Download Entropy Analysis of Pressure Driven Flow in a Curved Duct

Entropy Analysis of Pressure Driven Flow in a Curved Duct
V. K. Narla 1,a) and Vijayasekhar Jaliparthi 1, b)
1
Department of Mathematics, GITAM University, Hyderabad, India.
a)
Corresponding author: [email protected]
b)
[email protected]
Abstract. This paper aims to present a theoretical model describing entropy generation analysis using second law of
thermodynamics. A two-dimensional, incompressible, viscous MHD fluid flow in a curved duct undergoing peristalsis
with prescribed wall motions in the presence of heat transfer is applied and demonstrated. In this problem, It is assumed
that the inertial effect is very small and the wall wave length is comparatively large with duct width. The velocity and
temperature fields are obtained analytically by solving momentum and energy equations. The entropy generation number
is calculated by utilizing velocity and temperature profiles. The influence of various physical parameters on entropy
generation are discussed numerically with the help of graphs.
INTRODUCTION
Peristaltic pumping is a kind of fluid transport that occurs when progressive wave of contraction and expansion
passed along a conduit containing liquid. The fluid mechanics of this phenomenon has been studied by several
researchers because of its application in biology and industry. Sato et al. [1] developed a peristaltic model in a twodimensional curved channel based on lubrication theory. In a curved channel with sinusoidally oscillating walls,
they demonstrate that the peristaltic transport phenomena are strongly influenced by the curvature. This model was
extended by Ali et al. [2] to capture heat transfer effects using the assumptions long wavelength and low Reynolds
number. Hayat et al. [3] examined the effect of the compliant wall properties and heat transfer on the peristaltic flow
in a curved channel. Ramanamurthy et al. [4] studied unsteady peristaltic transport in curved channels to capture
spacial and temporal effects of flow phenomena. The impact of entropy generation due to heat transfer and fluid
friction for the peristaltic flow in a tube was investigated Akbar [5]. Recently, Narla et al. [6] discussed entropy
generation analysis for peristaltic flow in curved channels.
FIGURE 1: Schematic diagram of peristaltic transport in a two-dimensional curved channel.
MATHEMATICAL FORMULATION
The flow configuration is shown in Figure 1. The cartesian system ( ′, ′, ′) is related to
the intrinsic coordinates ( , , ) by the relations
′ = ( + ) ( ), ′ = ( + ) ( ), ′ = .
(1)
A two-dimensional peristaltic flow in a curved channel is filled with an incompressible viscous
fluid under the influence of radial magnetic field. The unperturbed width of the channel is 2
coiled in a circle with center and the radius of curvature is . The -axis lies along the center
line of the curved channel, -axis is normal to it and is measured from central line with scaling
factors ℎ =
, ℎ = 1 and ℎ = 1. There is no component in direction. The flow in the
channel gives velocity vector in the form = !( , , ")#̂ + %( , , ")#̂ . The fluid motion within
the channel is driven by two infinite trains of sinusoidal waves that are propagated along the
channel. The temperatures of the lower and upper walls are maintained at constant temperatures
(
&' and & respectively. An external magnetic field of strength ) is applied in radial direction as
shown in Figure 1. The fluid-wall interface is time-dependent and is given as follows:
= ±ℎ( , ") = ± ± + [ . ( − ")].
(2)
Here, is the axial distance,
the radius of the stationary curved channel, + is the wave
amplitude, 1 the wave length, " the time,
the velocity of the wave, and ℎ the radial
displacement of the wave from the centerline.
The mass conservation and the fluid momentum equations at the axial and radial
directions, which are given respectively by:
23
2
23
27
+ ( ⋅ ∇)! +
3:
+ ( ⋅ ∇)% −
27
3>
2
2
2<
= − ;(
2:
+
)2
{( + )%} = 0,
3
+ =[∇ ! − (
)>
2<
= − ; 2 + =[∇ % − (
2B
C
(3)
2:
+(
:
)>
)> 2
)> 2
23
:
2:
23
2:
where
( ⋅ ∇) = (
3
2
)2
2
−
2
+ %2 ,∇ = (
3
23
−(
A< [ 27 + ( ⋅ ∇)&] = ; ∇ & + =[2{(2 ) + (
+( 2 +
] − ?@ (
2
+
()
) !,
],
(5)
) }
) ],
)
2>
2
>
(4)
(6)
2
+
2
2>
+ 2 >.
