Research article Open Access
Numerical Method for the Analysis of Thermal Radiation on Heat Transfer in Nanofluid
J.I. Oahimire1*, F.E Bazuaye2, Taylor S. Harry2
1Department of Mathematics, Michael Okpara University of Agriculture, Umudike, Nigeria
2Department of Mathematics and Statistics, University of Port Harcourt, Port Harcourt, Nigeria
*Corresponding author: Sergei Department of Mathematics, Michael Okpara University of Agriculture, Umudike, Nigeria, Email: @
Received: July 02, 2016; Accepted: July 23, 2016; Published: July 30, 2016
Citation: Oahimire JI, Bazuaye FE, Harry TS (2016) Numerical Method for the Analysis of Thermal Radiation on Heat Transfer in Nanofluid. Nanosci Technol 3(1): 1-4. DOI: http://dx.doi.org/10.15226/2374-8141/3/1/00135
AbstractTop
This present study investigates the effects of thermal radiation on heat transfer in nanofluid. The governing equations are formulated base on already existing model and are transformed to ordinary differential equations using stream function and similarity variables. The resulting dimension less equations are then solved numerically by Runge-Kuta Fehlberg method with shooting techniques using Maple software. With the help of graphs and tables, influences of the governing parameters are discussed. The result shows that the radiation has significance influences on heat transfer.

Keywords: Nanofluid; Runge Kuta-Fehlberg method; Shooting technique; Heat transfer; Radiation
Nomenclature
a = Constant; g = Acceleration due to gravity; k = Thermal Conductivity; Pr = Prandtl Number; T = Fluid Temperature; Tw = Surface Temperature; T = Free Stream Temperature; u,v = Velocity Components; x, y = Cartesian Coordinates; f(x) = Dimensionless Stream Function; Gr = Grashof Number; qr = Heat Flux Radiation; Bo = Magnetic Field of Constant Strength; R = Radiation Parameter; Ks = Rosseland Mean Absorption Coefficient; K = Thermal Conductivity Coefficient
Greek Symbols
β = Thermal Expansion Coefficient; μ = Dynamic Coefficient of Viscosity; θ (η) = Dimensionless Temperature; η = Similarity Variable; ρ = Fluid Density; ψ = Stream Function; σ' = Stefan- Bottzman Constant
Introduction
Nanofluid is formed when Nanoparticles such as Aluminium Oxide (Al2O3), Copper (Cu), Copper Oxide (CuO), Gold (Au), Silver (Ag), Silica particles, e.t.c are mixed with base fluids such as water, oil, acetone, ethylene, etc. The discovery of nanofluids in enhancing heat transfer in industrial processes has drawn the attention of both scholars and industrialists to make researches into this relatively new area.

Wang and Choi, [1] studied the thermal condutivity of nanoparticle fluid mixture containing Al2O3 and CuO nanoparticles and showed that the thermal conductivity of nanofluids increased with increasing volume fraction of the nanoparticles. Steve (2006) found out that a small size radiator filled with well prepared nanofluid containing monosized nanoparticles with 2nm nominal diameter will be okay for heavy vehicle instead of a very big radiator and this will save cost and space. And he also discovered that fuel efficiency will be reduced by 5% in these Heavy Trucks when nanofluids are introduced into their cooling system. John and Baldev, [2] carried out experiment and they concluded that by controlling the linear aggregation length from nano- to macron scales, the thermal conductivity of the nanofluid can be tuned from a low to very high value and that under repeated magnetic cycling, the thermal conductivity is reversible.

Khan and Ipop, [3] studied the boundary-layer flow of a nanofluid past a stretching sheet and their result shows that the heat transfer at fixed values of the Lewis number (Le), the Brownian motion parameter (Nb) and the thermophoresis parameter (Nt) increases with the Prandtl number (Pr). This is because a higher Pr fluid has relatively lower thermal conductivity which reduces conduction and thereby increases the heat transfer rate at the surface of the sheet.

Godson, et al. [4] in separate papers agreed that Nanofluids are the most option to enhance heat transfer and showed that a very small amount of nanoparticles suspended stably in base fluids can provide impressive improvement in the thermal property of such fluids.

Zeinali and Salehi, [5] studied experimentally, the effect of magnetic field of various strengths on the thermal performance of Silver/water nanofluid and their results showed that the thermal efficiency in the presence of magnetic field increased.

