Boundary layerMotion of particles in fluid (single particles)

1. It is related to fluid flow on flat plates1. It is related to fluid flow around spheres

2. Only friction force is working on plates. So, drag force is identical to friction force.2. Friction force and distribution of pressure enclose the spheres. So, drag force = friction force and pressure difference.

3. Laminar boundary layer: Re < 3 x 10^6Re = . U. x/.Reynolds number is based on incident fluid velocity parallel the plate.3a. Laminar boundary layer* Laminar flow (behind the sphere): Re' < 0.2* Turbulent flow (behind the sphere): Re' > 500-1000 to 2 x 10^5.* Transition flow: Re': 0.2 to 500-10003b. Turbulent boundary layerRe' > 2 x 10^5Re' = . U. dp/Reynolds number is based on incident fluid velocity whose magnitude not affected by particle shape.

4. Mass loss of fluid from main flow into the boundary layer is calculated using *.* = mass loss/(.U.w)4. Force balance for particles in terminal velocity:Drag force on particle surface = gravity force of particles - bouyancy force.Ro'. 1/4.. dp^2 = 1/6. .. dp^3 (p- )g.Ro' = drag force on particle surface/cross-section area of particle.

5. Momentum loss due to friction is calculated using = momentum loss/(.U2.w)5. Centrifugal separation:Re' = . dr/dt. dp/At terminal velocity: dr/dt = uo x r2/g. uo is the terminal velocity. So particle velocity depends on radius.

6. Boundary conditions in boundary layera. First derivative u/y is zero at y=b. Second derivatives 2u/y2 are zero at y= 0 and y=

ba6. Curve a is used to get terminal velocity uo from properties of fluid and particles, curve b to get particle diameter if uo is known.

Fixed bedFluidised bed

1. Force balance:Ac.e.(-P) = all friction forces working on particles.Ac.e.(-P) = R1. S.l.(1-e).AcR1 is the friction force on particle/particle surface area. Friction may occur between fluid-particles and particles-particles. Ac.e. is average fraction of column cross-sectional area occupied by fluid. Consequently, superficial velocity uc = u1/e (Dupuit theorem). Therefore Ac.(1-e) represents average fraction of column cross-sectional area occupied by particles.1. Force balance:A.(-P) = gravity force of particles - bouyancy force. The value of -P is almost constant at all porosity values during fluidisation.

2. Re1 = u1..dm'/ = uc./(S.(1-e).)Reynolds number is based on average velocity of fluid flowing through pores u1 and average diameter of interstitial area between particles dm'.a. Laminar flow: Re1< 2.b. Transition flow: 2 < Re1 < 100.c. Turbulent flow: Re1>100Carman Kozeny equation is for laminar flow.Ergun equation is for all types of flow (laminar, transition and turbulent flows).2. Re' = uc.dp./Reynolds number is based on superficial velocity uc.a. Laminar flow: Re' < 0.2b. Transition flow: 0.2 < Re' < 500-1000.c. Turbulent flow: Re' > 500-1000Limits of flows similar to those in single particles (see Motion of Particles in Fluid).

3. Relative velocity is the velocity felt by particles. So, in the case of fixed bed, u1 represents the relative velocity.If the particle velocity is us, the relative velocity is ur, and a. particles and fluid move at opposite direction, magnitude of fluid velocity close to particles uf = ur - usb. particles and fluid move at the same direction, magnitude of fluid velocity close to particles uf = ur + us3. At incipient fluidisation, fluidised bed can be treated as fixed bed. Therefore, it may use Carman-Kozeny or Ergun equations.In this case, the validity of the use of either of these equations is checked by calculating Re1 in fixed bed.

4. Porosity calculated by this stack of particles above where e = 0.48 is not real. The real porosity is usually less < 0.48. Don't use this if a fixed bed problem allows to calculate the real porosity.4. At the end of fluidisation, transport of particles occurs and particles are treated as single particles (see Motion of Particles in Fluid).

5. In fluidisation where porosity and uc are not known, the concurrent use of Carman-Kozeny or Ergun equations (depends on Re') and Richardson-Zaki equation can obtain the values of these variables.