1.0 Introduction
Turbomachine is a device that exchanges energy with a fluid using continuously flowing fluid and rotating blades. Two classifications of turbomachines are that if the device delivers energy to the fluid or extracting energy from the fluid. Extraction of energy from fluid is known as a turbine which will result in a decrease in pressure across the turbine while inducing energy to fluids is known as a pump depending on the type of fluids used which will result in an increase in pressure across the pump.
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For pumps based on gases, they are further broken down to the types of gases which are the fans, blowers and compressors. For fans, it has low pressure gradient and high volume flow rate. Examples include ceiling fans and propellers. For blowers, it has a medium pressure gradient and medium volume flow rate. Examples include centrifugal and squirrel-cage blowers found in furnaces, leaf blowers, and hair dryers. Finally for compressors, it has a high pressure gradient and a low volume flow rate. Some examples of compressors include air compressors for air tools, refrigerant compressors for refrigerators and air conditioners.
In this assignment, the centrifugal pump will be investigated by analyzing the general principles of the pump, derivation of the dimensionless parameters, selecting a pump to match the system requirements and deducing the power, efficiency and Head formulae which are stated in the objectives below.
1.1 Objectives
The objectives of the assignment are stated below.
To explain the general principles of Centrifugal Pump
Dimensionless parameters of the pump is to be derived in terms of:
Flow Coefficient
Head Coefficient
Power Coefficient
Specific Speed
Explanation of pump selection to match the system requirements
Formulae for power, efficiency and Head are to be deduced.
1.2 Centrifugal Pump
Centrifugal pumps are one of the most common radial-flow turbomachines. In general, centrifugal pumps are high discharge, low pressure generating pumps. In order to increase the pressure generated in the pump, it is setup in stages wherein pressure increases from one stage to another. The primary advantages of a centrifugal pump are simplicity, low cost, uniform flow, small floor space, low maintenance expense, quiet operation, and adaptability for use with a motor or a turbine drive. The centrifugal pump has two main components which are the impeller and a stationary casing as shown in figure 1 below.
Figure Schematic diagram of basic elements of a centrifugal pump (Munson, 2010)
The impeller consists of a number of blades, either open or close type, mounted on the shaft that projects outside of the casing. The axis of rotation of the impeller can be horizontal or vertical depending on the type of work is to be done. Closed type impeller is generally the most efficient type compared to the open type.
Figure (a) Open impeller, (b) enclosed (Munson, 2010)
The pump casing provides a pressure boundary for the pump and contains channels to properly direct the suction and discharge flow of the fluid. There are three general types of casings, but each consists of a chamber in which the impeller rotates, provided with inlet and exit for the liquid being pumped. The simplest form is the circular casing, consisting of an annular chamber around the impeller. The second type is the volute type casing. Volute casings take the form of a spiral increasing uniformly in cross-sectional area as the outlet is approached. The volute efficiently converts the velocity energy imparted to the liquid by the impeller into pressure energy. The third type of casing is used in diffuser-type or turbine pumps. In this type, guide vanes or diffusers are interposed between the impeller discharge and the casing chamber
The working principle of the centrifugal pump is that as the impeller rotates, fluid is sucked in through the center of the casing or the eye and flows radially outward. Through the rotating blades, energy is added to the fluid and there will be an increased pressure and velocity from the eye to the periphery of the blades as shown in figure 1 above. Work is done on the fluid by the rotating blades which create a large increase of in kinetic energy of the fluid flowing through the impeller. As the fluid flows from the impeller into the casing, the kinetic energy is converted into an increasing pressure.
2.0 Dimensional Analysis
Coefficient Method
In this section, the derivation of the dimensionless parameters of the centrifugal pump in terms of Flow Coefficient, Head Coefficient, Power Coefficient, and Specific Speed. The power of a centrifugal pump can be analyzed using the parameters below.
Where,
P= power
= density
N= speed
D= the characteristic diameter
= head change
Q= volume flow rate
g= gravitational constant
Following the table of basic primary and secondary dimension the number of variables and parameters are calculated to find the number of pi groups.
