Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

46, 1, pp. 69-83, Warsaw 2008

ENTROPY GENERATION DUE TO NON-NEWTONIAN

FLUID FLOW IN ANNULAR PIPE WITH RELATIVE

ROTATION: CONSTANT VISCOSITY CASE

Ali Kahraman

Technical Education Faculty, Selcuk University, Konya, Turkey

e-mail: alikahraman@selcuk.edu.tr

Muhammet Yürüsoy

Technical Education Faculty, Afyon Kocatepe University, Afyon, Turkey

e-mail: yurusoy@aku.edu.tr

Entropy generation due to Non-Newtonian fluid flow in an annular pipe
with relative rotation is investigated. A third grade fluid with constant
viscosity is accommodated in the analysis. Relative rotationalmotion is
present between inner and outer cylinders, which induces the flow. Ana-
lytical solutions forvelocityand temperaturedistributions arepresented,
and entropy generation number is computed for different dimensionless
values of non-Newtonian viscosity, Brinkman’s number and velocity ra-
tio. It is found that the increasing of dimensionless non-Newtonianvisco-
sity lowers the number entropy generation. This is more pronounced in
the region close to the annular pipe inner wall. The increasing of Brink-
man’s number enhances the number entropy generation, particularly in
the vicinity of the annular pipe inner wall.

Key words: non-Newtonian fluid, third grade fluid, entropy generation
number

1. Introductin

Flow through annular pipes with relative rotation finds many significant en-
gineering applications. In addition to heat transfer situations, the resulting
flow is particularly applicable to rotating electrical machines, swirl nozzles,
rotating disks, standard commercial rheometers, and other chemical and me-
chanical mixing equipment, see Maron and Cohen (1991). Considerable rese-
arch studies were carried out to investigate the non-Newtonian fluid flow. The



70 A. Kahraman, M. Yürüsoy

modeling of viscoelastic materials using differential constitutive equationswas
introduced by Peters and Baaijens (1997). Their approach was based on the
concept of a slip tensor and elastic behavior of structure. Dunn and Fosdick
(1974) performed a complete thermodynamic analysis for fluids of the second
grade,whileFosdick andRajagopal (1990) carriedout thermodynamicanalysis
for the third grade fluid. Moreover, Szeri and Rajagopal (1985) investigated
the flow of a third grade fluid between two heated horizontal plates. They
assumed that the shear viscosity was temperature dependent. The effects of
variable viscosity and viscous dissipation on the flow of a third grade fluid in
a pipe were investigated by Massoudi and Christie (1995). They showed that
as dimensionless non-Newtonian viscosity increased, themagnitude of velocity
and temperature decreased in the pipe. The approximate analytical solutions
for the flow of a third grade fluid in a pipe were presented by Yürüsoy and
Pakdemirli (2002). They showed that if certain criteria for the fluid flow we-
re met, the approximate analytical solutions agreed well with the previous
numerical results of Massoudi and Christie (1995).
Thermodynamic irreversibility occurring in a flow system gives insight in-

to losses associated with the system. In this case, two major losses can be
accounted – due to heat transfer and due to frictional losses. Heat transfer
losses alter the thermodynamic work potential of the system while frictio-
nal losses lower the pressure in it. Moreover, entropy generation within the
system quantifies thermodynamic irreversibility. Consequently, investigation
into thermodynamic irreversibility associated with the thermal system thro-
ugh entropy analysis is faithful. Entropy generation in rotational flow systems
results from fluid friction and heat transfer. Moreover, entropy analysis pro-
vides information for quantification of thermodynamic irreversibility in the
flow system. Consequently, entropy minimization lowers frictional and heat
transfer losses in the system. Considerable research studies were carried out
to examine entropy generation in thermal systems. Bejan (1995) examined the
entropy generation and minimization in thermal systems. He indicated that
entropy minimization could be used as an effective tool for designing thermal
systems. Convective heat transfer in an annular packed bed was investigated
by Demirel and Kahraman (2000). They indicated that the volumetric entro-
py generation map could be used to identify the excessive entropy generation
due to operating conditions or design parameters for a required task. The
Shannon entropy characteristics of two-phase flow systems were examined by
Zhang and Shi (1999). They found that the entropy generation of the bubble
flowwas the smallest, the slugflow entropy generationwas the largest, and the
entropy generation of the annular flow was between the bubble flow and slug
flow. Mahmud and Fraser (2002)] reported the inherent irreversibility of the



Entropy generation due to non-Newtonian fluid flow... 71

fluid flow and heat transfer for non-Newtonian fluids in a pipe and a channel
flow. They presented the average entropy generation number graphically for
both pipe and channel flows. A study of entropy generation in fundamental
convective heat transfer was carried out byBejan (1979). He showed that flow
geometric parametersmight be selected in order tominimize the irreversibility
associated with a specific convective heat transfer process. Another studywas
done by Sahin (1998) who introduced a second law analysis of viscous fluids
in a circular duct under an isothermal boundary condition. In a more recent
paper, Sahin (1999) presented the effect of variable viscosity on the entro-
py generation rate for a constant heat flux boundary condition for a circular
duct. The non-Newtonian fluid flow in an annular pipe without rotation was
investigated byYilbas et al. (2004). Theypresented analytical solutions for ve-
locity and temperature fields by considering a third grade fluid with constant
viscosity.

To investigate the thermodynamic irreversibility in the non-Newtonian an-
nular pipe flow with relative rotation, the present study is carried out. The
governing equations of non-Newtonian fluids in cylindrical coordinates are so-
lved using the perturbation method. The velocity and temperature fields are
presented analytically after considering the third gradefluidmodel.The closed
form solutions for entropy generation due to fluid friction and heat transfer
are obtained, and the entropy number is computed for various values of di-
mensionless non-Newtonian viscosity and different Brinkman’s numbers.

2. Velocity and temperature profiles

Consider the non-Newtonian fluid flow between two concentric cylinders as
shown in Fig.1. A non-dimensional form of equations of motion of a third
grade fluid in an annular pipe with relative rotation and heat transfer was
derived by Beard andWalters (1964)

µ
(

v′′+
v′

r
−
v

r2

)

+Λ
(

v′−
v

r

)2(

6v′′−
2v′

r
+
2v

r2

)

=0

(2.1)

θ′′+
θ′

r
+Γ
(

v′−
v

r

)2[

µ+Λ
(

v′−
v

r

)2]

=0

and

v(1)= 1 v(R)=n θ(1)= 0 θ(R)= 1 (2.2)



72 A. Kahraman, M. Yürüsoy

where r is the dimensionless radius, v is the dimensionless tangential velocity
component, θ is the dimensionless temperature and µ is the dimensionless
viscosity. The terms are related to dimensional ones through the following
relations

R=
ro

ri
r=
r

ri
v=

v

riωi

θ=
θ−θi

θo−θi
µ=
µ

µ∗
n=
roωo

riωi

(2.3)

where R is the radius ratio, ri is the dimensional radius of the inner cylin-
der, ro is the dimensional radius of the outer cylinder, dimensional angular
velocities are denoted by ωi and ωo for the inner and outer cylinders, θi and
θo are the inner outer cylinder dimensional temperatures, µ∗ is the reference
viscosity, n is the velocity ratio. For positive n, both cylinders rotate in the
same direction and for negative n they rotate in the opposite directions.

Fig. 1. A schematic view of the annular pipe

The dimensionless parameters involved in equations (2.1) are

Γ =
µ∗(riωi)

2

k(θo−θi)
Λ=
βω2i
µ∗

(2.4)

where Γ is the Brinkman number, Λ is the dimensionless parameter related
to the non-Newtonian behavior, β is the dimensional material constant for
the third grade fluid and k is the thermal conductivity.

In this Section, velocity profiles will be calculated approximately. Appro-
ximate solutions can be obtained by selecting Λ = ǫλ, where ǫ is our per-
turbation parameter, a small quantity, and λ is the ordered non-Newtonian
coefficient. The approximate velocity and temperature profiles can then be
written as

v= v0+ ǫv1 θ= θ0+ ǫθ1 (2.5)



Entropy generation due to non-Newtonian fluid flow... 73

Assuming constant viscosity, the non-dimensional viscosity can be taken as
one (µ=1). Substituting all terms into the original equations of motion and
separating at each order of ǫ, one has:

• order 1

v′′0 +
v′0
r
−
v0

r2
=0 θ′′0 +

θ′0
r
+Γ
(

v′0−
v0

r

)2

=0 (2.6)

and

v0(1)= 1 v0(R)=n θ0(1)= 0 θ0(R)= 1 (2.7)

• order ǫ

v′′1 +
v′1
r
−
v1

r2
=−λ

(

v′0−
v0

r

)2(

6v′′0 −
2v′0
r
+
2v0
r2

)

