Aero-Hydrodynamic Characteristics Asst. Dr. Muanmai Apintanapong

In handling and processing : air and water is used as carrier for transport or separation. Pneumatic separation and conveying has been in use for many years. Fluid flow or fluid mechanics find increasingly wide applications in handling and processing Knowledge of physical properties which affect the aero- and hydrodynamic behavior of agricultural products becomes necessary.

Mechanics of particle motion in fluids To describe, two properties need: Drag coefficient Terminal velocity

Drag Coefficient For particle movement in fluids, drag force is a resistance to its motion. Drag coefficient is a coefficient related to drag force. Overall resistance of fluids act to particle can be described in term of drag force using drag coefficient.

Comparing with fluid flow in pipe principle, drag coefficient is similar to friction coefficient or friction factor (f).

For drag coefficient:

Frictional drag coefficient For flat plate with a laminar boundary layer: For flat plate with a turbulent boundary layer

Frictional drag coefficient For flat plate with a transition region:

If a plate or circular disk is placed normal to the flow, the total drag will contain negligible frictional drag and does not change with Reynolds number (NR)

Sphere object At very low Reynolds number (<0.2), Stoke law is applicable. The inertia forces may be neglected and those of viscosity alone considered.

Terminal or Settling Velocity Settling velocity (vt): the terminal velocity at which a particles falls through a fluid. When a particle is dropped into a column of fluid it immediately accelerates to some velocity and continues falling through the fluid at that velocity (often termed the terminal settling velocity).

The speed of the terminal settling velocity of a particle depends on properties of both the fluid and the particle: Properties of the particle include: The size if the particle (d). The density of the material making up the particle (rp). The shape of the particle.

FG, the force of gravity acting to make the particle settle downward through the fluid. FB, the buoyant force which opposes the gravity force, acting upwards. FD, the “drag force” or “viscous force”, the fluid’s resistance to the particles passage through the fluid; also acting upwards.

Particle Settling Velocity Put particle in a still fluid… what happens? Speed at which particle settles depends on: particle properties: D, ρp, shape fluid properties: ρf, μ, Re Fg Fd FB

STOKES Settling Velocity Assumes: spherical particle (diameter = dP) laminar settling FG depends on the volume and density (rP) of the particle and is given by: FB is equal to the weight of fluid that is displaced by the particle: Where rf is the density of the fluid.

FD is known experimentally to vary with the size of the particle, the viscosity of the fluid and the speed at which the particle is traveling through the fluid. Viscosity is a measure of the fluid’s “resistance” to deformation as the particle passes through it. Where m (the lower case Greek letter mu) is the fluid’s dynamic viscosity and v is the velocity of the particle; 3pd is proportional to the area of the particle’s surface over which viscous resistance acts.

From basic equation, F = mg = resultant force: With v = terminal velocity or vt:

In the case of 0. 0001<NR<0 In the case of 0.0001<NR<0.2, terminal velocity can be determined by using CD =24/NR:

In the case of 0.2<NR<500, terminal velocity can be determined by using CD as:

Example: A spherical quartz particle with a diameter of 0.1 mm falling through still, distilled water at 20°C dP = 0.0001m rP= 2650kg/m3 rf = 998.2kg/m3 m = 1.005 ´ 10-3 Ns/m2 g = 9.806 m/s2 Under these conditions (i.e., with the values listed above) Stoke’s Law reduces to: For a 0.0001 m particle: vt = 8.954 ´ 10-3 m/s or » 9 mm/s

Stoke’s Law has several limitations: i) It applies well only to perfect spheres. The drag force (3pdmvt) is derived experimentally only for spheres. Non-spherical particles will experience a different distribution of viscous drag. ii) It applies only to still water. Settling through turbulent waters will alter the rate at which a particle settles; upward-directed turbulence will decrease vt whereas downward-directed turbulence will increase vt.

iii) It applies to particles 0.1 mm or finer. Coarser particles, with larger settling velocities, experience different forms of drag forces. Stoke’s Law overestimates the settling velocity of quartz density particles larger than 0.1 mm.

When settling velocity is low (d<0 When settling velocity is low (d<0.1mm) flow around the particle as it falls smoothly follows the form of the sphere. When settling velocity is high (d>0.1mm) flow separates from the sphere and a wake of eddies develops in its lee. Drag forces (FD) are only due to the viscosity of the fluid. Pressure forces acting on the sphere vary. Negative pressure in the lee retards the passage of the particle, adding a new resisting force. Stoke’s Law neglects resistance due to pressure.

iv) Settling velocity is temperature dependant because fluid viscosity and density vary with temperature. Temp. m r vt °C Ns/m2 Kg/m3 mm/s 0 1.792 ´ 10-3 999.9 5 100 2.84 ´ 10-4 958.4 30

Grain size is sometimes described as a linear dimension based on Stoke’s Law: Stoke’s Diameter (dS): the diameter of a sphere with a Stoke’s settling velocity equal to that of the particle. Set ds = dP and solve for dP.

Measurement of terminal velocity a) Direct measurement b) Estimating settling velocity based on particle dimensions.

vt a) Direct measurement Settling velocity can be measured using settling tubes: a transparent tube filled with still water. In a very simple settling tube: A particle is allowed to fall from the top of a column of fluid, starting at time t1. The particle accelerates to its terminal velocity and falls over a vertical distance, L, arriving there at a later time, t2. The settling velocity can be determined: vt

A variety of settling tubes have been devised with different means of determining the rate at which particles fall. Some apply to individual particles while others use bulk samples. Important considerations for settling tube design include: i) Tube length: the tube must be long enough so that the length over which the particle initially accelerates is small compared to the total length over which the terminal velocity is measured. Otherwise, settling velocity will be underestimated.

ii) Tube diameter: the diameter of the tube must be at least 5 times the diameter of the largest particle that will be passed through the tube. If the tube is too narrow the particle will be slowed as it settles by the walls of the tube (due to viscous resistance along the wall). iii) In the case of tubes designed to measure bulk samples, sample size must be small enough so that the sample doesn’t settle as a mass of sediment rather than as discrete particles. Large samples also cause the risk of developing turbulence in the column of fluid which will affect the measured settling velocity.

b) Estimating settling velocity based on particle dimensions. Settling velocity can be calculated using a wide variety of formulae that have been developed theoretically and/or experimentally. Stoke’s Law of Settling is a very simple formula to calculate the settling velocity of a sphere of known density, passing through a still fluid. Stoke’s Law is based on a simple balance of forces that act on a particle as it falls through a fluid.

Terminal velocity determination There are two methods for determination of terminal velocity. Terminal velocity from drag coefficient-Reynold’s number relationship. Terminal velocity from time-distance relationship.

Terminal velocity from drag coefficient-Reynold’s number relationship Both CD and NR include a velocity term, calculation of vt from CD and NR relationship require a trial-and-error solution. To eliminate trial-and-error solution, case of vt or dp is unknown, the term CDNR2 or CD/NR are first calculated and plot against NR.

For spherical particles:

CDNR2 is first calculated and plot against NR

CD/NR is first calculated and plot against NR

Terminal velocity from time-distance relationship In free fall of an object in still air, the net force on the object is the difference in the force of gravity (mg) and the resultant frictional or drag force (kv2). This net force is also equal to m(dv/dt) After integration this equation and determine velocity in term of displacement (distance) with time (vt =s/t)

In such cases, from linear portion Slope = s/t =vt Distance (s) Time (t) or calculate vt from equation: