1

On the existence of complex temperature induced by

nonequilibrium phase transitions in a fire dynamics model

A. M. Abourabia a,*

, A. M. Morad a

, W. S. Amer a, b

,

a Department of Mathematics and Computer Science, Faculty of Science, Me noufiya University,

32511, Egypt

b Department of Mathematics, Faculty of Science and Arts, Taibah Unive rsity, 41411Al-Ula,

Saudi Arabia

E-mail addresses:

[email protected]

[email protected], [email protected]

* Corresponding author: (+2 010) 01382826 (Cell) | (+2 048) 2235689 (Fax)

ABSTRACT

This paper focuses on the occurrence of complex temperature caused by the

nonequilibrium phase transitions in a fire system. The nonlinear compartment fire

model for the flow of a perfect gas is investigated by incl uding the adiabatic and

buoyancy effects. The tackled system of equations is solved by the reductive

perturbation method considered as one of the reliable technique s valid over a large

temperature scale. These governing equations are convert ed into a nonlinear

ordinary differential, which permits to obtain the stochast ic complex Ginzburg-

Landau equation (SCGL), in which the temperature gradien t term is considered as

a white noise correlator.

PACS classification codes : 02.30.Jr; 05.20.Jj; 05.70.Fh; 47.10.ad; 51.10.+y ;

51.30.+i

Keywords : Fire model equations; Reductive perturbation technique; St ochastic

complex Ginzburg-Landau equation; phase transitions.

1-Introduction

The study of compartment fires is an important area of fir e safety engineering. It is

known that the combustion processes had shed the interest of many outstanding

researchers throughout history. In spite of this interes t, it is still not possible to

foretell quantitatively yet, how a given condensed phase f uel will burn as a

function of the geometric and physical parameters needed to assign a particular

procedure. The hardness accompanied with analyses of combust ion phenomena go

back to the fact that the active combustion zone of a fire has two significant roles,

which comprise different length and time scales. The com bustion area of a fire is

defined as a region in which the local mixing of gasifie d fuel and air produces the

chemical energy release and radiant energy emission th at retains the fire. These

processes take place on comparatively small length scales 1-3.

2

Indoor fires, including buildings fires captured the attent ion of many scientists

and they do their efforts to control the fire flow. As part of these efforts, the

Building and Fire Research Laboratory (BFRL) of the Nation al institute of

Standards and Technology (NIST) developed computational model for simulating

fire phenomena 2,6.

A newer insight into the dynamics of indoor fire motion could be adopted; it is

attributed to the seminal work of Wei, Chen, Po and Liu 7 in 2014, who

considered that: at low temperature, a thermodynamic syste m undergoes a phase

transition when a physical parameter passes through a si ngularity point of the free

energy. This corresponds to the formation of a new order. At high temperature,

thermal fluctuations destroy the order. Correspondingly, the free energy is a

smooth function of the physical parameter and singularitie s only occur at complex

values of the parameter. Since a complex valued parameter is unphysical, no phase

transitions are expected when the physical parameter is varied. They showed that

the quantum evolution of a system, initially in thermal equilibrium and driven by a

designed interaction, is equivalent to the partition funct ion of a complex parameter.

Therefore, the complex singularity points of thermodynamic functions could be

accessed and phase transitions even at high temperature are observed. Furthermore,

such phase transitions in the complex plane are related t o the topological properties

of the renormalization group flows of the complex parameters. This result makes it

possible to study thermodynamics in the complex plane of physic al parameters.

Not much earlier in 2010, Bertola and Cafaro 8 stated that the stochastic

approaches view the fire growth as a percolation process, wh ere the transition from

non-propagating to propagating fire is described as a phase cha nge phenomenon.

The reductive perturbation method, or multiscale analysis , allows one to replace

the initial mathematical problem, nonlinear and not solvab le, by another one, linear

and solvable. For some applications, an intrinsically nonlinear physical system is

put in a special situation, for which a linear approximati on can be used, e.g. for

transistor in analog electronics. On the other hand the r eductive perturbation

technique is applicable as a method of solution no matter how large are the

magnitudes of the existing fields of different natures in the physical systems under

investigation. It is used in the study of electromagnetic waves in ferromagnetic

media, where the sample is immersed in a strong enough external magnetic fields,

and successfully introduced in plasma researches of ion a coustic waves, where the

values of temperature ranges between 10 eV to 30 eV, see 9 -17.

The purpose of this paper is to obtain formally a nonlinear par tial differential

equation from the governing equations, which describe the m otion of hot gases in

the presence of an energy source, by using a perturbation method.

