7)–(2.9) are very sensitive to even small variations in initial temperature *T*_{0}. Thus, the range of values for λ and *μ* may vary by orders of magnitude. Consequently, it is important to understand the qualitative behaviour of the ignition position *ζ**(λ) in the entire range of

In §3, we will give a full analysis of solutions for the problem and derive an approximate formula for the ignition position *ζ**(λ).

## 3. Analysis of the model

In this section, we will discuss properties of the solutions of problem (2.7)–(2.9). These properties are given by the following theorem.

### Theorem 3.1

*For each λ>0, there exists* *such that problem* (2.7)–(2.9) *has a unique classical solution θ(ξ,ζ) provided μ<ζ**.

*If μ≥ζ***, the solution of problem* (2.7)–(2.9) *blows up as ζ→ζ***. The blow up position, ζ***, is a bounded non-decreasing function of λ that obeys the following upper bound:*
*where, σ*_{m}*<1 is given by*

*Moreover, over the interval of its existence, the solution of problem* (2.7)–(2.9) *is non-negative and non-decreasing in both ξ and ζ.*

### Remark 3.2

Let us note that theorem 3.1 can be easily adopted for more general types of nonlinearities. In particular, for the case when the exponent on the right-hand side of equation (2.9) is replaced by a positive increasing convex *C*^{1} function *g* satisfying

A proof of theorem 3.1 is given in §4. Here, let us discuss the physical implications of theorem 3.1. The most important result of this theorem is that the autoignition position *ζ**(λ) is a bounded monotone function of its argument. The fact that *ζ**(λ) increases as λ increases is intuitively clear as a jet of larger radius can absorb more heat before ignition occurs. What is rather interesting is that even in the limit of infinite radius, the jet still ignites at a finite height. This means that with jets of substantial radius and height, which can absorb an extremely large amount of heat, the limiting factor is the heat transport from the surface of the jet to its interior. As is evident from our analysis, this heat transport is not sufficiently fast to prevent the autoignition of jets with height exceeding its critical value *σ*_{m}, regardless of the jet's radius.

Position of the autoignition point can be computed numerically and is presented on figure 2. This curve can be approximated with good accuracy by the following simple equation:

Let us now discuss the behaviour of this curve. One can see that for small values of λ, the ignition position *ζ**(λ) scales linearly with λ. This behaviour can be explained as follows. In the case when the radius of the jet is small enough, the temperature field is close to uniform in the entire jet, despite the fact that the reaction takes place only on the surface (figures 3 and 4). The substantial gradient of temperature is only seen for *ζ* close to *ζ**. Therefore, the average temperature of the cross section of the jet *Θ*(*ζ*) is practically equal to the temperature on the boundary for *ζ* away from *ζ**. As a result, upon integration of (2.7) and taking into account boundary conditions (2.9), we have (see step 5 of theorem 3.1 for more details)
*Θ*(0)=0, we obtain from (3.4)
*ζ**(λ) is reasonably close to a linear function for λ≤0.5, and thus (3.6) is applicable in this range of λ. Let us also note that equation (3.4) is formally identical to a classical Semenov model of thermal explosion in the absence of heat loss [2].

As the radius of the jet increases, the curve *ζ**(λ) starts to deviate substantially from the linear function. This is due to the fact that when λ increases, a sharp boundary layer starts to form near the surface of the jet. Consequently, the average temperature of the jet's cross section becomes substantially smaller than its maximal value attained at the jet's surface (figures 5 and 6). For large enough λ, the temperature is essentially zero except for some small (order of unity) vicinity of the jet's surface. As a result, when λ is sufficiently large, the solution of problem (2.7)–(2.9) becomes indistinguishable from the solution of the one-dimensional heat equation posed on the half line. That is, the solution of the following boundary value problem:

One can show using simple reflection arguments that the solution of this problem, evaluated at the boundary *v*(*ζ*)=*ϑ*(0,*ζ*), solves the following integral equation:
*ζ*<*σ*_{m} and blows up as *ζ*→*σ*_{m}. The value of *σ*_{m}≈0.28 was estimated numerically and as a result, for sufficiently large values of λ, we have
*ζ**(λ) for λ>3.

In the following section, we give a proof of theorem 3.1.

## 4. Proof of theorem 3.1

In this section, we give a proof of the main theorem stated in the previous section. Part of the proof of the theorem is based on a simple version of the classical parabolic comparison principle [18–20] which is given by the following proposition.

