Mechanical Operation


Mechanical Operation
Flow through a packed bed of solids
  • Packed–beds of catalyst–particles are extensively used in chemical industries. Packed beds of solid particles are also used in absorption, adsorption, and distillation columns to increase interfacial area of contact between gas and liquid. In the next couple of lectures, we will address pressure–drop in packed beds of solids. Before that there are some definitions:
a’ = Specific surface area of a particle 
where the subscript ‘sph’ refers to the sphere having the same volume as that of the particle 

Table
Particles
Spheres     1
Cubes, cylinder
Berl saddle0.3
Ranching rings 
a = specific area of the bed  
(Fig. 29a)


Drag: We have already introduced this term in the previous lecture. There are two types of drag: form drag and wall or shear drag. The former is because of the fluid–pressure on the solid–surface and acts perpendicular to the surface wall. Drag is because of shear–forces and acts parallel to the surface. 
(Fig. 29b)
  • horizontal plate parallel to fluid flow will experience drag only because of wall shear. 
form drag 
wall drag
 (already defined in previous lectures) 
 (Stoke’s Law for Rep < 1) 

  •  in general depends upon Reynolds number, but is also dependent on the shape and orientation of the particle with respect to the flow.  Charts are available to determine drag coefficient. 
(Fig. 29b)
  • Terminal or settling velocity of a single particle in fluid may be calculated as 
  • If the fluid moves–up with velocity , velocity of the particle with respect to a stationary observer,
     , so that the relative velocity or drag remains the same.
  •  depends on  which cannot be directly calculated  because the velocity (settling) is not known. Therefore, iteration is required to determine . In the previous lecture, we took-up such an example. However, there is a criterion to check if the settling is in the Stoke’s regime (creeping flow) or in Newton’s region (high flow when inertial effects are important), which is calculated independent of velocity:
Criteria for settling
  (Stoke’s regime)
  • (As an exercise, substitute  in the expression for terminal velocity with  , and use the criterion   to obtain the above expression,   for the Stoke’s regime)
Hindered settling 
I
f there are particles in the fluid, then the settling of a single particle will be influenced by the presence of the neighboring particles. In such case, the settling velocity is larger than that of a single–particle.
  where  is the volume–fractions (not bed–porosity) of fine–suspension of particles in fluid,
The viscosity of a suspension is also affected by the presence of the dispensed phase and should be accordingly used in the calculation of Reynolds number:
 = viscosity of suspension;  = viscosity of pure fluid    

Flow through packed bed (continued) 
Pressure–drop calculation: 
Notes 
  1. Actual or real packed beds are randomly packed with irregular size particles 
  2. The flow–path of a fluid though the packed bed is tortuous. 
  • For the theoretical analysis to calculate pressure–drop, actual flow channels are replaced with parallel cylindrical conduits of constant cross–section. Particles are assumed to be of the same size and shape having constant sphericity, 
  • Pressure–drop occurs due to inertial and viscous effects. At high Reynolds number, inertial effects prevail, whereas the viscous effects are important at low Reynolds number. In general,
For a packed–bed 
  
 (Propose) 
(Re-call: wall shear-stress in tubular laminar flow, 
Similarly, pressure–drop at high Reynolds number, . Therefore, Pressure-drop in packed beds is related to pressure–drop due to viscous and inertial effects via two empirical constants, 
 (multiply both numerator and denominator by L) 
  SO = cross sectional area of packed–bed 
  


Therefore, 
Substituting, 
Or, 
Setting  (based on experimental data) 
We obtain 
--------- Ergun's equation
One also defines  as the friction factor for a packed–bed 


Therefore, 
If 
 (Blake – Plummer equation) 
It is pointed out that the Ergun's equation is applied to calculate pressure- drop across packed bed consisting of small size particles 
Also, note that  is to be interpreted as energy loss due to drag or friction per unit mass of the fluid , so that the term can be substituted in the general mechanical energy balance equation, consisting of KE, gravitational and shaft-work heads. 
where  frictional–pressure drop 


Fluidization 
  • When a liquid or gas is passed at a relatively small velocity though a bed of solid particles, the particles do not move. Fluid moves through the voids between the particles; pressure–drop is calculated by Ergun's equation. If the flow rate is steadily increased, pressure–drop (or drag) increases. Eventually, particles tend to move and bed expands a little. A stage is reached when the pressure–drop balances the weight of solid particles and buoyancy. Now, the bed apparently seems to be boiling. Particles–movement increases; yet they do not leave the bed. Such bed is termed as ‘fluidized bed'. 
  • Mechanistically, the frictional force (drag) between particles and fluid just counterbalances the weight of the particles; the vertical component of compressive forces between particles disappear and  equates the effective weight. 
  • A simple experiment can be carried out to observe the movement of particles packed within a glass or Perspex made column. The height of the bed and the pressure–drop across the bed can be measured with accuracy: 
(fig. 32a)