Yohannes and Shankar, [6] examine the boundary layer flow of heat and mass transfer in MHD flow of nanofluids through porous media with viscous dissipation and chemical reaction. The governing boundary layer equations were formulated and reduced to a set of ordinary differential equations using similarity transformations and then solved numerically by an explicit finite differential scheme known as the Keller box method. Wan-Mohd, [7] applied Runge-kutta Fehlberg method and shooting technique for solving classical Blasius equation [7]. The well-known Blasius equation is governed by the third order nonlinear ordinary differential equation and then solved numerically using rungekutta Fehlberg method with shooting techniques. Hamad, [8] found analytical solution to the differential equations that model heat transfer in nanofliud. We extended the work of Hamad, [8] by incorporating radiation term to have differential equations that model the effect of thermal radiation on heat transfer of a nanofliud.
Mathematical formulation
Consider the steady laminar two-dimensional flow of an incompressible viscous nanofluid past a linearly semi-infinite stretching sheet under the influence of a constant magnetic field of strength B0 which is applied normally to the sheet. x'and y' are the coordinates along and perpendicular to the sheet while u' and v' are the velocity components in the directions respectively as shown in the figure below:
Figure 1:
A water based nanofluid containing different types of nanoparticles: Al2 O3, Cu, TiO2 and Ag is used with the assumption that both the fluid and the nanoparticles are in thermal equilibrium. Based on the already existing model of Hamad, [8], the governing boundary layer equations of continuity, momentum and heat equations formulated are:
The boundary conditions of the equations are
$u ′ = u w ' ( x ′ )=a x ′ , v ′ =0, T ' = T w ' at y ′ =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepu0dbbc8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qaceWG1bWdayaafaWdbiabg2da9iaadwhapaWaa0baaSqaa8qacaWG 3baapaqaa8qacaGGNaaaaOWaaeWaa8aabaWdbiqadIhapaGbauaaa8 qacaGLOaGaayzkaaGaeyypa0JaamyyaiqadIhapaGbauaapeGaaiil aiqadAhapaGbauaapeGaeyypa0JaaGimaiaacYcacaWGubWdamaaCa aaleqabaWdbiaacEcaaaGccqGH9aqpcaWGubWdamaaDaaaleaapeGa am4DaaWdaeaapeGaai4jaaaakiaadggacaWG0bGabmyEa8aagaqba8 qacqGH9aqpcaaIWaaaaa@4F7A@$
Where qr is the radiative heat flux, T′ is the temperature of the fluid, x' and y' are the co-ordinates along and perpendicular to the sheet while u' and v' are the velocity components in the x' and y' directions respectively and a is a constant. The effective density (ρnf), effective dynamic viscosity (μnf), heat capacitance (ρCp)nf and the effective thermal conductivity (knf) of the nanofluid, in that order, are given as
Where A is the solid volume fraction (A ≠ 1), μf is the dynamic viscosity of the base fluid, while ρf and ρs are the densities of the pure fluid and the nanoparticle respectively. The constants kf and ks are the thermal conductivities of the base fluid and the nanoparticle respectively.

Following Rosseland approximation Brewstar [9] the radiative heat flux qr is modeled as
Where σ' is the Stefan-Boltzman constant and k' is the mean absorption coefficient. Assuming that the difference in temperature within the flow is such that T '4 can be expressed as a linear combination of the temperature, we expand T '4 in Taylor's series about T as follows:
and neglecting higher order terms beyond the first degree in ( T' - T' ), we have
Differentiating equation (6) with respect to y' and using equation (8) to obtain
Then equation (3) becomes
By the introduction of the following variables
Equation (11) transform equation (1), (2) and (10) to the followings
Where $M=\frac{\sigma {\beta }_{0}}{a}$ is the magnetic field parameter,${p}_{r}=\frac{{v}_{f}}{{K}_{f}}$ is the Prandtl number,${R}_{d}=\frac{16\sigma \text{'}{T}_{\infty }^{\text{'}3}}{3{K}^{*}{K}_{f}}$ is the radiation parameter,${G}_{r}=\frac{g{\beta }_{t}\left({T}_{w}^{\text{'}}-{T}_{\infty }^{;}\right)x}{a}$ Is the Grashof number
And the boundary conditions becomes
$u=x,v=0, θ=1 at y=0 u→0, θ→0 as y→∞ (15) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepu0dbbc8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWG1b Gaeyypa0JaamiEaiaacYcacaWG2bGaeyypa0JaaGimaiaacYcacaaM c8UaeqiUdeNaeyypa0JaaGymaiaaykW7caaMc8Uaamyyaiaadshaca aMc8UaamyEaiabg2da9iaaicdaaeaacaWG1bGaeyOKH4QaaGimaiaa cYcacaaMc8UaeqiUdeNaeyOKH4QaaGimaiaaykW7caaMc8Uaamyyai aadohacaaMc8UaaGPaVlaadMhacqGHsgIRcqGHEisPcaaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaabIcacaqGXaGaaeynaiaabMcaaaaa@E8C5@$
By introducing the stream function ψ, which can be defined as
and using the similarity transforma t i o n s
We have
And the boundary conditions now becomes
$f(0)=0,​ f ′ (0)=1 at η=0 f→0 at η→∞. θ(0)=1 and θ(∞)=0, } (19) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepu0dbbc8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiGaaqaabe qaaiaadAgacaGGOaGaaGimaiaacMcacqGH9aqpcaaIWaGaaiilaiaa ygW7caaMc8UaaGPaVlaaykW7ceWGMbGbauaacaGGOaGaaGimaiaacM cacqGH9aqpcaaIXaGaaGPaVlaaykW7caWGHbGaamiDaiaaykW7caaM c8UaaGPaVlabeE7aOjabg2da9iaaicdacaaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7aeaacaWGMbGaeyOKH4QaaGimaiaaykW7caaM c8UaamyyaiaadshacaaMc8UaaGPaVlabeE7aOjabgkziUkabg6HiLk aac6caaeaacqaH4oqCcaGGOaGaaGimaiaacMcacqGH9aqpcaaIXaGa aGPaVlaaykW7caWGHbGaamOBaiaadsgacaaMc8UaaGPaVlabeI7aXj aacIcacqGHEisPcaGGPaGaeyypa0JaaGimaiaacYcaaaGaayzFaaGa aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caGG OaGaaGymaiaaiMdacaGGPaaaaa@B817@$
Numerical Solution
Table 1: Thermo physical properties of water and nanoparticles Hamad [8].
 Compound ρ (kg/ m3) Cp(J/ kgK) k (W/ mK Pure water 997.1 4179 0.613 Copper (Cu) 8933 385 401 Alumina (Al2O3 ) 3970 765 40 Silver (Ag) 10500 235 429 Titanium Oxide (TiO2) 4250 686.2 8.9538
An algorithm was written and run with Maple codes to plot graphs and generate tables for analysis.
The thermo physical properties of pure water and those of the nanoparticles as given in table 1 by Hamad [8] will now be substituted in the transformed coupled ordinary differential equations. These equations will now be solved using Runge- Kutta-Fehlberg (RKF) method with shooting technique. The algorithm of Runge-Kutta-Fehlberg (RKF) method is order 5 and it gives a better approximation of the solution than the general Runge – Kutta method of order 4.