(2.8)

θ′′1 +
θ′1
r
=−Γ

[

2v′0v
′

1+
2

r2
v0v1−

2

r
v0v
′

1−
2

r
v′0v1+

+λ
(

−
4

r3
v30v
′

0+v
′

0
4
+
v40
r4
−
4

r
v′0
3
v0+

6

r2
v′0
2
v20

)]

and

v1(1)= 0 v1(R)= 0 θ1(1)= 0 θ1(R)= 0 (2.9)

For the first order, solutions satisfying the boundary conditions are

v0 =
(R2−r2)+nR(r2−1)

r(R2−1)
(2.10)

θ0 =
lnr

lnR
+
ΓR2(R−n)2

(R2−1)2

[

1−
1

r2
−
lnr

lnR

(

1−
1

R2

)]

Substituting these solutions to equations of the order ǫ, one finally obtains

v1 =
8λ(R−n)3

3(R2−1)3

(r4(R4+R2+1)−r6(R2+1)−R4

r5R

)

(2.11)

θ1 =
A1

36

[ lnr

lnR

(

1−
1

R6

)

+
1

r6
−1
]

−
A2

4

[ lnr

lnR

(

1−
1

R2

)

+
1

r2
−1
]



74 A. Kahraman, M. Yürüsoy

where

A1 =
16ΓλR4(R−n)4

(R2−1)4
(2.12)

A2 =
64Γλ(R−n)4(R4+R2+1)

3(R2−1)4

Combining the solutions at each order of approximation and returning back
to the original dimensionless parameters, one finally has

v=
(R2−r2)+nR(r2−1)

r(R2−1)
+

+
8Λ(R−n)3

3(R2−1)3

(r4(R4+R2+1)−r6(R2+1)−R4

r5R

)

(2.13)

θ=
lnr

lnR
+
ΓR2(R−n)2

(R2−1)2

[

1−
1

r2
−
lnr

lnR

(

1−
1

R2

)]

+

+
ǫA1

36

[ lnr

lnR

(

1−
1

R6

)

+
1

r6
−1
]

−
ǫA2

4

[ lnr

lnR

(

1−
1

R2

)

+
1

r2
−1
]

The perturbation solution is valid if the correction terms are much smaller
than the leading terms. Since there are many physical parameters involved,
analytical formulas for validity criteria cannot be accomplished. For a simpler
case of the normal pipe flow, such criteria have already been presented by
Yürüsoy and Pakdemirli (2002). In our case, as well as in the simpler case of
Yürüsoy and Pakdemirli (2002), validity does not depend on one parameter,
but on a combination of parameters. In all numerical computations, the vali-
dity is ensured by making numerical values of correction terms much smaller
than the leading terms.

3. Viscous dissipation and entropy generation

The dimensional viscous dissipation term (φ) can be obtained from the equ-
ations of motion, i.e.

φ=µ
(dv

dr
−
v

r

)2

+2β
(dv

dr
−
v

r

)4

(3.1)

or ehen inserting the dimensionless quantities

φ=µ∗ω
2
i

(dv

dr
−
v

r

)2[

µ+2Λ
(dv

dr
−
v

r

)2]

(3.2)



Entropy generation due to non-Newtonian fluid flow... 75

The dimensional volumetric entropy generation is defined as Bejan (1995)

S′′′gen =
k

T
2

0

(dθ

dr

)2

+
φ

T0
(3.3)

where T0 is the reference temperature. The first term in equation (3.3) is the
volumetric entropy generation due to heat transfer and the second term is the
entropy generation due to viscous dissipation. Substituting equation (3.2) into
(3.3), expressing the terms in dimensionless forms, one finally obtains

NS =
(dθ

dr

)2

+T0Γ
(dv

dr
−
v

r

)2[

µ+2Λ
(dv

dr
−
v

r

)2]

(3.4)

where NS is the entropy generation number. It is defined by dividing the
dimensional volumetric entropy generation to the reference volumetric entropy
generation S′′′G. The relevant definitions are

NS =
S′′′gen

S′′′G
S′′′G =

k(θo−θi)
2

T
2

0r
2
i

T0 =
T0

θ0−θi
(3.5)

In equation (3.4), the first termdue to heat generation can be assigned as NS1
and the second term due to viscous dissipation as NS2, i.e.

