On numerical simulation of electromagnetic field effects in the combustion process

Abstract

This paper deals with a simplified model taking into account the interplay of compressible, laminar, axisymmetric flow and the electrodynamical effects due to Lorentz force’s action on the combustion process in a cylindrical pipe. The combustion process with Arrhenius kinetics is modelled by a single step exothermic chemical reaction of fuel and oxidant. We analyze non-stationary PDEs with 6 unknown functions: the 3 components of velocity, density, concentration of fuel and temperature. For pressure the ideal gas law is used. For the inviscid flow approximation ADI method is used. Some numerical results are presented.

Full text

Mathematical Modelling and Analysis http://mma.vgtu.lt
Volume 23, Issue 2, 327–343, 2018 ISSN: 1392-6292
https://doi.org/10.3846/mma.2018.020 eISSN: 1648-3510
On Numerical Simulation of Electromagnetic
Field Effects in the Combustion Process
Harijs Kalisa, Maksims Marinakia, Uldis Strautinsaand
Maija Zakeb
aInstitute of Mathematics and Computer Science of University of Latvia
Raina bulv¯aris 29, LV-1459 R¯ıga, Latvija
bInstitute of Physics, University of Latvia
32 Miera Street, LV-2169 Salaspils-1, Latvia
E-mail(corresp.): kalis@lanet.lv
E-mail: maksims.marinaki@lu.lv
E-mail: uldis.strautins @lu.lv
E-mail: mzfi@sal.lv
Received July 22, 2017; revised March 23, 2018; accepted March 25, 2018
Abstract. This paper deals with a simplified model taking into account the inter-
play of compressible, laminar, axisymmetric flow and the electrodynamical effects
due to Lorentz force’s action on the combustion process in a cylindrical pipe. The
combustion process with Arrhenius kinetics is modelled by a single step exothermic
chemical reaction of fuel and oxidant. We analyze non-stationary PDEs with 6 un-
known functions: the 3 components of velocity, density, concentration of fuel and
temperature. For pressure the ideal gas law is used. For the inviscid flow approxima-
tion ADI method is used. Some numerical results are presented.
Keywords: compressible, laminar, axisymmetric flow, Lorentz force, Arrhenius kinetics.
AMS Subject Classification: 65N06; 65N20; 65N40; 65N25; 34C55; 65N35.
1 Introduction
The use of swirling flows to control the combustion processes is actual and
essential, because of the hypothesis suggesting that the swirl flows allow the
enhanced mixing of the reactants in the flame reaction zone and stabilization of
the processes of fuel combustion and heat energy production. Within the last
years, biomass combustion for energy purposes has gained rising popularity.
The studies of Syred, Gupta and Lilley [14], [23], [20] provide analysis of the

Copyright c
2018 The Author(s). Published by VGTU Press
This is an Open Access article distributed under the terms of the Creative Commons Attribution
License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribu-
tion, and reproduction in any medium, provided the original author and source are credited.

328 H. Kalis, M. Marinaki, U. Strautins and M. Zake
effect of the swirl level (S) on the swirl flow dynamics and structure formation
of the flame reaction zone. Barmina et al. [5] the experimental study of the
formation of the flame composition and temperature profiles downstream the
swirling flame flows are carried out. The complex research of the swirl flow
formation has shown that the formation of the compact recirculation zone is
observed at S > 0.6.
Ahmed and Das [3] have obtained that the chemical reaction and the heat
sink have significant effects on the flow and on the heat and mass transfer
characteristics. Battaglija et al. [9] has described the mathematical models of
the hydrodynamics and the combustion; with the large-eddy simulation (LES)
he has presented the temperature fields, swirl velocity characteristics and the
heat release rate. Bayona and Kindelan [10] have simulated the premixed
laminar flames with different values of Lewis number in the open ducts with
the spectral finite difference metd. Mittal et al. [22] has used the laminar
burning velocity has used for modelling the premixed combustion process. The
quasi -gasdynamic system of equations with the mass force and heat source for
the perfect polytropic gas has been studied in [26]. In [25] Zlotnik and ˇ
Ciegis
have investigated the stability for the high order finite difference scheme.
In the paper of Choi, Rusak et al. [12] a numerical investigation of the
inviscid, axisymmetric, steady swirling flow in cylindrical pipe for low Mach
number approximation is developed (ρ=1
T,where ρ, T are the dimensionless
density and the temperature in compressible flow). For numerical simulation
the pressure-correction method is used.
The investigation continues the study of Kalis et al. [4], [7], [17], [18], where
the pressure is eliminate from the Navier-Stokes equations for viscous, incom-
pressible and ideal, compressible flow by introducing the stream function and
vorticity for the combustion process with simple exothermic chemical reaction.
The swirling flow with axial and azimuthal components of velocity is developed
in the pipe. The axial velocity from uniform stream in the central part of the
cylindrical pipe-inlet is formed. The azimuthal velocity with rotation of the
part of tube inlet is obtained.
In [18] the results of numerical study of viscous, incompressible, laminar,
axisymmetric swirling flow with axial uniform magnetic field are presented. The
interplay of two strategies - inducing swirl in the flow combined the application
of an external magnetic field is considered. The swirl number is introduced by
controlling the axial and azimuthal velocity components at the inlet. A similar
experiment is reported in [4]. The external axial magnetic field is modeled for
the Lorentz force. The flow field is influenced by gravitation, magnetic field,
heat of reaction and swirl number. Since the conductivity of hot gases such as
air, oxygen, hydrogen is very low, the MHD mechanisms are not very effective
in practical applications of gas combustion. This is in contract with conductive
liquid, for which external magnetic field can be effectively used to control the
vorticity distribution in the flow [16].
In [17] inviscid, compressible, laminar, axisymmetric steady swirling flow is
numerically investigated. From the results one can conclude that the increase
in value of axial velocity in terms of maximal temperature of reaction Tmax
leads to increase for Le > 1 and decrease for Le < 1, where Le =λ
cpDis Lewis

