BEng/MEng Part II Fluid Mechanics

Mech/Aero/Auto/Energy and Civil

Engineering Courses Lecturer: Dr. J M Nouri; Room: C113; E-mail: j.m.nouri@city.ac.uk Learning Method 1 – Lecture: two hours per week. 2 – Tutorial: one per week, starting from week 3. Assessment Module pass mark is 40% and has two components: 1 – Exam: carries 70% of the total mark; pass mark 40% 2 – Coursework: carries 30% of the total mark; pass mark 40% Coursework breakdown: weighting • One in November; submitting report 20% • Pump experiment - short report 20% • Class test in January [Exam type] 30% • Pipe experiment - full report 30%

Lecture Contents: Pump and Turbine: principle, types and performance Viscous flow in ducts and pipes Turbulent flow; turbulent shear stress, eddy viscosity, mixing length theory Losses in pipes – Moody diagram Hydraulic and energy grade lines Minor losses – expansion, contractions, bend, …. Pipe networks and the three reservoir problem Boundary layers – laminar, turbulent and transition Flow separation, adverse pressure gradients Recommended book FLUID MECHANICS. 1999, 4TH Edition By Frank M. White, McGraw-Hill Book Co.

School of Engineering and Mathematical Sciences

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TURBOMACHINERY - PUMPS AND TURBINES INTRODUCTION ► The design of fluid machinery, which can exchange energy with a fluid, is the most common practical engineering application. Based on the type of energy transfer, there are two distinct types of hydraulic devices

• 1- If a machine adds energy to the fluid, it is called a pump • 2- If it extracts energy, then it is called turbine

► Since both types are connected to a rotating shaft, and as turbo in Latin mean circular motion, hence such devices are commonly named as turbomachinery. ► The pump is probably the oldest fluid-energy-transfer device known, and can deliver liquids as the hydraulic energy which may be in the form of kinetic, pressure, potential energies.

• Pumps convert mechanical energy (from electrical motors or IC engines) into hydraulic energy, which can be either to increase fluid flow rate, Q, or to increase potential head, H.

• The overall efficiency, η, of the pumps are given by η = Power output/ Power input = Po/Pi

P i η

ρgQH =

The efficiency for pumps is always less than 1 due to losses such as friction, recirculation and secondary flows, blade-to-blade flow leakage…etc.

• Turbines are devices that extract energy from fluid in motion. The energy output from the turbines is usually used to drive generators to produce electricity. The overall efficiency, η, of the Turbines are given by

η = Power output/ Power input = Po/Pi

ρgQHη Po=

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• H is called NET HEAD and is measured at: The outlet flange for pumps The inlet flange for turbines

using Bernoulli’s Equation TURBINE PUMP

ρg H p1

= + + v1

2 z1 2g

ρg Hp2

= + + v2

2 z2 2g

NOTE: Bernoulli’s equation is based on conservation of energy and is usually express as:

p + ρgh + ρv2/2= constant …….. Unit [Pa or N/m2]

which states that the summation of pressure energy, potential energy and kinetic energy of a fluid system is always constant for steady flow, incompressible fluid and ideal flow, i.e no viscous effect . Another form of Bernoulli’s equation is when the above equation is express in terms of head by dividing it by ρg, and then we have: [Remember: in B. eqt., pressure is always a measured of gauge pressure and not the absolute.]

p/ρg + h + v2/2g = constant …….… Unit [m] which is the form that was used to express the net head above.

z2

v1 p1

v2 p2

z1

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PUMP CLASSIFICATION ► There are two basic types of pumps called Positive-Displacement and Dynamic Pumps.

• Positive Displacement Pumps, PDP, have a moving boundary, which forces the fluid along by volume changes. A cavity opens, and the fluid is admitted through an inlet. The cavity then closes, and the fluid is squeezed through an outlet.

• All PDPs deliver a pulsating or periodic flow as the cavity volume opens, traps and squeezes the fluid.

• A great advantage of PDPs is that they can deliver any fluid regardless of its viscosity.

• Since PDPs compress mechanically against cavity filled with fluid, a common feature is that they develop very high pressure. Thus especial design is required for safety in case of accidental outlet shot down; always a pressure relief valve is introduced.

► Dynamic pumps simply add momentum to the fluid by means of fast-moving blades or vanes.

• There is no close volume and the fluid gains momentum while moving through open passages and then converts its high velocity to pressure increase by expanding into a diffuser section.

• Based on the flow direction at the exit of the blades or vanes, which depending on their geometries, the dynamics pump are referred to as radial, axial, and mixed flows. Rotary Positive displacement pump Piston/Reciprocating Fluid (Low H & Q) Machinery Centrifugal (radial) Rotodynamic Pumps Mixed Axial

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► Dynamics pumps that we consider here uses liquid as the working fluids, but for machines which pump gases, there are three different terms that may be used depending upon the pressure rise achieved.

1) If the pressure rise is very small, a tenth of atmospheric pressure (or Δh <1m H2O) a gas pump is called a fan.

2) For pressures up to atmospheric (1m<Δh <10m H2O). It is called a blower.

3) For pressures above atmospheric (Δh >10m H2O) it is called a compressor.

