CHAPTER-5 CONVECTIVE HEAT TRANSFER WITH GRAPHENE NANOFLUIDS

Transcription

1 115 CHAPTER-5 CONVECTIVE HEAT TRANSFER WITH GRAPHENE NANOFLUIDS

2 116 Table of Contents CHAPTER-5 CONVECTIVE HEAT TRANSFER WITH GRAPHENE NANOFLUIDS Introduction Experimental studies on convective heat transfer Thermo physical properties of nanofluids Specific heat Density Base fluid viscosity Heat transfer coefficient augmentation Effect of thermal conductivity on fully developed heat transfer Study on modelling of nanofluid flow Governing equations Steady wall temperature boundary condition Constant wall heat flux boundary condition Heat transfer coefficient improvement Further study Local Nusselt number 151

3 117 CHAPTER - 5 CONVECTIVE HEAT TRANSFER WITH GRAPHENE NANOFLUID 5.1 Introduction Nanofluids are capable heat conveying fluids owing to their high thermal conductivity. Thermal conductivity of nanofluids is already discussed item wise in Chapter 4. In a bid to employ nanofluids in practical applications, their convective heat transfer characteristics are required to be understood. For that reason, an investigation is carried out on the convective heat transfer performance of nanofluids. In this section, the deliberation is aimed on forced convection of nanofluids inside circular tubes. For the study, laminar flow of the nanofluid inside a straight circular tube under both constant wall temperature and constant wall heat flux boundary conditions is considered. In addition to this, the effect of thermal conductivity of nanofluids on fully developed flow heat transfer is studied. There are several models in the literature related to the convective heat transfer with nanofluids. In this literature survey, the conversation is focused on Forced convection of Graphene nanofluids in circular tubes. The examination of nanofluid convective heat transfer is investigated by comparing the combined results with the experimental data in the literature.

4 Experimental Studies on convective heat transfer Experimental Set up Fig: 5.1: experimental set up to measure heat transfer Fig: 5.: Schematic view of experimental set up Fig; 5.3: Insulated pipe with Nichrome heater inside

5 119 Fig: 5.4: Asbestos Insulated pipe Fig: 5.5: Thermo generator Fig: 5.6: DAQ system with 40 segments (KEITHLEY)

6 10 Fig: 5.7: DAQ connectors segments NOMENCLATURE SPECIFICATIONS Segments 40 Rating 50,60,400 Hz Voltage 8 VA MAX Temperature 00ºC max Simulation soft ware ARGUS DAQ-Manufacturer KEITHLEY 5..1 DESCRIPTION: The experimental set up to measure the convective heat transfer characteristics of the Graphene nanofluids is shown in the figure 5.1 It is planned to conduct the experiment in two conditions at constant wall heat flux and constant wall temperature, by experience and the literature survey it is impossible to maintain the constant wall temperature and constant wall heat flux for the same length of the pipe hence the pipe length and dimensions are changed for two different situations for steady wall temperature boundary conditions the tube dia is 5mm and the length is 1m and for steady wall heat flux boundary condition the tube dia is 4.57mm and length is m to achieve the same hydro dynamic boundary layer and thermal boundary layer for the two situations under this arrangement, Apart from this the remaining apparatus used in the experimental set up is a data acquisition system in

7 11 built with 40 segments(keithley) to collect the data, the copper-constantan thermo couples (-50ºc to 400ºc) are arranged at the entrance and exit of the pipe and also in between at interval of length, The thermo couples are arranged to measure the temperature of the fluid flow, in addition to this a syringe pump is used to pump the fluid into the tubes, The data given by the DAQ system is connected to a computer run by ARGUS soft ware, To check the parallel arrangement of the pipe bulls eye test (Spirit level) is conducted, Shah correlation is used to measure the constant wall heat flux boundary condition heat transfer enhancement ratio, Where as Sieder-Tate correlation is employed for constant wall temperature heat enhancement ratio, Generally the heat transfer enhancement is caused by two methods i.e., Thermal dispersion and Thermal diffusion, In the first technique, it is supposed that the occurrence of nanoparticles in the stream iluence the heat transfer only through the distorted thermo physical properties. Then the governing equations of fluid flow and heat convey for a conventional pure fluid can be used for the study of nanofluids by substituting the relevant thermo physical properties. This emphasizes that the established correlations of convective heat transfer for pure fluids can be utilized for nanofluids. In the second technique, the nanofluid is considered as a single segment fluid but the extra heat transfer improvement attained with nanofluids is measured by modelling the diffusion phenomenon. It was identified that thermal dispersion happens in nanofluid flow due to the irregular motion of nanoparticles. By allowing the fact that this irregular motion generates small perturbations in temperature and velocity, the effective thermal conductivity in the governing energy equation takes the form. keff k, kd (5.1)

