Scale/Analytical Analyses of Freezing and Convective Melting with Internal Heat Generation Page: 3 of 6
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thermochromic liquid crystals to visualize the motion of the
convection cells. The temperature profiles show that the high
temperature areas within the cell occur at the apexes. An
analytical study for natural convection in a cavity of different
aspect ratios with a uniform volumetric heat generation was
performed by Joshi et al. [6]. They found that the horizontal
component of the velocity is smaller than the vertical
component near the center and walls of the cavity. They
compared their results to computational results showing good
agreement. Approximate solutions of phase change with natural
convection were studied by Tien and Yen [7]. They found good
comparisons of their analytical solutions to numerical ones for
water-ice systems.
The scale analysis technique introduced by Bejan [8]
is used in this paper to analyze phase change and natural
convection with internal heat generation. The scale analysis
method evaluates order-of-magnitude estimates for the
behavior for physical systems. This method has also been used
and described by Astarita [9] to gain significant physical insight
to various problems. Zhang and Bejan [10] used scale analysis
to study time-dependent natural convection melting with
conduction in the solid. They found that in the conduction
dominated regime, the Nusselt number increases as the solid
subcooling parameter increases. Scale analysis was also used to
study turbulent heat transfer driven by buoyancy in a porous
layer with homogeneous heat sources by Kim and Kim [11].
They found a critical Rayleigh number for the onset of natural
convection.
In this paper, we model natural convection driven by
internal heat generation. This is an extension to numerical work
published previously [12].
The cylindrical geometry is given in Figure 1. Because
of the internal heat generation of the material, the center portion
of the cylinder melts first, and we assumed that the internal heat
generation is the same in both the liquid and solid phases. We
also assume that the material is pure, so that the material has a
single melting temperature, Tm, and there is no mushy zone. We
have performed scale analysis for both constant temperature
and constant surface heat flux boundary conditions along the
surface of the cylinder since both boundary conditions are
experienced based on the nuclear reactor.
z
~ liquidq,
T,
MELTING WITH CONSTANT HEAT FLUX BOUNDARY
CONDITIONS
We begin the scale analysis of melting in a cylindrical system
with internal heat generation and constant heat flux boundary
conditions using the conservation of mass, momentum and
energy equations [13],On Ow0
-+-= 0
Or Oz
Cw Cw (2w 1 owN 020
u-+w-= -gAT +v +--+
Or Oz yr2 rCr Z
ST ST q (52T 1 ST 92T
U+W-= pc + r2 r8r 8z2(1)
(2)
(3)Note that in this analysis, the continuity and momentum
equations are not used and the analysis solely relies on the
conduction equation and the assumed temperature profile in the
liquid region. Also the one dimensional analysis is performed
and origin of the z axis is not shown in Figure 1. We apply the
constant heat flux condition at the surface of the cylinder,
where r = ro,dT
- dr(4)
At the interface between the solid and liquid regimes, an energy
balance shows [14],dIT ds dTf
k ' + pAh - k s
dr dt d(5)
Equation (5) can be used to solve for the time rate of change of
the location of the solid-liquid phase front. In order to do so,
the temperature gradients in both the liquid and solid phases
must be determined. This is done by starting with the heat
diffusion equation in cylindrical coordinates, applied in the
solid phase, assuming a constant thermal conductivity and
conduction in the radial direction gives,-k drT +q =0
r s dr dr(6)
After integrating Eq. (6) twice and solving for the temperature,
we find,solid
To or q"T(r)= r +c, lnr+c2
2k, 2(7)
with boundary conditions given by Eq. (4) and, T(r = s) = Tm,
where Tm is the melting temperature of the solid.
After applying the two boundary conditions, we can solve for
the two constants of integration to get,Figure 1. Schematic diagram of phase change in a cylinder with
volumetric heat generation2
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Siahpush, Ali S.; Crepeau, John & Sabharwall, Piyush. Scale/Analytical Analyses of Freezing and Convective Melting with Internal Heat Generation, article, July 1, 2013; Idaho Falls, Idaho. (https://digital.library.unt.edu/ark:/67531/metadc839245/m1/3/: accessed April 25, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.