Excerpt
Contents
Introduction: ... 3
Phase-change: ... 3
Isotherm: ... 5
Stefan Problems: ... 6
Methods of Solution: ... 6
Melting Ice: ... 7
Isotherm Migration Equations: ... 8
Conclusion: ... 10
Acknowledgement: ... 10
References: ... 10
2
Introduction:
This article is aimed at presenting the definition and mathematical discussion of some
important thermodynamic terms such as phase change, isotherm and isotherm migration
method etc. It overviews some theoretical literature on these topics and presents their
interpretation. Phase-change, Isotherm, Stefan Problems, and Isotherm Migration method
are described with graphical or mathematical representation to clarify the concepts so that
one can get brief idea about them.
Phase-change:
The term phase transition or phase change is most commonly used to describe transitions
between solid, liquid and gaseous states of matter, and, in rare cases, plasma. A phase of a
thermodynamic system and the states of matter has uniform physical properties. During a
phase transition of a given medium certain properties of the medium change, often
discontinuously, as a result of the change of some external condition, such as temperature,
pressure, or others. For example, a liquid may become gas upon heating to the boiling point,
resulting in an abrupt change in volume. The measurement of the external conditions at
which the transformation occurs is termed the phase transition. Phase transitions are
common in nature and used today in many technologies.
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https://en.wikipedia.org/wiki/Phase_transition#/media/File:Phase_change_-_en.svg
https://upload.wikimedia.org/wikipedia/commons/3/34/Phase-diag2.svg
4
During a change in state the heat energy is used to change the bonding between the
molecules. In the case of melting, added energy is used to break the bonds between the
molecules. In the case of freezing, energy is subtracted as the molecules bond to one
another. These energy exchanges are not changes in kinetic energy. They are changes in
bonding energy between the molecules.
http://www.aplusphysics.com/courses/honors/thermo/phase_changes.html
If heat is coming into a substance during a phase change, then this energy is used to break
the bonds between the molecules of the substance. The example we will use here is ice
melting into water. Immediately after the molecular bonds in the ice are broken the
molecules are moving (vibrating) at the same average speed as before, so their average
kinetic energy remains the same, and, thus, their Kelvin temperature remains the same.
Isotherm:
The term Isotherm is derived from the Greek words,
isos
and
therm
, the former meaning
the equal and the latter heat.
Isotherm
thus means keeping
the same temperature at a
given time or on average over a given period.
5
Stefan Problems:
In mathematics and its applications, particularly to phase transitions in matter, a Stefan
problem is a particular kind of boundary value problem for a partial differential
equation (PDE), adapted to the case in which a phase boundary can move with time.
The classical Stefan problem aims to describe the temperature distribution in
a homogeneous medium undergoing a phase change, for example ice passing to water: this
is accomplished by solving the heat equation imposing the initial temperature distribution
on the whole medium, and a particular boundary condition, the Stefan condition, on the
evolving boundary between its two phases. Note that this evolving boundary is an
unknown surface: hence, Stefan problems are examples of free boundary problems.
A common-place physical phenomenon is the melting of a block of ice by raising its surface
to a temperature above 0 degree C. The two phases, water and ice, are separated by a
boundary on which melting occurs at 0 degree C and which moves further into the ice as
time progresses. In the mathematical treatment, the motion of the boundary has to be
determined and the usual equations of heat flow solved in the water and the ice. The
solutions and the boundary movement are dependent on each other. The melting of ice is
just one example of a whole class of problems commonly referred to as Stefan Problems.
They include the propagation of phase changes in metals diffusion with absorption and
processes controlled by discontinuous diffusion coefficients.
Methods of Solution:
In some cases analytical solutions can be obtained, according to Carslaw and Jaeger [1].
Several numerical methods, all based on finite-difference replacements of the original
partial differential equation, differ in the way they cope with the movement of the
boundary. As usual, in a one dimensional problem the region is covered by a grid of equally
spaced lines. The various numerical methods have really explored all possible ways of using
the grid. Special finite-difference formulas based on Lagrangian interpolation formula for
unequal intervals have been used in the neighborhood of the boundary when it falls
between two grid lines by Crank [2]. Unequal time intervals have been used, calculated so
that the boundary moves always from one grid time to the next in one time step by [3]. The
grid has been deformed so that the number of space intervals between the outer surface
and the moving boundary remains constant, with suitable transformation of the basis
equation by Murray and Landis [4].
Another method employs an apparent specific heat modified to include the latent heat in
the appropriate region, according to Albasiny [5]. Finally, the whole grid has been moved
with the velocity of the moving boundary in a method incorporating interpolating splines by
Crank and Gupta [6]. Recently a novel way of handling heat flow problems has been
proposed which is especially useful for tracking a moving boundary that occurs at a fixed
6
temperature. In the usual heat flow equation in one dimension the temperature is
expressed as a function of the independent space variable x, and time t i.e. u =u(x, t).
An alternative, however, is to seek a solution in which x is expressed as a function of u and t
i.e. x =x (u, t) so that x becomes the dependent variable. We calculate the positions of a
given temperature, which is of an isotherm at known times. Hence, the method is known as
the Isotherm Migration Method (IMM) [7]. Philip [8] dealt with a problem in concentration
dependent diffusion by making concentration an independent variable but he did not
transform the diffusion equation in the same way as Dix and Cizek. Rose [9] derived a
related transformed equation but did not develop a numerical method. The idea of tracking
the moving isotherms in media with phase changes was published by Chernoua'ko [10] in
1969, though the English version appeared only in 1970 [11].