In order to describe the fluid flow pattern in dimensionless form, the variables are scaled as
follows:
3
:
G
I
7
M
′ = . , ′ = D , !′ = E , %′ = FE , ℎ′ = D , H = D , J = D , "′ = B ∗ , L′ = .,
N′ =
D> <
, #=
OE.
EDF
P
R
T
, Q′ = DE , S = DE , U =
;PVW
C
,X = V
E>
W (BY ZB) )
BZB
, [ = B ZB) ,
Y
)
(7)
where # is Reynolds number, S is volume flow rate, J is the curvature parameter, \ is
D
Hartman number, ] = . is wavelength ratio, H is the amplitude ratio or the occlusion parameter,
L is the passage length of the channel, U is the Prandtl number and X is the Eckert number.
The quantities ! and % in equations (3)-(6) are related to the stream function Q as
follows:
2R
^ 2R
! = 2 ,% = − ^ 2 ,
(8)
Using (7) and (8), the governing equations (3)-(6) can be written (after eliminating the pressure
term) under the assumption of long wavelength (] ≪ 1) and small Reynolds number ( # → ∞)
approximation as
2
2
2 2R
2
2R
(
{( + J) } ) − ((1 + \ )
) = 0,
(9)
2
^2
2
2> b
2
2
2> R
2b
^(
^) 2
2R
+ ^ 2 + c ( 2 > − ^ 2 ) = 0,
(10)
2
where c = U X is the Brinkman number and primes are dropped for simplicity. The
corresponding dimensionless boundary conditions are obtained by:
2R
d
= 0, = ±ℎ( , "), Q = ± , = ±ℎ( , ")
(11)
>
2
[ = 0, = −ℎ( , "), [ = 1, = ℎ( , ")
(12)
SOLUTION OF THE PROBLEM
The obtained boundary value problem is solved for stream function with the boundary
conditions consisting of the equation (11):
>
>
Q( , ) = A + A ( + J) + A ( + J) Z√ f + Ag ( + J) √ f ,
(13)
where,
A =
A =
Here
ZdhgG^dY √
gh(^ > dk>
f> d> (
G> dl> )√
>
m
d(^ > ZG> ) Yno (
h(^ > dk>
S = (J − ℎ)
G> dl> )√
√
f>
f> )(G> i > )j
G^d> (
√
f> )(^dk Gdl )
f>
f> )j
G^d> (
+ (ℎ + J)
>
f> )j
,A =
f>
√
f>
>
, Ag =
gh(^ > dk>
d> f> d
>
>
G dl )√ f>
d(Z
h(^ > dk>
, S = (J − ℎ)
f> )[Z^dk Gdl ]
√
G> dl> )√
f> G^d> (
f>
− (ℎ + J)
√
,
f> )j
G^d> (
>
.
f> )j
√
f>
>
,
S = (J − ℎ)√ f − (ℎ + J)√ f , Sg = (J − ℎ)√ f + (ℎ + J)√ f .
The transformations between the wave and the laboratory frames, in dimensionless form, are
given by :
= − ", = , p = ! − 1, = %, q = S − 2ℎ, Ψ = Q − ,
(14)
where the left hand side parameters are in the wave frame and the right hand side parameters are
in the laboratory frame.
Averaging the volumetric flow rate along time period & gives
B
B
sB = B t' S( , ")u" = B t' (q + 2ℎ)u" = q + 2.
(15)
FIGURE 2:: Effect of curvature on temperature distribution.
FIGURE 3:: Effect of magnetic parameter on temperature distribution.
The solution of the energy equation (10) using equations (12) and (13) is obtained as:
[ = v + v ln( + J) −
Ag ( + J)
where,
√
v =
f>
f> )
g(
[4(1 + \ )A Ag ln ( + J) + A ( + J)
J Z
].
