Pumps
Rotodynamic Pumps
A rotodynamic pump is a device where mechanical energy is transferred from the rotor to the fluid by the principle of fluid motion through it. Therefore, it is essentially a turbine in reverse. Like turbines, pumps are classified according to the main direction of fluid path through them like (i) radial flow or centrifugal, (ii) axial flow and (iii) mixed flow types.
Centrifugal Pumps
The centrifugal pump, by its principle, is converse of the Francis turbine. The flow is radially outward, and the hence the fluid gains in centrifugal head while flowing through it. However, before considering the operation of a pump in detail, a general pumping system is discussed as follows.
General Pumping System and the Net Head Developed by a Pump
The word pumping, referred to a hydraulic system commonly implies to convey liquid from a low to a high reservoir. Such a pumping system, in general, is shown in Fig. 15.18. At any point in the system, the elevation or potential head is measured from a fixed reference datum line. The total head at any point comprises pressure head, velocity head and elevation head. For the lower reservoir, the total head at the free surface is 
and is equal to the elevation of the free surface above the datum line since the velocity and static pressure at A are zero. Similarly the total head at the free surface in the higher reservoir is (
) and is equal to the elevation of the free surface of the reservoir above the reference datum.
The variation of total head as the liquid flows through the system is shown in Fig. 15.19. The liquid enters the intake pipe causing a head loss 
for which the total energy line drops to point B corresponding to a location just after the entrance to intake pipe. The total head at B can be written as
As the fluid flows from the intake to the inlet flange of the pump at elevation 
the total head drops further to the point C (fig. 15.19) due to pipe friction and other losses equivalent to 
. The fluid then enters the pump and gains energy imparted by the moving rotor of the pump. This raises the total head of the fluid to a point D (Fig. 15.19) at the pump outlet (Fig. 15.18).
In course of flow from the pump outlet to the upper reservoir, friction and other losses account for a total head loss or 
down to a point E. At E an exit loss 
occurs when the liquid enters the upper reservoir, bringing the total heat at point F (Fig. 15.19) to that at the free surface of the upper reservoir. If the total heads are measured at the inlet and outlet flanges respectively, as done in a standard pump test then
Total inlet head to the pump = 
Total outlet head of the pump = 
where 
and 
are the velocities in suction and delivery pipes respectively.
Therefore, the total head developed by the pump,
(15.40)
The head developed H is termed as manometric head. If the pipes connected to inlet and outlet of the pump are of same diameter,
and therefore the head developed or manometric head H is simply the gain in piezometric pressure head across the pump which could have been recorded by a manometer connected between the inlet and outlet flanges of the pump. In practice, (
) is so small in comparison to 
that it is ignored. It is therefore not surprising o find that the static pressure head across the pump is often used to describe the total head developed by the pump. The vertical distance between the two levels in the reservoirs 
is known as static head or static lift. Relationship between
, the static head and H, the head developed can be found out by applying Bernoulli’s equation between A and C and between D and F (Fig. 15.18) as follows:
(15.41)
Between D and F,
(15.42)
substituting 
from Eq. (15.41) into Eq. (15.42), and then with the help of Eq. (15.40),
we can write
(15.43)
Therefore, we have, the total head developed by the pump = static head + sum of all the losses.
The simplest from of a centrifugal pump is shown in Fig. 15.20. It consists of three important parts: (i) the rotor, usually called as impeller, (ii) the volute casing and (iii) the diffuser ring. The impeller is a rotating solid disc with curved blades standing out vertically from the face of the disc. The tips of the blades are sometimes covered by another flat disc to give shrouded blades, otherwise the blade tips are left open and the casing of the pump itself forms the solid outer wall of the blade passages. The advantage of the shrouded blade is that flow is prevented from leaking across the blade tips from one passage to another.
As the impeller rotates, the fluid is drawn into the blade passage at the impeller eye, the centre of the impeller. The inlet pipe is axial and therefore fluid enters the impeller with very little whirl or tangential component of velocity and flows outwards in the direction of the blades. The fluid receives energy from the impeller while flowing through it and is discharged with increased pressure and velocity into the casing. To convert the kinetic energy or fluid at the impeller outlet gradually into pressure energy, diffuser blades mounted on a diffuser ring are used.
The stationary blade passages so formed have an increasing cross-sectional area which reduces the flow velocity and hence increases the static pressure of the fluid. Finally, the fluid moves from the diffuser blades into the volute casing which is a passage of gradually increasing cross-section and also serves to reduce the velocity of fluid and to convert some of the velocity head into static head. Sometimes pumps have only volute casing without any diffuser.
