Natural Convection of Nanofluid in a Porous Medium Filled Rectangular Cavity with Heat Loss and Viscous Dissipation

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Natural Convection of Nanofluid in a Porous Medium Filled Rectangular Cavity with Heat Loss and Viscous Dissipation

____________________________________________________________________________

Corresponding Author: G.N. RAMA KRISHNA

1. INTRODUCTION

The investigation of heat transfer and mixed convection flow in porous medium nooks of different shapes has gotten much consideration [4, 5 and 10]. Enthusiasm for these characteristic convection flow and heat transfer in porous medium has been persuaded by an expansive scope of utilizations, including geothermal frameworks, Crude oil creation, stockpiling of Nuclear waste materials, ground water contamination, fiber and granular protections, Solidification of castings, and so on. In an extensive variety of such issues, the physical framework can be displayed as a two-dimensional rectangular enclosure with vertical wall held at various temperatures and the connecting horizontal walls considered adiabatic. Convective heat transfer in a Rectangular porous duct whose vertical walls are kept up at two distinct temperatures and flat horizontal walls insulated, is an issue which has gotten consideration by numerous agents [18-20, 22, 23] some of these work incorporates numerical results by several authors [6-8,11,12].

The examination of heat transfer is fenced in enclosures containing porous media started with the test work of Verschoor and Greebler [31]. Verschoor and Greebler [31] were trailed by a few different specialists inspired by porous media heat transfer in rectangular enclosures [25-28].

There are various techniques for enhancing the heat transfer of fluids such as changing in C. Sulochana^#1 G.N. Ramakrishna^{, #2} O.D. Makinde ^#3

#1 Professor, Department of Mathematics, Gulbarga University, Gulbarga- 585106, India

#2 Research Scholar, Department of Mathematics, Gulbarga University, Gulbarga-585106, India. Mobile: +918332833401

#3 Professor, Faculty of Military Science, Stellenbosch University, South Africa

ABSTRACT

We consider Convective heat transfer flow of a nanofluid through a porous medium in rectangular duct with dissipation. The governing equations have been solved by using Galerkin finite element analysis with linear interpolation functions

with linear approximation functions. The effects of dissipation and heat source on all the flow characteristics have been discussed.

KEYWORDS: Nanofluid, Rectangular duct, Heat loss, Viscous Dissipation, Galerkin Method

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©2016 RS Publication, [email protected] Page 212 geometry, boundary conditions, or enhancing thermal conductivity of the fluid. Since the thermal conductivity of the solids are typically higher than that of liquids, by suspending nano- or larger sized solid particles in fluids, the thermal conductivity is increased. Many attempts in this field have been made to formulate the effective thermal conductivity and dynamic viscosity of nanofluids appropriately [14, 16, 32 and 33]. The results showed that suspending nanoparticles in the fluid improve thermal conductivity of the base fluid substantially. For example, Xuan and Li [33] presented a study on the thermal conductivity of a nanofluid consisting of copper nanoparticles. The measured data showed that adding 2.5- 7.5% copper oxide nanoparticles to the water increases its conductivity by about 24-78%. Also, there are some experimental and numerical works about the flow and heat transfer characteristics of the nanofluids in the literature.

Oztop and Abu-Nanda [17] numerically examined the impact of various nanofluids on natural convection flow field and temperature disseminations in somewhat warmed rectangular walled in areas. The impact of utilizing nanofluids on heat transfer and fluid flow qualities in a rectangular shaped micro channel heat sink [MCHS] was numerically examined by Mohammed et al. [15].

Zeinali Heris et al., [34] have tentatively examined constrained convective heat transfer through square cross-sectional channel under laminar flow administration utilizing CuO/water nanofluid and this exploration is a part of an incorporated examination task to study heat transfer qualities through non-circular ducts and by using numerous sorts of nanoparticles. Zeinali Heris et al., [35] have researched constrained convective heat transfer attributes of three distinctive nanofluids (Al2O3/water, CuO/water and Cu/water) flowing through a square cross-segment channel in laminar flow under constant wall temperature boundary conditions. Krishnan et al., [13] have defined the transport equations considering nanofluid as a homogeneous fluid utilizing two - dimensional finite volume technique. Ahmed H. Ali et al., [1] have explored the transient conduct of completely, laminar flow constrained convection with different groupings of nanoparticles and given Reynolds number on the heat transfer upgrade tentatively. Hejri et al., [9] have talked about laminar flow constrained convective heat transfer of Al2O3/water nanofluid intensive isosceles triangular cross sectional channel with steady divider heat flux was recreated numerically. Zeinali Heris et al., [36] have considered the impacts of kind of the nanoparticles (Al2O3/ CuO) and geometry (Square/Triangular pipe) on heat transfer are both researched at the same time, both square and triangular pipes are been with the same water powered width, and the recreation aftereffects of including diverse nanoparticles and distinctive geometry are concentrated all the more exhaustively. Additionally, the impact of nanoparticles sort on weight drop in conduits concentrated numerically, and in light of the writing this subject is once in a while examined.