Fixed bed 
Bed expands a little Incipiently or minimum fluidized bed 
(No movement of particles) Particles are unlocked and the bed expands from an height of 
Bed continues to expand
(for liquid only) till there is a carry-over of solid particles 
For gas–solid system, bubbly or segregated fluidization is observed, with large-size bubbles–formation .A qualitative graph showing the variation of packed–bed height and pressur–drop with the superficial velocity of the liquid is produced below: 
(Fig. 32b)
  1. Fixed bed of height  and porosity 
  2. The bed expands at  (minimum fluidization velocity) from 
  3. The bed continues to expand with increase in the bed porosity from  to higher porosity 
(Fig. 32c)
Very-often a hysteresis is observed, if the velocity is gradually decreased: 
(Fig. 32d)

  • Fluidized–bed has fluid–like behavior. It will appear boiling at the surface, with the particles moving up and down in the bed, especially on top of the surface. 
  • The minimum fluidization porosity, or the porosity at the minimum fluidization condition is particle–size and type–specific. Some examples are:
Size
Sharp sand, 
Adsorption carbon 
0.72 
0.69 
Fischer-Tropsch catalyst 
0.58 
  • For liquid, the state of fluidization past the minimum fluidization stage is called homogeneous/ smooth/particulate/non–bubbling fluidized bed, as the bed expands smoothly. At higher velocity, there is a carry-over of particles. Slurry flow ensues. 
  • For gases, the particulate or homogeneous fluidization occurs only for small (fine) particles. For large particles, bubbles are formed. At even higher velocity, vigorous fluidization occurs, with turbulent motion of solid clusters and bubbles. Such state is called “Fast Fluidized Bed”. There may be carryover/entrainment of particles with the outgoing gas. 
Minimum fluidization velocity 
The minimum fluidization velocity can be calculated by equating the pressure–drop across the fixed packed–bed, calculated from Ergun's equation to that from the expression for fluidized bed under particulate (smooth) conditions.
Let us calculate the pressure-drop from the 2nd expression:
Under fluidization conditions, pressure–drop equals effective weight of solid, as intraparticle forces disappear and solids float in the bed exhibiting ‘liquid–like ‘behavior. For a fluidized bed of length of L and bed-porosity of 
 Weight of solid-particles–buoyancy
Or
, etc. 
where 
R-call: 
(Fig. 33a)

At the minimum fluidization condition:

Apply Ergun's equation for ‘fixed–bed' at minimum fluidization condition or at the incipience of fluidization:
, where superficial average velocity at minimum fluidization state
Equate: 
The above-equation is quadratic on  (minimum fluidization velocity) and may be written in the following form: 
 , where 
For small particles 
For large particles 

To avoid or reduce carryover of particles form the fluidized bed, keep the gas velocity between . Recall 
Terminal velocity,  for low Reynolds number and, 
for high Reynolds number 
With the expressions for  and  known for small (viscous–flow) and large (inertial flow) particles or Reynolds number, one can take the ratio of  and :
For small 
For spherical particles,  and assuming  
Therefore, a bed that fluidizes at 1cm/s could preferably be operated with velocities < 50 cm/s, with few particles carried out or entrained with the exit gas.
For large 
Or, 
Therefore, operating safety margin in a bed of coarse particles is smaller and there is a disadvantage for the use of coarse particles in a fluidized bed.
However, make a note that the operating particle size is also decided by the other factors such as grinding cost, pressure-drop, heat and mass-transfer aspects.

