CONVECTIVE HEAT TRANSFER CHARACTERISTICS OF NANOFLUIDS. Convective heat transfer analysis of nanofluid flowing inside a

Transcription

1 Chapter 4 CONVECTIVE HEAT TRANSFER CHARACTERISTICS OF NANOFLUIDS Convective heat transer analysis o nanoluid lowing inside a straight tube o circular cross-section under laminar and turbulent conditions is taken up. The low and the thermal ield are assumed symmetrical with respect to the vertical plane passing through the main axis. Further there exists no ormulated theory known to date that could reasonably predict the low & heat transer behaviors o a nanoluid by considering it as multi-component model. Thereore, it has been suggested that the particles may be easily luidized and consequently, can be considered as conventional single-phase luid, which posses eective physical properties being unction o the properties o both constituents and their respective concentrations (Pak & Cho [44]; Xuan & Li [110]). As a result, a direct extension rom a conventional luid to nanoluid appears easible, and one may then expect that the classical theory developed or a conventional single-phase luid can be applied to nanoluid as well. Thus, all the equations o conservation (mass, momentum and energy) as well known or single-phase luid can be directly applied to nanoluids. u v 0 x y (4.1) 111

2 u v u v x y 1 dp dx 2 u ( uv) 2 y y (4.2) T u x T v y 2 T ( vt ) (4.3) 2 x y 4.1 Laminar Flow Wen & Ding [113] reported experimental results o the convective heat transer o Al2O3/ water nanoluid lowing through a copper tube in laminar regime. They compared their experimental results with Shah correlation or laminar low and ound the theoretical results are ar less than the experimental values. This may be due to considering the base luid properties instead o nanoluid properties. Heris et.al. [174] conducted experiments on laminar low o CuO/ water and Al2O3/ water nanoluids through copper tube and proposed Seider-Tate equation to compare experimental results and ound to be ar less than experimental results. This could be due to wrong estimation o thermal conductivity and viscosity o nanoluids. Thus, we propose a modiied Seider- Tate equation in the ollowing orm or Laminar low as ollows Nu D L b (Re Pr ) (4.4) The constant b is evaluated rom the regression analysis o the data obtained rom [44,113,174] and calculated as 1.98.Thus the correlation or calculation o Nusselt number or Laminar low is 112

3 developed with an average deviation o 6% and standard deviation o 7.4 % or all nanoluids. Nu D L (Re Pr ) (4.5) The correlation gives good agreement with the experimental results as shown in the Fig.4.1(a)-(b). The values o Re and Pr are to be calculated using the properties o nanoluids as developed & presented in sections 2 & 3. The variation o Nusselt number with respect to Reynolds number, volume raction, L/D ratio are presented in graphical orm as shown in Fig. 4.2 (a) (d). 113

4 Figure 4.1(a) Laminar low heat transer comparison 114

5 v Figure 4.1(b) Laminar low heat transer comparison or dierent L/D values 115

6 Figure 4.2 (a) Nusselt number vs Reynolds Number in Laminar low 116

7 Figure 4.2(b) Eect o Volume raction in Laminar low 117

8 Figure 4.2(c) Variation o Nusselt number with L /D in Laminar low 118

9 Figure 4.2(d) Heat transer in laminar low in dierent Nanoluids 119

10 4.2 Turbulent Flow Model 1 S B Maiga et.al. [120] presented the numerical study o ully developed turbulent low o Al2O3 + water nanoluid in circular tube at uniorm heat lux o 50 W/cm 2. The classical K- model was used or turbulence modeling and a new correlation or Nusselt number was derived as Re Pr Nu (4.6) Pak and Cho[44] or Pr 13.9;10 Re 5*10 and Li and Xuan [110] has developed 5 correlations o a orm similar to that o well known Dittus-Boelter ormula to characterize nanoluids heat transer. Pak & Cho[44] and Xuan & Li [110] proposed correlation treating it as a single phase luid or the calculation o nanoluid Nusselt number as given in Eqs.(4.7) & (4.8). In these correlations the Reynolds Number and Prandtl numbers are calculated by considering the base luid properties which will give estimated under results compared with experimental values Nu 0.021(Re ) (Pr ) (4.7) Nu d D p Re Pr Re Pr 4.8)