Electromagnetic Field Effects in the Combustion Process 329
number (λis the thermal conductivity, cp-the specific heat, D-the molecular
diffusivity). Similar results are obtained in [2] for second-order reaction kinetics
for fast and slow exothermic reactions in the combustion of CO and H2.
In [4] the electromagnetic force is generated by an electric current in a
coil electrode surrounding the flame at the inlet of the cylindrical pipe and
the azimuthal component of vector potential and Biot-Savart law were used
to investigate and calculate the distribution of the electromagnetic field axial
and radial components. It has been proved that the increase of electrodynamic
force increases the maximal velocity in the gas flow.
In [7] the magnetic field is induced by direct electric current in a coil, which
is placed at inlet of the combustor, close to the inner surface of the combus-
tor. The distribution of stream function, azimuthal component of velocity,
vorticity and formation of temperature profiles are calculated by varying the
electrodynamic force and swirl number.
In [6, 8, 15] a 2D axially-symmetric ideal, compressible swirling flow with
simple chemical raction is descibed by four Euler and two reaction-diffusion
equations. The perfect gas model p=ρT has been used (pis the dimension-
less pressure). The approximation is based on implicit finite-difference and
alternating direction(ADI) method of Douglas and Rachford [13]. The applied
electric field induces an electrical current between the positively based walls of
the combustor and the negative biased axially inserted electrode with different
length. The field influence on the flame length depends on the length of the
axially inserted electrode. The vortex breakdown at swirl number S > 0.9 in
the recirculation zone is obtained.
In [1] a nonlinear thermal conductivity model in two gypsum product layers
with different density and high temperature (T > 500◦C) is proposed. The
conservative averaging method (CAM) allows reducing the nonlinear 2-D heat
transfer initial-boundary problem to the initial value problem for system of
ODEs of first order.
In this paper we focus on a configuration in which a steady, low-speed
(0.1m
s),laminar flame exists in a straight pipe in the base state (in previ-
ous investigation the axial velocity has a uniform stream U0= 0.01[ m
s]). We
consider a simplified model taking into account the interplay of compressible,
laminar, axisymmetric flow and the electrodynamical effects due to Lorentz
force’s action on the combustion process in a cylindrical pipe.
Similarly to [15] the axial velocity from uniform stream in the central part
of the cylindrical pipe-inlet is formed, the azymuthal velocity with rotation of
the part from tube inlet is obtained. The fuel (propane) is injected axially into
the sectioned water-cooled channel, air swirl motion is generated through a
tangential air inlet. Therefore, the air swirl flow combines axial and azimuthal
motions. The exothermic chemical reaction is modelled by a single step of
fuel and oxidant. The rate of the reaction is given by the one-step first-order
Arrhenius kinetics. Our purpose is to understand how the stationary flame is
affected by the introduction of swirl and direct electric current, whish is fed to
discrete circular conductor-electrode or by an external magnetic field which is
induced by a permanent magnetic material wrapped around the cylinder.
The distribution of axial, radial and azimythal components of velocity, vor-
Math. Model. Anal., 23(2):327–343, 2018.

330 H. Kalis, M. Marinaki, U. Strautins and M. Zake
ticity, density, and temperature has been calculated using the implicit finite
difference method and ADI method. For 1D reaction-diffusion problem some
results are obtained with Matlab solvers ”pdepe” and ” bvp4c”.
2 The mathematical model
The combustion process with temperature T[K] and simple exothermic chem-
ical reaction is modelled by the first-order Arrhenius kinetics with mass frac-
tion Cof the reactant in time t, in the coaxial cylindrical pipe with radius
r0= 0.05[m] and length z0= 0.1[m] . The axial, radial and azimuthal compo-
nents of the velocity uz, ur, uθin the coaxial cylindrical pipe are formed.
Let T0= 300[K], ρ0= 1[ kg
m3], C0= 1 be the initial temperature, nominal
density, mass fraction of concentration of fuel and axial velocity with uniform
stream U0= 0.1[m
s] in the central part of the cylindrical pipe at inlet z= 0.
The boundary of the pipe (r=r0) is subject to a heat loss modelled by the
Newtonian cooling to the ambient surroundings at temperature T0and with
heat transfer coefficient h= 0.1[ J
s m2K].
Direct electric current with the meridian components of the vector den-
sity j= (jz, jr,0)[ A
m2] and current I[A] is fed to axially-symmetric conductor-
electrodes: L1={(z, r0, φ),0≤z≤z1< z0,0≤φ≤2π}(the part of walls of
the pipe) and L2={(z, r∗, φ),0≤z≤z2< z0,0≤φ≤2π}(the central part
of the bottom of the pipe r∗=r0/10).
From the Maxwell’s equations and Ohm’s law it follows [11], that the merid-
ian components of the vector density jz, jrcreate the azimuthal component Bφ
of the induced magnetic field (jz=1
µr
∂(rBφ)
∂r , jr=−∂ Bφ
µ∂z ) in the ionized gas,
which creates the axial Fz=Bφjrand radial Fr=−Bφjzcomponents of the
electromagnetic force, where µis the magnetic permeability in medium.
We analyze the 2D axially-symmetric nonstationary physical model for the
inviscid, compressible, swirling flow with the meridian components of the vector
velocity ur, uz, circulation v=ruφ, simple chemical reaction and electromag-
netic force in cylindrical pipe, which can be described with 4 Euler’s and 2
reaction-diffusion equations:
∂ρ
∂t +ur
∂ρ
∂r +uz
∂ρ
∂z +ρ
r
∂(rur)
∂r +ρ∂uz
∂z = 0,
∂ur
∂t +ur
∂ur
∂r +uz
∂ur
∂z −v2
r3=−∂p
ρ∂r +Fr/ρ,
∂uz
∂t +ur
∂uz
∂r +uz
∂uz
∂z =−∂p
ρ∂z +Fz/ρ,
∂v
∂t +ur
∂v
∂r +uz
∂v
∂z = 0,
∂T
∂t +ur
∂T
∂r +uz
∂T
∂z =λ
ρcp∇2T+˜
B
cp
A C exp(−E
RT ),
∂C
∂t +ur
∂C
∂r +uz
∂C
∂z =D∇2C−A C exp(−E
RT ),

