Two flow tests were conducted. Placebo product (a suspension with no active ingredient, target viscosity of 2500 cP) was pumped from a Manufacturing Tank to a Receiver. Here are the observed results:
Parameter  Test 1  Test 2 
Pump Speed  30%  40% 
Temperature  35 C  34 C 
Flow rate, based on load cells  220 kg/min  160 kg/min 
Pressure at pump discharge  125 kPa  83 kPa 
Pressure at receiver  28 kPa  28 kPa 
The observed values, along with knowledge of the physical piping, can be used to calculate the “apparent viscosity” of the placebo product. The piping from the pump to the Hold Tank is 50 DN (47.5 mm inside diameter). It includes straight piping, elbows, 45° bends, one divert valve and one diaphragm valve. The total equivalent length of this piping and valves is 85 m. Equivalent length is used to calculate pressure drop. The pressure gauge on the pump discharge is 1 meter higher than the one beneath the Receiver; the pressure drop due to friction is therefore P1 – P2 + 1 meter head, where P1 = pressure at pump discharge, and P2 = pressure beneath Receiver.
Estimation of Apparent Viscosity
There are three steps to calculating the pressure drop due to friction through the pipe. The first step requires fluid viscosity as an input. To calculate the viscosity, initially assume a value, proceed through the three steps, and compare the calculated pressure drop with the observed results. Iterate by changing the viscosity assumption until the calculation and observed result match.
For these calculations, a value for the placebo product density must also be known. This wasn’t measured, but plant operators stated that it is from 1.2 to 1.3; we calculated it to be 1.26.
Step 1: Reynolds Number^{i}
Eq. (1) 
ρ = density of liquid, kg/m^{3} or lb/ft^{3}
D = pipe diameter, m or ft
U = average fluid velocity, m/s or ft/s = G / ρ
G = mass flux, kg/sm^{2} or lb/sft^{2} = W / (3600 A)
A = crosssectional area of pipe, m^{2} or ft^{2}
μ = fluid dynamic viscosity, Pas or lb/fth (1 cP = 2.42 lb/fth)
Step 2: Friction Factor
Eq. (2) 
Where:
Eq. (3) 
Eq. (4) 
ε = surface roughness, m or ft
Surface roughness is a piping characteristic. The limiting value is “smooth,” and defined to be 0.0000015 m or 0.000005 ft. For this calculation, use a value of 0.000002 m.
Step 3: Pressure Drop Due to Friction
Eq. (5) 
L = pipe equivalent length, m or ftg_{c} = conversion factor, 1 m/s^{2} or 32.17 ft/s^{2}
Parameter  Test 1  Test 2 
Specific Gravity^{ii}  1.26  1.26 
Observed pressure drop  110 kPa  68 kPa 
Viscosity  39 cP  47 cP 
Calculated pressure drop  110 kPa  68 kPa 
Viscosity Measurements
Pseudoplastic compounds follow a power curve, with viscosity decreasing with shear rate. Viscosity measurements were performed with a Brookfield Viscometer, Model LVDVII+. Using the #31 spindle and Sample Adapter 13R cup, shear rate is determined by multiplying the rotational speed by 0.34^{iii}. This gives the following raw data for the placebo product made for the tests:
RPM 
Viscosity (cP) 
Torque (%) 
Shear Rate, sec^{1} 
0.5 
8338 
13.9 
0.17 
1.0 
5159 
17.2 
0.34 
1.5 
3919 
19.6 
0.51 
3.0 
2449 
24.6 
1.02 
6.0 
1540 
30.8 
2.04 
12.0 
977.3 
39.1 
4.08 
20.0 
706.3 
47.1 
6.8 
30.0 
551.9 
55.2 
10.2 
45.0 
435.2 
65.3 
15.3 
60.0 
370.4 
74.1 
20.4 
100.0 
283.7 
94.6 
34 
150.0 
EEEE 
EEEE 

Measurements made at 25°C
Here are the data charted in Excel with the trend line and trend line equation superimposed:
Figure 1: Graph of Viscosity Data with Trend Line 
The power law equation for apparent viscosity is:
μ_{a} = K γ^{n1}  Eq. (6) 
K = flow consistency index, Pasn = flow behavior index, dimensionless
Viscosity is in units Pas. Since our data is in cP (=mPas), the values of K and n are 2.53 and 0.356.
The viscometer’s measurement range is limited, but extrapolation is acceptable with decreased accuracy.
Using the trend line equation, the viscosity at any shear rate is estimated. But this is for a temperature of 25°C. Since data at only one temperature is given, the Lewis and Squires temperature correlation is used to estimate the viscosity at the actual flowing condition.
Figure 2: Lewis and Squires Temperature Correlation 
The equation for this chart is^{iv}:
Eq. (7) 
Shear Rate in Pipeline
The shear rate at the wall of a circular pipe is calculated with:
Eq. (8) 
And here are the results:
Parameter 
Test 1 
Test 2 
Flow rate 
220 kg/min 
160 kg/min 
Density 
1.26 
1.26 
Pipe diameter 
47.5 mm 
47.5 mm 
Velocity 
1.65 m/s 
1.20 ft/s 
Shear rate 
277 sec^{1} 
202 sec^{1} 
Viscosity at 25°C 
68 cP 
83 cP 
Corrected viscosity, 34°C 
42 cP 
53 cP 
Compare with calculation 
39 cP 
47 cP 
Discussion and Conclusions
After correction for shear rate and temperature, viscosity measurements from the Brookfield viscometer result in reasonably good estimates of pipeline pressure drop. It is important to ensure that the sample has reached the target temperature of 25° before recording the reading.
The limitations in this study include:
References
^{i}Hall, S.M., Rules of Thumb for Chemical Engineers, ButterworthHeinemann (2012)
^{ii}Specific gravity determined from the pump speed (40% x 449 rpm max) and capacity (0.96 l/rev). This gives volumetric flow rate. From the measured mass flow rate, the specific gravity is calculated. Results from the two flow tests were averaged to obtain a value of 1.26.
^{iii}Brookfield Engineering Labs, More Solutions to Sticky Problems, downloaded from http://www.brookfieldengineering.com/
^{iv}Poling, B.E., Prausnitz, J.M., O’Connell, J.P., Properties of Gases and Liquids, Fifth Edition, McGrawHill (2001).
]]>Step #1: Define the System
Step #2: Define a maximum allowable pressure drop for the valve
Step #3: Calculate the valve characteristic
Step #4: Preliminary valve selection
Step #5: Check the Cv and stroke percentage at the minimum flow
Step #6: Check the gain across applicable flowrates
Flow (gpm)  Stroke (%)  Change in flow (gpm)  Change in Stroke (%) 
25  35  11025 = 85  7335 = 38 
110  73  
150  85  150110 = 40  8573 = 12 
Other Notes
Selecting a Valve Type
Gate Valves
Globe Valves
Ball Valves
Butterfly Valves
Other Valves
References
Despite all the care in operation and maintenance, engineers often face the statement "the pump has failed i.e. it can no longer be kept in service". Inability to deliver the desired flow and head is just one of the most common conditions for taking a pump out of service. {parse block="google_articles"}There are other many conditions in which a pump, despite suffering no loss in flow or head, is considered to have failed and has to be pulled out of service as soon as possible. These include seal related problems (leakages, loss of flushing, cooling, quenching systems, etc), pump and motor bearings related problems (loss of lubrication, cooling, contamination of oil, abnormal noise, etc), leakages from pump casing, very high noise and vibration levels, or driver (motor or turbine) related problems.
The list of pump failure conditions mentioned above is neither exhaustive nor are the conditions mutually exclusive. Often the root causes of failure are the same but the symptoms are different. A little care when first symptoms of a problem appear can save the pumps from permanent failures. Thus the most important task in such situations is to find out whether the pump has failed mechanically or if there is some process deficiency, or both. Many times when the pumps are sent to the workshop, the maintenance people do not find anything wrong on disassembling it. Thus the decision to pull a pump out of service for maintenance / repair should be made after a detailed analysis of the symptoms and root causes of the pump failure. Also, in case of any mechanical failure or physical damage of pump internals, the operating engineer should be able to relate the failure to the process unit's operating problems.
Any operating engineer, who typically has a chemical engineering background and who desires to protect his pumps from frequent failures must develop not only a good understanding of the process but also thorough knowledge of the mechanics of the pump. Effective troubleshooting requires an ability to observe changes in performance over time, and in the event of a failure, the capacity to thoroughly investigate the cause of the failure and take measures to prevent the problem from reoccurring.
The fact of the matter is that there are three types of problems mostly encountered with centrifugal pumps:
The present article is being presented in three parts, covering all aspects of operation, maintenance, and troubleshooting of centrifugal pumps. The article has been written keeping in mind the level and interests of students and the beginners in operation. Any comments or queries are most welcome.
Working Mechanism of a Centrifugal Pump
A centrifugal pump is one of the simplest pieces of equipment in any process plant. Its purpose is to convert energy of a prime mover (a electric motor or turbine) first into velocity or kinetic energy and then into pressure energy of a fluid that is being pumped.
The energy changes occur by virtue of two main parts of the pump, the impeller and the volute or diffuser. The impeller is the rotating part that converts driver energy into the kinetic energy. The volute or diffuser is the stationary part that converts the kinetic energy into pressure energy.
All of the forms of energy involved in a liquid flow system are expressed in terms of feet of liquid i.e. head.Generation of Centrifugal Force
Figure 1: Liquid Flow Path Inside a Centrifugal Pump 
The process liquid enters the suction nozzle and then into eye (center) of a revolving device known as an impeller. When the impeller rotates, it spins the liquid{parse block="google_articles"} sitting in the cavities between the vanes outward and provides centrifugal acceleration. As liquid leaves the eye of the impeller a lowpressure area is created causing more liquid to flow toward the inlet. Because the impeller blades are curved, the fluid is pushed in a tangential and radial direction by the centrifugal force. This force acting inside the pump is the same one that keeps water inside a bucket that is rotating at the end of a string. Figure A.01 below depicts a side crosssection of a centrifugal pump indicating the movement of the liquid.
Conversion of Kinetic Energy to Pressure Energy
The key idea is that the energy created by the centrifugal force is kinetic energy. The amount of energy given to the liquid is proportional to the velocity at the edge or vane tip of the impeller. The faster the impeller revolves or the bigger the impeller is, then the higher will be the velocity of the liquid at the vane tip and the greater the energy imparted to the liquid.
This kinetic energy of a liquid coming out of an impeller is harnessed by creating a resistance to the flow. The first resistance is created by the pump volute (casing) that catches the liquid and slows it down. In the discharge nozzle, the liquid further decelerates and its velocity is converted to pressure according to Bernoulli's principle.
Therefore, the head (pressure in terms of height of liquid) developed is approximately equal to the velocity energy at the periphery of the impeller expressed by the following wellknown formula:
where:  Eq. 1 
A handy formula for peripheral velocity is:
where:  Eq. 2 
This head can also be calculated from the readings on the pressure gauges attached to the suction and discharge lines.
One fact that must always be remembered: A pump does not create pressure, it only provides flow.
Pressure is a just an indication of the amount of resistance to flow.
Pump curves relate flow rate and pressure (head) developed by the pump at different impeller sizes and rotational speeds. The centrifugal pump operation should conform to the pump curves supplied by the manufacturer. In order to read and understand the pump curves, it is very important to develop a clear understanding of the terms used in the curves. This topic will be covered later.
General Components of Centrifugal Pumps
A centrifugal pump has two main components:
A rotating component comprised of an impeller and a shaft
A stationary component comprised of a casing, casing cover, and bearings.
The general components, both stationary and rotary, are depicted in Figure 2. The main components are discussed in brief below. Figure3 shows these parts on a photograph of a pump in the field.
Figure 2: General Components of Centrifugal Pump  Figure 3: General Components of Centrifugal Pump 
Stationary Components
Casings
Figure 4: CutAway of a Pump Showing Volute Casing 
Casings are generally of two types: volute and circular. The impellers are fitted inside the casings.
Volute Casings
Volute casings build a higher head; circular casings are used for low head and high capacity. A volute is a curved funnel increasing in area to the discharge port as shown in Figure 4. As the area of the crosssection increases, the volute reduces the speed of the liquid and increases the pressure of the liquid. One of the main purposes of a volute casing is to help balance the hydraulic pressure on the shaft of the pump. However, this occurs best at the manufacturer's recommended capacity. Running volutestyle pumps at a lower capacity than the manufacturer recommends can put lateral stress on the shaft of the pump, increasing wearandtear on the seals and bearings, and on the shaft itself. Doublevolute casings are used when the radial thrusts become significant at reduced capacities.
Circular Casings
Circular casing have stationary diffusion vanes surrounding the impeller periphery that convert velocity energy to pressure energy. Conventionally, the diffusers are applied to multistage pumps.
Figure 5: Solid Casing 
The casings can be designed either as solid casings or split casings. Solid casing implies a design in which the entire casing including the discharge nozzle is all contained in one casting or fabricated piece. A split casing implies two or more parts are fastened together. When the casing parts are divided by horizontal plane, the casing is described as horizontally split or axially split casing. When the split is in a vertical plane perpendicular to the rotation axis, the casing is described as vertically split or radially split casing. Casing Wear rings act as the seal between the casing and the impeller.
Suction and Discharge Nozzles
The suction and discharge nozzles are part of the casings itself. They commonly have the following configurationstwo of which are shown in Figure 6:
Figure 6: Suction and Discharge Nozzle Locations 
End Suction/Top Discharge
The suction nozzle is located at the end of, and concentric to, the shaft while the discharge nozzle is located at the top of the case perpendicular to the shaft. This pump is always of an overhung type and typically has lower NPSHr because the liquid feeds directly into the impeller eye
Top Suction/Top Discharge
The suction and discharge nozzles are located at the top of the case perpendicular to the shaft. This pump can either be an overhung type or betweenbearing type but is always a radially split case pump.
Side Suction/Side Discharge
The suction and discharge nozzles are located at the sides of the case perpendicular to the shaft. This pump can have either an axially or radially split case type.
Seal Chamber and Stuffing Box
Figure 7: Parts of a Simple Seal Chamber 
Seal chamber and Stuffing box both refer to a chamber, either integral with or separate from the pump case housing that forms the region between the shaft and casing where sealing media are installed. When the sealing is achieved by means of a mechanical seal, the chamber is commonly referred to as a Seal Chamber. When the sealing is achieved by means of packing, the chamber is referred to as a Stuffing Box. Both the seal chamber and the stuffing box have the primary function of protecting the pump against leakage at the point where the shaft passes out through the pump pressure casing. When the pressure at the bottom of the chamber is below atmospheric, it prevents air leakage into the pump. When the pressure is above atmospheric, the chambers prevent liquid leakage out of the pump. The seal chambers and stuffing boxes are also provided with cooling or heating arrangement for proper temperature control. Figure7depicts an externally mounted seal chamber and its parts.
Glands
The gland is a very important part of the seal chamber or the stuffing box. It gives the packings or the mechanical seal the desired fit on the shaft sleeve. It can be easily adjusted in axial direction. The gland comprises of the seal flush, quench, cooling, drain, and vent connection ports as per the standard codes like API 68
Throat Bushing
The bottom or inside end of the chamber is provided with a stationary device called throat bushing that forms a restrictive close clearance around the sleeve (or shaft) between the seal and the impeller.
Throttle Bushing{parse block="google_articles"}
The throttle bushing refers to a device that forms a restrictive close clearance around the sleeve (or shaft) at the outboard end of a mechanical seal gland.
Internal Circulating Device
The internal circulating device refers to device located in the seal chamber to circulate seal chamber fluid through a cooler or barrier/buffer fluid reservoir. Usually it is referred to as a pumping ring.
Mechanical Seal
Mechanical seals will be discussed further in part two of this article series.
Bearing Housing
The bearing housing encloses the bearings mounted on the shaft. The bearings keep the shaft or rotor in correct alignment with the stationary parts under the action of radial and transverse loads. The bearing house also includes an oil reservoir for lubrication, constant level oiler, jacket for cooling by circulating cooling water.
Rotating Components
Impeller
The impeller is the main rotating part that provides the centrifugal acceleration to the fluid. They are often classified in many ways:
Based on mechanical construction (Figure 8)
Closed: Shrouds or sidewall enclosing the vanes.
Open: No shrouds or wall to enclose the vanes.
Semiopen or vortex type.
Figure 8: Impeller Types 
Closed impellers require wear rings and these wear rings present another maintenance problem. Open and semiopen impellers are less likely to clog, but need manual adjustment to the volute or backplate to get the proper impeller setting and prevent internal recirculation. Vortex pump impellers are great for solids and "stringy" materials but they are up to 50% less efficient than conventional designs. The number of impellers determines the number of stages of the pump. A single stage pump has one impeller only and is best for low head service. A twostage pump has two impellers in series for medium head service. A multistage pump has three or more impellers in series for high head service.
Wear ring provides an easily and economically renewable leakage joint between the impeller and the casing. clearance becomes too large the pump efficiency will be lowered causing heat and vibration problems. Most manufacturers require that you disassemble the pump to check the wear ring clearance and replace the rings when this clearance doubles.
Shaft
The basic purpose of a centrifugal pump shaft is to transmit the torques encountered when starting and during operation while supporting the impeller and other rotating parts. It must do this job with a deflection less than the minimum clearance between the rotating and stationary parts.
Figure 9: Shaft Sleeve 
Shaft Sleeves
Pump shafts are usually protected from erosion, corrosion, and wear at the seal chambers, leakage joints, internal bearings, and in the waterways by renewable sleeves. Unless otherwise specified, a shaft sleeve of wear, corrosion, and erosionresistant material shall be provided to protect the shaft. The sleeve shall be sealed at one end. The shaft sleeve assembly shall extend beyond the outer face of the seal gland plate. (Leakage between the shaft and the sleeve should not be confused with leakage through the mechanical seal).
Coupling
Couplings can compensate for axial growth of the shaft and transmit torque to the impeller. Shaft couplings can be broadly classified into two groups: rigid and flexible. Rigid couplings are used in applications where there is absolutely no possibility or room for any misalignment. Flexible shaft couplings are more prone to selection, installation and maintenance errors. Flexible shaft couplings can be divided into two basic groups: elastomeric and nonelastomeric
Auxilliary Components
Auxiliary components generally include the following piping systems for the following services:
Auxiliary piping systems include tubing, piping, isolating valves, control valves, relief valves, temperature gauges and thermocouples, pressure gauges, sight flow indicators, orifices, seal flush coolers, dual seal barrier/buffer fluid reservoirs, and all related vents and drains.
All auxiliary components shall comply with the requirements as per standard codes like API 610 (refinery services), API 682 (shaft sealing systems) etc.
Definition of Important Terms
The key performance parameters of centrifugal pumps are capacity, head, BHP (Brake horse power), BEP (Best efficiency point) and specific speed. The pump curves provide the operating window within which these parameters can be varied for satisfactory pump operation. The following parameters or terms are discussed in detail in this section.
Capacity
Head
NPSH
Power (Brake Horse Power, B.H.P) and Efficiency (Best Efficiency Point, B.E.P)
Specific Speed (Ns)
Affinity Laws
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 (m^{3}/hr). The capacity usually changes with the changes in operation of the process. For example, a boiler feed pump is an application that needs a constant pressure with varying capacities to meet a changing steam demand.
The capacity depends on a number of factors like:
For a pump with a particular impeller running at a certain speed in a liquid, the only items on the list above that can change the amount flowing through the pump are the pressures at the pump inlet and outlet. 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:  Eq. (3) 
Head
Significance of Using the Term "Head" Instead of the Term "Pressure"
The pressure at any point in a liquid can be thought of as being caused by a vertical column of the liquid 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. Imagine a pipe shooting a jet of water straight up into the air, the height the water goes up would be the head.
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 pump's 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 pump's head and forget about the pressure.
So a centrifugal pump's performance on any Newtonian fluid, whether it's heavy (sulfuric acid) or light (gasoline) is described by using the term 'head'. The pump performance curves are mostly described in terms of head.
A given pump with a given impeller diameter and speed will raise a liquid to a certain height regardless of the 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:
Eq. (4) 
Newtonian liquids have specific gravities typically ranging from 0.5 (light, like light hydrocarbons) to 1.8 (heavy, like concentrated sulfuric acid). Water is a benchmark, having a specific gravity of 1.0.
This formula helps in converting pump gauge pressures to head for reading the pump curves.
The various head terms are discussed below.
Note: The Subscripts 's' refers to suction conditions and 'd' refers to discharge conditions.
Static Suction Head (h_{s})
Head resulting from elevation of the liquid relative to the pump center line. If the liquid level is above pump centerline, h_{S} is positive. If the liquid level is below pump centerline, h_{S} is negative. Negative h_{S} condition is commonly denoted as a "suction lift" condition.
Static Discharge Head, (h_{d})
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.
Friction Head (h_{f})
The 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 pipefittings, flow rate, and nature of the liquid.
Vapor Pressure Head (h_{vp})
Vapor pressure is the pressure at which a liquid and its vapor coexist in equilibrium at a given temperature. The vapor pressure of liquid can be obtained from vapor pressure tables. When the vapor pressure is converted to head, it is referred to as vapor pressure head, h_{vp}. The value of h_{vp} 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.
Pressure Head (h_{p})
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 h_{p}, 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, h_{p} equals atmospheric pressure head.
Velocity Head (h_{v})
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.
Total Suction Head (H_{s})
The suction reservoir pressure head (hp_{S}) plus the static suction head (h_{S}) plus the velocity head at the pump suction flange (h_{VS}) minus the friction head in the suction line (hf_{S}).
H_{S}= hp_{S} + h_{S} + hv_{S}  hf_{S}  Eq. (5) 
The total suction head is the reading of the gauge on the suction flange, converted to feet of liquid.
Total Discharge Head (H_{d})
The discharge reservoir pressure head (hp_{d}) plus static discharge head (h_{d}) plus the velocity head at the pump discharge flange (hv_{d}) plus the total friction head in the discharge line (hf_{d}).
H_{d}= hp_{d} + h_{d} + hv_{d} + hf_{d}  Eq. (6) 
The total discharge head is the reading of a gauge at the discharge flange, converted to feet of liquid.
Total Differential Head (H_{T})
It is the total discharge head minus the total suction head or
H_{T} = H_{d} + H_{S} (with a suction lift)  Eq. (7) 
H_{T} = H_{d}  H_{S} (with a suction head)  Eq. (8) 
NPSH
When discussing centrifugal pumps, the two most important head terms are NPSHr and NPSHa.
Net Positive Suction Head Required, NPSHr
NPSH is one of the most widely used and least understood terms associated with pumps. Understanding the significance of NPSH is very much essential during installation as well as operation of the pumps.
Pumps Can Only Pump Liquids, Not Vapor{parse block="google_articles"}
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 ft^{3} of water at room temperature becomes 1700 ft^{3} 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
The manufacturer usually tests the pump with water at different capacities, created by throttling the suction side. When the first signs of vaporization induced cavitation occur, the suction pressure is noted (the term cavitation is discussed in detail later). This pressure is converted into the head. This head number is published on the pump curve and is referred as the "net positive suction head required (NPSHr) or sometimes in short as the NPSH. Thus the Net Positive Suction Head (NPSH) is the total head at the suction flange of the pump less the vapor pressure converted to fluid column height of the liquid.
NPSHr is a function of pump design
NPSH required is a function of the pump design and is determined based on actual pump test by the vendor. As the liquid passes from the pump suction to the eye of the impeller, the velocity increases and the pressure decreases. There are also pressure losses due to shock and turbulence as the liquid strikes the impeller. The centrifugal force of the impeller vanes further increases the velocity and decreases the pressure of the liquid. The NPSH required is the positive head in feet absolute required at the pump suction to overcome these pressure drops in the pump and maintain the majority of the liquid above its vapor pressure.
The NPSH is always positive since it is expressed in terms of absolute fluid column height. The term "Net" refers to the actual pressure head at the pump suction flange and not the static suction head.
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. Pump manufacturer's curves normally provide this information. The NPSH is independent of the fluid density as are all head terms.
Net Positive Suction Head Available, NPSHa
NPSHa is a function of system design
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.
NPSHa calculation
The formula for calculating the NPSHa is stated below:
NPSHa_{s} = hp_{s} + h_{s}  hvp_{s} + hf_{s} where: hp_{s} = Head pressure or the barometric pressure of the vessel converted to head  Eq. (9) 
NPSHa in a Nutshell
In a nutshell, NPSH available is defined as:
NPSHa = Pressure head + Static head  Vapor pressure head of your product  Friction head loss in the piping, valves and fittings. where all terms are in absolute feed of head.  Eq. (10) 
In an existing system, the NPSHa can also be approximated by a gauge on the pump suction using the formula:
NPSHa = hp_{S } hvp_{S }Â± hg_{S} + hv_{S} where: hp_{S} = Barometric pressure in feet absolute  Eq. (11) 
Significance of NPSHr and NPSHa
The NPSH available must always be greater than the NPSH required for the pump to operate properly. It is normal practice to have at least 2 to 3 feet of extra NPSH available at the suction flange to avoid any problems at the duty point.
Power and Efficiency
Brake Horse Power (BHP)
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)  Eq. (12) 
where: Q = Capacity in gallons per minute (GPM)  Eq. (13) 
The constant 3960 is obtained by dividing the number or footpounds for one horsepower (33,000) by the weight of one gallon of water (8.33 pounds). BHP can also be read from the pump curves at any flow rate. Pump curves are based on a specific gravity of 1.0. Other liquids' specific gravity must be considered. 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.
Eq. (14) 
Best Efficiency Point (BEP)
The H, 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. All points to the right or left of BEP have a lower efficiency.
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.
BEP as a measure of mechanically stable operation
The impeller is subject to nonsymmetrical forces when operating to the right or left of the BEP. These forces manifest themselves in many mechanically unstable conditions like vibration, excessive hydraulic thrust, temperature rise, and erosion and separation cavitation. Thus 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 as an important parameter in calculations
BEP is an important parameter in that many parametric calculations such as specific speed, suction specific speed, hydrodynamic size, viscosity correction, head rise to shutoff, 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 (N_{s}) is a nondimensional 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 sizefactor 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 in gallons per minute (GPM)  Eq. (15) 
The understanding of this definition is of design engineering significance only, however, and specific speed should be thought of only as an index used to predict certain pump characteristics.
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 onefoot head.
Specific Speed as a Measure of the Shape or Class of the Impellers
The specific speed determines the general shape or class of the impellers. As the specific speed increases, the ratio of the impeller outlet diameter, D2, to the inlet or eye diameter, D1, decreases. This ratio becomes 1.0 for a true axial flow impeller. Radial flow impellers develop head principally through centrifugal force. Radial impellers are generally low flow high head designs. Pumps of higher specific speeds develop head partly by centrifugal force and partly by axial force. A higher specific speed indicates a pump design with head generation more by axial forces and less by centrifugal forces. An axial flow or propeller pump with a specific speed of 10,000 or greater generates its head exclusively through axial forces. Axial flow impellers are high flow low head designs.
Specific speed identifies the approximate acceptable ratio of the impeller eye diameter (D1) to the impeller maximum diameter (D2) in designing a good impeller.
Ns: 500 to 5000;D1/D2 > 1.5 radial flow pump
Ns: 5000 to 10000;D1/D2 < 1.5 mixed flow pump
Ns: 10000 to 15000; D1/D2 = 1  axial flow pump
Specific speed is also used in designing a new pump by sizefactoring a smaller pump of the same specific speed. The performance and construction of the smaller pump are used to predict the performance and model the construction of the new pump.
Suction Specific Speed (Nss)
Suction specific speed (Nss) is a dimensionless number or index that defines the suction characteristics of a pump. It is calculated from the same formula as Ns by substituting H by NPSHr.
In multistage pump the NPSHr is based on the first stage impeller NPSHR. Nss is commonly used as a basis for estimating the safe operating range of capacity for a pump. The higher the Nss is, the narrower is its safe operating range from its BEP. The numbers range between 3,000 and 20,000. Most users prefer that their pumps have Nss in the range of 8000 to 11000 for optimum and troublefree operation.
The Affinity Laws
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. According to the Affinity Laws:
Capacity (Q) changes in direct proportion to impeller diameter D ratio, or to speed N ratio:
Q_{2} = Q_{1} x [D_{2}/D_{1}]  Eq. (16) 
Q_{2} = Q_{1} x [N_{2}/N_{1}]  Eq. (17) 
Head (H) changes in direct proportion to the square of impeller diameter D ratio, or the square of speed N ratio:
H_{2} = H_{1} x [D_{2}/D_{1}]^{2}  Eq. (18) 
H_{2} = H_{1} x [N_{2}/N_{1}]^{2}  Eq. (19) 
BHP changes in direct proportion to the cube of impeller diameter ratio, or the cube of speed ratio:
BHP_{2} = BHP_{1} x [D_{2}/D_{1}]^{3}  Eq. (20) 
BHP_{2} = BHP_{1} x [N_{2}/N_{1}]^{3}  Eq. (21) 
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:
Q_{2} = Q_{1} x [(D_{2 }x N_{2})/(D_{1 }x N_{1})]  Eq. (22) 
H_{2} = H_{1} x [(D_{2 }x N_{2})/(D_{1 }x N_{1})]^{2}  Eq. (23) 
BHP_{2} = BHP_{1} x [(D_{2 }x N_{2})/(D_{1 }x N_{1})]^{3}  Eq. (24) 
These equations are used to handcalculate the impeller trim diameter from a given pump performance curve at a bigger diameter.