Reddaiah et al., [24] have analyzed the effect of viscous dissipation on convective heat and mass transfer flow of a viscous fluid in a duct of rectangular cross section by employing Galerkin finite element analysis. Shanthi et al., [29] examined double diffusive e flow in a rectangular cavity with the help of Darcy model. She has examined the influence of dissipation and radiation on the double diffusive flow of a viscous fluid in the rectangular cavity. Chamka et al., [2] have examined the Hydro magnetic double –diffusive convection in a rectangular duct with opposing buoyancy forces. Chamka et al., [3] have discussed Heat and mass transfer in a porous medium filled rectangular duct with Soret and Dufour effects under inclined magnetic field. Umadevi et al., [30] have studied Finite element analysis of double-diffusive heat transfer flow in rectangular duct with thermo-diffusion and radiation effects under inclined magnetic field. Recently Rao et

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In this paper an attempt has been made to investigate the convective heat transfer flow of nanofluid in a rectangular duct with dissipative effects in the presence of heat sources. The velocity and temperature have been discussed for a different variation of governing parameters.

2. PROBLEM FORMULATION

We consider the mixed convective heat transfer flow of a viscous incompressible fluid in a soaked porous medium limited in the rectangular duct (Fig.1) whose base length is "a" and height "b". The heat flux on the base and top walls is maintained constant. The Cartesian coordinate system O (x, y) is picked with origin on the central axis of the duct and its base parallel to x-axis.

Fig.1 Schematic Diagram of the Problem The governing equations are given by

0

 





  y v x

u (1)

nf

k p

u μ x

 

 

     (2)

' nf nf

k p

v ρ g

μ y

  

      (3)

2 2

2 2 2

2 2 0

( _{p nf}) T T _nf T T ( ) _nf 2 u 2 v u v

c u v k Q T T

x y x y x y y x

 ^ ^ ^ ^ ^^ ^^ ^^ ^^   ^ ^^ ^^  ^^ ^^{ }   ^ ^^ ^^ ^ (4)

₀ T^h 2T^c

T 

 (5)

Where u and v are Darcy velocities along (x, y) direction. T, p and g are the temperature, pressure and acceleration due to gravity, Tc, and Th are the temperature on the cold and warm side walls respectively. , _f , _f , and  are the density, coefficients of viscosity, kinematic viscosity and thermal expansion of the fluid, k is the permeability of the porous medium, Kf is the thermal conductivity, C_p is the specific heat at constant pressure , Q is the strength of the heat source. The effective density of the nanofluid is given by

(1 )

nf f s

     (6)

Where  is the solid volume fraction of nanoparticle Thermal diffusivity of the nanofluid is

y

T=Tc u g T=Th v

x

(4)

( )

nf nf

p nf

k

 C

  (7)

Where the heat capacitance Cp of the nanofluid is obtained as

(C_{p nf})  (1  )( C_p)_f  ( C_{p s}) (8)

And the thermal conductivity of the nanofluid k_nf for spherical nanoparticles can be

Written as ⁽ ² ^{) 2 (} ⁾

( 2 ) ( )

nf s f f s

f s f f s

k k k k k



  

    (9)

The thermal expansion coefficient of nanofluid can determine by

()_nf  (1  )( )_f  ( )_s (10)

Also the effective dynamic viscosity of the nanofluid given by (1 )2.5

f nf

 

 

 , (11) Where the subscripts nf, f and s represent the thermo physical properties of the nanofluid, base fluid and the nanosolid particles respectively and  is the solid volume fraction of the nanoparticles. The thermo physical properties of the nanofluid are given in Table 1 (See Oztop and Abu-Nada [17]).