Electromagnetic Field Effects in the Combustion Process 331
where ∇2q=∂2q
∂z2+1
r
∂
∂r (r∂ q
∂r ), q=T , C,D= 5.10−5[m2
s] is the molecular
diffusivity, λ= 5.10−5[J
s,m K ] is the thermal conductivity, cp= 1000[ J
kg K ]
is the specific head at constant pressure, ˜
B= 1.5 106[J
kg ], A= 104[1
s], E=
2.5104[J
mol ] are the specific heat release, the reaction-rate pre-exponential factor
and the activation energy, Ris the universal gas constant. The azimuthal
component of the induced magnetic field can be obtained from the following
equation of conjugate Laplace operator
∂2Bφ
∂z2+r∂
∂r 1
r
∂Bφ
∂r = 0.
For the permanent magnet has a different form of the radial and axial com-
ponents of the Lorentz force. The FEMM software provides a finite element
solution of the Maxwell equations formulated for the magnetostatic case in two
spatial dimensions [21]:
∇ × H= 0,∇ · B= 0,B=µ(H+M),
where His the magnetic field intensity, Bis the magnetic flux density, M
stands for magnetization.
The software considers the vector potential formulation in the case of two
dimensions. Zero vector potential boundary condition was applied to the sym-
metry part of the boundary for the case of axial symmetry. For magnetic flux
density, two magnetization directions, perpendicular to the flow direction and
aligned with the flow direction, were considered (see Figure 1). The flux den-
sity varies between 0.001 Tesla and 0.1 Tesla. For the calculations, a the mean
value of 0.03 Tesla has been considered.
(a) the magnetization direction
perpendicular to the flow direction
(b) the magnetization direction
aligned with the flow direction
Figure 1. Magnetic flux density plot.
After the field Bwith the two components Br, Bzhas been obtained, we
use the Matlab built-in differentiation commands to calculate the Lorentz force
terms in the following form:
Fr= (∇ × B)φBz, Fz= (∇ × B)φBr,(∇ × B)φ=∂Br
∂z −∂Bz
∂r .
Math. Model. Anal., 23(2):327–343, 2018.

332 H. Kalis, M. Marinaki, U. Strautins and M. Zake
For the pressure we use the ideal gas law : p0=RT0ρ0/M0[N
m2],where M0=
0.0032[ kg
mol ] is the molar mass for O2[24].
The equations were put in the dimensionless form scaling all the lengths
to r0,the density to ρ0,the velocities ur, uzto U0,the circulation vto V0r0
(V0> U0is the tangential air component), the pressure pto p0=ρ0U2
0,the
temperature to T0,the magnetic induction Bφto B0=µI
2πr0[T esla], T esl a =
N
A.m ,the electric current densities jr, jzto j0=I
2πr2
0
[A
m2],the electromagnetic
forces Fr, Fzto F0=j0B0[N
m3], the special heat release ˜
Bto cp
T0,the reaction-
rate pre-exponential factor Ato U0
r0,the activation energy Eto R
T0.
The following parameters are used: P e =r0U0
D,Le =λ
cpDρ0are Peclet and
Lewis mumbers, P1=Le
P e , P2=1
P e ,S=V0
U0is the swirl number, Pe=F0r0
ρ0U2
0
is
the electrodynamical force parameter, β=˜
B
cpT0, δ =E
R T0are the scaled heat
release and the activation energy.
For the dimensionless parameters t,r,x=z/r0,ρ,u=ur/U0,w=uz/U0,
vwe have the following equations
∂ρ
∂t +u∂ρ
∂r +w∂ρ
∂x +ρ
r
∂(ru)
∂r +ρ∂w
∂x = 0,
∂u
∂t +u∂u
∂r +w∂u
∂x =S2v2
r3−∂p
ρ∂r +PeF0
r/ρ,
∂w
∂t +u∂w
∂r +w∂w
∂x =−∂p
ρ∂x +PeF0
x/ρ,
∂v
∂t +u∂v
∂r +w∂v
∂x = 0,
∂T
∂t +u∂T
∂r +w∂T
∂x =P1
ρ∇2T+βA C exp(−δ
T),
∂C
∂t +u∂C
∂r +w∂C
∂x =P2∇2C−A C exp(−δ
T),(2.1)
where p=ρT , F 0
r, F 0
xare the dimensionless forces. We introduce the stream
function Ψand the vorticity ζwith the following expressions:
rρw =∂Ψ
∂r , rρu =−∂Ψ
∂x , ζ =∂u
∂x −∂w
∂r .
Then we have the following equation for the stream function Ψ:
∂Ψ
∂t =∂
∂x (ρ−1∂Ψ
∂x ) + r∂
∂r (1
ρr
∂Ψ
∂r ) + rζ, (2.2)
where the equation for the numerical simulation is transformed to non-steady.
The approach seeks the steady solution as the limit of solutions of the unsteady
equations.
The boundary conditions for Bφon the electrode are the following:
1) On L1from total current condition
I=Z2π
0
r0Zz1
0
jr(z, r0)dzdφ =−2πr0
µ(Bφ(r0, z1)−Bφ(r0,0))