A more detail of pump classification is given in appendix A

• Here we shall focus on the rotary design, sometimes called rotodynamic pumps. Other design of the dynamic pumps and positive displacement can be found in the literature especially those of: T G Hicks and T W Edwards, Pump application engineering, MacGraw-Hill, New York, 1971. R H Warring, Pump: Selection, Systems, and Application, 2nd ed., Gulf Pub., Houston, TX, 1984. I J Karassik et al., Pump Handbook, 2nd ed., MacGraw-Hill, New York, 1985. ► Comparison between PDPs and Dynamic Pumps:

• Dynamic pumps generally provide higher flow rate than PDPs and a much steadier discharge, but ineffective in handling high viscosity liquids and multiphase-flows, in particular solid-liquid flows.

• Dynamic pumps generally needs priming, that is to say that if the pumps are filled with gas, they can not suck up a liquid from

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below into their inlet, while the PDPs are self-priming for most applications.

• Although the dynamic pumps provide very high flow rate (up to 40,000 lit/min) but the pressure rise with these pumps are moderate (a few bars). In contrast, a PDP can operate up to high pressure (around 300 bars), but at low flow rate of up to 25 Lit/min.

• There are major differences in the relative performance of these two types of the pumps as shown in figure below where curves of the Δp versus Q are presented. It shows that, at a constant shaft speed, the PDP produces nearly constant flow rate and virtually unlimited pressure rise, and that the effect of viscosity with these types of pumps is negligible. Figure 1 Comparison of performance curves of PDPs and dynamic pumps; (FM White p.714).

• In contrast, the dynamic pumps give maximum pressure rise, Δp, at zero flow (shutoff condition) and zero Δp at maximum flow rate. It is also evident that high viscosity liquid down-grad the performance of the pump greatly.

Pressure rise or Head increase

Discharge, Q,

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ROTODYNAMIC PUMPS 1) Radial flow

• In radial centrifugal pumps the impeller rotation causes the fluid to move radially from inlet to outlet of the impeller. As a result, water at the outlet of the impeller is under higher pressure and greater velocity than at inlet. • A part of the kinetic energy of the flow is converted into pressure energy within the casing of the pump before exiting into the discharge pipe.

• This conversion is made either in a gradually expanding volute chamber or between the diffusion vanes of the diffuser chamber.

• Radial centrifugal pumps are formidable devices and are able to deliver very high heads and reasonable flow rate with excellent efficiency, and can be matched many system requirements. 2) Axial flow

• With the axial flow pump the pressure is developed by the flow of liquid over the blades of an aerofoil section just as the wings of an aeroplane produce the lift.

• Axial flow pumps are design to produce very large quantity of water at low head and are ideally suited for irrigation purposes. 3) Mixed flow

• This type of pump has an impeller (modified radial flow impeller) capable of pumping large quantities of liquid. Mixed flow pumps are mostly used for irrigation purposes and other application where large quantity of water at low head is required.

• Flow through the impeller as suggested by the name is a combination of radial and axial flow and the shape of the impeller resemble the propeller of a ship.

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THE CENTRIFUGAL PUMP ► Let us examine the characteristics of a centrifugal pump, as shown in figure 2, consists of an impeller rotating within a casing (volute).

Figure 2 Cutaway schematic of a typical centrifugal pump (FM White p. 714). ► The liquid enters axially through the eye of the casing at 1 and then is diverted tangentially and radially outward by the action of the impeller blades until it leaves through circumferential parts of the impeller into the scroll or diffuser of the casing; the liquid exits the casing at point 2. ► The liquid gains both velocity and pressure while passing through the impeller, but in the scroll the flow decelerates and increases the pressure further. ► The blades are usually backward-curved, as shown in figure 2, but there are other types like radial and forward-curved designs, which affect the change of pressure output. ► The blades may be open which will have a very narrow gap from the front of the casing and shrouded on the back by impeller wall. Or closed blades, which are shrouded from the casing on both sides by an impeller wall. ► The diffuser may be vaneless, as in figure 2, or fitted with fixed vanes to help guide the flow toward the exit.

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► Assuming steady flow, the pump will increases the total head of the flow between the inlet, point 1, and the exit, point 2. Using Bernoulli energy equation and neglecting viscous work and heat transfer then the total head change between 1 and 2 is

ρg H p2

= + +v2

2 z2 2g ρg

p1 + +

v12

z12g

Or H = hp-hs Where hp is the pump head available at the outlet and hs is the head available at the inlet (suction) of the pump. The net head H is a primary output parameter for any turbo machine. The above equation is for incompressible flow and must be modified for gas compressor where large density changes take place. ► For centrifugal pumps the values of velocities at point 1 and 2 are similar, and also the height difference in no more than a meter for large pumps. Therefore the pump head is effectively due to change in pressure head

ρg H

Δp ≈

p2-p1

ρg =

The power delivered to the fluid is simply equals the specific weight times the discharge times the net head, which is Pw = ρgQH This is called the water horsepower. The unit is referred to as horsepower, hp, which is the same as 746 W or 550 ft.lbf/s. ► The power input that is required to drive the pump is called the brake horsepower, bhp, and is given by the product of shaft torque, T, and the shaft angular velocity, ω, as bhp = Pi = ωT ► The efficiency of the pump is defined as

bhp η

P o =

ωT

ρgQH =

10

The main objective in designing a good pump is to make η as high as possible over a large range of flow rate. ► The efficiency can be break down into three parts namely the volumetric, hydraulic and mechanical efficiencies. The volumetric efficiency is

Q ηv

Q-Q L =

where QL is the loss of fluid due to leakage in the impeller-casing clearance. The hydraulic efficiency is

hs ηh

hs-hf =

where hf is the total hydraulic loss and has three parts (a) shock loss at the eye due to imperfect matching between inlet flow and blade entrances, (b) friction losses in the blade passage, and (c) circulation loss within blade passages especially towards the exit. And finally the mechanical efficiency is

bhp ηm

bhp-Pf =

where Pf is the power loss due to mechanical friction in the bearing and any other contact points in the machine.

► The total efficiency is, by definition, simply the product of all three parts as

η =ηvηhηm To optimise the pump performance, the designer has to work in all three areas to reduce losses at a minimum cost and manufacturing practicality.

► EXAMPLE A pump draws water from reservoir A and lifts it to tank B. The diameter of the suction and discharge pipes are 75 and 60 mm, respectively. If the gauge pressures at the inlet and outlet of the pump are 17.95 and 1063.9 kN/m2, respectively, and if the discharge rate is 0.02 m3/s determine the power output of the pump. [28.45 hp]

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PUMP PERFORMANCE

► We have established from dimensional analysis the following dimensionless groups.

ρN3D 5

Pi = Φ

ND3

Q

N 2D 2

gH ,

ρND 2

μ , η ,

P* (CP) φ (CQ) , Ψ (CH) , 1/Re , η Power Flow , Head , , Efficiency Coefficient Coefficient Coefficient (Overall)

η Pp*

φΨ =

P t* ηφΨ =

where the subscript p and t are referred to as pump and turbine, respectively. ► Plots of P*, Ψ, and η against φ provide characteristic curves for turbomachine. These curves are obtained by conducting tests on pumps/turbines at constant speed. At each speed, Q is varied and then the resulting H is measured; P*, Ψ, η and φ are subsequently calculated. From these curves the performance of the machines can be deduced, as shown in figure 3 for a pump.

-The head is almost constant at low flow and then drops to almost zero at maximum flow.

-The efficiency increases from zero at no flow to a maximum of 80 to 90% at a flow of 0.6Qmax

-The horsepower require for driving the pump Rise with flow

Rated point (or BEP) Figure 3 Characteristic curves for typical centrifugal pump.

► The point of the maximum efficiency is called the rated point or the best effiiency point (BEP). It is desirable to have a flat curve for efficiency at the rated point so that a wide range of efficient operation can be achieved.

φ

Ψ

P*

η

And

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SIMILARITY LAWS ► For pumps belonging to one family and operating under same dynamic conditions (i.e. same Reynolds number, Re), then their dimensionless group parameters (P*, φ, Ψ and η) will be equal. Thus for two machines which are geometrically similar, their performance characteristics are correlated as (P*)1=(P*)2 φ1=φ2 Ψ1=Ψ2 η1=η2

With these relations the performance characteristics under new operating condition can be obtained. Example A centrifugal pump, impeller D=0.5m, when running at 750 rpm gave on the test the following performance characteristics:

Q (m3/min) 14 42 H (m) 40.4 25.6 η (%) 60 74

Predict the performance of a geometrically similar pump with D=0.35 and running at N=1450 rpm.

PUMP SELECTION

► Best efficiency point (BEP): is where the efficiency is maximum (ηmax) and the flow rate, head and power at this point are refereed to as QBEP, HBEP and PBEP. ► Specific speed (Ns): (also called type number or shape number) is such an effective parameter to indicate both the performance and efficiency at BEP and to help the selection of the type of the pump (centrifugal, mixed and axial). ► Duty point (DP): or operating point, is the point on the performance curves (QDP, HDP) where pump usually operates at which the head output (HDP) is exactly the same as the load imposed on it; this will be discussed further later on. This point may or may not be the same as BEP but a good design will have the duty point as close to maximum efficiency point.

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SPECIFIC SPEED ► This is accomplished, for pump, by eliminating the diameter from flow and head coefficients at the maximum efficiency only, i.e. BEP point, as follow:

Ψ 3/4

N s φ1/2

=

(gH BEP)3/4

ω(Q BEP)1/2

=

In SI units N=rad/s, QBEP=m3/s, HBEP=m. This means that the Ns is the speed required to produce a head of (1/g) m at a flow rate of 1 m3/s. ► In US customary units it represented as

(H BEP)3/4

N(QBEP)1/2

= Nsd

where N=rev/min, QBEP=gal/min, HBEP=ft. The conversion factor between the two units is Nsd=2733Ns. ► The chart shown below, provide the variation of specific speed for different type (shape) of the pumps. To select type of the pump from Ns we have:

1) Low specific speed, Ns<0.5: the flow is low and head is high and is best suited for centrifugal (radial) pump; in US units Nsd<1370.

2) Middle range specific speed, 0.6<Ns<2.5: is ideal for mixed flow pump; in US units 1640<Ns<6832.

3) High specific speed, Ns>3.0: the flow is high and head is low and is best suited for axial pump; in US units Ns>8200.

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SUCTION HEAD ► Pump creates a vacuum in suction pipe and the atmospheric pressure forces water up to fill vacuum. This will introduce the required suction head. For example at see level, where Patm is 1 bar or 105 N/m2 the suction head is h=p/ρg which gives a head of about 10 m.

∇ hs ► This is the maximum suction head without any losses, but due to friction losses in the pipe the suction head would be less than 10 m. Also, if the supply tank is fixed up in higher altitude (mountain) then the suction head would be less. ► Centrifugal pumps can not create sufficient vacuum to suck out air, due to inadequate seals, thus the pump and the suction pipe must be filled with water before pump is started; priming can be done by using a manual pump.

DELIVERY HEAD ► The total head required from a pump is called pumping head and includes the suction head, friction loss in the suction, delivery head and friction loss in delivery pipeline. ► The delivery head (lift head) is the elevation change between pump and the point of supply. For flow control there are other components that contribute to loss like pressure gauges, delivery valve to control the flow, and non-return valve to allow water to flow in one direction and prevent damages when the pump stopped suddenly. ► If HD and HS are the overall delivery and suction heads the the total head rise across the pump is Hacross pump= [HD + (VD)2/2g]-[HS + (VS)2/2g]

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CAVITATION

► When a liquid flow into a region where its pressure drop to vapor pressure, the liquid boils and vapor pockets develop. The vapor bubbles are carried along with the liquid until a region of higher pressure is reached, where they suddenly collapse. This process is called cavitation. ► Water boils at around 100 oC at atmospheric pressure (1 bar). By reducing the pressure, boiling can occur at temperature well below 100 oC, for example, if the pressure reduces to 0.5 bars and 0.015 bar then the boiling take place at 81.3 oC and 13 oC, respectively. ► If the vapor bubbles are near or in contact with a solid surface when they collapse, the forces exerted by the liquid rushing into the cavities and create very high localised pressures that cause pitting of the solid surface. This phenomenon is accompanied by noise and vibration and a drop in pump head and discharge. ► A useful parameter in characterising the susceptibility of the system to cavitate is the Cavitation Parameter, σ, defined as

0.5ρv2

σ p-pv

=

where p is the absolute pressure at the point of interest and pv is the vapor pressure of the liquid. The cavitation initiates when σ=0, and two geometrically similar system would have the same degree of cavitation for the same value of σ. ► Experiments indicate pressures of order of Gpa exposed to cavitation, and with this type of magnitude the observed mechanical damage caused by cavitation is not surprising. ► Protection against cavitation should start with hydraulic design of the system in order to avoid the low pressures if practical. Otherwise special cavitation material of coating may be effective.

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► There are three undesirable conditions with cavitation: (a) lower efficiency, (b) damage to flow passages, and (c) noise and vibration. Curved vanes are particularly susceptible to cavitation on their convex sides near the low-pressure side of the impeller and may have localised area where cavitation causes pitting or failure.

Figure 4 Cavitation on the surface of a marine propeller. (FM White p. 33).

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NET POSITIVE SUCTION HEAD (NPSH) ► NPSH is a useful and frequently used parameter in specification of minimum suction head required for a tourbomachine to avoid cavitation. If e is the point of minimum pressure, the energy equation for a pump or turbine applied between e and downstream liquid surface can be written as

ρg

pe + +

Ve2

hs2g pa

ρg = hl ±

where pa and pe are the atmospheric and absolute pressures, hl is

the total head loss in suction pipe and hs is the suction head.

► For cavitation to occur at e the pressure must be equal to or less than pv, the vapor pressure. The NPSH is defined as the kinetic head energy gained at the suction point and is given by

ρg

pv = =

Ve2

hs2gpa

ρg − hl −NPSH ±

Note: for a pump the sign of hl is negative. This is the minimum required head for cavitation and will be provided by manufacturer. In practice for satisfactory operation one must ensure that (NPSH)available is greater that the required value. ► EXAMPLE The required NPSH of a centrifugal pump is given by the manufacturer to be 3.3m together with information about losses like entrance loss=0.001m, bend losses=0.15m, and head losses in pipes=1.5m. Determine the allowable suction head; allow 15% higher value for NPSH for satisfactory operation. [4.2 m]

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Figure 5 Performance curves of a centrifugal pump. (FM White p. 722).

► Figure 5 present the actual performance data within the range of the operating condition near BEP. The head curves H vs Q are shown, but the break-horsepower, bhp, and the efficiency curve are superimposed as contour plots; a plot of NPSH is also given. Conversion: 1 US gallon=3.785x10-3 m3.

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MATCHING PUMP TO PIPELINE LOAD ► Consider the pumping system shown below where the water is drawn from a reservoir, with a suction head of hs, to be delivered to another reservoir, with a delivery head of hd.

Figure 6 Schematic of a pumping system. ► The matching is done by obtaining the total system load as a function of flow rate and plots it with pump head characteristic against the flow rate in the same graph. The intersection of the two curves is the physical matching point between the pump and the operating-system load as shown below.