Filtration 
  • Removal of solids from fluid (gas or liquid) by a filtering medium on which solid particles are deposited. 
  • For filtration, external force is applied to a  (gas or liquid + solid) mixture to make it flow through the medium. 
  • Filtration, when applied to gas cleaning, usually refers to the removal of fine particles  like dust from air or flue gas. In such case, a polymeric fiber or cloth is wrapped over a pretreated metallic cylinder, capable of capturing micron size particles, including soot and fly-ash. 
  • Very large size ceramic based filters for high temperature applications are also commercially available. 
In this and the next lectures we will confine our discussion to liquid – solid filtration . 
  • The liquid–solid filtration is often called “cake–filtration”, because the separation of solids from the slurry by the filtering medium is effective during the initial stages of filtration. Later, the ‘cakes' or deposits collected over the medium act as the filter. Therefore, cake thickness increases during filtration and the resistance (hydraulic) offered by the cake–material is larger than that by the filtering medium. 
  • There are two types of operation: 
    1. Constant-pressure 
    2. Constant filtering rate
In the 1st case, filtering rate varies with time, whereas in the 2nd case, pressure–drop increases with time. 
  • For ideal cake filtration, cake should be stable and large porosity. There are two common types of filters: 
    1. The plate and frame press 
    2. Rotary-drum filter 
The plate and frame press filter 
  • Consists of series of plates and frames sandwitched alternatively; cakes are built-up inside the frame–clamber. Cloth, filtering medium, is supported on a corrugated material. There are slurry and filtrate ports. 
(Fig. 35a)

While designing the plate and frame press filter, dismantling and re–assembling times, removal of cake from each frame, and other operations such as washing and drying of cakes should also be taken into consideration. 
Rotary Filter 
(Fig. 35b) 
See the schematic above. The portion of the cylinder (rotary drum) submerged in the trough is subjected to vacuum. A layer of solids builds upon the drum as the liquid is drained through cloth, slots, compartments, pipe to the tank, which collects the filtered water.
In the washing/drying zone; vacuum is removed; cakes are removed by scrapping it off with a knife, doctor blade. The process is continuous whereas the plate and frame press filter is a batch process.

Principles of Filtration (cake) 
(Fig. 35c)
  • Pressure–drop is applied across the filter: 
Assuming that the flow of filtrate is under laminar conditions (low Re and viscous flow), one can apply the Ergun's equation, neglecting the inertial forms: 
Consider a differential thickness of cake =  at a distance of  from the filter-medium. 
where,  viscosity of the filtrate 
 bed-porosity or porosity of the cake 
 surface area and volume of the cake-particles (solids of the slurry), respectively 
(Important to note is the time–change of pressure and cake–thickness) 
 superficial velocity of filtrate

Principles of filtration (continued ) 
Case 1: Constant Pressure-drop Filtration 
differential mass of the cake 
Substituting, 
Assuming, incompressible cake (Const 
 (pressure–drop) through cake 
, where,  total mass of cake. 
upstream-pressure of filter–media
Define, 

= property of cake

 Pressure–drop through filter medium
 hydraulic resistance of filter medium

Now ‘C' as the mass of the particles deposited in the filter per unit volume of the filtrate, 
It can be shown that 
 where, 
If turns out that 
Replacing  in the expression for 
This is the working equation for cake filtration. 

 Case 1: constant–pressure filtration 
(Fig. 36a)
Therefore,  (one can calculate  form the initial filtration-data when resistance due
to cake = 0)
One can also write,
= constant (known) 
(Fig. 36b)

 on integration
The above expression can be integrated to develop an expression for the amount of cake formed over time ‘t' or the production rate of cake for the rotary- drum filter: 
(Fig. 36c)

A = Total area of filtration

Case 2: Constant Rate Filtration 
(Fig. 36d)
Or 
 (neglecting )
 here, v is constant.
 varies linearly with time.
(Such operation is difficult to run, i.e, keeping volumetric flow rate constant)

Agitation of liquids 
  • The unit operation is used to prepare liquid–mixture by bringing in contact two liquids in a mechanically agitated vessel or container. 
  • Agitation refers to the induced motion of liquid in some defined may, usually in circulatory pattern and is achieved by some mechanical device. 
Why agitation? 
  • Dispenses a liquid which is immiscible with the other liquid by forming an emulsion or suspension of few drops. 
  • Suspends relatively lighter solid particles 
  • Promotes heat transfer between the liquid in the think or container and a coil or jacket surrounding the container 
  • Blends miscible liquids 
(Fig. 40a)

  • The equipment consists of a tank with an insulated jacket, baffles, shaft with motor, impeller, and other accessories such as thermometer and dip- leg. 
The role of baffles is to remove stratification in the radial direction and improve mixing, 
(Fig. 40b)
  • Typical configuration-dimensions are:

  • Two types of impellers:
    • Radial flow impellers (flow is induced in radial or tangential directions) 
    • Axial flow impellers (currents are parallel to the axis of impeller shaft) 
  • Two types of geometrical configurations:
(Fig. 40c)

