Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

53, 3, pp. 505-518, Warsaw 2015
DOI: 10.15632/jtam-pl.53.3.505

APPLICATION OF THE METHOD OF FUNDAMENTAL SOLUTIONS TO

THE ANALYSIS OF FULLY DEVELOPED LAMINAR FLOW

AND HEAT TRANSFER

Krzysztof Mrozek
Poznan University of Technology, Institute of Mechanical Technology, Poznań, Poland

e-mail: krzysztof.mrozek@put.poznan.pl

Magdalena Mierzwiczak
Poznan University of Technology, Institute of Applied Mechanics, Poznań, Poland

e-mail: magdalena.mierzwiczak@wp.pl

In this study, fully developed laminar flow and heat transfer in an internally longitudinally
finned tubeare investigatedthroughapplicationof themeshlessmethod.Theflow is assumed
to be both hydrodynamically and thermally developed, with a uniform outside-the-wall
temperature.The governing equationshavebeen solvednumericallybymeans of themethod
of fundamental solutions in combination with the method of particular solutions to obtain
the velocity and temperature distributions. The advantage of the proposed approach is that
it does not requiremesh generation on the considered domain or its boundary, but uses only
a cloud of arbitrarily located nodes. The results, comprising the friction factor as well as
the Nusselt number, are presented for varied length values and fin numbers, as well as the
thermal conductivity ratio between the tube and the flowing fluid. The results show that
the heat transfer improves significantly if more fins are used.

Keywords: laminar flow, finned tube, heat transfer enhancement, method of fundamental
solutions

1. Introduction

At present, many thermal engineering researchers are investigating new heat transfer enhan-
cement methods between surfaces and the surrounding fluids. Heat transfer enhancement is of
particular importance to the intensification of cooling of injection molds working in a cycle of
dynamic temperature changes and equipped with cooling inserts. The aforesaid injections are
the most important facet of the plastic injection molding process and affect the shape, aesthe-
tics, technical properties and utility of compacts (Benitez-Rangel et al., 2010). One of themajor
problems encountered during the injection process is to ensure the most efficient and uniform
heat transfer from the cooled material so as to avoid generation of excessive stress causing de-
formation of themolds. It should be emphasized that the cooling process canmake up to 70%of
the time cycle, and is one of themost important stages of the injection process. So far, themost
commonmethod of heat removal through application of cooling channels relies on conventional
drilling. In order to improve the cooling efficiency, we propose the use of a finned cooling channel
whosemain task is to increase the active surface area of the heat exchange between the injection
mold and the cooling fluid.
A comprehensive report on recent advances in heat transfer enhancements was presented

by Siddique et al. (2010), while the classification of heat transfer enhancement techniques was
documented byBergles (1998). Themechanisms of enhancing heat transfer that require external
power, dubbed activemethods, comprise, for example, application of stirring in the fluid vessels
or surface vibration, as presented by Nesis et al. (1994). The passive enhancement methods are



506 K.Mrozek,M.Mierzwiczak

those that do not require external power to sustain enhancement characteristics and rely on the
use of: treated surfaces, rough surfaces, extended surfaces, displaced enhancement devices, swirl
flow devices, coiled tubes, surface tension devices, additives for fluids andmany others.

Most of the heat transfer augmentation methods presented in the literature employ fins.
Laminar flow heat transfer in internally finned tubes is of particular importance in many en-
gineering industries relying on the heating or cooling of viscous liquids or oils, and specifically
including heating of the circulating fluid in solar collectors and heat transfer in heat exchan-
gers. Internally finned tubes are commonly used in engineering applications as efficient means
to improve convective heat transfer while maintaining a small size and low weight. To provide
additional heat transfer surfaces, various types of internal fins are utilized. However, when an
array of fins is used to enhance heat transfer, the presence of fins may increase the pressure
drop in the tube and reduce the mass flow rate. For this reason, the prime engineering focus is
to optimize the geometry of fins that will maximize the heat transfer rate under space and cost
constraints.

Extensive work has been carried out by different researchers, e.g., Rout et al. (2012), to
analyze a laminar heat exchanger with fins of various shapes and sizes. Experimental investiga-
tions show that the heat transfer characteristics and flow friction are greatly influenced by the
fin spacing, size, and shape. Soliman and Feingold (1977) obtained an analytical solution for a
fully developed laminar flow, encompassing an extensive range of finparameters (varying height,
width and number). The resulting equation of velocity distribution was rendered in the form
of infinite series involving arbitrary constants evaluated by equating the velocity and its radial
derivative at the boundary. In contrast, Soliman et al. (1980) presented numerical analysis of the
momentum and energy equations using a finite difference approach. Along with dimensionless
velocity, the authors determined the temperature field and defined theNusselt number. Further-
more, the fully developed laminar flow and convective heat transfer in an internally finned tube
heat exchanger were investigated numerically through application of an explicit finite-difference
scheme by Tien et al. (2012). The authors conducted additional experiments in a closed-loop
device to verify the numerical results. So far, the laminar flow heat transfer problem hasmostly
been resolved through utilization of the Finite DifferenceMethod (FDM) providing the solution
in a discrete form; both, the differential equation and the boundary conditions are fulfilled only
in an approximatemanner. ThemeshlessMethod of Fundamental Solutions (MFS) is free of the
disadvantages of the abovementionedmethod. InMFS, the approximated solution is convenien-
tly rendered as a continuous functionwith continuous derivatives. In thismethod, the governing
equation is fulfilled exactly and the approximation lies in the fulfillment of the boundary condi-
tions. For the homogenous differential problem of the maximum principle, the maximum error
is achieved on the boundary and can be controlled by the appropriate value of the method pa-
rameters. The foundations of this dynamically developed meshless procedure were given in the
1960s by Kupradze and Aleksidze (1964). However, the modern, computerized version of the
method was proposed a decade later by Mathon and Johnston (1977). The research conducted
since then, and presented by Chen et al. (2008), allowed expanding its scope and successful
application in solving the inhomogeneous differential equations, nonlinear problems, transient
problems, or inverse problems. In recent years, it has become increasingly popular due to its
simplicity of implementation. In such cases, the solution is approximated by linear combinations
of fundamental solutionswith singularities placed on a fictitious boundary lying outside the con-
sidereddomain.MFSwas successfully applied to resolve thepotential flowproblemsbyJohnston
and Fairweather (1984), the Helmholtz problems byTsai et al. (2009), the biharmonic equation
byKarageorghis and Fairweather (1987), the elliptic boundary value problems byKarageorghis
and Fairweather (1998), the Poisson equation by Golberg (1995), the Stokes flow problems by
Alves and Silvestre (2004), and the elasticity problems byTsai (2007). A comprehensive review
of MFS was presented by Golberg and Chen (1999).



Application of the method of fundamental solutions to... 507

This study considers two-dimensional heat conduction through fins with a fixed volume.
Both, thevelocity and temperaturefieldvalues are determinedbymeans ofMFSand theMethod
of Particular Solutions (MPS). For the velocity field, the analytical solution could be obtained
with a defined and acceptable accuracy. In contrast, the temperature field problem is tackled
iteratively, using the Radial Basis Function (RBF) and monomials to determine the particular
solution, and MFS to work out the homogenous solutions of each iterative step. It is assumed
that both, the fins and the fluid flow, are subjected to constant wall temperature conditions.
The parameters of thickness, length, and number of fins as well as the thermal conductivity
ratio of the fin to the working fluid are varied to obtain the friction factor as well as the Nusselt
number values in the internally finned tube.

2. Analytical formulation

Figure 1 shows the cross section of the internally finned tube considered in this paper.A variable
number of straight fins are evenly distributed around the circumference of the tube. Due to
geometric symmetry of the flow domain, as shown in Fig. 1, the solution to the governing
equations is sought only for a half of the region between the center lines of two consecutive
fins Ω∗f; i.e. between θ =0 and θ = γ.

Fig. 1. Geometry of (a) the cross section of the tube and (b) a circular repeated part of the tube

2.1. Determination of velocity

This analysis is applicable to a steady, laminar, and fully developed flow with a uniform
outside-the-wall temperature. Moreover, it is assumed that the fluid is Newtonian and has uni-
formproperties, and the viscous dissipationwithin the fluid is neglected. On these assumptions,
the momentum equation is reduced to

∇2w∗ =
1

µ

dp

dz
in (x∗,y∗)∈ Ω∗f (2.1)

where w∗ is the velocity along the tube; µ – dynamic viscosity; dp/dz – gradient of pres-
sure in the direction z. Using dimensionless variables, r = r∗/r∗0, x = x

∗/r∗0, y = y
∗/r∗0,

w = w∗/[−(1/µ)(dp/dz)r∗0], where r
∗

0 is the inner radius of the tube, equation (2.1) can be
written in a dimensionless form as

∇2w =−1 in (x,y)∈ Ωf (2.2)

The boundary condition assumes the following form

w =0 on (x,y)∈ (BC ∪CD∪DE) (2.3)