11 Hence the above considered single phase luid approach is adopted with a modiication that instead o base luid properties the thermophysical properties o the Nanoluid itsel are considered while evaluating the non dimensional numbers. Thus the Nusselt number is deined and evaluated as h D Nu 4.9) k The heat transer coeicient o turbulent low through circular tube can be developed more appropriately in the ollowing orm Nu a (Re ) (Pr ) (4.10) Where Re and Pr as deined as ollows: u D Re (4.11) C Pr (4.12) k The Nusselt number data o the nanoluids obtained rom the experimental works o Pak et al.[44] and Xuan and Li[110] is subjected to non-linear regression analysis and the constant a is obtained as or Al2O3 + H2O or Cu + H2O 121

12 Thus the correlations or calculation o Nusselt number are developed as ollows with an average deviation o 5% and standard deviation o 6.4% For circular tube Nu (Re ) (Pr ) or Al2O3+ H2O (4.13) Nu 0.027(Re ) (Pr ) or Cu + H2O (4.14) The above equation takes care o diameter o the nanoparticle, concentration and temperature eects. The correlations give good agreement with the experimental results as shown in the Fig.4.3 (a&b) & 4.4.(a&b) The values o Re and Pr are to be calculated using the properties o nanoluids as developed & presented in eqs.(2.1),(2.2),(2.8) & (3.11). 122

13 Figure 4.3(a) Turbulent heat transer data comparison or =

14 Figure 4.3(b) Turbulent heat transer data comparison or =

15 Figure 4.4(a) Turbulent heat transer data comparison =1% 125

16 Figure 4.4(b) Turbulent heat transer data comparison or =1.5% 126

17 Model 2 The earlier developed correlation is valid only when >0.01 and purely base on experimental data. To overcome this limitation another model is proposed in this section which will not only be helpul to evaluate heat transer rate but also or riction actor under dierent conditions. The model was developed by considering the experimental data o Xuan and Li [110] with nanoparticles o copper and data o Pak & Cho [44] or Aluminum in water medium as coolant. The data o Xuan and Li [110] and Pak &Cho [44] is recast into the dimensionless parameters, Nu and Re with the properties o the base luid water being employed in the computations. A single correlation or both Cu + H2O & Al2O3 + H2O nanoluids is developed and comparison with the experimental data are shown in Fig. 4.5 with the predictions rom the correlation eq (4.15) with average deviation o 2.9% and standard deviation o 3.5%. Nu / Re Pr (1 ) (4.15) 127

18 Figure 4.5 Validation o correlation with Experimental data 128

19 However, the variation o physical parameters o the base luid with as a variable is as ollows. For Al2O3 dispersion in the base luid water, the property relations are as ollows: Density ( Pak & Cho [44]): p 1) ( (4.16) Dynamic Viscosity ( Pak & Cho [44]): 2 ( ) (4.17) Thermal conductivity ( Pak & Cho [44]): k 2 k ( ) (4.18) Speciic heat ( Pak & Cho [44]): C C (1 ) C p p (4.19) Similarly or Cu dispersion in the base luid water, the property relations Density and Speciic heat is calculated by using the above eqs (4.16) & (4.19) and Dynamic Viscosity (Chen et al. [45]): 2 ( ) (4.20) Thermal conductivity ( Chen et al. [45]): k 2 k ( ) (4.21) 129

20 4.3 Evaluation o eddy momentum diusivity with nanoparticle dispersion in the base luid As observed by Buongiorno [175] treating the dispersion o nanoparticles as a continuum medium ollowing Newtonian luid properties, the shear stress distribution or turbulent low conditions can be assumed by Boussinesq approximation or turbulent lows. In addition, the experimental observation is that the presence o nanoparticles does not prooundly alter the momentum characteristics. Though dispersion o nanoparticles in the medium might signiicantly alter the viscous properties, it is an observed act that or viscous or non-viscous turbulent lows, the Darcy s riction actor dependence on the Reynolds number is uniquely deined by the relationship = /Re In other words, in the estimation o riction losses, the low volumetric percentage concentration o the nanoparticles can be totally ignored. Hence, it can be treated totally as Newtonian base luid ignoring the presence o nanoparticles in the medium. The basis or urther development o the model is that the universal nature o the velocity proile cannot be iluenced by the volumetric percentage concentration o the nanoparticles in the base medium. However, the inclusion o nanoparticles would aect the temperature proiles adjacent to the wall. 130