8 1 Where k is the thermal conductivity of the nanofluid. k d is scattered thermal conductivity and was proposed to be measured by using the following expression. k c( c ) u d r, (5.) d p x p o In this expression, c p is specific heat, ρ is density, Ø is particle volume fraction, u x is axial velocity, r 0 is tube radius and d p is nanoparticle diameter. C is an empirical constant that should be calculated by matching experimental data. Subscript n f represents the nanofluid. In one more study, based on this thermal diffusion model, following correlation was proposed by Li and Xuan [73] for the prediction of Nusselt number. Nu c c pe (5.3) m 1 m m (1 d )Re Pr, The correlation was developed for forced convection of nanofluid flow inside circular tubes. Pe d is particle Peclet number, which is defined as Pe d ud m p, (5.4) α is nanofluid thermal diffusivity and u m is average flow velocity. Re and Pr are the conventional Reynolds and Prandtl numbers; however the thermo physical properties of the nanofluid have to be used in the connected calculations. c 1, c, m 1, m and m 3 are empirical constants that should be determined by using experimental data. For the appropriate examination of nanofluid heat transfer, precise evaluation of thermo physical properties of the nanofluid is significant. In order to emphasis the significance of this problem, Combined equations are continuity equation for nanofluid, continuity equation for nanoparticles, energy equation for nanofluid and momentum equation for nanofluid. The variation in the fully developed velocity

9 13 profile in a circular tube due to the variation of viscosity in radial direction enhances convective heat transfer of nanofluids. Throughout the analysis, fully developed condition was studied for both laminar and turbulent flow cases and nanofluids were taken as single phase fluids. Several expressions used in the literature for the purpose of dynamic viscosity, specific heat and thermal conductivity were accessed and difference between them was illustrated Thermo physical Properties of Nanofluids In the investigation of convective heat transfer of nanofluids, precise determination of the thermo physical properties is a major issue. Measurements of specific heat and density of nanofluids are comparatively straight forward, However when it comes to thermal conductivity and viscosity, there is considerable Dis agreement in both theoretical models and experimental results presented in the literature Specific Heat There are two expressions for evaluating the specific heat of nanofluids [10, 7]: c c (1 ) c, (5.5) p, p, p p, f ( c ) ( c ) (1 )( c ), (5.6) p p p p f It is said that Eq. (5.6) is theoretically more dependable since specific heat is a mass specific quantity whose impression depends on the density of the ingredients of a mixture Density Density of nanofluids can be calculated by using the following expression [10]. (1 ), (5.7) p f

10 14 Here, Ø is particle volume fraction and subscripts, p and f correspond to nanofluid, particle and base fluid, respectively. Pak and Cho [10] experimentally proved that Eq. (5.5) is a precise expression for concluding the density of nanofluids Base fluid viscosity Nanofluid viscosity is a significant constraint for practical applications since it precisely iluences the pressure drop in forced convection. Therefore, for extensive use of nanofluids in practice, the amount of viscosity increase for nanofluids in comparison to pure fluids must be carefully observed. In order to measure the viscosity of base fluid and pure fluid Brookfield viscometer is used widely; calibrated red wood viscometer has been used for measuring the viscosity of base fluid as well as Graphene nanofluids, Fig.5.8: Red wood Viscometer to measure viscosity

11 viscosity(cp) 15 Table: 5.1 Temperature Vs Viscosity Pure Fluid and Base Fluid S.No Temperature (ºc ) Viscosity of Water (CP) Viscosity of Base Fluid(CP) viscosity vs Temperature water 70% water+30%eg Temperature (ºC) Fig; 5.9: Graph of Temperature Vs Viscosity To calculate the Graphene (Cn)/Water + EG nanofluid viscosity one should use equation of Nguyen et al. [77] (5.10) ( ) f, MODEL CALCULATION FOR VISCOSITY OF GRAPHENE NANOFLUID: μ = (1+.5 x x (0.04) ) x1.43=1.916