An algorithm was written and run with Maple codes to plot graphs and generate tables for analysis.
Results
Results are shown in figures 2- 3 and table 2
From derivation, the the Nuselt number can be written as follows;
$Nu=− θ ′ (0) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepu0dbbc8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOtaiaadw hacqGH9aqpcqGHsislcuaH4oqCgaqbaiaacIcacaaIWaGaaiykaiaa ykW7caaMc8oaaa@3FFD@$
Discussion of Results
Figure 2:Effect of M on velocity distribution profiles.
Figure 3:Effect of Rd on Temperature distribution profiles.
Figure 4:
Table 2: Values of −θ′(0) for various Rd, A When Gr = 0.1 , M = 0.5 and Pr = 6.2 .
 Rd A -θ'(O) Cu Ag Al2CO3 TiO2 0.5 0.30 1.31430 1.23109 1.28489 1.28961 0.35 1.33022 1.23834 1.29732 1.30086 0.40 1.34938 1.24650 1.31206 1.31409 0.45 1.37892 1.25349 1.33263 1.33158 2 0.30 0.85353 0.79483 0.83119 0.82915 0.35 0.86695 0.80360 0.84275 0.84011 0.40 0.87137 0.80646 0.84654 0.84371 0.45 0.87576 0.80929 0.85030 0.84727 4 0.30 0.60425 0.55941 0.58668 0.58351
We have formulated the differential equations modeling the effect of radiation on heat transfer in nanofluid based on the already existing model. Numerical evaluation of the numerical solutions reported in the previous sections was performed and the results are presented in graphical and tabular forms. This was done to illustrate the influence of some parameter involved.

The effect of magnetic field parameter on velocity distribution profiles across the boundary layer is presented in Figure 1. It is obvious that the effect of increasing values of the magnetic field parameter M results in a decreasing velocity distribution. This is due to the fact that the introduction of transverse magnetic field normal to the flow direction has a tendency to create a drag force due to Lorentz force and hence results in retarding the velocity profiles Figure 2 shows the influence of the radiation parameter on the temperature distribution profiles. The temperature increases as the radiation increases. Figure 3 presents the temperature distribution profiles for different values of the Prandtl number (Pr). The results show that the effect of increasing values of the Prandtl number results in a decrease in the temperature.

In table 2, as the radiation parameter increases, the nuselt number of the nanoparticles decreases, that is to say the effect of increasing the radiation parameters is to decrease the rate of heat transfer of the nanoparticles.Also, as the fraction volume parameter (A) increases, nuselt number increases.
Conclusion
Numerical solution was applied to the differential equation that model the effect of thermal radiation on heat transfer on nanofluid using runge-kutta Fehlberg method. It was found that the effect of high radiation is to decrease the rate of heat transfer.
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