Total Losses=∑hlf + ∑hlp Static Head= hd + hs

Duty point Figure 7 Characteristic curves for pump head and operating load to obtain the operating

duty point.

Q

H Hsys

η

HP

HDuty

QDuty

hd

hs

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► The total system head includes the static-elevation head, i.e. the sum of delivery and suction heads hd + hs, plus all the losses in the system due to friction and pipe fittings and is given as

Hsys = hd + hs +∑hlf + ∑hlp Where the frictional losses can be expressed by Darcy equation and the pipe fitting losses can be expressed in terms of kinetic head as

d

∑ hlf L

= λ . v2

2g

∑ hlf = K

v2

2g Since V is proportional to flow rate, Q, then the total head can be also expressed in terms of flow rate. ► Superimposing the efficiency curve on the same plot provide information that whether the selected DP is acceptable or not. If the DP point is well away from the maximum efficiency, i.e. from BEP, then the system is at off design and required changes to improve the system either changing the pump size or its speed. But if the duty point is around the BEP then the system design is acceptable. EFFECT OF PIPE DIAMETER ► If the total system load is represented in terms of pipe line diameter, d, using the continuity relation then a series of the system head curves can be obtained for different d from which a proper selection can be made to give DP as close to the BEP as possible. For example, for a given pumping system like in figure 6, the static head can be considered as a constant and then the losses can be expressed in terms of d. Since V=Q/A where the pipe cross-section area is A= πd2/4, thus V2 is proportional to d-4. ► Therefore head losses due to friction are proportional to d-5, and due to pipe fitting is d-4 and consequently the total system load has similar proportionality. Figure 8 presents the system dead variation with pipe diameter.

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Duty point= 1 2 3 4

Figure 8 Characteristic curves for pump head and operating load for different pipe

diameter.

COST IMPLICATION ► The capital cost for small pump is low but the running cost is higher due to higher friction losses (i.e. higher energy lost for the same flow rate). A reverse effect is true for large pumps. A reasonable cost valuation will involve the estimate of the capital cost plus running cost over a period of time, for example 10 years. ► The running cost can be estimated by using power input equation as Pin=(ρgQH)/η. Then the energy used per day can be estimated by multiply power input by 24 to give kW.hr per day, from which the running cost can be estimated by multiplying this to the cost of electricity (usually £/kW.hr).

PUMPS COMBINED IN SERIES AND PARALLEL ► When a pump provides the right discharge but little head, the head can be enhanced by considering to use another similar pump in series with the out put of the first pump fed into the second one directly. ► The physical principle is that the two heads at the same flow rate will be added to give the combined performance curve. The two pumps need not to be identical and they can be even run at different speeds, although normally both are driven by the same shaft.

Q

H Hsys

η

HP

d4

d3

d2 d1

d4 > d3 > d2 > d1

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► When a pump provides the right head but little discharge then a parallel combination of two similar pumps can be used with a common suction. ► The physical principle is that the total discharge flow rate, Q(1+2), in the combined system is the sum of the flow rate from either pumps at the same head. ► In Both cases the combined efficiency of the system is

ω1T1+ ω2T2

ρgQ (1+2)H(1+2)= η

EXAMPLE The following performance curve data were obtained from a test on centrifugal pump with a 21.6 cm impeller and at a constant speed of 1350 rpm.

Q (m3/min) 0 0.454 0.905 1.36 1.81 2.27 2.72 3.2 H (m) 12.2 12.8 13.1 13.4 13.4 13.1 12.2 11.0 η (%) 0 26 46 59 70 78 78 74

The pump is installed in a pipe network. The pumping head required is given by Hsys=5.0 + 1.0Q2, where Q is the flow rate in m3/min. Find (a) the pump head, capacity and efficiency at the duty point, (b) the power input required to operate the pump at these condition.

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ELEMETARY PUMP THEORY ► To predict head, power, efficiency, and the flow rate of a pump from flow motion there are two possible theoretical approaches. The first is a simple approach using one-dimensional flow, and the second one is a complex 3D model, which account for viscosity and three dimensionality. ► To construct an elementary theory of pump performance we assume one-dimensional flow and combine the idealised fluid velocity vector through impeller and considering the change of angular momentum from inlet to outlet of the impeller. ► The idealised velocity diagram are shown in figure 11. The flow enter the impeller at r=r1 (control volume is the angular region between sections 1 and 2 where the flow passes through blades), and exit at r=r2 with tangential velocities to the blade of w1 and w2, respectively. u1 and u2 are the blade tip velocities and equal to ωr1 and ωr2, respectively, and V1 and V2 are the absolute entrance velocity obtained from velocity diagram as the vector sum of w1 & u1 and w2 & u2, respectively.

