illustration of this problem is given in Fig. 1. A long ..... 'Beater W~ll. temPer~tuts and average ..... in Eqn. ~2.l8), notinq "tut u-v-O in' the solid, and introducinq.
.!' ,
•
o
EXPERIMENTAL AND NUMERICAL INVESTIGATION OF MELTING IN THE PRESENCE OF NATURAL CONVBCTION
! t
l
l 1.
i
)
\
AIIboke Bo•• _
., A The.l. S~tted to ~. Paculty of Gra4uate S~u4L••
,1
_ and . . . . .rcb ln, Partlal Fulfl1l.JDent of the Il
•
~ l
,
'
Requir..anta for the Degree of
1
. . .ter of Bnqln. .r1ng '-,
1
1
,,
./ 1
,
--
~
1
~t
of -.cbartlcal BnCZin_ring-
MaG!11 Univer.lty lIont.real, Canada
t;lJuly 1983
1 ,
, ~
\
1
(
\1
90
5.1.1
The Cases Studied o'
5.1. 2
Inner and Outer Tube Temperatures
5.1. 3
Visual and Photographie Observations of the Solid-Liquid Interface
93
'5.1. 4
Solid-Liquid Interface Evolution Plots
98
5.1. 5
Variation of Melt Fraction With Time
90
92 0-
· ··
5.1. 6
Concluding Remarks r
5.2
····
•
102
· · ·· ··· ··
104
NUMERICAL SIMULATION OF NATURAL CONVECTION IN A NARROW-GAP ANNULAR REGION . ,., 5.2.1 Problem Description
· ······ · ······
105
· ······ Computational Details ··· F1uid Flow Results ······· Heat Transfer Results ··· 'Con cl uding Remarks ··· ··
106
~
5.2.2
""
5-.2.3 5.2.4 5.2.5 5.2.6 5.3
PrQb1em Parameters
NUMERICAL,SIMU~TION
" 1
108 110
111
5.3.3
Variation of Me1t Fraction With Time
114
5.3.4
Solid-Liquid Interface Evolution Plots
115
5.3.5
Streamline Plots
5.3.2
(
107
··· ···· Problem Description and Parameters ··· Computational Details · ·· · · · ··
5.3.1
'.
107
OF MELTING IN 'l'HE PRESENCE
OF NATURAL CONVECTION
,,"'
105
J
········• ·
110 0
112
116 '
viii
,'
-PAGE 5.3.6 5.3~7
6.
Beat Transfer Coefficients on the Inner Surface • • • • • . • • • •
Tube
/'-'
.
Conc1uding Remarks
cONCLUSION
. 119
•
.
•
.
.
6.1
REVIEW OF THE THESIS·.
6.2
SUGGESTIONS FOR EXTENSIONS AND IMPROVEMBNTS TBIS'wo~ , • , •
FIGURES
--"
TABL!S
/
.
APPENDIX REFEREt4CES ~r
...
•
. •
.
• • • •
•
~
•
• •
..
•
.." . .
121 121
•
or .
.
124 127
•
•
116
. ...
164
. ..
·1
•
168
..
173
"
•
f
..
,
" J
..
':
0/;'
....~Il'I!.~'* ........'_'...
I .... ,_ .... _-....___ " • __...:."'..,.. ....... HH'O~~_
..
~
_
:~_
'...,
-
__
lx
,
(
LIST 0' PlGURBS ./
,DESCRIPTION
FIGURE , ,
.. 1
Schematic 0/ the two-dimensional-meltinq problem studied in the complementary exper1mental and numerical investigations • • • • • • • • • • 128
2
A typical multiply-connected calculation domain: (a) solid-liquid interfaces; (b) control volume in a single-phase region; (c) control . volume containing a portion of an interface • • • 129 view of a control volume containinq' the solid-liquid interface, and related nomenclature
De~ailed
3
Main-qri~ discretization of the calcqlation ,'"domain: (a) grid and control volumé's; (b)' typical control volume, and associated nomenclature - • • • • • • • • • • • • • • • . Staggered-qrid discretization of the calculation domain: '(a) locations of the u,{+) and v{t) . velocity camponents; (b) a typical x-momentum, .control volume: and (c) a typical y-momentum control volume, and as'sociated nomenclature •
.. 130
131
,
4
5
6 7
Photographs of the_ test cell: (b) side VCew • . : • • • • •
132
(a) fr'ont view;'
..... ..
133
Longitudinal (a) and transve~âe (b) sectional dr'awinqs of the test ce Il • • • • • • • •
134
Longitudinal (a) and transverse (b) sectional drawings of the inner-tube heater assembly •• •
135
~.
" 8 ,?
'\
9
!
L
Positions of the thermocouple-beads inside the inner-tube heater assembly: (a) top and bottom surfaces; (b) left and right sur'faces . • • . •
136
Photographs of (a) the cartridge hèaters and the double-pipe heat exchangers, and (b) the backend of the test cell with ~he cartridge heaters in place • • • • • • • • • • • • • • • •.• .'. •
137
t
,."t__ .