The derivation in this study is different from that used by the authors of 1-6,

which followed the behavior of fires by numerical simulati on methods of solution

for the model equations under some assumptions, while here u se is made by an

3

analytical model to describe the fluid flow and construct a partial differential

equation (PDE) from the equations of motion. This paper is organized as follows. In section 2, the problem is described

macroscopically, and the governing equations of the compartme nt fire flow model

are introduced. In section 3, we first derive the model equa tion by using the

reductive perturbation technique, and then use is made by s uitable transformations

to get its solutions leading a single nonlinear ordinary di fferential equation (OPD) .

In section 4, the necessary conclusion is introduced.

2-The physical problem and its mathematical model

The fluid is considered as a non-heat conducting gas, at an enough large Reynolds

number ( Re ~ 4

10 ). The motion considered is solely due to localized addition of

heat to an otherwise quiescent fluid, as a percolation proce ss, in the presence of

gravity. The magnitude, temporal and spatial variations of the heat source are taken as

known. The investigation of the flame structure of real fires is considered as a

complicated model, which can be avoided through specifying the source of heat. In

addition, the inertia and buoyancy forces, as the two fact ors that dominate the

buoyant flow stratification, are correlated through the Ri chardson and the Froude

numbers (see for example, Ref. 14).

Our approach is based upon two main facts, the first is tha t the temperature of both

the working substances (in gas phase) and fire environmen t is sufficiently high, the

second is that the particles occupation numbers (density) a re sufficiently small,

which permits the transformation of the quantum statisti cal distributions to the

classical microcanonical, canonical or generally the gr and canonical classical

distributions (Boltzmann distribution) , this tackling is i nspired by the works

18,19.

The equations of motion for a perfect gas in the presence of an energy source of

strength Q have the form 1-3

0

)

( =

+

u

x t

,

(1)

0

=

+

? ?

?

??

?

+

gn

x

P

x

u

u

t

u

, (2)

Q

x

P

u

t

P

x

T

u

t

T

C p =

? ?

?

??

?

+

??

?

??

?

+

,

(3)

T

R

P = , (4)

in which, the introduced symbols have the usual meaning of the fluid dynamical

and physical meanings:

refers to density, u represents the velocity at any

position

x and time t, and P denotes the pressure. Also gn is the acceleration due

4

to gravity, pC indicates the constant-pressure specific heat, T expresses the

temperature, and

R defines the gas constant.

The strength of the energy source takes the form:

)

,

(

~

0

0 0t

t

L x

Q

t

E

Q =

. (5)

Thus 0

E

is a dimensional constant, characterizes the strength of the source. The

quantity Q

~ is a dimensionless function, which varies smoothly with respect to its

arguments. The independent variables

x and t are now made dimensionless with

respect to

L and 0

t

, respectively, where L and

0

t

are the length and time scales

describing the spatial extent and the temporal variation of the heat source. The

dependent variables are scaled as follow 1:

P

E

t

x

P

u

E

t

x

u

T

E C

t

x

T

t

x p

0

2

1

0

0

0

0

0 1

1

)

,

(

,

1

)

,

(

,

)

,

(

,

)

,

(

=

?

? ?

?

?

??

?

=

=

=

, (6)

,

,

1

00

2

Lx

x

E

c s =

=

and

0

t t

t =

.

The density is normalized with respect to an ambient leve l

0

, which occurs in

the absence of any heat addition. The temperature scaling used in Eq. (6) is chosen

so that the time derivatives in the energy equation alw ays are of the same

amplitude as the source term. The velocity is non-dimensi onalised with respect to a

thermal (sound) speed s

c

based on the temperature scale. The pressure scale then

follows from the equation of state. The specific heat rati o is denoted by

vpC

C

= , the

gas constant is

v

pC

C

R = .

Substituting Eq. (5) and (6) into Eqs. (1)-(4), the governing equations can be

rewritten, after dropping the bars, in the following dimens ionless form:

0

)

( =

+

u

x t

, (7)

1 2

0

L t

gn

x

P

x

u

u

t

u

=

+

?

? ?

?

?

??

?

? ?

?

??

?

+

, (8)

,

Q

x

P

u

t

P

x

T

u

t

T

~

1

=

? ?

?

??

?

+

??

?

??

?

+

(9)

T

P

= . (10)

The relevant nondimensional parameter

(of the same order as the Froude

number,

~

Fr ), which represents the distance based on the characteri

stic velocity

,

1

2

1

0 0

?

? ?

?

?

??

?

E

is defined by

12 0 0

0

1 . t E

L

? ? ? ?

? ? ? ? ?

= ? ? ? ?? ?

The definition of the Froude number implies the Boussine sq assumption for the

density and it can be applied in the density range in whic h that assumption is valid.

In addition, the thermal stratification of a fire depends u pon the effects of

5

buoyancy and inertia force. The Froude number (of

)1(

O in our study) and

Richardson number (

2 1 Fr

Ri =

) can be used to characterize the stratified flows

14.

Also, we can define the quantity

L

tg 2

0 as the Knudsen number

L Kn = , such that if

1