Singular perturbed vector field method applied to ...

Applied Mathematical Modelling 61 (2018) 604–617

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Singular perturbed vector field method applied to combustion in diesel engine: Continuous case with thermal runaway OPhir Nave a,∗, Shlomo Hareli b a b

Department of Mathematics, Jerusalem College of Technology (JCT), Israel Department of Mathematics, Azrieli College of Engineering, Israel

a r t i c l e

i n f o

Article history: Received 28 April 2017 Revised 2 May 2018 Accepted 14 May 2018 Available online 22 May 2018 Keywords: Polydisperse spray Model reduction Asymptotic analysis Multi-scale systems

a b s t r a c t In this study, we employ the well-known method of a singularly perturbed vector field (SPVF) and its application to the thermal runaway of diesel spray combustion. Given a system of governing equations, consisting of hidden multi-scale variables, the SPVF method transfers and decomposes such a system into fast and slow singularly perturbed subsystems. The resulting subsystem enables us to better understand the complex system and simplify the calculations. Powerful analytical, numerical, and asymptotic methods (e.g., the method of slow invariant manifolds and the homotopy analysis method) can subsequently be applied to each subsystem. In this paper, we compare the results obtained by the methods of slow invariant manifolds and SPVF, as applied to the spray (polydisperse) droplets combustion model. © 2018 Elsevier Inc. All rights reserved.

1. Introduction Mathematical models relating to various engineering applications are usually described by a large set of complex equations (differential equations). For the purpose of numerical, analytical, and qualitative analysis, it is often desirable to reduce the system to smaller subsystems with a comparatively small accuracy loss. Generally, a large set of differential equations describing a complex realistic phenomenon has a number of essentially different time scales (i.e., rates of change), which correspond to sub-processes. Given such systems, it is highly challenging to reveal the hidden hierarchy and implicit multitime scales of the original systems that govern the equations; hence, an asymptotic method cannot be applied. Discovering the hierarchical structures of systems requires considerable complicated numerical treatments; however, this known hierarchical structure enables the application of numerous asymptotic approaches for the analysis of their behaviors. Several asymptotic methods and numerical tools are available that can be applied to multi-scale systems. Examples include the method of slow invariant manifolds (MIM), which has been applied to the thermal ignition of diesel spray [1–4], the iteration method of Roussel and Fraser [5–8], the computational singular perturbation (CSP) method [9,10], geometric singular perturbation theory [11–13], and the intrinsic low dimensional method (ILDM), which is a numerical approach [14–17]. The ILDM method successfully locates slow manifolds in the considered system, but also exhibits several principal problems. The main constraints are as follows: the algorithm cannot be applied to phase space domains, where the RHS leading eigenvalues of the considered system Jacobian matrix are complex.

∗

Corresponding author. E-mail addresses: [email protected], [email protected] (O. Nave).

https://doi.org/10.1016/j.apm.2018.05.016 0307-904X/© 2018 Elsevier Inc. All rights reserved.

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In this case, the ILDM method does not produce any decomposition of the original system, or produces incorrect decomposition. Even in the case of an explicitly known decomposition, the ILDM cannot treat certain zones in the phase plane, such as the turning zones (manifolds); that is, zones where critical changes in the system behavior occur, and where the method numerical algorithm produces no additional relevant solutions to the system dynamics (ghost-manifolds). The transposition intrinsic low dimensional method (TILDM) is a modification of the ILDM method [18,19]. The TILDM is based on the geometrical approach to hierarchical systems of ordinary differential equations. Researchers in the combustion theory field have suggested a new method known as global quasi-linearization (GQL) [20–23] and the singularly perturbed vector field (SPVF) to solve the above problems [24–27]. The main concept of this method is to transfer the original system of governing equations with the hidden hierarchy (hidden multi-scale structure) to its explicit form as the singularly perturbed subsystem (SPS). When this transformation from a hidden hierarchy model into a model with standard SPSs occurs, the analysis of the original system can be treated by the very powerful machinery of the standard SPS theory for model reduction and decomposition, as mentioned previously. The global information regarding the model decomposition is very useful and enables us to provide the qualitative structure of the slow and fast manifolds. This paper deals with the SPVF method applied to diesel spray thermal runaway. The aim of this research is to expose the hierarchy of the combustion process in a diesel engine, and decompose the model into fast and slow subsystems. This decomposition enables us to apply the well-known asymptotic method of the invariant manifold. The reduction of the model into a subsystem simplifies the model and hence reduces the computation time. 2. Preliminaries to SPVF method Given a large and complex mathematical/physical/chemical model with nonlinear governing equations, the aim of researchers in this field is to reduce the number of equations and discover the hierarchy of the dynamic system variables; that is, to decompose the system into fast and slow motions of the dynamic system variables. In order to achieve this, one should search for new coordinates and represent the governing equations (original model) in the form of a SPS. Once these coordinates are identified, we are able to decompose the original system into slow and fast subsystems, which enables us to apply asymptotic analytical methods such as MIM, perturbation analysis, and the homotopy analysis method (HAM). We introduce the general framework theory of the SPVF, as presented in the papers [28–30]. Definition 2.1. A real vector bundle ζ over a connected manifold I ⊂ Rm consists of a set E ⊂ Rm (the total set), a smooth map ρ : E → I (the projection), which is onto, and each fiber Fxζ = ρ −1 (x ) is a finite dimensional affine subspace. These objects are required to satisfy the following condition: for each x ∈ I, there exists a neighborhood U of x in I, an integer k, and a diffeomorphism φ : ρ −1 (U ) → U × Rk , such that a homomorphism exists on each fiber φ . Definition 2.2. Refer to a domain V ⊂ Rn as a structured domain (or a domain structured by a vector bundle) if there exists a vector bundle ζ and diffeomorphism ψ : V → W onto an open subset W ⊂ E, where E is the total set of ζ . Let

