이 동 현 상 (Transport phenomena) 2009 년 숭실대학교 환경화학공학과

3.1. FLOW PAST IMMERSED OBJECTS AND PACKED AND FLUIDIXED BEDS 3.1A. Definition of Drag coefficient for Flow Past Immersed Objects skin drag (wall drag): form drag: stagnation point: Chap. 3 Principles of Momentum Transfer and Applications Stream line

For flow past immersed objects drag coefficient C D : the ratio of the total drag force per unit area to  v 0 2 /2 A p : p.123 for a sphere, A p =  D p 2 /4 for a cylinder, A p = LD p total drag force F D Reynolds number N Re DpDp L not a pipe diameter!

3.1B. Flow Past Sphere, Long Cylinder, and Disk in the laminar region for N Re < 1.0 (Stokes’ law) Ex 3.1-1

3.5. NON-NEWTONIAN FLUIDS 3.5A. Types of Non-Newtonian Fluids For Newtonian fluids and  =const. a plot of  vs. (-dv/dr): straight line (linear through the origin) The slope is . For Non-Newtonian fluids a plot of  vs. (-dv/dr): not linear through the origin - time-independent fluids: major - time-dependent fluids - viscoelastic fluids

3.5B. Time-Independent Fluids 1. Pseudoplastic fluids - majority of non-Newtonian fluids K: n: 2. Dilatant fluids 3. Bingham plastic fluids - a plot of  vs. (-dv/dr): linear but not through the origin - yield stress  0 : to initiate flow (n < 1) Ostwald-de Waele equation (n > 1) * n = 1  Newtonian fluid

3.5H. Velocity Profiles for Non-Newtonian Fluids 1. Pseudoplastic fluids and dilatant fluids

2. Bingham plastic fluids

3.6. DIFFERENTIAL EQUATION OF CONTINUITY 3.6A. Introduction - Now, we use a differential element for a control volume.  differential balance - equation of continuity: differential equation for the conservation of mass 3.6B. Types of Time Derivatives and Vector Notation 1. Partial time derivative of  2. Total time derivative of  3. Substantial time derivative of 

6. Differential operations gradient divergence Laplacian others

3.6C. Differential Equation of Continuity a mass balance through a stationary element volume  x  y  z (rate of mass acc.) = (rate of mass in) – (rate of mass out) (Equation of Continuity)

Incompressible fluids  constant density or (Equation of Continuity)

3.7. DIFFERENTIAL EQUATIONS OF MOMENTUM TRANSFER OR MOTION 3.7A. Derivation of Equations of Momentum Transfer - equation of motion: differential equation for the conservation of momentum p 참조 (Equation of motion) (x component) =

3.7B. Equations of Motion for Newtonian Fluids with Varying Density and Viscosity - for Newtonian fluids in rectangular coordinates (x component of equation of motion for varying density and viscosity)

3.7B. Equations of Motion for Newtonian Fluids with Constant Density and Viscosity - for Newtonian fluids in rectangular coordinates constant  and  (x component of Navier-Stokes equation) = (continuity equation)

3.8 USE OF DIFFERENTIAL EQUATIONS OF CONTINUITY AND MOTION 3.8A. Introduction The purpose and uses of the differential equations of motion and continuity - to apply these equations to any viscous-flow problem The strategy to solve a given specific problem (1) Simplification - to discard the terms that are zero or near zero in the equation of continuity and motion (2) Integration with boundary conditions and/or initial conditions - boundary conditions: at wall (ex: no slip  v=0) at center (ex: symmetry  dv/dx=0) - initial conditions: at the beginning *steady-state 

3.8B. Differential Equations of Continuity and Motion for Flow between Parallel Plates Ex C. Differential Equations of Continuity and Motion in Stationary and Rotating Cylinders Ex (=pp ) Ex 3.8-4

3.10. BOUNDARY-LAYER FLOW AND TURBULENCE 3.10A. Boundary Layer Flow - boundary layer: for laminar flow x: v  : Chap. 3 Principles of Momentum Transfer and Applications N Re < 2  10 5 : laminar N Re > 3  10 6 : turbulent

3.10. BOUNDARY-LAYER FLOW AND TURBULENCE 3.10B. Boundary-Layer Separation and Formation of Wakes - wake: