CONTINUITY EQUATION

Continuity equation

• The continuity equation is a mathematical expression for the law of conservation of mass.
• The equation of continuity is useful for calculating the velocity of a fluid flowing through pipes of different diameters.

It states that for a steady flow system, the rate of mass entering the flow system is equal to that leaving as accumulation is either constant or nil in the flow system under steady conditions

• Rate of mass entering the flow system = ρ1 u1 A1
• Rate of mass entering the flow system = ρ2 u2 A2
• According to the law of conservation of mass

Mass entering the tube = Mass leaving the tube

per unit time       per unit time

• This is the equation of continuity.
• It is applicable to compressible as well as incompressible fluids.
• when ρ1 = ρ2

where

Continuity equation

IN PLACE & HEAP LEACHING

In place leaching

• In place leaching is also called solution mining in which minerals are recovered through boreholes drilled into the deposit.
• In this process, the solvent is pumped through the borehole into the ore body.
• Then the solvent is circulated through the porous rock of dissolving ore.
• Then the leach solution is extracted via the second borehole.

Example

• For salt deposit fresh water used as the solvent, for copper deposit acid used as the solvent and for uranium ores acid or sodium bicarbonates used as the solvent.

Heap Leaching

• Heap leaching is an industrial mining process. 
• In this process, precious metals are extracted from ore via a series of chemical reactions that absorb specific minerals and then reseparate them after their division from earth metal.
• In this process, mined ore is usually crushed into the small lump and heaped on an impermeable plastic or clay lined leached pad.
• Then the solvent is passed through it or sprinkled on it to dissolve valuable materials.
• Then the leach solution is percolated through the heap and then collected, treated in the process plant.

Example

• Copper has leached from pyritic ores.

Types of reactor

Types of reactor

• The confines in which chemical reactions occur are called reactors.
• In chemical engineering chemical reaction or set of reactions is carried out in a reaction vessel that is in chemical reactors.
• All chemical processes are centered in a chemical reactor.
• The design of a chemical reactor is the most important factor in determining the overall process economics.
• While doing any operation we have to first decide what reactor type and reactor shape to select and whether it would be advantageous to operate in batch or continuous mode.
• Ideal reactors have three ideal flow or contacting patterns. They are main basic type of chemical reactor

1. Batch reactor
2. Flow reactor

• Continuous Stirred-Tank Reactor(CSTR)
• Plug Flow Reactor (PFR)

Batch reactor

• Whether the system does not exchanges mass with its surroundings, then the system is called a batch reactor.
• The batch reactor is simply a container with an agitator and an integral heating/cooling system.

• The reactants are initially charged into a container, are well mixed, and are left to react for a certain period.
• The resultant product mixture is then discharged.
• This is an unsteady-state operation where composition changes with time; however, at any instant the composition throughout the reactor is uniform
• During the operation, no addition or withdrawal is made.
• The experimental batch reactor is usually operated isothermally and at constant volume because it is easy to interpret the results of such runs.
• This reactor is a relatively simple device adaptable to small-scale laboratory set-ups, and it needs but little auxiliary equipment or instrumentation.
• Thus, it is used whenever possible for obtaining homogeneous kinetic data.

Advantages

• Suitable for small scale production
• Suitable for processes where a range of different products or grades is to be produced in the same equipment
• Suitable for reactions requiring long reaction times
• Suitable for reactions with superior selectivity

Limitation

• Not suitable for large batch sizes.
• It is a closed system in which once the reactants are added in the reactor, they will come out as products only after the completion of the reaction.

Application

• Batch processes are used in chemical (inks, dyes, polymers) and food industry.

Continuous Stirred-Tank Reactor (CSTR)

• The flow reactor is used primarily in the study of the kinetics of heterogeneous reactions.
• The first of the two ideal steady-state flow reactor is called the mixed reactor, the backmix reactor, the ideal stirred tank reactor, CSTR, or the CFSTR (constant flow stirred tank reactor)
• As its names suggest, it is a reactor in which the contents are well stirred and uniform throughout.
• In CSTR the reactants are fed to the reactor and the products or byproducts are withdrawn in between while the reaction is still progressing.

• The CSTR is normally run at steady-state and is usually operated so as to be quite well mixed.
• This type of flow is mixed flow, hence this reactor also called the mixed flow reactor or MFR.
• The exit stream from this reactor has the same composition and temperature as the fluid within the reactor.
• Continuous reactors are usually preferred for large scale production.

Advantages

• Highly flexible device
• By products may be removed in between the reaction.
• It is economically beneficial to operate several CSTRs in series or in parallel.
• Reaction can be carried out in horizontal as well as vertical reactors.

Limitation

• More complex and expensive than tubular units.
• All calculations performed with CSTRs assume perfect mixing.
• At steady state, the flow rate in must equal the flow rate out, otherwise the tank will overflow or go empty.

Application

• Chemical industry especially involving liquid/gas reactions.

Plug Flow Reactor (PFR)

• The other ideal steady-state flow reactors are variously known as the plug flow, slug flow, piston flow, ideal tubular, and unmixed flow reactor.
• It consists of a cylindrical pipe and is normally operated at a steady-state, as is the CSTR.

• This reactor is known as the plug flow reactor, or PFR because the flow in the reactor is highly turbulent and the flow field may be modeled by that of plug flow.
• It is characterized by the fact that the flow of fluid through the reactor is orderly with no element of fluid overtaking or mixing with any other element ahead or behind.
• Actually, there may be lateral mixing of fluid in a plug flow reactor; however, there must be no mixing or diffusion along the flow path.
• Hence in the tubular reactor, the reactants are continually consumed as they flow down the length of the reactor.
• The necessary and sufficient condition for plug flow is for the residence time in the reactor to be the same for all elements of fluid.

Advantage

• Higher efficiency than a CSTR of the same volume
• PFRs may have several pipes or tubes in parallel
• Both horizontal and vertical operations are common
• They can be jacketed
• Reagents may be introduced at locations even other then inlet

Limitation

• Not economical for small batches

Application

• The tubular reactor is especially suited to cases needing considerable heat transfer, where high pressures and very high or very low temperatures occur

AFFINITY LAWS OF PUMP

The Affinity Laws of pump

• The Affinity Laws of centrifugal pumps or fans indicates the influence on volume capacity, head (pressure) and/or power consumption of a pump or fan due to

1. change in speed of wheel - revolutions per minute (rpm)
2. geometrically similarity - change in impeller diameter

• Hence, the Affinity Laws are mathematical expressions that define changes in pump capacity,head, and BHP when a change is made to pump speed, impeller diameter, or both.
• Affinity laws are useful when an existing pump must be modified ti give a higher or lower head or different capacity
• The Affinity Laws are valid only under conditions of constant efficiency.

According to Affinity Laws:Capacity, Q: changes in direct proportion to impeller diameter D ratio, or to speed N ratio:

Head, H: changes in direct proportion to the square of impeller diameter D ratio, or the square of speed N ratio:

BHP: changes in direct proportion to the cube of impeller diameter ratio, or the cube of speed ratio:

• Where the subscript: 1 refers to initial condition, 2 refer to new condition
• If changes are made to both impeller diameter and pump speed the equations can be combined to:

POWER AND EFFICIENCY OF CENTRIFUGAL PUMP

Power and efficiency of centrifugal pump
Brake Horse Power (BHP) and Water horsepower (WHP)

• The work performed by a pump is a function of the total head and the weight of the liquid pumped in a given time period.

Pump input or brake horsepower (BHP) is the actual horsepower delivered to the pump shaft.

Pump output or hydraulic or water horsepower (WHP) is the liquid horsepower delivered by the pump.

• These two terms are defined by the following formulas.

where

• Q = Capacity in gallons per minute (GPM)
• HT = Total differential head, ft
• Specific gravity= Specific gravity of liquid
• Efficiency = Pump efficiency, %

• The constant 3960 is obtained by dividing the number or foot-pounds for one horsepower(33,000) by the weight of one gallon of water (8.33 pounds). 

The brake horsepower or input to a pump is greater than the hydraulic horsepower or output due to the mechanical and hydraulic losses incurred in the pump.

• Therefore the pump efficiency is the ratio of these two values.

Best Efficiency Point (BEP)

• The Head, NPSHr, efficiency, and BHP all vary with flow rate, Q.

Best Efficiency Point (BEP) is the capacity at maximum impeller diameter at which the efficiency is highest.

Significance of BEP

• BEP as a measure of optimum energy conversion
• When sizing and selecting centrifugal pumps for a given application the pump efficiency at design should be taken into consideration.
• The efficiency of centrifugal pumps is stated as a percentage and represents a unit of measure describing the change of centrifugal force (expressed as the velocity of the fluid) into pressure energy.
• The B.E.P.(best efficiency point) is the area on the curve where the change of velocity energy into pressure energy at a given gallon per minute is optimum; in essence, the point where the pump is most efficient.
• The operation of a centrifugal pump should not be outside the furthest left or right efficiency curves published by the manufacturer.
• Performance in these areas induces premature bearing and mechanical seal failures due to shaft deflection, and an increase in temperature of the process fluid in the pump casing causing seizure of close tolerance parts and cavitation.
• BEP is an important parameter in that many parametric calculations such as specific speed, suction specific speed, hydrodynamic size, viscosity correction, head rise to shut-off, etc. are based on capacity at BEP.
• Many users prefer that pumps operate within 80%to 110% of BEP for optimum performance.

