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On a numerical approach to Stefan problem
Pavel Kotalík
To cite this version:
Pavel Kotalík. On a numerical approach to Stefan problem. Journal de Physique III, EDP Sciences, 1993, 3 (11), pp.2113-2120. �10.1051/jp3:1993264�. �jpa-00249070�
Classification Physics Abstracts
02.60 44.10 52.75
On a numerical approach to Stefan problem
Pavel Kotal%
Institute of Plasma Physics~ Czech Academy of Sciences, Za Slovankou 3, P-O- BOX IT 182 11 Praha 8~ Czech Republic
(Received ii May1993, revised 30 August 1993, accepted I September 1993)
Abstract. A one-dimensional Stefan problem for a spherical particle moving in a plasma jet is solved by approximating the enthalpy formulation of the problem by C° piecewise linear
finite elements in space combined with a semi-implicit scheme in time.
Introduction.
In iii a new numerical approach to bidimensional Stefan-like problems was proposed. Our aim is to show efficiency of this approach by modifying the resultant numerical scheme to
a solution of a one-dimensional (radial) heat transfer equation (HTE) in a spherical particle with temperature dependent material parameters. As an example, we consider the particle of
alumina (A1203) moving in a water plasma jet.
When speaking about Stefan problem we mean that the particle melts and resolidifies and thus a sharp moving solid-liquid boundary occurs. Two additional conditions have to be satisfied on this boundary: the temperature T equals a melting temperature Tm and the heat flow exhibits a jump corresponding to the absorption or liberation of a latent heat of fusion Lm (Stefan condition) [2~ 3].
The finite difference schemes of the temperature formulated problem usually involve both these conditions as new equations in addition to the HTE and use deforming grids in order to follow the interphase boundary [4, 5]. Nevertheless, it is well-known that the numerical
modelling of the temperature formulated problem can be simplified by avoiding these difficul- ties; it only suffices to redefine suitably the specific heat in a neighbourhood of the melting point [6]. An alternative approach is to use the enthalpy formulation of the problem in which
case no explicit conditions on the temperature and the heat flow at the solid-liquid interface need to be accounted for and the fixed grid can be used as well. However, there are first-order discontinuities of the thermal parameters of particle at melting point and one is led to a weak formulation of both the temperature and enthalpy formulated problem 11, 2].
The enthalpy formulation in iii is based on a physically nice fact that the temperature is an
2114 JOURNAL DE PHYSIQUE III N°11
T
Tb
Tm
ui uz u3 u4 u
Fig-I- Temperature as a function of enthalpy.
increasing function of enthalpy~ T
= p(u), whose graph has a flat part at melting point Tm, and for our purposes similarly at boiling point Tb, as shown in figure i. The differences12
vi u4 u3 correspond with the latent heat of fusion and evaporation Lm and Lv~ respectively.
The weak formulation is approximated by continuous piecewise linear finite elements in space combined with a semi-implicit scheme in time.
Before presenting the numerical study we point out that the behaviour of solid particles in
high pressure (thermal) plasmas has been extensively studied. We refer the reader to overviews [7,8] for details and to the Appendix where the necessary transport coefficients are summarized.
From now on, we let p to denote the density, the thermal conductivity, cp the specific heat
at constant pressure, ~1 the viscosity, u the kinematic viscosity, r the radial coordinate and
rp(t) the (time dependent) particle radius. The subscripts g, v are let to denote "gas" and
"vapour", respectively.
Weak formulation of the problem.
The HTE is written in terms of the enthalpy
u =
f~ cp(T')dT' (which is defined up to an additive constantj see also Fig. I) as followsi ~
p~
= V ()VT)
= V (A~~ Vu). (I)
We suppose that the initial temperature of particle is equal to a constant To which is different from melting or boiling temperature and we set the initial condition
u(.,0) # uo " fl~~ (To) on no
" (0,rp(0)]. (2)
We assume the following boundary and symmetry conditions:
AVnT = hT (Tg T) e a (T~ Tj) j n[Lv + cpv (Tg T)]8(u2) (3)
+ RHS at r = rp(t),
VT=o at T=o. (4)
Here n denotes the outer normal to the surface of the particle, e the surface emissivity~ a the Stefan-Boltzmann constant, Ta the temperature of an environment surrounding the jet and e the Heaviside function. The heat transfer coefficient hT and the vapour mass flow density j
are defined in the Appendix. The three terms in RHS describe the heat flow due to forced convection, the radiative and the evaporative heat losses, respectively. The particle vapours
are assumed to be heated to the temperature Tg of the carrier gas.
Moreover, we suppose that the temperature dependent parameters p, cp, are strictly positive and that the function p is piecewise C~ function satisfying the conditions fl(f)
= Tm
(resp. Tb) for all f E [vi,u2) (resp. f E [u3,u4)) and 0 < fl'(f) < B, where B is some finite number, for all f f [vi,u2) U [u3,u4) see figure I.
Denoting (#,#)
= f)~ ##r~dr, (#,#) = f~ ##dS = ##r( (one-dimensionality of the prob- lem !) we get in a standard way the weak formulation of (1-4) (see iii and [2] for mathematical
details):
Find u such that
lp~",bt #) + (Afl'vu, Vi) = (RHS, ii, u(., 0) = uo on Ho (5)
for all suitable test functions # and almost every t E (0, tF), where tF is a fixed positive time.
Numerical scheme.
We solve (5) in a class of C° piecewise linear functions on fit m lo,rp(t)]. We let Ih to denote the decomposition of Qt into M + I intervals I, = [(I I)h,ih], I
= I,..
,
M + I, M > 0, and R, = irp(t)/(M + I) + ih, I
= 0,..
,
M + I, its equidistant nodal points.
We consider the finite element space
Vh " (ifi E C° (fit) ~fi(I is a polynomial function of degree
< I on all I E Ih and V~fi
= 0 at r = 0 )
with the basis generated by the set (~fi,)f(~ of "pyramidal" functions ~fi,(r) = (R,+i r)/h
on I,+i, ifii " (r R,-i)/h on I,, ~fi,
= 0 otherwise, which are orthogonal with respect to the discrete inner product
~~~~~~ ~ ~~~~~~~~~~ ~(
/
~(~X)(~~)i~kr~dr,~~~~ E h Rj
R( denote the nodal points belonging to the interval I and rh is the interpolation operator related to Vh.
Remark 1. Notice the possible time dependence of h which takes into account the decrease of particle due to vaporization or even due to intense evaporation at boiling point.
Setting the time step r m tF/N, where the integer N > 0, t"
= nr, and following iii we
obtain the space-time discretization of (5):
Find (U")$j/, U" E Vh such that
fun+i
p(U") ~n,#) + (a(U")VU"+~, V#)h " (RHS(U"), #) for all # E Vh,
T h
U°
m uo> (6)
where
2116 JOURNAL DE PHYSIQUE III N°11
and where the notation h a h"+~ is used for brevity.
If U" is constant, equal to vi, u2 or u3 on I E Ih, we can arbitrarily set fl'(U")
= fl'(ui 0) if U"
= vi, fl'(U") = fl'(u2 + 0) if U"
= u2 and fl'(U")
= fl'(u3 0) if U"
= u3.
The matrix form of (6) is obtained by putting # = ~fij, U"
=
£$(~ Uf~fi,, where Uf
+
U"(R,)~ and by remembering that US = U/ (symmetry at r = 0). The result is as followsi
~
U"+~ U"
~ ~ ~i ~
~
r
~ ~ ~ ~
or equivalently
(1 + r (D")~~R")U"+~
m U" + r (D")~~F",
n = 0,
,
N 1. (7)
The (M + I)-dimensional column vectors U" and F" have their components Uf and 6,jM+i) RHS"r((t"+~), respectively.
The (M + I) x (M + I) matrix D" is diagonal with strictly positive elements
dir " (P"(llo +lli),llo +lli)h
" Pi (Bo + Al + Bi)h~,
£l$ = (p"ffi,,ffi,)h " P~(A, + B,)h~~ " 2,..
,
M,
d~M+I)(M+I) " (P"i~M+I,i~M+I)h
" P~i+TAM+lh~.
The square matrix R" is symmetric tridiagonal with the elements
Tii " (a"V(llo +lli), V(~fio + ifii))h
" (ai,RBI + ai,LA2)h,
r$ = (a"i7ffi,,i7ffi;)h " (tl~-l~R~i-1 ~ ~~L~i ~ ~~R~'~ ~~+l,L~'+~~~' ~ ~' '~'
Tif+I,M+I " (a"i7ffiM+1,i7ffiM+1)h " (air,RBM + air+I,LAM+I)h,
rj,~ij = (anv~b,, v~b,+i)
= -(ajRB, + aj~i,LA;+ilh, i
= i,..
