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Coupled map lattice model for boiling
PHYSICS LETTERS A
Physics LettersA 165 (1992) 405—408 North-Holland
Coupled map lattice model for boiling Tatsuo Yanagita Institute of Statistical Mathematics, Graduate UniversityforAdvanced Studies, 4-6-7Minami-Azabu, Minato-ku, Tokyo 106, Japan Received 25 February 1992; accepted for publication 31 March 1992 Communicated by A.R. Bishop
A simple model for boiling is proposed. With increasing the temperature of a bottom plate, our minimal model shows three successive phases; conduction, nucleate and film boiling. In the nucleate regime the heat flux increases with the temperatureof the bottom plate, while it starts to decrease in the film boiling regime. In the boiling state, the maximum Lyapunov exponent is positive, implying that the boiling phenomena are spatiotemporal chaos.
The boiling transition is very important not only from the scientific but also from the engineering viewpoint [1]. With the increase ofthe heat flux fed into a system, the dynamics of boiling is known to show a transition from nucleate to film boiling at a burnout point [2]. In the nucleate boiling regime, each bubble is isolated, while a thin film vapor coyers the surface of the bottom plate and floats into pieces in the film boiling. In principle, the dynamics of boiling is described by the Navier—Stokes equation and the phase transition dynamics. However, it is difficult to simulate the boiling phenomena based on the Navier—Stokes equation because the boundary of the two phases is complex and varies in time. Although the Navier— Stokes equation should be used to capture the phenomena quantitatively, acoupled maplattice (CML) is one of the most powerful strategies to represent the qualitative features of dynamical phenomena in a spatially extended system. A CML is a dynamical system withIta discrete and space,byand a continuous state. has beentime investigated K.aneko extensivelyin various contexts [3,4]. The use ofCMLs as simulators for various physical phenomena have been proposed [5]. A successful example is the application of CMLs for spinodal decomposition by Oono and Pun [6]. Here we design a minimal CML model for boiling, which exhibits the transition from nucleate to film boiling,
Here we study two-dimensional boiling phenomena, taking a two-dimensional lattice (x, y) and a temperature field T~,, on it at time 1. The dynamics of boiling is decomposed into independent units (diffusion, creation and floating motion of bubbles, latent heat). We replace, each unit by the following simplest parallel dynamics on the lattice. (1) Thermal diffusion: — X,~ —
T’ +‘ ‘T’
+ T’
x+ I Y
X~Y
+ T~~~_ —
x.Y+ I
+ T’ X
I,y
(1)
4T~~) 1 .
(2) Creation and floating motion:
r~= T~
.
—
T’~,~[p( T~,~+1 ) —p( T~_1)]
,
p( T) = tanh [a ( T— Ta)].
(3) Latent heat effect: If T~’,~> T~and ~ then t-~-1 = —
r n(x,y)
<
T,~
(4)
n(x,y)
if ~
< 7’,
T’-4-’
—
n(x,y)
(2) (3)
—
and Ti.,> T~then
T” n(x,y)
+
5 ~1,
where n (x, y) denotes the nearest neighbor of the position (x, y). These three unit dynamics (“procedures”) are separated and successivelycarried out. The first pro-
0375-960l/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
405
Volume 165, number 5,6
PHYSICS LETTERS A
cedure is the thermal diffusion process, given by the
1 June 1992
(heating surface) boundary is fixed at T10~,TbottOm, respectively. The initial condition is chosen to be ~ ~( ~ + T50~)+ ö with a small uniform random number ö. We investigate the pattern of this model increasing the temperature T~ttom. (A) Below some critical value Tb.P. (—.‘ 9.77) > T~110,,,,heat is mainly transported by diffusion (procedure 1) from the bottom. If the temperature gradient EiT= T~ttom T500 is sufficiently small, the mean temperature field obeys the Fourier law for heat conduction (fig. 1 a). (B) Above the boiling point TbP, heat is no longer transferred efficiently by a diffusion process only and a large temperature gradient appears near the bottorn. Thus the floating procedure 2 is relevant to form bubbles (i.e. patches where T> Ta). Slightly above the boiling point, the heat supply from the bottom plate cannot be large enough to grow the bubble. The
discretized Laplace equation. The second one represents the creation of a bubble and its floating motion forced by buoyancy. The buoyancy depends on the density p ( T), where we assume local equilibrium. The hyperbolic tangent term represents the variation of density by the phase transition. Due to this term, the density decreases obviously to form a gas phase, if the temperature exceeds T~.The cell with a temperature beyond T. is regarded as a bubble, which floats upwards. The third one represents the latent heat effect. When the phase transition to gas occurs in a cell, the temperature of its nearest neighbors is decreased. Our model is described by the parameters e, a, a, ,~i, T~.We set these parameters as e=0.5, a= 10, a=0.3, ~=0.5, T~=10. We take a periodic boundary condition for the horizontal direction. The top (cooling surface) and the bottom
—
7 60
60
soJ
50
10
-
1
~
10
, -fl
~
..