The Affinity Laws are valid only under conditions of constant efficiency
Understanding Centrifugal Pump Performance Curves
The capacity and pressure needs of any system can be defined with the help of a graph called a system curve. Similarly the capacity vs. pressure variation graph for a particular pump defines its characteristic pump performance curve.
The pump suppliers try to match the system curve supplied by the user with a pump curve that satisfies these needs as closely as possible. A pumping system operates where the pump curve and the system resistance curve intersect. The intersection of the two curves defines the operating point of both pump and process. However, it is impossible for one operating point to meet all desired operating conditions. For example, when the discharge valve is throttled, the system resistance curve shift left and so does the operating point.
Figure 10: Typical System and Pump Performance Curves 
Developing a System Curve
The system resistance or system head curve is the change in flow with respect to head of the system. It must be developed by the user based upon the conditions of service. These include physical layout, {parse block="google_articles"}process conditions, and fluid characteristics. It represents the relationship between flow and hydraulic losses in a system in a graphic form and, since friction losses vary as a square of the flow rate, the system curve is parabolic in shape. Hydraulic losses in piping systems are composed of pipe friction losses, valves, elbows and other fittings, entrance and exit losses, and losses from changes in pipe size by enlargement or reduction in diameter.
Developing a Pump Performance Curve
A pump's performance is shown in its characteristics performance curve where its capacity i.e. flow rate is plotted against its developed head. The pump performance curve also shows its efficiency (BEP), required input power (in BHP), NPSHr, speed (in RPM), and other information such as pump size and type, impeller size, etc. This curve is plotted for a constant speed (rpm) and a given impeller diameter (or series of diameters). It is generated by tests performed by the pump manufacturer. Pump curves are based on a specific gravity of 1.0. Other specific gravities must be considered by the user.
Normal Operation Range
A typical performance curve (Figure 10) is a plot of Total Head vs. Flow rate for a specific impeller diameter. The plot starts at zero flow. The head at this point corresponds to the shutoff head point of the pump. The curve then decreases to a point where the flow is maximum and the head minimum. This point is sometimes called the runout point. The pump curve is relatively flat and the head decreases gradually as the flow increases. This pattern is common for radial flow pumps. Beyond the runout point, the pump cannot operate. The pump's range of operation is from the shutoff head point to the runout point. Trying to run a pump off the right end of the curve will result in pump cavitation and eventually destroy the pump.
By plotting the system head curve and pump curve together, you can determine:
1.Where the pump will operate on its curve?
2.What changes will occur if the system head curve or the pump performance curve changes?
Requirements for Consistent Operation
References
In Part I of this article, two basic requirements for trouble free operation and longer service life of centrifugal pumps are mentioned in brief:
The condition of cavitation is essentially an indication of an abnormality in the pump suction system, whereas the condition of low flow indicates an abnormality in the entire pumping system or process. The two conditions are also interlinked such that a low flow situation can also induce cavitation.
Cavitation is a common occurrence but is the least understood of all pumping problems. Cavitation means different things to different people. Some say when a pump makes a rattling or knocking sound along with vibrations, it is cavitating. Some call it slippage as the pump discharge pressure slips and flow becomes erratic. When cavitating, the pump not only fails to serve its basic purpose of pumping the liquid but also may experience internal damage, leakage from the seal and casing, bearing failure, etc.
In summary, cavitation is an abnormal condition that can result in loss of production, equipment damage and worst of all, injury to personnel.
The plant engineer's job is to quickly detect the signs of cavitation, correctly identify the type and cause of the cavitation and eliminate it. A good understanding of the concept is the key to troubleshooting any cavitation related pumping problem.
The Meaning of Cavitation
The term â€˜cavitation' comes from the Latin word cavus, which means a hollow space or a cavity. Webster's Dictionary defines the word â€˜cavitation' as the rapid formation and collapse of cavities in a flowing liquid in regions of very low pressure.
In any discussion on centrifugal pumps various terms like vapor pockets, gas pockets, holes, bubbles, etc. are used in place of the term cavities. These are one and the same thing and need not be confused. The term bubble shall be used hereafter in the discussion.
In the context of centrifugal pumps, the term cavitation implies a dynamic process of formation of bubbles inside the liquid, their growth and subsequent collapse as the liquid flows through the pump.Generally, the bubbles that form inside the liquid are of two types: Vapor bubbles or Gas bubbles.
Both types of bubbles are formed at a point inside the pump where the local static pressure is less than the vapor pressure of the liquid (vaporous cavitation) or saturation pressure of the gas (gaseous cavitation).
Vaporous cavitation is the most common form of cavitation found in process plants. Generally it occurs due to insufficiency of the available NPSH or internal recirculation phenomenon. It generally manifests itself in the form of reduced pump performance, excessive noise and vibrations and wear of pump parts. The extent of the cavitation damage can range from a relatively minor amount of pitting after years of service to catastrophic failure in a relatively short period of time.
Gaseous cavitation occurs when any gas (most commonly air) enters a centrifugal pump along with liquid. A centrifugal pump can handle air in the range of Â½ % by volume. If the amount of air is increased to 6%, the pump starts cavitating. The cavitation condition
is also referred to as Air binding. It seldom causes damage to the impeller or casing. The main effect of gaseous cavitation is loss of capacity.
The different types of cavitation, their specific symptoms and specific corrective actions shall be explored in the next part of the article. However, in order to clearly identify the type of cavitation, let us first understand the mechanism of cavitation, i.e. how cavitation occurs. Unless otherwise specified, the term cavitation shall refer to vaporous cavitation.
Important Definitions
To enable a clear understanding of mechanism of cavitation, definitions of following important terms are explored:
Static Pressure (P_{s})
The static pressure in a fluid stream is the normal force per unit area on a solid boundary moving with the fluid. It describes the difference between the pressure inside and outside a system, disregarding any motion in the system. For instance, when referring to an air duct, static pressure is the difference between the pressure inside the duct and outside the duct, disregarding any airflow inside the duct. In energy terms, the static pressure is a measure of the potential energy of the fluid.
Dynamic Pressure (P_{d})
A moving fluid stream exerts a pressure higher than the static pressure due to the kinetic energy (Â½ mv^{2}) of the fluid. This additional pressure is defined as the dynamic pressure. The dynamic pressure can be measured by converting the kinetic energy of the fluid stream into the potential energy. In other words, it is pressure that would exist in a fluid stream that has been decelerated from its velocity â€˜v' to â€˜zero' velocity.
Total Pressure (P_{t})
The sum of static pressure and dynamic pressure is defined as the total pressure. It is a measure of total energy of the moving fluid stream. i.e. both potential and kinetic energy.
Relationship Between P_{s}, P_{d}, and P_{t}
In an incompressible flow, the relation between static, dynamic and total pressures can be found out using a simple device called Pitot tube (named after Henri Pitot in 1732) shown in Figure 1.
The Pitot tube has two tubes:
Figure 1: Sketch of a Pilot Tube 
The two tubes are connected to the legs of a manometer or equivalent device for measuring pressure.
The relation between p_{s}, p_{d} and p_{t }can be derived by applying a simple energy balance.
Potential Energy + Kinetic Energy = Total Energy (Constant)  Eq. (1) 
As mentioned earlier, in the case of a fluid or gas the potential energy is represented by the static pressure and the kinetic energy by dynamic pressure. The kinetic energy is a function of the motion of the fluid, and of course it's mass. It is generally more convenient to use the density of the fluid (ρ) as the mass representation.
K.E = p_{d} = Â½ m v^{2 }= Â½ρ v^{2}  Eq. (2) 
The corresponding pressure balance equation is^{:}
Eq. (3) 
In place of the pressure terms as used above, it is more appropriate to speak of the energy during pumping as the energy per unit weight of the liquid pumped and the units of energy expressed this way are footpounds per pound (Newtonmeters per Newton) or just feet (meters) i.e. the units of head. Thus the energy of the liquid at a given point in flow stream can be expressed in terms of head of liquid in feet.
The pressure term can be converted to head term by division with the factor â€˜ρ g'. For unit interconversions the factor â€˜ρ g/g_{c}'. should be used in place of â€˜ρg'.
Static Pressure Head
The head corresponding to the static pressure is called as the static pressure head.
Eq. (4) 
Velocity Head
The head corresponding to dynamic pressure is called the velocity head.
Eq. (5) 
From the reading h_{m}of the manometer velocity of flow can be calculated and thus velocity head can be calculated. The pressure difference, dP (P_{t } P_{s)} indicated by the manometer is the dynamic pressure.
Eq. (6)  
Eq. (7) 
Vapor Pressure (P_{v})
Vapor pressure is the pressure required to keep a liquid in a liquid state. If the pressure applied to the surface of the liquid is not enough to keep the molecules pretty close together, the molecules will be free to separate and roam around as a gas or vapor. The vapor pressure is dependent upon the temperature of the liquid. Higher the temperature, higher will be the vapor pressure.
Mechanism of Cavitation
The phenomenon of cavitation is a stepwise process as shown in Figure 2.
Figure 2: Steps in Cavitation 
The bubbles form inside the liquid when it vaporises i.e. phase change from liquid to vapor. But how does vaporization of the liquid occur during a pumping operation?