Figure 15.21 shows an impeller of a centrifugal pump with the velocity triangles drawn at inlet and outlet. The blades are curved between the inlet and outlet radius. A particle of fluid moves along the broken curve shown in Fig. 15.21.
Let 
be the angle made by the blade at inlet, with the tangent to the inlet radius, while 
is the blade angle with the tangent at outlet. 
and 
are the absolute velocities of fluid at inlet an outlet respectively, while 
and
are the relative velocities (with respect to blade velocity) at inlet and outlet respectively. Therefore, according to Eq. (15.3),
Work done on the fluid per unit weight = 
(15.44)
A centrifugal pump rarely has any sort of guide vanes at inlet. The fluid therefore approaches the impeller without appreciable whirl and so the inlet angle of the blades is designed to produce a right-angled velocity triangle at inlet (as shown in Fig. 15.21). At conditions other than those for which the impeller was designed, the direction of relative velocity 
does not coincide with that of a blade. Consequently, the fluid changes direction abruptly on entering the impeller. In addition, the eddies give rise to some back flow into the inlet pipe, thus causing fluid to have some whirl before entering the impeller. However, considering the operation under design conditions, the inlet whirl velocity 
and accordingly the inlet angular momentum of the fluid entering the impeller is set to zero. Therefore, Eq. (15.44) can be written as
Work done on the fluid per unit weight = 
(15.45)
We see from this equation that the work done is independent of the inlet radius. The difference in total head across the pump [given by Eq. (15.40)], known as manometric head, is always less than the quantity 
because of the energy dissipated in eddies due to friction.
The ratio of manometric head H and the work head imparted by the rotor on the fluid 
(usually known as Euler head) is termed as manometric efficiency
. It represents the effectiveness of the pump in increasing the total energy of the fluid from the energy given to it by the impeller. Therefore, we can write
(15.46)
The overall efficiency 
of a pump is defined as
(15.47)
where, Q is the volume flow rate of the fluid through the pump, and P is the shaft power, i.e. the input power to the shaft. The energy required at the shaft exceeds 
because of friction in the bearings and other mechanical parts. Thus a mechanical efficiency is defined as
(15.48)
so that
(15.49)
Slip Factor
Under certain circumstances, the angle at which the fluid leaves the impeller may not be the same as the actual blade angle. This is due to a phenomenon known as fluid slip, which finally results in a reduction in 
the tangential component of fluid velocity at impeller outlet. One possible explanation for slip is given as follows.
In course of flow through the impeller passage, there occurs a difference in pressure and velocity between the leading and trailing faces of the impeller blades. On the leading face of a blade there is relatively a high pressure and low velocity, while on the trailing face, the pressure is lower and hence the velocity is higher. This results in a circulation around the blade and a non-uniform velocity distribution at any radius. The mean direction of flow at outlet, under this situation, changes from the blade angle at outlet 
to a different angle 
as shown in Fig. 15.22. Therefore the tangential velocity component at outlet 
is reduced to 
, as shown by the velocity triangles in Fig. 15.22, and the difference 
is defined as the slip. The slip factor 
is defined as
With the application of slip factor
, the work head imparted to the fluid (Euler head)
becomes
. The typical values of slip factor lie in the region of 0.9.
Losses in a Centrifugal Pump
It has been mentioned earlier that the shaft power P or energy that is supplied to the pump by the prime mover is not the same as the energy received by the liquid. Some energy is dissipated as the liquid passes through the machine. The losses can be divided into different categories as follows:
Lecture -35
Characteristics of a Centrifugal Pump
With the assumption of no whirl component of velocity at entry to the impeller of a pump, the work done on the fluid per unit weight by the impeller is given by Eq. (15.45). Considering the fluid to be frictionless, the head developed by the pump will be the same san can be considered as the theoretical head developed. Therefore we can write for theoretical head developed 
as
(15.50)
From the outlet velocity triangle (Fig. 15.21).
(15.51)
where Q is rate of flow at impeller outlet and A is the flow area at the periphery of the impeller. The blade speed at outlet 
can be expressed in terms of rotational speed of the impeller N as
Using this relation and the relation given by Eq. (15.51), the expression of theoretical head developed can be written from Eq. (15.50) as
where, 
and
For a given impeller running at a constant rotational speed. 
and 
are constants, and therefore head and discharge bears a linear relationship as shown by Eq. (15.52). This linear variation of 
with Q is plotted as curve I in Fig. 15.24.
