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This book presents a theoretical study of heat transfer due to laminar natural convection of nanofluids, using Al2O3-water nanofluid as an example. An innovative method of similarity transformation of velocity fields on laminar boundary layers is applied for the development of a mathematical governing model of natural convection with actual nanofluids, and a novel model of the nanofluid's variable thermophysical properties is derived by a mathematical analysis based on the developed model of variable physical properties of fluids combined with the model of the nanofluid's thermal conductivity and viscosity. Based on these, the physical property factors of nanofluids are produced, which leads to a simultaneous solution for deep investigations of hydrodynamics and heat transfer of nanofluid's natural convection.

The book also proposes novel predictive formulae for the evaluation of heat transfer of Al2O3-water nanofluid’s natural convection. The formulae have reliable theoretical and practical value because they are developed by rigorous theoretical analysis of heat transfer combined with full consideration of the effects of the temperature-dependent physical properties of nanofluids and the nanoparticle shape factor and concentration, as well as variations of fluid boundary temperatures. The conversion factors proposed help to turn the heat transfer coefficient and rate of fluid natural convection into those of nanofluid natural convection. Furthermore, several calculation examples are provided to demonstrate the heat transfer application of the proposed predictive formulae.

### Chapter 1. Introduction

Using Al2O3-water nanofluid as an example, this book presents a theoretical study on heat transfer of nanofluid’s laminar natural convection. An innovative method of similarity transformation of velocity fields on laminar boundary layers is applied for development of a mathematical governing model of natural convection with actual nanofluids. An innovative model of nanofluid’s variable thermophysical properties is developed by a mathematical analysis based on the developed model of variable physical properties of fluids combined with the model of nanofluid’s thermal conductivity and viscosity. Based on the innovative method of similarity transformation and the innovative model of nanofluids variable thermophysical properties, the physical property factors of nanofluids are produced. It leads to a simultaneous solution for deep investigations of hydrodynamics and heat transfer of nanofluids natural convection. Innovative predictive formulae are proposed for the evaluation of heat transfer of Al2O3-water nanofluid’s natural convection. The predictive formulae have reliable theoretical and practical value because they are developed by rigorous theoretical analysis of heat transfer combined with comprehensive consideration of effects of temperature-dependent physical properties of nanofluids and the nanoparticle shape factor and concentration, as well as variations of fluid boundary temperatures. The conversion factors proposed help to turn the heat transfer coefficient and rate of fluid natural convection to those of nanofluid natural convection. Furthermore, several calculation examples are provided to demonstrate the heat transfer application of the proposed predictive formulae.
De-Yi Shang, Liang-Cai Zhong

### Chapter 2. Conservation Equations of Fluid Flow

The purpose of this chapter is to derive fluid’s three dimensional conservation equations through mathematical and physical analysis on fluid dynamics. The derived partial differential equations describe fluid’s continuity, momentum and energy transportation regulations. They are taken as the basis of the two-dimensional mass, momentum, and energy conservation equations of nanofluid’s natural convection obtained by boundary layer analysis.
De-Yi Shang, Liang-Cai Zhong

### Chapter 3. Partial Differential Equations of Boundary Layer of Conventional Fluid’s Natural Convection

In this chapter, first, the two-dimensional partial differential equations of fluid’s natural convection are presented, including mass, momentum, and energy conservation equations. They are obtained through a magnitude method of physical variable, to transform the three-dimensional conservation equations of fluid’s flow into the partial differential equations of boundary layer of conventional fluid’s natural convection . Then, the corresponding two-dimensional equations of nanofluid’s natural convection are demonstrated in order for a convenient study of heat transfer of natural convection.
De-Yi Shang, Liang-Cai Zhong

### Chapter 4. Ordinary Differential Equations of Boundary Layer of Nanofluid’s Natural Convection

The partial differential equations of natural convection boundary layer with conventional fluid’s flow are equivalently transformed to the related ordinary differential equations. A feasibility is demonstrated to describe the mass, momentum and energy conservation of nanofluid’s natural convection by using those of nanofluid’s natural convection. By an innovative similarity transformation, the partial differential equations of nanofluid’s natural convection boundary layer are equivalently transformed to the related ordinary differential equations. On this basis, the two-dimensional ordinary differential equations of nanofluid’s natural convection are determined for extensive exploration of heat transfer of nanofluid’s natural convection.
De-Yi Shang, Liang-Cai Zhong