Electromagnetic Field Effects in the Combustion Process 333
and from Bφ(r0, z) = 0, z ∈[z1, z0] it follows that Bφ(r0,0) = B0. From the
uniform distribution of current density jr=const on the electrode we get
that Bφ(r0, z) = −µjrz+Bφ(r0,0) and Bφ(r0, z) = B0(1 −z
z1), z ∈[0, z1].
2) Similarly on L2we get: Bφ(r∗, z) = 0, z∈[z2, z0], Bφ(r∗, z) = B0(1 −
z/z2)r0/r∗,z∈[0, z2].
On the inlet z= 0 we have jz= 0 and Bφ(r, 0) = B0r0/r. The other
dimensionless BCs are the following:
1) along the wall r=r∗=r∗
r0−u=v= 0, Bφ(r∗, x) = (1 −x
x2)1
r∗, x ∈[0, x2],
∂T
∂r =∂C
∂r =∂ρ
∂r =∂w
∂r = 0, Ψ = 0,
2) at the wall r= 1 −u=v= 0, Bφ(1, z) = B0(1 −x/x1), x∈[0, x1],
∂T
∂r +Bi(T−1) = 0,∂w
∂r =∂ρ
∂r =∂C
∂r = 0, Ψ =q,
3) at the pipe outlet x=x0=z0
r0−u=Bφ= 0,∂s
∂r = 0, s =ρ;P si;T;C;w;v,
4) at the pipe inlet x= 0 −u= 0, ρ = 1, Bφ(r, 0) = 1
rfor r∈[0,1] and
w= 1, T =C= 1, v = 0, Ψ = 0.5r2for r∈[0, r1] and w= 0, T = 1, C = 0,
Ψ=q, v = 4(r−r1)(1−r)
(1−r1)2for r∈[r1,1]; (we have the uniform jet flow for
r < r1and and the rotation for r≥r1with maximal azimuthal velocity 1
when r= (1 −r1)/2).
Here q=r2
1/2 is the dimensionless fluid volume, Bi =hr0/λ is the Biot number,
x1=z1/r0,x2=z2/r0. Figure 2 represents the sketch of the pilot device (a)
and the computational domain (b).
Figure 2. a: the pilot device for experimental studies - 1. biomass gasifier; 2.
water-cooled sections of the combustor; 3. primary axial air supply; 4. secondary swirling
air supply; 5. propane flame supply; 6., 7. orifices for diagnostic tools; 8. positively biased
electrode, b: principal computational domain.
3 The numerical approximations
We use the uniform grid in space ((M)×(N+ 1)): {(ri, xj), ri= (i−1)hr+hr,
xj= (j−1)hx}, i = 1, M , j = 1, N + 1, M hr= 1, M1hr=r1, N hx=x0.
Math. Model. Anal., 23(2):327–343, 2018.

334 H. Kalis, M. Marinaki, U. Strautins and M. Zake
For time we use the discrete time moments tn=nτ, n = 0,1, . . .. Subscripts
(i, j, n) refer to indices r, x, t with mesh spacing and for the approximation of
function u(t, r, x) we use the grid function with values un
i,j ≈u(tn, ri, xj).
For invisced flow we have PDE of hyperbolic type. In this case we use
the implicit FDS in time with the upwind differences in space. Here it is an
example for v-equation of (2.1):
(vn+1
i,j −vn
i,j )/τ + (uδrv)n+1
i,j + (wδxv)n+1
i,j = 0,
where
(uδrv)i,j =(|ui,j |+ui,j
2hr
(vi,j −vi−1,j )−(|ui,j | − ui,j
2hr
(vi+1,j −vi,j ),
(wδxv)i,j =(|wi,j |+wi,j
2hx
(vi,j −vi,j−1)−(|wi,j | − wi,j
2hx
(vi,j+1 −vi,j ).
From Taylor expression it follows, that for the upwind approximation
(uδrv) = u∂v
∂r −|u|hr
2
∂2v
∂r2+O(h2
r),(wδxv) = w∂v
∂x −|w|hx
2
∂2v
∂x2+O(h2
x)
and for this approximation we use the artificial viscosities |u|hr
2,|w|hx
2.From
maximum principle follows that this approximation is stable. The second order
derivatives are approximated with central differences. The derivatives at the
boundary with the finite differences of first order are approximated.
The reaction-diffusion equations are discretized in the following way:
Tn+1
i,j −Tn
i,j
τ+(uδrT)n+1
i,j +(wδxT)n+1
i,j =P1(ρ−1∆hT)n+1
i,j +βACexp −δ
Tn
i,j ,
Cn+1
i,j −Cn
i,j
τ+(uδrC)n+1
i,j +(wδxC)n+1
i,j =P2(∆hC)n+1
i,j −ACexp −δ
Tn+1
i,j ,
where
(∆hg)i,j =1
rih2
r
(ri+0.5(gi+1,j −gi,j )−ri−0.5(gi,j −gi−1,j ))
+1
h2
x
(gi,j+1 −2gi,j +gi,j−1), g =T, C.
For flow with small Mach number we take ρi,j = 1/T n
i,j .
4 The stability for finite difference approximation and
numerical method
For stabilization of the calculations the first equation of (2.1) is approximated
by using backward differences formula:
(ρn+1
i,j −ρn
i,j )/τ + (uδrρ)n+1
i,j + (wδxρ)n+1
i,j +ρi,j ((wi,j −wi,j−1)/hx
+ ((ru)i,j −(ru)i−1,j )/(hrri,j )) = 0,
but the pressure terms ∂ p
∂r ,∂ p
∂z , (p=ρT ) are approximated by the forward
differences: ((ρT )i+1,j −(ρT )i,j )/hr, ((ρT )i,j+1 −(ρT )i,j )/hx. In this case we
obtain a fast iteration pocess.

Electromagnetic Field Effects in the Combustion Process 335
4.1 The stability analysis for model equations
We consider the model equations with constant coefficients for the first three
PDEs of (2.1) in the following linear form:
∂ρ
∂t +u0
∂ρ
∂r +w0
∂ρ
∂x +ρ0
∂u
∂r +ρ0
∂w
∂x = 0,
∂u
∂t +u0
∂u
∂r +w0
∂u
∂x +T0
ρ0
∂ρ
∂r = 0,
∂w
∂t +u0
∂w
∂r +w0
∂w
∂x +T0
ρ0
∂ρ
∂x = 0,
where u0, w0, ρ0≥0, T0≥0 are given constants. Here the periodical boundary
conditions in space are used. Similar system of two equations is considered
in [19]. We use the following implicit finite difference scheme:
ρn+1
i,j −ρn
i,j
τ+(u0δrρ)n+1
i,j +(w0δxρ)n+1
i,j +ρ0(δru)n+1
i,j +ρ0(δxw)n+1
i,j =0,
un+1
i,j −un
i,j
τ+ (u0δru)n+1
i,j + (w0δxu)n+1
i,j +T0
ρ0
(˜
δrρ)n+1
i,j = 0,
wn+1
i,j −wn
i,j
τ+ (u0δrw)n+1
i,j + (w0δxw)n+1
i,j +T0
ρ0
(˜
δxρ)n+1
i,j = 0,(4.1)
where (δru)i,j = (ui,j −ui−1.j )/hr,(δxw)i,j = (wi,j −wi,j−1)/hxare the finite
backward differences, (˜
δrρ)i,j = (ρi+1,j −ρi,j )/hr,(˜
δxρ)i,j = (ρi,j+1 −ρi,j )/hx
are the forward differences. Using the spectral method for the stability investi-
gation of (4.1), the solution is represented as sn
i,j =Csλnexp(ı(krhr+kxhx)),
where Csare unknown constants, s= (ρ;u;w), kr, kxare the wave numbers,
ıis the imaginary unit. From (4.1) we get the characteristic equations for ob-
taining the roots of λin the form of 3rd-order matrix-determinant det A= 0,
where
A=