12 Viscosity(cp) 16 Table: 5. Temperature Vs Viscosity of Graphene nanofluid S.No Temperature (ºc ) Viscosity of Base Fluid(CP) Viscosity of Graphene nanofluid(cp) Base fluid Graphene nano fluid Temperature(ºc) Fig; 5.10: Graph of Temperature Vs Viscosity Experimental Studies regarding viscosity Compared to the argument of thermal conductivity, there is a considerable Inconsistency in the experimental results concerning the viscosity of nanofluids. However, the common inclination is that the rise in the viscosity by the addition of nanoparticles to the conventional base fluid is important. The viscosity of Graphene

13 17 (Cn) /Water + ethylene glycol nanofluid is determined at room temperature. For the fragment volume fraction of 3.5%, 40% increment in viscosity was noticed. It is recognized that nanofluid viscosity depends on many points such as; particle size, particle volume fraction, extent of clustering and temperature. Raising particle volume fraction improves viscosity and this was supported by many studies [63, 77, 78, 79]. Einstein [1906] suggested an expression for calculating the dynamic viscosity of mitigate suspensions that hold spherical particles. In this case, the relations among the particles are deserted. The linked expression is as follows. (1.5 ), (5.8) f Brinkman [80] proposed the following equation. 1,.5 f (5.9) (1 ) In particular studies, the exchanges between particles were taken into consideration. These chances widened the applicability choice of the models in terms of particle volume fraction. Nguyen et al. [77] suggested viscosity equation for Al O 3 /water + EG nanofluids are: (5.10) ( ) f, The preceding relationship is suitable for the nanofluids with a particle size of 0-30 nm. Tentative studies show that nanoparticle size is a significant constraint that iluences the viscosity of nanofluids. Conversely at current, it is complicated to attain a dependable set of tentative data for nanofluids that covers a wide range of particle size and particle volume fraction. Therefore, for the time being, Eq. (5.10) can be used as estimation for Graphene (Cn)/Water+ Ethyl glycol nanofluids with

14 18 different volumetric ratios. When it comes to temperature dependence of viscosity, Nguyen et al. [77] concluded that for particle volume fractions below 4%, viscosity enrichment ratio (viscosity of nanofluid divided by the viscosity of base fluid) does not considerably change with temperature. Precise study of convective heat transfer of nanofluids is significant for reasonable usage of nanofluids in thermal devices. The methods propose that the study can be made by considering the nanofluid flow as single phase flow because the nanoparticles are very small and they mix easily [84]. The heat transfer accomplishment of nanofluids can be elaborated by using traditional correlations developed for the calculation of Nusselt number for the flow of pure base fluids. In the calculations, one should replace the thermo physical properties of the nanofluid to the associated expressions. This line of attack is a very sensible way of measuring convective heat transfer of nanofluids [80, 81]. The afore said methods of using conventional correlations was proposed by Shah [80] and Sieder-Tate [81] to calculate constant wall temperature and constant wall heat flux temperature is deliberated. 5.4 Heat Transfer Coefficient Augmentation The speculative study of heat transfer obtained with nanofluids is presented in two sections, according to the type of the boundary condition, namely; constant wall heat flux boundary condition and constant wall temperature boundary condition Constant Wall Heat Flux Boundary Condition The more frequently used empirical correlations for the calculation of convective heat transfer in laminar flow method inside circular tubes is the Shah

15 19 correlation [80]. The linked expression for the calculation of local Nusselt number is as follows. Nu 1.30x 1forx 5 10 x 1/3 5 * * 1.30x 0.5 for510 x , (5.11) 1/3 5 3 * * x e forx x e 3 * * X * is defined as follows x / d x / d 1 x *, RePr Pe GZ (5.1) x Pr, Re and Pe are Prandtl number, Reynolds number and Peclet number, respectively. d is tube diameter and x is axial position. At this spot, it must be prominent that for the same tube diameter and same flow velocity, Pe f and Pe are unlike. Pe f k f cp., Pe k c (5.13) f f p. f Improvement in density and specific heat increases Pe where as thermal conductivity enhances results in a diminishing in Pe. Therefore, when the pure working fluid in a system is replaced with a nanofluid, the flow velocity should be adapted in order to activate the system at the same Peclet number. In this section of the examination, heat transfer development attained with nanofluids are calculated by comparing the linked results with the conventional pure fluid case for the same flow velocity and tube diameter, so that the effect of the change in Peclet number is also examined In order to study the heat transfer augmentation obtained with nanofluids for the same flow velocity, axial position and tube diameter, by using Eq. (5.11) and obtain the local Nusselt number enhancement ratio.