NS1 =
(dθ

dr

)2

NS2 =T0Γ
(dv

dr
−
v

r

)2[

µ+Λ
(dv

dr
−
v

r

)2]

(3.6)

Assuming constant viscosity, the non-dimensional viscosity can be taken as
one (µ=1). Since the temperature and velocity profiles are known functions,
they can be taken fromequations (2.13) and then inserted into equations (3.4)
and (3.6) for final evaluations of the entropy generation numbers.

4. Results and discussions

The non-Newtonian fluid flow in an annular pipe with relative rotation is
considered in the paper. Entropy generation in the flow field due to fluid
friction and heat transfer is formulated. The influence of Brinkman’s number
and dimensionless non-Newtonian viscosity on the entropy generation number
is examined. In the analysis, the constant viscosity case is assumed.
Figure 2 shows velocity profiles along the radial distance in the annular

pipe for different dimensionless values of the non-Newtonian viscosity. In the



76 A. Kahraman, M. Yürüsoy

Fig. 2. Velocity profiles along the pipe radius for different dimensionless
non-Newtonian viscosities Λ: n=0, Γ =1

case of a Newtonian fluid (Λ=0), the velocity profile decays gradually along
the radial distance. However, the velocity increases with an increase in the
dimensionless non-Newtonian viscosity (Λ).

In Fig.3 the velocity distribution for different velocity ratios n in plotted.
The velocity decreases along the radial direction and exhibits the minimum
value at the outer cylinder up to n ≈ 0.5. For 0.5 < n ¬ 1, the minimum
velocity occurs inside the annular gap.

Fig. 3. Velocity profiles along the pipe radius for different velocity ratios n: Γ =1,
Λ=0.03

Figures 4 and 5 show temperature distribution in the annular pipe for dif-
ferent dimensionless non-Newtonian viscosities and Brinkman’s numbers Γ ,
respectively. In Fig.4, growth of the dimensionless non-Newtonian viscosity
reduces the temperature in the flow field. A plot of the radial distance ver-



Entropy generation due to non-Newtonian fluid flow... 77

sus temperature is given in Fig.5 for different values Γ . As Γ increases, the
temperature inside the radial direction increases due to the dissipation effect.

Fig. 4. Temperature profiles along the pipe radius for different dimensionless
non-Newtonian viscosities Λ: n=0, Γ =1

Fig. 5. Temperature profiles along the pipe radius for different Brinkman’s
numbers Γ : n=0,Λ=0.01

Figure6 showstheentropygenerationnumberduetoheat transfer fordiffe-
rent dimensionless non-Newtonian viscosities. The entropy generation number
attains high values in the vicinity of the annular pipewall, which ismore pro-
nounced in the region close to the innerwall of the pipe (seeFig.6). Reduction
of the dimensionless non-Newtonian viscosity increases the entropy generation
rate, in which case the rate of heat transfer enhances.
In Fig.7 for n = 0, Λ = 0.03 the entropy generation number versus the

radial distance is plotted for different Brinkman’s numbers. The entropy ge-
neration number is high in magnitude near the inner cylinder due to high
gradient of temperature. The entropy generation number falls then exponen-
tially along the radial distance and approaches an asymptote near the outer
cylinder. The entropy generation number increases with growth ofBrinkman’s
number.



78 A. Kahraman, M. Yürüsoy

Fig. 6. Entropy generation number due to heat transfer along the pipe radius for
different dimensionless non-Newtonian viscosities Λ: n=0, Γ =1

Fig. 7. Entropy generation number due to heat transfer along the pipe radius for
different Brinkman’s numbers Γ : n=0,Λ=0.03

Figures 8 and 9 show the entropy generation number due to fluid friction
for different non-Newtonian viscosities andBrinkman’s numbers, respectively.
The entropy generation number attains high values in the inner pipe wall
region. Reduction of the dimensionless non-Newtonian viscosity increases the
magnitude of entropy generation number in this region (Fig.8̇). This is because
of the fluid strain, which is high close to the annular pipe wall. Moreover, the
maximum entropy generation number moves away from the pipe wall for a
dimensionless non-Newtonian viscosity less than 0.03. In the case of Fig,9,
the increasing Brinkman number increases the entropy generation number,
particularly in the region close to the annular pipe inner wall.
The effect of the velocity ratio n on the total entropy generation number

is plotted in Fig.10 for Λ = 0.03, T0 = 1.07, Γ = 1. The inner wall still



Entropy generation due to non-Newtonian fluid flow... 79

Fig. 8. Entropy generation number due to fluid friction along the pipe radius for
different dimensionless non-Newtonian viscosities Λ: n=0, Γ =1, T0 =1.07

Fig. 9. Entropy generation number due to fluid friction along the pipe radius for
different Brinkman’s numbers: n=0,Λ=0.03, T0 =1.2

Fig. 10. Entropy generation number due to heat transfer and fluid friction along the
pipe radius for different velocity ratios n: Λ=0.03, T0 =1.07, Γ =1



80 A. Kahraman, M. Yürüsoy

acts as a strong concentrator of irreversibility, but now the magnitude of NS
significantly drops at the inner wall for higher n due to lower temperature
and velocity gradient.