Vaporization of any liquid inside a closed container can occur if either pressure on the liquid surface decreases such that it becomes equal to or less than the liquid vapor pressure at the operating temperature, or the temperature of the liquid rises, raising the vapor pressure such that it becomes equal to or greater than the operating pressure at the liquid surface. For example, if water at room temperature (about 77 Â°F) is kept in a closed container and the system pressure is reduced to its vapor pressure (about 0.52 psia), the water quickly changes to a vapor. Also, if the operating pressure is to remain constant at about 0.52 psia and the temperature is allowed to rise above 77 ^{Â°}F, then the water quickly changes to a vapor.
Just like in a closed container, vaporization of the liquid can occur in centrifugal pumps when the local static pressure reduces below that of the vapor pressure of the liquid at the pumping temperature.
The vaporisation accomplished by addition of heat or the reduction of static pressure without dynamic action of the liquid is excluded from the definition of cavitation. For the purposes of this article, only pressure variations that cause cavitation shall be explored. Temperature changes must be considered only when dealing with systems that introduce or remove heat from the fluid being pumped.Step One: Formation of Bubbles
To understand vaporization, two important points to remember are:
Thus, the key concept is  vapor bubbles form due to vaporization of the liquid being pumped when the local static pressure at any point inside the pump becomes equal to or less than the vapor pressure of the liquid at the pumping temperature.
The reduction in local static pressure at any point inside the pump can occur under two conditions:
Pressure Reduction in the External Suction of the Pump
Figure 3: External Suction System 
A simple sketch of a pump external suction system in shown in Figure 3. The nomenclature used for this figure is as follows:
ρ  Liquid density in lb_{m} / ft^{3}
G  Acceleration due to gravity in ft / s^{2}
Psn  p refers to local static pressure (absolute). Subscript s refers to suction and subscript n refers to the point of measurement. The pressure at any point can be converted to the head term by division with the factor  ρ g
p_{s1}  Static pressure (absolute) of the suction vessel in psia
hp_{s1}  Static pressure head i.e. absolute static pressure on the liquid surface in the suction vessel, converted to feet of head (p_{s1}/ ρ g/g_{c}). If the system is open, hp_{s1} equals the atmospheric pressure head.
v_{s1}  Liquid velocity on the surface in the suction vessel in ft/s
hv_{s1 } Velocity head i.e. the energy of a liquid as a result of its motion at some velocity â€˜v_{s1}'. (v^{2}_{s1} / 2g). It is the equivalent head in feet through which the liquid would have to fall to acquire the same velocity, or the head necessary to accelerate the liquid to velocity v_{s1}. In a large suction vessel, the velocity head is practically zero and is typically ignored in calculations.
h_{s}  Static suction head. . . . i.e. head resulting from elevation of the liquid relative to the pump centerline. If the liquid level is above pump centerline, h_{S} is positive. If the liquid level is below pump centerline, h_{S} is negative. A negative h_{S} condition is commonly referred to as "suction lift".
hf_{s}  Friction head i.e. the head required to overcome the resistance to flow in the pipe, valves and fittings between points A and B, inclusive of the entrance losses at the point of connection of suction piping to the suction vessel (point A in Figure 1). The friction head is dependent upon the size, condition and type of pipe, number and type of fittings, valves, flow rate and the nature of the liquid. The friction head varies as the square of the average velocity of the flowing fluid.
p_{s2}  Absolute static pressure at the suction flange in psia
hp_{s2}  Static pressure head at the suction flange i.e. absolute pressure of the liquid at the suction flange, converted to feet of head  p_{s2} / ρ g/g_{c}
v_{s2}  Velocity of the moving liquid at the suction flange in ft/s. The pump suction piping is sized such that the velocity at the suction remains low.
hv_{s2 } Velocity head at suction flange i.e. the energy of a liquid as a result of its motion at average velocity â€˜v_{s2}' equal to v^{2}_{s2} / 2g.
p_{v}  Absolute vapor pressure of the liquid at operating temperature in psia.
hp_{v } Vapor Pressure head i.e. absolute vapor pressure converted to feet of head (p_{v} / ρ g/g_{c}).
H_{s } Total Suction Head available at the suction flange in ft.
Note: As pressure is measured in absolute, total head is also in absolute.
The pump takes suction from a vessel having a certain liquid level. The vessel can be pressurised (as shown in the Figure 3) or can be at atmospheric pressure or under vacuum.
Calculation of the Total Suction Head, H_{s}
The external suction system of the pump provides a certain amount of head at the suction flange. This is referred to as Total Suction Head (TSH), H_{s}.
TSH can be calculated by application of the energy balance. The incompressible liquid can have energy in the form of velocity, pressure or elevation. Energy in various forms is either added to or subtracted from the liquid as it passes through the suction piping. The head term in feet (or meters) is used as an expression of the energy of the liquid at any given point in the flow stream.
As shown in Figure 3, the total suction head, H_{s}, available at the suction flange is given by the equation,
H_{s }= hp_{s1 }+ hv_{s1 }+ h_{s}  hf_{s }+ hv_{s2}  Eq. (8) 
For an existing system, Hs_{ }can also be calculated from the pressure gauge reading at pump suction flange,
H_{s }= hp_{s2 }+ hv_{s2}  Eq. (9) 
Equations8 and9 above include the velocity head terms hv_{s1 }and_{ }hv_{s2},_{ }respectively.
Velocity Head
There is a lot of confusion as to whether the velocity head terms should be added or subtracted in the head calculations. To avoid any confusion remember the following:
Just like a static tube of Pitot, a pressure gauge can measure only the static pressure at the point of connection. It does not measure the dynamic pressure as the opening of the gauge impulse pipe is parallel to the direction of flow and there is no velocity component perpendicular to its opening.
Figure 4: Measuing Static Pressure 
In Figure 4 below, flow through a pipe of varying cross section area is shown. As the cross section at point B reduces, the velocity of flow increases. The rise in kinetic energy happens at the expense of potential energy. Assuming that there are no friction losses, the total energy (sum of potential energy and kinetic energy) of fluid at point A, B and C remains constant. The pressure gauges at point A, B and C measure only the potential energy i.e. the static pressures at respective points. The drop in static pressure from 10 psi (point A) to 5 psi (point B') occurs owing to rise the dynamic pressure by 5 psi i.e. increase in velocity at point B. However the gauge at point B records only the static pressure. The velocity decreases from point B to C and the static pressure is recovered again to 10 psi.
At a particular point of flow, the total pressure is the sum of the static pressure and the dynamic pressure. Thus, theoretically, the velocity head terms must always be added and not subtracted, in calculating Total Suction Head (TSH), H_{s}. However, practically speaking, the value of these terms is not significant in comparison to the other terms in the equation.
Therefore, neglecting the velocity head terms, Equations8 and9 simplify to:
H_{s }= hp_{s1 }+ h_{s}  hf_{s }  Eq. (10) 
Â
H_{s }= hp_{s2 }  Eq. (11) 
Two important inferences can be drawn from the above equations:
Therefore the pump's external suction system should be designed such that the static pressure available at the suction flange is always positive and higher than the vapor pressure of the liquid at the pumping temperature.
For no vaporization at pump suction flange,
(p_{s2 }> p_{v)} or_{ }(p_{s2 } p_{v }) or (hp_{s2 } hp_{v} ) > 0  Eq. (12) 
As the liquid enters the pump, there is a further reduction in the static pressure. If the value of p_{s2 }is not sufficiently higher than p_{v}, at some point inside the pump the static pressure can reduce to the value of p_{v}. In pumping terminology, the head difference term corresponding to Equation 5 (hp_{s2 } hp_{v}) is called the Net Positive Suction Head or NPSH. The NPSH term shall be explored in detail in the next part of the article. For now, the readers should focus only on how the static pressure within the pump may be reduced to a value lower than that of the liquid vapor pressure.
Pressure Reduction in the Internal Suction System of the Pump
The pressure of the fluid at the suction flange is further reduced inside the internal suction system of the pump.
Figure 5: Internal Pump Locations  Figure 6: Internal Pump Nomenclature 
The internal suction system is comprised of the pump's suction nozzle and impeller. Figures 5 and 6 depict the internal parts in detail. A closer look at the graphic is a must in understanding the mechanism of pressure drop inside the pump.
In Figure 7, it can be seen that the passage from the suction flange (point 2) to the impeller suction zone (point 3) and to the impeller eye (point 4) acts like a venturi i.e. there is gradual reduction in the crosssection area.
Figure 7: Pump Internal Suction System 
In the impeller, the point of minimum radius (r_{eye}) with reference to pump centerline is referred to as the "eye" of the impeller (Figure 8).
According to Bernoulli's principle, when a constant amount of liquid moves through a path of decreasing crosssection area (as in a venturi), the velocity increases and the static pressure decreases. In other words, total system energy i.e. sum of the potential and kinetic energy, remains constant in a flowing system (neglecting friction). The gain in velocity occurs at the expense of pressure. At the point of minimum crosssection, the velocity is at a maximum and the static pressure is at a minimum.
The pressure at the suction flange, p_{s2} (Point 2) decreases as the liquid flows from the suction flange, through the suction nozzle and into the impeller eye. This decrease in pressure occurs not only due to the venturi effect but also due to the friction in the inlet passage. However, the pressure drop due to friction between the suction nozzle and the impeller eye is comparatively small for most pumps. However the pressure reduction due to the venturi effect is very significant as the velocity at the impeller increases to 15 to 20 ft/s. There is a further drop in pressure due to shock and turbulence as the liquid strikes and loads the edges of impeller vanes. The net effect of all the pressure drops is the creation of a very lowpressure area around the impeller eye and at the beginning of the trailing edge of the impeller vanes.
Figure 8: Impeller Eye  Figure 9: Pressure Profile in a Pump 
The pressure reduction profile within the pump is depicted in Figure 9.