dx = F (x, δ ), dt

(2.1)

for δ > 0 be a dynamical family of ODE systems. Definition 2.3. Suppose that V is a domain structured by a vector bundle ζ and diffeomorphism ψ . For any point x ∈ V, refer to Mx := ψ −1 (ρ −1 (ρ (ψ (x ))) ∩ W ) a fast manifold associated with point x. Refer to the set of all fast manifolds Mx a family of fast manifolds of V. Denote dimMx = n f ast and refer to it as the fast dimension of F (x, δ ), and T Mx as the tangent space to Mx at point x. Definition 2.4. A family of fast manifolds Mx is linear if there exists a fixed linear subspace Lfast of Rn , such that Mx = {x} + L f ast for any x ∈ V . Refer to Lfast as a fast subspace. Definition 2.5. A parametric family F (x, δ ) : V → Rn of vector fields defined in a domain V, structured by a vector bundle ζ and diffeomorphism ψ , is an asymptotic SPVF if limδ →0 F (x, δ ) ⊂ T Mx for any x ∈ V, and the structure of domain V is minimal for the vector field F (x, δ ) in the following sense. There is a proper vector sub-bundle ζ 1 of the vector bundle ζ , such that F (x, δ ) is not an asymptotic SPVF in domain V structured by the sub-bundle ζ 1 and the same diffeomorphism ψ . This minimality property means that it is not possible to reduce further the dimension of fast manifolds Mx using subbundles. The next step is to introduce the method for decomposing the SPVF. Following the notations above, the vector field F (x, δ ) can be decomposed as the following sum:

F (x, δ ) = Ff ast (x, δ ) + Fslow (x, δ ),

(2.2)

Ff ast (x, δ ) = P r f ast F (x, δ ),

(2.3)

where

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Fslow (x, δ ) = F (x, δ ) − P r f ast F (x, δ ).

(2.4)

Here, P r f ast F (x, δ ) is the projection of F (x, δ ) onto the tangent space T Mx of the fast manifold Mx , and Fslow (x, δ ) is the projection of F (x, δ ) onto the linear subspace T Mx of the slow motions, which is transversal to T Mx . Definition 2.6. Refer to an asymptotic SPVF F (x, δ ) as a uniformly asymptotic SPVF (or simply a uniform vector field) if





lim supx∈V P rslow F (x, δ ) = 0.

δ →0

   (x, δ ) := P r f ast F (x, δ ) is the fast part of the original vector field F (x, δ ), and Denote  := supx∈V P rslow F (x, δ ). Then, H G(x, δ ) :=

Prslow F (x,δ ) supx∈V |Prslow F (x,δ )|

is the slow part of the original vector field F (x, δ ). Hence, the vector field F (x, δ ) is formally