G(λ)ρ0λd0ρ0λd0
−T0
ρ0λd0G(λ) 0
−T0
ρ0λd00G(λ)

.
Here G(λ) = λ−1 + λd0(|u0|+|w0|), d0=τ
h(1 −cos(kh) + ısin(kh)) and
d0=τ
h(1 −cos(kh)−ısin(kh)) is the complex conjugate expression. For these
calculations we have assumed, that kr=kx=k, hx=hr=h. The eigenvalues
are: λ1= 1/(1+d0(|u0|+|w0|)) with |λ1|2= 1/(1+4 τ
hsin2(kh/2)(|u0|+|w0|)(1+
τ
h(|u0|+|w0|)) ≤1, λ2,3= 1/(1+d0(|u0|+|w0|)±ı√2T0|q|) with |λ2,3|2= 1/((1+
2τ
hsin2(kh/2)(|u0|+|w0|))2+ ( τ
hsin(kh)(|u0|+|w0|)±√2T0|q|)2)≤1, where
q= 2 τ
hsin(kh/2). Therefore the finite difference scheme (4.1) is unconditionally
stable.
If the backward differences are used for the approximation of the derivatives
in term T0
ρ0,then in the matrix Awe have −d0=d0and λ2,3= 1/(1 + d0(|u0|+
|w0|)±√2T0d0).In this case |λ2,3|2= 1/(1 + 4b0(1 + b2
0) sin2(kh/2)),where
b0=τ
h(|u0|+|w0| ± √2T0).If √2T0≤(|u0|+|w0|) the the approximation is
unconditionally stable, but for √2T0>(|u0|+|w0|) (the velocities are smaller
Math. Model. Anal., 23(2):327–343, 2018.

336 H. Kalis, M. Marinaki, U. Strautins and M. Zake
than the acustic velocity [19]) we have the conditionally stability with the
condition τ≥h/(√2T0−(|u0|+|w0|)) (a non-standard stability condition).
4.2 The alternating-direction implicit (ADI) method
For solving the discrete problem
Un+1 −Un
τ= (Λx+Λr)Un+1 +fn, n ≥0
we use the ADI method of Duglas and Rachford [13] in the form
Un+0.5−Un
τ=ΛxUn+0.5+ΛrUn+fn,
Un+1 −Un+0.5
τ=Λr(Un+1 −Un).
Here the vector Uof six components (ρ, u, w, v, T , C) and the scalar function
Ψis used for solving the discrete equations (2.1) and the equation (2.2). Λx, Λr
are corresponding differential operators, containing the first and second order
derivatives for Uwith respect to xand r,fncontains all the other functions
and derivatives in the PDEs. For solving Un+0.5and Un+1 we use the Tomas
algorithm in xand rdirections respectively. By eliminating the half time step
we obtain the previous discrete problem with an approximation error O(τ2).
5 Some numerical results
In the equations (2.1) the minimal value of the flow density, maximal val-
ues of the flow velocity components, the temperature, the reaction rate R∗=
A C exp(−δ
T), the pressure gradient and the stream function have been calcu-
lated.
5.1 Some results for the full problem
For the modelling of the full problem, we choose the following parameters:
S= 3, x0= 2, r1= 0.75, r∗=hy= 0.025, P1= 0.1; 0.01, P2= 0.1; 0.01,
β= 5, δ = 10, A = 50000, Bi = 0.1, Pe= 0; 0.05; 0.1, τ = 0.0008, I t ≤18000
(number of time steps), N= 80, M = 40.
The influence of the molecular diffusivity and thermal conductivity on the
main characteristics of the undisturbed flame flow is observed for Pe= 0 and
Pe= 0.05. These results show that when the molecular diffusivity Dis constant
a decrease in thermal conductivity λ(P1= 0.01, Le = 0.1) leads to an increase
in maximal values of the flow velocity components (w), temperature, reaction
rates, flow vortices with a decrease in flow density, but for constant thermal
conductivity λthe decrease in molecular diffusivity D(P2= 0.01, Le = 10)
results in an increase of maximum density and in a decrease of velocity, pressure
gradient, temperature, reaction rate and flow vortices.
The results of the numerical simulation show that the electric field influence
on the main flame characteristics is determined by the length of the axially

Electromagnetic Field Effects in the Combustion Process 337
inserted electrode (see Figure 3). For x1=x2= 1, P1=P2= 0.01, Pe= 0
we have obtained Figure 3 (a) and the following numerical results (It = 1723):
p∈[0.153,4.40], w=∈[0,5.797], u∈[0,2.786], ρ∈[0.0305,1], T∈[1,5.994],
R∗
max = 298.76, Ψmax = 0.2924 (a small vortex), Ta= 5.1927 (Tais the
averaged value of temperature, q= 0.2812).
22
2
44
4
55
5
5
5.4 5.4
5.4
5.4
5.4
z/r0
r/r0
0 0.5 1 1.5 2
0.2
0.4
0.6
0.8
1
(a) for Pe= 0
2
2
2
4
4
4
5
5
5
5
5
5.4
5.4
5.4
5.4
5.4
z/r0
r/r0
0 0.5 1 1.5 2
0.2
0.4
0.6
0.8
1
(b) for Pe= 0.05
Figure 3. Temperature levels.
If Pe= 0.05 (see Figure 3 (b)) then p∈[0.162,4.32], w ∈[0,11.21],
u∈[−0.25,2.718], ρ ∈[0.0321,1], T ∈[1,5.995], R∗
max = 298.92, Ψmax =
0.2924, Ta= 5.1986.
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2.4
2.4
2.4
2.4
2.4
2.4
z/r0
r/r0
0 0.5 1 1.5 2
0.2
0.4
0.6
0.8
1
(a) for Pe= 0
−0.2
0
0
0
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2.4
2.4
2.4
2.4
2.4
2.4
z/r0
r/r0
0 0.5 1 1.5 2
0.2
0.4
0.6
0.8
1
(b) for Pe= 0.05
Figure 4. Levels of radial velocity.
In Figure 4 we can see that the electric field in the flame reaction zone
(Pe= 0.05) disturbs the stream lines, initiates the formation of small vortices
and the radial velocity changes the sign.
In Figure 5, 6 we represent the levels of the electric current (x1= 0.25, x2=
2) and maximal temperature time dependence for β= 0.5, P1= 0.01, P2= 0.1;
the flame temperature (t = 0.2s) rapidly increases to its maximum value.
5.2 Some numerical experiments with the reaction-diffusion equa-
tion
For the fixed values of velocity u=v= 0, w = 1 and P1=P2= 0.1, r1= 0.5,
the heat-reaction problem is solved numerically in two ways: ρ= 1/T (small
Math. Model. Anal., 23(2):327–343, 2018.