1
'. -......-..._ ...................
~
-
-
~~t Tm) and maintained constant throughout the experiment. As a result, the solid PCM melts, and this
.
melting process is influenced by buoyancy-driven'natural convection in the liquid phase. a
The evolution of the solid-
liquid interface is monitored photographically, and this data is used to calculate the variation of the melt fraction with time. The
aforement~oned
experimental investigation is
limited to melting problems in which the solid phase is maintained at a temperature just below the melting temperature of the PCM.
Problems involving subcooled' solid regions and
,freezing are not within its scope.
The aim of this
experimental investigation is not to study ahy particular engineerinq, problem.
Rather, its goal is to study a
s~ple
proble~ with well-defined boundary and initi~l conditio~s, and obtain reliable and accurate data which can be used a~ checks in ~~e validation and performance evaluation of
nwnerica 1 methods., The development of a computer program incorporating "~he 1
\
•
o'
.
proposed numerical method formed an
part of .
this work.
The progN1l\ 1.8 writ"tt.en ,in the FOR'l'RAN language,
in a fairly
u~er-orien~ed
format.
applied to several test problems. ~~pplication
(
im~rtant
It has been successfully The results of it's
to the two-dimensional melting problem
8chernatically depicted in Fig., 1 were cOIl\pared wi th those
j
,
~
5
of the
afor~tnentioned
experimental" investigation.
The se
compàrisons and their implications are diacussed in this \
,
lA
thesl.s.
~
'1.
. 1.2
SYNOPSIS OF RELATED INVESTIGATIONS Many of the early studies"of heat transfer in
m~terials with solid-liquid phase~change are based on
mathematical analyses, and they are limited tq situation$ , in which conduction is the only mode of heat transfer. Eckert and Drake [9] and Carslaw and Jaeqer [10] have presented analytical solutions to several idealized onedimensional heat conduction problems involving solid-liquid -
phase change1 a particular case is the well-known Stefan problem.
One-dimensional meltinq of' a sami-infinite solid
with a constant heat-flux boundary condition has been studied by Goodman [111 usinq an,approximate series solution and an integral approach.
In qeneral, such an41ytical and
semi-analytical studies are not applicable to the problema of
~nterest
in'tnis thesia.
Detailed information on many
of 'these methods is available in textbooks [9,10], and a review of tbese methods is available in a paper by Muehlbauèr and Sunderland [1]. Many numerical and heat transfer ,in
,
~xperimental
solid-~iquid
.
investigations of
phase change problems have
been repàrted in the p}lblished ,it~rature.
1
..
~
The experimental
1
biU
.~a
4,...,,·. .')1:1"4#$.
lt.
•
/
~--..,
~
__..;....,.~...,,~.. ~~,,_v
• "',., ~ _
.
,
1 .. _.. _
...
_.-".,...
~-4tt~~_tf' -,.....,>1
# ...._ _ " ' ........
,t"""~ ~~'-f\~~f'l!!i'V~If:l)~~~-~'!·· ....
(
6
..
.,
.
,
investiqatîons show clearly'that buoyancy-driven natural
•
convection plays a significant role in a majority of,metting ~d
freezing
phènomen~,
encountered on ,arth.
Nevertheleas,
most of the available DUmerical methods ,for solid-liquid phase change problems are limited to situations -in which conduction is the sole mode of heat transfer.
ft
reviews of numerical
~ethods
Detailéd
for unsteady heat conduction
in materials with or without phase change can be found in [1-3,5,12,13].
Bere, only
..
a
few of these methods, oonsidered
relevant to .this thesis, will be discussed briefly.
A fev
numerical methods for ,phase chànqe problems involving buoyancy-driven natural convection are available in the literature, and they will be
revie~ed
in this section.
In
addition,- a review of some of the important expertmental investigations of these problema will be presented. The
followin~discus.ion
ia
·spli~
in the first par,!:, soma of the avai1able
into two numeric~l
part~:
methods
for sol:i:d': liquid phaSe change prob1ema are reviewed; and
">
/
"
'eXP,8r~enta~
in the second
1.2.1,
investigations of such problemyare surveyed p~.
Numerical Methods Numer ica1 methods for heat tran.fer in phase-change
probleins can be grouped into two cateqori.ea: 'models and enthalpy modela [5,13].
temperature
In nuaerical methoda
based on temperature PIOde1a, aa the n _ iaplie., the
'.
.
"""""'''''''M\IIIIlIl!pIU",;IIIIISIIIII"._IIllM$l1li1.....""'_''''''''''''''"_
......._.. _ _ _ _ _ ...._ _ •_ _ ,
7
.J temperature of the PeM is one of the dependent variables; anerqy conservation .equations are written separately for the solid and
l~quid
phases; and the requirements of continuity of
temperature and energy conservation are used to match the solutions at the interfaces, or phase boundaries.