Specific Speed

Specific speed as a measure of the geometric similarity of pumps

• Specific speed (Ns) is a non-dimensional design index that identifies the geometric similarity of pumps.
• It is used to classify pump impellers as to their type and proportions.
• Pumps of the same Ns but of different size are considered to be geometrically similar,one pump being a size- factor of the other.

Specific speed Calculation

• The following formula is used to determine specific speed:

where

• Q = Capacity at best efficiency point(BEP) at maximum impeller diameter, GPM(gallons per meter)
• H = Head per stage at BEP at maximum impeller diameter, ft
• N = pump speed, RPM

• As per the above formula, it is defined as the speed in revolutions per minute at which a geometrically similar impeller would operate if it were of such a size as to deliver one gallon per minute flow against one-foot head.
• Specific speed should be thought of only as an index used to predict certain pump characteristics.
• The specific speed determines the general shape or class of the impellers.

NET POSITIVE SUCTION HEAD

Net Positive Suction Head (NPSH)

• Power supplied to the pump depend on the difference in the pressure between discharge and suction and is independent of the pressure level.
• There is no effect on pump when suction pressure is below atmospheric pressure or well above it, as long as the fluid remains liquid.
• But if suction pressure is only slightly greater than the vapor pressure, some liquid may flash to vapor inside to pump, a process called cavitation, which greatly reduces the pump capacity.
• If the suction pressure is actually less than the vapor pressure, there will be vaporization in the suction line, and no liquid can be drawn into the pump.
• To avoid cavitation, the pressure at the pump inlet must exceed the vapor pressure by certain value called net positive suction head (NPSH)

Hence, Net Positive Suction Head or NPSH for pumps can be defined as the difference between liquid pressure at pump suction and liquid vapor pressure, expressed in terms of height of liquid column.

• Net positive suction head (NPSH) may refer to one of two quantities in the analysis of cavitation.

1. The Available NPSH (NPSHA): A measure of how close the fluid at a given point is to flashing, and so to cavitation.
2. The Required NPSH (NPSHR): The head value at a specific point (e.g. the inlet of a pump) required to keep the fluid from cavitating.

Net Positive Suction Head Required, NPSHr

• Pumps can pump only liquids, not vapors
• The satisfactory operation of a pump requires that vaporization of the liquid being pumped does not occur at any condition of operation.
• This is so desired because when a liquid vaporizes its volume increases very much.
• For example, 1 ft3 of water at room temperature becomes 1700 ft3 of vapor at the same temperature.
• This makes it clear that if we are to pump a fluid effectively, it must be kept always in the liquid form.
• Rise in temperature and fall in pressure induces vaporization
• The vaporization begins when the vapor pressure of the liquid at the operating temperature equals the external system pressure, which, in an open system is always equal to atmospheric pressure.
• Any decrease in external pressure or rise in operating temperature can induce vaporization and the pump stops pumping.
• Thus, the pump always needs to have a sufficient amount of suction head present to prevent this vaporization at the lowest pressure point in the pump.
• NPSH as a measure to prevent liquid vaporization
• NPSH required is a function of the pump design and is determined based on actual pump test by the vendor.
• NPSHr increases as capacity increases
• The NPSH required varies with speed and capacity within any particular pump.
• The NPSH required increase as the capacity is increasing because the velocity of the liquid is increasing, and as anytime the velocity of a liquid goes up, the pressure or head comes down.
• The NPSH is independent of the fluid density as are all head terms.

Net Positive Suction Head available, NPSHa

• Net Positive Suction Head Available is a function of the system in which the pump operates.
• It is the excess pressure of the liquid in feet absolute over its vapor pressure as it arrives at the pump suction, to be sure that the pump selected does not cavitate.
• It is calculated based on system or process conditions.
• The formula for calculating the NPSHa is as below

where

hps= pressure head i. e. absolute pressure at surface of reservoir converted into head
hs= static suction head
hvps= vapor pressure head
hfs= friction head 

Significance of NPSHr and NPSHa

• The NPSH available must always be greater than the NPSH required for the pump to operate properly.

PERFORMANCE PARAMETER OF CENTRIFUGAL PUMP

Performance parameter of centrifugal pump
The key performance parameters of centrifugal pumps are 

1. Capacity
2. Head
3. BHP(Brake horse power)
4. BEP (Best efficiency point)
5. Specific speed

Capacity

Capacity means the flow rate with which liquid is moved or pushed by the pump to the desired point in the process. 