### Proposition 4.1 (Parabolic comparison principle)

*Let*
*where α*=*α*(*ξ*,*ζ*) *and β*=*β*(*ξ*,*ζ*,*w*) *are some given functions*.

*Assume that there exist continuous bounded functions* *and* *with continuous first derivatives in ζ and continuous first and second derivatives in ξ that satisfy differential inequalities*
*and*
*respectively, where* *w*_{0} *is a given function*.

*Then*,

Existence of functions

Now we turn to the proof of theorem 3.1.

### Proof of theorem 3.1

The proof proceeds in several steps.

*Step 1*. *Short time existence and uniqueness and finite height blow up.*

Existence of a unique classical solution for problem (2.7)–(2.9) with *μ* sufficiently small, as well as the fact that the solution of this problem blows up provided *μ* is large enough, were established in [21] (see also [22] for a more general framework).

*Step 2*. *Non-negativity and monotonicity of solution.*

Let us now show that on the entire interval of its existence, the solution of problem (2.7)–(2.9) satisfies *θ*,*θ*_{ζ},*θ*_{ξ}≥0.

Assume that the classical solution of problem (2.7)–(2.9) exists for *ζ*∈[0,*μ*_{1}) for some *μ*_{1}>0. The fact that *θ*≥0 is a direct consequence of parabolic comparison principle, as *θ*=0 is a sub-solution for problem (2.7)–(2.9). We now turn to the monotonicity properties of *θ*. We start with the monotonicity in *ζ*. Let *ψ*(*ξ*,*ζ*)=*θ*(*ξ*,*ζ*+*τ*). As *ψ*(*ξ*,0)=*θ*(*ξ*,*τ*)≥0, we have by the comparison principle *w*(*ξ*,*ζ*)≥*θ*(*ξ*,*ζ*) for *ξ*∈[0,λ], *ζ*∈(0,*μ*_{1}−*τ*). Thus, for any *τ*>0 , we have (*ψ*(*ξ*,*ζ*)−*θ*(*ξ*,*ζ*))/*τ*≥0. Taking the limit as *τ*→0 , we have *θ*_{ζ}≥0. To show the monotonicity in *ξ*, we integrate equation (2.7) with respect to *ξ* that gives
*θ*_{ζ}≥0, we have *θ*_{ξ}≥0.

*Step 3*. *Monotonicity of blow up position ζ* in λ*.

Now let us show that *ζ**(λ) is a non-decreasing function of its argument. Let λ_{2}>λ_{1} and set *ϕ*_{1}(*ξ*,*ζ*):=*θ*_{λ1}(*ξ*,*ζ*) and *ϕ*_{2}(*ξ*,*ζ*):=*θ*_{λ2}(*ξ*+λ_{2}−λ_{1},*ζ*), *η*(*ξ*,*ζ*):=*ϕ*_{1}(*ξ*,*ζ*)−*ϕ*_{2}(*ξ*,*ζ*). Assume that both solutions *θ*_{λ1} and *θ*_{λ2} are classical for *ζ*∈[0,*μ*_{2}) for some *μ*_{2}>0. It then follows from equation (2.7) that
*ϕ*_{1} and *ϕ*_{2}, and

In a view of the fact that (*ϕ*_{2})_{ξ}≥0, we conclude that *η*=0 is a sub-solution for problem (4.8)–(4.10) and thus *ϕ*_{1}(*ξ*,*ζ*)≥*ϕ*_{2}(*ξ*,*ζ*) for all *ξ*∈[0,λ] and *ζ*∈[0,*μ*_{2}). By monotonicity of solutions of problem (2.7)–(2.9) established in Step 2 of this theorem, we have
*θ*_{λ2} cannot blow up before *θ*_{λ1}.

*Step 4*. *Limiting behaviour of solution as*

Observe first that since *θ* solves the heat equation (2.7) in a disc of radius λ, we can represent the solution of this equation on a boundary of this disc in an integral form
*B*_{λ} and *dS*(**r ^{′}**) are circle and element of length of a circle of radius λ,

**r**and

**r**are position vectors of an arbitrary fixed and arbitrary point on a circle of radius λ,

^{′}*ν*is an outward unit normal to a circle and

Equation (4.12) follows from the standard representation formula for the solution of a heat equation [23], ch. 4 and the jump condition on the normal derivative of the heat kernel [23], lemma 4.3.2. After straightforward computations, we obtain, from equation (4.12) with boundary condition (2.9), the following integral equation:
*I*_{n} are modified Bessel functions of the first kind [24]. One can verify that
*ζ* where the classical solution of problem (2.7)–(2.9) exists, we have *θ*(λ,*ζ*)→*v*(*ζ*) as *v*(*ζ*) solves an integral equation

*Step 5*. *Upper bound on the blow up position*.

