Etiket Arşivleri: FE 222

Flow Passed Immersed Objects

the flow of fluids over bodies that are immersed in a fluid, called external flow, with emphasis on the resulting lift and drag forces.
Many applications in food and chemical engineering
-flow passed spheres (settling)
-packed beds( drying, fluidising, filtration)
-flow passed tubes( heat exchangers)
Sometimes a fluid moves over a stationary body (such as the wind blowing over a building), and other times a body moves through a quiescent fluid (such as a car moving through air).
These processes are equivalent to each other; what matters is the relative motion between the fluid and the body.
Drag Force on Solid Particles in Fluids
Force in the direction of flow exerted by the fluid on the solid is called drag.
Due to turbulence, the pressure on the downstream side of the sphere will never fully recovered to that on the upstream side, and there will be a form drag( pressure drag) to the right of the sphere. (For purely laminar flow, the pressure recovery is complete, and the form drag is zero.)
In addition, because of the velocity gradients that exist near the sphere, there will also be a net viscous drag (also called as skin drag, wall drag, friction drag) to the right (In potential flow there is no wall drag). The sum of these two effects is known as the (total) drag force.
When a fluid moves over a solid body, it exerts pressure forces normal to the surface and shear forces parallel to the surface along the outer surface of the body.
The component of the resultant pressure and shear forces that acts in the flow direction is called the drag force, and the component that acts normal to the flow direction is called the lift.
The friction drag coefficient is analogous to the friction factor in pipe flow discussed.
The pressure drag is proportional to the frontal area and to the difference between the pressures acting on the front and back of the immersed body.
Therefore, the pressure drag is usually dominant for blunt bodies, negligible for streamlined bodies
The experimental results of the drag on a smooth sphere may be correlated in terms of two dimensionless groups – the drag coefficient CD and particles Reynolds number NRe:
Creeping Flow, Stokes Law, Settling of Particles
consider a spherical solid particle settling in a large volume of fluid.
FZ=FD+FB (buoyancy force)
FD=CDЛR2 (1/2)ρv02 FB=4/3ЛR3ρg
For NRE<1 FD=6Лμv0 R
So terminal velocity v0 =2/9* R2 (ρp-ρ)ρ/μ
Settling time t=h/v0
Lift Generated by Spinning
Golf, soccer, and baseball players utilize spin in their games. The phenomenon of producing lift by the rotation of a solid body is called the Magnus effect after the German scientist Heinrich Magnus (1802–1870), when the cylinder is rotated about its axis, the cylinder drags some fluid around because of the no-slip condition.
The average pressure on the upper half is less than the average pressure at the lower half because of the Bernoulli effect, and thus there is a net upward force (lift) acting on the cylinder.
Hindered settling
If large number of particles exist, suraounding particles interfere with the individual motion of particles.
Due to Upward movement of surrounding fluid so relative velocity decreases.
Settling will be slower than one calculated from Stokes law. (true drag force is greater)
μm= effective viscosity ,μ=actual viscosity
μm= μ/ψP where ψP =1/101.82(1-ε)
where ε=volume fr. of liq in slurry
ρm= ερ+(1-ε)ρP ρm: slurry density

v0 =2/9* (R2 (ρp-ρ)ρm/μ)* (ε2 ψP )
correction factor
Packed Beds
Consider a porous medium consisting of sand or some porous rock or glass beads or cotton cloth contained in a pipe.
At any one cross section perpendicular to the flow, the average velocity may be based on the entire cross sectional area of pipe, in which case it is called the superficial velocity Vs
Or it may be based on the area actually open to the flowing fluid, in which case it is called the interstitial velocity VI
Non-Spherical Particles
For non-spherical particles: an equivalent diameter is defined.
The equiv. D of a non-spherical particle is defined as a sphere having the same volume as the particle.
Sphericity is the ratio of surface area of this sphere to the actual surface area of particle. (table3.1-1)
The formula for sphericity is reduced to Фs = 6vp/ (DpSp)
Where vp is the volume of particle, Dp is the characteristic dimension of particle, and Sp is the surface area of particle.
For non-spherical particle Ergun equation is given by,

