Variational Solution of Axisymmetric Fluid Flow in Tubes with Surface Solidification

Final year project dealing with
obtaining the approximate analytical
solution to the nonlinear, two-
dimensional free-boundary problem of
axisymmetric heat conduction with
internal surface solidification in the
inlet regions of the tube under the
guidance of
Variational
Solution of
Axisymmetric Fluid
Flow in Tubes with
Surface
Solidification
Variational Solution of Axisymmetric Fluid
Flow in Tubes with Surface Solidification
Submitted by:
Santosh Kumar Verma
07/ME/52
Department of Mechanical Engg.
National Institute of Technology Durgapur
India
Dr. A K PRAMANICK

Variational Solution of Axisymmetric Fluid Flow in Tubes with Surface Solidification 1
4
 Boundary Conditions………………………………………………………………………….…16
 Continuity Equation……………………………………………………………………………..17
 Basic principle………………………………………………………………………….17
 Application to the problem……………………………………………………...18
 Equation of motion………………………………………………………………………….…..19
 Basic principle………………………………………………………………………….19
 Application to the problem……………………………………………………...21
 Energy equation in solid phase……………………………………………………………..22
 Basic principle………………………………………………………………………….22
 Application to the problem………………………………………………………23
 Energy equation in liquid phase……………………………………………………………24
 Basic principle………………………………………………………………………….24
 Application to the problem………………………………………………………25
NOMENCLATURE
ACKNOWLEDGEMENT
SPECIAL FEATURES
CERTIFICATE
Title Page
ABSTRACT
INTRODUCTION
PREVIOUS WORKS
ANALYSIS
5
6
8
9
10
11
13

 Variational energy equation, liquid region……………………………………………….26
 Variational energy equation, solid region…………………………………………………30
 Basic principle………………………………………………………………………………………….32
 Application to the problem……………………………………………………………………..33
 First solution……………………………………………………………………………….33
 Second solution…………………………………………………………………………..35
 Third solution………………………………………………………………………………36
 Integral energy equation, liquid region…………………………………………………….40
 Basic principle………………………………………………………………………………………….42
 Application to the problem……………………………………………………………………..43
 First solution……………………………………………………………………………….43
 Second solution…………………………………………………………………………..45
 Comparison of methods predicting the solidification history of water………47
 Comparison of methods for predicting the axial distribution of ice layers…49
 Comparison of limiting transient solutions with available non-flow data….51
 Comparison of limiting solution with available steady state data…………….53
 Comparison of variational solution based on different profiles………………..55
 Cylindrical coordinates…………………………………………………………………………….57
 Euler’s equation for variational calculus…………………………………………………..59
VARIATIONAL SOLUTION BASED ON PROFILE (V)
STEADY STATE SOLUTION (V)
SHORT TIME SOLUTION (V)
VARIATIONAL SOLUTION BASED NUSSELT NUMBER AND PROFILE (VN)
STEADY STATE SOLUTION (VN)
GRAPHICAL INTERPRETATION
APPENDIX
LIMITATIONS
26
32
38
42
46
57
60

FUTURE SCOPE
REFERENCES
60
61

NATIONAL INSTITUTE OF TECHNOLOGY DURGAPUR
WEST BENGAL (INDIA) 713209
This is to certify that the project work titled “Variational Solution of
Axisymmetric Fluid Flow in Tubes with Surface Solidification” is a
bonafide work done by Santosh Kumar Verma, Roll no 07/ME/52, of
Mechanical Engineering Department of National Institute of
Technology Durgapur under the curriculum of the institute for the final
year students during 7th and 8th semester.
Dr. Achintya Kumar Pramanick
Professor
Department of Mechanical Engineering
Place: Signature of the Guide:
Date:

I, Santosh Kumar Verma, a student of National Institute of Technology
Durgapur, have done this project for partial fulfilment of my B.Tech. graduation
degree at the institute under the curriculum programme for B.Tech. final year
students of Mechanical Engineering.
I am indebted towards Dr. Achintya Kumar Pramanick, my project guide,
for providing me with this opportunity to undertake the project, and to work
under his profound guidance and support.
I take this opportunity to thank Mr. Pinaki Pal and Dr. Seema Mondal
Sarkar from the Department of Mathematics for endowing me with their
knowledgeable help to undertake this project & for their kind cooperation.
I would like to take this opportunity to thank all my friends for being so
kind and cooperative at each and every step.
A special thanks to the management of National Institute of Technology
Durgapur for being supportive during this whole year in order to complete the
project report.
Santosh Kumar Verma
March 2011, India

Tube radius
Constants used in the definition of Nusselt Number
Generalized coordinate in the solid region
Specific heat
Generalized coordinate in the liquid region
Heat transfer coefficient
Functional integral
Thermal conductivity
Thermal conductivity ratio =
Dimensionless length of tube =
Latent heat
Lagrangian density
N Nusselt number =
Pressure
Dimensionless pressure =
Peclet number =
Prandtl number =
Radial coordinate
Interface position
Dimensionless radial coordinate =
Dimensionless interface position =
Reynolds number = /
Thickness of solid phase =
Dimensionless thickness of solid phase =
Time coordinate
Temperature
Mean inlet velocity
Velocity
Dimensionless velocity
Hyper volume in the n-dimensionless space
Set of coordinates in the n-dimensional space
Axial coordinate
Notations

Dimensionless axial coordinate
Thermal diffusivity ratio
Function defined in the asymptotic solution
Variational operator
Dimensionless temperature =
Set of field variables
Latent to sensible heat ratio
Viscosity
Dimensionless density difference
Dimensionless time coordinate =
Based on tube diameter
Fusion front conditions
Liquid conditions
Outlet conditions
Mean conditions
Field variable
Radial component
Solid conditions -th coordinate
Wall conditions
Axial component
Inlet conditions
Conditions at which short-time and asymptotic solutions
Steady-state conditions
Conjugate variable used with the Lagrangian density
Subscripts
Superscripts
s

Additional information that has been used in different
sections of this paper for different purposes have been
placed in the Solid Box
The solutions obtained by solving different characteristic
equations for the purpose of comparison with the solutions
obtained by the author have been put in a Dash Box
Original solutions to the problems that were obtained by
the author and have been included in the research paper
are boxed in the Long Dash Dot Box

The problem of axisymmetric heat conduction with internal surface solidification in
the regions of tube is discussed. An approximate analytical solution is presented to this
nonlinear, two dimensional free boundary problem. The analysis employs a variational
technique which extends the Lagrangian formalism to treat the internal flow, two-
dimensional moving-interface problems. The solution is expressed in the terms of the short-
time and steady-state components. Two forms of the variational solution are presented. One
has limited validity in the entrance region of the tube, and the other, while less general , is
more accurate.

The problem considered is that of a general class of nonlinear free boundary
problems, such as those characterized by moving boundaries whose motion is not known a
priori but must be determined as part of the solution.
Specifically, the problem is concerned with axisymmetric fluid flow in tubes with
surface solidification. Initially, the fluid is flowing in a tube with a fully developed velocity and
a uniform temperature distribution. A segment of the tube is then given a step input in the
wall temperature to a constant sub-fusion value. As a result, a two dimensional solidification
start at the wall. The interface between the solid and the liquid phases moves inward. During
freezing, the liquid floe rate into the cooled section is maintained constant. The inlet velocity
and the temperature remain fully maintained constant. The inlet velocity and temperature
remain fully developed and uniform respectively. However, the flow field in the cooled
section is characterized by a boundary layer flow in the entrance region, and a fully
developed flow further downstream. The inherent difficulty in the free-boundary problem is
a nonlinear boundary condition that must be satisfied at the moving interface.
Fig. No. 1

A lot of research and scientific work has been done and established in the field in the
time gone by. The present state of work is very steady. To make note of some of the authors
and scientists who have put there remarkable hard in the field are too many.
Excellent literature reviews are given by Boley and Muehlbeuer and Suderland. Most
of these problems deal with a phase change without fluid flow or with external flow. Non-
flow problems usually are based on two coupled conduction equations to be satisfied in the
solid and liquid regions. The external flow problems ordinarily can be uncoupled, since the
field variables of the external phase are not significantly affected by the motion of the free
boundary.
Limited work has been done on problems involving internal flow with surface
solidification. In such systems, the dynamic and thermal response of liquid phase is directly
affected by the interface motion. Therefore, the field equations in both phases cannot be
uncoupled unless one of the phases is assumed to be at fusion temperature.
Grigorian has considered a special one-dimensional problem of melting due to
friction between two moving solid bodies. The problem was formulated in terms of the
equations of continuity, motion and energy in both phases. The problem has a self-similar
solution; therefore, an exact solution of the interface position was determined to within a
constant which was evaluated approximately for some limiting conditions.
Bowley and Coogan considered melting of two parallel quarter-infinite solids due to
an internal fluid flow between the solids. Bowley’s major restriction was that the solid region
be maintained at the melting temperature throughout. This allowed uncoupling of the
equations for the two regions. An integral method was used to transform the Cartesian field
equations of continuity, momentum and energy to a set of first-order nonlinear partial
differential equations which were then solved by quadrature.

Zerkle and Sunderland considered a steady-state case of fluid flow in tubes with
surface solidification. Experimental results were obtained and used to develop a semi-
empirical solution. A steady-state analytical solution was also determined. At steady state,
the interface is stationary. Zerkle made use of this and transformed the convection equation
to the classical Graetz form by assuming a parabolic velocity distribution. The coefficients of
the series solution were evaluated numerically.
Ozisik and Mulligan obtained a quasi-static solution to the freezing of liquids in
forced flow inside tubes. The problem was formulated in terms of a steady-state one-
dimensional conduction equation in the solid region, and a transient two-dimensional
convection equation in the liquid. The method of solution was based on the integral
transforms which could be used only with the assumption of slug velocity. According to
the authors, the applicability of their solution is restricted to the regions where the
rate of change of thickness of the frozen layers is small with respect to both time and
distance, along the tube (close to steady state and away from the entrance region).
Few free boundary problems have been solved exactly. Most solutions have been
obtained numerically or by approximate analytical methods. Of interest here are the
approximate variational methods. These methods, based on the minimum principle, have
been successfully applied in optics, dynamics, wave, mechanics, quantum mechanics and
Einstein’s law of gravitation. Helmholtz was probably the first to attempt to apply the
variational principles to thermodynamics; however, the minimum principles were not
directly applicable to the dissipative systems. Biot developed a method based on the
principle of minimum rate of entropy production and applied it to several one-
dimensional external flow problems. The method has also been applied by Lapadula
and Mueller to an external flow problem involving freezing over a flat plate. A more
general formulation of the variational principle, known as Lagrangian formalism, is usually
presented without reference to any specific system. The Lagrangian formalism may be
specialized to solve a diffusion or conduction equation. The variational solution presented in
this paper is based on the Lagrangian formalism. The, application of the method is
extended to solve the free-boundary problems involving internal flow.

The problem can be formulated in cylindrical coordinates in terms of a complete set
of field equations in the liquid and solid region; both of these regions being coupled by a set
of nonlinear boundary conditions to be satisfied at the moving liquid-solid interface. An order
of magnitude analysis of such a set shows that the axial conduction, axial viscous shear,
dissipation, body forces and radial pressure gradient may be neglected under the usual
conditions of the boundary layer flow.
Two variational solutions of the above problem are presented in this report. The first
variational solution, abbreviated as (V), is less accurate than the second, (VN), solution. The
less accurate solution (V) is presented because it is more general and also because its
examination permits the evaluation of several aspects of the physical problem.
Also author has used numerical solutions to solve the problem to compare the solution
obtained with that of the solutions obtained from variational formulations. Authors have
used this numerical solution tom plot various graphs to show different characteristics of the
problem. But these numerical solutions have not been included in the paper. Also, because of
the complexity of these solutions no attempt has been made to obtain them in this report.
Problem Statement
s

Fig. No. 2
A 3D representation of the flow through the tube along with the surface solidification because
of presence of temperature gradient
Fig. No. 3
A 3D representation of the flow through the tube along with the surface solidification, by a
cross-section of the tube by cutting it along its axis, because of presence of temperature
gradient

Fig. No. 4
Configuration of the problem well explained by different notations, showing both – solid phase
as well liquid phase
The tube has been shown by brown colour and the portion inside the tube which has got
solidified because of variation in temperature present inside, has been shown by blue colour.
The rest of the pipe has water in liquid state.