 (x, δ ) +  G  (x, δ ). represented as a linear combination of its fast and slow subfileds; that is, F (x, δ ) = H 2.1. Connection between SPVF method and SPS system In this subsection, we introduce the method of splitting a given system into  fast and slow subsystems. Given a vector field F (x, δ ), suppose that the fast subspace Lfast does not depend on x and dim L f ast = n f ast . Then, we can rewrite the vector field F (x, δ ) as F (x, δ ) = Fslow (x, δ ) + Ff ast (x, δ ), where Fslow (x, δ ) := F (x, δ ) − P r f ast F (x, δ ) and Ff ast (x, δ ) := P r f ast F (x, δ ). Here, P r f ast F (x, δ ) is the projection of the given vector field onto the fast subspace, and Fslow (x, δ ) is the corresponding projection of the given vector field onto the linear slow subspace. Furthermore, suppose that z1 = P r f ast x and z2 = P rslow x are the fast and slow variables representing a new coordinate system, with dimensions nfast and nslow = n − n f ast , respectively. Define  (z1 , z2 , δ ) is a representation of H  (x, δ ) := P r f ast F (x, δ ) in the new the small parameter  as  = supx∈V |P rslow F (x, δ )|. Then, H  (z1 , z2 , δ ) is a representation of G  (x, δ ) := coordinate system (z1 , z2 ), and G

(z1 , z2 ). Hence, the system (2.1) exhibits the standard SPS form:

Prslow F (x,δ ) supx∈V |Prslow F (x,δ )|

dz1  (z1 , z2 , δ ) =H dτ dz2  (z1 , z2 , δ ), = G dτ

in the new coordinate system

(2.5)

in the new coordinate system (z1 , z2 ).  (x ) +  G  (x ). For The next step is to describe how to determine the fast subspace, given a uniform SPVF; that is, F (x ) = H this purpose, let us fix an arbitrary n points {x1 , . . . , xn } ∈ V . We define the columns of the n × n matrix to be the images of these points under the vector field F ; that is, T := {F (x1 ), . . . , F (xn )} a quasi-linearization matrix of the vector field F associated with the points {x1 , . . . , xn }. When all of these points are in a general position, the matrix T exhibits the linear form T (x1 , . . . , xn , δ ) := {F (x1 , δ ), . . . , F (xn , δ )}. According to the SPVF decomposition, the matrix T can be decomposed to the sum:

T (x1 , . . . , xn , δ ) = T f ast (x1 , . . . , xn , δ ) +  Tslow (x1 , . . . , xn , δ ), where









(2.6)

 ( x1 , δ ), . . . , H  ( xn , δ ) , T f ast (x1 , . . . , xn , δ ) = H  ( x1 , δ ), . . . , G  ( xn , δ ) . Tslow (x1 , . . . , xn , δ ) = G

(2.7)

Here, Tf and Ts are known as the fast and slow matrices, respectively. In the case where  → 0, then δ → 0 and then T (x1 , . . . , xn , 0 ) = T f ast (x1 , . . . , xn , 0 ), and according to the SPVF theory, the T column space defines the fast subspace Lfast . Asymptotically, when  is very small, and for a suitable choice of the coordinate {x1 , . . . , xn }, the matrix T has two groups f ast f ast of eigenvalues: nfast large numbers {λ1 , . . . , λn f ast }, the eigenvectors of which asymptotically form the basis for the fast subspace Lfast , and nslow small eigenvalues {λslow , . . . , λslow nslow }, which asymptotically tend towards zero when  → 0. 1 Following this decomposition of the matrix T, and using the invariant subspaces of the eigenvectors corresponding to these two groups of the eigenvalues, T exhibits the form:





T = slow , f ast ·



Bslow 0

0 B f ast





· slow , f ast

−1

,

(2.8)

where Bslow , Bfast are block diagonal matrices with the corresponding eigenvalues. The slow and fast invariant eigenspaces −1 define the reduced system dimensions, because the columns and rows of the matrices slow and slow represent the basis of the right and left invariant subspaces corresponding to the nslow eigenvalues with the smallest real parts Bslow . Moreover, the matrices fast and −1 consist of vectors spanning the right and left invariant subspaces, including nfast eigenvalues f ast

Bfast with the largest negative real parts, respectively (n = n f ast + nslow ). Hence, Bfast asymptotically defines the basis of the fast subspace Lfast , while Bslow represents the slow one.

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In numerous practical engineering applications, the physical models are provided in a dimensional form (the hierarchy of the processes is hidden in most cases), and therefore cannot be used without transforming the model into a nondimensional form. The following procedure overcomes this problem. Let F (x1 , . . . , xn ) be a vector field that is singularly perturbed and where the small parameter is hidden; that is, the vector field depends on a hidden small parameter. Select n linearly independent points {x1 , . . . , xn } (that are not close to one another and far from the slow manifold, because in the slow manifold neighborhood no motion decomposition occurs), in such a manner that the vectors {F (x1 ), . . . , F (xn )} are linearly independent. Build the following column matrices T

⎡ ⎤ x1  T := F (x1 ), . . . , F (xn ) · ⎣ ... ⎦. 