Dimension
Parameters
P
ML2T-3
ML-3
N
T-1
D
L1
M0L2T-2
Q
M3T-1
No. of variables = n = 6 (
No. of fundamental dimensions = m = 3 (M, L, T)
groups = n – m = 3
Therefore, there are six quantities and three dimensions, which results in three dimensionless groups, 1,2,3. Exponent method is used to find the pi-groups which will be shown below.
For group 1,
Equating the exponents to the values,
Mass
Time
Length
Substituting the value of the exponents to the equation,
For group 2,
Time -1= -b b=1
Mass 0 = a
Length
Therefore,
For group 3,
Mass
Time
Length
Therefore,
Finally, the complete equation from the derivation above is shown below.
Step by step method
3.0 Specific Speed
Specific speed is one of the first parameters that a centrifugal pump designer looks at when evaluating a pump application. Specific speed may be used to rapidly determine the feasible designs for the service conditions. Specific speed can be determined by the three coefficients that have been derived in section 3. These three coefficients are the power coefficient, flow coefficients and head coefficient respectively. To get specific speed, the assumption of similar machines is operating at similar conditions such that the three coefficients must have the same value for each size.
Let,
Equating in terms of D, yields
Moving all the constant variables to the right hand side yields,
Equating for N yields,
Moving the constants to the right hand side yields,
Finally the specific speed is determined by,
The units that are involved in the equation of specific speed are rev/min for speed (N), for flow rate (Q) and meters for head ( . Specific speed itself is a dimensionless quantity. From the specific speed equation, it is found that given at a speed, the specific speed is low for large heads and small flow and high specific speed is due to low heads and large flows.
4.0 Matching Pump to System Requirement
Figure Typical Q-H-Efficiency Relationship of Centrifugal Pump (Munson,2010)
Figure 3 above describes a typical Head vs. Flow of a centrifugal pump. The QH-curve above shows the head, which the pump is able to perform at a given flow. The relationship between the head and flow is that low flow results in high head while high head results in low flow. This is the type of curve that all pump manufacturers publish for each model pump for a given operating speed.
Using the specific speed formula, various pumps can be compared easily. Pump designers have a way to compare the efficiency of their designs across a large range of pump model and types while pump users can know what the expected efficiency of a particular pump design is. The efficiency of pumps with the same specific speed can be compared providing the user or the designer a starting point for comparison or as a benchmark for improving the design and increase the efficiency.
Whenever pumps are used, the size of a pump should match the size of the system in order to most efficiently utilize energy. If the pump is too small for the job, then the pump will be overloaded and will break down. On the other hand, if the pump is too large for the job, it will use excess energy and will result in high operation costs. A properly matched pump will minimize both operation costs and maintenance costs.
5.0 Selecting Pump Size
It is unlikely for pump users will find the best size of pump that is perfectly suited for the system requirements. In order to find the right size of pump, a geometrically similar pump is chosen which can produce the required flow rate and head at optimal conditions. For similar conditions and optimal points, the geometrically similar pump will have the same specific speed NS. The right size of the unknown pump will be determined by its diameter.
Specific speed pump A:
Specific speed pump B:
Since they are the same, the diameter of pump B can be determined by equating using either pump’s flow coefficients or head coefficients.
Using flow coefficients:
Using head coefficients:
A worked example problems will be given below to show how to select the right size of pumps which can produce the same flow rate and head of the system requirements.
Efficiency
Qa
Ha
0
0
69.5
48
0.015
75
66
0.0285
68.8
66
0.045
54.1
45
0.061
23.2
The following table shows the operating characteristic of a pump given at speed of 1500 rev/min and rotor diameter of 130mm.
Flow rate = 0.0200 m3/s
Total head = 40m
The correct size of pump is to be determined in order to produce the flow rate and total head given.
First, the operating characteristic of the pump is plotted.
Figure QH curve for example pump
From the graph above, it is found that point A is the optimal flow and head of 0.045m3/s and 54.1m
Specific speed Ns at point A:
Speed for geometrically similar pump:
The diameter of the pump is calculated below,
Therefore, in order find the suitable pump which can run following the system requirement, the pump must have a diameter of 201.36 mm and must be running at 1793.86 rev/min.
6.0 Running with the wrong size
Although the right size of the pump has been determined to match the system requirement, it is uncertain whether the pump will be available at that size. Therefore, the nearest size available will be used. In order to get the required flow and head, the chosen pump’s speed will be adjusted. Assume B is the point of the required operating point, and A is the optimal point for the wrong size pump. The flow and head coefficients will be the same with some other point C on the operating curve shown in figure below.
Figure Wrong size QH performance curve
Equating flow rate point B and point C, —equation 1
Equating head coefficients for point C and B, —equation 2
Substituting equation 1 into 2 to eliminate the speed,
A worked example will be shown below to illustrate how to find the speed at which a pump must run giving only one size is available. The example will relate to the worked example shown before this.
Available size, 130 mm pump, , m3/s
Figure Running with wrong size plotted graph
In order to determine, the graph is plotted and it is found that = 76m , 0.028 m3/s
To find speed at B, the flow coefficients is equated which yields,
From the graph also, it is found that the efficiency is 63%
If using the head coefficients formula,
Therefore the required specific speed of the given pump needs to be adjusted to 1071.43 rev/min in order to match the required flow and head requirements
7.0 Deduce formulae for power, efficiency and Head
7.1 Efficiency
The overall efficiency of a centrifugal pump is simply the ratio of the water (output) power to the shaft (input) power and is illustrated by the equation below.
η = PW / PS (Munson, 2010)
Where,
η = efficiency (%)
PW = water power (output power)
PS = shaft power (input power)
7.2 Head
The head of a pump can be describe as a pump’s vertical discharge “pressure-head” is the vertical lift in height measured in meters at which a pump can no longer exert enough pressure to move water. At this point, the pump may be said to have reached its “shut-off” head pressure.
The equation for head is given below.
where,
h = total head developed (m)
p2 = pressure at outlet (N/m2)
p1 = pressure at inlet (N/m2)
Ï = density (kg/m3)
g = acceleration of gravity (9.81) m/s2
v2 = velocity at the outlet (m/s)
7.3 Power
The power of a centrifugal pump can be derived using head (H), density (Ï), gravitational constant (g) and volumetric flow rate (Q).
This derivation comes from the vertical outlet of the pump moving the fluid upwards against gravity at a certain height which is the head (H). For the work done,
Power is the rate of doing work, therefore,
Finally,
8.0 Discussion
The objectives of the assignment are to analyze and design a centrifugal pump. Firstly a dimensional analysis is done to derive the flow coefficient, head coefficient, power coefficient and specific speed. Before finding the right pump, the pump has to match the system requirement. In the system requirement, the relationship between the flow rate and head is stated and that for high flow rate results in low head whereas for low flow rate results in high head. The specific speed of the pump is used to select the suitable pump. After finding the specific speed, the size of the pump is to be determined using derivations of the specific speed.
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Although the size of the pump has been determined, it is unlikely to find the size in the market, therefore the pump with the closest value to the desired size is to be selected which the speed is adjusted to suit the flow and head requirements of the system desired. In order to increase the efficiency of the pump, while selecting the pump it is best to match the flow rate of the pump to the best efficiency point (B.E.P) around 80% to achieve the maximum efficiency possible. The best efficiency point (B.E.P.) this is the point at which the pump is the most efficient and operates with the least vibration and noise
9.0 Conclusion
In this paper, the principle workings of a centrifugal pump are stated. After that the dimensional analysis is done to find the flow, head, power and specific speed. The specific speed is used to find the size required to match the system requirements. Also, analysis is done to determine the required specific speed of the centrifugal pump while using the wrong pump size. It is found that the efficiency of the pump is around 63%.
Also, there is a unique relationship between the actual pump head gained by the fluid and the flow rate. It is vital for the pump selection process to utilize the system curve as determined by the system equation and the pump performance curve.
Finally, whenever pumps are to be used, the size of a pump should match the size of the system in order to most efficiently utilize energy. If the pump is too small for the job, then the pump will be overloaded and will break down. On the other hand, if the pump is too large for the job, it will use excess energy and will result in high operation costs. A properly matched pump will minimize both operation costs and maintenance costs.
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