21 Thus in view o (Buongiorno [175]) m u 1 (4.22) y Thus, the above equation can be converted into dierential orm with assumption that the shear distribution is linear over the tube cross section. y 1 w R (4.23) Further it can be written in dimensionless orm as ollows: u y y 1 R m 1 (4.24) The boundary condition is that at y + =0, u + =0 The eddy momentum parameter m can be deined as per the existing experiences or turbulent orced momentum exchange mechanisms and we can employ the ollowing Van Driest expression [176]. 1 exp( y / A 2 m By ) (4.25) Where B is considered as unction o transitional Reynolds number and A + =26. Its valve is determined as a dependent unction o transitional Reynolds number. 131

22 4.3.1 Numerical Procedure to evaluate B in Equation (4.25) For the present case o internal lows in a tube, the value o B can be estimated theoretically ollowing the numerical procedure as outlined in steps: 1. Obtain u + =(y + ) by solving equation (4.25) or an assumed value o R + and B. 2. Calculate Reynolds number rom the ormula Re 4 R 0 u y 1 R dy 3. Calculate cal R 8 Re Calculate Blasius Re 5. Compare Blasius with Cal and check whether cal Blasius Blasius I the criterion o accuracy is satisied, the assumed value o B is accepted as correct approximation. Thus the numerical procedure the value o B is obtained close to 0.5 or 8000< Re< 80,000. The theoretical prediction o riction coeicient is valid with Blasius relationship as shown in Fig.4.6. Hence the momentum eddy diusivity or nanoparticle inclusion 132

23 can be obtained by substituting the values o B=0.5 and A + =26 in eq (4.25) with reasonable accuracy as 1 exp( y / A 2 m 0.5y ) (4.26) 4.4 Evaluation o Eddy thermal diusivity For turbulent convective heat transer without nanoparticle inclusion, the ollowing expression is taken or heat transer calculations ( Incropera et al. [167]): H m Pr orpr 1 (4.27) In the present model it is thought that the presence o nanoparticles will greatly iluence the thermal eddy diusivity and such a modiication can be considered by modiication o the term as ollows H m Pr where ( ) (4.28) Thus, hereater the analysis is devoted to evaluate the unctional relationship between ξ and. The reason or such a postulation is based on the conclusion by Buongiorno [175] that the thermal penetration thickness is greatly iluenced by the particles leading to enhancement o heat transer in the presence o nanoparticles. 133

24 For thermally developed low conditions, the temperature proile over the cross section can be obtained rom the dierential equation 1 H T 0 y y (4.29) As per the thermal model, the equation can be written as ollows in dimensionless orm y 1 m (Pr) T 0 y (4.30) where T T T w w T T c The boundary conditions are as ollows: At y + =0, T + =0 At y + =R +, T + = Numerical procedure to evaluate ξ This program is to be conjointly worked out with the earlier one in which the velocity proiles are already obtained as u + = F [y +, R + ] where R + =F [Re] 1. For the known values o R + and Re, evaluate the temperature proiles i.e. T + rom equation (4.30) with an assumed value o ξ or given value o volumetric concentration. 134

25 For given value o R + and rom the computer run satisying the boundary conditions equation (4.30), obtain 0 y y T 3. Calculate the theoretical value o Nusselt number rom the equation : b w c w y T T T T T y T R Nu 0 2 (4.31) where R R c w b w dy R y u dy R y u T T T T T Compare the theoretical value o NuT rom the equation (4.31) with the value calculated rom correlation or nanoparticle in the medium i.e. equation (4.15). 5. I the result o Nusselt number in step 3 is same as the one in step 4, then the assumed value o ξ, the exponent to Prandtl number is considered as valid magnitude. 6. I the disagreement in the NuT and Nu correlation is not within the limits o prescribed accuracy, then a proper iteration technique is employed to achieve convergence.

