An Overview of Phase Change, Isotherm, and Isotherm Migration Method

Essay , 2017 , 10 Pages , Grade: 1

Physics - Thermodynamics

Phase change, isotherm and isotherm migration method are very important terms to learn in thermodynamics. 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 offers an overview of 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.

This study focuses on different types of discussion about the phase change, isotherm, and isotherm migration method. From this article one can get a brief idea about the phase change and isotherm migration method. The essay offers definitions of the terms ("phase change", "isotherm"), explains the Stefan problems and offers methods of solution.

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

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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

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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.

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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

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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)

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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)

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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

.

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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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Frequently asked questions

What is the purpose of this document?

This document provides a comprehensive language preview, including the title, table of contents, objectives and key themes, chapter summaries, and key words for a text. It aims to present the definition and mathematical discussion of some important thermodynamic terms.

What are the key thermodynamic terms discussed?

The key terms discussed include Phase-change, Isotherm, Stefan Problems, and Isotherm Migration method.

What is Phase-change?

Phase transition or phase change describes transitions between solid, liquid, gaseous states of matter, and plasma. During a phase transition, certain properties change due to changes in external conditions like temperature or pressure.

What is an Isotherm?

The term Isotherm means keeping the same temperature at a given time or on average over a given period.

What are Stefan Problems?

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. It aims to describe temperature distribution during a phase change (e.g., ice to water).

What are the methods of solution for Stefan Problems?

Methods of solution include analytical solutions and numerical methods based on finite-difference replacements of the original partial differential equation. Examples include special finite-difference formulas, unequal time intervals, grid deformation, modified specific heat, and moving the entire grid with the boundary.

What is the Isotherm Migration Method (IMM)?

The Isotherm Migration Method (IMM) is a method for tracking a moving boundary that occurs at a fixed temperature. It involves expressing the space variable 'x' as a function of temperature 'u' and time 't', so that x = x(u, t), and calculating the positions of a given temperature (isotherm) at known times.

How is melting ice described in the document?

The melting ice example describes a plane sheet of ice initially occupying a region and being melted by the application of constant temperature. It provides a mathematical framework for describing the heat flow and boundary conditions during the melting process. Equations are derived to relate heat flow, latent heat of fusion, and temperature gradients.

What are the initial and boundary conditions for the melting ice problem?

The initial condition is that the ice is at 0 degrees. Boundary conditions include a constant temperature applied at the surface, and conditions at the moving boundary separating the water from the ice that relate the heat flow to the latent heat of fusion.

What is the numerical approximation used for the Isotherm Migration Method?

The derivatives in the equations are approximated by finite differences to obtain an explicit expression for the value of X at a future time step in terms of values already available at previous time steps.

What is the stability criterion for the finite difference scheme in IMM?

The criterion is related to the coefficient of temperature gradient, is less than 1/8 (the difference in the value of x across 2 steps - value of x at a current step) squared, this criterion allows to change as the solution proceeds.

Where can I find the references for these concepts?

A list of references is provided at the end of the document, citing various articles and books on conduction of heat, finite difference methods, and the Isotherm Migration Method.

• Grade: 1
• Author: Khairuzzaman Mamun (Author)
• Publication Year: 2017
• Pages: 10
• Catalog Number: V376165
• ISBN (eBook): 9783668538146
• ISBN (Book): 9783668538153
• Language: English
• Tags: phase change