Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved International Review of Mechanical Engineering, Vol. 5, N. 2 Special Issue on Heat Transfer
289 As mentioned earlier, we performed detailed mass
balance calculations to monitor the flow of metallic and silicate melts through the asteroid. The results presented
in Figs. 1 have precision limited by the use of the numerical technique involving the finite difference
method with explicit approximations [10]. Further, numerous thermodynamical properties, e.g., specific
heats, latent heats, thermal diffusivities, the temperature ranges of the melting of silicate and metal [5], etc., are
involved in the simulations. The uncertainties in their assumed values would also influence the precision of the
simulations. These uncertainties cannot be quantified presently. Nonetheless, we do not anticipate any major
source of error that would significantly influence the generic nature of the major conclusions drawn from the
simulations.
The simulations were run with T
Onset
time of either 0.5 or 1 Myr. for the onset of the accretion of the asteroid
subsequent to the time of the condensation of the earliest solar system dust grains that defines the beginning of the
solar system [2]. The asteroids gradually acquired their final radii over an accretion duration of 1 Myr. The
radiogenic heat production as well as the thermal conduction within the asteroid commence from the onset
of asteroid accretion. The asteroids lose their heat from the surface to the interplanetary space. The thermal
profiles of the asteroids at some chosen epochs are presented in the main figure of Figs. 1. The initial
thermal gradients in the thermal profiles are essentially due to the gradual accretion of the asteroids, the earlier
accreted regions being comparatively hotter. Subsequent to substantial melting at temperature 1,600 K, that
commences from the center, the asteroids achieve convection in the interiors. The convective zone
gradually enhances towards the outer regions during the course of further evolution. The onset of convection is
marked by almost homogeneous thermal profiles through-out the asteroids as convection is more efficient
mode of heat transfer compared to thermal conduction. As mentioned earlier, the convection leads to thermal
homogeneity and rapid cooling of the asteroids. The latter aspect is reflected by the lowering of asteroids
temperature with time as the heat is transferred to the surface from where it is radiated away to space.
The inset figures in Figs. 1 show the gradual growth of the metallic iron-nickel core in asteroids on account of
gravity assisted metal-silicate segregation as deduced from our simulations. A sketch is also presented to
schematically show the differentiated asteroid with a metallic iron-nickel core and a convective molten silicate
magma-ocean as a mantle. The asteroids acquired their final metallic core sizes within the initial couple of
million years in the beginning of the solar system. This time is consistent with the isotopic records of the
meteorites [4].
In none of the present thermal models we observed the global cooling of the molten silicate magma-ocean
below 1,500 K within the initial 10 Myr. in spite of the fact that we used the thermal diffusivity of standard non-
porous rocks at the time of asteroid accretion. This inability raises serious doubts on the substantial ~80
crystallization of the molten convective silicate magma- ocean within the initial 10 million years. Hence, it would
be difficult to form the volcano generated basaltic crust in the stipulated timescales [4] within the theoretical
framework of this proposed hypothesis. Based on the present work on the thermal evolution of the non-porous
asteroids it seems unlikely that the major inferences drawn regarding the production of the basaltic crust
within the initial ~6 Myr. have been significantly altered from the simulations with asteroids with an initial
porosity [7].
As mentioned earlier, the observational records deciphered from meteorites by several analytical studies
indicate the formation of basaltic crust within the initial ~6 Myr. of the early solar system [4]. The disagreement
between the theoretical and observational data can be resolved by conceptualizing an alternative differentiation
scenario. It could be possible that the basaltic crust was produced by another not yet identified differentiation
scenario on contrary to the one simulated in the present work. Alternatively, it is quite possible that the routine
bombardment of the asteroids by small planetesimals destroyed the insulating crust, thereby, rapidly cooling
the body.
The insulating regions constituting the planetary surface and beneath controls the rate of flow of heat from
the asteroid interiors to the surface. An efficient removal of this insulation would directly expose the silicate
magma-ocean to the interplanetary space, thereby, rapidly cooling the magma-ocean for crystallization. The
residual melts in this scenario could produce an early basaltic crust.