In
numerical methods based on enthalpy models, the specifie
Ok
enthalpy of the PCM is used as a dependent variable Along with the temperature: the phase boundary position and shape are not explicitly determined, and the problem formulation ,
resembles that of
na~-linear ~eat
transfer without
pha~e
change [5,13]; and afèhough the phase boundary is not explicitly, its
positian_ca~'be
tracke~
determined approximately
by an examnation of the enthalpy distribution. In all numerical methods based on
temperat~re
modela,
'one of the main diffiGUlties, is the handling of the time-
"
v~ing phase boundary, and a n~er of different ~echniques
have been devised to solve it [2,31.
One approach is based
on the so-called quasi-steady assumptian:
over a small
t~e
interval, the phase boundary, which is at the saturation or melting temperature of the PCM, is assumed to be 'fixed i~
space, and the temperature distributions in the liquid
and salid phases are solved; then the phase boundary is relaxed, or adjusted, ta a new position determined by an energy balance condition; and the procedure is repeated • • ,t
until the entire tinte period of interest i . cx>vered. \ Often, a coordinate
tran.formation~.
use4 to fix the phase boundary'
\
!
8
and map the domain on to a regular-shaped region, if required. The quasi-steady approach is a tedious method, especially in multi-dimensional problems:
it usually requires adjust-
ments of the computational grid at each time step, and this, in turn, necessitates the interpolation of the dependent variables.
Furthermore, methods wpich' rely on transformations
to fix the phase boundary become very complicated, or fail completely, when multiply-connected domains are encountered. Examples of the aforementioned quasi-steady approach and transformation. techniques can be found in the works of Kroeger and Ostrach [14J, Spaid et aL. [16], Saitoh [17], Sparrow et aL
[19],
~ieger
[15], Duda et aL.
[18], Ramachandran et al..
[20], and Ettouney and Brown [21].
et aL.
The
methods proposed in [15-17] are finite-difference methods based on the Landau transformation [22], and they are limited to situations in which conduction ls the sole mode The work of Sparrow et aL. [18] can hand~e
of heat transfer.
phase-change problems involving buoyancy-driven natural convection:
it uses a finite difference method based on a
simple transformation technique suitable for singly-connected domains; the fluid flow in the liquid phase is handled by the
~emi-!mplicit-Method-for-~ressure-~inked-!quations
(SIMPLE) [61. comings:
Their method, however, has a number of short-
the computational grid in the transformed plane ,
'
"
is non-orthogonal, but it'is treated as orthogonal by dropping terms related 'to the curvature of the phase bound&ry1
l.
1
.....
9
1
i ,
J
and it is limited to situations in which the solid phase is ~
maintained at the saturation, or melting, temperature.
The
,method proposed by Ramachandran et aL. [19] is patterned after
th~t ~f sp~row
et dL. , but it can solve problems
.
,
with subcooled solid regions._
The work of Reiger et al.
[20J
uses a finite difference,method based on a non-orthogonal boundary-fitted
coordina~e
system [23]; the buoyancy-driven
f1uid flow in the 1iquid phase is solved using a streamfunction-vorticity formulation [6].
They have successfully
,solved a problem invo1vinq melting around a heated horizontal
cir~u1ar
cylinder, but their method is 1imited
to situations in which the ·solid phase is maintained at the me1ting temperature.
Ettouney and Brown [21] have
investigated four Galerkin finite element methods for steady solidification problems; all four of the se methods are 1imited to situations in which conduction is the sole mode of heat transfer. A variation of the Landau ,transformation technique
-
[22] is the Isotherm-Migration-Method
-
Crank et al.
.
[24,25].
-
(r.MM)
developed by
In this method, temperature is
interchanged with a spatial. variable and treated as an independent variable.
The warks in [24,25] are
limite~
to
problems in which the peM properties are constant and conduction ia the only mode of, ' heat transfer. Another technique to handle ~e moving phase boundarY -in ,numerical methods basad on the temperature mOdel i8 to ,
10 model the latent heat effect as a large
(.
the~al
capaci~y
over a small temperature range, which includes the saturatton temperaturè, and smooth out the values of thermal' conductivity and specifie heat.
This technique has been used in:the
numerical methods proposed by Bonacina and Dilpare [27], Comini et al. and Gartling [30].
e~
al.
[26], Talwar
[28], Morgan et aL
[291,
Of these methods, only that of Gartling
is capable of solving buoyancy-driven naturai convection in the liquid phase; the others are limited to situations in which conduction is the sole mode of heat transfer.
A
1
basic limitation of the se methods is that they can only
1
handle problerns in which the phase-change takes place over a smaii but fini te temperature interval; they are incapable
J.
of solving problems with a discrete, or singie-valued, phase change ternperature. A control-volume finite-difference scheme basedo on the tempe rature model has been proposed by Hsu, Sparrow, and Patankar [311 for the solution of moving boundary problems.
In their,method, the moving
~dary
is
immobilized only by a non-orthogonal coordinate transformation. The quasi-steady assumption is not involved.
Therefore,
with respect' to a given control volume in the new coordinate • system, mass appears to pass through the control surface which bounds the volume, and this mase movement brings about a convection-like transport of energy. equation
~or
The energy
a moving, non-orthogonal control volume 18
>,
Il Q
derived in general and'then specialized to the transformed coordinate system.
A
fully-tmplicit~
upwind-type [6],
finite difference scheme is used to discretize this energy equation.
The method has been successfully applied only
to a/two-~imensional phase change problem in which conduction is the sole mode of heat transfer [32].
Another limitation
of this method is that tt has" been developed for singlyconnected domains, and its extension to multiply-connected domains is not straightforward.
Nevertheless, an extension
of this method to phase change problems involving natural convection in the melt would be a useful undertaking. Numerical methods based on enthalpy models have been proposed by Shamsundar and Sparrow [4,5], Crowley [33], Ronel and Baliga [13,34], and Rolph and Bathe [35].
In the
methods proposed in {4,s,33], finite difference approximations of the governing equations are solved by iteratlve procedures akin to the Gauss-Seidel technique [6]. solution procedures, however, are limited to
These or
fre~zing,
solidification, problems in which the entire liquid phase is initially in a saturated state.
In a subsequent paper,
Shamsundar and Srinivasan [36] have
ext~nded
[4] to account for superheat effects: èondu~ion,
the method in
the single phase
.
or natura1 convection, problem in the super-
heated liquid phase is solved using a suitable method, and the time required for the initiation of solidification is determined:
at this stage, aIl re.idual superheat effec••
1 l
:l
•
ij
,
'
l~
12 ..,
(
...
are ignored, ànd the phase chanqe'problem is solved usinq the method described in [4].
The method in [36] is not
~
. applicable to melting problems, and it could get quite inaccurate in freezing
pro~ems
c~oli~~~ates ~re
and high
if significant superheat
involved.
The methods proposed
\
in [34,35] are extensions of the method in 14] in that they are capable of sol ving ~
\
m~l ting
and freezing pr.oblems wi,th
superheated.liquid regions and subcooled solid regions. In
add~tion,
the methods in [34,35] are based on finite 1
element'formulations, so they are
well-suite~
for the
solution of problems vith irregular domain shapes.
An
enthalpy method which yields more accurate solutions of moving boundary problems than the aforementioned enthalpy methods has been proposed by Voller and Cross [37].
Their
method, however, has only been applied to one-dimensional problems, and its extension to multi-dimensional problems . seems quite difficult. As of the
a~oncluding
afor~entioned
note, it should he noted that all
enthalpy methods are limited to
situations in which conduction ia the only mode of heat transfer.
To the best knowledge of the author, none of the
enthalpy methoda in the published literature have been applied to phase change problems involving buoyancy-driven natural convection in the liquid phase.
The,method proposed in
,
~his
thesis represents an effort to overcome this limitation
of available enthalpy methods.
\
"
13 1. 2.2
Experimental Investigations ,"
1
A
r~view
of the literature shows man y experimental
ïnvestigations of solid-liquid phase change phenomena involving natural convection in the liquid phase.
One of
the earliest of such studies under controlled laboratory conditions seems to be that of Boger and Westwater [38]. They studied the unidirectional solidification of a column of water under the influence of Bénard convection, but a generalization of their results was complicated by the density inversion of water
~t
4°C. 'Some of the other early
experimental investigations were motivated by a need to understand
prob~ems encounte~ed
in metallurgical engineering.
Szekely and Chhabra [39] considered the effect of laminar
•
natural convection on the controlled solidification of lead,.
j
Prates et al. [40] investigated heat flow parameters affecting the unidirectional solidification of
~e
metals.
Natural convection heat transfer rates during the solidification and melting of metals and alloy systems have been experimentallyinvestigated by Chiesa and Guthrie [41]. , Over the ,last decade, however, the motivation behind a majority of such experimental investigations has!been their relevance and applicability to thermal energy storage systems, solar energy systems, and thermal control devices used in spacecrafts and defense energy systems.
These
investigations are reviewed in the remainder of this sub(
l
section.
14
Experiments on the raIe of natural convection in the melting of solids have been performed by Sparrow, Schmidt and Ramsey [42].
In their experiments, a eutectic
,
mixture of sodium nitrate and sodium hydroxide was used as the PCM.
1
The test chamber was a cubical mild-steel tank
l
t
fitted with a horizontal heating cylinder of circular crQss-section.
1
The PCM in its solid state was initially
brought to its fusion temperature by external heating.
Then
1
the horizontal heating cylinder embedded in the solid was
1
,
energized to give a steady rate of heat transfer.
The
1
cylinder was instrumented with thermocouples, and, in addition, an array of 92 thermocouPle~ was deployed throughout the PCM.
"
1
The cylinder-mounted thermocouples facilitated
the evaluation of instantaneous heat transfer coefficients, ---....
while those in the PCM enabled the position of the solidliquid
inter~cè~o
he identified •
./
'
//
The results of Sparrow, Schmidt and Ramsey [42]' showed that meltlng primarily occurred above the heated ~_.....-
cylinder, with
ve~
little melting below.
.
These and other
results conveyèa'strong evidence of thè dominant role played by natqral convection in the melting of a solid due to an embedded heat source.
~portant
and interesting qualitat1ve
information on the melting procas. ha. been provided by 1
these results, but they ar~ of little use as quantitative checks on the results of numerical methods.
This is becauae
complete and precise boundary éondition information has not
.1
"
15
been provided, the heat
ca~acity
of the heating
cyli~qer
has not been adequately accounted for in sorne of the results, and reliable property values are not available for the
'0
sodïum nitrate-sodium hydroxide eutectic used as the PCM. " Melting experirnents similar ta those in [42] have
also been perforrned by Goldstein and Ramsey [43].
They
used naphthalene as the PCM, and the emphasis was on determining the liquid-solid interface and the local heat transfer
\
rate at,this interface.
Their apparatus permitted a
continuous visual observation of the melting process. Photographs of the molten region were used to evaluatè the local progression of the
solid~liquid
interface and
thus the local heat transfer rates at the interface.
1
In
addition', the horizontal heating cylinder contained 'thermo.couples which allowed the measurement of
time~varyinq
wall temperatures and heat transfer coeffi'cients. \
heater
In all
experiments, the temperature of the solid was maintained
-
within O.2°C of the meltinq temperature of naphthalene.
1
Their results are qualitatively similar to thpse reported in [42]. ,.
'
The
ia doubtful:
"-
~titative
accuracy of the resulta, however,
different melt shapes were often obaerved
in similar tests, and there is no clear indication of wbat
-\
triqgered these discrepanciès. Bathelt, Viskanta and Leidenfrost [441 and Bathelt ~
and Viskanta [451 have also inveatiqated natural convection in the melted G
~eqion
around à heated horizontal cylinder. •
..
j 1
1
1
16
The PCM's used in these experiments were n-octadecane and n-heptadecane.
f1
In [44], the shape of the solid-liquid
,
interface was measured photographically, and local heat transfer coefficients on the h~ated cylinder were measured using a shadowgraph technique.
Beat transfer at the solid-
liquid interface was the focus ,of the study in [45]:
the
instantaneous shape of the melt volume was recorded photot
.
graphically and local heat trarléfer coefficients at the sOlid-liquid interface were determined from those solid PCM's were maintained just below the in both investigations.
da~a.
The,
fusio~perature
These studies have provi~d
-important information on the fundamental heat transfer processes active in melting problems •. K~ink
and Sparrow [46] have experimentally studie?
melting about a vertical cylinder with ana without subcooling of the solid phase and for open and closed containment. ,The PCM used was 99' pure n-eicosane.
Great care was taken
in designing the experimental apparatus and performing the experiment.
As a result, their data seems accurate and
detailed enough to serve as quantitative. checks for \
nume~c~predictions. 'Beater W~ll. temPer~tuts and average ~
heat transfer coefficients are presented as f
ctions of 1 \
'time in [46].
Other
exper~ntal
1
studies of melting lin the presence
of natural'convection lnclude the works of.~rey, Sparrow (
and Varejao [47], Bathelt, Viskanta and
Leiden~rost
.\ 1
..
\1
[48],
17
G,
Hale and Viskanta [49], and Van Buren and Viskanta [50].
In j"
[47], melting about a horizontal row of heated cylinders ••
1
was investigated, and rnelting from an array of three ,
staggered, ele~trically heated, horizOntal cylinders was , . studied in [48]. Photographie observations and ~nter
..
i
1
ferometric measurernents of solid-liquid interface motion and heat transfer during rnelting from a vertical surface are
1
1
reported in [49,50].
Melting in a finned phase-change
energy storage device has been studied by Henze and Humphrey [51].
1
du~ing
Heat transfer ana interface motion
melting and solidification around a finned heat source and ~
sink have been ipvestigated by Bathelt and Viskanta [52]. Experimental studies of freezing in the presence of natural convection have been conducted by Sparrow, • J
Ramsey and Kémink [53], Sparrow, Ramsey and Harris [54], and Van Buren and Viskanta [55].
Freezing under conditions
~here the liquid phase is superhea~ed was studied in [53J; ,
'
the liquid was housed in a vertical cylindrical cnntainrnent vessel, and the freezing occurred on a cooled vertical t~be
positioned along the axis of the containment vessel:
the variations of freezing patterns and frozen mass with time are presented.
