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water hybrid nanofluid
Accepted Manuscript Heat transfer enhancement with Ag-CuO/water hybrid nanofluid Tanzila Hayat, S. Nadeem PII: DOI: Reference:
S2211-3797(17)30846-X http://dx.doi.org/10.1016/j.rinp.2017.06.034 RINP 750
To appear in:
Results in Physics
Please cite this article as: Hayat, T., Nadeem, S., Heat transfer enhancement with Ag-CuO/water hybrid nanofluid, Results in Physics (2017), doi: http://dx.doi.org/10.1016/j.rinp.2017.06.034
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Heat transfer enhancement with Ag-CuO/water hybrid nanouid Tanzila Hayat1 and S. Nadeem Department of Mathematics, Quaid-i-Azam University, Islamabad 44000, Pakistan Abstract: Nanouids are of great importance to researchers as they have signi cant uses industrially due to their high heat transfer rates. Recently, a new class of nanouid, "hybrid nanouid" is being used to further enhance the heat transfer rate. This new model in 3D is employed to examine the impact of thermal radiation, heat generation and chemical reaction over stretching sheet in the presence of rotation. It is concluded from the current research that even in the presence of radiation, heat generation and chemical reaction the heat transfer rate of Hybrid nanouid is higher than the simple nanouid.
Keywords: Hybrid nanouid, Three dimensional ow, Heat and mass transfer, Radiation, Heat generation, Chemical reaction, Stretching sheet.
1
Introduction
Nanouids are a classi cation of heat transfer uids which are engineered suspension nanoparticles(1-100nm) being dispersed in the uid. Usually base uids incorporate water, organic uids (e.g. ethylene, triethylene and so on) engine oil, polymeric solutions, bio-uids and other basic uids. Medium normally utilized as nanoparticles encompass carbon in di¤erent structures (e.g. carbon nanotubes, graphite, diamond) metals (e.g. copper, silver, gold), metal oxides (e.g titania, zirconia) and functionalized nanoparticles. Utilization of nanouids has found an extensive variety of potential applications. Choi was the rst one to study enhancement of thermal conductivity in nanouids [1]. As indicated by applications, nanouids are listed as heat transfer uids, bio and pharmaceutical nanouids, medicinal nanouids, enviromental nanouids etc. Numerous analysts contemplated how the size, concentration, shape and other properties inuence the heat transfer rate of uid. The fusion of specialized uids that are designed towards enhancing the performance of heat exchangers has turned out to be progressively appealing lately. So this topic has pulled in collossal interest from researchers because of interesting properties and applications[2-18]. Chen et al. [19] presented an approach to predict the thermal conductivity of uid containing nano-sized particles, based on their rheological properties. Zhou et al. [20] studied the viscosity and thermal conductivity of various kinds of surfactant mixtures. Heat transfer enhancement and thermal conductivity in nanouids had been studied by [21]. Kabeel et al. [22] examined the performance of heat transfer of plate heat exchanger with alumina-water nanouid and water-water uids. Signi cance and importance of nanouids to increase the heavy duty engine and automative cooling rates was explained by Peyghambarzadeh et al. [23]. The improvement in heat transfer with the help of nanoparticles concentration and ow conditions were addressed in [24]. Duangthongsuk and Wongwises [25] exposed the di¤erence 1
Corresponding author: [email protected]
1
among the experimental data and computed thermophysical properties of nanouids on heat transfer phenomena. Quiet recently numerous experiments have been done with two types of nanoparticles suspended in base uid named as "Hybrid Nanouid", the cutting edge nanouid. The principle preferred standpoint of utilizing hybrid nanouid is by choosing a proper combination of nanoparticles, positive features can be improved and inconveniences can be covered due to their synegistic e¤ect. These hybrid nanouids are reasonably a new class of nanouids which have numerous conceivable applications in all the elds of heat transfer e.g. micro uidics, manufacturing, transportation, defence, medical, naval structures, acoustics etc. When nano-sized particles are dispersed appropriately, hybrid nanoparticles o¤er collosal bene t having exceptional high e¤ective thermal conductivity. Particularly, nanouid ow is well-known for high heat transfer as compared to normal uid. To improve it even more, the hybrid nanouid is instigated. Many experimental research articles have been published with concept of hybrid nanouid. Momin [26] carried out an experimental study of mixed convection with (Al2 O3 Cu=H2 O) hybrid nanouid for laminar ow in an inclined tube. Study on synthesize (Al2 O3 Cu=H2 O) of hybrid nanouid was examined by Suresh et al [27]. Suresh et al. [28] studied e¤ects of (Al2 O3 Cu=H2 O) hybrid nanouid in heat transfer. Turbulent heat transfer and pressure drop characteristics of dilute water based (Al2 O3 Cu=H2 O) hybrid nanouids was examined by suresh et al. [29]. Since no numerical study has yet been done for the heat transfer characteristics for boundary layer ow of 3D rotating hybrid nanouid over a stretching sheet with radiation, heat generation and chemical reaction e¤ects. So the present study is centered to examine these e¤ects. The governing equations of phase ow hybrid nanouid model are simpli ed with the help of similarity transformations and nally the reduced equations are solved numerically. Various important parameters are discussed at the end of the paper.
2
Problem formulation
Let us consider the three-dimensional, rotating, hybrid nanouid (Ag CuO=water) ow past a linearly stretching surface at z = 0. The uid occupy the half space at z 0: We have considered Copper Oxide (CuO) and Silver(Ag) nano-size particles with base uid as a water. Initially, CuO( 1 ) nanoparticle of 0:1 volume fraction (which is not changed in the given problem) is scattered into the base uid to make nanouid (CuO water). Thus, to develop the targetd "hybrid nanouid (Ag CuO=water)", Silver ( 2 ) with di¤erent volume fractions is dispersed in nanouid (CuO water). The rotation of nanouid (CuO water) is taken about vertical axis so that the uids angular velocity ! is constant. Using these assumptions, the governing equations can be written in the following form as :
2
Figure 1: Geometry of the problem
@u @v @w + + = 0; @x @y @z
u
(1)
u
@u @u @u +v +w @x @y @z
@2u ; @z 2
(2)
u
@v @v @2v @v +v +w + 2 u = hnf 2 ; @x @y @z @z
(3)
2 v = hnf
@T @T @T @ 2T Q +v +w = hnf 2 + (T @x @y @z @z (Cp )hnf u
@c @2c @c @c +v +w = hnf 2 @x @y @z @z
1 (c
T1 )
@qr ; @z
c1 ) n :
(4) (5)
Where hnf ; hnf and hnf are momentum, thermal and concentration di¤usivities of hybrid nanouid respectively. The dimensional heat generation / absorption coe¢cient is denoted by Q and constant rate of 1st order chemical reaction is denoted by 1 : Using Roseland approximation [30-32], the radiation ux qr is given by
4 @T 4 qr = ; 3 @z
(6)
where and are the "Stefan-Boltzman constant" and mean absorption co-e¢cient respectively. Now expanding the Tailor series about T1 we have 3 T 4 4T T1
4 3T1
The related boundary conditions for three dimensional ow are given by 3
(7)
u = Uw = ax; v = Vw = by; w = 0; T = Tw ; c = cw ; at z = 0; u ! 0;
v ! 0;
T ! T1 ; c ! c1 as z ! 1:
(8) (9)
The given issue can be stated in a more straightforward form by using the suitable similarity transformation de ned as
0
p
0
u = axA (); v = ayB (); w = C () =
T Tw
a f (A() + B()); = z
r
a ; f
c c1 T1 ; D () = : T1 cw c1
(10)
With the help of above transformations, Eq. (1) is identically satis ed while Eqs. (2-9) transformed into following coupled nonlinear di¤erential equations.
