Algebraically Explicit Analytical Solutions Of Energy Utilization

Analytical solutions have irreplaceable theoretical vaule. Many analytical solutions played key roles in the early development of fluid mechanics and heat conduction. Besides their theoretical meaning, analytical solutions can also be applied to check the accuracy, convergence and effectiveness of various numerical computation methods and to improve their differencing schemes, grid generation ways and so on. This is due to they can express the detailed analytical situation under certain initial and boundary conditions. The analytical solutions are therefore very useful even for the newly rapidly developing computational fluid dynamics and heat transfer. Especially, algebraically explicit analytical solutions are more suitable for theoretical research or to be used as benchmark solutions to check numerical calculations. Nevertheless, there have been few algebraically explicit analytical solutions reported in open literature up to now, due to mathematical difficulties. Supported by the National Natural Science Foundation of China (No.50246003, No.50576097) and the Major State Basic Research Development Program of China (No.G20000263), the dissertation researches algebraically explicit analytical solutions of energy utilization, concerning fields such as heat conduction, convection, mass transfer, non-Newtonian fluid flow and so on. The main contents are as follows:Hyperbolic (thermal wave) equation has been widely concerned in the research community as a typical non-Fourier heat conduction model. Some analytical solutions with arbitrary functions for 2-D and 3-D hyperbolic heat conduction equation are presented. Especially, for 2-D case, 8 arbitrary functions are included in the final solution. For 3-D case, infinite arbitrary functions are included in the final solution. Some special boundary and initial conditions are discussed. It is pointed out that when the internal heat source attenuates with special manner, it would have no effect on the temperature distribution. This is a special feature of the hyperbolic heat conduction equation.Chen-Holmes equation may be the optimum perfusion heat transfer model up to now. In order to further expand the understanding of this model and to enrich the theoretical research, an analytical general solution under certain kind of relationship between thermal parameters is presented for the non-Fourier Chen-Holmes model. It is possible to get special solutions with heat wave effect from it. The physiology meaning of the heat wave solutions is that for biotissue with high blood perfusion rate such as tumour and cerebrum, because of non-Fourier effect, the temperature may fluctuate around equilibrium point.Pennes equation is the most widely used bioheat transfer model nowadays. An analytical general solution under certain kind of relationship between thermal parameters is presented for the non-Fourier Pennes model based on the abovementioned general solution of Chen-Holmes equation. The solution depicts the temperature distribution in biotissue medium for any boundary conditions and initial conditions discovered by the non-Fourier Pennes equation.The porous medium Wang model is a representative bioheat transfer model. It is more general since its derivation has no relationship with Darcy Law and its application is not confined to Newtonian fluid. This dissertation presents an analytical general solution for the non-Fourier Wang equation with special internal heat production to further expand the understanding of the highly complex bioheat transfer phenomena. The solution depicts the temperature distribution in biotissue medium for any boundary conditions and initial conditions discovered by the non-Fourier Wang equation with special internal heat production.When the thermal conductivity and volumetric specific heat are functions of temperature, the governing equation of non-Fourier heat conduction is nonlinear. Therefore it is difficult to find its analytical solutions. Several algebraically explicit analytical solutions of nonlinear non-Fourier heat conduction are derived, both to develop theory and to serve as benchmark solutions for numerical calculations.Brinkman model is one of the improvements of Darcy model. It can reflect some anisotropic and non-Darcy effects (such as the no slip boundary effect). For the sake of more strictly and accurately studying the regularity reflected by Brinkman model, some algebraically explicit analytical solutions of the governing equation set are derived and presented. The physical feature of the first solution is a uniform temperature Brinkman flow in a channel between two infinite long solid walls parallel with y coordinate. The second solution represents a Brinkman natural convection flow in porous media between two infinite long permeable walls parallel to x coordinate. The third solution can represent a Brinkman natural convection in porous media between two infinite long solid walls parallel to y coordinate, and the temperature distribution is linear.Two algebraically explicit analytical solutions of axisymmetric steady laminar natural convection are derived to develop the theoretical understanding. The first analytical solution represents the natural convection in a semi-infinite space with boundary suction along an infinitely long vertical cold moving down porous tube. The second solution describes the natural convection between two infinite concentric vertical solid tubes.A thermodynamically strict discussion is given about the convection. It is pointed out that the convection is not a rigorous kind of "heat" transfer but mainly internal energy transfer by the movement of particles. Based on the discussion of convection, the concept of field synergy—the best convection "heat" transfer is the case where the velocity vectors are always perpendicular to the isothermal surfaces—is easy to understand. For further theoretically developing the field synergy principle and researching the artificial measures to accomplish field synergy, different kinds of algebraically explicit analytical solutions are derived and given, including solutions with heat source, with mass source, full field synergy solutions and boundary field synergy solutions.Convective heat transfer enhancement is one of the research hot spots nowadays in the research community. Some analytical solutions for 2-D convective heat transfer between two parallel penetrable walls are presented and analysed using field synergy principle in this study. These results are valuable to inspire the methods to improve or weaken field synergy in practice. The influences of some factors on heat transfer and field synergy number are also discussed. It is demonstrated that the field synergy degree might have different influences on the heat transfer conditions of different walls for this kind of flow. It is also pointed out that the local field synergy degree might be more meaningful than the field synergy degree in the whole domain in some cases.In regard to the governing equations, unsteady convection problems are more complex than steady ones. Generally speaking, steady convection problems can be regarded as simple special cases of unsteady ones. Some analytical solutions for unsteady 2-D convective heat transfer between two parallel walls are presented. And also some meaningful conclusions can be drawn form them. For example, the first solution demonstrates that the unsteady convective heat transfer could degenerate into an unsteady heat conduction problem under some conditions. Then the fluid flow would contribute nothing to the heat transfer. The third solution demonstrates that the unsteady fluid motion could be utilized to weaken the heat transfer.Two algebraically explicit analytical solutions of the equation set describing non-Fourier and non-Fick heat and mass transfer in capillary porous media are reported. These analytical solutions are useful to deepen the understanding of non-Fourier and non-Fick heat and mass transfer in the rapid drying process of porous media. They can also be applied as benchmark solutions to check the numerical computation results. Therefore, these analytical solutions are of high value.The governing equation set for the double diffusive convection is rather complicated. It is a nonlinear mathematical 3-D equation set. Two algebraically explicit analytical solutions for the double diffusive convection are derived. One is for the double diffusive convection in an infinite long cylindrical tube. Another one is for the double diffusive convection in an infinite long circular tube with porous wall. They are meaningful for the theory of heat and mass transfer.There are many different kinds of rheological complicated fluids with non-Newtonian constitutive equations. The understanding of the flow phenomena of these fluids is helpful for many disciplines. Two algebraically explicit analytical solutions are derived for the incompressible unsteady rotational flow of a rheologic flow of Oldroyd-B type in an annular pipe. Even for complicated rheologic governing equation, the solutions are very simple. In addition, a similar solution is also derived for the generalized second grade rheologic fluid flow.