As shown in Figure 9, the impeller eye is the point where the static pressure is at a minimum, p_{4. }During pump operation, if the local static pressure of the liquid at the lowest pressure becomes equal to or less than the vapor pressure (p_{v})_{ }of the liquid at the operating temperature, vaporization of the liquid (the formation of bubbles) begins i.e. when p_{4 }<= p_{v.}
It is at the beginning of the trailing edge of the vanes near the impeller eye where the pressure actually falls to below the liquid vapor pressure. The region of bubble formation is shown in Figure 10.
Figure 10: Impeller Cavitation Regions 
In summary, vaporization of the liquid (bubble formation) occurs due to the reduction of the static pressure to a value below that of the liquid vapor pressure. The reduction of static pressure in the external suction system occurs mainly due to friction in suction piping. The reduction of static pressure in the internal suction system occurs mainly due to the rise in the velocity at the impeller eye.
Step Two: Growth of Bubbles
Unless there is no change in the operating conditions, new bubbles continue to form and old bubbles grow in size. The bubbles then get carried in the liquid as it flows from the impeller eye to the impeller exit tip along the vane trailing edge. Due to impeller rotating action, the bubbles attain very high velocity and eventually reach the regions of high pressure within the impeller where they start collapsing. The life cycle of a bubble has been estimated to be in the order of 0.003 seconds.
Step Three: Collapse of Bubbles
Figure 11: Collapse of Vapor Bubbles 
As the vapor bubbles move along the impeller vanes, the pressure around the bubbles begins to increase until a point is reached where the pressure on the outside of the bubble is greater than the pressure inside the bubble. The bubble collapses. The process is not an explosion but rather an implosion (inward bursting). Hundreds of bubbles collapse at approximately the same point on each impeller vane. Bubbles collapse nonsymmetrically such that the surrounding liquid rushes to fill the void forming a liquid microjet. The micro jet subsequently ruptures the bubble with such force that a hammering action occurs.Bubble collapse pressures greater than 1 GPa (145x10^{6} psi) have been reported. The highly localized hammering effect can pit the pump impeller. The pitting effect is illustrated schematically in Figure 11.
After the bubble collapses, a shock wave emanates outward from the point of collapse. This shock wave is what we actually hear and what we call "cavitation". The implosion of bubbles and emanation of shock waves (red color) is shown in a small video clip shown below.
In nutshell, the mechanism of cavitation is all about formation, growth and collapse of bubbles inside the liquid being pumped. But how can the knowledge of mechanism of cavitation can really help in troubleshooting a cavitation problem. The concept of mechanism can help in identifying the type of bubbles and the cause of their formation and collapse. The troubleshooting method shall be explored in detail in the next part of the article.
Next let us explore the general symptoms of cavitation and its affects on pump performance.
Video: Cavitation in a Centrifugal Pump 
Perceptible indications of the cavitation during pump operation are more or less loud noises, vibrations and an unsteadily working pump. Fluctuations in flow and discharge pressure take place with a sudden and drastic reduction in head rise and pump capacity. Depending upon the size and quantum of the bubbles formed and the severity of their collapse, the pump faces problems ranging from a partial loss in{parse block="google_articles"} capacity and head to total failure in pumping along with irreparable damages to the internal parts. It requires a lot of experience and thorough investigation of effects of cavitation on pump parts to clearly identify the type and root causes of cavitation.
A detailed description of the general symptoms is given as follows.
Reduction in Capacity of the Pump
The formation of bubbles causes a volume increase decreasing the space available for the liquid and thus diminish pumping capacity. For example, when water changes state from liquid to gas its volume increases by approximately 1,700 times. If the bubbles get big enough at the eye of the impeller, the pump "chokes" i.e. loses all suction resulting in a total reduction in flow. The unequal and uneven formation and collapse of bubbles causes fluctuations in the flow and the pumping of liquid occurs in spurts. This symptom is common to all types of cavitations.
Decrease in the Head Developed
Bubbles unlike liquid are compressible. The head developed diminishes drastically because energy has to be expended to increase the velocity of the liquid used to fill up the cavities, as the bubbles collapse. As mentioned earlier, The Hydraulic Standards Institute defines cavitation as condition of 3 % drop in head developed across the pump. Like reduction in capacity, this symptom is also common to all types of cavitations.
Figure 12: Pump Performance Curves 
Thus, the hydraulic effect of a cavitating pump is that the pump performance drops off of its expected performance curve, referred to as break away, producing a lower than expected head and flow. The Figure 12 depicts the typical performance curves. The solid line curves represent a condition of adequate NPSHa whereas the dotted lines depict the condition of inadequate NPSHa i.e. the condition of cavitation.
Abnormal Sound and Vibration
It is movement of bubbles with very high velocities from lowpressure area to a highpressure area and subsequent collapse that creates shockwaves producing abnormal sounds and vibrations. It has been estimated that during collapse of bubbles the pressures of the order of 10^{4} atm develops.
The sound of cavitation can be described as similar to small hard particles or gravel rapidly striking or bouncing off the interior parts of a pump or valve. Various terms like rattling, knocking, crackling are used to describe the abnormal sounds. The sound of pumps operating while cavitating can range from a lowpitched steady knocking sound (like on a door) to a highpitched and random crackling (similar to a metallic impact). People can easily mistake cavitation for a bad bearing in a pump motor. To distinguish between the noise due to a bad bearing or cavitation, operate the pump with no flow. The disappearance of noise will be an indication of cavitation.
Similarly, vibration is due to the uneven loading of the impeller as the mixture of vapor and liquid passes through it, and to the local shock wave that occurs as each bubble collapses. Very few vibration reference manuals agree on the primary vibration characteristic associated with pump cavitation. Formation and collapsing of bubbles will alternate periodically with the frequency resulting out of the product of speed and number of blades. Some suggest that the vibrations associated with cavitation produce a broadband peak at high frequencies above 2,000 Hertz. Some suggest that cavitation follows the vane pass frequency (number of vanes times the running speed frequency) and yet another indicate that it affects peak vibration amplitude at one times running speed. All of these indications are correct in that pump cavitation can produce various vibration frequencies depending on the cavitation type, pump design, installation and use. The excessive vibration caused by cavitation often subsequently causes a failure of the pump's seal and/or bearings. This is the most likely failure mode of a cavitating pump,
Damage to Pump Parts
Cavitation Erosion or Pitting
Figure 13: Photographic Evidence of Cavitation 
During cavitation, the collapse of the bubbles occurs at sonic speed ejecting destructive micro jets of extremely high velocity (up to 1000 m/s) liquid strong enough to cause extreme erosion of the pump parts, particularly impellers. The bubble is trying to collapse from all sides, but if the bubble is lying against a piece of metal such as the impeller or volute it cannot collapse from that side. So the fluid comes in from the opposite side at this high velocity and bangs against the metal creating the impression that the metal was hit with a "ball pin hammer". The resulting longterm material damage begins to become visible by so called
Pits (see Figure 11), which are plastic deformations of very small dimensions (order of magnitude of micrometers). The damage caused due to action of bubble collapse is commonly referred as Cavitation erosion or pitting. The Figure 13 depicts the cavitation pitting effect on impeller and diffuser surface.
Cavitation erosion from bubble collapse occurs primarily by fatigue fracture due to repeated bubble implosions on the cavitating surface, if the implosions have sufficient impact force. The erosion or pitting effect is quite similar to sand blasting. High head pumps are more likely to suffer from cavitation erosion, making cavitation a "highenergy" pump phenomenon.
The most sensitive areas where cavitation erosion has been observed arethe lowpressure sides of the impeller vanes near the inlet edge. The cavitation erosion damages at the impeller are more or less spread out. The pitting has also been observed on impeller vanes, diffuser vanes, and impeller tips etc. In some instances, cavitation has been severe enough to wear holes in the impeller and damage the vanes to such a degree that the impeller becomes completely ineffective. A damaged impeller is shown in Figure 14.
Figure 14: Cavitation Damage on Impellers 
The damaged impeller shows that the shock waves occurred near the outside edge of the impeller, where damage is evident. This part of the impeller is where the pressure builds to its highest point. This pressure implodes the gas bubbles, changing the water's state from gas into liquid. When cavitation is less severe, the damage can occur further down towards the eye of the impeller. A careful investigation and diagnosis of point of the impeller erosion on impeller, volute, diffuser etc. can help predict the type and cause of cavitation.
The extent of cavitation erosion or pitting depends on a number of factors like presence of foreign materials in the liquid, liquid temperature, age of equipment and velocity of the collapsing bubble.
Mechanical Deformation
Apart from erosion of pump parts, in bigger pumps, longer duration of cavitation condition can result in unbalancing (due to unequal distribution in bubble formation and collapse) of radial and axial thrusts on the impeller. This unbalancing often leads to following mechanical problems:
These mechanical deformations can completely wreck the pump and require replacement of parts. The cost of such replacements can be huge.
Cavitation Corrosion
Frequently cavitation is combined with corrosion. The implosion of bubbles destroys existing protective layers making the metal surface permanently activated for the chemical attack. Thus, in this way even in case of slight cavitation it may lead to considerable damage to the materials. The rate of erosion may be accentuated if the liquid itself has corrosive tendencies such as water with large amounts of dissolved oxygen to acids.
Cavitation The Pump Heart Attack
Thus fundamentally, cavitation refers to an abnormal condition inside the pump that arises during pump operation due to formation and subsequent collapse of vapor filled cavities or bubbles inside the liquid being pumped. The condition of cavitation can obstruct the pump, impair performance and flow capacity, and damage the impeller and other sensitive components. In short, Cavitation can be termed as "the heart attack of the pump".
References
{parse block="google_articles"}Practical applications of this topic include sizing relief valve outlet laterals and lowpressure compressor suction lines. These pose a special challenge as the velocities and pressure changes are high.