Figure 9 Inlet and exit velocity diagrams for an idealised pump impeller; F White p. 717

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► Knowing the absolute flow velocities, V1 and V2, in and out of the impeller, then the absolute circumferential and normal (radial) velocity components, Vt and Vn respectively, of the flow can be obtained as shown. It is the Vt, which contributes to the change of angular momentum from inlet to outlet of impeller. The flow through the impeller can also be obtained from the product of Vn and the flow passage cross-section area at inlet and outlet. ► For a steady and incompressible flow, the contribution of pressure to the torque is zero since the pressure acts radially through centre. The applied torque is, therefore, the change of angular momentum of the fluid going from 1 to 2 and is given by T = ρQ (r2Vt2 – r1Vt1) (1) Where Vt2 and Vt1 are the absolute circumferential velocity components of the flow. The power delivered to fluid by this torque is P = ωT = ρQ (ωr2Vt2 – ωr1Vt1) Or P = ρQ (u2Vt2 – u1Vt1) (2) Since for an ideal situation where there is no loss the power is given by P = ρgQH. Thus the ideal head generated by the applied torque will be H = (u2Vt2 – u1Vt1)/g (3) These equations are called the Euler turbomachine equations showing that the torque, power and ideal head are functions of the rotor tip velocities u1,2 and the absolute fluid tangential velocities Vt1,2. ► Another useful form of equation (3) can be produced by using the geometrical property of the velocities triangle in figure 9. From cosine rules we have

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V2 = w2 + u2 – 2uwcos β and wcos β = u- Vt Or uVt = (V2 + u2 - w2)/2 (4) Substituting this into equation (3) give H = [(V2

2-V12)+(u2

2-u12)-(w2

2-w12)]/2g (5)

The ideal head is, therefore, related to the absolute (V) plus the rotor tip (u) kinetic energy change minus the rotor tangential (w) kinetic energy change. ► The power can also be related to the normal (radial) velocity, Vn. From the velocity triangle in figure 11 we have Vn = Vt tanα. Thus the input power is P = ρQ (u2Vn2 cot α2 - u1Vn1 cot α1) (6) Vn can also be related to flow rate as Vn2 = Q/(2π r2b2) And Vn1 = Q/(2π r1b1) (7) Where b1 and b2 are the width of the blade at inlet and outlet. ► A special design flow rate, Q, is commonly estimated by assuming that the flow enters normal to the impeller that is when α1= 90o And Vn1 = V1 (8) And is referred to as design point. Actual measured values showed that this simple analysis predicts the head, power, and discharge of a pump to within ±25%. EXAMPLE- The following data are given for a commercial centrifugal water pump: r1 = 100 mm, r2 = 177 mm, β1 = 30o, β2 = 20o, N = 1440 rpm. Estimate (a) the design-point discharge, (b) the water horsepower, and (c) the head if b1 = b2 = 45 mm.

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TURBINES ► A turbine is a device that extract energy from fluid in motion, which possesses high head. The way that this high head energy is converted to mechanical energy defines the type of turbines and there are basically two types of turbines, reaction and impulse. (I) Reaction: Here the fluid fills the blade passages, and as

the fluid passes through the impeller the pressure drop occurs and. Like in pump, there are three designs for reaction turbine, the radial-flow, mixed-flow and axial flow, which are dynamic devices for extracting momentum from high-head energy fluid.

(II) Impulse: High-head energy is first converted to a high

velocity jet through a nozzle. The jet flow strikes the blades at one position as they pass by. Since the impeller passages are not filled by fluid the pressure at the blade is essentially constant. Also for this reason the impulse turbines are larger than reaction turbines.

TURBINE PERFORMANCE ► The dimensionless forms of head, Ψ, flow, φ, and power, P*, obtained from dimensional analysis are the same as those for the pump as shown below.

ρN3D5

P = Φ

ND3

Q

N2D2

gH ,

ρND2

μ , η ,

P* (CP) φ (CQ) , Ψ (CH), 1/Re , η Exception is that the power output with turbine is the shaft brake horsepower and power input is fluid head while the reverse is true for pumps. The overall efficiency for turbine is

φΨ η t

P* =

ρgHQ Po

=

P* ηp

φΨ =

Po

ρgHQ =

While for the pump it is

27

SIMILARITY LAWS ► Similar to the pumps, turbines belonging to one family and operating under same dynamic conditions (i.e. same Reynolds number, Re), have the same dimensionless group parameters (P*, φ, Ψ and η). Thus for two machines which are geometrically similar, their performance characteristics are correlated as (P*)1=(P*)2 φ1=φ2 Ψ1=Ψ2 η1=η2

SPECIFIC SPEED (Ns) ► Also called type number or shape number. The specific speed is used to help the selection of the type of the pump, impulse or reaction; in case of reaction turbines (radial, mixed or axial). Ns is obtained by using a combination of dimensionless coefficients so that the impeller diameter can be eliminated; for turbine it is done as

ψ 4/5

2/1*PN s= OR

)()(

4/52/1

2/1

gHPN

N sρ

=

In SI units N=rad/s, P=Watt, H=m. In US customary units it represented as

(H)3/4

N(P)1/2

= Nsd

where N=rev/min, P=hp, H=ft. The conversion factor between the two units is Nsd=43.48Ns. Note that Ns and Nsd (like for pumps) are defines with respect to BEP and has a single value for a given turbine family.

Figure 10 Optimum efficiency of turbine designs.

Nsd

Reaction Turbines Radial- and mixed-flow Axial-flow

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► The chart shows the variation of maximum efficiency with the specific speed for different types (shape) of the turbines and provides the range of specific speed for different turbines as:

4) Impulse turbines perform well at low specific number ranging 1<Nsd<10 (0.023<Ns<0.23)

5) Middle range specific speed 10<Nsd<110 (0.23<Ns<2.5) is good for radial and mixed flow turbines (Francis).

6) In the range of 100<Nsd<250 (2.3<Ns<5.7) axial (propeller) turbines provides good performance.

► The chart also shows that the efficiency of the reaction turbines is higher than those with impulse turbines. CAVITATION

► Like in pumps cavitation can happen in turbines where the liquid pressure drops to vapour pressure. It takes place in regions where the pressure drops considerably like the discharge region (or the eyes of the turbine) since a large pressure drop happened as the flow passes through the impeller passages. Curved vanes are particularly susceptible to cavitation on their convex sides near the low-pressure side of the impeller. Hence, cavitation causes undesirable conditions like lowering efficiency, damaging to flow passages, producing noise and vibration. NET POSITIVE SUCTION HEAD (NPSH) ► It has been shown that for cavitation to occur at the eye the pump or turbine the pressure must be equal to or less than pv, the vapour pressure of the liquid. The NPSH is defined as the kinetic head energy gained at the suction point and is given by

ρg

pv = =

Ve2

hs2gpa

ρg − hl −NPSH ±

► This is the minimum required head for cavitation and will be provided by manufacturer. For a turbine the sign of hl is positive.

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EXAMPLE: A 1/5th scale model of a full-scale turbine develop 3kW at 360 rpm under a net head of 1.8m. Find the speed and power of the full size turbine when operating under a head of 5.8 m. Assume that both are operating at maximum efficiency. REACTION TURBINES ► In general, they are low-head, high-flow devices with a flow motion, opposite to that in a pump, entering at the larger section and discharging through the eye giving up most of its energy to impeller. ► The early designs had no stationary guide vanes to direct the flow into the impeller passage smoothly and were very inefficient. The first efficient reaction turbine was made in 1949 by J B Francis, a U.S. engineer, and all radial- and mixed-flow designs are called Francis turbine. A purely axial flow turbine, with lower head, is called propeller turbine, with either fixed-blade or adjustable (Kaplan type). Figure 11 Reaction turbines: (a) Francis, radial type; (b) Francis, mixed-flow type; (a) propeller, axial type. (FM White p. 743).

Ns=0.46 Ns=1.38 Ns=3.22

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RADIAL TURBINE THEORY ► The Euler turbomachine formulas can also be applied to energy-extracting machines. Like with the pump, assume one-dimensional frictionless flow through the blades. Figure 15 show a radial turbine runner with an adjustable guide vane to ensure the inlet flow to the blade with minimum directional-mismatch or shock loss. ► Using the vector velocity diagram at the inlet and outlet and also the application of angular momentum, similar to that in the pump, will yields an idealized formula for the power extracted from the liquid by the runner can be obtained.

Figure 12 Inlet and outlet velocity diagrams for an idealized radial-flow reaction turbine runner. (FM White p. 743).

► The extracted power is

T = ρQ(r2Vt2 – r1Vt1)

P = ωT = ρQ(ωr2Vt2 – ωr1Vt1) = ρQ(u2Vt2 – u1Vt1)

Or P = ρQ (u2V2 cos α2- u1V1cos α1)

Where Vt2 and Vt1 are the absolute inlet and outlet circumferential velocity components of the flow. The equations are identical for

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pumps and turbine except the shape of the blades are not the same. ► The absolute normal inlet velocity, Vn2, is proportional to the flow rate, Q, and is given by Vn2 = V2 cos α2. ► The design of the large-scale power-generating turbine system is a major engineering project. It involves inlet and outlet ducts, guide vanes, spiral case, generator with cooling coils, bearing and transmission gear, runner blades, and automatic controls. As an example is the awesome 1000 MW hydropower designs when viewed on a human scale, as shown below for a Francis turbine unites at Grand Coulee Dam of Columbia River. ► This implies the economic advantages of the small-scale model testing for the design of the large-scale turbines.

Figure 13 large scales Francis turbine unit at Grand Coulee Dam (FM White p. 747).

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IMPULSE TURBINES ► In general, with impulse machine there is no change of static pressure across the runner. In the impulse turbine all the available energy of the flow is converted by a nozzle into kinetic energy at atmospheric pressure before the fluid contacts the moving blades. Thus they are ideal for high head and relatively low power, i.e. low Ns. ► The schematic diagram below shows the available energy, or the total head Ha, at the base of the nozzle. The total head is given

ρg Ha p1

= + V1

2

2g For a perfect nozzle, the entire available head would be converted to jet velocity, Vj. In practice, there are losses of order of 2 to 8% and this is expressed with velocity coefficient, Cv. Thus the jet velocity is related to this available energy through the nozzle as Vj = Cv √(2g Ha) Or V2

j /2g= Cv2Ha

The head lost in the nozzle is Ha - V2

j/2g= Ha - Cv2Ha = Ha(1- Cv

2)

Figure 14 Schematic diagram of an impulse turbine.