### Chapter 5. Mathematical Model of Variable Physical Properties of Nanofluids

Mathematical model of nanofluid’s variable physical properties is provided for extensive study on heat transfer of nanofluid’s natiral convection. Through consideration of base fluid’s variable physical properties, the nanofluid’s variable physical properties involved are treated. In the mathematical model of variable physical properties of nanofluids, the nanofluid’s variable physical properties are turned to the related physical properties factor. They are organically coupled with the governing ordinary differential equations for a in-depth theoretical research of nanofluid’s natural convection.
De-Yi Shang, Liang-Cai Zhong

### Chapter 6. Numerical Solutions of Velocity and Temperature Fields

The purpose of this chapter is to conduct numerical calculation on nanofluid’s natural convection. The Al2O3-water nanofluids are taken as an example, and systems of numerical solutions of velocity and temperature fields of boundary layer are obtained with consideration of nanofluid’s variable physical properties. They will become the basis of further investigation on hydrodynamics and heat transfer of nanofluid’s natural convection.
De-Yi Shang, Liang-Cai Zhong

### Chapter 7. Skin-Friction Coefficient

Local and average characteristic coefficients of skin friction of nanofluid’s natural convection are proposed with consideration of nanofluid’s variable physical properties. Meanwhile, a defined characteristic similarity velocity is defined. The formulae of the local and average characteristic coefficient of skin friction of nanofluid’s natural convection are obtained, with the defined characteristic similarity velocity whose value is set to be a unit. The system of the rigorous numerical solutions of the skin velocity gradient is provided for Al2O3-water nanofluid’s natural convection under consideration of variable physical properties. With the further analysis on the system of the rigorous numerical solutions of the skin velocity gradient, the effects of some physical variables and parameters are found on skin velocity gradient for Al2O3-water nanofluid’s natural convection. With increase of the fluid bulk temperature, the skin velocity gradient, and the skin friction coefficient will decrease. With increase of the wall temperature, the skin velocity gradient and the skin friction coefficient will increase. With increase of the Al2O3 nanoparticle’s volume fraction, the skin velocity gradient and the skin friction coefficient will increase as a linear function. However, the effect of the nanoparticle’s volume fraction from 0 to 0.1 is not obvious compared with the effect of the flow boundary temperatures.
De-Yi Shang, Liang-Cai Zhong

### Chapter 8. Predictive Formula of Wall Temperature Gradient

Systems of numerical solutions are obtained on wall temperature gradient of Al2O3-water nanofluid’s natural convection. On this basis, a correlation of the wall temperature gradient is developed. Such correlation has significant theoretical and practical value on heat transfer application, because it is based on the comprehensive consideration of physical properties and parameters, including nanofluid’s variable thermophysical properties, nanoparticle’s shape factor and fluid’s boundary temperatures. Then, it is found that comprehensive consideration of fluid’s variable thermophysical properties is importance for theoretical research of nanofluid’s convection heat transfer. The calculation examples demonstrate that Boussinesq approximation method will lead to an absurd calculation result of the related convection heat transfer research. Such absurd calculation result lies in that by using Boussinesq approximation method, as long as the flow average temperature is same, the calculation results, such as, the wall temperature gradient are same. However, such calculation results could be quite different according to the present calculation examples with consideration of fluid’s variable physical properties. Thus, proper consideration of fluid’s variable physical properties is necessary for exact theoretical research of convection heat transfer. The wall temperature gradient investigated in this Chapter is the key issue for research of convection heat transfer coefficient in actual fluids. This Chapter demonstrates that this issue can be resolved theoretically with comprehensive consideration of variable physical properties and parameters. In this way, the theoretical research result will have a significant theoretical and practical value on heat transfer application.
De-Yi Shang, Liang-Cai Zhong