16 130 Nu Nu 1.30x 1 1/3 x, *, 1/3 x, f x*, f for 5 x * 5 10, 1.30x 0.5 1/3 *, 1/3 *, f 1.30x 0.5 for 510 x , (5.14) 5 3 * x x x*, *, e x*, f *, f e for x * , From the above equations it is evident that the value of the term x *, increases with an increase in the density and specific heat of the nanofluid, which subsequently increases local Nusselt number enhancement ratio. On the other hand, enhancement in the thermal conductivity of the nanofluid reduces values of the terms with x *, and the local Nusselt number augmentation ratio reduces as a corollary. By integrating the numerator and denominator of Eq. (5.14) along the tube and attain the average Nusselt number augmentation ratio for a specialized case. The values of thermo physical properties on average Nusselt number augmentation ratio are qualitatively the same as the local Nusselt number case. For a pure fluid, average heat transfer coefficient can be defined as follows. h f Nu fk f, (5.15) d For a nanofluid, average heat transfer coefficient becomes the following expression. h Nu k, (5.16) d By using Eqs (5.15, 5.16), one can gain the average heat transfer coefficient enrichment ratio as follows.

17 131 h k Nu, (5.17) h k Nu f f f Eq. (5.14) can be used to execute the related integrations to arrive at average Nusselt number augmentation ratio through Eq. (5.17). Although the enhancement in thermal conductivity of the nanofluid reduces Nusselt number augmentation ratio, due to the multiplicative result of thermal conductivity in the characterization of heat transfer coefficient, it should be accepted that the growth in the thermal conductivity of the nanofluid increases heat transfer coefficient augmentation ratio. The present work is to study the convective heat transfer of Graphene (Cn)/water + EG nanofluids for different particle volume fractions between 0.5% and %. Reynolds number was varied between 800 and 00. At the steady wall heat flux boundary condition, the heat transfer development of nanofluids improves. The average Nusselt number for the forced convection of nanofluids inside circular tubes is estimated using the thermal diffusion model as given below. Nu c c Pe (5.18) m 1 m m (1 d )Re Pr, Pe d is particle Peclet number which is defined as ud m p Pe d, (5.19) u m is mean flow velocity. Re and Pr are the Reynolds and Prandtl numbers of the nanofluids, but the thermo physical properties of the nanofluid must be used. For steady wall heat flux boundary condition, Li and Xuan [73] suggested the empirical constants c 1, c, m 1, m and m 3 depend on their tentative study and combined expression is the following.

18 13 Nu ( Ped ) Re Pr, (5.0) It can be believed that Eq. (5.0) is applicable in the series of the experimental data [73]; 800 < Re < 300 and 0.5% < φ < %. It might also be distinguished that tube diameter is 1 cm and tube length is 1 m in the experiments. MODEL CALCULATIONS: Table: 5.3: Heat transfer coefficient enhancement ratio values with Re for different volume fractions of Graphene (Cn)/ water + EG S.No %Volume added Re f h /h f

19 h/hf Ref 0.50% 1.00% 1.50%.00% Figure: 5.11 Variation of average heat transfer coefficient enhancement ratio with Reynolds number for different particle volume fractions of the Graphene (Cn) 0nm/water+EG nanofluid Constant Wall Temperature Boundary Condition The method followed in the earlier for constant wall heat flux boundary condition is recurring in this segment for steady wall temperature boundary condition. To conclude the average Nusselt number, one can use the classical Sieder-Tate correlation [81] b Nu 0.07 ReD Pr w 0.7 Pr 16700; Re 10, 000 : L 60 D D (5.1) By integrating the above equation d b Nu 1.86 Pe, (5.) L w Here L is tube length. μ w is dynamic viscosity at the wall temperature where as μ b is 1/3 dynamic viscosity at the bulk mean temperature which is defined as:

20 134 Ti To Tb, (5.3) T i is inlet temperature and T o is outlet temperature. Neglecting the dissimilarity of viscosity augmentation ratio of the nanofluid (μ / μ f ) with temperature and using Sieder-Tate correlation (Eq. 5.) for a pure base fluid and nanofluid, following equation can be obtained. 1/3 1/3 Nu Pe k f c p, Nu f Pe f k f c p, f, (5.4) Using Eq. (5.17), heat transfer coefficient augmentation ratio becomes the following. 1/3 /3 h c p, k, (5.5) h f f c p, f k f By exploring this equation, it would be seen that the improvements in the thermo physical properties of the nanofluid; specific heat, density and thermal conductivity, develops the heat transfer coefficient. It should be eminent that the effect of thermal conductivity development is more marked when compared to density and specific heat. Their analysis part consists of a straight circular tube with an inner diameter of 5 mm and length of 1 m. The nanoparticles used in the nanofluid have a diameter of 0 nm. Peclet number was limited between 500 and 6500 and the heat transfer measurements were considered for different nanofluids with particle volumes fractions changing between 1.0% and.5%.the related under forecast shows that there should be added augmentation mechanisms associated to the convective heat transfer of nanofluids which extended to improve the heat transport.