Adiabatic Flow of a Compressible Fluid Through a Conduit
Flow through pipes in a typical plant where line lengths are short, or the pipe is well insulated can be considered adiabatic. A typical situation is a pipe into which gas enters at a given pressure and temperature and flows at a rate determined by the length and diameter of the pipe and downstream pressure. As the line gets longer friction losses increase and the following occurs:
The question is "will the velocity continue to increasing until it crosses the sonic barrier?" The answer is NO. The maximum velocity always occurs at the end of the pipe and continues to increase as the pressure drops until reaching Mach 1. The velocity cannot cross the sonic barrier in adiabatic flow through a conduit of constant cross section. If an effort is made to decrease downstream pressure further, the velocity, pressure, temperature and density remain constant at the end of the pipe corresponding to Mach 1 conditions. The excess pressure drop is dissipated by shock waves at the pipe exit due to sudden expansion. If the line length is increased to drop the pressure further the mass flux decreases, so that Mach 1 is maintained at the end of the pipe.
Analyzing the adiabatic flow using energy and mass balance yields the following analyses along with this nomenclature:
Variable  Definition  Variable  Definition 
h  enthalpy/unit mass  hst  stagnation enthalpy 
v  velocity  Ma  Mach number 
g  gravitational constant  M  molecular weight 
z  elevation  T  temperature 
Q  heat flow  P  pressure 
Ws  shaft work  R  gas constant 
Cp  specific heat (constant pressure)  Z  compressibility 
r  density  g  Cp/Cv 
G  mass flux  Â  Â 
Analysis One
This analysis derives the relationship between the stagnation temperature, flowing temperature, and the Mach number for a flowing ideal gas. Stagnation temperature is the temperature a flowing gas rises to when it is brought isentropically to rest, thereby converting its kinetic energy into enthalpy.
Conservation of energy requires that the energy balances:
Eq. (1) 
For adiabatic flow, no shaft work and for gases: Q=0, Ws=0 and dz=negligible....or:
Eq. (2) 
Enthalpy per unit mass of an ideal gas is defined H = C_{p} T
The gas, at rest, has no kinetic energy and is at its stagnation temperature (Tst), while the moving gas has kinetic energy and is at another temperature (T). The energies are therefore:
energy at rest, per unit mass = 0 + C_{p} T_{st}energy in motion, per unit mass = v^{2}/2 + C_{p} T
Equating the energy at rest and in motion:
hst= h+v^{2}/2  Eq. (3) 
or
h= hstv^{2}/2  Eq. (4) 
or
Eq. (5) 
This implies:
A useful way of looking at this relationship is by fanno lines. The fanno lines are lines of constant mass flux plotted on an enthalpy/entropy diagram:
Figure 1: Subsonic Flanno Flow 
C_{p} T_{st} = v^{2}/2 + C_{p} T  Eq. (6) 
To make this equation useful, we must replace C_{p }and v by terms containing only constants and the Mach number.
Also for an ideal gas:
Eq. (7) 
and
Eq. (8) 
Substituting yields:
Eq. (9) 
or
Eq. (10) 
Thus we see that for an ideal gas the temperature decreases as velocity increases.
If the gas is flowing adiabatically, then no energy has been added or subtracted from it and Tst is constant along the length of the pipe. Knowing Tst, then the above equation can be used to find the flowing temperature from the Mach number, (or vice versa) at any position along the pipe.
Analysis Two
This analysis uses the principles of conservation of energy and mass to derive a relationship between pressure and Mach number at up and downstream conditions, for adiabatic flow in a pipe of constant crosssectional area.
The conservation of mass requires the mass flux to be the same at any position along a pipe. Mass flux at any of these positions can be expressed in terms of density and velocity :
Eq. (11) 
For an ideal gas:
Eq. (12) 
and
Eq. (13) 
Substituting for density and velocity, we obtain Equation14 which relates Mach number, mass flow rate and flowing pressure and temperature:
Eq. (14) 
or
Eq. (15) 
Substituting for T from Equation 10:
Eq. (16) 
G is same at inlet (1) and outlet(2), so:
Â
Eq. (17) 
which leads to:
Eq. (18) 
This implies that pressure decreases as the Mach number increases. A similar analysis for temperature gives:
Eq. (19) 
This implies that temperature decreases as the Mach number increases. However, this is true for ideal gases only. For real gases temperature may increase!
Analysis Three
Now the momentum equation is introduced to incorporates the losses due to friction. The derivation is available in any standard textbook for compressible flow In summary the final result is:
Eq. (20) 
where
f= Average Darcy friction factor
L= Equivalent length of line
d= I.D. of the line
Thus this equation relates losses due to friction to inlet and outlet velocities. Solving for the unknown parameter requires a trial and error approach and is suitable for an Excel spreadsheet using the "Goal Seek" or "Solver" tools. Depending on the number of unknowns one or all three of the following equations need to be solved simultaneously:
Mass balance Equation 11
Energy balance Equation18 or 19
Momentum balance Equation 20.
In cases where the outlet velocity is defined as Mach 1, then the equation can be solved for the maximum length, which can be used to flow a certain amount of fluid through a line of known diameter. Beyond this length choked flow condition occurs and, as explained above, any further increase in pipe length will cause the flow to decrease in such a manner that velocity at the end of the pipe is still sonic ( Mach=1). This particular application is of considerable practical use in sizing blowdown lines or relief valve outlet lines relieving to the atmosphere.
Recall that the above equations have assumed that the gas is ideal. One can compensate for nonideality to an extent by incorporating the Z factor. A rigorous approach implies solving simultaneously the momentum, energy, and mass balance equation numerically. An analytical approach, as given above for ideal gases, is useful most of the time and the results are valid for engineering purpose.
Isothermal Flow
In isothermal flow, the temperature of the gas remains constant. This simplifies matters considerably. Starting with the mechanical energy equation:
Eq. (21) 
Multiplying both sides by ?^{2}:
Eq. (22) 
Eq. (23) 
Rearranging and integrating gives:
Eq. (24) 
When the temperature change over the conduit is small EquationÂ 24 can be used instead of the adiabatic Equation 20.Â Adibatic flow below Mach 0.3 follows EquationÂ 24 closely.
If EquationÂ 24 is differentiated with respect to ?_{bÂ Â }to obtain a maximum G then:
Eq. (25) 
and the exit Mach number is:
Eq. (26) 
This apparent choking condition for isothermal flow is not physically meaningful, as at these high speeds, and rates of expansion, isothermal conditions are not possible.
References
These properties are entered into a computer program or spreadsheet along with some pipe physical data (pipe schedule and roughness factor) {parse block="google_articles"}and out pops a series of line sizes with associated Reynolds Number, velocity, friction factor and pressure drop per linear dimension. The pipe size is then selected based on a compromise between the velocity and the pressure drop. With the line now sized and the pressure drop per linear dimension determined, the pressure loss from the inlet to the outlet of the pipe can be calculated.
Calculating Pressure Drop
The most commonly used equation for determining pressure drop in a straight pipe is the Darcy Weisbach equation. One common form of the equation which gives pressure drop in terms of feet of head is given below:
Eq. (1) 
The term is commonly referred to as the Velocity Head.
Another common form of the Darcy Weisbach equation that is most often used by engineers because it gives pressure drop in units of pounds per square inch (psi) is:
Eq. (2) 
To obtain pressure drop in units of psi/100 ft, the value of 100 replaces L in Equation 2.
The total pressure drop in the pipe is typically calculated using these five steps.
See any problems with this method?
Relationship Between K, Friction Factor, and Equivalent Length
The following discussion is based on concepts found in reference 1, the CRANE Technical Paper No. 410. It is the author's opinion that this manual is the closest thing the industry has to a standard on performing various piping calculations. If the reader currently does not own this manual, it is highly recommended that it be obtained.
As in straight pipe, velocity increases through valves and fittings at the expense of head loss. This can be expressed by another form of the Darcy equation similar to Equation 1:
Eq. (3) 
When comparing Equations 1 and 3, it becomes apparent that:
Eq. (4) 
K is called the resistance coefficient and is defined as the number of velocity heads lost due to the valve or fitting. It is a measure of the following pressure losses in a valve or fitting:{parse block="google_articles"}
Pipe friction in the inlet and outlet straight portions of the valve or fitting is very small when compared to the other three. Since friction factor and Reynolds Number are mainly related to pipe friction, K can be considered to be independent of both friction factor and Reynolds Number.^{ }Therefore, K is treated as a constant for any given valve or fitting under all flow conditions, including laminar flow. Indeed, experiments showed^{1} that for a given valve or fitting type, the tendency is for K to vary only with valve or fitting size. Note that this is also true for the friction factor in straight clean commercial steel pipe as long as flow conditions are in the fully developed turbulent zone. It was also found that the ratio L/D tends towards a constant for all sizes of a given valve or fitting type at the same flow conditions. The ratio L/D is defined as the equivalent length of the valve or fitting in pipe diameters and L is the equivalent length itself.
In Equation 4, f therefore varies only with valve and fitting size and is independent of Reynolds Number. This only occurs if the fluid flow is in the zone of complete turbulence (see the Moody Chart in reference 1 or in any textbook on fluid flow). Consequently, f in Equation 4 is not the same f as in the Darcy equation for straight pipe, which is a function of Reynolds Number. For valves and fittings, f is the friction factor in the zone of complete turbulence and is designated f_{t}, and the equivalent length of the valve or fitting is designated L_{eq}. Equation 4 should now read (with D being the diameter of the valve or fitting):
Eq. (5) 
The equivalent length, L_{eq,} is related to f_{t}, not_{ }f, the friction factor of the flowing fluid in the pipe. Going back to step four in our five step procedure for calculating the total pressure drop in the pipe, adding the equivalent length to the straight pipe length for use in Equation 1 is fundamentally wrong.
Calculating Pressure Drop, The Correct Way
So how should we use equivalent lengths to get the pressure drop contribution of the valve or fitting? A form of Equation 1 can be used if we substitute f_{t} for f and L_{eq} for L (with d being the diameter of the valve or fitting):
Eq. (6) 
The pressure drop for the valves and fittings is then added to the pressure drop for the straight pipe to give the total pipe pressure drop.
Another approach would be to use the K values of the individual valves and fittings. The quantity of each type of valve and fitting is multiplied by its respective K value and added together to obtain a total K. This total K is then substituted into the following equation:
Eq. (7) 
Notice that use of equivalent length and friction factor in the pressure drop equation is eliminated, although both are still required to calculate the values of K^{1}. As a matter of fact, there is nothing stopping the engineer from converting the straight pipe length into a K value and adding this to the K values for the valves and fittings before using Equation 7. This is accomplished by using Equation 4, where D is the pipe diameter and f is the pipeline friction factor.
How significant is the error caused by mismatching friction factors? The answer is, it depends. Below is a real world example showing the difference between the Equivalent Length method (as applied by most engineers) and the K value method to calculate pressure drop.
An Example
The fluid being pumped is 94% Sulfuric Acid through a 3", Schedule 40, Carbon Steel pipe:
Mass Flow Rate, lb/hr  63,143 
Volumetric Flow Rate, gpm  70 
Density, lb/ft^{3}  112.47 
S.G.  1.802 
Viscosity, cp  10 
Temperature, ^{o}F  127 
Pipe ID, in  3.068 
Velocity, ft/s  3.04 
Reynold's No  12,998 
Darcy Friction Factor, (f) Pipe  0.02985 
Pipe Line ?P/100 ft  1.308 
Friction Factor at Full Turbulence (f_{t})  0.018 
Straight Pipe, ft  31.5 
Â
Fittings  L_{eq}/D^{1}  L_{eq}^{2, 3}  K^{1, 2} =  Quantity  Total L_{eq}  Total K 
90^{o} Long Radius Elbow  20  5.1  0.36  2  10.23  0.72 
Branch Tee  60  15.3  1.08  1  15.34  1.08 
Swing Check Valve  50  12.8  0.90  1  12.78  0.90 
Plug Valve  18  4.6  0.324  1  4.60  0.324 
3" x 1" Reducer^{4}  None^{5}  822.68^{5}  57.92  1  822.68  57.92 
Total  Â  Â  Â  Â  865.63  60.94 
Notes:
Â  Typical Equivalent Length Method  K Value Method 
Straight Pipe ?P, psi  Not Applicable  0.412 
Total Pipe Equivalent Length ?P, psi  11.734  Not Applicable 
Valves and Fittings ?P, psi  Not Applicable  6.828 
Total Pipe ?P, psi  11.734  7.240 
The line pressure drop is greater by about 4.5 psi (about 62%) using the typical equivalent length method (adding straight pipe length to the equivalent length of the fittings and valves and using the pipe line fiction factor in Equation 1).
One can argue that if the fluid is water or a hydrocarbon, the pipeline friction factor would be closer to the friction factor at full turbulence and the error would not be so great, if at all significant; and they would be correct. However hydraulic calculations, like all calculations, should be done in a correct and consistent manner. If the engineer gets into the habit of performing hydraulic calculations using fundamentally incorrect equations, he takes the risk of falling into the trap when confronted by a pumping situation as shown above.
Another point to consider is how the engineer treats a reducer when using the typical equivalent length method. As we saw above, the equivalent length of the reducer had to be backcalculated using equation 5. To do this, we had to use f_{t }and K.
Why not use these for the rest of the fittings and apply the calculation correctly in the first place?
Final Thoughts on K Values
The 1976 edition of the Crane Technical Paper No. 410 first discussed and used the twofriction factor method for calculating the total pressure drop in a piping system (f for straight pipe and f_{t} for valves and fittings). Since then, Hooper^{2 }suggested a 2K method for calculating the pressure loss contribution for valves and fittings. His argument was that the equivalent length in pipe diameters (L/D) and K was indeed a function of Reynolds Number (at flow rates less than that obtained at fully developed turbulent flow) and the exact geometries of smaller valves and fittings. K for a given valve or fitting is a {parse block="google_articles"}combination of two Ks, one being the K found in CRANE Technical Paper No. 410, designated K_{Y}, and the other being defined as the K of the valve or fitting at a Reynolds Number equal to 1, designated K_{1}. The two are related by the following equation:
K = K_{1 }/ N_{RE }+ K_{?} (1 + 1/D)
The term (1+1/D) takes into account scaling between different sizes within a given valve or fitting group. Values for K_{1} can be found in the reference article^{2} and pressure drop is then calculated using Equation 7. For flow in the fully turbulent zone and larger size valves and fittings, K becomes consistent with that given in CRANE.
Darby^{3} expanded on the 2K method. He suggests adding a third K term to the mix. Darby states that the 2K method does not accurately represent the effect of scaling the sizes of valves and fittings. The reader is encouraged to get a copy of this article.
The use of the 2K method has been around since 1981 and does not appear to have "caught" on as of yet. Some newer commercial computer programs allow for the use of the 2K method, but most engineers inclined to use the K method instead of the Equivalent Length method still use the procedures given in CRANE. The latest 3K method comes from data reported in the recent CCPS Guidlines^{4} and appears to be destined to become the new standard; we shall see.
Conclusion
Consistency, accuracy and correctness should be what the Process Design Engineer strives for. We all add our "fat" or safety factors to theoretical calculations to account for realworld situations. It would be comforting to know that the "fat" was added to a basis using sound and fundamentally correct methods for calculations.
Questions and Answers
Question #1
Could you please give me in layman terms a better definition for K values. I know that K is defined as "the number of velocity heads lost"...But what exactly does that mean???
Mr. Leckner's reply to this question:
Well, I'll try to give you the Chemical Engineer's version of the layman answer. Velocity of any fluid increases through pipes, valves and fittings at the expense of pressure. This pressure loss is referred to as head loss. The greater the head loss, the higher the velocity of the fluid. So, saying a velocity head loss is just another way of saying we loose pressure due to and increase in velocity and this pressure loss is measured in terms of feet of head. Now, each component in the system contributes to the amount of pressure loss in different amounts depending upon what it is. Pipes contribute fL/D where L is the pipe length, D is the pipe diameter and f is the friction factor. A fitting or valve contributes K. Each fitting and valve has an associated K. 
Â
Question #2
It appears that the K values in CRANE TP410 were established using a liquid (water) flow loop. Is this K value also valid for compressible media systems? (Can a K value be used for both compressible and incompressible service?)
Mr. Leckner's reply to this question:
Crane also tested their system on steam and air. Now, this is where things get sticky. As per CRANE TP410, K values are a function of the size and type of valve or fitting only and is independent of fluid and Reynolds number. So yes, you can use it in ALL services, including twophase flow.Â However, as I point out towards the end of my article, there is now evidence that shows using a single K value for the valve and fitting is not correct and that K is indeed a function of both Reynolds number and fitting/valve 
Â
Question #3
When answering my first question, you stated:Â 'Velocity of any fluid increases through pipes, valves and fittings at the expense of pressure.'Â When you say this, you are talking about compressible (gas) flow right?Â For example, in a pipe of constant area, the velocity of a gas would increase as the fluid traveled down the pipe (due to the decreasing pressure). Â However, the velocity of a liquid would remain constant as it traveled down the same pipe (even with the decreasing pressure).Â Is this a correct statement?
Mr. Leckner's reply to this question:
Sorry for the confusion. Yes to both of your questions. If you look at the Bernoulli equation, you will see that velocity cancels out for a liquid as long as there is no change in pipe size along the way and pressure drop is only a function of frictional losses and a change in elevation. 
Â
Question #4
I'm reading the Crane Technical Paper #410 and I have the following
questions/comments:
Page 28 of TP 410 states that:
"Velocity in a pipe is obtained at the expense of static head".Â This makes sense and Equation 21 shows this relationship where the static head is converted to velocity head.Â However, there is no diameter associated with this.Â So is it correct to say based on equation 21 that if you had a barrel of water with a short length of pipe attached to the bottom that discharged to atmosphere, and in this barrel you had 5 feet of water (5' of static head), the resulting water velocity would be 17.94 ft/sec (regardless
of the pipe diameter).
Maybe the real question is how do you use equation 21.Â Do you have to know the velocity and then you can calculate the headloss?Â And why does equation 21 and equation 23 seem to show headloss equaling two different things?
Also, why does it say that a diameter is always associtated with the K value, when as I mentioned above there is no diameter associated with equation 21?
Maybe I'm trying to read into all of this too deeply, but I still do not feel that I fully grasp what page 28 is trying to reveal.
Mr. Leckner's reply to this question:
You need a diameter to get velocity. Velocity is lenght/time (for example, feet/sec). Flow is usually given in either mass units (weight/time or lb/hr for example) or in volumetric units (cubic feet per minute for example). To get velocity, you need to divide the volumetric flow by a cross sectional area (square feet). To get an area, you need a diameter. So the velocity is always based on some diameter. 
Nomenclature
D  =  Diameter, ft 
d  =  Diameter, inches 
f  =  Darcy friction factor 
f_{t}  =  Darcy friction factor in the zone of complete turbulence 
g  =  Acceleration of gravity, ft/sec^{2} 
h_{L}  =  Head loss in feet 
K  =  Resistance coefficient or velocity head loss 
K_{1}  =  K for the fitting at N_{RE} = 1 
K_{?}  =  K value from CRANE 
L  =  Straight pipe length, ft 
L_{eq}  =  Equivalent length of valve or fitting, ft 
N_{RE}  =  Reynolds Number 
?P  =  Pressure drop, psi 
n  =  Velocity, ft/sec 
W  =  Flow Rate, lb/hr 
?  =  Density, lb/ft^{3} 
References