Table 1 Physical

properties

Fluid phase

CuO (Copper)

Al₂O₃ (Alumina)

TiO₂

(Titanium dioxide) C_p(j/kg K)

ρ(kg m³) k(W/m K) βx10^-5 1/k)

4179 997.1 0.613 21

385 8933

400 1.67

765 3970

40 0.63

686.2 4250 8.9538

0.85 The boundary conditions are

u = v = 0 on the boundary of the duct T = Tc, on the side wall to the left

T = Th , on the side wall to the right (12)

0



  y

T , on the top (y = 0) and bottom

0

v

u Walls(y = 0) which are insulated.

We now introduce the following non-dimensional variables

x = ax ; y = by ; h = b/a

' (v^f )

u u

 a , ^' (v^f )

v v

 a ,

2 '

(v^f2 )

p p

a

  , T = T₀ +  (T_{h –}T_c) (13)

1 2.5

1

(1 )

A ^  ; ₂ (1 ) ( ^s )

f

A   

    ; ₃ 1 ⁽ ⁾

( )

p s

p f

A C

C

  

    ;

4

( )

1 (

( )

s

f

A   

    ; ₅ ^nf

f

A k

 k

The governing equations in the non-dimensional form are

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2 1

k p

u a A x

  

    (14)

4

2 2 2

1

( _h _c) kag A T T k p kag

v a A y v v

  

    

 (15)

 

2 2

5 2 2 C

P u v A E u v

x y x y

  ^   ^ 

     

      

     

    (16)

In view of the equation of continuity we introduce the stream function  as v x

u y





 

  

;

(17)

Eliminating p from the equation (14) and (15) and making use of (18) the equations in terms of  and  are

1 2

1 4( )

D GA A

x

 

   

 (18)

2 2

3 5 2 2 1

r r C

P A A P E A

y x x y x y y x

    ^   ^  ^   ^

             

              

      

(19)

Where

 

3 2

( )

Grashofnumber

h c

g T T a

G v

 

 P = C / k_r  _p _f(Prandtl number)

 = Qa^z/k_f (Heat source parameter)

2

1 a

D k

  (Inverse Darcy Parameter)

Ec

2

( ^f )

kCp T



 (Eckert number) The boundary conditions are

1 , 0 0

,

0  



 



 on x

y x



 (20)

0

1 

 on x



1 ,

0 

 on x

 (21)

3. FINITE ELEMENT ANALYSIS AND SOLUTION OF THE PROBLEM

The region is divided into a finite number of three node triangular elements, in each of which the element equation is derived using Galerkin weighted residual method. In each element f_i the approximate solution for an unknown f in the variation formulation is expressed as a linear combination of shape function.

 

N_kⁱ k1,2,3, Which are linear polynomials in x and y.

If ⁱ, ⁱ and C^{i be} are approximate values of , and C in an element _i. then

i i i i i i

i N₁ ₁ N₂ ₂ N₃ ₃

    (22a)

i i i i i i

i N₁ ₁ N₂ ₂ N₃ ₃

    (22b)

Substituting the approximate value ⁱ and ⁱ for  and respectively in (19) and (21) the error residuals are

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2 2

1 5( 2 2 ) 1

i i i i i i

i

r C

E A P E A

x y y x x y y x

  ^     ^  ^^  ^  ^

        

                   

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Under Galerkin method this error is made orthogonal over the domain of ei to the respective shape functions (weight functions) where

0

1  



ei

i k

i N d

E ,

2 2

5 2 2 1

( 0

z i z i i i i i

i

k r c

ei

N A P E A d

x y y x x y y x

        



 

               

                

     



⁽²⁴⁾

Using Green’s theorem we reduce the surface integral (24) without affecting  terms and obtain

5

2 2

1

i i i i i i i i

k k

r k

i k ei

C

N N

A P N

x x y y y x x y

N d

E A y x

      

 

           

   

             

 

       

      

        

 



=





 



 







 



i

i y i x

i i

k n d

n y

N x 

(25)

Where I is the boundary of ei

Substituting L.H.S. of (22a) - (22b) for ⁱ , ⁱ in (25) We get

5

1 1

i i i i i i i i

k L L k i m L m L

r m

ei ei

N N N N N N N N

A d P d

x x y y  y x x y

            

           

   

   

2 2

1 i

l i C

ei k ei

N N d A E d

y x

 

  ^^^ ^ ^^ ^ ^



 

 



       







 



















 

i

i k i y i

x i i

k n d Q

n y

N x 

(l, m, k = 1,2,3) (26)

WhereQ_kⁱ Q_kⁱ₁Q_kⁱ₂Q_kⁱ₃,Q_kⁱ’s being the values of Q_kⁱ on the sides s = (1, 2, 3) of the element ei. The sign ofQ_kⁱ’s depends on the direction of the outward normal w.r.to the element.

Choosing differentN_kⁱ’s as weight functions and following the same procedure we obtain matrix Equations for three unknowns (Q ) viz., ⁱ_p

) ( ) )(

(aⁱ_p _pⁱ  Q_kⁱ (27)

Where (aⁱ_pk) is a 3 x 3 matrix, (ⁱ_p),(Q_kⁱ) are column matrices.

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Introducing the global coordinates and global values for _pⁱand making use of inter element continuity and boundary conditions relevant to the problem the above stiffness matrices are assembled to obtain a global matrix equation. This global matrix is r x r square matrix if there are r distinct global nodes in the domain of flow considered.

Similarly substituting ⁱ,ⁱ and ⁱ in (20) and defining the error

1 2

3 4 A1

x

Ei D^ GA ^^ ^ (28)

and following the Galerkin method we obtain

3 0





d

Eⁱⁱ_j (29)

Using Green’s theorem (29) reduces to

1

4 1

( )

i i i i i

k k i k

N N N

D GA A d

x x y y x

  





       

  

   

        

 



i i

i i i

k x y i k x i

N n n d N n d

x y

  

 

  





      



 ⁽³⁰⁾

In obtaining (30) the Green’s theorem is applied w.r.to derivatives of  without affecting  terms.

Using (22 a) and ( 22 b) in (30 ) we have

1

i

4 1 k

( N

i i i i

k m m k

i

m i

m i L

L i

L

N N N N

D d

x x y y

GA A N d

x









      

  

       

   

 

   

  

 

 

i i

i i i i

k x y i k i k

N n n d N d

x y

  

 

  





      



   ⁽³¹⁾

In the problem under consideration, for computational purpose, we choose uniform mesh of 10 triangular elements (Fig. 2). The domain has vertices whose global coordinates are (0, 0), (1,0) and (1,c) in the non-dimensional form. Let e1, e2…..e10 be the ten elements and let 1, 2, …..10

be the global values of  and 1, 2,……10 be the global values of  at the ten global nodes of the domain (Fig. 2).

The global matrix equations are coupled and are solved under the following iterative procedures. At the beginning of the first iteration the values of (i) are taken to be zero and the global equations (26) &

(31) are solved for the nodal values of . These nodal values (_i) obtained are then used to solve the global equation (31) to obtain (_i).In the second iteration these (i) values are obtained are used in (26) to calculate (i) and vice versa. The two equations are

(1, h)

(2/3, 2h/3)

(y=2h/3)

(1/3, h/3)

(y=h/3) (2/3, h/3)

(x=0) (x=1/3) (x=2/3) (x=1)

Fig.2 Schematic Diagram of the Problem

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The dimensionless Nusselt numbers (Nu) on the non-insulated boundary walls of the rectangular duct are calculated using the formula Nu = (

x



) x=1 and

Nusselt number on the side wall x=1in different regions are given by

     

1 3 4 5

5.1 x 5.2 x 5.3 x

Nu  n   n   n  , (0yh/3)

     

2 6 5 9

6.1 x 6.2 x 6.3 x

Nu  n   n   n  , , ( ² )

3 3

h h

 y

     

3 8 9 10

9.1 x 9.2 x 9.3 x

Nu  n   n   n  , ,(² ) 3

h  y h And the corresponding expressions are

Nu₁=3-3₃, (0 yh/3); Nu₂=3-3₅, (h/3 y2h/3) ; Nu₃=3-3₇, (2h/3 yh) Comparison:

In this analysis, it should be mentioned that the results obtained herein are compared with the results of Rao et al., (21) as shown in Table 2 and 3 in the absence of Dissipation (E_c=0) and Heat sources (α=0) in CuO-water Nanofluid and Al₂O₃-water Nanofluid.

Table-2

Comparison of the present results (Ec=0 and α=0) with Rao et al., in Al₂O₃-water Nanofluid

Rao et al.,(21) Present (Ec=0 and α=0) CuO-water CuO-water

 0.1 0.3 0.7 0.1 0.3 0.7

Nu₁ 2.5678 2.6067 2.6206 2.5677 2.6066 2.6207 Nu2 2.5897 2.6167 2.6312 2.5896 2.6168 2.6311 Nu3 2.6045 2.6338 2.6798 2.6044 2.6339 2.6799

Table-3

Comparison of the present results (E_c=0 and α=0) with Rao et al., in Al2O3-water Nanofluid

Rao et al.,(21) Present (E_c=0 and α=0) Al2O3-water Al2O3-water

 0.1 0.3 0.7 0.1 0.3 0.7

Nu1 2.4897 2.4765 2.4532 2.4898 2.4766 2.4533 Nu₂ 2.3912 2.3440 2.3123 2.3913 2.3341 2.3124 Nu3 2.3315 2.3123 2.3076 2.3316 2.3122 2.3077

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In this analysis we investigate the effect of dissipation and heat source on convective heat transfer flow of an electrically conducting Nano fluid through a porous medium in a rectangular cavity. The equations of the flow of heat transfer have been solved by employing Galerken Finite element analysis with triangular elements. The approximation functions are taken as are linear in x and y. The velocity and concentration at different levels has been discussed for different parametric values.

Effects of parameters on temperature profiles:

Figures 3-6 represents θ with heat source parameter α. It can be seen from the profiles that in the presence of heat source, energy is generated in the thermal boundary layer and as a consequence the temperature rises, in the case of heat sink, energy absorbed in the layer. It is also observed that the values of the actual temperature in CuO-water nanofluid are lesser than those values in Al2O3-water nanofluid at all levels.

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4

0.666 0.716 0.766 0.816 0.866 0.916 0.966

I II III IV V VI VII VIII

x



-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2

0.333 0.433 0.533 0.633 0.733 0.833 0.933

x



Fig. 3 Variation of  with  at y = 2h/3 Fig. 4 Variation of  with  at y = h/3

CuO-water Al2O3-water CuO-water Al2O3-water I II III IV V VI VII VIII I II III IV V VI VII VIII

 2 4 -2 -4 2 4 -2 -4  2 4 -2 -4 2 4 -2 -4

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4

0.666 0.716 0.766 0.816 0.866 0.916 0.966 I II III IV V VI VII VIII

x



-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2

0.333 0.433 0.533 0.633 0.733 0.833 0.933

x



Fig.3 Variation of  with  at y = 2h/3 Fig.4 Variation of  with  at y = h/3 CuO-water Al2O3-water CuO-water Al2O3-water I II III IV V VI VII VIII I II III IV V VI VII VIII

 2 4 -2 -4 2 4 -2 -4  2 4 -2 -4 2 4 -2 -4

-1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6

0 0.05 0.1 0.15 0.2 0.25 0.3 I

II III IV V VI VII VIII

y



-1.5 -1 -0.5 0 0.5 1

0 0.1 0.2 0.3 0.4 0.5 0.6

y



Fig. 5 Variation of  with  at x = 1/3 Fig. 6 Variation of  with  at x = 2/3 CuO-water Al2O3-water CuO-water Al2O3-water I II III IV V VI VII VIII I II III IV V VI VII VIII

 2 4 -2 -4 2 4 -2 -4  2 4 -2 -4 2 4 -2 -4

The variation of θ with Eckert number Ec is represented in figs. 7-10. We observe from these profiles that higher the dissipative heat larger the actual temperature at all levels for both types of nanofluid. This is due to the fact that the thermal energy is reserved in the fluid on account of frictional heating; hence the actual temperature raises at all horizontal and vertical levels.

-1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6

0 0.05 0.1 0.15 0.2 0.25 0.3 I

II III IV V VI VII VIII

y



-1.5

-1 -0.5 0 0.5 1

0 0.1 0.2 0.3 0.4 0.5 0.6

y



Fig. 5 Variation of  with  at x = 1/3 Fig. 6 Variation of  with  at x = 2/3 CuO-water Al2O3-water CuO-water Al2O3-water I II III IV V VI VII VIII I II III IV V VI VII VIII

 2 4 -2 -4 2 4 -2 -4  2 4 -2 -4 2 4 -2 -4

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

x



0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

x



Fig.7 Variation of  with Ec at y = 2h/3 Fig.8 Variation of  with Ec at y = h/3

CuO-water Al2O3-water CuO-water Al2O3-water I II III IV V VI VII VIII I II III IV V VI VII VIII

Ec 0.1 0.3 0.5 0.7 0.1 0.3 0.5 0.7 Ec 0.1 0.3 0.5 0.7 0.1 0.3 0.5 0.7

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

x



0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

x



Fig.7 Variation of  with Ec at y = 2h/3 Fig.8 Variation of  with Ec at y = h/3

CuO-water Al2O3-water CuO-water Al2O3-water I II III IV V VI VII VIII I II III IV V VI VII VIII

Ec 0.1 0.3 0.5 0.7 0.1 0.3 0.5 0.7 Ec 0.1 0.3 0.5 0.7 0.1 0.3 0.5 0.7

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-0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2

0 0.05 0.1 0.15 0.2 0.25 0.3



-0.1 0 0.1 0.2 0.3 0.4 0.5

0 0.2 0.4 0.6

y



Fig.9 Variation of  with Ec at x = 1/3 Fig.10 Variation of  with Ec at x = 2/3

CuO-water Al2O3-water CuO-water Al2O3-water I II III IV V VI VII VIII I II III IV V VI VII VIII Ec 0.1 0.3 0.5 0.7 0.1 0.3 0.5 0.7 Ec 0.1 0.3 0.5 0.7 0.1 0.3 0.5 0.7 -0.15

-0.1 -0.05 0 0.05 0.1 0.15 0.2

0 0.05 0.1 0.15 0.2 0.25 0.3



-0.1 0 0.1 0.2 0.3 0.4 0.5

0 0.2 0.4 0.6

y



Fig.9 Variation of  with Ec at x = 1/3 Fig.10 Variation of  with Ec at x = 2/3

CuO-water Al2O3-water CuO-water Al2O3-water I II III IV V VI VII VIII I II III IV V VI VII VIII

Ec 0.1 0.3 0.5 0.7 0.1 0.3 0.5 0.7 Ec 0.1 0.3 0.5 0.7 0.1 0.3 0.5 0.7

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

x



0 0.1 0.2 0.3 0.4 0.5 0.6

x



Fig.11 Variation of  with  at y = 2h/3 Fig.12 Variation of  with  at y = h/3 CuO-water Al2O3-water CuO-water Al2O3-water I II III IV V VI VII VIII I II III IV V VI VII VIII  0.1 0.3 0.5 0.7 0.1 0.3 0.5 0.7  0.1 0.3 0.5 0.7 0.1 0.3 0.5 0.7

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

x



0 0.1 0.2 0.3 0.4 0.5 0.6

0.333 0.433 0.533 0.633 0.733 0.833 0.933

x



Fig.11 Variation of  with  at y = 2h/3 Fig.12 Variation of  with  at y = h/3 CuO-water Al2O3-water CuO-water Al2O3-water I II III IV V VI VII VIII I II III IV V VI VII VIII

 0.1 0.3 0.5 0.7 0.1 0.3 0.5 0.7  0.1 0.3 0.5 0.7 0.1 0.3 0.5 0.7

-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2

0 0.05 0.1 0.15 0.2 0.25 0.3

y



-0.1

0 0.1 0.2 0.3 0.4 0.5

0 0.1 0.2 0.3 0.4 0.5 0.6

y



Fig.13 Variation of  with  at x = 1/3 Fig.14 Variation of  with  at x = 2/3 CuO-water Al2O3-water CuO-water Al2O3-water

I II III IV V VI VII VIII I II III IV V VI VII VIII  0.1 0.3 0.5 0.7 0.1 0.3 0.5 0.7  0.1 0.3 0.5 0.7 0.1 0.3 0.5 0.7 -0.2

-0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2

0 0.05 0.1 0.15 0.2 0.25 0.3

y



-0.1

0 0.1 0.2 0.3 0.4 0.5

0 0.1 0.2 0.3 0.4 0.5 0.6

y



Fig.13 Variation of  with  at x = 1/3 Fig.14 Variation of  with  at x = 2/3 CuO-water Al2O3-water CuO-water Al2O3-water

I II III IV V VI VII VIII I II III IV V VI VII VIII

 0.1 0.3 0.5 0.7 0.1 0.3 0.5 0.7  0.1 0.3 0.5 0.7 0.1 0.3 0.5 0.7 Figures 11-14 shows the variation θ with nanoparticle volume fraction. An increase in  enhances the actual temperature at y= and reduces at y= , x= and x= levels for both types of nanofluid. The volume fraction parameter  is found to be of significance in this problem, which has non negligible effect on the improvement of the heat characteristics of the fluid. Also the values of actual temperature in CuO-water nanofluid are relatively smaller than those values of actual temperature in Al2O3-water nanofluid.