Let us first give a global upper bound on *ζ**(λ). As follows from the result of the previous step, as *g*(0)=0, we conclude
*g*(*ζ*) and therefore *v*(*ζ*) blows up at a point *ζ**<1. As *ζ**(λ) is a non-decreasing function in λ, we conclude that *ζ**<1 is a global upper bound for all λ.

Let us show that the solution of problem (2.7)–(2.9) blows up at the position *ζ**(λ)≤λ/2. Integrating equation (2.7) in *ξ* and taking into account boundary condition (2.9), we have
*θ* over a circular cross section of the jet. By maximum principle and monotonicity of *θ* in both variables, we have that *θ*(λ,*ζ*)≥*Θ*(*ζ*), which in combination with (3.2) gives
*Θ*(0)=0, we have from equation (4.26) that
*ζ*≤λ/2, and in a view of non-negativity and monotonicity of *θ*, we then have *ζ**(λ)≤λ/2. After combining the results of this step, we then have equation (3.1). ▪

### Remark 4.2

It follows directly from the proof of theorem 3.1 that analysis of problem (2.7)–(2.9) in the limiting cases of small and large λ reduces to analysis of the following equation:
*v* is temperature on the surface of the jet, *s* is a scaled coordinate along the jet, *α*=1, *s*=2*ζ*/λ in the limit of small λ and *s*=*ζ* in the limit of large λ. This, in particular, implies that near the blow up position (height of ignition) *s** the temperature of the surface of the jet asymptotically behaves as follows:

## 5. Concluding remarks

In this paper, we have developed and fully analysed a simple model for autoignition of a laminar jet injected into ambient at rest. The motivation for this study was a renewed interest in hydrothermal flames in the SCWO environment. As a result of our analysis, we have derived a sharp condition for autoignition of such flames. With a good accuracy, this condition in dimensional form reads

Equation (5.1) connects the principal physical and geometric parameters of the problem considered in this paper, showing that the autoignition occurs when the height of the jet is sufficiently large. Additionally, our results show that the temperature of the jet increases from the inlet position to an ignition point along the axial length of the flow. Given sufficient energy release from the ignition kernel a stable flame will develop and then become elevated from the inlet point. These observations are in qualitative agreement with experimental observations of ignition and oxidation in a supercritical environment [14,26,27]. Equation (5.1) may be used for assessing appropriate experimental parameters (e.g. flow rates, bulk fluid temperature and reactant concentrations) in future experimental work on hydrothermal flames. Additionally, this work provides a first step in understanding the linkages between the dominant fundamental physical processes upon which more complicated first-order analytical models will be developed. Future work will use a similar modelling approach to develop a flame ignition model using a co-flow burner configuration.

## Data accessibility

The paper is theoretical and contains no data.

## Authors' contributions

P.V.G., D.J.G, U.G.H., M.C.H., M.J.K. and G.I.S. formulated the model and analysed the results. P.V.G., M.C.H. and G.I.S. drafted the manuscript. P.V.G. performed mathematical and numerical analysis of the model. All authors gave final approval for publication.

## Competing interests

We have no competing interests.

## Funding

The work of D.J.G, U.G.H., M.C.H. and M.J.K. was supported by the NASA Space Life and Physical Sciences Research and Applications Program. The work of P.V.G. and G.I.S. was supported, in part, by the US-Israel Binational Science Foundation under grant no. 2012057. G.I.S. also acknowledges support of the Israel Science Foundation via grant no. 335/13. P.V.G. was also supported by grant no. 317882 from Simons Foundation.

## Acknowledgements

Part of this work was done when P.V.G. was visiting NASA Glenn Research Center under the NASA Glenn Faculty Fellowship Program (NGFFP 2014). He thanks the Combustion Physics and Reacting Processes Branch of NASA Glenn Research Center for their hospitality. P.V.G. also thanks V. Moroz and C.B. Muratov for very valuable discussions and J. Coleman for creating excellent work environment.

- Received January 28, 2015.
- Accepted June 1, 2015.

- © 2015 The Author(s) Published by the Royal Society. All rights reserved.