IV. Applications in Engineering
Apart for the broad applications in the field of planetary sciences, the numerical techniques associated
with the thermal modeling implemented in the present work could have wide-ranging applications in
engineering. Some of these applications could involve theoretically understanding:
i the processes associated with the melting and cooling
of materials, e.g., the natural mineralsores during metallurgy, during the manufacturing of alloys.
ii the segregation of immiscible fluids assisted by gravity or centrifuges.
iii the thermal effects of nuclear radiation in nuclear reactors and nuclear fall-out.
iv the processes associated with thermal conduction and convection, with the incorporation of temperature
dependent thermal diffusivity, specific heat and porosity.
v the future space explorations of the differentiated asteroids for commercial mining of metals and other
mineral resources [12].
Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved International Review of Mechanical Engineering, Vol. 5, N. 2 Special Issue on Heat Transfer
290
V. Conclusion
We present results of the thermal modeling of the planetary differentiation of asteroids that were linearly
accreted as non-porous compacted bodies. We have incorporated the detailed physico-chemical processes
that include radiogenic heating and melting of asteroids. This was followed by the gravity assisted metal-silicate
segregation, and the cooling of the convective silicate magma-ocean and the molten metallic iron-nickel core.
Our results indicate that the deduced timescales of the initial couple of million years for the segregation of
metallic melt from the molten silicate magma-ocean are consistent with the observational records of the
meteorites that are essentially the survived samples of the planetary differentiated bodies. However, it would not be
possible to produce basaltic volcanic crust on a differentiated asteroid within the initial 10 million years
of the solar system. This is in contradiction with the meteoritic records that suggest an early basaltic crustal
growth. This could imply that either an alternative complex mechanism was involved in planetary
differentiation, or the insulating surface layer of the asteroid was efficiently removed by continuous
bombardment of other small planetesimal to expose the silicate magma-ocean to the interplanetary surface for
rapid cooling and crystallization.
Acknowledgements
We are extremely grateful for the numerous suggestions made by the referees. This work was
supported by a research grant from the Planetary Science and Exploration program of the Indian Space Research
Organization.
References
[1] J. S. Lewis, Physics and chemistry of the solar system 2nd edition, Elsevier academic press, 2004.
[2] S. Sahijpal, J. N. Goswami, A. M. Davis, R. S. Lewis, L. Grossman, A stellar origin for the short-lived nuclides in the early
solar system, Nature, 391, pp. 559-561, 1998. [3] S. Sahijpal, G. Gupta, The plausible sources of
26
Al in the early solar system: A massive star or the X
‐wind irradiation scenario? Meteoritics and Planetary Science, 44, pp. 879–890, 2009
[4] T. Kleine, M. Touboul, B. Bourdon, F. Nimmo, K. Mezger, H. Palme, S. B. Jacobsen, Q. –Z. Yin, A. N. Halliday, Hf–W
chronology of the accretion and early evolution of asteroids and terrestrial planets, Geochimica et Cosmochimica Acta, 73, pp.
5150–5188, 2009. [5]
T. J. McCoy, D. W. Mittlefehldt, L. Wilson, Asteroid differentiation, In D. S. Lauretta, H. Y. McSween Ed.,
Meteorites and the early solar system II, Tucson: University of Arizona Press, 2006, 733–745
[6] S. Sahijpal, P. Soni, G. Gupta, Numerical simulations of the planetary differentiation of accreting planetesimals with
26
Al and
60
Fe as the heat sources, Meteoritics and Planetary Science, 42, pp. 1529–1549, 2007.