10
1
(a)
40
S
—
~
-
60
(b)
(c)
~rIOIII~ 10
(e)
20
30
40
(d)
I
50
60
X
10
20
30
40
50
(f)
Fig. 1. Snapshot pattern. The lattice size is 64 x 64. White means high temperature and black means low temperature. Allparameters are the same, except for the bottom temperature T~,,,ti0m.Above the burnout point Tb,, -~9.91 a film vapor appears and detaches itself from the bottom simultaneously. (a) T~,,,t~m=9.75, (b) Ti,,,ttom9.8, (c) Ti,,,tu,m9.85, (d) T~ttom9.9, (e) T~,,,ttom=9.925,(f) T~ttom=9.975.
406
Volume 165, number 5,6
PHYSICS LETI’ERS A
1 June 1992
bubbles eventually disappear before floating and emit the latent heat around the cells. After repeating this process a few times, heat is accumulated till a bubble of Tbottom, bubbles are released from the bottom constarts to grow and floats upward. With the increase stantly. Due to the thermal diffusion most of them diminish in size and eventually disappear (figs. ib— ld). (C) At the burnout point TbO—.9.9l, bubbles are joined to form a film vapor. In our model, this film vapor detaches itself from the bottom plate simultaneously. Thus we observe a stripe pattern in figs. 1 e, 1f. This pattern is stable because our model has no hydrodynamic and surface effects. To include such an effect, we must add a variable for the density on the lattice and also have to take into account the dynamics of the phase transition. In order to describe the boiling transition quantitatively, Nukiyama measured the temperature of the heater as a function of the heat flux fed into the system [2]. This relationship is called a boiling characteristic curve. With this curve he found that heat flux increases with heating temperature and reaches a maximum value (critical heat flux) at the burnout point. Above the burnout point, the heat flux decreases with increasing the heating temperature. In our simulation, we estimate the heat flux by
q 0.14
7( ~
~a(T~x,I)—Tbottom)
0.1
/
//
bO.08
I
06 0. ~44
1
_______________________________ 9.8
TBottOm
9.85
TB. P.
9
9.95
TBO
Boiling characteristic curve. The temperature of the heating surface as a function ofthe heat flux fed into the system. Each point of the heat flux q is averaged over 1000 time steps. The Fig. 2.
critical values Tb.P 9.8 and Tb.,, and burnout point, respectively.
9.91 indicate the boiling point
Lyapunov exp.
1IL
q=
0.12
•Ij
(6)
1~
\x=I
0.25
.1
I ~
where < >, means the average over many time steps and L is the size of the bottom plate. We have numerically calculated the heat flux q by varying Ttyjttom. Through this numerical calculation of the
I
t
~
• • . i
S
heat flux q, we have obtained the boiling characteristic curve (fig. 2). Our curve in fig. 2 is in good qualitative agreement with the experimental observation [2]. Chaotic dynamical systems are usually characterized by Lyapunov exponents [7], a measure for the divergence rates of nearby trajectories. Here we calculated the maximum Lyapunov exponent using the method of Benettin et al. [8]. In fig. 3, the maximum Lyapunov exponent starts to take a positive value at the boiling point. Thus the boiling phenomena are temporally chaotic. At the burnout point, the maximum Lyapunov exponent shows a large de-
0.2
I
~ O,.U I.e 0.1
S
0.05
S
TBOttom
~
9
TB. p.