(2.9)

xn From a geometrical point of view, T is the linear transformation that transforms the points {x1 , . . . , xn } into {F (x1 ), . . . , F (xn )}. In order to obtain the fast subspace Lfast , we take the matrix TT∗ , where T∗ is the conjugate matrix to T. The reason for using this matrix is that it has real eigenvalues. Then, the small parameter can be estimated as follows:



 = min

 λi (T ) , λi+1 (T )

(2.10)

where i = nslow and n f ast = n − nslow for order eigenvalues i < j⇔|λi | ≥ |λj | of T. 2.2. Algorithm for SPVF The above procedure for the SPVF method depends on the selection of the coordinate linear independent points

{x1 , . . . , xn }. The selection of these points as well as of the domain V is a crucial point of the above algorithm and should be adapted to every particular model. The following steps are implemented. 1: Select linear points = {x1 , . . . , xN }, where N > > n, uniformly distributed in the domain V, using quasi-stochastic distribution.

2: Compute the mean value of the vector field over the points from step 1: F¯ = N1 N i=1 F (xi ).   3: Define the so-called control set (separated set), as follows: xi ∈ : F (xi ) > F¯ , i = 1, . . . , k · n , where k > > n. 4: Build the approximation of Ti for i = 1, . . . , k based on the control set from step 3: x∗i = {x(i−1 )·n+1 , . . . , xi·n }.   5: Select only the reference set from step 4 that has Det (x∗i ) above the average level over all subsets:  =   1 k  x∗i ), and denoted by {xi : xik ∈ : |Det (xi )| ≥ , i = 1, . . . , k}, the control set of ordered subsets of length i=1 Det ( k n from set . 6: Compute the eigenvalues of Ti∗ ; that is, λ j (Ti∗ ), j = 1, . . . , n. 7: The final reference sequence xi∗ = {x(i∗ −1 )·n+1 , . . . , xi·n } and approximation of T = Ti∗ are determined simultaneously:   T = Ti∗ = F (x(i∗ −1 )·n+1 ), . . . , F (xi∗ ·n ) (xi∗ )−1 by the maximum gap for the given dimension of the reduced model ns ,





as: i∗ :  = mini λns +1 (Ti∗ )/|λns (Ti∗ | )

−1

.

Comments 1: Steps 1 to 3 guarantee that the set of points γ will be far from the slow system manifold. Steps 4 to 5 guarantee that the points from set γ will not be too close to one another. Steps 6 to 7 guarantee that the values of the vector field F (xi ) should represent a different behavior for different i values. 2: The eigenvalues λj (Ti ), j = 1, . . . , n are arranged in increasing order by absolute values. 3. Physical assumptions and governing equations The physical assumptions of the model are as follows. We consider an infinite medium filled with a combustible gas mixture (oxidant and gaseous fuel) and liquid fuel droplets. The adiabatic approach is applied, owing to the extremely short ignition period (during which almost no heat transfer takes place out of the system). The pressure is negligible according to Semenov [31] and Frank-Kamenetskii [32]. The pressure change in the reaction volume is negligible in its influence on the combustion process. The heat flux from the burning gas to droplets is assumed to consist of two components: convection and radiation fluxes. The liquid phase thermal conductivity is greater than that of the gas phase. Thus, the heat transfer coefficient in the liquid gas mixture is defined by the thermal properties of the gas phase. The fuel droplets are assumed to be semitransparent, their surfaces are gray, and the radiation heat fluxes at these surfaces are described by the Stefan–Boltzmann law, and Siegel and Howell [33], with a given emissivity at the droplet surface. The quasi-steady state approximation is valid for the vaporizing droplets, according to William [34]. The droplet boundaries are assumed to be on a saturation line (i.e., the liquid temperature is constant and equal to the liquid saturation temperature). The chemistry is modeled as a one-step, highly exothermic chemical reaction. The chemical reaction order is rather general; the reaction rate contains the multiple of non-integer powers of the fuel and oxidizer concentrations in addition to the usual Arrhenius exponential term. The pre-exponential factor is a very important thermophysical parameter in the kinetic theory of gases,