[7] G. Gupta, S. Sahijpal, Differentiation of Vesta and the parent bodies of other achondrites, Journal of Geophysical Research
Planets, 115, E08001, 2010. [8] S. Mostefaoui, G. W. Lugmair, P. Hoppe,
60
Fe: A heat source for planetary differentiation from a nearby supernova explosion, The
Astrophysical Journal, 625, pp. 271–277, 2005. [9]
K. Yomogida, T. Matsui, Physical properties of ordinary chondrites, Journal of Geophysical Research, 88, pp. 9513–9533,
1983. [10] L. Lapidus and G. F. Pinder, Numerical solutions of partial
differential equations in science and engineering Wiley
‐Interscience, 1982. [11] C. T. Russell et al., Dawn mission to Vesta and Ceres, Earth,
Moon, and Planets, 101, pp. 65–91, 2007. [12] J. S. Lewis, Mining the Sky: Untold Riches from the Asteroids,
Comets, and Planets Addison-Wesley Publications, 1997.
Authors’ information
Department of Physics, Panjab University,
Chandigarh, India 160014. E-mail:
sandeeppu.ac.in
Sandeep Sahijpal is an Associate Professor at the Department of Physics, Panjab University,
Chandigarh, India. He is M.S. and Ph. D. in Physics. He is developing theoretical models and
numerical codes of the various processes associated with Astrophysics and Planetary
Sciences. This includes the origin and the evolution of the galaxy, the origin and the early
evolution of the solar system, the planetary differentiation and aqueous alterations of asteroids and planetesimals. He has also performed
numerical simulations of the irradiation environment associated with the infant Sun in the early solar system. He has also worked on the mass
spectroscopic analysis of meteorites.
Special Issue on Heat Transfer, February 2011
Manuscript received and revised January 2011, accepted February 2011 Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved
291
Analysis of Temperature Field Inside the Fuel Rod of the VVER 440 Fuel Assembly
Miriama Sa čková
1,2
, Branislav Hatala
1
, Vladimír Ne čas
2
Abstract – The presented work is oriented to analysis of temperature field inside the fuel rod of the VVER 440 fuel assembly. The project is based on overview of physical phenomena, which
influence the thermal conductivity of the gap between fuel and cladding and temperature field inside the fuel rod. By analyzing the physical causes, the most important boundary conditions
were identified that determine fuel temperature values. The work analyses the temperature field inside the fuel rod in dependence on parameters, which have significant impact on the maximum
temperature inside the fuel. The work is focused on the evaluation of the impact of changes of analysed parameters on the maximum temperature inside the fuel. Copyright © 2011 Praise
Worthy Prize S.r.l. - All rights reserved. Keywords:
Fuel Rod, RELAP5, Temperature Field
Nomenclature
α Heat transfer coefficient
[Wm
2
K] α
G
Heat transfer coefficient in the gap [Wm
2
K] δ
G
Thickness of gap [mm]
δ
C
Thickness of cladding [mm]
λ
F
Thermal conductivity of fuel [WmK]
g
1
,g
2
Temperature jump distance terms for fuel andcladding
[-] q
F
´´´ Heat flux density
[Wm
3
] q
C
´´´ Heat flux density in coolant
[Wm
3
] n
Number of a circumferential segment
[-] N
Total number of circumferential segments 8
[-] r
F
Surface roughness of the fuel [m]
r
C
Surface roughness of the cladding [m]
R
F
Radius of fuel pellet [mm]
t
n
width of gap at the midpoint of the n-th circumferential segment
[m] t
G
Circumferentially averaged fuel- cladding gap width
[m] t
o
As-fabricated fuel-cladding gap width
[m] T
S0
Temperature inside the fuel [K]
T
S1
Temperature on the edge of the fuel
[K] T
S2
Temperature on the inner edge of cladding
[K] T
S3
Temperature on the outer edge of cladding
[K] T
ref
Reference coolant temperature [K]
u
F
Radial displacement of the fuel pellet surface
[m] u
C
Radial displacement of cladding inner surface
[m] u
TF
Radial displacement due to thermal expansion
[m] u
r
Radial displacement due to uniform fuel relocation
[m] u
S
Radial displacement due to fission gas induced fuel swelling and
densification [m]
u
TC
Radial displacement due to thermal expansion
[m] u
CC
Radial displacement due to cladding creepdown
[m] u
e
Radial displacement due to elastic deformation
[m]
I. Introduction