8
9.85
9 9~
9.95
TBO
Fig. 3. Maximum Lyapunov exponent. The lattice size is 64 X64. Each point of the maximum Lyapunov exponent is averaged over 1000 time steps.
407
Volume 165, number 5,6
PHYSICS LETTERS A
I June 1992
crease. This decrease is related with the formation of an ordered stripe pattern. In conclusion, we have proposed a minimal CML model for boiling. This model shows the transition from nucleate to film boiling. The model reproduces some qualitative features of this transition. In particular, the boiling characteristic curve agrees with experimental observations qualitatively.
References
I would like to thank Professor Y. Itoh for his continual encouragement and interest in the boiling model. I am also grateful to Professor K. Kaneko and his group for discussions. Enjoyable discussion with and criticism from Y. Iba are very much appreciated.
[5] K. Kaneko,in: Formation, dynamics and statistics ofpatterns, VoL 1, eds. K. Kawasaki, M. Suzuki and A. Onuki (World Scientific, Singapore, 1990) pp. 1—50. [6] Y. Oono and S. Pun, Phys. Rev. A 38 (1988) 434. [7] I. Shimada and T. Nagashima, Prog. Theor. Phys. 61(1979) 1605. [8] G. Benettin, L. Galgani and J.M. Strelcyn, Phys. Rev. A 14
[1] S. van Stralen and R. Cole, Boiling phenomena, Vol. 1
(Hemisphere, Washington, DC, 1979). [2] S. Nukiyama, J. Soc. Mech. Eng. Japan 37 (1934) 367 [in Japanese]. [3] K. Kaneko,Collapse of tori and genesis ofchaos in dissipative systems (World Scientific, Singapore, 1983). [4] J.P. Crutchfield and K. Kaneko, in: Directions in chaos, ed. B.-L. Hao (World Scientific, Singapore, 1987) pp. 272—3 53.
(1976) 2338.
408
Physics LettersA 165 (1992) 405—408 North-Holland
Coupled map lattice model for boiling Tatsuo Yanagita Institute of Statistical Mathematics, Graduate UniversityforAdvanced Studies, 4-6-7Minami-Azabu, Minato-ku, Tokyo 106, Japan Received 25 February 1992; accepted for publication 31 March 1992 Communicated by A.R. Bishop
A simple model for boiling is proposed. With increasing the temperature of a bottom plate, our minimal model shows three successive phases; conduction, nucleate and film boiling. In the nucleate regime the heat flux increases with the temperatureof the bottom plate, while it starts to decrease in the film boiling regime. In the boiling state, the maximum Lyapunov exponent is positive, implying that the boiling phenomena are spatiotemporal chaos.
The boiling transition is very important not only from the scientific but also from the engineering viewpoint [1]. With the increase ofthe heat flux fed into a system, the dynamics of boiling is known to show a transition from nucleate to film boiling at a burnout point [2]. In the nucleate boiling regime, each bubble is isolated, while a thin film vapor coyers the surface of the bottom plate and floats into pieces in the film boiling. In principle, the dynamics of boiling is described by the Navier—Stokes equation and the phase transition dynamics. However, it is difficult to simulate the boiling phenomena based on the Navier—Stokes equation because the boundary of the two phases is complex and varies in time. Although the Navier— Stokes equation should be used to capture the phenomena quantitatively, acoupled maplattice (CML) is one of the most powerful strategies to represent the qualitative features of dynamical phenomena in a spatially extended system. A CML is a dynamical system withIta discrete and space,byand a continuous state. has beentime investigated K.aneko extensivelyin various contexts [3,4]. The use ofCMLs as simulators for various physical phenomena have been proposed [5]. A successful example is the application of CMLs for spinodal decomposition by Oono and Pun [6]. Here we design a minimal CML model for boiling, which exhibits the transition from nucleate to film boiling,
Here we study two-dimensional boiling phenomena, taking a two-dimensional lattice (x, y) and a temperature field T~,, on it at time 1. The dynamics of boiling is decomposed into independent units (diffusion, creation and floating motion of bubbles, latent heat). We replace, each unit by the following simplest parallel dynamics on the lattice. (1) Thermal diffusion: — X,~ —
T’ +‘ ‘T’
+ T’
x+ I Y
X~Y
+ T~~~_ —
x.Y+ I
+ T’ X
I,y
(1)
4T~~) 1 .