508 K.Mrozek,M.Mierzwiczak

and

∂w

∂n
=0 on (x,y)∈ (AB ∪EA) (2.4)

and the dimensionless bulk velocity is obtained from the equation

wb =
1

Af

∫

Af

w dAf (2.5)

where Af = A
∗

f/r
∗

0
2 is the dimensionless flow area of the tube; A∗f = π∗r

∗

0
2−Mβ(r∗0

2−r∗w
2) –

the total flow area; M – number of fins in the tube; β – half of the angle subtended by one fin;
rw = r

∗

w/r
∗

0 – dimensionless radial coordinate at the tip of the fin, L = L
∗/r∗0 – dimensionless

fin height, L∗ – fin height.
The friction factor and the Reynolds number are defined as

f =
π2ρr∗0

5

ṁ2

(
−
dp

dz

)
Re=

2ṁ

πr∗0µ
(2.6)

where ṁ = ρA∗fw
∗

b is the mass flow rate of the fluid; ρ – density of the fluid.
The product fRe can be expressed in the dimensionless form

fRe=
π

Af

2

wb
(2.7)

2.2. Determination of temperature

For a fully developed temperature profile, the dimensionless temperature (Tw−T)/(Tw−Tb),
where T is the fluid temperature, Tb is the bulk mean temperature, and Tw is the tube-wall
temperature, does not depend on z, that is

∂

∂z

(Tw−T
Tw−Tb

)
=0 (2.8)

After somemathematical manipulations, (2.8) gives

∂T

∂z
=
dTw
dz
−
Tw−T

Tw−Tb

dTw
dz
+
Tw−T

Tw−Tb

dTb
dz

(2.9)

Considering the constant-wall-temperature boundary condition, dTw/dz =0, equation (2.9) can
be reduced to

∂T

∂z
=
Tw−T

Tw−Tb

dTb
dz

(2.10)

If the flow is thermally developed and there is no axial conduction, the energy equation for the
fluid flow takes the following form

kf∇
2T = ρCpw

∗
∂T

∂z
in (x∗,y∗)∈ Ω∗f (2.11)

where kf is the thermal conductivity of the fluid; Cp – specific heat at constant pressure.
Introducing the dimensionless temperature of the fluid, Θ(x,y) = (T − Tw)/[qw(z)r

∗

0/kf],
where qw = Q/(2πr

∗

0) is the average heat flux at outer tube wall; Q – total heat transfer rate at
the solid-fluid interface, and employing (2.7) and (2.10) into equation (2.11), yields

∇2Θ =
ṁCp
2πqwr

∗

0

fRew
Θ

Θb

dTb
dz

(2.12)



Application of the method of fundamental solutions to... 509

For the energy balance of a small element ∆z in the axial direction of the tube, one obtains

qw(2πr
∗

0)∆z = ṁCp
(
Tb
∣∣∣
z+∆z

−Tb
∣∣∣
z

)
(2.13)

As ∆z approaches zero, (2.13) can be simplified to

dTb
dz
=
qw2πr

∗

0

ṁCp
(2.14)

Substituting (2.14) into (2.12) gives

∇2Θ = fRew
Θ

Θb
in (x,y)∈ Ωf (2.15)

and for the solid fin, the dimensionless energy equation becomes

∇2Θs =0 in (x,y)∈ Ωs (2.16)

The dimensionless boundary conditions for the fluid and the fin are

∂Θ

∂n
=0 in (x,y)∈ (AB ∪EA)

∂Θ

∂n
= k

∂Θs
∂n

∧ Θ = Θs ∈ (x,y)∈ (CD∪DE)

(2.17)

and

∂Θs
∂n
=0 in (x,y)∈ EF

Θs =0 in (x,y)∈ FB

Θ =0 in (x,y)∈ BC

(2.18)

where k = βks/kf represents the ratio of thermal conductivity of the fin to the fluid, and ks is
the thermal conductivity of the fin.
The dimensionless bulk mean temperature Θb and the Nusselt number Nu for the flow are

derived as

Θb =

∫

Af

w(x,y)Θ(x,y) dAf

∫

Af

w(x,y) dAf
Nu=

2r∗0qw(z)

kf(Tw−Tb)
=−

2

Θb
(2.19)

3. Numerical solution procedure

To solve the boundary value problem for the velocity (2.2)-(2.4) and for the temperature (2.15)-
-(2.18), we propose to use theMFS.
The particular solution to (2.2) has the following form

wp =−
1

4
(x2+y2) (3.1)

and for theMFS, the homogenous solution can be represented as

wh =
N∑

n=1

cn lnr
2
n (3.2)



510 K.Mrozek,M.Mierzwiczak

where rn =
√
(x− x̃n)2+(y− ỹn)2 and{(x̃n, ỹn)}

N
n=1 are the coordinates of sourcepoints placed

outside the considered region on a fictitious contour at a distance s from the boundary (see
Fig. 2). The unknown coefficients {cn}

N
n=1 are obtained through fulfilling boundary conditions

(2.3) and (2.4) in the collocation points {(xi,yi)}
Mc
i=1

N∑

n=1

cn ln
(
(xi− x̃n)

2+(yi− ỹn)
2
)
=
1

4
(x2i +y

2
i ) on (xi,yi)∈ (BC ∪CD∪DE)

N∑

n=1

cn
∂

∂n
ln
(
(xi− x̃n)

2+(yi− ỹn)
2
)
=
1

2
(nxxi+nyyi) on (xi,yi)∈ (AB ∪EA)

(3.3)

where n = [nx,ny] is the unit outward normal vector at the boundary.