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but is often assumed to be constant. It characterizes the total number of molecule collisions at the average thermal velocity, which in turn affects the reaction rates. We assume that the pre-exponential factor is a function of the gas temperature; that is, A(Tg ) = aTgm , according to Ajadi and Nave [35]. In our model, the volumetric phase content α g is constant and can be assumed as equal to 1 in our calculations. The explanation for this assumption is as follows: let Vg and VL be the volume of the gas and liquid phases, respectively; then: αg = Vg /(Vg + VL ) = 1/(1 + (VL /Vg )). Because VL < < Vg ; for example, α g is a dimensionless quantity and it does not matter what happens with it, we can assume that α g ≈ 1, as the leading order of the asymptotic expansion of the function (1 + x )−1 is 1. In order to simplify the analysis, the evaporation droplet temperature is taken to be equal to the saturation temperature Ts . This assumption is well known and valid, with a high accuracy level, as explained by Goldfarb et al. [36] and Sazhin et al. [37]. Hence, it follows that the entire heat flux delivered to the droplet surface is fully spent for evaporation; that is,

jm =

qc + qr , L

(3.1)

where L is the evaporation latent heat. The assumption that the pressure variations are negligible implies that the state equation for the gaseous medium can be simplified to the form: ρg Tg = ρg0 Tg0 = constant. According to the film model, the density of the convective heat flux to the droplet surfaces in terms of the effect of evaporation on the heat transfer coefficient is represented as [38]

qc =

c pg jm (Tg − Ts ) , −1

e(2 jm Rc pg /Nuλg )

(3.2)

where Nu is the Nusselt number, disregarding the evaporation effect, and can be calculated by the equation:

Nu = 2 + 0.6 ·

√ √ 3 Re · P r,

(3.3)

where Re, Pr are the Reynolds and Prandtl numbers, respectively. In order to simplify the calculations, we use the assumption that the effect of the transverse flow of matter owing to the evaporation process is negligible; that is, 2jm Rcpg /Nuλg < < 1. Moreover, we assume that the contribution of the convection to the heat transfer, compared to the thermal conductivity, may be ignored for Re < < 1; hence, Eq. (3.2) takes the form of

qc =

λg (Tg − Ts ) R

.

(3.4)

The temperature dependence of the thermal conductivity coefficient is taken to be

 1 λg = λg0 Tg /Tg0 2 .

(3.5)

In general, the flux density of the thermal radiation to the droplet surfaces is a function of temperature, droplet radii, and their concentration. Moreover, it follows from the Mie theory [39] for large semitransparent droplets (R > > λ) that the absorption coefficient for diesel fuel takes the form [40,41] of Qa = 1 − e−2Ka R , where Ka is the spectral absorptivity of fuel and Ka R represents the optical particle thickness. Applying the above assumptions, we can write the following expression for the radiative heat flux qr at the surface of each droplet as:  

qr = 1 − e−2κ1 R

σ · (Tg4 − Ts4 ).

(3.6)

A formalization of the transition from the discrete to continuous model was presented in our previous work, namely Nave et al. [42]. We use an interpretation of the sum in the discrete model as a Riemann sum for the integral in the continuous model; the size distribution of droplets is taken to be continuous, and characterized by the PDF PR , which is normalized as [43]:   ∞

0

PR = nd , αl =

4π 3

∞

R3 PR dR,

(3.7)

0

where nd is the number of droplets per unit volume, R is the droplet radius, and α l is the disperse phase volume concentration. The evolution of the size distribution of droplets owing to the evaporation process is described using kinetic energy:

where

∂ PR ∂  Jm  = P , ∂t ∂ R ρl R

(3.8)

∞ jm R2 PR dR Jm = 0 ∞ 2 . 0 R PR dR

(3.9)

The integro-differential Eq. (3.8) has a similar solution that satisfies the initial distribution PR0 (R), as follows:

PR = PR0 (R + δ ), where

δ=



t 0

Jm

ρl

dt,

(3.10) (3.11)

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and δ characterizes the initial size of the droplets that evaporate by moment of time t. Under the above assumptions, the mathematical/physical model takes the form of [43]:

αg ρg c pg dδ = dt

E dTg b = aTgmC af Cox αg μ f Q f e(− BTg ) − dt

λg (Tg − Ts ) L

∞ 0 ∞ 0

RPR dR

R2 PR dR

+



∞ 0

4π R2 (qc + qr )PR dR,

  σ Tg4 − Ts4

E dC f b (− BTg ) = −aTgm ν f C af Cox e + dt

∞

1−

L 

1

αg μ f L

·

∞ 0

0

(3.12)



R2 e−2κ1 R PR dR ∞ , 2 0 R PR dR

4π R2 (qc + qr )PR dR,

E dCox b (− BTg ) = −aTgm ν0C af Cox e . dt

(3.13)

(3.14)

(3.15)

The initial conditions for the continuous model are:

t = 0 : Tg (t = 0 ) = Tg0 , C f = C f 0 , Cox = Cox0 , δ (0 ) = δ0 .