If slip is taken into account, the theoretical head will be reduced to 
. Moreover the slip will increase with the increase in flow rate Q. The effect of slip in head-discharge relationship is shown by the curve II in Fig. 15.24. The loss due to slip can occur in both a real and an ideal fluid, but in a real fluid the shock losses at entry to the blades, and the friction losses in the flow passages have to be considered. At the design point the shock losses are zero since the fluid moves tangentially onto the blade, but on either side of the design point the head loss due to shock increases according to the relation
(15.53)
where 
is the off design flow rate and 
is a constant. The losses due to friction can usually be expressed as
(15.54)
where, 
is a constant.
Equation (15.53) and (15.54) are also shown in Fig. 15.24 (curves III and IV) as the characteristics of losses in a centrifugal pump. By subtracting the sum of the losses from the head in consideration of the slip, at any flow rate (by subtracting the sum of ordinates of the curves III and IV from the ordinate of the curve II at all values of the abscissa), we get the curve V which represents the relationship of the actual head with the flow rate, and is known as head-discharge characteristic curve of the pump.
Effect of blade outlet angle
The head-discharge characteristic of a centrifugal pump depends (among other things) on the outlet angle of the impeller blades which in turn depends on blade settings. Three types of blade settings are possible (i) the forward facing for which the blade curvature is in the direction of rotation and, therefore, 
(Fig. 15.24a), (ii) radial, when
(Fig. 15.25b), and (iii) backward facing for which the blade curvature is in a direction opposite to that of the impeller rotation and therefore, 
(Fig. 15.25c). The outlet velocity triangles for all the cases are also shown in Figs. 15.25a, 15.25b, 15.25c. From the geometry of any triangle, the relationship between 
and 
can be written as.
which was expressed earlier by Eq. (15.51).
In case of forward facing blade, 
and hence cot 
is negative and therefore 
is more than
. In case of radial blade, 
and 
In case of backward facing blade, 
and 
Therefore the sign of 
, the constant in the theoretical head-discharge relationship given by the Eq. (15.52), depends accordingly on the type of blade setting as follows:
For forward curved blades 
For radial blades 
For backward curved blades 
With the incorporation of above conditions, the relationship of head and discharge for three cases are shown in Fig. 15.26. These curves ultimately revert to their more recognized shapes as the actual head-discharge characteristics respectively after consideration of all the losses as explained earlier (Fig. 15.27).
For both radial and forward facing blades, the power is rising monotonically as the flow rate is increased. In the case of backward facing blades, the maximum efficiency occurs in the region of maximum power. If, for some reasons, Q increases beyond 
there occurs a decrease in power. Therefore the motor used to drive the pump at part load, but rated at the design point, may be safely used at the maximum power. This is known as self-limiting characteristic. In case of radial and forward-facing blades, if the pump motor is rated for maximum power, then it will be under utilized most of the time, resulting in an increased cost for the extra rating. Whereas, if a smaller motor is employed, rated at the design point, then if Q increases above 
the motor will be overloaded and may fail. It, therefore, becomes more difficult to decide on a choice of motor in these later cases (radial and forward-facing blades).
Lecture-36
Flow through Volute Chambers
Apart from frictional effects, no torque is applied to a fluid particle once it has left the impeller. The angular momentum of fluid is therefore constant if friction is neglected. Thus the fluid particles follow the path of a free vortex. In an ideal case, the radial velocity at the impeller outlet remains constant round the circumference. The combination of uniform radial velocity with the free vortex (
=constant) gives a pattern of spiral streamlines which should be matched by the shape of the volute. This is the most important feature of the design of a pump. At maximum efficiency, about 10 percent of the head generated by the impeller is usually lost in the volute.
Vanned Diffuser
A vanned diffuser, as shown in Fig. 36.1, converts the outlet kinetic energy from impeller to pressure energy of the fluid in a shorter length and with a higher efficiency. This is very advantageous where the size of the pump is important. A ring of diffuser vanes surrounds the impeller at the outlet. The fluid leaving the impeller first flows through a vaneless space before entering the diffuser vanes. The divergence angle of the diffuser passage is of the order of 8-10° which ensures no boundary layer separation. The optimum number of vanes are fixed by a compromise between the diffusion and the frictional loss. The greater the number of vanes, the better is the diffusion (rise in static pressure by the reduction in flow velocity) but greater is the frictional loss. The number of diffuser vanes should have no common factor with the number of impeller vanes to prevent resonant vibration.