If the number of collocation points Mc is equal to the number of unknown coefficients,
{cn}

N
n=1Mc = N, the system of algebraic equations (3.3) can be solved by the Gaussian elimi-

nation method. Otherwise, if Mc > N, system (3.3) is overdetermined and is solved through
application of the least squares approach.

The dimensionless velocity profile can be expressed as

w =
N∑

n=1

cn lnr
2
n−
1

4
(x2+y2) (3.4)

Similarly, the solution to (2.16) can bewritten as a linear combination of fundamental solutions
for the Laplace operator

Θs(x,y) =
Nd∑

n=1

dn lnrdn (3.5)

where rdn =
√
(x− x̃dn)

2+(y − ỹdn)
2 and {(x̃dn, ỹdn)

}Nd
n=1
are source points located around the

fin area.

The homogenous solution to (2.15) can be written as

Θh(x,y)=
Nf∑

n=1

fn lnrfn (3.6)

where rfn =
√
(x− x̃fn)

2+(y − ỹfn)
2 and {(x̃fn, ỹfn)

}Nf
n=1
are source points located around

the fluid area.

To obtain the particular solution to (2.15), we propose the use of RBF andmonomials. The
solution can be expressed as

Θp(x,y)=
Mi∑

m=1

amψ(rm)+
K∑

j=1

bjqj(x,y) (3.7)

where rm =
√
(x− x̂m)2+(y− ŷm)2 and {(x̂m, ŷm)}

Mi
m=1 are interpolation points located in the

fluid area, ψ(rm) is the particular solution of the RBS ϕ(rm) for the Laplace operator, and
qj(x,y) is the particular solution of the monomials pj(x,y) for the Laplace operator

∇2ψ(rm)= ϕ(rm) m =1, . . . ,Mi

∇2qj(x,y)= pj(x,y) j =1, . . . ,K
(3.8)



Application of the method of fundamental solutions to... 511

The coefficients {am}
Mi
m=1 and {bj}

K
j=1 are calculated by interpolation of the right hand side of

equation (2.15)

Mi∑

m=1

amϕ(rmi)+
K∑

j=1

bjpj(x̂i, ŷi)= fRew(x̂i, ŷi)
Θ(x̂i, ŷi)

Θb
i =1, . . . ,Mi

M∑

m=1

ampj(x̂m, ŷm)= 0 j =1, . . . ,K

(3.9)

The coefficients {dn}
Nd
n=1 and {fn}

Nf
n=1 are obtained through fulfilling the boundary conditions

(2.17) and (2.18) in the collocation points

Nf∑

n=1

fn
∂ lnrfni
∂n

=−
Mi∑

m=1

am
∂ψ(rmi)

∂n
−
K∑

j=1

bj
∂qj(xi,yi)

∂n

{(xi,yi)}
M1+M5
i=1 ∈ (AB ∪EA)

Nf∑

n=1

fn lnrfni =−
Mi∑

m=1

amψ(rmi)−
K∑

j=1

bjqj(xi,yi) {(xi,yi)}
M2
i=1 ∈ BC

Nf∑

n=1

fn
∂ lnrfni
∂n

+
Nd∑

n=1

dn
∂ lnrdni
∂n

=−
Mi∑

m=1

am
∂ψ(rmi)

∂n
−
K∑

j=1

bj
∂qj(xi,yi)

∂n

{(xi,yi)}
M3+M4
i=1 ∈ (CD∪DE)

Nf∑

n=1

fn lnrfni +
Nd∑

n=1

dn lnrdni =−
Mi∑

m=1

amψ(rmi)−
K∑

j=1

bjqj(xi,yi)

{(xi,yi)}
M3+M4
i=1 ∈ (CD∪DE)

Nd∑

n=1

dn
∂ lnrdni
∂n

=0 {(xi,yi)}
M3
i=1 ∈ EF

Nd∑

n=1

dn lnrdni =0 {(xi,yi)}
M4
i=1 ∈ CF

(3.10)

The solution to (2.15) can be written as a sum

Θ(x,y)=
Nf∑

n=1

fn lnrfn +
Mi∑

m=1

amψ(rm)+
K∑

j=1

bjqj(x,y) (3.11)

Since the right hand sideof governing equation (2.15) dependson the temperatureΘ(x,y) aswell
as the bulk temperature value Θb, we developed the following iterative procedure ascertaining
its successful solution. First, we assumed uniform temperature conditions throughout the area
Θ/Θb = 1. The results are substituted into the right hand side of equation (2.15), which is
then numerically solved for new values of Θ. Based on these, a new value Θb is calculated and
all results are again substituted into the right hand side of equation (2.15). The calculations
are repeated until the values of Θ converge to acceptable tolerance values and carried out in
accordance with the presented algorithm.

Step 0 Input of the data M, β, k, and rw

Step 1 Determination of the optimal parameter s characterized by the smallest error of the
boundary value of temperature



512 K.Mrozek,M.Mierzwiczak

Step 2 Determination of the velocity value (2.2)-(2.4) byMFS (3.1)-(3.4)

Step 3 Computation of the bulk velocity wb, (2.5), and the product fRe, (2.7)

Step 4 Assumption that Θ0/Θ0b =1, i =1

Step 5 Solution of interpolation problem (2.15) by RBF andmonomials (3.9)

Step 6 Determination of temperature (2.15)-(2.18) by MFS (3.10) and (3.11)

Step 7 Calculation of the bulkmean temperature Θib, (2.19)1, and theNusselt number, (2.19)2

Step 8 Convergence verification:

if δΘ = ‖Θi−Θi−1‖¬ tol STOP,

else, take i = i+1 and go to Step 5

4. Results and discussion

In numerical experiments, as in those employing RBF, we use the thin-plate spline function

ϕ(rm)= r
2
m lnrm (4.1)

for which the particular solution for the Laplace operator has the form

ψ(rm)=
r4m lnrm
16

−
r4m
32

(4.2)

and six monomials presented in Table 1.

Table 1.Monomial functions and their particular solutions

j pj(x,y) qj(x,y)

1 1 (x2+y2)/4

2 x x(x2+y2)/8

3 y y(x2+y2)/8

4 xy xy(x2+y2)/12

5 x2
(
x4+x2y2−

y4

6

)
/14

6 y2
(
y4+x2y2− x

4

6

)
/14

An example of the distribution of collocation, the source and the interpolation points for the
fluid area Ωf and for the fin area Ωs is shown in Fig. 2.
We calculate the number of the collocation and source points by means of the following

formulas

M1 =111 N1 =M1/3

M2 =4α/γM1/M +1 N2 =(N2+N4)/3

M3 = M1(1−rw)+1 N3 =M3/3

M4 =4β/γM1/M +1 N4 =M4/2

M5 = M1rw+1

(4.3)

To interpolate the right side function in (2.15), 378 evenly located points in the considered
fluid area (x̂, ŷ) ∈ Ωf are used. The parameter s proved to have a substantial impact on the



Application of the method of fundamental solutions to... 513

Fig. 2. Exemplary distribution of � Mc – collocation,× N – source, and • Mi – interpolation points
(a) in the fluid area and (b) in the solid fin area

accuracy of the presented method. Therefore, at the beginning of the calculation procedure,
the optimal value of s is determined, for which the maximum error of the boundary value of
temperature in the control points is the lowest.

In order to verify the accuracy of the proposed algorithm, as the first test example, we have
consitered a smooth tube without fins. The results of our calculations for different values of γ
are presented in Table 2 and are consistent with the literature, see e.g. Soliman et al. (1980).
This allowed us, even at this early stage, to confirm the effectiveness of the proposed algorithm.

Table 2.Numerical results of the smooth tube investigation for different γ values

M γ [◦] sOPT fRe Nu δΘ

4 45 0.2121 16 3.6787 5.35E-06

8 22.5 0.1913 16 3.6773 5.47E-06

12 15 0.1812 16 3.6735 5.44E-06

16 11.25 0.1756 16 3.6707 5.41E-06

20 9 0.1721 16 3.6688 5.42E-06

24 7.5 0.1697 16 3.6676 5.41E-06

28 6.429 0.1679 16 3.6667 5.39E-06

32 5.625 0.1666 16 3.6662 5.32E-06

Furthermore, comparison between this study and previous work of Soliman and Feingold
(1977) and Soliman et al. (1980) has been made to validate the postulated method. Figures
3a and 3b show the comparison of Nu values obtained in this work and those of Soliman et
al. (1980) for k = 1,5, and 10 and for M = 4 and 8. The maximum discrepancies in Nusselt
numbers proved lower than 9% for M = 4 and 12% for M = 8. This is presumably due to
the hereby assumed two-dimensional heat transfer, which differs from the one-dimensional fin
conductance in the tube.