### Chapter 9. Predictive Formulae on Heat Transfer of Al2O3-Water Nanofluid’s Natural Convection

In view of a lack of study on heat transfer coefficient of nanofluid’s convection, the correlations on heat transfer coefficient of Al2O3-water nanofluid’s natural convection are reported in this chapter. Such correlations are based on the theoretical equations with heat transfer analysis of nanofluid’s natural convection, and contain the wall temperature gradient, the only unknown variable for prediction of heat transfer of nanofluid’s natural convection. The correlation of wall temperature gradient is obtained according to systems of the related rigorous numerical solutions on Al2O3-water nanofluid’s natural convection. These systems of the numerical solutions on wall temperature gradient are based on comprehensive consideration of nanofluid’s variable thermophysical properties, nanopartical shape factor and concentration, and fluid’s boundary temperatures. Thus, the reported correlations on heat transfer rate and coefficient of Al2O3-water nanofluid’s natural convection have theoretical and practical value for heat transfer application. Furthermore, correlations on heat transfer coefficient of water natural convection are obtained with the nanoparticle’s volume fraction $$f_{p} = 0$$.
De-Yi Shang, Liang-Cai Zhong

### Chapter 10. Calculation Examples by Using the Predictive Formulae on Heat Transfer

The predictive formulae on heat transfer developed in the previous Chapter are applied in the present calculation examples to evaluate heat transfer of Al2O3-water nanofluid’s natural convection. Their theoretical and practical value in heat transfer application is attributed to that they are developed based on the comprehensive consideration of the effects of nanofluid’s variable thermophysical properties, nanoparticle’s shape factor and concentration, and fluid’s boundary temperature. It demonstrates that it is feasible to conduct theoretical study on convection heat transfer with actual nanofluids. In this theoretical study, two key works are performed. The first one is to develop advanced theory and method for the challenging research, and the second one is to develop advanced model for comprehensive consideration of various physical variables and parameters, including the variable thermophysical properties. By using the present correlations, four calculation Examples are provided to evaluate heat transfer rate with respective physical conditions. It is found that the present correlation on heat transfer can be used to rigorously evaluate heat transfer rate of Al2O3-water nanofluid’s natural convection. It is seen from the calculation results that with same fluid’s average temperature level, different fluid’s boundary temperatures still lead to different evaluated results on heat transfer coefficient. The calculation results also show that with increase of the fluid average temperature level, the heat transfer coefficient will increase. It reveals the charm of the present correlations developed based on the various physical variables and parameters including nanofluid’s variable thermophysical properties.
De-Yi Shang, Liang-Cai Zhong

### Chapter 11. Conversion Factors on Heat Transfer of Nanofluid’s Natural Convection

The concept of conversion factors on heat transfer of nanofluid’s natural convection are proposed. The conversion factors on heat transfer coefficient, and local and total heat transfer rates of nanofluid’s natural convection are equal, and proportional to the thermal conductivity ratio at wall temperature, Grashof number ratio and wall temperature gradient ratio of the nanofluid’s natural convection, respectively. Through further formula analysis, the predictive formulae of the wall temperature gradient ratio at wall temperature, Grashof number ratio and wall temperature gradient ratio are respectively obtained. Then, the predictive formulae of the conversion factors on heat transfer coefficient, local and total heat transfer rates, as well as Nusselt number are developed for Al2O3-water nanofluid’s natural convection. These predictive formulae are actually based on comprehensive consideration of nanofluid’s variable thermophysical properties, nanoparticle’s shape factor and concentration, and fluid’s boundary temperatures, they have reliable theoretical and practical value for heat transfer application.
De-Yi Shang, Liang-Cai Zhong