A000 ()
2 )2:5 [(1
2 )f(1
[(A0 ())2
A00 ()(A() + B())
B 000 ()
1 )2:5 (1
(1
[(B 0 ())2
(
1 )2:5 (1
(1
2 )2:5 [(1
1 ) + 1 (
s1 )g + 2 ( s2 )] f f
2 B 0 ()] = 0;
2 )f(1
1 ) + 1 (
(11)
s1 )g + 2 ( s2 )] f f
B 00 ()(A() + B()) + 2 A0 ()] = 0;
(12)
(Cp )s1 (Cp )s2 Khnf 4 + R)C 00 ()+Pr[(1 2 )f(1 1 )+ 1 ( )g+ 2 ( )](A()+B())C 0 ()+C() = 0; Kf 3 (Cp )f (Cp )f (13) Sc D00 () + [(A() + B())D0 () Rc D() = 0; (14) (1 1 )(1 2 ) A = 0; A0 ! 0;
A0 = 1; B 0 ! 0;
B = 0;
B 0 = ;
C = 1; D = 1;
at = 0;
C ! 0; D ! 0 as ! 1;
(15)
where ; ; Pr; R; ; Sc and Rc are the rotation parameter, stretching ratio parameter, Prandtl number, radiation parameter, heat generation parameter, Schmidt number and chemical reaction constraint respectively and are de ned as 3 b f (Cp )f 4T1 Q
; = ; Pr = ; R= ; = ; a a Kf Kf a(Cp )f (c c1 )n 1 f ; Rc = 1 : Sc = f a
=
4
(16)
The physical quantities of the given problem are, the "Skin-friction" along x and y axis Cf x , Cf y ; the "local Nusselt number" N ux and the "Sherwood number" Shx ; de ned by Cf x Shx or
hnf ( @u hnf ( @v ) ) @z z=0 @z z=0 = ; C = ; N ux = fy 2 f (ax) f (ax)2 @c xKhnf : = Kf (cw c1 ) @z z=0 1
1
Re 2 Cf x =
(1 Re
1 1 2
)2:5 (1
N ux =
)2:5
A00 (0);
2 Khnf 0 C (0) ; Re Kf
1
xKhnf Kf (Tw T1 )
1
Re 2 Cf y = 1 2
Shx =
@T , @z z=0
1 )2:5 (1
(1 1 Khnf 0 D (0) Kf
2 )2:5 .
(17)
B 00 (0); (18)
Where Re = Uw x= f is the restricted Reynolds number.
3
Numerical solution
The coupled non-linear ordinary di¤erential equations (11 14) together with their boundary conditions given in equation (15) are solved numerically using BVP-4C technique [33-34] invoking shooting process. In this procedure, rstly the system of equations (11 14) escorted with boundary conditions are reduced to rst order equations. Then appropriate initial guesses are opted which satisfy the boundary conditions. The results obtained depict the impact of various dimensionless parameters such as rotation parameter, stretching ratio parameter, Schmidt number, chemical reaction parameter, radiation and heat generation on velocity, temperature and concentration pro les. For attaining the convergence criterion of 10 6 shooting methodology is reiterated. Solutions to the given problem are given in graphical and tabular form.
4
Results and discussion
Numerical evaluation of the non-linear di¤erential equations has been carried out to get a better understanding of the problem. The inuence of pertinent physical parameters namely rotation parameter, stretching ratio parameter, heat generation parameter, radiation parameter, chemical reaction parameter and Schmidt number on velocity, temperature and concentration pro le are presented graphically in F igs: (2 14): Table. 1 gives us the standard thermophysical properties of nanouid whereas in Table. 2 we can see these properties at 25 C. From Table. 3 we observe that the magnitude of skin friction coe¢cient of hybrid nanouid is enhanced with nanoparticle volume fraction in both x and y directions whereas decreases when we increase stretching ratio parameter. Chemical reaction and Schmidt number have no impact on skin friction coe¢cient. Increment in rotation reduces the magnitude of skin friction coe¢cient in the x-direction whereas it intensi es in the y-direction. From 5
Table. 4 we conclude that changes in stretching ratio parameter, rotation, Schmidt number and chemical reaction have no inuence on heat transfer rate. The magnitude of local Nusselt number ampli es in the presence of nanoparticle volume fraction. We learnt that due to hybrid nanouid (Ag CuO=water) the heat transfer rate was further augmented. The mass transfer rate diminishes when we enhance rotation parameter and nanoparticle volume fraction. While it elevates with an increment in stretching ratio parameter, chemical reaction parameter and Schmidt number. To validate our present numerical structure, comparisons of uid friction A00 (0) with previous published literature for = 0 = 1 = 2 = R is made in Table. 5. A satisfactory settlement is declared to corroborate the numerical scheme.
4.1
Comparison of velocity and temperature pro les
The comparison of velocity pro le for H2 O, CuO water and Ag CuO=water is displayed in F ig:(2). Since no magnetic eld is being applied in the present study which accelerates nanoparticles, hence the hybrid nanouid (Ag CuO=water) reduces the uid velocity. There is also a decrease in uid velocity due to density and dynamic viscosity which rise because of hybridity and so there is a decline in velocity. We also observe that the velocity of hybrid nanouid (Ag CuO=water) is less than nanouids velocity. The reason being obvious that including further massive particles hurdles the normal uid ow. F ig:(3) depicts the comparison of temperature pro le amid hybrid nanouid (Ag CuO=water), CuO water and H2 O. It is visualized that under same circumstances and equivalent total quantities of volume particle fraction, the hybrid nanouid (Ag CuO=water) reaches higher temperature than nanouid (CuO water). A sudden rise in temperature is a result of hybrid nanouid (Ag CuO=water).
1
1
Pure w ater
0 .9
CuO-w a te r
Ag-CuO/w ater
0 .8
0 .8
0 .7
Ag-CuO/w ater
0 .7
0 .6
0 .6
C( η)
A'( η)
Pure w ater
0 .9
CuO-w ater
0 .5 0 .4
0 .5 0 .4
0 .3 0 .3
0 .2 0 .2
0 .1 0 .1
0 0
1
2
3
4
0
5
0
η
1
2
3
4
η
Figure 2: Comparison of A0 ()
Figure 3: Comparison of C ()
6
5
4.2
Impact of rotation parameter ()
It is demonstrated through F igs:(4 5) the inuence of rotation over the velocity distribution A0 () and B0 () in x and y direction respectively. We observe that for higher values of rotation parameter both the ows in x and y direction decelerate as well as the associated boundary layer thickness. The temperature distribution for Ag CuO=water and CuO water is displayed in F ig: (6). From this gure it is learnt that the rotation ampli es the thermal boundary layer thickness. Moreover the instant rise in temperature is due to the hybrid nanouid Ag CuO=water. In F ig:(7) the concentration, D(), has been plotted to see the e¤ects against rotation. Rotation boosts the concentration.
