This document summarizes an article from the International Journal of Mechanical Engineering and Technology (IJMET) that examines the effect of particle size and chemical reaction on convective heat and mass transfer of magnetohydrodynamic (MHD) nanofluid flowing in a cylindrical annulus filled with porous material. The article presents governing equations to model the coupled heat and mass transfer, which are solved numerically. Parameters like the Darcy number, Grashof number, Hartmann number, and Nusselt number are defined. The study aims to provide insights into how particle size and chemical reactions impact convective transport for applications like nuclear technology and space systems.

1. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –
INTERNATIONAL JOURNAL OF MECHANICAL ENGINEERING
6340(Print), ISSN 0976 – 6359(Online), Volume 5, Issue 2, February (2014), pp. 26-35, © IAEME
AND TECHNOLOGY (IJMET)
ISSN 0976 – 6340 (Print)
ISSN 0976 – 6359 (Online)
Volume 5, Issue 2, February (2014), pp. 26-35
© IAEME: www.iaeme.com/ijmet.asp
Journal Impact Factor (2014): 3.8231 (Calculated by GISI)
www.jifactor.com
IJMET
©IAEME
EFFECT OF PARTICLE SIZE AND CHEMICAL REACTION ON
CONVECTIVE HEAT AND MASS TRANSFER OF A MHD NANO FLUID IN
A CYLINDRICAL ANNULUS
*1G.V.P.N. SRIKANTH,
4
2
Dr. G. SRINIVAS,
3
B. SURESH BABU,
Dr. B. TULASI LAKSHMI DEVI
1,2,4
3
Department of Mathematics, Guru Nanak Institute of Technology, Hyderabad, A.P, India
Department of Mathematics, Sreyas Institute of Engineering & Technology, Hyderabad, A.P, India.
ABSTRACT
The present work deals with the effect of size of the nano-particle and the liquid like layer
formed duo to the natural chemical reaction of the liquid with the metical particle. The particle size
and the layer around the particle certainly alter the heat and mass transfer. We have study the effects
of particle size and the layer on convective heat and mass transfer of MHD nano-fluid with heat
source and chemical reaction when the nano-fluid flows in a cylindrical annulus filled with porous
metrical. The coupled governing equations are solved using method of lines in Mathematical
package.
Key words: MHD Nano - Fluid, Annulus, Porous Material, Chemical Reaction.
1.
INTRODUCTION
The investigation for enhancing the heat and mass transfer through a fluid has given birth to
the nano-fluid duo to the presents of metical particle in the fluid enhances the heat and mass transfer
widely. The enough literature is providing an excellent support for the enhancement of heat and mass
transfer.
Xuan and Li [10] presented a study on the thermal conductivity of a nano-fluid consisting of
copper nano-particles. The measured data showed that adding 2.5-7.5% copper oxide nano particles
to the water increases its conductivity by about 24-78%. Rafee [4] has concluded that for a constant
heat transfer rate, by increasing the length ratio of the annulus (L/Dh), firstly the entropy generation
ratio will decrease until reaches its minimum value.
26

2. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –
6340(Print), ISSN 0976 – 6359(Online), Volume 5, Issue 2, February (2014), pp. 26-35, © IAEME
Free convection flow and heat transfer in hydro magnetic case is important in nuclear and
space technology (Singh KR 1963[6], Yu CP 1969[8], Yu CP 1970[9]). In particular, such
convection flow in a vertical annulus region in the presence of radial magnetic field has been studied
by Sastry and Bhadram (1978)[5].Nanda and Purushotham (1976)[2] have analyzed the free
convection of a thermal conducting viscous incompressible fluid induced by traveling thermal waves
on the circumference of a long vertical circular cylindrical pipe. Whitehead (1972)[7], Neeraja
(1993)[3] has made a study of the fluid flow and heat transfer in a viscous incompressible fluid
confined in an annulus bounded by two rigid cylinders. The flow is generated by periodical traveling
waves imposed on the outer cylinder and the inner cylinder is maintained at constant temperature.
Recently Ali J. Chamkha [1] has studied the Heat and Mass Transfer from MHD Flow over a
Moving Permeable Cylinder with Heat Generation or Absorption and destructive Chemical Reaction.
He found that the diffusion decreases with increase in chemical reaction.
The above studies motivated to investigate theoretically the effects of particle size, chemical
reaction and heat source on convective heat and mass transfer in a nano - fluid while passing through
horizontal porous annulus in the presence of magnetic field.
2. FORMULATION OF THE PROBLEM
We consider free and force convection flow of Cu-water nano-fluid through a porous medium
in a circular cylindrical annulus in the presence of heat source, whose inner and outer walls are
maintained at a constant temperature also the concentration is constant on the both walls. The flow
velocity, temperature and concentration in the fluid to be fully developed. The fluid region has
constant physical properties and the flow is a mixed convection flow taking place under thermal and
molecular buoyancies and uniform axial pressure gradient. The boussenissque approximation is
invoked so that the density variation is confined to the thermal and molecular buoyancy forces. In the
momentum, energy and diffusion are coupled and non-linear. Also the flow in is unidirectional along
the axial cylindrical annulus. Making use of the above assumptions the governing equations are
∂
(r u) = 0
∂r
(1)
g ( ρ β T ) (T − T0 ) g ( ρ βc ) ( c − c0 )
µ n f  ∂2 w 1 ∂ w ∂2 w  µ n f
σ B02
nf
nf
+
+ 2 −
w−
w+
+
= 0
 2
ρ nf  ∂r
ρ nf
ρ nf
ρ nf
r ∂r ∂ z  ρ nf k
(2)
α
∂T
∂z
w
D nf ∂  ∂c 
∂c
=
r
 + kl ( c − c0 )
∂z
r ∂r  ∂r 
=
nf
r
∂
∂r
 ∂T 
r
−
 ∂r 
w
Q
(T − T0 )
( ρ cp ) n f
(3)
(4)
27

3. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –
6340(Print), ISSN 0976 – 6359(Online), Volume 5, Issue 2, February (2014), pp. 26-35, © IAEME
The appropriate initial and boundary conditions for the problem are given by
T = T0 , c = c0 , w = 0 on r = a
(5)
T = Tm , c = cm , w = 0 on r = a + s
Thermo-Physical properties are related as follows:
k
ρ
= (1 − φ ) ρ
n f
n f
(ρ c )
p n f
+φρ , α
=
f
s
n f
(ρ c )
= (1 − φ ) ( ρ c ) + φ ( ρ c )
p nf
p f
p s
(ρ β )
µ
nf
where
= (1 − φ ) ( ρ β )
f
+ φ (ρ β )
s




= 1 + 2.5 φ + 4.5 

 h
d
 p

n f
µ
k
nf
f
= k
1



 2 + h  1 + h 

d 
d 
p 
p 

2










d
f
1− φ ) + β k φ + c
k Re 2 d pr φ
(
f
1 p
1 d
p
p
p
(6)
β = 0.01 is a cons tan t for considering the kapitza resis tan ce per unit area
1
c = 18 × 106 is a proportionality cons tan t
1
d
Re d
p
=
γ
p
f
κT
3π µ d l
f
p f
=
1.381×1023 T
γ 3π µ ( 0.738)
f
f
ν
d
l
f
f
= 0.384 nm for water , pr = Pr andtl number =
= Mean free path = 0.738 , κ
=
α
f
f
Boltzman n cons tan t , T = 300 k
28

4. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –
6340(Print), ISSN 0976 – 6359(Online), Volume 5, Issue 2, February (2014), pp. 26-35, © IAEME
The thermo-physical properties (values) of the materials used are as follows.
Table 1:
Physical
Properties
Water
Copper(Cu)
ρ ( kg / m )
κ (W / m K )
4,179
997.1
0.613
385
8,933
400
β ×10−5 (1/ K)
21
1.67
β × 106 (m2/ h)
298.2
3.05
µ
8.94 × 10 −4
-----
C p ( J / kg K )
3
T
c
We introduce the following dimensionless variables:
R=
T − T0
c − c0
r
z
w
u
, Z = , W =
, U =
, θ =
,C =
νf
νf
a
a
Tm − T0
cm − c0
(7)
Using equations 5, 6, 7 the Eqs. 2, 3 & 4 can be written in the following dimensionless form:











   ∂2 W 1 ∂ W 
1
1
1 + 2.5 φ + 4.5 
  2 +

2   ∂ R
R ∂R 

 