Flow patterns in agitated vessels 
There are three principal currents in the vessel during agitation: (a) radial (perpendicular to the shaft) (b) tangential (tangential to the circular path) (c) longitudinal (parallel to the shaft)
  1. Radia!
  2. Longitudinal 
  3. Swirling 
(Fig. 40d)
Notes: 
  • Tangential component induces vortex and swirling, which in turn create stratification responsible for non–uniform mixing. In such case fluid particles are followed by another fluid particle. 
  • At relatively higher rpm, the center of vortex may reach impeller and air may be sucked in. This may not be desirable. 
  • Swirling can be minimized by placing the shaft slightly away from the center of the vessel, or by putting baffles. In the latter–configuration, tangential streamlines will also be reduced.
Power requirement
Dimensional analysis is used to determine the power requirement. Variables are

Relatively larger viscous fluid requires high power for mixing. Similarly, high density fluid–mixture also require large power for mixing:
From Buckingham  theorem, no of independent dimensionless groups can be formed. For (6+m) variables, there will be (3 + m) groups: 
  1. Power number,  , 
  2. Reynolds number, , where  is the tangential velocity of the tip of the impeller or 
  3. Froude number  
The other groups are 
(Power number is analogous to friction factor and equals drag force on an unit area of impeller per KE of unit-fluid-volume ) 
Or,

(Here, Reynolds number is based on peripheral speed and diameter of impeller)

Graphical results are available for different types of impellers to calculate power number: 
(Fig. 40e)
(Slop is -1 on log–log plot for 
  • As in the case of tubular flow flow, viscous effects are predominant and density of fluid is not important at low Reynolds number.

(Tables are available to calculate 
Or
  • At high Reynolds number , power number is independent of the Reynolds number and viscosity is not important. Flow is fully turbulent. 

Or
 (Tables are available to calculate P)

Cyclone (Centrifugal settler) 
  • The equipment separates solid particles from a gas (eg. dust laden flue gas), based on the principle of centrifugal force, which is much stronger than gravitational force. Cyclone works relatively more efficiently at high gas flow rates. 
  • The equipment requires large flow rates/velocity to create a swirling movement inside the column. Cyclone, as such, does not have moving parts but may require a blower upstream to impart KE to the gas laden with particles. 
(Fig. 41a)


(Fig. 41b)


  1. The real trajectory of gas and particles is difficult to analyze. The particles laden gas enters the cyclone from the sideway (see top view) at a high flow rate and moves downward in a swirling/ spiral path. 
  2. Solid particles are thrown outward radially due to centrifugal force. They strike the walls of cyclone and settle down. Gas, on the other hand, will move radially inward, then upward through the least hydrodynamically resistance – path to the exit. 
  3. Gas moving in spiral reaches the apex of the cone, then moves upward in a smaller spiral
    () path to the exit at the top, as the opening at the bottom is filled with solid particles. For the gas, the least resistance – path is the exit at the top. For the particles, the least resistance- path is the exit at the bottom. 
  4. Mechanistically, if the centrifugal force acting on the particles is larger than the drag (inward) by the gas, the particles will strike the walls and settle down, else they will move inward alongwith the gas. At a radius r, where these two forces are equal, particle will rotate in equilibrium and move downward till they hit the slant walls and are collected. Gas on the other hand has a very high upward flow rate at the center, typically in the core-diameter of . Any particle in the zone will be carried upward. 
(Fig. 41c)

  • Theoretical ‘cut-size' of a cyclone is the particle size above which all particles will be collected. A theoretical expression considering drag and centrifugal forces on a particle, has been obtained to estimate the ‘cut size' of cyclone. The calculation takes into account the experimental observation that the equilibrium rotation-radius of all captured particles in cyclone is  do, where do is the diameter of the nozzle at the top of the cyclone though which the gas exits. 
  • The settling velocity of captured particles, 
where,
  


  


Form , the theoretical cut–diameter, dp is determined from the settling velocity equation:

(Note that it is assumed that particles settle in Stoke's regime)

  • All particles having diameter  will have equilibrium radius within 0.5 do so that they will be carried away with the gas.
    All particles having diameter  will be captured in cyclone. 
  • Cyclones are very effective in removing particles from gas. Disadvantages are large flow rate required and large pressure–drop because of the tortuous path of the gas. 
  •  gas velocity at the inlet 
  • Separation factor of a cyclone, s is defined as 
  • Cyclones are effective typically for particle size 
  • Efficiency (capturing) of cyclone, 
Design graphs are available to calculate the efficiency. 
(Fig. 41d)

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