26 Figure 4.6 Variation o riction coeicient with Reynolds Number 136

27 4.5 Velocity & Temperature proiles The universality in velocity proiles or the ranges o 0.3% < < 3% is uniquely depicted in Fig. 4.7 and the proiles are iluenced by the Reynolds number irrespective o the act whether the base luid is homogenized with the nanoparticles or not. This observation can be considered valid at least or the ranges o the concentrations o nanoparticles investigated. Typical plots depicting the temperature proiles near to the wall i.e. y R 0.12 are shown plotted in Fig Evidently, the wall temperature gradients are ound to vary steeply with concentration o the nanoparticles. The proound iluence o the concentration is more clearly depicted or dierent values o versus Re (see Fig. (4.9).The results o the plot can be comprehensively correlated with the help o the equation as ollows dt dy y (1 ) Re 0 (4.32) The wall temperature gradients are shown as unctions o Reynolds number or dierent values o in Fig The equation obtained by regression analysis or 72 points rom this analysis is ound to possess the ollowing limits o accuracy o a standard deviation o 0.73% and an average deviation o 0.59%. 137

28 Figure 4.7 Variation o velocity proile with Reynolds number 138

29 Figure 4.8 Eect o on temperature Proiles or dierent Re 139

30 Figure 4.9 Variation o Temperature gradients at the wall with Re 140

31 Figure 4.10 Variation o exponent to Prandtl number with volume raction 141

32 4.6 Thermal Eddy Diusivity In Fig. 4.10, the variation o ξ, the exponent to Prandtl number o the base luid is shown plotted as a unction o Re. The plot reveals that ξ assumes asymptotic values or high Reynolds number. The data o ξ rom the theory can be represented by the relationship (1 ) Re (4.33) However, rom the nature o independency o ξ with Re (or high magnitudes), ξ can be ound to be purely a unction o and independent o Re. The asymptotic expression or the evaluation o ξ in the eddy thermal diusivity is given by the relationship as ollows: (1 ) 0.036(1 ) (4.34) The magnitude o H is ound to be substantially aected with the concentration in the base luid (see Figure 4.11). The reason or such increase in the magnitude o eddy thermal diusivity is due to enhanced mixing eects resulting rom the Brownian motion o the nanoparticles in the medium. The inclusion o nanoparticles leads to proound change in thermal conductivities o the dispersion relative to the base luid. 142

33 Figure 4.11 Variation o eddy thermal diusivity y + / R + with Volume raction as a parameter 143

34 4.7 Conclusion This chapter is mainly ocused on convective heat transer o nanoluids lowing inside a circular cross section tube. Correlations to predict Nusselt number or nanoluids both in laminar and turbulent have been developed separately. In laminar low to calculate Nusselt number or nanoluids a modiied Seider- Tate correlation as been developed by considering nanoluids properties as Nu 1.98 (Re Pr D ) L The Nusselt number thus calculated by considering the above correlation or nanoluids in Laminar low ound to be an average deviation o 6% and standard deviation o 7.4 % with the experimental data. In Turbulent low two models are developed to calculate Nusselt number or Al2O3 + water and Cu + water Nanoluids. In irst model, correlation is developed by considering single phase luid approach. The correlation thus obtained is Nu (Re ) (Pr ) or Al2O3+ H2O Nu 0.027(Re ) (Pr ) or Cu + H2O with an average deviation o 5% and standard deviation o 6.4% with the experimental data. However, this correlation is valid when >0.01 as thermal conductivity used valid or >

35 In second model, a combined correlation or Al2O3 + water and Cu + water Nanoluids is developed by considering base luid properties. The correlations thus obtained is Nu / Re Pr (1 ) with average deviation o 2.9% and standard deviation o 3.5% with experimental data. The correlations can be valid or Further, velocity and temperature proiles or nanoluids are developed by considering turbulent orced momentum exchange Van Driest expression below as been used. 1 exp( y / A 2 m By ) Where B is considered as unction o transitional Reynolds number and A + =26 is calculated by numerical method and ound as 0.5.Thereore the momentum and thermal eddy diusivities are The Momentum eddy diusivity 1 exp( y / A 2 m 0.5y ) The Thermal eddy diusivity, where A + =26. H m Pr orpr (1 ) 0.34 Re