(2) Creation and floating motion:
r~= T~
.
—
T’~,~[p( T~,~+1 ) —p( T~_1)]
,
p( T) = tanh [a ( T— Ta)].
(3) Latent heat effect: If T~’,~> T~and ~ then t-~-1 = —
r n(x,y)
<
T,~
(4)
n(x,y)
if ~
< 7’,
T’-4-’
—
n(x,y)
(2) (3)
—
and Ti.,> T~then
T” n(x,y)
+
5 ~1,
where n (x, y) denotes the nearest neighbor of the position (x, y). These three unit dynamics (“procedures”) are separated and successivelycarried out. The first pro-
0375-960l/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
405
Volume 165, number 5,6
PHYSICS LETTERS A
cedure is the thermal diffusion process, given by the
1 June 1992
(heating surface) boundary is fixed at T10~,TbottOm, respectively. The initial condition is chosen to be ~ ~( ~ + T50~)+ ö with a small uniform random number ö. We investigate the pattern of this model increasing the temperature T~ttom. (A) Below some critical value Tb.P. (—.‘ 9.77) > T~110,,,,heat is mainly transported by diffusion (procedure 1) from the bottom. If the temperature gradient EiT= T~ttom T500 is sufficiently small, the mean temperature field obeys the Fourier law for heat conduction (fig. 1 a). (B) Above the boiling point TbP, heat is no longer transferred efficiently by a diffusion process only and a large temperature gradient appears near the bottorn. Thus the floating procedure 2 is relevant to form bubbles (i.e. patches where T> Ta). Slightly above the boiling point, the heat supply from the bottom plate cannot be large enough to grow the bubble. The
discretized Laplace equation. The second one represents the creation of a bubble and its floating motion forced by buoyancy. The buoyancy depends on the density p ( T), where we assume local equilibrium. The hyperbolic tangent term represents the variation of density by the phase transition. Due to this term, the density decreases obviously to form a gas phase, if the temperature exceeds T~.The cell with a temperature beyond T. is regarded as a bubble, which floats upwards. The third one represents the latent heat effect. When the phase transition to gas occurs in a cell, the temperature of its nearest neighbors is decreased. Our model is described by the parameters e, a, a, ,~i, T~.We set these parameters as e=0.5, a= 10, a=0.3, ~=0.5, T~=10. We take a periodic boundary condition for the horizontal direction. The top (cooling surface) and the bottom
—
7 60
60
soJ
50
10
-
1
~
10
, -fl
~
..
10
1
(a)
40
S
—
~
-
60
(b)
(c)
~rIOIII~ 10
(e)
20
30
40
(d)
I
50
60
X
10
20
30
40
50
(f)
Fig. 1. Snapshot pattern. The lattice size is 64 x 64. White means high temperature and black means low temperature. Allparameters are the same, except for the bottom temperature T~,,,ti0m.Above the burnout point Tb,, -~9.91 a film vapor appears and detaches itself from the bottom simultaneously. (a) T~,,,t~m=9.75, (b) Ti,,,ttom9.8, (c) Ti,,,tu,m9.85, (d) T~ttom9.9, (e) T~,,,ttom=9.925,(f) T~ttom=9.975.