Effects of parameters on Nusselt number:

The rate of heat transfer (Nusselt Number) is shown in the table-4 for α, Ec and . The variation of Nu with heat source parameter α shows that the Nusselt number in all the three quadrants reduces with increase in the strength of heat generating source while the Nusselt number enhances with increase in the strength of heat absorbing source (α<0) in both types of nano fluids. The variation of Nu with Eckert number Ec exhibits a decreasing tendency with increasing Ec at all the three quadrants in both types of nanofluid. The variation of Nu with volume fraction parameter  shows that the Nusselt number in the lower and middle quadrants enhances in CuO-water nanofluid and reduces in Al2O3-water nano fluid while the Nusselt number on the upper quadrant reduces with increase in  in both types of nanofluid.

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Nusselt Number (Nu)

Parameters CuO-water Al2O3-water

Nu1 Nu2 Nu3 Nu1 Nu2 Nu3

 2 4 -2 -4

2.9085 2.5056 3.7350 4.1610

2.6226 1.9074 4.0800 4.8270

2.4321 1.5090 4.3110 5.2710

2.5951 1.7081 4.4396 5.4005

2.4829 1.6109 4.2904 5.2286

2.5374 1.8826 3.8843 4.8773 Ec 0.1

0.3 0.5

2.9085 2.9068 2.8986

2.6226 2.6223 2.6220

2.4321 2.4312 2.4306

2.5951 2.5926 2.5901

2.4829 2.4818 2.4802

2.5374 2.5360 2.5347

 0.1 0.3 0.7

2.9085 2.9145 2.9236

2.6226 2.6250 2.6286

2.4321 2.4318 2.4283

2.5951 2.5866 2.5610

2.4829 2.4740 2.4500

2.5374 2.5245 2.4587

5. CONCLUSIONS:

The effects of dissipation, heat source, nano concentration on convective heat and mass transfer flow of a chemically reacting nano fluid through a porous medium in a rectangular duct. The important conclusions are

 In the presence of heat source, energy is generated in the thermal boundary layer and as a consequence the temperature rises. In the case of heat sink, energy is absorbed in the layer. It is also observed that the values of the actual temperature in CuO-water nanofluid are lesser than those values in Al2O3-water nanofluid at all levels.

The actual temperature enhances with increase in the heat generating source and reduces with that of heat absorbing source at all levels. It is found that the actual concentration in CuO-water nanofluid at y= , x= and x= levels is greater than those values in Al2O3- water nanofluid for all α. At y= level the actual concentration in CuO-water nanofluid is relatively greater than those values in Al2O3-water nanofluid in heat generating case. A reversed effect is observed in the case of heat absorbing.

The Nusselt number in all the three quadrants reduces with increase in the strength of heat generating source while the Nusselt number enhances with increase in the strength of heat absorbing source (α<0) in both types of nanofluids.

 Higher the dissipative heat larger the actual temperature at all levels for both types of nanofluid. This is due to the fact that the thermal energy is reserved in the fluid on account of frictional heating; hence the actual temperature raises at all horizontal and vertical levels.

 An increase in nano particle volume fraction φ enhances the actual temperature at y=

and reduces at y= , x= and x= levels for both types of nanofluid. The volume fraction parameter  is found to be of significance in this problem, which has non negligible

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©2016 RS Publication, [email protected] Page 222 effect on the improvement of the heat characteristics of the fluid. Also the values of actual temperature in CuO-water nanofluid are relatively smaller than those values of actual temperature in Al₂O₃-water nanofluid.

The Nusselt number in the lower and middle quadrants enhances in CuO-water nanofluid and reduces in Al2O3-water nano fluid while the Nusselt number on the upper quadrant reduces with increase in  in both types of nanofluid.

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