Further numerical results of fRe for β = 3◦, and for varying fin numbers and lengths, are
presented in Fig. 4. The value of fRe increases with the increase in M for all values of L. The
effect of M is much more appreciable for longer fins.

TheNusselt number, defined by (2.19)2, was used as ameasure of the overall performance of
any heat transfer surface; in other words, Nu reflects the influence of the internal finning on the
overall heat transfer performance. The calculated values of Nu corresponding to the same value
of the half of the angle of one fin, β =3◦, are listed in Table 3 for k = {1,5,10,100} illustrating



514 K.Mrozek,M.Mierzwiczak

Fig. 3. Comparison of the results obtained in the present study with those presented by Soliman et al.
(1980) for the solution of fRe for (a) M =4 and (b) M =8

Fig. 4. The friction factor fRe for β =3◦

different tube geometries. Obviously, the magnitude of heat transfer enhancement depends on
M, L, and k. It can be observed that for any tube geometry, the value of Nu increases as k
increases. For a given value of L and k, in most cases, the maximum value of Nu is obtained
for the tube with eight fins (M =8). The results depicted in Table 3 also show that the effect
of k is more visible for longer fins (L ­ 0.7) when there are 4, 8, or 16 of them. For the tube
with eight fins and 0.8 in length, for k = 1 Nu = 16.719 and for k = 100 the Nusselt number
was more than two times greater (Nu=35.254). Comparing these results to those obtained for
the smooth tube, the Nusselt number increases almost five times while, at the same time, the
resistance increases more than 14 times.

A similar numerical experiment has been performed for constant values of the dimensionless
flow area of the tube, Af =2.7 (Table 4). For such a geometry of the tube, which changes the
angle of the fins, β =(π−Af)/[ML(2−L)], the largest value of Nu is obtained for M =16 and
L =0.8.

However, in this case, we deal with a high resistance value, fRe= 671.92. Considering the
same example, for a fixed value of β = 3◦ (Table 3),we observe that the resistance is much
higher, fRe = 975.96 (since Af = 2.337 is smaller), and the value of the Nusselt number is
smaller (although β =3◦ is larger). In the case of short fins (L ¬ 0.5), the largest reinforcement
of theheat transfer (Nu) is obtained for the tubewith fourfins (Table 4).This is advantageous, as
in tubeswith fewer fins the resistance fRe is lower than in theirmulti-finned counterparts. From
the numerical experiments, it appears that in order to intensify heat transfer, it is preferable to
use tubeswith slimfins, since, for M =16 and L =0.8, better results are obtained for β =1.647
and Af =2.7 (Table 4) than for β =3 and Af =2.337 (Table 3).

Figures 5 and 6 illustrate the effect of the angle β of the fins of length L = 0.8 on the
enhancement of heat conduction, Nu and resistance to flow fRe. The results show that fRe and
the width of the fins also play significant roles in enhancing the heat transfer.



Application of the method of fundamental solutions to... 515

Table 3.Overall heat transfer results for β =3◦ and Af = π−MβL(2−L)

M =4 M =8 M =16 M =24) M =32

L k fRe Nu fRe Nu fRe Nu fRe Nu fRe Nu

0.2

1
19.87
Af =
3.066

3.729
24.39
Af =
2.991

3.734
31.88
Af =
2.840

3.642
35.74
Af =
2.689

3.587
37.48
Af =
2.538

3.575
5 3.793 3.843 3.739 3.662 3.662
10 3.803 3.852 3.751 3.695 3.672
100 3.812 3.864 3.773 3.701 3.672

0.4

1
31.47
Af =
3.008

4.172
55.50
Af =
2.874

4.100
93.89
Af =
2.605

3.647
110.85
Af =
2.337

3.480
117.90
Af =
2.069

3.487
5 4.596 4.617 3.933 3.696 3.650
10 4.685 4.681 3.981 3.737 3.684
100 4.745 4.758 4.018 3.771 3.693

0.5

1
42.04
Af =
2.985

4.737
89.79
Af =
2.827

4.737
179.13
Af =
2.513

3.801
223.23
Af =
2.199

3.435
242.01
Af =
1.885

3.392
5 5.615 5.663 4.255 3.748 3.637
10 5.796 5.809 4.319 3.820 3.678
100 5.983 5.960 4.394 3.884 3.719

0.6

1
56.12
Af =
2.966

5.816
141.01
Af =
2.790

6.433
354.37
Af =
2.438

4.429
503.75
Af =
2.086

3.649
575.44
Af =
1.734

3.397
5 7.754 8.354 5.158 4.047 3.741
10 8.245 8.715 5.281 4.121 3.784
100 8.776 9.088 5.400 4.190 3.826

0.7

1
71.60
Af =
2.951

7.541
198.43
Af =
2.760

11.219
655.25
Af =
2.379

7.005
1228.03
Af =
1.998

4.559
1663.91
Af =
1.617

3.567
5 11.603 17.660 8.654 5.352 4.025
10 12.859 18.983 8.942 5.497 4.092
100 14.363 20.276 9.243 5.593 4.149

0.8

1
83.13
Af =
2.941

8.845
233.49
Af =
2.739

16.719
975.96
Af =
2.337

20.063
2762.44
Af =
1.935

8.843
5628.56
Af =
1.533

5.000
5 13.380 29.398 28.323 11.347 5.858
10 14.614 32.372 29.694 11.682 5.975
100 15.960 35.254 31.272 11.977 6.067

Further, it can be observed that fRe varies linearly both for M = 4 and M = 8. However,
when it comes to Nu, for M = 4 we have found some regularity, and for M = 8 a significant
increase inNuhasbeenobtained forβ =1.8 (the result comparable to that obtained forβ =2.8)
and β =2.0 (the result comparable to that obtained for β =3.0). For both angles β we observed
enhancement of heat conduction while maintaining lower resistance fRe.

Fig. 5. The effects of the fin angle β on the Nusselt number Nu and the product of the friction factor
and the Reynolds number fRe for k =10, L =0.8, and M =4



516 K.Mrozek,M.Mierzwiczak

Table 4.Overall heat transfer results for Af =2.7 and β =(π−Af)/[ML(2−L)]

M =4 M =8 M =16 M =24) M =32

L k fRe Nu fRe Nu fRe Nu fRe Nu fRe Nu

0.2

1
26.33
β =
17.57

3.526
30.11
β =
8.785

3.611
34.12
β =
4.393

3.597
35.46
β =
2.928

3.578
36.45
β =
2.196

3.602
5 3.781 3.761 3.711 3.682 3.660
10 3.822 3.800 3.717 3.690 3.675
100 3.860 3.827 3.736 3.696 3.686

0.4

1
41.53
β =
9.883

3.992
63.82
β =
4.942

3.987
90.81
β =
2.471

3.710
101.76
β =
1.647

3.597
107.48
β =
1.235

3.571
5 4.634 4.484 4.014 3.795 3.736
10 4.767 4.562 4.055 3.825 3.775
100 4.892 4.634 4.098 3.876 3.791

0.5

1
54.83
β =
8.434

4.582
99.61
β =
4.217

4.539
163.91
β =
2.108

3.940
194.40
β =
1.406

3.683
208.19
β =
1.054

3.604
5 5.758 5.507 4.410 4.044 3.920
10 6.011 5.655 4.491 4.096 3.958
100 6.278 5.779 4.585 4.126 3.991

0.6

1
71.69
β =
7.530

5.720
155.77
β =
3.765

6.241
302.24
β =
1.883

4.823
393.19
β =
1.255

4.146
450.78
β =
0.941

3.856
5 8.128 8.243 5.664 4.712 4.260
10 8.742 8.551 5.775 4.784 4.328
100 9.473 8.838 5.964 4.857 4.367

0.7

1
89.15
β =
6.951

7.505
207.91
β =
3.475

11.068
505.72
β =
1.738

8.221
782.40
β =
1.158

6.069
1015.07
β =
0.869

5.241
5 12.255 17.470 10.333 7.254 5.992
10 13.775 18.660 10.842 7.435 6.039
100 15.625 19.885 11.043 7.497 6.039

0.8

1
100.42
β =
6.589

8.730
242.48
β =
3.294

16.685
671.92
β =
1.647

24.696
1248
β =
1.098

17.056
1821.94
β =
0.824

12.921
5 13.507 29.539 36.785 21.135 15.438
10 14.785 32.512 38.967 21.879 15.737
100 16.117 35.340 40.993 22.791 15.858

Fig. 6. The effects of the fin angle β on the Nusselt number Nu and the product of the friction factor
and the Reynolds number fRe for k =10, L =0.8 and for M =8

5. Conclusions

In this paper, we have employed MFS with RBF to investigate a fully developed laminar
flow and convective heat transfer in an internally finned tube. The presented method is ve-
ry easy to implement, even in the case of highly challenging domains, because requires a clo-
ud of points only. The hereby presented numerical results, pertaining to diverse experimen-
tal data, show that MFS is an accurate and reliable numerical technique generating solutions
comparablewith the literature. The proposed scheme is a competitive alternative to the existing



Application of the method of fundamental solutions to... 517

methods of heat transfer investigation. The numerical analysis shows that the usage of finned
cooling channels can not only contribute to the improvement of cooling effectiveness, but also,
through proper fin placement, the enable manipulation of heat conduction in the mold. This
finding is very significant from the technological point of view, as it will allow decreasing the
stress strain of the mold, thereby improving its quality andmechanical properties.