7
1
0 .5
CuO-w a te r
CuO-w ater
0 .9
Ag-CuO/w ater
0 .8
Ag-CuO/w ater
0 .4
0 .7
0 .3
B'( η)
A'( η)
0 .6 0 .5 0 .4
ε = 0.0, 0.5, 0.8
0 .2
ε = 0.0, 0.5, 0.8
0 .1
0 .3 0
0 .2 0 .1
-0 .1
0 0
1
2
3
4
-0 .2
5
0
1
2
η
3
Figure 4: variation of on A0 ()
1
CuO-w a te r
0 .9
CuO-w a te r Ag-CuO/w ater
0 .9
Ag-CuO/w ater 0 .8
0 .8
0 .7
0 .7
0 .6
0 .6
D( η)
C( η)
5
Figure 5: variation of on B0 ()
1
0 .5
ε = 0.0, 0.5, 0.8
0 .5
ε = 0.0, 0.5, 0.8
0 .4
0 .4
0 .3
0 .3
0 .2
0 .2
0 .1
0 .1
0 0
1
2
3
4
5
0
0
1
η
2
3
4
η
Figure 6: variation of on C()
4.3
4
η
Figure 7: variation of on D()
Impact of stretching ratio parameter ()
F ig:(8) elucidate the impact of stretching ratio parameter on velocity distribution in ydirection. Increment in stretching ratio parameter correlates with an increase in the rate of stretching along y-axis so it is obvious there is a rise in the velocity eld and the momentum boundary layer thickness. The e¤ect of stretching ratio parameter on temperature distribution is depicted through F ig:(9). It is discovered from the graph that with an increase in stretching ratio parameter there is a reduction in the temperature pro le. Through F ig:(10) we observe the variation of concentration with respect to stretching 8
5
ratio parameter : Increase in stretching ratio parameter declines the concentration pro le.
0 .9
1
CuO-w a te r
CuO-w a te r
0 .9
Ag-CuO/w ater
Ag-CuO/w ater
0 .7
0 .8
0 .6
0 .7 0 .6
0 .5
C( η)
B'( η)
0 .8
0 .4
0 .4
λ = 0.2, 0.5, 0.9
0 .3
0 .5
0 .3 0 .2 0 .2
λ = 0.2, 0.5, 0.9
0 .1 0 .1 0 0 0
1
2
3
4
5
0
1
2
η
3
4
η
Figure 8: variation of on B0 ()
Figure 9: variation of on C()
1
CuO-w a te r Ag-CuO/w ater
0 .9 0 .8 0 .7
D( η)
0 .6 0 .5 0 .4 0 .3 0 .2
λ = 0.2, 0.5, 0.9
0 .1 0
0
1
2
3
4
5
η
Figure 10: variation of on D()
4.4
Impact of heat generation parameter ()
F ig:(11) illustrates the nature of temperature pro le with the variation of heat generation parameter. Increase in prompts an increase in the temperature eld since energy is produced at thermal boundary layer.
9
5
1
CuO-w ater
0 .9
Ag-CuO/w ater 0 .8 0 .7
C( η)
0 .6 0 .5
δ = -0.3, 0.0, 0.3
0 .4 0 .3 0 .2 0 .1 0
0
1
2
3
4
5
η
Figure 11: variation of on C()
4.5
Impact of radiation parameter (R)
The impact of radiation parameter over the temperature pro le is demonstrated in F ig:(12 (a)). We analyze that an increase in radiation R causes the temperature to in K crease. Physically, the quantity 4T f3 in radiation parameter measures the thermal radiation 1 transfer relative to the conduction heat transfer. Therefore, higher values of this quantity exhibit that thermal radiation is dominate over conduction. Hence a great amount of heat energy due to radiation is being released in the system giving a rise to temperature.The dual temperature pro le in the case of shrinking sheet for various values of radiation parameter R in the absence of nanoparticle volume fraction and rotation is presented in F ig:(12 (b)): From F ig:(12 (b)) it can be observed that the temperature pro les increase with the increase of the radiation parameter R for both solutions: the e¤ect of the radiation parameter R causes an increase in the radiative heat ux.
10
1
CuO-w ater
0 .9
1
Ag-CuO/w ater 0 .8 0 .8
0 .7
C( η)
C( η)
0 .6 0 .5
R = 0.0, 0.5, 1.0
0 .4
0 .6
R =1, 2, 3
0 .3 0 .2
0 .2 0 .1 0
fi rst so l u ti o n
0 .4
se co n d so l u ti o n
0
0
1
2
3
4
5
0
1
2
η
Figure 12(a): variation of R on C()
4.6
3
4
Figure 12(b): variation of R on dual C()
Impact of chemical reaction parameter (Rc )
F ig:(13) demonstrate the e¤ects of chemical reaction parameter Rc on the concentration pro le. The increasing values of chemical reaction lead to decline the concentration of the uid. Consequently the concentration boundary layer thickness get increased.
1
CuO-w ater
0 .9
Ag-CuO/w ater 0 .8 0 .7
D( η)
0 .6 0 .5 0 .4
R = 1.0, 2.0, 3.0 c
0 .3 0 .2 0 .1 0
0
1
2
3
4
5
η
Figure 13: variation of Rc on D()
4.7
5
η
Impact of Schmidt number (Sc )
The e¤ects of Schmidt number on the concentration pro le is demonstrated in F ig:(14). As the Schmidt number is the quantitative relation between momentum to mass di¤usivity. 11
Due to Schmidt number, the di¤usivity is decreases consequently the concentration of the uid decreases.