Cavitation in Centrifugal Pump
Cavitation is likely to occur at the inlet to the pump, since the pressure there is the minimum and is lower than the atmospheric pressure by an amount that equals the vertical height above which the pump is situated from the supply reservoir (known as sump) plus the velocity head and frictional losses in the suction pipe. Applying the Bernoulli’s equation between the surface of the liquid in the sump and the entry to the impeller, we have
(15.55)
Where, 
is the pressure at the impeller inlet and 
is the pressure at the liquid surface in the sump which is usually the atmospheric pressure, Z1 is the vertical height of the impeller inlet from the liquid surface in the sump, 
is the loss of head in the suction pipe. Strainers and non-return valves are commonly fitted to intake pipes. The term 
must therefore include the losses occurring past these devices, in addition to losses caused by pipe friction and by bends in the pipe.
In the similar way as described in case of a reaction turbine, the net positive suction head ‘NPSH’ in case of a pump is defined as the available suction head (inclusive of both static and dynamic heads) at pump inlet above the head corresponding to vapor pressure.
Therefore,
(15.56)
Again, with help of Eq. (15.55), we can write
The Thomas cavitation parameter s and critical cavitation parameter 
are defined accordingly (as done in case of reaction turbine) as
(15.57)
and
(15.58)
We can say that for cavitation not to occur,
In order that s should be as large as possible, z must be as small as possible. In some installations, it may even be necessary to set the pump below the liquid level at the sump (i.e. with a negative vale of z) to avoid cavitation.
Lecture-37
Axial Flow or Propeller Pump
The axial flow or propeller pump is the converse of axial flow turbine and is very similar to it an appearance. The impeller consists of a central boss with a number of blades mounted on it. The impeller rotates within a cylindrical casing with fine clearance between the blade tips and the casing walls. Fluid particles, in course of their flow through the pump, do not change their radial locations. The inlet guide vanes are provided to properly direct the fluid to the rotor. The outlet guide vanes are provided to eliminate the whirling component of velocity at discharge. The usual number of impeller blades lies between 2 and 8, with a hub diameter to impeller diameter ratio of 0.3 to 0.6.
Matching of Pump and System Characteristics
The design point of a hydraulic pump corresponds to a situation where the overall efficiency of operation is maximum. However the exact operating point of a pump, in practice, is determined from the matching of pump characteristic with the headloss-flow, characteristic of the external system (i.e. pipe network, valve and so on) to which the pump is connected.
Let us consider the pump and the piping system as shown in Fig. 15.18. Since the flow is highly turbulent, the losses in pipe system are proportional to the square of flow velocities and can, therefore, be expressed in terms of constant loss coefficients. Therefore, the losses in both the suction and delivery sides can be written as
(15.59a)
(15.59b)
where, 
is the loss of head in suction side and 
is the loss of head in delivery side and f is the Darcy’s friction factor, 
and 
are the lengths and diameters of the suction and delivery pipes respectively, while 
and 
are accordingly the average flow velocities. The first terms in Eqs. (37.1a) and (37.1b) represent the ordinary friction loss (loss due to friction between fluid ad the pipe wall), while the second terms represent the sum of all the minor losses through the loss coefficients 
and 
which include losses due to valves and pipe bends, entry and exit losses, etc. Therefore the total head the pump has to develop in order to supply the fluid from the lower to upper reservoir is
(15.60)
Now flow rate through the system is proportional to flow velocity. Therefore resistance to flow in the form of losses is proportional to the square of the flow rate and is usually written as
= system resistance = 
(15.61)
where K is a constant which includes, the lengths and diameters of the pipes and the various loss coefficients. System resistance as expressed by Eq. (15.61), is a measure of the loss of head at any particular flow rate through the system. If any parameter in the system is changed, such as adjusting a valve opening, or inserting a new bend, etc., then K will change. Therefore, total head of Eq. (15.60) becomes,
(15.62)
The head H can be considered as the total opposing head of the pumping system that must be overcome for the fluid to be pumped from the lower to the upper reservoir.
The Eq. (15.62) is the equation for system characteristic, and while plotted on H-Q plane (Fig. 15.30), represents the system characteristic curve. The point of intersection between the system characteristic and the pump characteristic on H-Q plane is the operating point which may or may not lie at the design point that corresponds to maximum efficiency of the pump. The closeness of the operating and design points depends on how good an estimate of the expected system losses has been made. It should be noted that if there is no rise in static head of the liquid (for example pumping in a horizontal pipeline between two reservoirs at the same elevation), 
is zero and the system curve passes through the origin.
Effect of Speed Variation
Head-Discharge characteristic of a given pump is always referred to a constant speed. If such characteristic at one speed is know, it is possible to predict the characteristic at other speeds by using the principle of similarity. Let A, B, C are three points on the characteristic curve (Fig. 15.31) at speed
.
For points A, B and C, the corresponding heads and flows at a new speed 
are found as follows:
From the equality of 
terms [Eq. (15.14)] gives
(since for a given pump D is constant) (15.63)
and similarly, equality of 
terms [Eq. (15.14)] gives
(15.64)
Applying Eqs. (15.63) and (15.64) to points A, B and C the corresponding points 
and 
are found and then the characteristic curve can be drawn at the new speed 
Thus,
and 
which gives
or
(15.65)
Equation (15.65) implies that all corresponding or similar points on Head-Discharge characteristic curves at different speeds lie on a parabola passing through the origin. If the static lift 
becomes zero, then the curve for system characteristic and the locus of similar operating points will be the same parabola passing through the origin. This means that, in case of zero static life, for an operating point at speed 
, it is only necessary to apply the similarity laws directly to find the corresponding operating point at the new speed since it will lie on the system curve itself (Fig. 15.31).
Lecture-38
Variation of Pump Diameter
A variation in pump diameter may also be examined through the similarly laws. For a constant speed,
and
or
(15.66)
Pumps in Series and Parallel
When the head or flow rate of a single pump is not sufficient for a application, pumps are combined in series or in parallel to meet the desired requirements. Pumps are combined in series to obtain an increase in head or in parallel for an increase in flow rate. The combined pumps need not be of the same design.
Figures 38.1 and 38.2 show the combined H-Q characteristic for the cases of identical pumps connected in series and parallel respectively. It is found that the operating point changes in both cases. Fig. 38.3 shows the combined characteristic of two different pumps connected in series and parallel.
Specific Speed of Centrifugal Pumps
The concept of specific speed for a pump is same as that for a turbine. However, the quantities of interest are N, H and Q rather than N, H and P like in case of a turbine.
For pump
(15.67)
The effect of the shape of rotor on specific speed is also similar to that for turbines. That is, radial flow (centrifugal) impellers have the lower values of 
compared to those of axial-flow designs. The impeller, however, is not the entire pump and, in particular, the shape of volute may appreciably affect the specific speed. Nevertheless, in general, centrifugal pumps are best suited for providing high heads at moderate rates of flow as compared to axial flow pumps which are suitable for large rates of flow at low heads. Similar to turbines, the higher is the specific speed, the more compact is the machine for given requirements. For multistage pumps, the specific speed refers to a single stage.
Problems
(Ans. 72.78m, 65.87m)
(Ans. 58.35%, 54.1%, 1.83m of water)
(Ans. 0.55m, 0.48m)
? If the pump is to give the same total head and discharge in location where the normal atmospheric pressure is 622mm of Hg and the vapour pressure of water is 830 Pa, by how much must the height of the pump above the supply level be reduced?(Ans. 0.084, 1.65m)
Solutions
(given)
Work input per unit weight of 
Water = 
=72.78m
Under ideal condition (without loss), the total head developed by the pump = 72.78 m
Absolute velocity of water at the outlet
=23.28 m/s
At the whirlpool chamber,
The velocity of water at delivery = 0.5 ´ 23.28m/s
Therefore the pressure head at impeller outlet
=72.78 - 
= 65.87m
Hence, we theoretical maximum lift = 65.87m
Module-6
Lecture-39
Fans and blowers (Fig. 39.1) are turbulent which deliver air at a desired high velocity (and accordingly at a high mass flow rate) but at a relatively low static pressure. The total pressure rise across a fan is extremely low and is of the order of a few millimeters of water gauge. The rise in static pressure across a blower is relatively higher and is more than 1000 mm of water gauge that is required to overcome the pressure losses of the gas during its flow through various passages.
A large number of fans and blowers for relatively high pressure applications are of centrifugal type. The main components of a centrifugal blower is shown in Fig. 39.2. It consists of an impeller which has blades fixed between the inner and outer diameters. The impeller can be mounted either directly on the shaft extension of the prime mover or separately on a shaft supported between two additional bearings. Air or gas enters the impeller axially through the inlet nozzle which provides slight acceleration to the air before its entry to the impeller. The action of the impeller swings the gas from a smaller to a larger radius and delivers the gas at a high pressure and velocity to the casing. The flow from the impeller blades is collected by a spiral-shaped casing known as volute casing or spiral casing. The casing can further increase the static pressure of the air and if finally delivers the air to the exit of the blower.