(3.16)

4. Analysis and results In this section, we apply the SPVF and MIM methods to the physical model. 4.1. MIM method The set of Eqs. (3.12)–(3.15) contains the first integral, which represents the law of energy conservation. In order to obtain the first integral, we combine Eq. (3.12) with Eq. (3.14), and obtain the following expression:

αg μ f L

  E dC f dTg b (− BTg ) + αg ρg c pg = aTgmC af Cox e · αg μ f Q f − ν f . dt dt

(4.1)

We divide Eq. (4.1) by Eq. (3.15) and obtain the following expression:

αg μ f LdC f + αg ρg c pg dTg dCox

=

αg μ f Q f − ν f . ν0

(4.2)

Integrating Eq. (4.2) with respect to the initial conditions (3.16), we obtain an explicit expression for the fuel concentration:

Cf = Cf0 +

 αg μ f Q f − ν f ρg c pg  T −T . (Cox − Cox0 ) − ν0 αg μ f L μ f L g g0

(4.3)

This procedure enables us to reduce the system from four to three differential equations. The reduced system exhibits a multi-scale form, as the temperature is the fast system variable compared to the radius and oxygen concentration. The MIM method allows us to reduce the study of the multi-scale system to an investigation into the so-called slow subsystem. According to this classic method, the fast system variable is the temperature (heat-emitting process), while the slow system variables are the radius and oxygen concentration. When applying the MIM method, we restrict ourselves to a zero-order approximation. According to the MIM method, the dynamics of the system can be extracted from the so-called slow surface in the 3D space of Tg = Tg (δ, Cox ). In Fig. 4.1, we present the level curves of this surface. This figure illustrates the effect of the distribution of droplets on the slow surface obtained by applying the MIM method. According to these results, it can be observed that, as the distribution parameter increases, the oxygen increases until the ignition of droplets, following which the oxygen decreases to zero. 4.2. Application of SPVF The data for the SPVF algorithm are as follows. The considered model is 4D; hence n = 4, and we select N = 20 0 0 0 arbitrary points as N > > n according to the algorithm. The results of the SPVF algorithm; that is, the eigenvalues and corresponding eigenvectors, are summarized below.

λ1 = 4.86098 ←→ v1 = (−0.0879, 0.0 0 05, 0.8756, 0.6534 )T λ2 = 0.003323 ←→ v2 = (0.8976, −0.9087, −0.7640, −0.7865 )T λ3 = 2.9878 ←→ v3 = (0.7649, −0.0 0 07, 0.6545, 0.8765 )T λ4 = 0.0067 ←→ v3 = (−0.6543, 0.3273, 0.0986, 0.6754 )T

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Fig. 4.1. Level curves of model slow surface obtained by applying MIM method. The slow surface is the function Tg = Tg (Cox , δ ).

According to the SPVF theory, the system hierarchy is as follows: the eigenvector v1 , corresponding to the largest eigenvalue λ1 , indicates the fast direction of the system. Then, v3 , and finally, λ2 and λ4 correspond to the slow direction of the system, as they are at approximately the same order. The next step is to rewrite the model in new coordinates, using the above eigenvectors. Therefore, let us write:

⎛ ⎞

⎛

x1 −0.0879 ⎜x2 ⎟ ⎜ 0.8976 ⎝x ⎠ = ⎝ 0.7649 3 x4 −0.6543

0.0 0 05 −0.9087 −0.0 0 07 0.3273

0.8756 −0.7640 0.6545 0.0986

⎞ ⎛

⎞

0.6534 Tg −0.7865⎟ ⎜ δ ⎟ · 0.8765 ⎠ ⎝ C f ⎠ 0.6754 COx

(4.4)

Following the above multiplication, the aim of the next step is to write the gas temperature, initial size of droplets (δ ), concentration, and oxygen as a function of the new coordinates; that is, Tg = Tg (x1 , x2 , x3 , x4 ), δ = δ (x1 , x2 , x3 , x4 ), C f = C f (x1 , x2 , x3 , x4 ), Cox = Cox (x1 , x2 , x3 , x4 ). From Eq. (4.4), we obtain the following system:

dx1 dt dx2 dt dx3 dt dx4 dt

dC f dTg dδ dCox + 0.0 0 05 · + 0.8756 · + 0.6534 · , dt dt dt dt dC f dTg dδ dCox = 0.8976 · − 0.9087 · − 0.7640 · − 0.7865 · , dt dt dt dt dC f dTg dδ dCox = 0.7649 · − 0.0 0 07 · + 0.6545 · + 0.8765 · , dt dt dt dt dC f dTg dδ dCox = −0.6543 · + 0.3273 · + 0.0986 · + 0.6754 · , dt dt dt dt = −0.0879 ·