1
CuO-w a te r
0 .9
Ag-CuO/w ater 0 .8 0 .7
D( η)
0 .6 0 .5
Sc = 0.5, 1.0, 1.5 0 .4 0 .3 0 .2 0 .1 0 0
1
2
3
4
5
η
Figure 14: variation of Sc on D()
Table 1: Thermophysical Properties of CuO water and Ag CuO=water Properties Nanouid (CuO water) nf = (1 )f + s Density () f nf = (1 ) 2:5 Viscosity () (Cp )nf = (1 )(Cp )f + (Cp )s Heat Capacity (CP ) Knf K +(n 1)Kf (n 1) (Kf Ks ) Thermal Conductivity (K) = s Ks +(n 1)K Kf f + (Kf Ks ) Properties Density () Viscosity () Heat Capacity (CP ) Thermal Conductivity (K)
Hybrid Nanouid (Ag CuO=water) hnf = [(1 2 )f(1 1 )f + 1 s1 g] + 2 s2 f hnf = (1 )2:5 (1 2 )2:5 1 (Cp )hnf = [(1 2 )f(1 1 )(Cp )f + 1 (Cp )s1 g] + 2 (Cp )s2 K +(n 1)Kbf (n 1) 2 (Kbf Ks2 ) Kbf K +(n 1)Kf (n 1) 1 (Kf Ks1 ) Khnf ; Kf = s1Ks1 +(n 1)K = s2Ks2 +(n 1)K Kbf bf + (Kbf Ks2 ) f + (Kf Ks1 ) 2
Table 2: Thermophysical Properties of nanoparticles and base uid Properties CuO Ag Base uid (water) 997:1 6320 10500 Cp 4179:0 531:80 235 0:6130 K 76:50 429 6:20 Pr
12
1
Table 3: E¤ects of (Ag CuO=water)
2
Rc
A00 (0),
Sc
B 00 (0) for nanouid (CuO
1 A00 (0) (1 1 )2:5
0.1 0.5 0.07 0.5 0.5 1.91658 0.3 1.80930 0.5 1.77585 0.3 0.2 1.88714 0.5 1.80930 0.9 1.74547 0.05 1.68693 0.07 1.80930 0.1 1.99913 0 1.80930 1 1.80930 2 1.80930 0.5 1.80930 1.0 1.80930 1.5 1.80930 Table 4: E¤ects of (Ag CuO=water)
2
Rc
(1 1 ) 2:5 00 A (0) (1 2 )2:5
2.29784 2.16922 2.12912 2.26254 2.16922 2.09269 1.91773 2.16922 2.60156 2.16922 2.16922 2.16922 2.16922 2.16922 2.16922
water) and hybrid nanouid
1 B 00 (0) (1 1 )2:5
1.06164 1.49397 1.88764 1.00655 1.49397 2.44057 1.39306 1.49397 1.65059 1.49397 1.49397 1.49397 1.49397 1.49397 1.49397
C 0 (0) and -D0 (0) for nanouid (CuO
Sc
Knf 0 C (0) Kf
0.1 0.5 0.07 0.5 0.5 1.32460 0.3 1.32460 0.5 1.32460 0.3 0.2 1.32460 0.5 1.32460 0.9 1.32460 0.05 1.32460 0.07 1.32460 0.1 1.32460 0 1.32460 1 1.32460 2 1.32460 0.5 1.32460 1.0 1.32460 1.5 1.32460
Khnf 0 C (0) Kf
1.22440 1.22440 1.22440 1.22440 1.22440 1.22440 1.15700 1.22440 1.33120 1.22440 1.22440 1.22440 1.22440 1.22440 1.22440
13
(1
(1 1 ) 2:5 00 B (0) (1 2 )2:5
1.27283 1.79116 2.26314 1.20679 1.79116 2.92606 1.58365 1.79116 2.14799 1.79116 1.79116 1.79116 1.79116 1.79116 1.79116
water) and hybrid nanouid
1 )D0 (0)
0.65733 0.64297 0.62757 0.60094 0.64297 0.68567 0.64230 0.64297 0.64486 0.38047 0.82157 1.08770 0.64297 0.957580 1.209200
(1
1 )(1
0.61131 0.59796 0.58364 0.55887 0.59796 0.63767 0.61019 0.59796 0.58038 0.35384 0.76406 1.01150 0.59796 0.89055 1.12460
2 )D0 (0)
Table 5: Comparision of ( = 0 = 1 = 2 = R)
5
A00 (0) for various values of stretching ratio parameter when
Wang [35] Arial [36] Butt et al [37] Present Results
0.0 0.1 0.2 0.3 0.4 0.5
1 1.020902 1.041804 1.062705 1.083607 1.104509
1 1.017027 1.034587 1.052470 1.070529 1.088662
1 1.020260 1.039495 1.057955 1.075788 1.093095
1 1.02137 1.0404 1.05871 1.07643 1.09364
Conclusion
The three dimensional steady rotating ow of "hybrid nanouid (Ag CuO=water)" with thermal radiation, heat generation and chemical reaction is examined on a linearly stretching surface. The main conclusion of the work is as follows: Hybridity boosts the temperature distribution as well as the heat transfer rate at surface. The thermal boundary of "hybrid nanouid (Ag menting the heat generation parameter .
CuO=water)" increases by incre-
There is an enhancement in the rate of mass transfer at surface by increasing the Schmidt number Sc and chemical reaction Rc : There is an increment in concentration pro le with the increase in rotation parameter but it () decreases the concentration at surface.
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[5] S. U. Rehman, R. Ellahi, S. Nadeem, Q. M. Zaigham Zia, Simultaneous e¤ects of nanoparticles and slip on Je¤rey uid through tapered artery with mild stenosis, J Mol Liq., Vol. 218, (2016), 484-493. [6] M. Akbarzadeh, S. Rashidi, M. Bovand, R. Ellahi, A sensitivity analysis on thermal and pumping power for the ow of nanouid inside a wavy channel, J Mol Liq. Vol. 220, (2016), 1-13. [7] K. M. Shirvan, M. Mamourian,S. Mirzakhanlari, R. Ellahi, Two phase simulation and sensitivity analysis of e¤ective parameters on combined heat transfer and pressure drop in a solar heat exchanger lled with nanouid by RSM, J Mol Liq., Vol, 220, 888-901 (2016). [8] N. Shehzad, A.Zeeshan, R.Ellahi, K.Vafai, Convective heat transfer of nanouid in a wavy channel: Buongiornos mathematical model, J Mol Liq. Vol. 222, 446-455 (2016). [9] R. Ellahi, A. Zeeshan, M. Hassan, Particle shape e¤ects on marangoni convection boundary layer ow of a nanouid, Int. J. Numer. Methods Heat Fluid Flow, Vol. 26 (7), 2160-2174 (2016) [10] M. M. Bhatti, A. Zeeshan, R Ellahi, Endoscope analysis on peristaltic blood ow of sisko uid with titanium magneto-nanoparticles, Comput. Biol. Med, Vol. 78, 2941 (2016). [11] M. Sheikholeslami, Q. M. Zaigham Zia, R. Ellahi, Inuence of induced magnetic eld on free convection of nanouid considering Koo-Kleinstreuer (KKL) correlation, J. appl. sci, Vol. 6,324 (1-11), (2016). [12] R. Ellahi, M. Hassan, A. Zeeshan, A study of heat transfer in power law nanouid, J Therm Sci , Vol. 20 ( 6) 2015-2026 (2016). [13] M. M. Bhatti, A.Zeeshan, R.Ellahi, Simultaneous e¤ects of coagulation and variable magnetic eld on peristaltically induced motion of Je¤rey nanouid containing gyrotactic microorganism, Microvasc Res, Vol. 110 (2017) 3242. [14] k. M. Shirvan, R. Ellahi , M. Mamourian , M. Moghiman, E¤ect of Wavy Surface Characteristics on Heat Transfer in a Wavy Square Cavity Filled with Nanouid: Int. J. Heat Mass Transfer. Vol. 107 (2017) 11101118. [15] R. Ellahi, M.H. Tariq, M. Hassan, K. Vafai, On boundary layer magnetic ow of nanoFerroliquid under the inuence of low oscillating over stretchable rotating disk, J Mol Liq. Vol. 229 (2017) 339345. [16] K. M. Shirvan, M. Mamourian, S. Mirzakhanlari, R. Ellahi, Numerical Investigation of Heat Exchanger E¤ectiveness in a Double Pipe Heat Exchanger Filled With Nanouid: A Sensitivity Analysis by Response Surface Methodology, Power Technology, Vol. 313 (2017) 99111. 15
[17] J. A. Esfahani , M. A. Sokkeh , S. Rashidi , M.A. Rosen , R. Ellahi, Inuences of wavy wall and nanoparticles on entropy generation in a plate heat exchanger, Int. J. Heat Mass Transfer, Vol 109, 1162-1171 (2017). [18] S. Rashidi, J. A. Esfahani, R. Ellahi, Convective heat transfer and particle motion in an obstructed duct with two side-by-side obstacles by means of DPM model, J. appl. sci, 7, 431, 2017. [19] H. Chen, S. Witharana, Y. Jina, C. Kim, Y. Ding, Predicting thermal conductivity of liquid suspensions of nanoparticles (nanouids) based on rheology, Particuology 7 (2009) 151157. [20] M. Zhou, G. Xia, J. Li, L. Chai, L. Zhou, Analysis of factors inuencing thermal conductivity and viscosity in di¤erent kinds of surfactant solutions, Exp Therm Fluid Sci 36 (2012) 2229. [21] K. Kwak, C. Kim, Viscosity and thermal conductivity of copper oxide nanouid dispersed in ethylene glycol, Korea Aust Rheol J Journal 17 (2005) 3540. [22] A.E. Kabeel, T.A.E. Maaty, Y.E. Samadony, The e¤ect of using nano-particles on corrugated plate heat exchanger performance, Appl. Therm. Eng. 52 (2013) 221229. [23] S.M. Peyghambarzadeh, S.H. Hashemabadi, S.M. Hoseini, M.S. Jamnani, Experimental study of heat transfer enhancement using water/ethylene glycol based nanouids as a new coolant for car radiators, Int J Heat Mass Transf. 38 (2011) 12831290. [24] S.M. Peyghambarzadeh, S.H. Hashemabadi, M. Naraki, Y. Vermahmoudi, Experimental study of overall heat transfer coe¢cient in the application of dilute nanouids in the car radiator, Appl. Therm. Eng. 52 (2013) 816. [25] W. Duangthongsuk, S. Wongwises, Comparison of the e¤ects of measured and computed thermophysical properties of nanouids on heat transfer performance, Exp Therm Fluid Sci 34 (2010) 616624. [26] G. G. Momin, Experimental investigation of mixed convection with water-Al2 O3 & hybrid nanouid in inclined tube for laminar ow, IJSTR. 2 (2013) 195-202. [27] S. Suresh, K. Venkitaraj, P. Selvakumar and M. Chandrasekar, Synthesis of Al2 O3 Cu/water hybrid nanouids using two step method and its thermo physical properties. Colloids Surf., A 388 (2011) 41- 48. [28] S. Suresh, K. Venkitaraj, P. Selvakumar and M. Chandrasekar, E¤ect of Al2 O3 Cu/water hybrid nanouid in heat transfer, Exp Therm Fluid Sci., 38 (2012) 54 60. [29] S. Suresh, K. P. Venkitaraj, M. S. Hameed, and J. Sarangan, Turbulent Heat Transfer and Pressure Drop Characteristics of Dilute Water Based Al2O3Cu Hybrid Nanouids. J. Nanosci. Nanotechnol. 14 (2014) 25632572. 16