,
M.
In the above formulae, the strictly positive coefficients A, and B, are given by the equations
1 j4 (j 1)4 f3 (j 1)3
~~ h3
/,
'~~~~~~4 ~~ ~~ 3 ' ~ ~' ~ ~ ~'
Bj = ~fi,r~dr = (I + 1)~~ ~ ~~~ ~~ ~~ ~ ~~~ ~~, i
= 0~..
,
M.
~l
,+i
3 4
The a/~ and a]~ denote the (positive) al's with the derivatives fl' taken at the point R, from the rigit and frim the left, respectively.
The matrix R" is easily shown to be diagonally dominant (one has r] > 0, rf~
~i < o), Thus the semi-implicit scheme (6) is positive and it is shown to be stable whenever (in iondimensional units) rB/h~ < c, where c is an arbitrary positive constant. Moreover, the solution of (6)
converges strongly (in a corresponding space) to the solution of problem (5) whenever r and h tend to zero in such a way that r/h~ tends to zero iii.
Numerical results.
To set the correct space and time steps in (7), it is convenient to proceed to the dimen-
sionless variables r*
= r/rp(0)~ t*
= Fo Ste~ u*
= u/Lm, T*
= (T Tm)/(Tm To)-
Fo
= A~t/(p~cp~r((0)) and Ste = cp~(Tm To)/Lm are the Fourier and Stefan number, re- spectively. We also define the dimensionless density p* = p/p~~ the specific heat capacity c( = cp/cp~ and the heat conductivity A* = A/As (the subscript s refers to solidus). The only
change in equation (7) is that all the parameters in matrices D" and R" are replaced by their nondimensional equivalents, U" is replaced by U*"~ R" is divided by Ste and F" is divided by
ToAs(Tm To).
RemaTk 2. As the information about whether the particle is melted or not does not depend
on the units used, we plot the results by means of nondimensional variables.
We set Tm
= 2326 K~Tb = 3800 K, To " Ta = 300 K, Lm = 1.065 MJ/kg~ Lv = 24.7 MJ/kg,
pa = 3900 kg/m3, pi
" 2750 kg/m3~ cps = 1.419 kJ/(kgK), cpv
= 0.967 kJ/(kgK), As
=
9.66 J/(msK), vi = 0, vi = I, vi + 2.96, u( + 26.16. The Stefan number Ste + 2.7. To mimic
experimental conditons, we suppose that the jet temperature and velocity decrease rapidly from 12000 K and 600 mIs at the jet origin x = 0.0 m to 1000 K and 240 mIs at x = 0.3 m.
We present results for particles of the initial diameter 30 and 120 ~lm and of the initial velocity 10 m/s. In both cases, we set the space step h
= 1/25 and the time step r
= h/20.
The results were obtained by solving the set of linear algebraic equations (7) for U"+~. Our program was written in Fortran and its flow diagram is similar to that in [5]. The particle mo-
tion is supposed to be one-dimensional, parallel to that of the gas in which case the equation of motion is integrated analyticallyj provided that the gas velocity and temperature are constant
over the short time step r. The moving grid is only used if the particle decreases by vapor- ization or evaporation. The intense evaporation is taken into account by "switching-ofi" the
'',
-
_ , , / ,
~e
,
_
, ~~jocdv
z,0
/
~ /
~
m
~
E~
izo
o.o -l.0
0.0
Fig.2. Particle velocity~ radius~ surface (thick line) and central (thin line) temperature vs. the distance from the jet origin. Label 30 (120) particle diameter 30 (120) ~Lm. Unlabelled lines gas.
2118 JOURNAL DE PHYSIQUE III N°11
I
iq~
ij~
~~
$~ q lj
~i
o
~~
'i j~
~ o~
~ °
il
'o~~
l'o~~
'~ 0j %~
~ ~, ~
~~ q
~ /~p
~~ q
_~~~$~~~ j~ ~
~~ C~~ ~ ~
ks~ ~ ~
Fig.3. Enthalpy (above) and temperature (below) history of an A1203 particle diameter 30 ~Lm.
Time 1 + 4.8 x 10~~s.
mass flow density j whenever the particle surface reaches the boiling point and by "cutting-ofi"
the thin surface layer whose temperature exceeds Tb. The vaporization is negligible unless the surface temperature is above 3700 K.
Both the enthalpy and temperature histories of the particles were calculated. The depen-
dence of some remarkable parameters of the particles on a distance from the jet origin is shown in figure 2. To this purpose, the dimensionless velocity v*
= v/100 m/s and the x-axis co- ordinate x*
= x/lm were defined. We conclude that while the smaller particle is partially evaporated and resolidified at z* = 0.3~ the greater one is melted and unevaporated.
Comparison of the enthalpy and temperature history of the particle is shown in figure 3. The
melting or solidification front is defined at time t" as the set of points (r E Qtn vi < U" < ~t2).
The enthalpy history is seen to be advantageous for determining where this front lies see figure
3.
As pointed out in iii, the choice of a(U") is made in order to avoid small oscillations of the solution to problem (6). It is argued that the oscillations observed near the melting point
are caused by the discontinuity of the solution to problem (5) at this point, while in (6) the continuity and piecewise linearity are required. Our numerical tests show that the oscillations of the solution of (6) are actually smaller if the recommended a(U") is used. Moreover, the
two solutions of (6) with and without the corrected form of a(U") are shown to be identical away from the melting point.
Conclusion.
The straightforward modification to different one or twc-dimensional problems (including those with nonlinear source term~ nonhomogeneous Dirichlet or Neumann boundary conditions, so- lidification of alloys etc.), the economy in numerical modelling requirements (no moving grids, continuity, piecewise linearity) and the behaviour of our results make the numerical approach
in iii useful to all those who are concerned with Stefan-like problems.
Appendix.
The heat transfer coefficient hT is given by the semi-empirical formula
f 0.15
~~ ~p ~~ ~ ~'~~~~~~~~~~~ )
'
where
Pr = ~fl~
g
and
2Tpfl~(Vg v(
Rep =
llg
are the Prandtl and Reynolds number~ respectively. The bar over the gas parameters means
that they are calculated at the mean gas-surface temperature (Ts + Tg)/2.
To determine the particle velocity and position in a gas, we assume the equation of motion
dv ~_
~j §~~PPg(Vg V((Vg V)CD.
JO~RNAL DE PHYS,QUE ~H -T 3. N'IJ NO~EMBER ~993 76
2120 JOURNAL DE PHYSIQUE III N°11
The drag coefficient CD is given for Rep E 11,1000] by the formula
~~ p 0.15
CD = ~(l + °.l5Rel'~~~)
(~)
P g
The mass flow density j is defined by the equation j = h~(ps p~)n,
where ps and pm are the vapour density at the surface and away from the surface, respectively.
The mass transfer coefficient hM is given by the formula for hT with f replaced by the vapour- gas diifusivity b and with Pr replaced by the Schmidt number
s~ $
p~fl'
When combined with the equation of state of an ideal gas and with the Clausius-Clapeyron's equation of first-order phase transitions the equation for j can be rewritten in terms of the partial pressures of vapours.
References
ill Amiez G.~ Gremaud P.-A.~ Numer. Math. 59 (1991) 71.
[2] Ladyzenskaya O.~ Solonnikov V.~ UraYceva N.~ Linear and Quasilinear Equations of Parabolic
Type (Nauka~ Moscow, 1967) (in Russian).
[3] Carslaw H. S.~ Jaeger J. C., Conduction of Heat in Solids (Oxford Univ. Press, Oxford~ 1959).
[4] Yoshida T.~ Akashi K., J. Appt. Phys. 48(1977) 2252.
[5] Fiszdon J., Int. J. Heat Mass 7Fansfer 22 (1979) 749.
[6] Bonacina C.~ Comini G.~ Fasano A.~ Primicerio M.~ Int. J. Heat Mass Transfer16 (1973) 1825.
[7] Fauchais P.~ Bourdin E.~ Coudert J. F.~ McPherson R.~ Topics in Current Chemistry, F. L. Boschke Ed., Vol. 107 (Springer~ Berlin~ Heidelberg, New York, 1983) p. 59.
[8] Pfender E.~ Plasma Chem. Plasma Process. (suppl.) 9 (1989) 167.