Melting Ice:
Suppose a plane sheet of ice initially occupies the region 0 x a and is being melted by the
application of a constant temperature,
0
, on the surface x = 0. At any time, t, let the
moving boundary separating the water from the ice be at
0
(). The region then consists of
water with specific heat,
0
0
() density and thermal conductivity denoted by , C
and K respectively. The temperature, u, of the water satisfies the heat flow equation
=
2
2
(1)
Where
= , the heat diffusivity.
We take the ice to be initially at
0 throughout. Otherwise we should have an equation
similar to (1) for the temperature in the ice phase and containing appropriate heat
parameters.
At the melting boundary,
0
(), the heat flowing per unit area from the water into the ice in
a short time,
, is -(
). If the boundary moves a distance
0
in time
, the heat
required to melt the mass
0
of ice per unit area is
0
where L is the latent heat of
fusion for ice.
Equating these two amounts of heat and proceeding to the limit
=0, we see that the
first condition to be satisfied on the boundary is
0
=-
(2)
7
A second condition, since ice melts at
0 is
=0,=
0
,0 (3)
Commonly used variables
=
, =
2
,
0
=
0
, =
(4)
Lead to the following system of equations
=
2
2
, 0<<
0
, 0 (5)
0
=-
, =
0
, 0 (6)
=0,=
0
,0 (7)
=
0
,=0,0 (8)
=0, 0<<1, =0 (9)
Isotherm Migration Equations:
We wish to re-write the heat flow equation (5) so that X is expressed as a function of u and
T. Since the temperature u is constant along an isotherm, we have-
=
+
=0
=>
=-
(
)(
)
=-
(10)
Substituting (5) in (10) and dropping suffices we obtain
=-
2
2
(11)
Now
2
2
=
-1
=-
2
2
(
)
-3
(12)
8
Therefore,
=(
)
-2
2
2
(13)
This equation represents X as a function of u and T.
The other equations (6) - (8) become
0
=-(
)
-1
, =0,0 (14)
=0,=
0
,0 (15)
=
0
,=0,0 (16)
We can approximate the derivatives in (13) by finite difference, in; the usual way and
obtain an explicit expression for
+1
, the value of X at
= , =(+1) in terms
of values already available at
(,).
We find
+1
=
+4
+1
-2
+
-1
-1
-
+1
2
(17)
The corresponding finite-difference replacement of (14) is
(
0
)
+1
=(
0
)
-
0
-
1
(18)
A rigorous analysis of the stability of the non-linear finite difference scheme has not been
attempted. Dix and Cizek [7] consider instead the coefficient of
in (14). In order that the
isotherms should move with time in the manner expected
+1
should increase with
i.e.
the coefficient of
must be positive. This leads to the criterion
<1/8(
-1
-
+1
)
2
(19)
Which allows
to change as the solution proceeds. The truncation error [7] of the IMM is
proportional to
and ()
2
.
9
Conclusion:
Phase change, isotherm and isotherm migration method are very important terms to learn
in thermodynamics. Therefore, I have tried to give a short overview on the Phase change,
isotherm and isotherm migration method. From this article one can get brief idea about the
phase change and isotherm migration method.
Acknowledgement:
From the core of my heart I would like to convey my earnest admiration, loyalty and
reverence to the almighty Allah, the most merciful, for keeping everything in order and
enabling me to complete this article successfully. Then let me express my profound
gratitude to my tutor for his inspiration. It would be impossible for me to finish this article
without his enthusiasm and motivation. It has also been a pleasure to have worked with
him.
References:
[1] Carslaw, H.S., and Jaeger, J.C. Conduction of Heat in Solids, O.U.P., 1959, Chap.XI .
[2] Crank J., Quart J., Mech App.Math., 10 (1957) 220.
[3] Douglas J., and Gallie T.M., Duke Math. J., 22 (1955) 557.
[4] Murray W.D., and Landis F. Trana. ASME 81 (1959) 106.
[5] Albasiny, E.L.: Proc.I.E.E. 103 (1956) Series B, Parts 1-3, 158.
[6] Crank,J., and Gupta,R.S., JIMA (in the press)
[7] Bix, R.C., and Cizek, J. Heat Transfer 1970, Vol.1, 4th International Heat Transfer
Conference, Paris Versailles, Elsevier, 1970.
[8] Philip, J.R. Trans.Faraday Soo 51 (1955) 885.
[9] Rose, M.E. SIAM J.of Applied Maths.15 (1967) 495.
[10] Chernous'ko, F.I, Zh.Prikl.Mekh.i,Tekhn. Fis.No.2 (1969) 6.
[11] Chernoua'ko, F.L. International ChenuEng.10, No.1, (l970) 42.
[12] Goodman, T.R Adavances-5 in Heat Transfer, Vol.1 Academic Press, New York, 1964.
[13] Coldrey, T.J. M.Tech Dissertation, Brunel University, 1966.
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- Khairuzzaman Mamun (Author), 2017, An Overview of Phase Change, Isotherm, and Isotherm Migration Method, Munich, GRIN Verlag, https://www.grin.com/document/376165
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