Acknowledgments

The work has been supported by LIDER/006/143/L-5/13/NCBR/2014Grant.

References

1. Alves C.J.S., Silvestre A.L., 2004, Density results using Stokeslets and a method of funda-
mental solutions for the Stokes equations, Engineering Analysis with Boundary Elements, 28, 10,
1245-1252

2. Benitez-Rangel J.P., Trejo-Hernandez M., Morales-Hernandez L.A., Dominguez-
-Gonzalez A., 2010, Improvement of the injection mold process by using vibration through a
mold accessory,Materials and Manufacturing Processes, 25, 7, 577-580

3. Bergles E., 1998.,Handbook of Heat Transfer, McGraw-Hill, NewYork, NY, USA, 3rd edition

4. ChenC.S.,KarageorghisA., SmyrlisT.S. (ed.), 2008,TheMethod of Fundamental Solutions
– A Meshless Method. Dynamic Publishers, Inc., Atlanta

5. Golberg M.A., 1995, The method of fundamental solutions for Poisson’s equation, Engineering
Analysis with Boundary Elements, 16, 3, 205-213

6. GolbergM.A., ChenC.S., 1999,Themethod of fundamental solutions for potential, Helmholtz
and diffusion problems, [In:]Boundary integral methods: numerical and mathematical aspects. So-
uthampton: Computational Mechanics Publications, GolbergM.A. (Edit.), 103-176

7. Johnston R.L., Fairweather G., 1984, The method of fundamental solutions for problems in
potential flow,Applied Mathematical Modelling, 8, 4, 265-270

8. Karageorghis A., Fairweather G., 1987, Themethod of fundamental solutions for the nume-
rical solution of the biharmonic equation, Journal of Computational Physics, 69, 2, 434-59

9. Karageorghis A., Fairweather G., 1998, The method of fundamental solutions for elliptic
boundary value problems,Advances in Computational Mathematics, 9, 69-95

10. KupradzeV.D.,AleksidzeM.A., 1964,Themethodof functional equations for the approximate
solution of certain boundary value problems,USSRComputational Mathematics andMathematical
Physics, 4, 4, 82-126

11. MathonR., JohnstonR.L., 1977,The approximate solutionof elliptic boundary-valueproblems
by fundamental solutions, SIAM Journal on Numerical Analysis, 14, 4, 638-650

12. Nesis E. I., Shatalov A. F., Karmatskii N. P., 1994, Dependence of the heat transfer coef-
ficient on the vibration amplitude and frequency of a vertical thin heater, Journal of Engineering
Physics and Thermophysics, 67, 1/2, 696-698

13. Rout S.K., Mishra D. P., Thatoi D. N., Acharya A. K., 2012, Numerical analysis of mi-
xed convection through an internally finned tube, Advances in Mechanical Engineering, Vol.2012,
Article ID 918342, p. 10

14. SiddiqueM., KhaledA.-R.A., Abdulhafiz N.I., BoukharyA.Y., 2010,Recent advances in
heat transfer enhancements: a review report, International Journal of Chemical Engineering, vol.
2010, Article ID 106461, p. 28, doi:10.1155/2010/106461

15. Soliman H.M., Chau T.S., Trupp A.C., 1980, Analysis of laminar heat transfer in internally
finned tubes with uniform outside wall temperature, Journal of Heat Transfer, 102, 598-604

16. Soliman H.M., Feingold A., 1977, Analysis of fully developed laminar flow in longitudinal
internally finned tubes,Chemical Engineering Journal, 14, 119-128



518 K.Mrozek,M.Mierzwiczak

17. TienW.-K.,YehR.-H.,Hsiao, J.-C., 2012,Numerical analysis of laminarflowandheat transfer
in internally finned tubes,Heat Transfer Engineering, 33, 11, 957-971

18. TsaiC.C., 2007,Themethodof fundamental solutions for three-dimensional elasto-staticproblems
of transversely isotropic solids,Engineering Analysis with Boundary Elements, 31, 7, 586-594

19. TsaiC.C.,YoungD.L.,ChiuC.L.,FanC.M., 2009,Numerical analysis of acousticmodesusing
the linear least squaresmethodof fundamental solutions,The Journal of Sound andVibration,324,
3/5, 1086-1100

Manuscript received January 14, 2014; accepted for print March 11, 2014