The work reported in [54] involved an
experimental setup similar to that in [53]
1
but attention
was focused on the transition from natural-convectioncontrolled freezing to conduction-controlled freezing. '.
[55], interferornetric observations of natural convection
,
In
1 J
!
i
18.
c
,
during freezing from a vertical f1at plate are presented. In a11 of the aforementioned experiments on freezing, it was clear1y evident that natural convection slows down the freezing process.
This is in marked contra st to me1ting
problems in which natural convection augmepts the me1ting rate.
~.
In the absence of liquid superheating, conduction 1
was the sole mode of heat transfer, but the experimental1y measured freezing rates were higher than those predicted ,
numeric~11y, because of the presencê of whi~ker-1ike 1
dendrites on the freezing surface' [54]. The experimenta1 work reported in this thesis complements and extendS investigations.
th~ aforementioned published
As was mentioned ear1ier, its main objective
is to obtain complete and reliable data which can be used \
as quantitative'checks on numerica1 predictions. ' Furthermore, the geometrica1 configuration, used in this "experiment is, different from those studied previous1y, and it, is very , weIl suited w~ich
~o
the testing of finite difference methods
use formulations ,based on the Cartesian coordinate
system.
/ 1. 3
J
SURVEY OF THE THESIS
There are a total of six chapters in thi. thesis. The aims of the thesis and a survey of related investigations have already been presented/ in this chapter.
The organizatlon
of ithe other five chapters is summarized in the follovlnq
,u ';,'
'.
-
19
\
paragraph. In Chapter II, the mathèmatical mqdels of the ~hase change problems considered in this thesis are
p~esented,
and the equivalence of temperature-based and 'entha~py based formulations ds demonstrated.
The complète formulation
of the proposed' control-volume based
finLt~
,method is presented in Chapter III. considerations which
w~nt
différence
In Chapter IV, the
into the design and construction
of the experimental apparatus are 'discussed, and descriptions 1
of the supporting equipment, instrumentation, and
calibrat~on
and
~esti~
procedu;es are
p~e~en\t.d.
The
results of the experimental and numerical ~nvesti9ations ft
are presented and discussed in Chaptèr V.
In Chapter VI,
the concluding chapter of this thesis, the main contributions ,
(j,
of this thesis are reviewed, and, some sU9gestions for the extension and improvement of
thi~
work are preaented.
r
, \.
.
20
c
CRAP'rBR 2I
.. 2.1
PROBLtM S'tATEMBRT AND ASSOMPTIONS
The mathematical formulation of two-dimensional· - .. "
solid-lihuid phase-change problems with or without ~uoyancy-driven natural convection in the~liquid 'reqions is
the subject of this chapter.
An example of- snch problems
is depiçted schematically in Fig. 2a.
The calculation
1
dQmain could be 'regular or irregular in s,hape, and it could be sinqly- or multiply-connected.
.-
The problems of
interest could involve melting and·freezinq with superheated
1
, liquid ragions and subcooled tplid ragions. ' In the proposee! mathematical model of the problems of interest, the fOllov!nq assumptiona are made:
(a) equi-
librium melting and solidification with a smooth
interface~
(b) density difference between the liquid and is negligible;
(c) the
li~id
so~id
phases
phase is an incompressible
Newtonian fluid~ an~ (d) thermopby.~cal prôperties are constant i~ each phase,'except for thr denSi~y in the buoyancy term
(Bou•• iDeaq condition).
The validity of th...e asSulnpt.ions
is briefly.discu. . .d in the fOllowinq paragraphe Supercoolin~, or . .tastable ~tates, are po.8ihle~
durinq the freezinq of SOlDe PCM' s
[54], but such
are not camaonly encountered durinq m.elting.
-
-
- - - -"---
1_ _-
pr~bleJlls
Purthermore,
-.
'''''''''''!II!a""6II1II1""""_ _ 0 ... '*_Lç_.............II
~iffusion
Thus the velocity components u and v
could becobtained by interpreting the yariable
~
~
r -
l.l
as
gene~al ~ependent
"
one of these velocity components, and setting ~
~
and S - (Su - dP/ÔX) or (Sv -ôp/ôy).
In this sense,
a calculation procedure for '"flûid flow has already been o
described.
Of course, the moni9ntum equatiqps are non-linear
,, ; , (
46· r
and coupled, but these complications can be handled by i teration.
c
The pressure
f~eld,
and none of the Eqns.
." .
however, is not known a
pr~or~,
(3.26)-(3.28) allows the explicit
determination of pressure.
It
~s
specified indirectly:
a
correct pressure fièld, when substituted into the momentum equations, yields a velocity field which also satisfies the continuity equation.
This indirect
s~cification
of
pressure is computationally inconvenient; a direct method for determining pressure 1S des1rable. Furthermore, if the velocity components and the pressure are calculated for the same grid points, physically unrealistic checkerboard-type velocity and pressure fields 1
may result/io].