338 H. Kalis, M. Marinaki, U. Strautins and M. Zake
0.01
0.01
0.01
0.01
0.01
0.2
0.2
0.2
0.2
0.4
0.4
0.4
0.6
0.6
0.6
0.8
0.8
0.8
z/r0
r/r0
0 0.5 1 1.5 2
0.2
0.4
0.6
0.8
1
Figure 5. Levels of electric
current for x1= 0.25, x2= 2.
0 2 4 6
1
1.5
2
2.5
t[s]/5
maxT[K]/300
Figure 6. Maximal
temperature depending on time t
by β= 0.5, P1= 0.01, P2= 0.1.
Mach numbers for compressible fluid) and ρ= 1 (incompressible flow). If ρ=
1/T , we have the maximal and averaged values of temperature: Tmax = 2.957,
Ta= 1.827 and the value of R∗
max = 252.20.For ρ= 1 we have Tmax = 5.969,
Ta= 3.319, R∗
max = 593.31.The calculation results are presented in Figure 7.
For r1= 0.75 we have the corresponding Tmax = 6.806, Ta= 5.168, R∗
max =
593.51 (ρ= 1), Tmax = 3.320, Ta= 2.890, R∗
max = 255.84 (ρ= 1/T ).
0 1 2
1
1.5
2
2.5
3
Temp.,Tmax=2.9569
x
T
r=0
r=0.1
r=0.5
r=0.75
r=1
(a) for ρ= 1/T
0 0.5 1 1.5 2
0
2
4
6
Temp.,Tmax=5.9689
x
T
r=0
r=0.1
r=0.5
r=0.75
r=1
(b) for ρ= 1
Figure 7. Profiles of temperature depending on x.
5.3 Solving 1-D heat-reaction problem with MATLAB solver ”pdepe”
Using the two simple reactions from (2.1) we obtain following 1D reaction-
diffusion problem for temperature T(x, t) and 2 reaction concentrations
C1(x, t), C2(x, t) (ρ= 1, u = 0):
∂T
∂t +w∂T
∂x =P1
∂2T
∂x2+β1A1C1exp(−δ1
T) + β2A2C2exp(−δ2
T),
∂C1
∂t +w∂C1
∂x =P2
∂2C1
∂x2−A1C1exp(−δ1
T),(5.1)
∂C2
∂t +w∂C2
∂x =P2
∂2C2
∂x2−A2C2exp(−δ2
T),
t∈(0, tf), x ∈(0, L), T (0, t) = 1, C1(0, t)=0.8, C2(0, t)=0.2,
∂(s(L, t)
∂x = 0, s =T , C1, C2, T (x, 0) = 1, C1(x, 0) = C2(x, 0) = exp(−αx),

Electromagnetic Field Effects in the Combustion Process 339
where A1=A= 5.104,A2= 5.105,β1=β= 5, β2= 1, δ1=δ= 10,
δ2= 15, tf= 1, w= 0,1,2,3,4, P1= 0.1,0.01, P2= 0.1,0.01,0.001, α =
0,1,2,3,4,5,6, L = 2,4, α ∈[0,6] is the parameter for the initial fuel amount
in the combustion.
The results of calculation using 2 reactions (L= 2, α= 6, P1=P2= 0.1
with Tmax = 5.269, w = 1 (T(2,1) = 1.248) and w= 4 (T(2,1) = 5.200)) are
represented in the Figure 8.
0 0.5 1 1.5 2
1
2
3
4
5
6
Temp. of x,Tmax=5.2690
x/x0
T/T0
t=1.001
t=0.351
t=0.026
t=0.0005
t=0.101
(a) for w= 1, α = 6, P1=P2=
0.1
0 0.5 1 1.5 2
1
2
3
4
5
6
Temp. of x,Tmax=5.2685
x/x0
T/T0
t=0.101
t=0.026
t=0.351
t=1.001
t=0.0005
(b) for w= 4, α = 6, P1=P2=
0.1
Figure 8. Profiles of temperature depending on xin fixed time t.
Using one reaction (C2(0, t)=0, C1(0, t) = 1): for w= 1, Tmax = 6.0881,
T(2,1) = 1.295 and for w= 4, Tmax = 6.114, T (2,1) = 1.248. In Figures 9,
10 (one reaction) we can see the surface in (x, t) plane at w= 4, L = 4, α =
6P1=P2= 0.1,(Tmax = 6.281, T (4,1) = 4.085) and profile of temperature for
w= 3, L = 2, P1= 0.01, P2= 0.001 (Tmax = 5.690, T (2,1) = 4.000).
0
2
4
0
0.5
1
0
5
10
x
Surface T(x,t)
t
Figure 9. Temperature
depending on (x, t) for w= 4, L =
4, α = 6, P1=P2= 0.1.
0 0.5 1 1.5 2
1
2
3
4
5
6
Temp. of x,Tmax=5.6963
x/x0
T/T0
t=0.026
t=0.101 t=0.351
t=1.001
Figure 10. Profile of
temperature depending on xin
fixed time tfor w= 3, L = 2, α =
6, P1= 0.01, P2= 0.001.
5.4 Some results obtained with MATLAB solver ”pdepe”
For P1=P2= 0.1, L = 2 we obtain the following results for the temperature
T(2, tf) in outlet (x= 2) (see Table 1):
1) for constant initial conditions for C with (α= 6, tf= 1) an increase in
velocity wleads to an increase in the value of temperature T(2,1),
Math. Model. Anal., 23(2):327–343, 2018.