( fl
g(
( fl
f>
√
+
(16)
f> )
[4(1 + \ ))A Ag ln (J − ℎ) + A (J − ℎ)Z
−
z{(^ZG)
|n}
z{(
)
}~|
v2 =
( fl
[1 − g(
|n}
z{(
)
}~|
[1
1−
f> )
( fl
g(
√
f>
+ Ag (J − ℎ)
√
(4(1 + \ )A Ag • + A • + Ag S )],,
f> )
(4(1 + \ )A Ag • + A • + Ag S )],
)
• = ln (J − ℎ) − ln (ℎ + J), • = (J − ℎ)Z
√
f>
− (ℎ + J)Z
FIGURE 44: Effect of curvature on entropy.
√
f>
.
f>
]
FIGURE 55: Effect of magnetic parameter on entropy.
ENTROPY GENERATION
Entropy generation is closely associated with thermodynamic irreversibility, which is
encountered in all practical heat transfer processes. Different sources are responsible for entropy
generation such as the heat transfer in the presence of temperature di
difference
fference and the viscous
dissipation. The volumetric rate of entropy generation is defined as
^
O
XŒ = B > (∇&) + B Φ + S “ 0.
(17)
•Ž•Ž•
•Ž•Ž•
)
)
‘'
‘'
The non-dimensional
dimensional entropy generation equation under long wavelength approximation can be
expressed as follows (Narla et al. [[6]):
2b
X€ = X• + Xd = ( 2 ) +
where, Eƒ =
„…
„†
‡> ˆ>) „…
= ‰(ˆ
ˆ) )>
Y Zˆ
(
”
2> R
[( 2
>
−
2R
i2
) +\ (
2R
^2
) ],
]
(18)
is non dimensional entropy generation, BrΩ
Br Z is viscous
dissipation parameter, EŠ is entropy generation by heat transfer due to transverse heat
conduction and E‹ is entropy generation due to the fluid friction.
RESULTS AND DISCUSSION
A graphical analysis of the solutions of temperature distribution and entropy generation
is presented. Figure 2 shows the influence of channel curvature on the dimensionless temperature
at fixed values of the parameters H = 0.4, c = 2, = 0.2, " = 0.5, sB = 1.5, \ = 1. The fluid
temperature decreases with increasing curvature parameter J.. The temperature distribution
distrib
is
parabolic nature. This is due to the presence of viscosity dissipation that increases the fluid’s
temperature in the central part of the channel. This means the temperature distribution in the
curved channel is very high than in the straight channel. The variation of magnetic parameter \
on temperature is presented in figure 3 at fixed parameters H = 0.4, J = 3, c = 2, = 0.4, " =
0.5, sB = 1.5. The temperature is a decreasing function of magnetic parameter. This means that
the high magnetic strength decreases the temperature of the fluid in a curved channel. The
distribution of entropy generation for different values of curvature parameter J is plotted in
figure 4 at fixed parameters H = 0.6, c = 2, = 0.2, " = 0.5, sB = 1, \ = 1, c /Ω = 1.5. The
increasing channel curvature decreases the entropy generation rate at upper wall and increases
much at lower wall. For the decreasing channel curvature (as J → ∞) the entropy generation rate
variation is not much at upper and lower walls. This shows that the entropy generation rate is
strongly depends on curvature of the channel. Figure 5 illustrates the variation of X€ with
magnetic parameter \ at fixed values c = 1, H = 0.6, J = 3, = 0.3, " = 0.5, sB = 1, c /Ω =
1. It is observed that the entropy generation rate increases with an increase in the magnitude of
magnetic parameter.
REFERENCES
1.
2.
3.
4.
5.
6.
H. Sato, T. Kawai, T. Fujita, and M. Okabe, Trans. Jpn. Soc. Mech. Eng. Ser. B 66, 679–685 (2000).
N. Ali, M. Sajid, T. Javed, and Z. Abbas, Int. J. Heat Mass Transfer 53, 3319–3325 (2010).
T. Hayat, M. Javed, and A. A. Hendi, Int. J. Heat Mass Transfer 54, 1615–1621 (2011).
J. V. Ramanamurthy, K. M. Prasad, and V. K. Narla, Phys. Fluids 25, 091903(1–20) (2013).
N. S. Akbar, Energy 82, 23–30 (2015).
V. K. Narla, K. M. Prasad, and J. V. R. Murthy, Journal of Engineering Physics and Thermophysics 89,
441–448 (2016).