and substitute the expressions of

dTg dδ dC f , , dt dt dt

, and

dCox dt

(4.5)

from the model in (3.12)–(3.15) into Eq. (4.5); that is, substitute F1 (Tg ,

δ , Cf , Cox ), F2 (Tg , δ , Cf , Cox ), F3 (Tg , δ , Cf , Cox ), and F4 (Tg , δ , Cf , Cox ) from Eqs. (3.12)–(3.15) instead of the expressions

dTg dδ dC f , , dt dt dt

,

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Fig. 4.2. Solution profiles of gas temperature of original model in (3.12)–(3.15) for different PDF functions: 1: Nukiyama–Tanasawa distribution, 2: Rosin– Rammler distribution, and 3: log-normal distribution, with σ = 1, μ = 0.

and dCdtox , correspondingly, into Eq (4.5). We express Tg , δ , Cf , and Cox as a function of (x1 , x2 , x3 , x4 ) from Eq. (4.4), and then substitute into F1 , F2 , F3 , and F4 above, and obtain the spray model in the new coordinates (x1 , x2 , x3 , x4 ) in the following form (let x = (x1 , x2 , x3 , x4 ),  = (Tg , δ, C f , Cox )):

dx1 dt dx2 dt dx3 dt dx4 dt

= −0.0879 · F1 ((x ) ) + 0.0 0 05 · F2 ((x ) ) + 0.8756 · F3 ((x ) ) + 0.6534 · F4 ((x ) ), = 0.8976 · F1 ((x ) ) − 0.9087 · F2 ((x ) ) − 0.7640 · F3 ((x ) ) − 0.7865 · F4 ((x ) ), = 0.7649 · F1 ((x ) ) − 0.0 0 07 · F2 ((x ) ) + 0.6545 · F3 ((x ) ) + 0.8765 · F4 ((x ) ), = −0.6543 · F1 ((x ) ) + 0.3273 · F2 ((x ) ) + 0.0986 · F3 ((x ) ) + 0.6754 · F4 ((x ) ).

(4.6)

Data for the diesel engine:

n = decane

ρL = 730(kgm−3 ); ρg = 0.712(kgm ); μ f = 142(kgkmol −1 ); Q f = 4.42 · 107 (Jkg−1 ); E = 1.26 · 108 (Jkg−1 ); λg = 0.0193(W m−1 K −1 ); αg = 1(dimensionless ); δ0 = 0(m ); A = 1.91 · 107 (s−1 ); L = 3.21 · 105 (Jkg−1 ); Tg0 = 450(K ); ν f = 0.127; ν0 = 0.329; C f 0 = 5 · 10−4 (kmolm−3 ); a = 0.25; b = 1.5; σ = 5.67 · 10−8 (W m−2 K −4 ).

c pg = 1050(Jkg−1 K −1 ); −3

The initial conditions are obtained from Eq. (4.4) when substituting t = 0. The above model, (4.6), is a system of ordinary linear differential equations that can be solved using simple code in Matlab. The results of the original model in (3.12)–(3.15)

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Fig. 4.3. Solution profiles of δ of original model in (3.12)–(3.15) for different PDF functions: 1: Nukiyama–Tanasawa distribution, 2: Rosin–Rammler distribution, and 3: log-normal distribution, with σ = 1, μ = 0.

Fig. 4.4. Solution profiles of oxygen concentration of original model in (3.12)–(3.15) for different PDF functions: 1: Nukiyama–Tanasawa distribution, 2: Rosin–Rammler distribution, and 3: log-normal distribution, with σ = 1, μ = 0.

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Fig. 4.5. Solution profiles of fuel concentration of original model in (3.12)–(3.15) for different PDF functions: 1: Nukiyama–Tanasawa distribution, 2: Rosin– Rammler distribution, and 3: log-normal distribution, with σ = 1, μ = 0.

are presented in Figs. 4.2–4.5 for different PDF functions: Nukiyama–Tanasawa distribution, Rosin–Rammler distribution, and log-normal distribution, with σ = 1, μ = 0. Fig. 4.2 illustrates the solution profile of the gas temperature Tg . According to this figure, the gas temperature begins at the initial condition, increased for ≈ 3.04 t, following the thermal runaway with radiation occurs for the Nukiyama–Tanasawa distribution. At ≈ 3.085 t, thermal runaway with radiation occurs for the Rosin–Rammler distribution, and at ≈ 3.92 t, thermal runaway with radiation occurs for the log-normal distribution. Following these times, the temperature increases to a large number. When comparing these physical dynamic variables to other model variables δ , Cf , and Cox we can assume that this is the fast system variable. Fig. 4.3 illustrates the solution profile of the mass of the droplets δ for different PDF functions. It can be observed that the mass of the droplet spray decreases at the same time when thermal runaway with radiation occurs with the gas temperature for the different PDF functions. The same analysis can be conducted for the physical dynamic variables Cox and Cf of the original model; that is, a correlation exists between the solution profiles of the oxygen and gas temperature for the times of the thermal runaway with radiation. Figs. 4.6–4.9 present the solution profiles of the model in the new coordinates x1 , . . . , x4 for different PDF functions. Every variable is a combination of old variables in the original model, according to the equations in (4.4). According to the SPVF theory, the fast physical dynamic variable belongs to the largest eigenvalues. In our case, these are λ1 and λ3 , and the corresponding variables of the new model are x1 and x3 . These variables are a combination of the gas temperature, mass, concentration, and oxygen, in the form of: x1 = −0.0879 · Tg + 0.0 0 05 · δ + 0.8756 · C f + 0.6534 · Cox and x3 = 0.7649 · Tg − 0.0 0 07 · δ + 0.6545 · C f + 0.8765 · Cox . One can observe that the fast variables of x1 and x3 are of the same approximate magnitude, whereas in x3 , the first coordinate corresponding to the temperature is of a dominant nature (a dominant variable refers to a variable that affects the fastest trajectory. While the temperature is classically the fastest variable, it also appears in the combination of variables along the fast trajectory, and exerts a significant influence on the trajectory. Hence, the classic point of view does not reveal the true nature of the balance between the variables on the fast trajectory, yet it detects one of the governing variables affecting the fast trajectory. The dominance of temperature corresponds to the classic analysis of the physical variables. The new approach reveals a novel combination that indicates the faster combination that cannot be determined in the classic point of view. This means that, if we sum the solution profiles of the gas temperature, mass, concentration, and oxygen with the relevant coefficients point by point, we obtain the graph of x1 , and Fig. 4.6 is indeed coherent with this analysis. The same analysis can be conducted for x2 , x3 , x4 for the relevant combination and different PDF functions. As we discovered, the system has a fast trajectory when the variables exist in an equilibrium that allows for this trajectory, whereas another equilibrium of the variables yields a slow trajectory. The equilibrium is represented mathematically

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Fig. 4.6. Solution profiles of x1 in transformed model after changing coordinates.

Fig. 4.7. Solution profiles of x2 in transformed model after changing coordinates.

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Fig. 4.8. Solution profiles of x3 in transformed model after changing coordinates.

Fig. 4.9. Solution profiles of x4 in transformed model after changing coordinates.

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by a linear combination of the variables with appropriated coefficients (weights). There is no classic meaning of adding different variable types, yet the trajectory is dictated by the ratio of the different physical variables. It should be noted that we avoid using non-dimensional variables that allow for addition, as we wish to emphasize the new concept of the equilibrium (linear combination) of variables comprising the fast trajectory. The new fast variables describe the fast linear manifold. Slow variables are constants in the fast manifolds (during fast motions). The physical meaning can be understood as energy integrals on the fast parts of the original system trajectories. On the slow parts of the trajectories, the slow variables gradually change. 5. Conclusions In our research, we have presented a theoretical framework and algorithm for the SPVF method and its application to the problem of the thermal runaway of the polydisperse spray model with convection and radiation fluxes. In general, in physical problems and particularly combustion processes, the hierarchy is hidden. By applying the SPVF method, we have rewritten the spray model in new coordinates, which reveals the system hierarchy. This enables us to decompose the system dynamics of the spray model into so-called slow and fast motions. Our analysis included a comparison between the MIM method, which is asymptotic, and the SPVF method. The energy integral procedure enabled us to reduce the system from four nonlinear ordinary differential equations to only three differential equations. Thereafter, we applied the MIM method using the assumption that the temperature is the fast variable of the physical model, while the distribution parameter and oxygen are the slow system variables; that is, the slow subsystem of the physical model. This assumption enabled us to investigate the model as a multi-scale system and hence to apply the MIM method. According to this method, the system in this model has a slow surface, from which all the analyses and physical insights can be extracted. In applying the SPVF method, we presented the original model using the eigenvectors in new coordinates, which are a combination of the original system variables. This may be useful for engineering applications. For example, we found that the x1 and x3 -coordinates in the asymptotic case are a combination of the gas temperature, concentration, and oxygen. This combination can be used as an optimal combination for optimal combustion of the spray droplets. 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