Fig. 11. Entropy generation number due to heat transfer and fluid friction versus
Brinkman’s number for different radial locations in the pipe: n=0,Λ=0.03,

T0 =1.2

Fig. 12. Entropy generation number due to heat transfer and fluid friction versus
dimensionless non-Newtonian viscosity for different radial locations in the pipe:

n=0, Γ =1, T0 =1.07

Figures 11 and12 show the total entropy generation number atdifferent lo-
cations in theannularpipe fordifferentBrinkman’snumbersanddimensionless
non-Newtonian viscosities, respectively. The increasing of Brinkman’s number
increases the total entropy generation number, particularly in the region close
to the annular pipe inner wall. In the case of dimensionless non-Newtonian
viscosity (Fig.12), the total entropy generation number increases with gro-
wing dimensionless non-Newtonian viscosity at r=1 in the annular pipe, but



Entropy generation due to non-Newtonian fluid flow... 81

at the radial location corresponding to r = 1.5, the total entropy generation
number decreases with growth of the dimensionless non-Newtonian viscosity.
The total entropy generation number is presented against the velocity ratio n
in Fig.13 where n is kept between −1 to 1 for convenience.

Fig. 13. Entropy generation number due to heat transfer and fluid friction versus
velocity ratio for different radial locations in the pipe: Λ=0.03, T0 =1.07, Γ =1

5. Conclusions

Anon-Newtonian fluid flow in an annular pipe and the influence of dimension-
less non-Newtonian viscosity is considered in the paper. Brinkman’s number
and velocity ratio on the entropy generation number due to fluid friction and
heat transfer are examined. The flow inside the annular gap is induced by re-
lative rotation between the inner and outer cylinders. Near the inner cylinder,
the entropy generation rate is higher due to higher temperature and velocity
gradient. It is found that reduction of the dimensionless non-Newtonian vi-
scosity increases the total entropy generation number in the inner pipe wall
region. The profile of the entropy generation number shows asymptotic be-
havior near the outer cylinder. A growth in Brinkman’s number increases the
total entropygeneration number,particularly in the region close to theannular
pipe inner wall.

References

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Cambridge Philos. Soc., 60, 667-674



82 A. Kahraman, M. Yürüsoy

2. BejanA., 1979,A study of entropy generation in fundamental convective heat
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4. Demirel Y., Kahraman R., 2000, Thermodynamic analysis of convective
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Entropy generation due to non-Newtonian fluid flow... 83

Generacja entropii przy przepływie indukowanym względnym ruchem

obrotowym ścian pierścieniowego przewodu – przypadek przepływu

o stałej lepkości

Streszczenie

W pracy zbadano zagadnienie generacji entropii obserwowanej podczas przepły-
wu nieniutonowskiej cieczy przez przewód pierścieniowy, którego ścianki obracają się
względemsiebie.Doanalizyprzyjętopłyn trzeciegostopniao stałej lepkości.Przepływ
czynnika jest indukowany względnym ruchem obrotowym zewnętrznego i wewnętrz-
nego cylindra tworzącego ścianki przewodu. Rozwiązania analityczne zaprezentowano
dla rozkładu prędkości i temperatury płynu, a liczbę generacyjną entropii wyznaczo-
no dla różnychwartości lepkości nieniutonowskiej cieczy, liczbyBirnkmana i stosunku
prędkości obwodowej cylindrów. Potwierdzono, że zwiększenie bezwymiarowej lepko-
ści obniża liczbę generacyjną entropii. Ten efekt jest szczególnie wyraźny w obszarze
bliskim ścianywewnętrznego cylindra.Wzrost liczbyBirnkmana powiększa liczbę ge-
neracyjną entropii, także w pobliżu ściany wewnętrznej przewodu.

Manuscript received February 21, 2007; accepted for print July 16, 2007