21 135 Table: 5.4: Heat transfer coefficient enhancement ratio with Pe number for different particle volume fractions S.No %Volume added Pe f h /h f

22 h/hf % 1.50%.00%.50% Pef Figure 5.1: Variation of average heat transfer coefficient enhancement ratio with Peclet number for different particle volume fractions Of the Graphene (Cn)/water+EG nanofluid 5.5 Effect of Thermal Conductivity on Fully Developed Heat Transfer In this supplementary segment, effect of thermal conductivity of nanofluids on fully developed heat transfer coefficient values is studied. Parallel to the examination in the previous sections, the nanofluid is treated as a pure fluid with improved thermo physical properties. While this approach is shown to underrate the tentative results in the preceding sections, yet it will be used to get a good understanding about the consequence of thermal conductivity on heat transfer due to its plainness. As a consequence of the Graetz solution for parabolic velocity profile under steady wall temperature and constant wall heat flux boundary conditions. Nusselt number can be obtained as follows [8]: Nu x hd x k n0 n0 n n Ae A e n n n, For constant wall temperature (5.6)

23 137 Nu x 1 n x 11 1 e 4 48 n1 An n hd k, For constant wall heat flux (5.7) Where ξ= x/ (r o Pe). These terms are justifiable for the thermal admission region of a circular pipe with hydro dynamically fully developed laminar nanofluid flow under the assumption of treating nanofluids as pure fluids. Below the fully developed conditions, Nusselt number becomes: Nu fd o ( ) 3.657, for constant wall temperature, (5.8) Nu 4.364, for constant wall heat flux. (5.9) fd In organize to emphasize the importance of the exact strength of the thermal conductivity of nanofluids, heat transfer coefficient of the laminar flow of Graphene (Cn)/water+ ethyl glycol nanofluid inside a circular tube is studied by using the exceeding mentioned asymptotic values of Nusselt number. Nanoparticles are supposed to be spherical with a diameter of 0 nm. Different temperatures are considered in the study at room temperature. Flow is both hydro dynamically and thermally fully developed. Tube diameter is selected as 1 cm. For the determination of the thermal conductivity of the nanofluids at room temperature, Hamilton and Crosser [18] model (Eqs., 3) is utilize and fully developed heat transfer coefficients are resolute. A sample computation of this study is available in Appendix. For the 4% vol Graphene (Cn)/water+EG nanofluid. In Table 5.1, results of 1 and 4% Vol. Graphene (Cn)/water+ ethyl glycol nanofluids are compared with pure water. As seen from the table, due to the description of the Nusselt number (Nu=hd/k), the improvement in thermal

24 138 conductivity by the use of nanofluids openly results in the increment in heat transfer coefficient. Thermal conductivity of the nanofluids is resolute by using the model of Jang and Choi [65]. Table.5.5.Thermal conductivity and heat transfer coefficient values for pure water and Graphene (Cn)/Water + ethyl glycol nanofluid at room temperature K[W/mK] (Enhancement) h fd for constant wall temp[w/m k] (Enhancement) h fd for constant wall Heat flux [w/m k] (Enhancement) Pure Water+EG (-) 1.6 (-) 64.5 (-) 1 %vol Graphene(Cn)/Water+ ethyl glycol 1.63 (.7%) 37.6 (.7%) (.7%) The percentage values indicated are according to the expression 100(K K f )/K f 4 %vol Graphene(Cn)/Water+ ethyl glycol (11.1%) 367. (11.1%) 39.8 (11.15%) Investigational data presented in the theoretical study part of Chapter 4 concludes that the convective heat transfer improvement of nanofluids overcomes the augmentation expected due to the enhancement in the thermal conductivity. There are various techniques newly projected to elucidate this bonus augmentation in convective heat transfer; such as, thermal dispersion [7] and particle migration [74]. At this contemporary, there is disagreement about the comparative implication of these mechanisms. Hence, further investigations are needed for the amplification of this condition Stable Wall Temperature Boundary Condition For steady wall temperature boundary condition, the planned view of the arrangement is shown in Fig.5.13 In the study tube diameter is 5 mm and tube length is 1 m. In the theoretical examination, depending on Peclet number, the field is sometimes chosen to be longer than 1 m for getting thermally fully developed