406
Volume 165, number 5,6
PHYSICS LETI’ERS A
1 June 1992
bubbles eventually disappear before floating and emit the latent heat around the cells. After repeating this process a few times, heat is accumulated till a bubble of Tbottom, bubbles are released from the bottom constarts to grow and floats upward. With the increase stantly. Due to the thermal diffusion most of them diminish in size and eventually disappear (figs. ib— ld). (C) At the burnout point TbO—.9.9l, bubbles are joined to form a film vapor. In our model, this film vapor detaches itself from the bottom plate simultaneously. Thus we observe a stripe pattern in figs. 1 e, 1f. This pattern is stable because our model has no hydrodynamic and surface effects. To include such an effect, we must add a variable for the density on the lattice and also have to take into account the dynamics of the phase transition. In order to describe the boiling transition quantitatively, Nukiyama measured the temperature of the heater as a function of the heat flux fed into the system [2]. This relationship is called a boiling characteristic curve. With this curve he found that heat flux increases with heating temperature and reaches a maximum value (critical heat flux) at the burnout point. Above the burnout point, the heat flux decreases with increasing the heating temperature. In our simulation, we estimate the heat flux by
q 0.14
7( ~
~a(T~x,I)—Tbottom)
0.1
/
//
bO.08
I
06 0. ~44
1
_______________________________ 9.8
TBottOm
9.85
TB. P.
9
9.95
TBO
Boiling characteristic curve. The temperature of the heating surface as a function ofthe heat flux fed into the system. Each point of the heat flux q is averaged over 1000 time steps. The Fig. 2.
critical values Tb.P 9.8 and Tb.,, and burnout point, respectively.
9.91 indicate the boiling point
Lyapunov exp.
1IL
q=
0.12
•Ij
(6)
1~
\x=I
0.25
.1
I ~
where < >, means the average over many time steps and L is the size of the bottom plate. We have numerically calculated the heat flux q by varying Ttyjttom. Through this numerical calculation of the
I
t
~
• • . i
S
heat flux q, we have obtained the boiling characteristic curve (fig. 2). Our curve in fig. 2 is in good qualitative agreement with the experimental observation [2]. Chaotic dynamical systems are usually characterized by Lyapunov exponents [7], a measure for the divergence rates of nearby trajectories. Here we calculated the maximum Lyapunov exponent using the method of Benettin et al. [8]. In fig. 3, the maximum Lyapunov exponent starts to take a positive value at the boiling point. Thus the boiling phenomena are temporally chaotic. At the burnout point, the maximum Lyapunov exponent shows a large de-
0.2
I
~ O,.U I.e 0.1
S
0.05
S
TBOttom
~
9
TB. p.
8
9.85
9 9~
9.95
TBO
Fig. 3. Maximum Lyapunov exponent. The lattice size is 64 X64. Each point of the maximum Lyapunov exponent is averaged over 1000 time steps.
407
Volume 165, number 5,6
PHYSICS LETTERS A
I June 1992
crease. This decrease is related with the formation of an ordered stripe pattern. In conclusion, we have proposed a minimal CML model for boiling. This model shows the transition from nucleate to film boiling. The model reproduces some qualitative features of this transition. In particular, the boiling characteristic curve agrees with experimental observations qualitatively.
References
I would like to thank Professor Y. Itoh for his continual encouragement and interest in the boiling model. I am also grateful to Professor K. Kaneko and his group for discussions. Enjoyable discussion with and criticism from Y. Iba are very much appreciated.
[5] K. Kaneko,in: Formation, dynamics and statistics ofpatterns, VoL 1, eds. K. Kawasaki, M. Suzuki and A. Onuki (World Scientific, Singapore, 1990) pp. 1—50. [6] Y. Oono and S. Pun, Phys. Rev. A 38 (1988) 434. [7] I. Shimada and T. Nagashima, Prog. Theor. Phys. 61(1979) 1605. [8] G. Benettin, L. Galgani and J.M. Strelcyn, Phys. Rev. A 14
[1] S. van Stralen and R. Cole, Boiling phenomena, Vol. 1
(Hemisphere, Washington, DC, 1979). [2] S. Nukiyama, J. Soc. Mech. Eng. Japan 37 (1934) 367 [in Japanese]. [3] K. Kaneko,Collapse of tori and genesis ofchaos in dissipative systems (World Scientific, Singapore, 1983). [4] J.P. Crutchfield and K. Kaneko, in: Directions in chaos, ed. B.-L. Hao (World Scientific, Singapore, 1987) pp. 272—3 53.
(1976) 2338.
408