### Chapter 12. Numerical Simulation of Conversion Factors on Heat Transfer

The predictive formulae of conversion factor on heat transfer of nanofluid’s natural convection with the predictive formulae of nanofluid’s thermal conductivity ratio, Grashof number ratio, and wall temperature gradient ratio obtained in the previous chapter are applied to explore the variation of the conversion factor on heat transfer of nanofluid’s natural convection. It is seen that the conversion factor on heat transfer is proportional to nanofluid’s thermal conductivity ratio, fourth power of local Grashof number ratio, and wall temperature gradient ratio. The nanofluid’s thermal conductivity ratio increases with increasing the nanoparticle’s shape factor and concentration, and leads to increase of the conversion factor on heat transfer. The local Grashof number of nanofluid’s natural convection depends on the nanoparticle’s concentration and the fluid’s bulk temperature, and decreases with increase of the nanoparticle’s concentration and the fluid’s bulk temperature. It will lead to a decrease of the conversion factor on heat transfer of nanofluid’s natural convection. The wall temperature gradient of nanofluid’s natural convection depends on the nanoparticle’s concentration and the fluid’s bulk temperature, and decreases with increase of the nanofluid’s concentration and fluid’s bulk temperature. It will lead to a decrease of conversion factor on heat transfer of nanofluid’s natural convection. The coupled effect of the nanofluid’s thermal conductivity, Grashof number and wall temperature gradient ratios is attributed to those of the nanoparticle’s shape factor and concentration, and the fluid’s bulk temperature. The conversion factor on heat transfer will increase with increasing the nanoparticle’s shape factor and concentration. The effect of the fluid’s bulk temperature on the conversion factor on heat transfer is smaller than that of the nanoparticle’s shape factor and concentration. In order to increase the heat transfer coefficient, to increase the nanoparticle’s shape factor and concentration will be an importance choice, and especially, to increase the nanoparticle’s shape factor is the first choice. However, If the nanoparticle’s concentration increases to reach a limitation, the nanoparticles will be accumulated in the fluids. Such critical limitation of nanoparticle’s concentration need to be determined by experiment.
De-Yi Shang, Liang-Cai Zhong

### Chapter 13. Conversion Formulae on Heat Transfer of Al2O3-Water Nanofluid’s Natural Convection

Based on the predictive formulae of the conversion factor on heat transfer obtained in Chap. 9, the related conversion formulae are developed for evaluation of heat transfer coefficient and rate of Al2O3-water nanofluid’s natural convection. Then, the calculation of heat transfer of Al2O3-water nanofluid’s natural convection can be realized conveniently by using the predictive formula of the base fluid’s natural convection. The present predictive formulae are developed based on comprehensive consideration of effects of nanofluid’s variable thermophysical properties, nanoparticle’s shape factor and concentration, and fluid boundary temperatures, thus, have reliable theoretical and practical value for heat transfer application of Al2O3-water nanofluid’s natural convection.
De-Yi Shang, Liang-Cai Zhong

### Chapter 14. Calculation Examples on Heat Transfer by Using Conversion Formulae

The predictive and conversion formulae on heat transfer respectively reported in Chaps. 9 and 13 are innovative achievement of the theoretical research of Al2O3-water natural convection. Their theoretical and practical value lies in that they are developed by theoretical analysis and numerical calculation based on comprehensive consideration of effects of nanofluid’s variable thermophysical properties, nanoparticle’s shape factor and concentration, and fluid’s boundary temperatures. These calculation examples show that the heat transfer increases with increase of fluid’s average temperature of nanofluid’s natural convection, due to increasing nanofluid’s thermal conductivity level. If the fluid boundary temperatures are interchanged, the heat transfer coefficient will be changed, and the higher the fluid bulk temperature, the larger the heat transfer rate. It demonstrates the theoretical flaws and practical inaccuracies of Boussinesq approximation, and it is important to comprehensively consider the effects of fluid’s variable thermophysical properties in the study of heat transfer of natural convection. All these show the charm of the present formulae on heat transfer application. Of course, these developed formulae on heat transfer prediction need to be verified by the experiment. There are two different groups of the predictive formulae on heat transfer of Al2O3-water nanofluid’s natural convection. Group 1 is for predictive formulae and group 2 is for conversion formulae on heat transfer prediction. Since the formulae of group 2 is developed based on those of group 1, the calculation results of group 1 are coincident to those of group 2. However, the conversion formulae in group 2 are more convenient to be used than those in group 1 for heat transfer prediction of Al2O3-water nanofluid’s natural convection.
De-Yi Shang, Liang-Cai Zhong

### Chapter 15. Postscript

This book explored calculation approach on heat transfer of nanofluid’s laminar natural convection, and proposed two groups of formulae, the predictive and conversion formulae of heat transfer coefficient. Such two groups of different formulae on heat transfer coefficient are consistent in predictive results. Especially, these formulae are developed based on comprehensive consideration of effects of nanofluid’s variable physical variables, variation of nanoparticle’s shape factor and concentration, and variation of fluid’s boundary temperatures, which ensure their theoretical and practical significance for heat transfer application. In the sense of heat transfer, this is a successful study.
De-Yi Shang, Liang-Cai Zhong