340 H. Kalis, M. Marinaki, U. Strautins and M. Zake
2) for constant velocity (w= 1, tf= 1) a decrease in parameter α(the initial
fuel amount in combustion is increased) leads also to an increase in the value
of temperature T(2,1), item for constant α= 6, w = 0 an increase in the
time segment tfleads to an increase in temperature T(2, tf).
Table 1. The values T(2, tf) depends on w, α, tf
tf= 1, α = 6 w=tf= 1 w= 0, α = 6
wT(2,1) αT(2,1) tfT(2, tf)
0 1.002 6 1.295 1 1.002
1 1.295 3 1.694 10 2.630
2 4.396 1 3.204 20 4.177
3.5 6.000 0 6.000 100 6.000
The maximal temperature depends on molecular diffusivity and thermal
conductivity (P1, P2), an example for L= 2, tf= 1, α= 6, w= 1: P1=
P2= 0.1, Tmax = 6.09, T(2,1) = 1.295, P1=P2= 0.01, Tmax = 6.27,
T(2,1) = 1.000, P1= 0.1, P2= 0.01, Tmax = 4.46, T(2,1) = 1.196, P1= 0.01,
P2= 0.001, Tmax = 5.69, T(2,1) = 1.020.
5.5 Solving 1-D stationary reaction-diffusion problem with MAT-
LAB solver ”bvp4c”
For stationary reaction-diffusion equation 5.1 (A1= 5.104,A2= 5.108,δ1= 10,
δ2= 20, ρ = 1, w= 0, T=T(x), C1=C1(x), C2=C2(x), x∈[0,2]) with BCs
T(0) = 1, C1(0) = 0.8, C2(0) = 0.2, T(2)0=C0
1(2) = C0
2(2) = 0,by multiplying
both second equations by β1, β2and summing them, we get the equations
LeT 00(x) + β1C00
1(x) + β2C00
2(x)=0,
Le(T(x)−1) + β1(C1(x)−0.8) + β2(C2(x)−0.2) = 0,
LeT 0(x) = −β1C0
1(x)−β2C0
2(x).
In the limit case x→2 it follows that the maximal temperature is Tmax =
T(2) = 1 + β10.8+β20.2
Le ,Le =P1
P2. We have obtained in Matlab, that the
increase in the axial velocity wleads to an increase for Le > 1 and the desrease
for Le < 1 in Tmax. For Le = 1, Tmax does not depend on w; increase in w
leads to the descrease in thickness of the boundary layers T0(0) and C0(0) (see
Table 2 for one reaction and β= 1).
In the Figure 11 the profile of temperature and concentration for one re-
action we can see depending on xfor w= 0.5, P1= 0.005, P2= 0.01 and
β= 0.1; 0.2; 0.3; 0.4.
6 Conclusions
•The stability for discreet problem of the model equations has been inves-
tigated.

Electromagnetic Field Effects in the Combustion Process 341
Table 2. The values of Tmax ,T0(0), C0(0) depend on P1,P2,w
P1P2wTmax T’(0) C’(0)
0.01 0.01 0.0 2.00 68.6 -68.6
0.01 0.01 1.0 2.00 9.15 -9.15
0.02 0.01 0.0 1.50 17.6 -35.3
0.02 0.01 1.0 1.83 12.9 -9.02
.005 0.01 0.0 3.00 98.1 -78.3
.005 0.01 0.5 2.54 84.9 -69.5
0 1 2
0
0.5
1
1.5
P1=0.005,P2=0.01,Le=0.50
x
T,C
β=0.1β=0.2
β=0.3
β=0.4
Figure 11. Profile of temperature and concentration depending on xfor
β= 0.1; 0.2; 0.3; 0.4.
•For the diffusion-reaction problem, the influences of maximal temperature
in pipe and temperature at outlet are obtained.
•The maximal temperature depends on molecular diffusivity and thermal
conductivity.
•The maximal axial velocity depends on Lewis number value.
•Increase in electrodynamical force parameter Peleads to an increase in
maximal velocity and in temperature.
Acknowledgement
This work was partially supported by the grant 623/2014 of the Latvian Council
of Science and ERAF project Nr.1.1.1.1/16/A/004.
References
[1] A. Aboltins, H. Kalis, K. Pulkis, J. Skujans and I. Kangro. Mathematical mod-
elling of heat transfer problem for two layered gypsum board products exposed
to fire. In Latvia J. Palabinskis, Latvia University of Agriculture(Ed.), Proc. of
of the 16-th Int. Conf. on Engineering for Rural Development (ERDev17), Jel-
gava, Latvia, May. 24-26, 2017, pp. 1369–1376, Latvia University of Agriculture,
Faculty of Engineering, 2017. https://doi.org/10.22616/ERDev2017.16.N312.
Math. Model. Anal., 23(2):327–343, 2018.