• It is commonly measured in either gallons per minute(gpm) or cubic meters per hour (m3 /hr). 
• The capacity usually changes with the changes in operation of the process. 
• The capacity depends on a number of factors like: 

1. Process liquid characteristics i.e. density, viscosity 
2. Size of the pump and its inlet and outlet sections 
3. Impeller size 
4. Impeller rotational speed RPM 
5. Size and shape of cavities between the vanes 
6. Pump suction and discharge temperature and pressure conditions

• The effect on the flow through a pump by changing the outlet pressures is graphed on a pump curve.
• As liquids are essentially incompressible, the capacity is directly related with the velocity of flow in the suction pipe. 
• This relationship is as follows:

where

• Q= capacity
• V = velocity of flow
• A = area of pipe

Head

The pressure at any point in a vertical column of the liquid can be caused due to its weight. The height of this column is called the static head and is expressed in terms of feet of liquid. 

• The same head term is used to measure the kinetic energy created by the pump.

In other words, head is a measurement of the height of a liquid column that the pump could create from the kinetic energy imparted to the liquid. 

• The head is not equivalent to pressure. 
• Head is a term that has units of a length or feet and pressure has units of force per unit area or pound per square inch. 
• The main reason for using head instead of pressure to measure a centrifugal pumps energy is that the pressure from a pump will change if the specific gravity (weight) of the liquid changes, but the head will not change. 
• Since any given centrifugal pump can move a lot of different fluids, with different specific gravities, it is simpler to discuss the pumps head and forget about the pressure.
• A given pump with a given impeller diameter and speed will raise a liquid to a certain height regardless weight of the liquid. 

Pressure to Head Conversion formula 

• The static head corresponding to any specific pressure is dependent upon the weight of the liquid according to the following formula

There are different type of head, which are as follows


1. Static Suction Head (hS ) :

Head resulting from elevation of the liquid relative to the pump centerline is called static suction head. 

• If the liquid level is above pump centerline, hS is positive. 
• If the liquid level is below pump centerline, hS is negative. 
• Negative hS condition is commonly denoted as a “suction lift” condition 

2. Static Discharge Head (hd):

It is the vertical distance in feet between the pump centerline and the point of free discharge or the surface of the liquid in the discharge tank.

3. Friction Head (hf):

This head required to overcome the resistance to flow in the pipe and fittings. 

• It is dependent upon the size, condition and type of pipe, number and type of pipe fittings, flow rate, and nature of the liquid.

3. Vapor Pressure Head (hvp):

• Vapor pressure is the pressure at which a liquid and its vapor co-exist in equilibrium at a given temperature. 

When the vapor pressure is converted to head, it is referred to as vapor pressure head, hvp. 

• The value of hvp of a liquid increases with the rising temperature and in effect, opposes the pressure on the liquid surface, the positive force that tends to cause liquid flow into the pump suction i.e. it reduces the suction pressure head.

4. Pressure Head (hp):

Pressure Head must be considered when a pumping system either begins or terminates in a tank which is under some pressure other than atmospheric. 

• The pressure in such a tank must first be converted to feet of liquid. 
• Denoted as hp, pressure head refers to absolute pressure on the surface of the liquid reservoir supplying the pump suction, converted to feet of head. 
• If the system is open, hp equals atmospheric pressure head.

5. Velocity Head (hv):

Velocity head refers to the energy of a liquid as a result of its motion at some velocity ‘v’. 

• It is the equivalent head in feet through which the water would have to fall to acquire the same velocity, or in other words, the head necessary to accelerate the water. 
• The velocity head is usually insignificant and can be ignored in most high head systems.
• However, it can be a large factor and must be considered in low head systems.

6. Total Suction Head (HS ):

• The suction reservoir pressure head (hpS ) plus the static suction head (hS ) plus the velocity head at the pump suction flange (hVS) minus the friction head in the suction line (hfS ). 

HS = hpS + hS + hvS – hfS 

The total suction head is the reading of the gauge on the suction flange, converted to feet of liquid.

7. Total Discharge Head (Hd):

• The discharge reservoir pressure head (hpd ) plus static discharge head (hd) plus the velocity head at the pump discharge flange (hvd ) plus the total friction head in the discharge line (hfd). 

Hd = hpd + hd + hvd + hfd 

The total discharge head is the reading of a gauge at the discharge flange, converted to feet of liquid. 

8. Total Differential Head (HT):

It is the total discharge head minus the total suction head  

HT = Hd + HS (with a suction lift) 

HT = Hd - HS (with a suction head)