25 139 situation at the exit, but only the 1-m part is measured in the purpose of heat transfer parameters. Fig 5.13: Schematic view of the problem considered in the numerical study. Boundary condition is stable wall temperature. Shaded region is the solution domain Steady wall heat flux boundary condition For stable wall heat flux boundary condition, the proposed view of the coiguration is shown in Figure.5.14 In order to get a proper assessment in experimental data, tube size is chosen to be 4.57 mm and tube length is m. In the numerical study, depending on Peclet number, the domain is sometimes chosen to be longer than m for achieving thermally fully developed condition at the exit, but only the -m part is measured in the calculation of heat transfer parameters. Fig 5.14: Schematic view of the problem considered in the numerical study. Boundary condition is stable wall heat flux. Shaded region is the solution domain.

26 Study on modelling of Nanofluid Flow Single phase approach In the literature, there are mostly two techniques for the construction of nanofluid flow. In the first method, the nanofluid is assumed as a single phase fluid due to the reason that the particles are very small and they mix easily [75]. In this method, the iluence of nanoparticles can be considered into account by using the thermo physical properties of the nanofluid in the governing equations. In the second method, the problem is studied as a two-phase flow and the relations between nanoparticles and the liquid matrix are constructed [76]. In the current study, the nanofluid is taken as a single phase fluid. Such an attempt is an additional realistic way of studying heat transfer of nanofluids. Conversely, the legality of the single phase guess needs authentication. It should be noted that exclusively considering the thermo physical properties of the nanofluid to the governing equations is not much diverse than using the conventional correlations of convective heat transfer with thermo physical properties of the nanofluid. In the theoretical examination part the single phase method needs some changes in order to consider the additional enhancement. For this reason, the thermal diffusion model proposed by Xuan and Rotzel [7] is used Governing Equations For the study of heat transfer in the current problem, the governing equations are the continuity, momentum and energy equations. For cylindrical coordinates, incompressible and steady continuity equation is as follows [8]. ur ur 1 u u x 0, (5.30) r r r x u x, u r and u θ are axial, radial and tangential parameters of flow velocity, respectively.

27 141 In the problem considered, the flow is hydro dynamically fully developed. Therefore, the velocity of the flow does not vary in x direction and derivatives of the velocity components in x-direction are zero. Moreover, while the flow is axisymmetric, all terms with is also zero. Then the continuity equation becomes the following. ur ur 0, r r Noting that where r 0 is the tube radius, it can be obtained (5.31) ur 0, (5.3) r-momentum, θ-momentum and x-momentum equations in the lack of body forces for cylindrical coordinates are as follows, respectively [8]. u u r r ur u ur u ur ux r r r x 1 p 1 1 ur u u v r ( ru r ), r r r r r r x (5.33) u u u u u u r ux r r r x 1 p 1 1 u ur u v ru, r r r r r r x (5.34) u r u u u u x x x ux r r x 1 p 1 ux 1 ux u v x r, r x r r r r x (5.35) ν is kinematic viscosity of the nanofluid and p is pressure. It should be remembered that pressure does not change with θ due to axisymmetric. Applying the simplifications concerning the x- and θ-derivatives to Eq. (5.34): 1 0 ( ru ), r r r (5.36) Noting that, it can be obtained ur θ =0

28 14 u 0, (5.37) Eqs (5.33, 5.35) can also be re written by applying the calculations about the x- and θ- derivatives and substituting Eqs. (5.3, 5.37): 1 p 0, r (5.38) 1 p 1 ux 0 v r, r x r r r (5.39) u 0 at r r0, (5.40) r u x u x x 0 Where u m is mean flow velocity at r 0 (5.41) r um1, r0 (5.4) After the resolution of the velocity distribution in the field, heat transfer in the system can be discussed by allowing the energy equation which is as follows [8]. DT cp. keff T q, (5.43) Dt Where D u ur ux, (5.44) Dt t r r x 1 T 1 T T. keff T rkeff k, eff keff r r r r x x (5.45) ur 1 x 1 1 x u u u u ur r r x x r 1 ux ur 1 1 ur u r, r x r r r (5.46) Is the volumetric heat generation rate and Φ is the dissipation function. It should be noted that the thermal conductivity term in the energy equation is changed by the effective thermal conductivity (k eff ) according to the thermal diffusion model.

29 143 Applying the calculations about the x- and θ-derivatives to Eq. (5.43) and substituting Eqs. (5.3, 5.37, 5.4, ) T r T 1 T T ux cp um 1 k, eff r keff t r0 x r r r x x r (5.47) The term with the time derivative is conserved in the energy equation since the numerical solution technique used reaches the steady-state solution by marching in time. Hence the problem is considered as transient, for the proper study of the problem, non dimensionalization should be applied to Eq. (5.47). Non dimensionalizations for steady wall temperature boundary condition and constant wall heat flux boundary condition are a little different. Hence the connected discussion is presented in two different sections Steady Wall Temperature Boundary Condition For steady wall temperature boundary condition, following non dimensional parameters are defined: T TW, Ti TW (5.48) * x x, r (5.49) r t * * 0 r, (5.50) r 0, (5.51) t, b r0 k Pe k * eff, T, (5.5) k, b ud m, (5.53), b

30 144 Br, bum k T T, b i W, (5.54) T i and T w are inlet and wall temperatures, respectively and d is tube diameter. By using these non dimensional parameters, Eq. (5.47) becomes: * 1 * * * * Pe * 1 r k r k 16 r Br, * * * * * * t x r r r x x (5.55) Br is the nanofluid Brinkman number, which is a gauge of viscous effects in the flow. For the current flow conditions, Br is on the order of 10-7, therefore viscous dissipation is negligible. As a consequence, the final form of the energy equation is : * 1 * * * Pe * 1 r k r k, * * * * * * t x r r r x x (5.56) * 0 at r 0, * r (5.57) * 0 at r 1, (5.58) 1 at * X 0, (5.59) Constant Wall Heat Flux Boundary Condition For steady wall heat flux boundary condition, the expressions for the non dimensional parameters, x *, r *, t *, k * and Pe are similar to the steady wall temperature condition, which are distinct from Eqs ( ) correspondingly. But the expressions for θ and Br differ as given below " qw r0 um " qw r0 k T Ti, (5.60) Br, (5.61) It is optimistic when heat is transferred to the working fluid. Diligence of non dimensionalization to the energy equation results in precisely the same differential

31 145 equation as in the condition of constant wall temperature (Eqs. 5.55, 5.56). The boundary conditions are as follows: 0 * r at 1 * * r k at 0 at * r 0, (5.6) * r 1, (5.63) * x 0, (5.64) For both stable wall temperature and stable wall heat flux boundary conditions, all of the thermo physical properties are measured at the bulk mean temperature of the flow, which is shown by the subscript b in the linked expressions, excluding the thermal conductivity. Nondimensional thermal conductivity, k *, is clear as the effectual thermal conductivity at the local temperature divided by the nanofluid thermal conductivity at the bulk mean temperature. Bulk mean temperature is: T b Ti To, (5.65) It should be noted that k * is a function of temperature and local axial velocity due to Eqs. (5.5) Stable Wall Heat Flux Boundary Condition Comparable to the earlier case, the results of stable wall heat flux boundary condition are compared with the predictions of the connected Graetz solution [83] for parabolic velocity profile, which According to Graetz solution, local Nusselt number is as follows [84]: 1 n hd x 11 1 e Nux 4 k 48 n1 An n, (5.66) x ro Pe (5.67) ξ is given by Eq. (5.67) and A n, β n values are got from Siegel et al. [84]. Fig provides the linked assessment in terms of the discrepancy of local Nusselt number in

32 146 axial direction for the flow of pure water for Pe = 500, 4500 and When the figure is observed, it is concluded that there is perfect coormity between the numerical results and the predictions of the Graetz solution. Table: 5.6: Experimental calculations of Graetz solution S.No Pe Nu x X * =X/r o

33 147 Table: 5.7: Numerical values of Graetz solution S.No Pe Nu x X * =X/r o

34 Nux Nux Pe=500,Graetz Pe=4500,Graetz Pe=6500,Graetz x*=x/ro Pe=500,numerical Pe=4500,numerical Pe=6500,numerical X*=X/r0 Fig 5.15: dissimilarity of local Nusselt number in axial direction according to the numerical results and Graetz solution for 0nm (Cn) of 1%, 1.5%, %,.5% Vol Stable wall heat flux condition. The discussion is presented in two main heading, stable wall temperature boundary condition and stable wall heat flux boundary condition, correspondingly Stable Wall Temperature Boundary Condition For the Stable wall temperature boundary condition, the results are first discussed in terms of the standard heat transfer coefficient augmentation ratio (heat