1− φ + φ

 h  2 + h  1 + h   




d 
d   d  

p 
p  


 p

1
−

1− φ + φ


W
∂θ
∂z
∂C
∂z
ρs 

ρf 

MW +
=



 θ




1  ∂2 C 1 ∂ C 
+

 + KC
S c  ∂ R2
R ∂R
29
1
W

ρs 
1− φ + φ


ρf 


1
1
Gr θ +
Gc θ = 0


ρs 
ρs 
1− φ + φ

1− φ + φ



ρf 
ρf 







ρ 2c
k
k
1 
f
pf
p
p
− 52
=
1 − φ + 0.01 φ
+
φ
28632.9991 ×10
2
3
Pr 
k
k
d µ

f

f
p f






1
1
−
QH 
(ρ c )
Pr

p s
1 − φ + φ
(ρ c )

p f

W











1
1
− 1 + 2.5 φ + 4.5 

2
ρs  D 
 

 


 h  2 + h  1 + h   
ρf 


 d  d   d  

p 
p  


 p









1


(ρ c )  
p s 
1 − φ + φ

(ρ c )  
p f 


 ∂2 θ
1 ∂θ 
 ∂ R2 + R ∂ R 



5. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –
6340(Print), ISSN 0976 – 6359(Online), Volume 5, Issue 2, February (2014), pp. 26-35, © IAEME
Where the corresponding boundary conditions (5) can be written in the dimensionless form as:
W = 0 , θ = 0 , c = 0 on R = 1
W = 0 , θ = 1, c = 1 on R = 1 + s
Here pr is the Prandtl number, M is the magnetic parameter(Hartmann number), QH is the heat
source parameter, S c is the Schmidt number, K is the chemical Reaction parameter, D −1 is the Darcy
number, G r is the Grashof number, G c is the Molecular Grashof number, which are defined as:
νf
νf
σ B0 2
k
Q
1
a
a , QH =
a , Sc =
=
Pr =
, M =
, K = l a,
kf
Dn f
D
k
αf
µf
νf
Gr =
g (ρ β
T
)( T m − T0 )
ν
2
, Gc =
f
g (ρ β
c
)( cm − c0 )
ν
2
f
The local Nusselt number Nu in dimension less form:
k nf
Nu = −
θ ' (1 + s )
kf
3. SOLUTION OF THE PROBLEM
The cross section of the cylinder has considered which appears in the annulus form for
numerical computations. The governing equations are solved for momentum (w), temperature ( θ )
and diffusion (C) by using method of lines with the help of Mathematica package across the
cylindrical annulus subject to the boundary conditions.
4. RESULTS
The profiles of momentum, temperature and diffusion are drawn at Pr = 7 and for constant
axial temperature and axial concentration gradients.
From Figs. 1 - 11 the flow is maximum in the mid region of the annulus. The momentum
decreases with solid volume fraction ( φ ).If the amount of Cu nano-particles increases the
momentum decreases due to Brownian motion. The width of the annulus affects the flow very much.
The flow is maximum for maximum width of the annulus (s). The velocity enhances with increase of
the thickness of the layer (h) around the nano-particle due to increase in friction. The velocity is
maximum for 10nm or more and almost constant for 2nm or less thickness. The flow has obstructed
by the Cu nano-particle very much. As the Particle size (dp) increases from 20nm to 100nm the
velocity is decreased due to low Brownian motion. The velocity is maximum for small particles at
about 20nm and the velocity is almost constant at about 100nm or more. The velocity increases with
increase of both molecular (Gc) and thermal Grashof numbers (Gr). The velocity decreases with
increase in the Hartmann number (M) and the velocity found maximum in the absence of the
magnetic field. The velocity decreases with increase in heat source (QH). The velocity increases with
increase in Schmidt number (Sc). The velocity increases with increase in chemical reaction
parameter. The velocity is more for destructive (K > 0) chemical reaction than the generative (K < 0)
chemical reaction. The flow increases with increase in the porosity ( D −1 ).
30

6. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –
6340(Print), ISSN 0976 – 6359(Online), Volume 5, Issue 2, February (2014), pp. 26-35, © IAEME
From Figs. 12 – 18 the heat transfer takes place from inner cylinder to outer cylinder linearly
for all variations of φ, s, h, dp, QH, K and D-1. The temperature enhances with increase in volume
fraction ( φ ) of Cu particles. Temperature increases rapidly with increase of the width (s) of the
annulus. The temperature is maximum in the annulus when the inner and outer cylinders are of same
radius. The layer (h) around the nano-particle reduces the temperature due to less thermal
conductivity of CuOH when compared with Cu. The temperature enhances with increase in size of
the particle (dp). The temperature decreases with increase in heat source (QH). The temperature is
more when the chemical reaction is generative (K<0) and less when the chemical reaction is
destructive (K>0). The temperature reduces with the increase in porosity ( D −1 ).
From Figs. 19 – 22, the diffusion is linear for all variations of φ, s, Sc and K. The diffusion
increases with increase of φ, s, Sc and K. the Diffusion is rapidly increasing with the increase in the
width (s) of the annulus and is very high for same size of inner and outer cylinders of annulus. The
diffusion more during the destructive (K>0) chemical reaction and the diffusion is less during the
generative (K<0) chemical reaction with the Cu-particle.
9.00E-03
2.50E-02
8.00E-03
2.00E-02
7.00E-03
6.00E-03
1.50E-02
5.00E-03
4.00E-03
1.00E-02
φ = 0,0.02,0.04,0.05
s = 0.2, 0.5, 0.8, 1
3.00E-03
5.00E-03
2.00E-03
1.00E-03
0.00E+00
0.00E+00
1 2 3 4 5 6 7 8 9 101112131415161718192021
-1.00E-03 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5
-5.00E-03
Fig.1 Variation of w with φ
Fig.2 Variation of w with s
0.025
0.018
0.016
0.02
0.014
0.015
0.012
h = 2,4,6,8,10
0.01
0.01
dp = 20,40,60,80,100
0.008
0.006
0.005
0.004
0.002
0
0
-0.002 1
1 1.05 1.1 1.2 1.25 1.3 1.35 1.4 1.45 1.5
-0.005
Fig.3 Variation of w with h
1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5
Fig.4 Variation of w with d p
31

7. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –
6340(Print), ISSN 0976 – 6359(Online), Volume 5, Issue 2, February (2014), pp. 26-35, © IAEME
0.012
0.012
0.01
0.01
0.008
0.008
Gc = 2,5,8,10
0.006
0.006
0.004
0.004
0.002
Gr = 2,5,8,10
0.002
0
0
1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5
1 1.051.1 1.15 1.2 1.251.3 1.35 1.4 1.45 1.5
-0.002
-0.002
Fig.5 Variation of w with G c
Fig.6 Variation of w with G r
0.009
0.009
0.008
0.008
0.007
0.007
0.006
0.006
0.005
0.005
0.004
M = 0,5,10
0.004
0.003
0.002
0.002
0.001
QH = 2,5,8,10
0.003
0.001
0
0
-0.001 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5
-0.001 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5
Fig.8 Variation of w with QH
Fig.7 Variation of w with M
0.009
0.009
0.008
0.008
0.007
0.007
0.006
0.006
0.005
Sc = 0.3,0.6,1.3
0.005
0.004
0.004
0.003
0.003
0.002
0.002
0.001
K = -0.5,-0.2,0,0.2,0.5
0.001
0
-0.001 1 1.051.11.151.21.251.31.351.41.451.5
0
-0.001 1 1.051.1 1.151.2 1.251.3 1.351.4 1.451.5
Fig.9 Variation of w with Sc
Fig.10 Variation of w with K
32

8. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –
6340(Print), ISSN 0976 – 6359(Online), Volume 5, Issue 2, February (2014), pp. 26-35, © IAEME
0.009
1
0.008
0.9
0.007
0.8
0.006
0.7
0.005
0.6
0.5
0.004
1/D = 2,5,8,10
φ = 0,0.02,0.04,0.05
0.4
0.003
0.3
0.002
0.2
0.001
0.1
0
0
-0.001 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5
1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5
Fig.11 Variation of w with D−1
Fig.12 Variation of θ with φ
1
1
0.9
0.9
0.8
0.8
0.7
0.7
s = 0.2,0.5,0.8,1
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
h = 2,4,6,8,10
0
0
1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5
1 2 3 4 5 6 7 8 9 101112131415161718192021
Fig.13 Variation of θ with s
Fig.14 Variation of θ with h
1
1
0.9
0.9
0.8
0.7
0.8
0.7
dp = 20,40,60,80,100
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
QH = 2,5,8,10
0
1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5
1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5
Fig.15 Variation of θ with d p
Fig.16 Variation of θ with QH
33

9. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –
6340(Print), ISSN 0976 – 6359(Online), Volume 5, Issue 2, February (2014), pp. 26-35, © IAEME
1
1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
1 / D = 2,5,8,10
0.5
0.5
K = -0.5,-0.2,0,0.2,0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5
1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5
Fig.18 Variation of θ with D−1
Fig.17 Variation of θ with K
1
1
0.9
0.9
0.8
0.8
0.7
0.6
s = 0.2,0.5,0.8,1
0.7
φ = 0,0.02,0.04,0.05
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5
1 2 3 4 5 6 7 8 9 101112131415161718192021
Fig.19 Variation of c with φ
Fig.20 Variation of c with s
1
1
0.9
0.9
0.8
0.8
0.7
0.6
0.7
Sc = 0.3,0.6,1.6
K = -0.5,-0.2,0,0.2,0.5
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5
1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5
Fig.21 Variation of c with Sc
Fig.22 Variation of c with K
34

10. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –
6340(Print), ISSN 0976 – 6359(Online), Volume 5, Issue 2, February (2014), pp. 26-35, © IAEME
Nusselt Number:
Table-2
s = 0.5, h = 4, G c = 5, G r = 5, M = 5, S c = 0.6, D −1 = 5
K = 0.2,
d p = 40,
K = 0.2,
d p = 40,
K = 0.2,
d p = 40,
K = 0.2,
d p = 20,
K = 0.2,
d p = 100,
K = − 0.2,
d p = 40,
K = 0,
d p = 40,
QH = 0
QH = 5
QH = 10
QH = 5
QH = 5
QH = 5
QH = 5
0.02
-1.64838
-2.35051
-2.96791
-2.35512
-2.34821
-2.35049
-2.3505
0.04
-1.64798
-2.29046
-2.86112
-2.29467
-2.28836
-2.29045
-2.29046
0.05
-1.64781
-2.26412
-2.81397
-2.26814
-2.2621
-2.26411
-2.26411
߶
The variation of Nusselt number for different values of volume fraction, K, dp and QH are
depicted in the Table-2 on the surface of the outer cylinder. The magnitude of heat transfer decreases
as the volume fraction increases on the surface of the outer cylinder. The magnitude of heat transfer
increases as the heat source increases. The magnitude of heat transfer decreases with increase in
particle size. The magnitude of heat transfer increases from generative to destructive chemical
reaction.
REFERENCES
[1]
Ali. J. Chamkha, Heat and Mass Transfer from MHD Flow over a Moving Permeable
Cylinder with Heat Generation or Absorption and Chemical Reaction. Communications in
Numerical Analysis, 2011 doi:10.5899/2011.
[2] Nanda RS and Prushotham R (1976) : Int.dedication seminar on recent advances on maths
and applications, Vaeanasi.
[3] Neeraja,G(1993):Ph.D thesis, S.P.Mahila University, Tirupathi, India.
[4] R. Rafee,Entropy Generation Calculation for Laminar Fully Developed Forced Flow and
Heat Transfer of Nanofluids inside Annuli, Journal of Heat and Mass Transfer Research xxx
(2013).
[5] Sastri VUK and Bhadram CVV(1978) : App,Sci.Res,V.34,2/3.p.117
[6] Singh KR and Cowling TJ (1963) : Quart.J.Maths.Appl.Maths,V.16.p.1.
[7] Whitehead JA (1972):Observations of rapid means flow produced mercury by a moving
heater, Geophys. Fluid dynamics, V.3, pp.161-180.
[8] Yu CP and Yong,H(1969): Appl.Sci.Res,V.20,p.16.
[9] Yu CP (1970):Appl.Sci.Res, V.22, p.127.
[10] Y.M. Xuan, Q. Li, Heat transfer enhancement of nanofluids, Int. J. Heat Fluid Flow, 21,
58–64, (2000).
[11] Dr P.Ravinder Reddy, Dr K.Srihari and Dr S. Raji Reddy, “Combined Heat and Mass
Transfer in MHD Three-Dimensional Porous Flow with Periodic Permeability & Heat
Absorption”, International Journal of Mechanical Engineering & Technology (IJMET),
Volume 3, Issue 2, 2012, pp. 573 - 593, ISSN Print: 0976 – 6340, ISSN Online: 0976 – 6359.
[12] Kavitha T, Rajendran A, Durairajan A, Shanmugam A, “Heat Transfer Enhancement Using
Nano Fluids And Innovative Methods - An Overview”, International Journal of Mechanical
Engineering & Technology (IJMET), Volume 3, Issue 2, 2012, pp. 769 - 782, ISSN Print:
0976 – 6340, ISSN Online: 0976 – 6359.
35