The packings are made with clay, porcelain, plastics or metals. The following table gives the different packing materials and their approximate void fraction.
Requirements of a tower packing are:
It must be chemically inert .
It must be strong without excessive weight.
It must contain adequate passages without excessive pressure drop.
It must provide good contact
It should be reasonable in cost.
Non-Uniform Size Particles
Fluidised Bed
As v increases settling particles will start rising and moving( fluidisation will start, onset of fluidisation) because pressure drop will increase (according to Ergun equation ) and upwards pressure force and buoyancy forces will be larger than gravitational force.
At this point height of particles called Lmf, velocity is called as v’mf,, and porosity is called as εmf.
As the velocity increase further, to be able to keep the pressure drop constant particles will occupy more volume, means L and ε increases untill they occupy whole bed.
Then there is no space to expend further so particles will be entrained on top.
At this point velocity is called as entrainment velocity v’t
LA(1- ε )=total volume of solids L2,ε2 =constant L1,ε1
L1A(1- ε1 )= L2A(1- ε2 )
Combining these equations and replacing Dp with Фs Dp
İf εmf and Фs are not known, as an approximation ( by Wen and Yu)
Фs εmf 3≈1/14 and 1- εmf / Фs2 εmf 3 ≈11
Then above eq becomes
NRe,mf =[(33.7)2 +0.0408Dp3ρ(ρp –ρ)g/μ2]-33.7
Valid for 0.0011000
Advantages and Disadvantages of fluidization:
the solid is vigorously agitated by the fluid passing through the bed
no temperature gradients in the bed even with quite exothermic or endothermic reactions.
Erosion of vessel internals
Attrition of solids. the size of the solid particles is getting reduced and possibility for entrapment of solid particles with the fluid are more.

Non-Newtonian Fluids

Newtonian Behavior
the shear rate is directly proportional to the shear stress, and the plot begins at the origin.
Typical Newtonian foods:
-containing compounds of low molecular weight (e.g., sugars)
-do not contain large concentrations of either dissolved polymers (e.g., pectins, proteins, starches)
-containing insoluble solids.
Examples: water, sugar syrups, most honeys, most carbonated beverages, edible oils, filtered juices, and milk.
t(Shear stress)= -μ (dv/dr) ( shear rate)
μ constant ( independent of shear rate)
All other types of fluid foods are non-Newtonian,
– shear stress-shear rate plot is not linear
-and/or the plot does not begin at the origin
-or exhibits time-dependent rheological behavior
Flow behavior may depend only on shear rate and not on the duration of shear (time independent) or may depend also on the duration of shear (time dependent).
Foods that exhibit time-dependent shear-thinning behavior are said to exhibit thixotropic flow behavior.
Most of the foods that exhibit thixotropic behavior( reversible decrease in shear stress with time at constant shear rate)are heterogeneous systems containing a dispersed phase
( foods such as salad dressings and soft cheeses where the structural adjustments take place in the food due to shear until an equilibrium is reached. )
Time-dependent shear-thickening behavior is called antithixotropic( Formerly, it was called rheopectic) behavior. Reversible increase in shear stress with time at constant shear rate) ( ie gypsum suspension,bentonite suspension)
Time-Independent Behavior
Shear-Thinning Behavior ( pseudoplastic)
t = K.(-dv/dr)n Power law eq.
t = K.(-dv/dr)n-1(-dv/dr)=μa(-dv/dr)
μa=apparent viscosity
K=consistency index (
n=flow behavior index, dimensionless
may be due to breakdown of structural units in a food due to the hydrodynamic forces generated during shear.
Most non-Newtonian foods exhibit shear-thinning behavior, including biological fluids, xanthan gum soln, starch suspensions many salad dressings and some concentrated fruit juices.
Shear-Thickening Behavior (Dilatant) n>1
This type of flow has been encountered in partially gelatinized starch dispersions.
shear-thickening should be due to increase in the size of the structural units as a result of shear.
Bingham plastic behavior
The flow of some materials may not commence until a threshold value of stress,
the yield stress (ty), is exceeded
t= -μ.(dv/dr) +ty where ty is yield stress
Shear-thinning with yield stress behavior is exhibited by foods such as tomato concentrates, tomato ketchup, mustard, and mayonnaise.
If the shear rate-shear stress data follow a straight line with a yield stress, the food is said to follow the Bingham plastic model.
Flow behavior of fluids under shear-stress
1-Flow: permenant deformation of fluid
Existance of flow mens: fluid does not recover its originl shape even we remove applied force
2- elastic behavior: all the applied force is stored as bond energy when force is removed it takes its original shape.
If some bonds are broken, so only some bonds are formed to store some of the applied force …. viscoelastic behavior
3-Yield stress(ty): elastic property changed at ty
(when you squeeze tooth paste,it comes out as a cylinder because t is max at wall so first wall side stars flowing)
4- n measures resistance to deformation
n=1 ( Newtonian)…deforms readily under force
As n 0 it becomes elastic( jells shows elastic behavior untill ty, after ty jell breaks down to deform fluid
Complex rheological models
( containing more than 2 parameters)
Herschel-Bulkey Model
t= ty+K(-dv/dr)n (-dv/dr)=γ
μa= t/γ= ty / γ + K.γn-1
as γ increases μa decreases then μ becomes constant
When n =1 behaves like Bingham
When n 0 μa=(ty +K)/ γ so as γ increases μa decreaes
Cassan Model
If μ constant for very low and very high γ values and obeys Power law for intermediate γ values,
Then cassan is a sutable model
(μ)1/2 =(μ∞ )1/2 +(ty/γ)1/2
Where μ∞ is viscosity at very high γ
Example: chocolate, cocoa soln
Viscoelastic fluids: exhibits elastic recovery from the deformation that occur during flow. They show both viscous and elastic properties.
Part of deformation is recovered upon removal of the stress.
Example : flour dough, polymer melts
Commonly capillary tube viscometers are used to determine properties of the fluids.
ΔP for a given flowrate q is measured in a straigt tube length L and diameter D at different q values.
For a power law fluid:
Plot of log (tw) vs log(8V/D) yields:
Slope n’ and intercept K’
For most fluids K’ and n’ are constant over wide range of D.ΔP/4.L or 8V/D For some fluids they will vary
In many cases rotational viscometers are used to determine fluid properties.
When the flow propertis are constant over a Range of shear stresses that ocurs for many fluids following equations hold;
n’=n K’=K((3n’+1)/4n’)n’
Generalised viscosity :γ=K’8n’-1
Table 3.5-1 gives n’, K’ and γ values for some foods.
To calculate pressure drop or mech energy loss due to friction in straight pipe
1- ΔP= (K’4L/D)/(8V/D)n’ ..Ff= ΔP/ρ
2- NRe,Gen for power law fluid
ΔP=4fρ(L/D)(v2/2)…. Ff= ΔP/ρ
Friction loss in contractons, expansions, fittings for non-newtonian fluid, laminar flow
1- kinetic energy correction factor:

2-Losses in contraction and fittings:Calculate α as in 1 and use.

3-losses in sudden expansion
Non-Newtonian turbulent flow
1- kinetic energy correction factor α=1
2-contractions for fittings α=1
3-expansion treat as newtonian
4-straight pipe:
For 0.36<n’<1 and upto NRe =3.5x104and smooth pipe use fig 3.5-3
For non-Newtonian, turbulent and rough tube use Moody with NRe,Gen as an approximation
Emprical testing instruments for foods
Measuring viscosity
Since viscosity is related with molecular momentum transportation, t, measurement of viscosity should be performed under laminar conditions( resistance is mainly due to viscosity)
There are three major group of viscometers
1- Capillary tube viscometer ( poiseuille flow)
Falling sphere (ball)viscometers
Rotational viscometers