For constant properties in each phase the boundary conditions imposed to the problem can
be given as:
* ( ) +
* ( ) +
Boundary Conditions
Statement

Basic Principle
In steady flow, the mass flow per unit time passing through each section does not
change, even if the pipe diameter changes. This is the law of conservation of mass.
For the pipe shown here whose diameter decreases between sections 1 and 2, which
have cross-sectional areas A1, and A2 respectively, and at which the mean velocities are
and and the densities and respectively,
= or
= constant
If the fluid is incompressible, e.g. water, with being effectively constant, then
= constant
Continuity Equation
Fig. No. 5

Application to the Problem
Continuity equation in cylindrical coordinates can be presented as,
Considering density to be constant ), the above equation becomes
Since there is no vortex formation and the flow is irrotational, the situation can be reduced to
Converting and to dimensionless quantities, by using
We get the final equation as,
𝟏
𝑹
𝑹𝑽 𝑹
𝝏
𝝏𝒁
𝑽 𝒁 𝟎

Basic Principle
The dynamic behavior of fluid motion is governed by a set of equations, known as
equations of motion. These equations are obtained by using the Newton’s second law, which
may be written as
where is the net force acting in the x-direction upon a fluid element of mass producing
an acceleration of in the x-direction.
The forces which may be present in the fluid flow problems are: gravity force, pressure force,
force due to viscosity, force due to turbulence, and the force due to compressibility of fluid.
When volume changes are small, the force due to compressibility is negligible , and the
general equation of motion in the x-direction using previous equation may be written as
Similar expressions for y and z- directions may also be written. When we substitute the
expressions for various quantities in this equation, the resulting equations are known as
Reynolds equations.
For flow at low Reynolds number, the force due to turbulence is of no significance and,
therefore, pressure force and the viscous force is
together with similar expressions for y and z-directions are known as the Navier-Stokes
equations.
Equation of Motion

Thus Navier-Stokes equation in Cartesian form can be written as
( ) { ( )} { ( )}
Similarly expressions for y and z-directions can be obtained.

Equation of motion along z-axis in cylindrical coordinates is given as
( ) * ( ) +
Since the given flow condition is irrotational and the body forces have been assumed to be
zero, thus
( ) * ( ) +
( ) * + ( )
Also, axial viscous shear is zero, and thus the last term can be put to zero resulting into
( ) * +
After transforming and into dimensionless quantities and making substitution using,
We get the final result as
𝟏
𝜶𝑷𝒆
𝝏𝑽 𝒁
𝝏𝝉
(𝑽 𝑹
𝝏𝑽 𝒁
𝝏𝑹
𝑽 𝒁
𝝏𝑽 𝒁
𝝏𝒁
)
𝝏𝑷
𝝏𝒁
𝟏
𝑹𝒆
𝟏
𝑹
𝝏
𝝏𝑹
(𝑹
𝝏𝑽 𝒁
𝝏𝑹
)

Basic Principle
For solid phase, the energy equation when combined with Fourier’s Law of heat
conduction, becomes
If the thermal conductivity can be assumed to be independent of the temperature and
position, then above equation becomes
in which is the thermal diffusivity of the solid.
Energy Equation in Solid Phase

The basic equation for energy in solid phase can be written as
[ ( )]
Differentiating the equations given below with respect to respectively,
we get
Putting these above given differential equations in the initial equation, and reducing the thus
obtained equation, results into the final equation as shown below
[ ]
And the final solution is,
𝝏𝜽 𝑺
𝝏𝝉
𝟏
𝑹
[
𝝏
𝝏𝑹
(𝑹
𝝏𝜽 𝑺
𝝏𝑹
)]

Basic Principle
While considering the liquid phase, the velocity effects of the liquid will come into
the picture. Thus, making slight amendments will give us the energy equation in liquid phase.
The desired equation in Cartesian form can be given as,
( ) ( )
Addition of axial velocity and radial velocity will serve our purpose in order to obtain the
energy equation in cylindrical coordinates.
Energy Equation in Liquid Phase

Energy equation in liquid phase is as shown below,
( ) ( )
Dividing the above equation by throughout, we obtain
( ) ( )
Changing to dimensionless quantities, we obtain
( )
( )
The resulting equation after simplification is,
𝟏
𝜶𝑷𝒆
𝝏𝑻
𝝏𝝉
𝑽 𝑹
𝝏𝑻
𝝏𝑹
𝑽 𝒁
𝝏𝑻
𝝏𝒁
𝟏
𝑷𝒆
𝟏
𝑹
𝝏
𝝏𝑹
(𝑹
𝝏𝑻
𝝏𝑹
)

The differential energy equations in the liquid and solid regions are identical to the
Euler-Lagrange of the Variational principle. Thus, the differential equations can be used to
formulate the Variational statement in the liquid and the solid regions. The two regions are
coupled at the moving interface by the nonlinear boundary condition. Thus, the variational
statement of the problem consists of the variational liquid and solid equations, as shown
below,
∫ ∫ ∫
∫ ∫ ∫ ( )
∫ ∫ ∫ ( )
The equation can be rearranged as shown below,
∫ ∫ ∫ [ ( ) ( )]
Variational Energy Equation, Liquid Region

If varies invariably, then the term in square brackets can be put to zero i.e.
[ ( ) ( )]
The profile which has been used for solving this physical problem is,
using which different parts of the equation can be simplified as,
{ }
( ) { }
( )
It is to be noted that here and .
Adding above and equating to zero and further reduction gives,
𝑓 𝑥 𝑦 𝑧 𝛿𝑉𝑑𝑥𝑑𝑦𝑑𝑧
𝑓 𝑥 𝑦 𝑧
For a given integration as given below,
If the functional 𝛿𝑉 varies without any restriction for all values of x, y, z ; then only
option we are left are with is that

Now subjecting the above equation to the following initial conditions,
(at )
The equation becomes,
Finally we obtain the resulting equation as,
Now comparing the result with that obtained by the authors of the paper, we see that
assigning and will serve the purpose.
Hence, the final result will be,
( )
𝝏𝑪
𝝏𝝉
𝟑𝑪
𝑹 𝑭
𝝏𝑹 𝑭
𝝏𝝉
𝟔𝜶𝑪
𝑹 𝑭
𝟐
𝜶𝑷𝒆 (
𝟗
𝟐
𝑪
𝑹 𝑭
𝟑
𝝏𝑹 𝑭
𝝏𝒁
𝟑
𝟐
𝟏
𝑹 𝑭
𝟐
𝝏𝑪
𝝏𝒁
)

But the actual result as obtained by the authors, is
𝝏𝑪
𝝏𝝉
𝟑𝑪
𝑹 𝑭
𝝏𝑹 𝑭
𝝏𝝉
𝟔𝜶𝑪
𝑹 𝑭
𝟐
𝜶𝑷𝒆 (
𝟑𝑪
𝑹 𝑭
𝟑
𝝏𝑹 𝑭
𝝏𝒁
𝟑
𝟐
𝟏
𝑹 𝑭
𝟐
𝝏𝑪
𝝏𝒁
)

∫ ∫ ∫ ∫ ∫ ∫ ( )
The equation can be rearranged as show below,
∫ ∫ ∫ [ ( )]
If varies invariably, then the term in square brackets can be put to zero i.e.
[ ( )]
The profile which has been used for solving this physical problem,
𝑓 𝑥 𝑦 𝑧 𝛿𝑉𝑑𝑥𝑑𝑦𝑑𝑧
𝑓 𝑥 𝑦 𝑧
For a given integration as given below,
If the functional 𝛿𝑉 varies without any restriction for all values of x, y, z; then only
option we are left are with is that
Variational Energy Equation, Solid Region

using which different parts of the above expression can be simplified as,
* +
( )
Thus,
* +
Now rearranging the above equation we obtain,
But the expression obtained by the authors can be shown as below,
𝝏𝑩
𝝏𝝉
𝜽 𝒘
𝝏𝑹 𝑭
𝝏𝝉
[
𝟏
𝟏 𝑹 𝑭
𝟐 𝑹 𝑹 𝑭
] 𝑩 (
𝟏
𝟏 𝑹 𝑭
)
𝝏𝑹 𝑭
𝝏𝝉
[
𝜽 𝒘
𝑹 𝑹 𝟏 𝟏 𝑹 𝑭 𝑹 𝑹 𝑭
] 𝑩 [
𝟒𝑹 𝑹 𝑭 𝟏
𝑹 𝑹 𝟏 𝑹 𝑹 𝑭
]
𝝏𝑩
𝝏𝝉
𝜽 𝒘
𝝏𝑹 𝑭
𝝏𝝉
*
𝟐 𝟑𝑹 𝑭
𝟏 𝑹 𝑭
𝟑 𝟏 𝑹 𝑭
+ 𝑩
𝝏𝑹 𝑭
𝝏𝝉
*
𝟐 𝟑𝑹 𝑭
𝟏 𝑹 𝑭 𝟏 𝑹 𝑭
+
[
𝟏𝟎𝜽 𝒘
𝟏 𝑹 𝑭 𝟏 𝑹 𝑭
𝟑
] [
𝟏𝟎𝑩
𝟏 𝑹 𝑭
𝟐
]

Basic Principle
If the fluid and flow characteristics such as density, velocity, pressure, acceleration
etc., at a point do not change with time, the flow is said to be steady, thus for steady flow
( )
( )
( )
and so on.

For steady state solution, the characteristic equations for fluid flow conditions in
concerned problem will have all the time derivatives equal to zero, i.e.
Considering the first solution obtained from the Variational energy equation in liquid region,
( )
and putting all time derivatives equal to zero, in order to obtain a steady state solution, we
get
( )
( )
( )
Now by using separation of variable technique,
( )
𝜕
𝜕𝜏
First Solution

On integrating we obtain,
√
where, represents constant of integration.
Simplification of the result,
And its comparison with the actual one shows that the constant of integration has a
value .
𝑪∞ 𝒁
𝟏
𝝃𝑹 𝑭∞
𝟐
𝒆𝒙𝒑 (
𝟒𝒁
𝑷𝒆
)
𝑪∞ 𝒁
𝟏 𝟓
𝑹 𝑭∞
𝟐
𝒆𝒙𝒑 (
𝟒𝒁
𝑷𝒆
)

In order to obtain the second steady state solution, we equate all time derivatives equal to
zero in the first solution obtained using the Variational energy equation in solid region.
Simplifying the above equation we obtain the final result as,
𝑩∞ 𝒁
𝜽 𝑾
𝟏 𝑹 𝑭∞
𝟐
Second Solution

Now moving to the 2nd
solution obtained from the Variational energy equation in the solid
region, we again employ the same strategy in order to obtain the steady state solution.
[ ] * ( ) +
[ ] * ( ) +
Seeing above,
[ ]
since [ ( ) ] will always be positive and greater than zero.
Now substituting the expressions for B and C, from the steady state solutions obtained above,
the equation transforms into
( )
This transforms into a quadratic equation,
where
( )
( )
Third Solution

Solving this quadratic equation we obtain the roots of the equation as,
( ) , [ ( )] -
Neglecting the negative sign in the above, final solution will be
𝑹 𝑭
𝜽 𝑾
𝟑𝑲
𝒆𝒙𝒑 (
𝟒𝒁
𝑷𝒆
) ,𝟏 [
𝜽 𝑾
𝟑𝑲
𝒆𝒙𝒑 (
𝟒𝒁
𝑷𝒆
)]
𝟐
-
𝟏 𝟐

In order to obtain the very first short time solution of variational solutions based on
profiles (V), we consider the equation,
[ ] * ( ) +
The first solution is based on zero convection and linear .
Thus,
, since has been considered to be linear
, zero convection
, when there is no convection taking place, the interface position will not
change with respect to axial distance
Hence, the equation reduces to,
( )

Now integrating the above equation, we obtain
√( )
So, the resulting solution obtained is
This solution is applicable for relatively short time, when the solid phase thickness
∞ .
𝑹 𝑭 𝟏 √(
𝟐𝜽 𝑾
𝝀
𝝉)

The Variational solution obtained here is based on Nusselt number. The profiles for
and is same as that used previously. However, the profile for is replaced by a mean
liquid temperature and a Nusselt number .Thus, the Variational statement of
the problem remains same except that Variational energy equation in the liquid region is
replaced by an integral energy equation in terms of and .
∫ ∫ ∫ ∫ ∫ ( )
Bringing all the terms on the left hand side,
∫ ∫ ∫ ∫ ∫ ( )
Or,
∫ ∫ ( )
Integral Energy Equation, Liquid Region

Substituting and * + in the above expression, we obtain
∫ ∫ ( * + )
Integration of the above equation with respect to R, will give us
* +
Rearrangement of the above terms, will result into
The same solution as obtained by the authors is,
𝝏𝜽 𝑴
𝝏𝝉
𝟐𝜶𝑵𝜽 𝑴
𝑹 𝑭
𝟐
𝟐 𝑷𝒆
𝑹 𝑭
𝟐
𝝏𝜽 𝑴
𝝏𝒁
𝝏𝜽 𝑴
𝝏𝝉
𝟐𝜶𝑵𝜽 𝑴
𝑹 𝑭
𝟐
𝟒 𝑷𝒆
𝑹 𝑭
𝟐
𝝏𝜽 𝑴
𝝏𝒁
[
𝟏
𝟐
𝟏
𝟑𝑹 𝑭
]

Basic Principle
If the fluid and flow characteristics such as density, velocity, pressure, acceleration
etc., at a point do not change with time, the flow is said to be steady, thus for steady flow
( )
( )
( )
and so on.

For steady state solution, the characteristic equations for fluid flow conditions in
concerned problem will have all the time derivatives equal to zero, i.e.
First steady state solution can be obtained by equating time derivatives in the equation
equal to zero. Thus, we have
The expression for Nusselt number in terms of dimensionless axial coordinate, Peclet number
and other constants can be given as
[ ]
𝜕
𝜕𝜏
First Solution

Hence substituting the expression of in the above equation, we get
[ ]
[ ]
[ ]
* ( ) +
* ( ) +
* ( ) +
Thus, rearranging the above, final result will be as shown below
𝜽 𝑴∞
𝒁 *𝟏
𝟏
𝒄
(
𝒁
𝑷𝒆
)
𝟐 𝟑
+
𝟑𝒂𝒃
𝒆𝒙𝒑 [ 𝟐𝒂 (
𝒁
𝑷𝒆
)]

Similarly the second steady state solution can be obtained from the equation
[ ] * ( ) +
[ ] * ( ) +
Since,
* ( ) +
thus,
[ ]
Substituting the expression for B in the above equation gives us
It is a quadratic equation, whose roots will lend us the required results.
Hence, the solution is
𝑹
𝜽 𝑾
𝑲𝑵𝜽 𝑴
*𝟏 (
𝜽 𝑾
𝑲𝑵𝜽 𝑴
)
𝟐
+
𝟏 𝟐
Second Solution

In order to compare and analyze the problem graphically, the authors have used three
different solutions to obtain the results and have them plotted for short time, asymptotic and
steady state composite parts.
For the research paper originally three solutions were obtained, which are as follows:
 Variational solution based on profiles
 Variational solution based on profiles and Nusselt no.
 Numerical solution
But only two solutions have been given in the paper from the above. No data or expressions
used regarding Numerical solution have been included. Also, there is no expression explaining
the relationship of dimensionless thickness of solid phase with dimensionless
time .
The validity of variational solution (V) is limited only in the entrance region. The (VN) solution
is generally more accurate and therefore is preferred to the variational solution (V).
In spite of these limitations, it has been tried to explain the graphs, when and wherever
possible.

.
Fig. No. 6
Legend
(V), Variational solution based on profiles
(VN), Variational solution based on profiles and Nusselt No
(N), Numerical solution
Comparison of Methods for Predicting the
Solidification History of Water
𝜏Type equatio here
Dimensionless time (𝝉)
Dimensionlessthickness
ofthesolidphase(S)

Fig. No. 7
Fig. No. 8
ofthesolidphase(S)
ofthesolidphase(S)

Fig. No. 9
Fig. No. 10
Comparison of Methods for Predicting the
Axial Distribution of Ice Layer
Normalized distance from entrance region (Z/Pe)
ofthesolidphase(S)
ofthesolidphase(S)

ofthesolidphase(S)
Fig. No. 11
Legend
(N), Numerical solution

Comparison of Limiting Transient
Solutions with Available Non-flow Data
Dimensionlessthicknessofthesolidphase(S)
Fig. No. 12

Legend
Numerical solution, 𝑅𝑒 𝐷 𝜌 𝑇 𝑇𝐹 ℉
Short time solution based on zero convection and linear 𝜃𝑆 profile
Short time solution based on zero convection and non-linear 𝜃𝑆 profile
Poots integral solution-Karman method
Poots integral solution-Tani method
Allen and Severn numerical solution
(Based on initial 𝜃 𝐿 𝜆 𝑊
𝐿
𝐶 𝑆 𝑇 𝐹 𝑇 𝑊
𝜌
𝜌 𝐿 𝜌 𝑆
𝜌 𝐿
)

Comparison of Limiting Solutions with
Available Steady-State Data
Normalizedinterfaceposition𝑹𝑭
𝑲𝜽𝑾
Fig. No. 13

Legend
(N), Numerical solution 𝑅𝑒 𝐷
Zerkle’s analytical steady state solution
Zerkle’s semi-empirical steady state data, 𝑅𝑒 𝐷
Ozisik-Mulligan steady state solution

Comparison of Variational Solutions
Based on Different Profiles
Dimensionlesssteady-statethicknessofthesolidphase(S)
Fig. No. 14

Legend
(V), Numerical solution 𝑅𝑒 𝐷
(V), based on 2-parameters 𝜃 𝐿
(V), based on slug 𝑉𝑍
(V), based on linear 𝜃𝑆

 The effect of natural convection cannot be fully evaluated here since it is not
considered in any solution presented here.
 The study can be used to make modifications in the current scenario of cold storage.
 It will beneficial to those countries where there is serious problem of solidification of water
pipe lines and water in engine radiators.

A cylindrical coordinate system is a three-dimensional coordinate system that
specifies point positions by the distance from a chosen reference axis, the direction from the
axis relative to a chosen reference direction, and the distance from a chosen reference plane
perpendicular to the axis. The latter distance is given as a positive or negative number
depending on which side of the reference plane faces the point.
The origin of the system is the point where all three coordinates can be given as zero. This is
the intersection between the reference plane and the axis.
The axis is variously called the cylindrical or longitudinal axis, to differentiate it from the polar
axis, which is the ray that lies in the reference plane, starting at the origin and pointing in the
reference direction.
The distance from the axis may be called the radial distance or radius, while the angular
coordinate is sometimes referred to as the angular position or as the azimuth. The radius and
the azimuth are together called the polar coordinates, as they correspond to a two-
dimensional polar coordinate system in the plane through the point, parallel to the reference
plane. The third coordinate may be called the height or altitude (if the reference plane is
considered horizontal), longitudinal position, or axial position.
Cylindrical coordinates are useful in connection with objects and phenomena that have some
rotational symmetry about the longitudinal axis, such as water flow in a straight pipe with
round cross-section, heat distribution in a metal cylinder, and so on.
Cylindrical Coordinates

Definition
The three coordinates (ρ, φ, z) of a point P are defined as:
 The radial distance ρ is the Euclidean distance from the z axis to the point P.
 The azimuth φ is the angle between the reference direction on the chosen plane and the line
from the origin to the projection of P on the plane.
 The height z is the signed distance from the chosen plane to the point P.
Coordinate system conversions into Cartesian coordinates
For the conversion between cylindrical
and Cartesian coordinate systems, it is
convenient to assume that the reference
plane of the former is the Cartesian x–y
plane (with equation z = 0), and the
cylindrical axis is the Cartesian z axis.
Then the z coordinate is the same in both
systems, and the correspondence
between cylindrical (ρ, φ) and Cartesian
(x, y) are the same as for polar
coordinates, namely
os
si
in one direction, and
√
{
si ( )
si ( )
Fig. No. 15

The basic problems in Variational calculus consist of determining, from among
functions possessing certain properties, that functions for which a given integral (functional)
assumes it maximum or minimum value. The integrand of the integral in question depends on
the function and its derivatives.
Consider the many values of the integral
∫
where is the unknown, and
The special function for which reaches an extremum satisfies the Euler equation:
( )
Euler’s Equation for Variational Calculus

1. J. A. Bilenas and L. M. Jiji, “Variational Solution Of Axisymmetric Fluid Flow In Tubes
With Surface Modification”, Ph.D. thesis, City University of New York, New York, 1968.
2. Heat Transfer (2nd
edition), by Cengel.
3. Transport Phenomena (2nd
edition), by R. B. Bird, W. E. Stewart and E. N. Ligthfoot.
4. Fluid Mechanics, by Dr. A. K. Jain.
5. Higher Engineering Mathematics, by Dr. B. S. Grewal.
6. Wikipedia (free encyclopedia), http://en.wikipedia.org.
7. Wolfram Mathworld, http://mathworld.wolfram.com.

Variational Solution of Axisymmetric Fluid Flow in Tubes with Surface Solidification

Recommended

Recommended

More Related Content

What's hot

What's hot (18)

Similar to Variational Solution of Axisymmetric Fluid Flow in Tubes with Surface Solidification

Similar to Variational Solution of Axisymmetric Fluid Flow in Tubes with Surface Solidification (20)

More from Santosh Verma

More from Santosh Verma (10)

Recently uploaded

Recently uploaded (20)

Variational Solution of Axisymmetric Fluid Flow in Tubes with Surface Solidification