One way to avoid these difficulties is the
so-called staggered grid method 16], which is adopted in this thesis.
3.2.3
The staggered grid is described next.
The Staggered Grid Figure 4a shows a two-dimensional stagqered grid.
.The velocity components are shawn by short arcows: and t for v.
.J...
.. for u
In.the staggered grid, the velocity components
are computed at points lying on the main-grid control voluaae faces, shown by dashed linesi the pressure and aIl other variables are stored and calculated at the main grid points';; - Thus the grid for u is
dis~aced.~
the x direction vith
respect to the main grid, and the grid for v is diaplaced in
.
' ;
47
... the y _irection.
,
l /
Consequently, the control volumes for u
ahd v are staggered in the x and y directions, reapectiyely, as shown in Fig. 4b.
Furthermore, the pressure difference
.
betweên two adJacent grid points drives the velocity
\
"
component at the main-grid control-volume face .between them.
3.2.4
Momentum Riscretization Equations \
The appropriate control volumes for u and v are shown i.n Fig. '4b.
The discretization equations for
the$~
velocity components are obtaineQ by first integrating the appropriate momentum equations over the corresponding control volumes, and then deriving algebraic approximations to these 1ntegral momentum conservation equations.
In these
derivations, the unsteady terms and the convection-diffusion flux terms are
h~dled
by the procedures described in the
last section, except that the staqgered geometry of the control volumes must be taken intcJaccount in determining the diffusion conductances 0 and the flow rates F [6]. Integration of the pressure gradient (-
~/ax)
Qver the
control volume for ue ' shawn in Fig. 4b, gives:.
J . VUe
"-
"
1 (3.29) \.
•Similarly 1 the int8C)ration of the. pre.aure gradient (- -ap/~y). over the control vol,.. for "n' .bovn in.Pi9. 4b, givea:
..~.
48
•
J
Vv
, n
- .
-1
-
Pu.. . . +
p.""'.-':'" P,."a"",
(l ••5)
j
,1 1 "
I
-....
. .locLty
It ajlou14 lie DOted tbat vU:b & 91. . . , " ~
'J
4
.'
....
\
..
.
'
, #'
1
,j
·51 1
fièld, the p.eudovelocitie. can be computed via equation.
(
such as Eqna. (3.35) and (3.36), and then the corre.ponding pr.ssure ,
." ,
fiel~
cau be ,obtained by solving the set of
equations repreaented by Eqn.
(3.39)~
.
Pr••• ure-Correction·Bquation
3.2.7
Starting vith a gu••••d velocity
fi.~d,
the
corresponding pressure field can be calculated a. de.cribed in the la.t ••ction: l.t thia càlculat8d pr•••ure field be denoted by
p'.
With thi. calculated pres.ure fi~ld,'P'~ it
is possible to sol. va the --.ntum equations to obtain a
new velocity f~eld denoted by u!, v!, etc.
.uu·
...
• e'
-
This impli••
tanbu U ' + bU + nb
À• (~-p') !!
(3.46'
t~y!t, +
~(~-~)
(3.47)
and \
:
.;
.~y' ft ft
•
b
Y
+
t'be calculated velociti•• , u' ~ y', vill not, iD 9. . .81.
Atiafy the contaulty equatiOD.
To _aure
~t they
d:O
aatiaty continuity, th.y are correcteci via • pre.aur. correction p' applied to the ~iculated pra.aur. p'.' ft_
.
,
p
,
'1
-
p'- + p'
.
"
.. (3.48)
1
)
"
52 and, ae •
(
con~.,-
..
\
1
u
.
•
u' + u'
,(3.49) b
and
~
.... -~
'/'? /
V
,
• v' + v'
(3.50)
,
Equation. (3.31) , , (3.32) , (3.48) , (3.49) &Def (3. '0) can be uaed te obtain v.loci ty-correction foraul.. [6].
1l \
\\
Tb. . .
1
formul_ can he .iJDpiified if the ter-- ta~u:m and
~ j
f,,
ta~v:m are, caas:l.derild 'neqliqible, 'as they vi11 be vbeIl a
\
,
conver9ed aolution Jla obtaine4.
'1'be
,
~
res~ltin9
ve10city
correction, foraul..e are z
\,
•
u•
"
.
"
~
'.
u! + 4:,pP....,.)
(3.51)
.,
and
\
J
•
"n
.: + ,s:(,PP~)
. (3.52)
1
ne
1
au..,tlariqo bebM_ t.be_ eqaatiOlUl aD4 BqIUI. (3.33) anel
(3.34) aboulelJ be not:ecJ at thia ataeJe.
If foJ:1aUlae a11cb
.s eqna. (3.51) and (3.52) are aubatituted into the diacretiaed continuity equ,atiorl, Bqn. (3.38), an equation
.