Ha

Head-water

1 J

Surge tank

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PELTON WHEEL ► This is the only hydraulic turbine of the impulse type now in common use and is named after Lester A. Pelton (1829-1908) the American engineer. It is an efficient machine and particularly well suited to high head. The rotor consists of a circular disc with a number of spoon-shaped buckets evenly spaced round its periphery. Each bucket has a splitter ridge, which divides the oncoming jet into equal portions.

Figure 15 Schematic diagram of a Pelton impulse turbine: (a) side view of the wheel and the jet; (b) top view of the bucket; (c) Typical velocity diagram. (FM White p. 747). ► By considering the change of momentum between oncoming jet and outgoing jets, it can be shown that the force exerted by the fluid on the Pelton wheel is F = ρQ (Vj – u)(1-cos β) and the power extracted from the fluid is P = Fu= ρQu(Vj – u)(1-cos β) Where u is the tip velocity of the wheel and is equal to rω where ω is the wheel angular velocity and r is the pitch radius of wheel.

34

► A bucket angle β=180o produce the maximum power but it is physically impractical; at this angle the deflected jet flow will impact on the back of the following bucket causing more problem. In practice, β=165o will give 1-cos 165o =1.966 which is only 2 percent less than the maximum power. ► If the available head is kept constant then the jet velocity will be constant (i.e. Vj = constant= [Cv √(2g Ha)]). P = ρQu(Vj – u)(1-cos β) Or P = ρ Aj Vj u(Vj – u)(1-cos β) Hence the maximum power is when uVj(Vj – u) is maximum, and since the power equation gives a parabolic variation with u then the maximum power is when dP/du=0. Differentiating power with respect to u gives dP/du= d(uV2

j – Vj u2)/du = 0 V2

j – 2Vj u*=0 Thus u*= Vj/2 ► The theoretical impulse turbine efficiency now can be obtain by combining the power and efficiency equations as follow η=P/(ρgQHa) Substitute for P = ρQu(Vj – u)(1-cos β) Hence η=u(Vj – u)(1-cos β)/(gHa) By rearranging this equation we will get the following expression. η=2(1-cos β)φ(Cv – φ) Where Cv=√(2gHa) is the velocity coefficient and φ is the peripheral-velocity to jet-velocity factor and is defined as

φ = u/Vj

35

► Figure 19 present the variation of the efficiency, η, of a Pelton impulse turbine as a function of velocity factor, φ. In all cases the curves are in a parabolic form with the maximum efficiency at around φ ≈ 0.44 t0 0.48. The ideal efficiency curve, solid line, is for the case when β=180o and Cv=1.0. Under working conditions (design condition), β=160o and Cv=0.94, the typical curve would be the dashed line with a maximum efficiency of 85% at φ = 0.47.

Figure 16 Efficiency of a Pelton impulse turbine: solid curve is ideal case, i.e. β=180o and Cv=1.0; dashed curve is the actual, i.e β=165o and Cv=0.94; open circle is measured data, Pelton wheel=2m. (FM White p. 748). ► However the measured data, open circle, show somewhat less efficient performance, which can be due to windage, mechanical friction, back-splashing, and non-uniform bucket flow. In this case the maximum efficiency is around 80%. As it was discussed before, the impulse turbine in general are not as efficient as the Francis or axial (propeller) turbines at their BEP.

36

Bernoulli’s Equation Assumptions:

• Flow is steady and irrotational

• Fluid incompressible and non-viscous i.e. ideal fluid

Consider energy conservation in the flow of liquid through a pipe:

Work can be done by applying a

force F1 at position 1 (to a small

volume of liquid cross sectional area

A1, through a distance s1 at constant

velocity v1). At position 2 the force is

transformed into push F2 (to a small

volume of liquid cross sectional area

A2, through a distance s2 at constant

velocity v2).

Conservation of Energy equation

External work done= Internal energy exchange

W = ΔPE + ΔKE (1)

where W is the net work done, ΔPE the change of potential energy and

ΔKE is the change in kinetic energy between positions 1 and 2:

W = F1s1 – F2s2 and since pressure P=F/A then

W = P1A1s1 – P2 A2s2 = (P1 – P2)V (2)

where V = Volume displaced =V1 = A1s1 =V2= A2s2 (since incompressible)

Also the change in potential energy and kinetic energy can be expressed by:

Δ PE = Δ (mgh) = mgΔ h = ρ Vg(h2 - h1)

37

Δ KE = Δ (½mv2) = ½mΔ v2 = ½ρ V(v22 - v1

2)

Therefore from equations we have

(P1 – P2)V = ρ Vg(h2 - h1) + ½ρ V(v22 - v1

2)

Diving by V and rearranging the equation we get Bernoulli’s Equation:

P1 + ρgh1 + ½ρ v12 = P2 + ρgh2 + ½ρ v2

2

in terms of pressure unit [N/m2]. If we divide both side by ρg then the equation can be expressed in terms of Head unit [m] as:

P1 /ρg + h1 + v12/2g= P2 /ρg + h2 + v2

2/2g

Since the energy is conserved the it is often expressed as

P + ρgh + ½ρ v2 = constant

38

Appendix - I

39

40

41

42

Fig. 1 Positive Displacement Pumps

43

44

Fig. 2 Centrifugal Pump

45

46