[30] Mukhopadhyay. S., Gorla R.S.R., E¤ects of partial slip on boundary layer ow past a permeable exponential stretching sheet in presence of thermal radiation, Heat Mass Transfer., 2012; 48, 17731781. [31] Mukhopadhyay. S., Slip e¤ects on MHD boundary layer ow over an exponentially stretching sheet with suction/blowing and thermal radiation, Ain Shams Eng J., 2013; 4, 485491. [32] Mahmoodi M., Kandelousi Sh. Analysis of the hydrothermal behaviour and entropy generation in a regenerative cooling channel considering thermal radiation, Nucl Eng Des., 291, 2015; 277-286. [33] Shampine LF , Gladwell I , Thompson S . Solving ODEs with Matlab. Cambridge University Press: Cambridge; 2003 . [34] Shampine L.F., Reichelt M.W., Kierzenka J. Solving boundary value problems for ordinary di¤erential equations in MATLAB with bvp4c . http://www. mathworks.com/bvptutorial . [35] Wang CY., The three dimensional ow due to a stretching at surface, Phys Fluids., 1984; 27,1915-1917. [36] Ariel P. D., The three-dimensional ow past a stretching sheet and the homotopy perturbation method, Comput Math Appl., 2007; 54, 920925. [37] Butt A.S., Asif A., Investigation of entropy generation e¤ects in magnetohydrodynamic three-dimensional ow and heat transfer of viscous uid over a stretching surface, J Braz Soc Mech Sci Eng., 2015; 37, 211219.
17
S2211-3797(17)30846-X http://dx.doi.org/10.1016/j.rinp.2017.06.034 RINP 750
To appear in:
Results in Physics
Please cite this article as: Hayat, T., Nadeem, S., Heat transfer enhancement with Ag-CuO/water hybrid nanofluid, Results in Physics (2017), doi: http://dx.doi.org/10.1016/j.rinp.2017.06.034
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Heat transfer enhancement with Ag-CuO/water hybrid nanouid Tanzila Hayat1 and S. Nadeem Department of Mathematics, Quaid-i-Azam University, Islamabad 44000, Pakistan Abstract: Nanouids are of great importance to researchers as they have signi cant uses industrially due to their high heat transfer rates. Recently, a new class of nanouid, "hybrid nanouid" is being used to further enhance the heat transfer rate. This new model in 3D is employed to examine the impact of thermal radiation, heat generation and chemical reaction over stretching sheet in the presence of rotation. It is concluded from the current research that even in the presence of radiation, heat generation and chemical reaction the heat transfer rate of Hybrid nanouid is higher than the simple nanouid.
Keywords: Hybrid nanouid, Three dimensional ow, Heat and mass transfer, Radiation, Heat generation, Chemical reaction, Stretching sheet.
1
Introduction
Nanouids are a classi cation of heat transfer uids which are engineered suspension nanoparticles(1-100nm) being dispersed in the uid. Usually base uids incorporate water, organic uids (e.g. ethylene, triethylene and so on) engine oil, polymeric solutions, bio-uids and other basic uids. Medium normally utilized as nanoparticles encompass carbon in di¤erent structures (e.g. carbon nanotubes, graphite, diamond) metals (e.g. copper, silver, gold), metal oxides (e.g titania, zirconia) and functionalized nanoparticles. Utilization of nanouids has found an extensive variety of potential applications. Choi was the rst one to study enhancement of thermal conductivity in nanouids [1]. As indicated by applications, nanouids are listed as heat transfer uids, bio and pharmaceutical nanouids, medicinal nanouids, enviromental nanouids etc. Numerous analysts contemplated how the size, concentration, shape and other properties inuence the heat transfer rate of uid. The fusion of specialized uids that are designed towards enhancing the performance of heat exchangers has turned out to be progressively appealing lately. So this topic has pulled in collossal interest from researchers because of interesting properties and applications[2-18]. Chen et al. [19] presented an approach to predict the thermal conductivity of uid containing nano-sized particles, based on their rheological properties. Zhou et al. [20] studied the viscosity and thermal conductivity of various kinds of surfactant mixtures. Heat transfer enhancement and thermal conductivity in nanouids had been studied by [21]. Kabeel et al. [22] examined the performance of heat transfer of plate heat exchanger with alumina-water nanouid and water-water uids. Signi cance and importance of nanouids to increase the heavy duty engine and automative cooling rates was explained by Peyghambarzadeh et al. [23]. The improvement in heat transfer with the help of nanoparticles concentration and ow conditions were addressed in [24]. Duangthongsuk and Wongwises [25] exposed the di¤erence 1
Corresponding author: [email protected]
1
among the experimental data and computed thermophysical properties of nanouids on heat transfer phenomena. Quiet recently numerous experiments have been done with two types of nanoparticles suspended in base uid named as "Hybrid Nanouid", the cutting edge nanouid. The principle preferred standpoint of utilizing hybrid nanouid is by choosing a proper combination of nanoparticles, positive features can be improved and inconveniences can be covered due to their synegistic e¤ect. These hybrid nanouids are reasonably a new class of nanouids which have numerous conceivable applications in all the elds of heat transfer e.g. micro uidics, manufacturing, transportation, defence, medical, naval structures, acoustics etc. When nano-sized particles are dispersed appropriately, hybrid nanoparticles o¤er collosal bene t having exceptional high e¤ective thermal conductivity. Particularly, nanouid ow is well-known for high heat transfer as compared to normal uid. To improve it even more, the hybrid nanouid is instigated. Many experimental research articles have been published with concept of hybrid nanouid. Momin [26] carried out an experimental study of mixed convection with (Al2 O3 Cu=H2 O) hybrid nanouid for laminar ow in an inclined tube. Study on synthesize (Al2 O3 Cu=H2 O) of hybrid nanouid was examined by Suresh et al [27]. Suresh et al. [28] studied e¤ects of (Al2 O3 Cu=H2 O) hybrid nanouid in heat transfer. Turbulent heat transfer and pressure drop characteristics of dilute water based (Al2 O3 Cu=H2 O) hybrid nanouids was examined by suresh et al. [29]. Since no numerical study has yet been done for the heat transfer characteristics for boundary layer ow of 3D rotating hybrid nanouid over a stretching sheet with radiation, heat generation and chemical reaction e¤ects. So the present study is centered to examine these e¤ects. The governing equations of phase ow hybrid nanouid model are simpli ed with the help of similarity transformations and nally the reduced equations are solved numerically. Various important parameters are discussed at the end of the paper.
2
Problem formulation
Let us consider the three-dimensional, rotating, hybrid nanouid (Ag CuO=water) ow past a linearly stretching surface at z = 0. The uid occupy the half space at z 0: We have considered Copper Oxide (CuO) and Silver(Ag) nano-size particles with base uid as a water. Initially, CuO( 1 ) nanoparticle of 0:1 volume fraction (which is not changed in the given problem) is scattered into the base uid to make nanouid (CuO water). Thus, to develop the targetd "hybrid nanouid (Ag CuO=water)", Silver ( 2 ) with di¤erent volume fractions is dispersed in nanouid (CuO water). The rotation of nanouid (CuO water) is taken about vertical axis so that the uids angular velocity ! is constant. Using these assumptions, the governing equations can be written in the following form as :
2
Figure 1: Geometry of the problem
@u @v @w + + = 0; @x @y @z
u
(1)
u
@u @u @u +v +w @x @y @z
@2u ; @z 2
(2)
u
@v @v @2v @v +v +w + 2 u = hnf 2 ; @x @y @z @z
(3)
2 v = hnf
@T @T @T @ 2T Q +v +w = hnf 2 + (T @x @y @z @z (Cp )hnf u
@c @2c @c @c +v +w = hnf 2 @x @y @z @z
1 (c
T1 )
@qr ; @z
c1 ) n :
(4) (5)
Where hnf ; hnf and hnf are momentum, thermal and concentration di¤usivities of hybrid nanouid respectively. The dimensional heat generation / absorption coe¢cient is denoted by Q and constant rate of 1st order chemical reaction is denoted by 1 : Using Roseland approximation [30-32], the radiation ux qr is given by
4 @T 4 qr = ; 3 @z
(6)
where and are the "Stefan-Boltzman constant" and mean absorption co-e¢cient respectively. Now expanding the Tailor series about T1 we have 3 T 4 4T T1
4 3T1
The related boundary conditions for three dimensional ow are given by 3
(7)
u = Uw = ax; v = Vw = by; w = 0; T = Tw ; c = cw ; at z = 0; u ! 0;
v ! 0;
T ! T1 ; c ! c1 as z ! 1:
(8) (9)
The given issue can be stated in a more straightforward form by using the suitable similarity transformation de ned as
0
p
0
u = axA (); v = ayB (); w = C () =
T Tw
a f (A() + B()); = z
r
a ; f
c c1 T1 ; D () = : T1 cw c1
(10)
With the help of above transformations, Eq. (1) is identically satis ed while Eqs. (2-9) transformed into following coupled nonlinear di¤erential equations.
A000 ()
2 )2:5 [(1
2 )f(1
[(A0 ())2
A00 ()(A() + B())
B 000 ()
1 )2:5 (1
(1
[(B 0 ())2
(
1 )2:5 (1
(1
2 )2:5 [(1
1 ) + 1 (
s1 )g + 2 ( s2 )] f f
2 B 0 ()] = 0;
2 )f(1
1 ) + 1 (
(11)
s1 )g + 2 ( s2 )] f f
B 00 ()(A() + B()) + 2 A0 ()] = 0;
(12)
(Cp )s1 (Cp )s2 Khnf 4 + R)C 00 ()+Pr[(1 2 )f(1 1 )+ 1 ( )g+ 2 ( )](A()+B())C 0 ()+C() = 0; Kf 3 (Cp )f (Cp )f (13) Sc D00 () + [(A() + B())D0 () Rc D() = 0; (14) (1 1 )(1 2 ) A = 0; A0 ! 0;
A0 = 1; B 0 ! 0;
B = 0;
B 0 = ;
C = 1; D = 1;
at = 0;
C ! 0; D ! 0 as ! 1;
(15)
where ; ; Pr; R; ; Sc and Rc are the rotation parameter, stretching ratio parameter, Prandtl number, radiation parameter, heat generation parameter, Schmidt number and chemical reaction constraint respectively and are de ned as 3 b f (Cp )f 4T1 Q
; = ; Pr = ; R= ; = ; a a Kf Kf a(Cp )f (c c1 )n 1 f ; Rc = 1 : Sc = f a
=
4
(16)
The physical quantities of the given problem are, the "Skin-friction" along x and y axis Cf x , Cf y ; the "local Nusselt number" N ux and the "Sherwood number" Shx ; de ned by Cf x Shx or
hnf ( @u hnf ( @v ) ) @z z=0 @z z=0 = ; C = ; N ux = fy 2 f (ax) f (ax)2 @c xKhnf : = Kf (cw c1 ) @z z=0 1
1
Re 2 Cf x =
(1 Re
1 1 2
)2:5 (1
N ux =
)2:5
A00 (0);
2 Khnf 0 C (0) ; Re Kf
1
xKhnf Kf (Tw T1 )
1
Re 2 Cf y = 1 2
Shx =
@T , @z z=0
1 )2:5 (1
(1 1 Khnf 0 D (0) Kf
2 )2:5 .
(17)
B 00 (0); (18)
Where Re = Uw x= f is the restricted Reynolds number.
3
Numerical solution
The coupled non-linear ordinary di¤erential equations (11 14) together with their boundary conditions given in equation (15) are solved numerically using BVP-4C technique [33-34] invoking shooting process. In this procedure, rstly the system of equations (11 14) escorted with boundary conditions are reduced to rst order equations. Then appropriate initial guesses are opted which satisfy the boundary conditions. The results obtained depict the impact of various dimensionless parameters such as rotation parameter, stretching ratio parameter, Schmidt number, chemical reaction parameter, radiation and heat generation on velocity, temperature and concentration pro les. For attaining the convergence criterion of 10 6 shooting methodology is reiterated. Solutions to the given problem are given in graphical and tabular form.
4
Results and discussion
Numerical evaluation of the non-linear di¤erential equations has been carried out to get a better understanding of the problem. The inuence of pertinent physical parameters namely rotation parameter, stretching ratio parameter, heat generation parameter, radiation parameter, chemical reaction parameter and Schmidt number on velocity, temperature and concentration pro le are presented graphically in F igs: (2 14): Table. 1 gives us the standard thermophysical properties of nanouid whereas in Table. 2 we can see these properties at 25 C. From Table. 3 we observe that the magnitude of skin friction coe¢cient of hybrid nanouid is enhanced with nanoparticle volume fraction in both x and y directions whereas decreases when we increase stretching ratio parameter. Chemical reaction and Schmidt number have no impact on skin friction coe¢cient. Increment in rotation reduces the magnitude of skin friction coe¢cient in the x-direction whereas it intensi es in the y-direction. From 5
Table. 4 we conclude that changes in stretching ratio parameter, rotation, Schmidt number and chemical reaction have no inuence on heat transfer rate. The magnitude of local Nusselt number ampli es in the presence of nanoparticle volume fraction. We learnt that due to hybrid nanouid (Ag CuO=water) the heat transfer rate was further augmented. The mass transfer rate diminishes when we enhance rotation parameter and nanoparticle volume fraction. While it elevates with an increment in stretching ratio parameter, chemical reaction parameter and Schmidt number. To validate our present numerical structure, comparisons of uid friction A00 (0) with previous published literature for = 0 = 1 = 2 = R is made in Table. 5. A satisfactory settlement is declared to corroborate the numerical scheme.
4.1
Comparison of velocity and temperature pro les
The comparison of velocity pro le for H2 O, CuO water and Ag CuO=water is displayed in F ig:(2). Since no magnetic eld is being applied in the present study which accelerates nanoparticles, hence the hybrid nanouid (Ag CuO=water) reduces the uid velocity. There is also a decrease in uid velocity due to density and dynamic viscosity which rise because of hybridity and so there is a decline in velocity. We also observe that the velocity of hybrid nanouid (Ag CuO=water) is less than nanouids velocity. The reason being obvious that including further massive particles hurdles the normal uid ow. F ig:(3) depicts the comparison of temperature pro le amid hybrid nanouid (Ag CuO=water), CuO water and H2 O. It is visualized that under same circumstances and equivalent total quantities of volume particle fraction, the hybrid nanouid (Ag CuO=water) reaches higher temperature than nanouid (CuO water). A sudden rise in temperature is a result of hybrid nanouid (Ag CuO=water).
1
1
Pure w ater
0 .9
CuO-w a te r
Ag-CuO/w ater
0 .8
0 .8
0 .7
Ag-CuO/w ater
0 .7
0 .6
0 .6
C( η)
A'( η)
Pure w ater
0 .9
CuO-w ater
0 .5 0 .4
0 .5 0 .4
0 .3 0 .3
0 .2 0 .2
0 .1 0 .1
0 0
1
2
3
4
0
5
0
η
1
2
3
4
η
Figure 2: Comparison of A0 ()
Figure 3: Comparison of C ()
6
5
4.2
Impact of rotation parameter ()
It is demonstrated through F igs:(4 5) the inuence of rotation over the velocity distribution A0 () and B0 () in x and y direction respectively. We observe that for higher values of rotation parameter both the ows in x and y direction decelerate as well as the associated boundary layer thickness. The temperature distribution for Ag CuO=water and CuO water is displayed in F ig: (6). From this gure it is learnt that the rotation ampli es the thermal boundary layer thickness. Moreover the instant rise in temperature is due to the hybrid nanouid Ag CuO=water. In F ig:(7) the concentration, D(), has been plotted to see the e¤ects against rotation. Rotation boosts the concentration.
7
1
0 .5
CuO-w a te r
CuO-w ater
0 .9
Ag-CuO/w ater
0 .8
Ag-CuO/w ater
0 .4
0 .7
0 .3
B'( η)
A'( η)
0 .6 0 .5 0 .4
ε = 0.0, 0.5, 0.8
0 .2
ε = 0.0, 0.5, 0.8
0 .1
0 .3 0
0 .2 0 .1
-0 .1
0 0
1
2
3
4
-0 .2
5
0
1
2
η
3
Figure 4: variation of on A0 ()
1
CuO-w a te r
0 .9
CuO-w a te r Ag-CuO/w ater
0 .9
Ag-CuO/w ater 0 .8
0 .8
0 .7
0 .7
0 .6
0 .6
D( η)
C( η)
5
Figure 5: variation of on B0 ()
1
0 .5
ε = 0.0, 0.5, 0.8
0 .5
ε = 0.0, 0.5, 0.8
0 .4
0 .4
0 .3
0 .3
0 .2
0 .2
0 .1
0 .1
0 0
1
2
3
4
5
0
0
1
η
2
3
4
η
Figure 6: variation of on C()
4.3
4
η
Figure 7: variation of on D()
Impact of stretching ratio parameter ()
F ig:(8) elucidate the impact of stretching ratio parameter on velocity distribution in ydirection. Increment in stretching ratio parameter correlates with an increase in the rate of stretching along y-axis so it is obvious there is a rise in the velocity eld and the momentum boundary layer thickness. The e¤ect of stretching ratio parameter on temperature distribution is depicted through F ig:(9). It is discovered from the graph that with an increase in stretching ratio parameter there is a reduction in the temperature pro le. Through F ig:(10) we observe the variation of concentration with respect to stretching 8
5
ratio parameter : Increase in stretching ratio parameter declines the concentration pro le.
0 .9
1
CuO-w a te r
CuO-w a te r
0 .9
Ag-CuO/w ater
Ag-CuO/w ater
0 .7
0 .8
0 .6
0 .7 0 .6
0 .5
C( η)
B'( η)
0 .8
0 .4
0 .4
λ = 0.2, 0.5, 0.9
0 .3
0 .5
0 .3 0 .2 0 .2
λ = 0.2, 0.5, 0.9
0 .1 0 .1 0 0 0
1
2
3
4
5
0
1
2
η
3
4
η
Figure 8: variation of on B0 ()
Figure 9: variation of on C()
1
CuO-w a te r Ag-CuO/w ater
0 .9 0 .8 0 .7
D( η)
0 .6 0 .5 0 .4 0 .3 0 .2
λ = 0.2, 0.5, 0.9
0 .1 0
0
1
2
3
4
5
η
Figure 10: variation of on D()
4.4
Impact of heat generation parameter ()
F ig:(11) illustrates the nature of temperature pro le with the variation of heat generation parameter. Increase in prompts an increase in the temperature eld since energy is produced at thermal boundary layer.
9
5
1
CuO-w ater
0 .9
Ag-CuO/w ater 0 .8 0 .7
C( η)
0 .6 0 .5
δ = -0.3, 0.0, 0.3
0 .4 0 .3 0 .2 0 .1 0
0
1
2
3
4
5
η
Figure 11: variation of on C()
4.5
Impact of radiation parameter (R)
The impact of radiation parameter over the temperature pro le is demonstrated in F ig:(12 (a)). We analyze that an increase in radiation R causes the temperature to in K crease. Physically, the quantity 4T f3 in radiation parameter measures the thermal radiation 1 transfer relative to the conduction heat transfer. Therefore, higher values of this quantity exhibit that thermal radiation is dominate over conduction. Hence a great amount of heat energy due to radiation is being released in the system giving a rise to temperature.The dual temperature pro le in the case of shrinking sheet for various values of radiation parameter R in the absence of nanoparticle volume fraction and rotation is presented in F ig:(12 (b)): From F ig:(12 (b)) it can be observed that the temperature pro les increase with the increase of the radiation parameter R for both solutions: the e¤ect of the radiation parameter R causes an increase in the radiative heat ux.
10
1
CuO-w ater
0 .9
1
Ag-CuO/w ater 0 .8 0 .8
0 .7
C( η)
C( η)
0 .6 0 .5
R = 0.0, 0.5, 1.0
0 .4
0 .6
R =1, 2, 3
0 .3 0 .2
0 .2 0 .1 0
fi rst so l u ti o n
0 .4
se co n d so l u ti o n
0
0
1
2
3
4
5
0
1
2
η
Figure 12(a): variation of R on C()
4.6
3
4
Figure 12(b): variation of R on dual C()
Impact of chemical reaction parameter (Rc )
F ig:(13) demonstrate the e¤ects of chemical reaction parameter Rc on the concentration pro le. The increasing values of chemical reaction lead to decline the concentration of the uid. Consequently the concentration boundary layer thickness get increased.
1
CuO-w ater
0 .9
Ag-CuO/w ater 0 .8 0 .7
D( η)
0 .6 0 .5 0 .4
R = 1.0, 2.0, 3.0 c
0 .3 0 .2 0 .1 0
0
1
2
3
4
5
η
Figure 13: variation of Rc on D()
4.7
5
η
Impact of Schmidt number (Sc )
The e¤ects of Schmidt number on the concentration pro le is demonstrated in F ig:(14). As the Schmidt number is the quantitative relation between momentum to mass di¤usivity. 11
Due to Schmidt number, the di¤usivity is decreases consequently the concentration of the uid decreases.
1
CuO-w a te r
0 .9
Ag-CuO/w ater 0 .8 0 .7
D( η)
0 .6 0 .5
Sc = 0.5, 1.0, 1.5 0 .4 0 .3 0 .2 0 .1 0 0
1
2
3
4
5
η
Figure 14: variation of Sc on D()
Table 1: Thermophysical Properties of CuO water and Ag CuO=water Properties Nanouid (CuO water) nf = (1 )f + s Density () f nf = (1 ) 2:5 Viscosity () (Cp )nf = (1 )(Cp )f + (Cp )s Heat Capacity (CP ) Knf K +(n 1)Kf (n 1) (Kf Ks ) Thermal Conductivity (K) = s Ks +(n 1)K Kf f + (Kf Ks ) Properties Density () Viscosity () Heat Capacity (CP ) Thermal Conductivity (K)
Hybrid Nanouid (Ag CuO=water) hnf = [(1 2 )f(1 1 )f + 1 s1 g] + 2 s2 f hnf = (1 )2:5 (1 2 )2:5 1 (Cp )hnf = [(1 2 )f(1 1 )(Cp )f + 1 (Cp )s1 g] + 2 (Cp )s2 K +(n 1)Kbf (n 1) 2 (Kbf Ks2 ) Kbf K +(n 1)Kf (n 1) 1 (Kf Ks1 ) Khnf ; Kf = s1Ks1 +(n 1)K = s2Ks2 +(n 1)K Kbf bf + (Kbf Ks2 ) f + (Kf Ks1 ) 2
Table 2: Thermophysical Properties of nanoparticles and base uid Properties CuO Ag Base uid (water) 997:1 6320 10500 Cp 4179:0 531:80 235 0:6130 K 76:50 429 6:20 Pr
12
1
Table 3: E¤ects of (Ag CuO=water)
2
Rc
A00 (0),
Sc
B 00 (0) for nanouid (CuO
1 A00 (0) (1 1 )2:5
0.1 0.5 0.07 0.5 0.5 1.91658 0.3 1.80930 0.5 1.77585 0.3 0.2 1.88714 0.5 1.80930 0.9 1.74547 0.05 1.68693 0.07 1.80930 0.1 1.99913 0 1.80930 1 1.80930 2 1.80930 0.5 1.80930 1.0 1.80930 1.5 1.80930 Table 4: E¤ects of (Ag CuO=water)
2
Rc
(1 1 ) 2:5 00 A (0) (1 2 )2:5
2.29784 2.16922 2.12912 2.26254 2.16922 2.09269 1.91773 2.16922 2.60156 2.16922 2.16922 2.16922 2.16922 2.16922 2.16922
water) and hybrid nanouid
1 B 00 (0) (1 1 )2:5
1.06164 1.49397 1.88764 1.00655 1.49397 2.44057 1.39306 1.49397 1.65059 1.49397 1.49397 1.49397 1.49397 1.49397 1.49397
C 0 (0) and -D0 (0) for nanouid (CuO
Sc
Knf 0 C (0) Kf
0.1 0.5 0.07 0.5 0.5 1.32460 0.3 1.32460 0.5 1.32460 0.3 0.2 1.32460 0.5 1.32460 0.9 1.32460 0.05 1.32460 0.07 1.32460 0.1 1.32460 0 1.32460 1 1.32460 2 1.32460 0.5 1.32460 1.0 1.32460 1.5 1.32460
Khnf 0 C (0) Kf
1.22440 1.22440 1.22440 1.22440 1.22440 1.22440 1.15700 1.22440 1.33120 1.22440 1.22440 1.22440 1.22440 1.22440 1.22440
13
(1
(1 1 ) 2:5 00 B (0) (1 2 )2:5
1.27283 1.79116 2.26314 1.20679 1.79116 2.92606 1.58365 1.79116 2.14799 1.79116 1.79116 1.79116 1.79116 1.79116 1.79116
water) and hybrid nanouid
1 )D0 (0)
0.65733 0.64297 0.62757 0.60094 0.64297 0.68567 0.64230 0.64297 0.64486 0.38047 0.82157 1.08770 0.64297 0.957580 1.209200
(1
1 )(1
0.61131 0.59796 0.58364 0.55887 0.59796 0.63767 0.61019 0.59796 0.58038 0.35384 0.76406 1.01150 0.59796 0.89055 1.12460
2 )D0 (0)
Table 5: Comparision of ( = 0 = 1 = 2 = R)
5
A00 (0) for various values of stretching ratio parameter when
Wang [35] Arial [36] Butt et al [37] Present Results
0.0 0.1 0.2 0.3 0.4 0.5
1 1.020902 1.041804 1.062705 1.083607 1.104509
1 1.017027 1.034587 1.052470 1.070529 1.088662
1 1.020260 1.039495 1.057955 1.075788 1.093095
1 1.02137 1.0404 1.05871 1.07643 1.09364
Conclusion
The three dimensional steady rotating ow of "hybrid nanouid (Ag CuO=water)" with thermal radiation, heat generation and chemical reaction is examined on a linearly stretching surface. The main conclusion of the work is as follows: Hybridity boosts the temperature distribution as well as the heat transfer rate at surface. The thermal boundary of "hybrid nanouid (Ag menting the heat generation parameter .
CuO=water)" increases by incre-
There is an enhancement in the rate of mass transfer at surface by increasing the Schmidt number Sc and chemical reaction Rc : There is an increment in concentration pro le with the increase in rotation parameter but it () decreases the concentration at surface.
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