342 H. Kalis, M. Marinaki, U. Strautins and M. Zake
[2] M. Abricka, I. Barmina, R. Valdmanis, M. Zake and H. Kalis. Experimental and
numerical studies on integrated gasification and combustion of biomass. Chemical
Engineering Transactions,50:127–132, 2016.
[3] N. Ahmed and K.Kr. Das. Effects of heat absorption and chemical reaction on
a three dimensional MHD convective flow past a porous plate. Int. J.of Phys.
Sciences,8(39):1907–1922, 2013.
[4] I. Barmina, A. Kolmickovs, R. Valdmanis, M. Zake and H. Kalis. Experimen-
tal and numerical studies of electric field effects on biomass thermo-chemical
conversion. Chemical Engineering Transactions,50:121–126, 2016.
[5] I. Barmina, A. Komickova, R. Valdmanis and M. Zake. Combustion dynamics of
swirling flame at thermo chemical conversion of biomass. Chemical Engineering
Transaction,43:649–654, 2015.
[6] I. Barmina, R. Valdmanis, H. Kalis and M. Marinaki. Experimental and numeri-
cal study of the development of swirling flow and flame dynamics and combustion
characteristics at biomass thermo-chemical conversion. In Latvia J. Palabin-
skis, Latvia University of Agriculture(Ed.), Proc. of of the 16-th Int. Conf. on
Engineering for Rural Development (ERDev17), Jelgava, Latvia, May. 24-26,
2017, pp. 68–74, Latvia University of Agriculture, Faculty of Engineering, 2017.
https://doi.org/10.22616/ERDev2017.16.N013.
[7] I. Barmina, R. Valdmanis, M. Zake, H. Kalis, M. Marinaki and U. Strautins.
Magnetic field control of the combustion dynamics. Latvian Jour. of Physics
and Technical Sciences,53(4):36–47, 2016. https://doi.org/10.1515/lpts-2016-
0027.
[8] I. Barmina, M. Zake, H. Kalis and M. Marinaki. Electic field effects on gasifica-
tion / combustion at thermo-chemical conversion of biomass mixtures. In Latvia
J. Palabinskis, Latvia University of Agriculture(Ed.), Proc. of of the 16-th Int.
Conf. on Engineering for Rural Development (ERDev17), Jelgava, Latvia, May.
24-26, 2017, pp. 60–67, Latvia University of Agriculture, Faculty of Engineering,
2017. https://doi.org/10.22616/ERDev2017.16.N012.
[9] F. Battaglija, K. McGrattan, R.G. Rehm and H.R. Baum. Simul-
ing fire whirls. Combustion Theory and Modelling,4(2):123–138, 2000.
https://doi.org/10.1088/1364-7830/4/2/303.
[10] V. Bayona and M. Kindelan. Propagation of premixed lami-
nar flames in 3D narrow open ducts using PBF- generated finite-
differences. Combustion Theory and Modelling,17(5):789–803, 2013.
https://doi.org/10.1080/13647830.2013.801519.
[11] V.V. Boyarevitch, Y.Zh. Freiberg, E.I. Shilova and E.V. Therbinyn. Electro-
vortexed flows. Zinatne Press,Riga, 1985, in Russian.
[12] J.J. Choi, Z. Rusak and A.K. Kapila. Numerical simulation of premixed chemical
reactions with swirl. Combustion theorie and modelling,6(11):863–887, 2007.
[13] J. Douglas and R. Rachford. On the numerical sulution of heat conduction
problems in two and three space variables. Trans. Amer.Math.Soc.,82(2):421–
439, 1956. https://doi.org/10.1090/S0002-9947-1956-0084194-4.
[14] A.K. Gupta, D.G. Lilley and N. Syred. Swirl Flows. Abacus Press, 1984.
[15] H. Kalis, I. Barmina, M. Zake and A. Koliskins. Mathematical modelling and
experimental study of electrodynamic control of swirling flame flows. In Latvia
J. Palabinskis, Latvia University of Agriculture(Ed.), Proc. of of the 15-th Int.

Electromagnetic Field Effects in the Combustion Process 343
Conf. on Engineering for Rural Development (ERDev16), Jelgava, Latvia, May.
25-27, 2016, pp. 134–141, Latvia University of Agriculture, Faculty of Engineer-
ing, 2016.
[16] H. Kalis and M. Marinaki. Numerical study of 2D MHD convection around
periodically placed cylinders. Int. Jour. of Pure and Applied Mathematics,
110(3):503–517, 2016. https://doi.org/10.12732/ijpam.v110i3.10.
[17] H. Kalis, M. Marinaki, U. Strautins and Lietuvietis. On the numerical simulation
of the combustion process with simple chemical reaction. In Dmitri Meshumayev
and Bength Sunden(Eds.), Proc. of the 7-th Baltic Heat Transfer Conference ,
Tallinn, Estonia, Aug. 24-26, 2015, Baltic Heat Transfer Conference BHTC, pp.
175–180. Tallinn University of Technology, 2015. ISBN 978-9949-23-817-0.
[18] H. Kalis, M. Marinaki, U. Strautins and Lietuvietis. On the numerical simulation
of the vortex breakdown in the combustion process with simple chemical reaction
and axial magnetic field. Int. Jour. of Differential Equations and Applications,
14(3):235–250, 2015.
[19] V.M. Kovenya and N.N. Yanenko. The method of decomposition in gaz dynamical
problems. Nauka Press, Novosibirsk, 1981, in Russian.
[20] D.G. Lilley. Swirl flows in combustion: A review. AIAA Journal,15(8):1063–
1078, 1977. https://doi.org/10.2514/3.60756.
[21] D. Meeker. Finite Element Method Magnetics. Version 4.2, User’s Manual, 2015.
[22] V. Mittal, H. Pitsh and F. Egolfopoulos. Assessment of coun-
terflow to measure laminar burning velocities using direct numerical
simulation. Combustion Theory and Modelling,16(3):419–433, 2012.
https://doi.org/10.1080/13647830.2011.631033.
[23] N. Syred and J.M. Beer. Combustion in swirling flows: A review. Combustion
and Flame,23(2):143–201, 1974. https://doi.org/10.1016/0010-2180(74)90057-
1.
[24] A.B. Watathin, G.A. Lyubimov and S.A. Regirer. Magnetohydrodynamical flows
in the canalls. Nauka Press,Moscou, 1970, in Russian.
[25] A. Zlotnik and R. ˇ
Ciegis. A converse stability condition is necessary for a com-
pact higher order scheme on non-uniform meshes. Applied Mathematics Letters,
80:35–40, 2018. https://doi.org/10.1016/j.aml.2018.01.005.
[26] A. Zlotnik and V. Gavrilin. On quasi-gasdynamic system of equations with gen-
eral equations of state and its application. Mathematical Modelling ana Analysis,
16(4):509–526, 2011. https://doi.org/10.3846/13926292.2011.627382.
Math. Model. Anal., 23(2):327–343, 2018.