Advanced Topics in Heat, Momentum and Mass Transfer

1. Advanced Topics in Heat, Momentum and Mass Transfer Lecturer Payman Jalali, Docent Faculty of Technology Dept. Energy & Environmental Technology Lappeenranta University of Technology

2. What are the approaches for an engineer or scientist making a research or solving a problem? • Approach: Method of solution, method of scientific working. • There are 3 major approaches for any scientific or technological problem, as follows: • Experimental approach: The problem under consideration is totally analyzed in experimental facilities of the laboratory. • Analytical approach: The problem is modeled theoretically (formulated mathematically) and solved with a number of simplifications. • Computational approach: The problem is modeled theoretically but it is solved with no (or little) simplifications. • Computational approach solves the governing equations of physical phenomena accurately using computers. If the computational approach is used in fluid dynamics problems, it is called ’computational fluid dynamics (CFD)’. • The next slide simply draws the differences between the three approaches!

3. Experimentalist Theoretician Computerist Problem: Fluid flow and drag around a cylinder

4. What are advanced topics in heat, momentum and mass transfer? • They are topics related to some important phenomena such as diffusion, convection, radiation and advanced computational methods to deal with these phenomena in fluid mechanics and heat transfer. • The following tasks will be fulfilled in this course: • Review governing equations for the transport of mass, momentum and energy in fluids. • Numerical study of diffusion problems using CFD. • Investigate how convection will change the domain created by diffusion. • How can we transfer partial differential equations (PDE) into algebraic equations needed in CFD? • Developing codes in MATLAB to solve diffusion problems. • Using commercial software (FLUENT) to solve complex problems in fluid and heat flows.

5. C0 T0 U0 Gas (low diffusivity) = (weak transport of mass) Non-metalic (low conductivity) = (weak transport of heat) Liquid (low viscosity) = (weak transport of momentum) C0 T0 U0 Liquid (high diffusivity) = (strong transport of mass) Metal (very high conductivity) = (strong transport of heat) Solid (very high viscosity) = (strong transport of momentum) • What is transport phenomenon? • Transport phenomena are dealing with all physical processes which cause the movement or transportation of mass, momentum and thermal energy (heat). • Transport properties of substances are different and they are characterized by the coefficient of viscosity (for momentum), conductivity (for thermal energy), and diffusivity (for mass concentration).

6. What is ’diffusion’ and how related to transport phenomena? • ’Diffusion’ is the natural propagation mechanism of some physical quantities such as mass, thermal energy and momentum through a medium of certain state. • Diffusion of mass is driven due to concentration gradient  mixing of different species, or if there is 1 component it is called self-diffusion. • Diffusion of thermal energy (heat) is driven by temperature gradient  heat conduction. • Diffusion of momentum is caused by velocity gradient  shearing fluids. • The physics of diffusion can be associated with either random molecular motions or combined effects (such as molecular motion and turbulence etc.). Mathematically, the diffusion phenomenon is expressed as following: If we are talking about thermal energy (Fourier’s law): Heat flux (q”) = Thermal conductivity(k)x temperature gradient (dT/dx) In case of mass diffusion (Fick’s law): Heat flux (m”) = Thermal conductivity(D)x temperature gradient (dC/dx)

7. (I) and (II) • What is the mathematical equation for diffusion? • Consider that a species (for example O2 in air) with concentration C is distributed along x axis at time t. Writing the mass balance equation for this species gives: species (I) (II) This is diffusion equation when the diffusivity is taken a constant. Its extension to 3D will be as follows:

8. r R Flow direction Injection of dye Growing the width of the dye through advancing sections • Examples of diffusion problems • a) Transient diffusion of a dye in a medium. The growth of the width of the dye is a diffusion-type process, which can be formulated as: Here, C is the concentration of the dye, Dr is the radial diffusivity and r is radial position. For large enough values of R we can assume the following boundary conditions and initial conditions: The initial condition:C0 is the initial concentration at the origin x=0. Boundary condition 1: Symmetry boundary condition at the origin. Boundary condition 2: Zero concentration at infinity.

9. C The solution of diffusion equation under mentioned initial and boundary conditions is found as:

10. z Ct C0 b) Transient diffusion from a source to semi-infinite medium. We can solve diffusion equation analytically with the given boundary conditions as follows:

11. Error function The mass flux can be found by the Fick’s law:

12. Time progress Time progress Graphical plots of the above-mentioned solution is shown below: Concentration distribution in time Flux distribution in time

13. Why do we need computational fluid dynamics (CFD)? • The two examples given above showed that exact mathematical (analytical) solutions to governing equations of fluid mechanics may be too difficult. Therefore, computational approach can be the practical method to solve equations in fluid mechanics such as the diffusion equation in complex geometries under various boundary conditions. • Major steps in any CFD calculations: • a) Domain discretization: Create nodes and elements (small control volumes) in the domain of solution. It is also called mesh generation which sometimes obeys complicated mathematical processes and calculations. • b) Equations discretizations: The governing equations of fluid dynamics are algebraically discretized. It means that the governing equations which are partial differential equations (PDE) are written for the meshes (elements) produced from step a. Then we get a system of algebraic equations whose unknowns correspond to the nodes (elements). • c) Solution: Discretized algebraic equations are solved with considerations of initial and boundary conditions. • d) Postprocessing: The solution must be processed, visualized and interpreted.

14. How can we use CFD in practice? • a) Developing own codes: A CFD work can be done independently by performing all the above-mentioned steps in a series of codes developed by a group of researchers. Depending on the level of complexity of the problem, the software for programming and postprocesses can vary. In rather simple problems, MATLAB is an appropriate tool for solving and postprocessing in a CFD problem. For more complex and large systems, one must develop the codes in one of the languages such as FORTRAN and C++. • b) Commercial software: Various computer packages are developed for handling professional problems in CFD. For instance, FLUENT, FIDAP, CFX, Finflo are some examples of such CFD software packages. The package for mesh generation is usually given separately, for example, GAMBIT is one of the packages used for creating the geometry and discretizing it into various types of elements appropriate for the methods employed by the CFD packages.

15. REFERENCES • - J.D. Anderson, Computational Fluid Dynamics, McGraw-Hill, Inc. 1995. • - D.A. Anderson, J.C. Tannehill, R.H. Pletcher, Computational Fluid Mechanics and Heat Transfer, McGraw-Hill, Inc. 1984. • - J.H. Ferziger, M. Peric, Computational Methods for Fluid Dynamics, Springer-Verlag 1996. • - C. Hirsch, Numerical Computation of Internal and External Flows, Volume 1: Fundamentals of Numerical Discretization, John Wiley & Sons, 1988 • - J.M. Haile, Molecular Dynamics Simulation: Elementary Methods, John Wiley & Sons, Inc., 1992. • - R.B. Bird, W.E. Stewart, E.N. Lightfoot, Transport Phenomena, John Wiley & Sons, Inc.,1960. • - J. Crank, The Mathematics of Diffusion, Oxford University Press. 1975. • - MATLAB user manual. • - FLUENT user manual.