35 149 transfer coefficient of nanofluid divided by the heat transfer coefficient of corresponding conventional base fluid). Then the change of local Nusselt number in axial direction is studied for separate particle volume fractions. At last, iluence of particle size, heating and cooling is presented in terms of heat transfer coefficient augmentation ratio. 5.8 Heat Transfer Coefficient improvement The experimental data for heat transfer of nanofluid flow under stable wall temperature boundary conditions. The results of the current study considered the heat transfer characteristics of Graphene (Cn)/water+EG nanofluid in laminar flow. The flow is hydro dynamically developed and thermally developing. Nanofluid flows inside a circular tube with a diameter of 5 mm and length of 1 m. The discrepancy of standard heat transfer coefficient augmentation ratio with Peclet number for dissimilar particle volume fractions. Augmentation ratios are considered by comparing the nanofluid with the pure fluid at the same Peclet number in direct to concentrate on the special result of the improved thermal conductivity and thermal diffusion. In direct to emphasis the significance of the claim of thermal diffusion model,

36 150 Table: 5.8: Heat transfer enhancement ratio with Pe for different particle volume ratios of Graphene (Cn) 0nm /water + EG S.No %Volume added Pe f h /h f

37 h/hf Pe Blue 1.0% Red 1.5% Green.0% Violet.5% Fig 5.16: Variation of average heat transfer coefficient enhancement ratio with Peclet number for different particle volume fractions of the Graphene (Cn) 0nm/water+EG nanofluid. 5.9 Further study Local Nusselt Number In this chapter, the same flow arrangement is analyzed theoretically in the earlier sections is examined in terms of the axial variation of local Nusselt number. To know the fully developed Nusselt number as well; the flow stream within a longer tube is taken (5 m). Figure 5.17 shows the related results for the flow of pure water and Graphene (Cn)/water+EG nanofluid at a Peclet number of In the figure, it is observed that the local Nusselt number is higher for nanofluids throughout the tube. This is mostly due to the thermal diffusion in the flow. Thermal scattering results in a higher effective thermal conductivity at the middle of the tube which straightens the radial temperature profile. Flattening of temperature profile raises the temperature gradient at the tube wall and as a result, Nusselt number becomes superior when compared to the flow of pure water. This is owing to the truth that the iluence of thermal distribution becomes more prominent with growing particle volume fraction.

38 15 It must be noted that the fully developed nanofluid Nusselt number values are also superior to pure water case. Linked values for different particle volume fractions of the Graphene (Cn)/water+EG nanofluid are offered in Table It is observed that rising particle volume fraction raises the fully developed Nusselt number. The results existing in the table are for Pe = 6500 and since thermal diffusion is dependent relative on flow velocity (Eq. 5.), fully developed Nusselt number rises also with Peclet number for the case of nanofluids. In Table 5.10 fully developed heat transfer coefficient values are also presented. It should be noted that heat transfer coefficient augmentation ratios are larger than Nusselt number augmentation ratios since the previous shows the collective effect of Nusselt number augmentation and thermal conductivity enrichment with nanofluids.

39 153 Table: 5.9: Calculation of local Nusselt number with dimensionless axial position for pure water and Graphene (Cn) 0nm/water+EG nanofluid. Pe = Pe f = S.No Pure Water Nu x Graphene nanofluid Volume fraction (1.0%) Graphene nanofluid Volume fraction (.0%) X * =x/r o Nu x Nu x

40 Nux Blue.0 % Pure Water Green 1.0% X*= X/ro Fig 5.17: Dissimilarity of local Nusselt number with dimensionless axial position for pure water and Graphene (Cn)/water+EG nanofluid. Pe = Pe f = Table 5.10 : Fully developed Nusselt number and heat transfer co efficient values obtained from the numerical solution for pure water and Graphene(Cn) / Water + EG nanofluid with different particle volume fractions pe f = pe = 6500 Fluid Nu fd Nu Enhancement Ratio(Nu fd, /Nu fd,f ) h fd [W/m K] h Enhancement ratio(h fd, / h fd,f ) Water+EG Nanofluid 1.0 %vol %vol %vol Vol %