Fluid Particle Interactions Basics Outline u 1. Single sphere: Stokes and turbulent drag 2. Many spheres: Stokes flow 3. Many spheres: intermediate Re 4. Many spheres: terminal velocity?
1. Single sphere: Stokes and turbulent drag u
1. Single sphere: Stokes and turbulent drag Stokes drag (Re 0): Turbulent drag (Re > 10 3 ): General form of drag relation for a single sphere: with drag coefficient Re 0 Re > 10 3
1. Single sphere: drag coefficient
1. Single sphere: drag coefficient Re < 10-1 : Re > 10 3 : General Re: Most simple expression Dallavalle (1948) Turton & Levenspiel (1986)
1. Single sphere: drag coefficient
1. Single sphere: terminal velocity First application: terminal velocity v of a sphere Note: with For which v do the forces exactly balance? Solve: for Re
1. Single sphere: terminal velocity Solve: 1. Re < 10-1 : 2. Re > 10 3 : 3. General Re: Solve numerically
1. Single sphere: Galileo experiment Second application: Galileo Galilei, dropping a large and small iron ball from the tower of Pisa. Common conception: Balls hit the ground at the same time (= true without air effect) Actual observation of Galileo: when the larger has reached the ground, the other is short of it by two fingerbreadths
1. Single sphere: Galileo experiment Equation of motion for the sphere: Solving the differential equation for d = 0.22 m and d = 0.05 m gives that when the larger has reached the ground, the other is short of it by 96 cm!
2. Many spheres: Stokes flow u Dimensioneless drag force
2. Many spheres: Stokes flow What is known from theory? Kim & Russel, 1985: Accurate up to = 0.1 Brinkman, 1947: Diverges for = 0.667
2. Many spheres: Stokes flow
2. Many spheres: Stokes flow Pragmatic approach: make link with pipe flow Laminair flow through circular pipe, with pressure gradient L R P 1 P 2 volume of fluid wet surface (exact result) Laminair flow through arbitrary shape pipe L P 1 R P 2 Can be well described by the above expression
2. Many spheres: Stokes flow Flow through a network of pipes: u X k: kozeny constant Experiments for the pressure drop over a wide range of porous media shows that k = 5, independent of the type of medium: Darcy s law:
2. Many spheres: Stokes flow Flow through an array of spheres: u Carman-Kozeny equation Relation to F? volume of fluid wet surface
2. Many spheres: Stokes flow Relation pressure drop to drag force F d that fluid exerts on a particle Total force that fluid exerts on a particle: u P 1 P 2 Force that each particle exerts on the fluid: L S Steady fluid flow: total force on fluid is zero
2. Many spheres: Stokes flow Carman-Kozeny:
2. Many spheres: Stokes flow Kim & Russel, 1985: Accurate up to = 0.1 Brinkman, 1947: Diverges for = 0.667 Carman-Kozeny, 1937: Does not approach 1 for Van der Hoef, Beetstra & Kuipers, 2005:
2. Many spheres: Stokes flow
3. Many spheres: intermediate Re Dimensioneless drag force u
3. Many spheres: intermediate Re What is known from theory? Kaneda, 1986: Accurate for Re < < 0.01 Pragmatic approach: connection with pipe flow Pressure drop for turbulent flow through circular pipe For bed of spheres: Burke- Plummer eq.
3. Many spheres: intermediate Re Limit of low Re: Carman-Kozeny Re > 4000: Burke-Plummer: For general Re: try equation of the form Fit to 640 experimental data points: A = 150, B = 1.75 Ergun equation
3. Many spheres: intermediate Re
3. Many spheres: intermediate Re
3. Many spheres: overview Expressions for the dimensionless drag force Stokes flow in the limit Drag for single particle at finite Re given by, and Drag for low Re flow through dense random arrays Carman Ergun equation from pressure drop data General form of Ergun type equations General form of Wen & Yu type equations
4. Many spheres: terminal velocity Experimentally Richardson-Zaki exponent Steady state: force balance for one particle: (mass displaced suspension) (mass particle)
4. Many spheres: terminal velocity Note that the terminal velocity is a function of
4. Many spheres: terminal velocity Assuming a dimensioneless drag force of the form gives: Terminal velocity experiments are fitted to the form: Hence the exponents are related as:
3. Many spheres: terminal velocity Experimental results for the exponent n : Wen & Yu (1966): over entire Re range Di Felice (1994):
4. Many spheres: overview Expressions for the dimensionless drag force Stokes flow in the limit Drag for single particle at finite Re given by, and Drag for low Re flow through dense random arrays Carman Ergun equation from pressure drop data Wen & Yu equation from terminal velocity data
3. Many spheres: overview Expressions for the dimensionless drag force Stokes flow in the limit Drag for single particle at finite Re given by, and Drag for low Re flow through dense random arrays Carman Ergun equation from pressure drop data General form of Ergun type equations General form of Wen & Yu type equations
4. Many spheres: terminal velocity Experimentally Richardson-Zaki exponent Steady state: force balance for one particle: (mass displaced suspension) (mass particle)
4. Many spheres: terminal velocity Note that the terminal velocity is a function of
4. Many spheres: terminal velocity Assuming a dimensioneless drag force of the form gives: Terminal velocity experiments are fitted to the form: Hence the exponents are related as:
3. Many spheres: terminal velocity Experimental results for the exponent n : Wen & Yu (1966): over entire Re range Di Felice (1994):
4. Many spheres: overview Expressions for the dimensionless drag force Stokes flow in the limit Drag for single particle at finite Re given by, and Drag for low Re flow through dense random arrays Carman Ergun equation from pressure drop data Wen & Yu equation from terminal velocity data