The centrifugal fan impeller can be fabricated by welding curved of almost straight metal blades to the two side walls (shrouds) of the rotor. The casings are made of sheet metal of different thickness and steel reinforcing ribs on the outside. Suitable sealing devices are used between the shaft and the casing.
A centrifugal fan impeller may have backward swept blades, radial tipped blades or forward swept blades as shown in Fig. 39.3. The inlet and outlet velocity triangles are also shown accordingly in the figure. Under ideal conditions, the directions of the relative velocity vectors 
and 
are same as the blade angles at the entry and the exit. A zero whirl at the inlet is assumed which results in a zero angular momentum at the inlet. The backward swept blades are employed for lower pressure and lower flow rates. The radial tipped blades are employed for handling dust-laden air or gas because they are less prone to blockage, dust erosion and failure. The radial-tipped blades in practice are of forward swept type at the inlet as shown in Fig. 16.18. The forward –swept blades are widely used in practice. On account of the forward-swept blade tips at the exit, the whirl component of exit velocity 
is large which results in a higher stage pressure rise.
The following observations may be noted from figure 39.3.
, if 
, backward swept blades
, if 
, radial blades
, if 
, forward swept blades
Parametric Calculations
The mass flow rate through the impeller is given by
(16.26)
The areas of cross sections normal to the radial velocity components 
and 
are 
and 
(16.27)
The radial component of velocities at the impeller entry and exit depend on its width at these sections. For small pressure rise through the impeller stage, the density change in the flow is negligible and the flow can be assumed to be almost incompressible. For constant radial velocity
(16.28)
Eqs. (16.27) and (16.28) give
(16.29)
Work
The work done is given by Euler’s Equation (Eq. 15.2) as
(16.30)
It is reasonable to assume zero whirl at the entry. This condition gives
and hence, 
Therefore we can write,
(16.31)
Equation (16.30) gives
(16.32)
For any of the exit velocity triangles (Fig. 16.18)
(16.33)
Eq. (16.32) and (16.33)
(16.34)
where 
is known as flow coefficient
Head developed in meters of air = 
(16.35)
Equivalent head in meters of water = 
(16.36)
where 
and 
are the densities of air and water respectively.
Assuming that the flow fully obeys the geometry of the impeller blades, the specific work done in an isentropic process is given by
(16.37)
The power required to drive the fan is
(16.38)
Lecture-40
The static pressure rise through the impeller is due to the change in centrifugal energy and the diffusion of relative velocity component. Therefore, it can be written as
(16.39)
The stagnation pressure rise through the stage can also be obtained as:
(16.40)
From (16.39) and (16.40) we get
(16.41)
From any of the outlet velocity triangles (Fig. 16.18),
or,
(16.42)
or,
or,
(16.43)
Work done per unit mass is also given by (from (16.32) and (16.43)):
(16.44)
Efficiency
On account of losses, the isentropic work 
is less than the actual work.
Therefore the stage efficiency is defined by
(16.45)
Number of Blades
Too few blades are unable to fully impose their geometry on the flow, whereas too many of them restrict the flow passage and lead to higher losses. Most of the efforts to determine the optimum number of blades have resulted in only empirical relations given below
(16.46)
(16.47)
(16.48)For a detailed procedure on design, please refer to Stepanoff [2].
Impeller Size
The diameter ration 
of the impeller determines the length of the blade passages. The smaller the ratio the longer is the blade passage. The following value for the diameter ratio is often used by the designers
(16.49)
where 
The following relation for the blade width to diameter ratio is recommended:
(16.50)
If the rate of diffusion in a parallel wall impeller is too high, the tapered shape towards the outer periphery, Is preferable.
The typical performance curves describing the variation of head, power and efficiency with discharge of a centrifugal blower or fan are shown in Fig. 16.19.
Fan Laws
The relationships of discharge Q, head H and Power P with the diameter D and rotational speed N of a centrifugal fan can easily be expressed from the dimensionless performance parameters determined from the principle of similarly of rotodynamic machines as described in Section 15.3.2. These relationships are known as Fan Laws described as follows
(16.51)
(16.52)
where 
and 
are constants.
For the same fan, the dimensions get fixed and the laws are
and 
For the different size and other conditions remaining same, the laws are
and 
Exercise
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