~
for p' reaulta. ; 1
'fbe dari vation ia very aiailar to that of
the pre.sure equation [6].
'l'he presllUft-con:ect.ion equation
.'
, " \
•
••
t
;iQil(:pl~...........
,
S3
can he coepactly
t'
.~re.Hd
/
••
'~
-
(3.53)
Ar. 9i".
where the coefficient. ~, ~, ... ' as' and -Ap Eqna. (3.40)-(3_44). The te~ bP '• ia defined by
by ,
\
\
~.
\
(3.54'
The term b P,' repre.ent. a -ma.a-souree- pres.nt in the ~lculated
velocity field.
The task
o~
the pressure-
correction is to correct the calculated velocity field and remove thi. apparent -ma.s-source- [6].
3.2.8
B
The SIMPLER Procedure At each time step, the following iterative procedure,
called SIMPLER by
Pat~nka~
[61, is uaed to solve the
fluid flow problem. 1. Gue.s a velocity field, the known velocity field
at the previou. time, or the initial velocity field at the \
.tart , constitutes a qood guess.
2. Calculate the coefficients in the discretized momentum equations and ~alculate the pse~dovelocit~es û and u.inq
equa-~ion.
similar to Eqns. (3.35) and (3.36).
3. Calculate the coefficients
" "
in
the pressure
V,
1
equation. an~ ~lve th. . to obtain p'. ,4. u•• the calculated pre ••ure p' -!n the aa.Dt~
,-
equati9D., such a. Eqn. (3.31)
(3.32), to obta4n
• calculated -,ve10city field u' 5. Calcul.te the -mas.
in the pre••ure
correction equation, and sol ~ 6. U8in9 the p' field; correct the calculated
" (3.5~).
velocity field via equations 8uch as Bqns. (3."51' and Do
.
not correct" the pressure • ,
7. Solve the discretization equations for other .'s, if necesBary. 8. Return to step 2 vith the èorrected velocity
field and repeat the procedure until convergence. The rec~nded
followi~q
in
set:of under-relaxatioD par_ter.
this procedure:
and
Note that the pressure-correction
~ation
not satiàfy continuity requirement8 [-6].
(,
(3.55)
• O.)
must Dot be
under-relaxed, otherwise the èorrected velocity' field
,-
u.
w~ll
..
"
.1
55
3.3 '
Tbe aolution metboda for aingle-pbase probl. . . aeacribed in the laat ~\ sections are u . .d in thia section t.b
conatruct a Duaerical metbod for the' aolution of aolid-
liquJ,.d pba. . change . . probl. . ,.
\ Daaain D1acretization
3.3.1
All dependent variablea other tban the velocity coaponent~,
u and V, are stored and
~ut.d
vith reapect
ta the . . in grid pointa and control volu.ea, depictec1. in
Fig. 3.
~nent.
The u and v velocity
.
are coaaputed vith
respect to the ataggerec1 grida and control VOlUlll88
-
illustratec1 in Fig. 4.
Aa
tM..
. . qrids are .'
fixed
the propoaed metbod, tbe aolid-liquid
interfa~
uaed in i.
approxiJaated by a piecewise-linear curve made up of controlvol~ ~ac:ea.
l
,
An- exa.ple of thia approximation. ia 4epictec1
in! Pige 22.•
1 /
3.3.2
\
,Diaœet1a.tion !g,U!t1ou Pen.tub, te
..
!UoYUcY"':' _
Driven l'luid Flow iD the Lime! l!!9ion •
\'1
, ~ - The fluid flow iri the liquid ragian ia gov.nad by
•
Bqna. (2 .. 3) .1. (2.5)
~
•
they are the ,x- and y-mo_ntœa equations
and the continuity equation.
The .y-manentum equation bas
a bucyancy-force tera wbicb centains the fluid teaperature. field, or the latest t.-perature
•
1
< Q,' ~he node j "and the PCM
are iÏl the
-the Dode j ia at the saturation
a
.
1
.
~
solid r~ion: when 0 ~
e
~ l,
t"pera~~~ .Tm and the value \
.
-
1, j
59
of
e
represents the fraction of the PCM in the control volume
e
which is in the liquid state; wh en the peM in its
con~rol
volume are in
> l, ttre node j and
liquid region .
th~
. Discretization of the Enthalpy Equation
3.3.4
The discretization of Eqn.. (3.61) is presented in this section.
The first term in this equation is the ,
unsteady, or accumulation term, and it is discretized as fol19 WS :
e.-8.o '\,
=
p
(J
lit
J)V
(3.63)
j
The second and third terms in Eqn. (3.61) are similar to the convection and diffusion transport terms in, the general
1
control-volume convection-diffusion equation, Eqn.
1
~
(3.5)'.
a
They are discretized using procedures analogous to those described in Section 3.1.4. The final discretized forro of the enthalpy equ4tion for a typical node j can be ,cast in the fOll
