Selection of Drive System Components:

The group decided on a 'wedge belt' based 'overhung shaft' drive system.

Initial Calculations Necessary to start Drive System Design.

genrated 3 Generator rated Output (kW)

genspeed 1500 Rotation Running Speed of Generator (rpm)

geneff 0.7 Conservative Estimate Efficiency of Generator (%)

turbspeed 625 Rotational Speed of Turbine (rpm)

Calculation of Power Transmitted to Generator (kW)

powergen

genrated

geneff powergen 4.286

Note: This is the power required by the drive.

Calculation of Speed Ratio

Speedratio

genspeed

turbspeed Speedratio 2.4

The driving shaft rotates slower than the driven shaft and thus the system is a speed-increasing drive.

Position of Pulleys on Shafts

Belt tensioning applies a sideways load on the system shafts. That, is, the belt tension applies a radial load to the shaft. The radial load must be taken by the bearings which hold the shaft in place. Moments must be applied in order to calculate the radial load on bearings. It is therefore followed that the moment arms (distance from applied load to bearings) be kept to a minimum. That is, XT and XG in Figure below.Only two bearings should be used on one shaft. Never attempt to use extra bearing on shaft. Size bearings appropriately.

Calculation of torque at generator (Nm)

Use conversion factor 9550 which converts kW to W and rpm to radians per second.

conversion 9550

torquegen

conversionpowergengenspeed

N m torquegen 27.286 J

Note: 1J=1Nm

Selection of Appropriate Wedge Belts

Selection of the wedge belt will be from the J.H Fenner & Co. Limited, Belt Drive Catalogue. The catalogue is used in conjunction with the following calculations.A copy of the online catalogue can be found at;

http://www.fptgroup.com/index/index.asp

Fenner International Ltd are a leading manufacturer of belts for a variety of drive systems. Fenner international can be found in Europe, North America, Australia and South Africa.

South Africa is particularly important. South Africa is the main trading partner of Malawi. The group was informed at the beginning of the project to proceed under the premise that if the materials were available in South-Africa then they are also available in Malawi.

As Malawi is a developing country it is often hard to obtain information about such matters unless one has a contact in the country knowledgeable in the particular field.

Determining of Service and Duty Factor

servicefactor 1.11 This was obtained knowing the speed ratio and using Table 3 from the Fenner Belt Drive Catalogue.

dutyfactor 1.2 Micro Hydro system represented by electric motor driving a centrifugal pump, "Soft Starts" and operational for more than 16 hours a day. Class 1 light duty. Provides a duty factor of 1.2.

Calculation of the design power for belts (kW)

design power for belts=power to be transmitted X service and duty factors

designpower powergen servicefactor dutyfactor designpower 5.709

Selection of Belt SPA, SPB, SPC or SPZ, all industry standard sections.

This is dependent on the design power calculated above and the generator running speed. Table 2 Fenner Catalogue. At 5.7 kw at 1500 rpm follow graph reading instructions in catalogue.

The grade is SPZ. These are British standards for wedge belts. If operating conditions fall between area encompassed by two belts use the lower one.

Minimum Pulley Diameter (mm)

Again using the design power and the generator speed the minimum pulley diameter was found from Table 1. 80 mm. Checking this theoretical value against available stock of minimum pulley diameters at the desired speed ratio of 2.4, table page 43 SPZ belts, nearest value was 75mm.

Minpully 75

Approximation of Large Pulley Diameter (mm)

Largepully MinpullySpeedratio Largepully 180

If a pulley is not actually available at this size it will have an effect on the generator running speed. Over or under speed. There are expectable levels for this given for any generator.

Calculate the new speed ratio if necessary. Multiply by turbine speed and see if the over/under speed of generator is within range. Conclude if acceptable or not.

Approximation of Centre Distance (mm)

A rough approximation of the centre distance is as follows

centredistance Largepully Minpully

centredistance 255

Determination of Belt Length

Using A speed ratio=2.4, min & max pulley dia=75mm,180mm respectively and reading along to the right in this row until reached centre distance=255mm, table page 43, reading upwards to column titles for nearest stock belt length = 900mm. Moreover, this also gave a correction factor of 0.85, indicated by the colour of the column.

Basic Belt Power (kW)

This information was found using the minimum pulley diameter and speed ratio in the , power rating tables for SPZ wedge belts page 54.

basicbeltpower 1.65

Speed ratio Increment (kW/belt)

The increase in belt power due to the speed ratio per belt. Obtained using generator speed against speed ratio in the lower SPZ-belt power rating table, page 54

additionkwbelt 0.23

Corrected Power (kW)

The corrected power per belt= (basic belt power + speed ratio increment)X Correction factor

correctionfactor 0.85

correctedpower basicbeltpower additionkwbelt correctionfactorcorrectedpower 1.598

Number of Belts

numberbelts

designpower

correctedpower numberbelts 3.572

Use four belts, the next whole number as guided by selection manual.

numberbelts1 4

Belt Tension

The following belt tension formula was obtained form page 215, Harvey.A, Micro Hydro Design Manual A guide to small scale water power schemes.

Number of Belts numberbelts1 4

Force to depress belt 16mm per metre span (N). p 16.41

Obtained from force tables tables Harvey. A, page 207 and interpolating for shown value.

Belt Tension Td 32 p numberbelts1 N

Td 2.1 103 N

Shaft Sizing and Deflection for Turbine

Note : The following formulae and methods used to calculate shaft diameter and deflection are those advised by Micro Hydro Design Manual.

Taken from final runner_calcs Mcad. Required turbine shaft power to run generator at 3kW.Turbineshaft_power 3.722 10

3

Turbinetorque

Turbineshaft_powerN m

turbspeed 2 60 Turbinetorque 56.868 J

Drunner 0.246 m

Forcejet

Turbinetorque

0.5 Drunner Forcejet 462.341N

Belt Tension Force Td 2.1 103 N

X_X plane Radial Forces

L1 0.125 m L2 0.250 m L3 0.120 m L4 0.190 m

ltotal L4 L2 ltotal 0.44 m

Moments about B

Td 2.1 103 N Forcejet 462.341N

RaTd L1 Forcejet L2 L3

L2 Ra 1.735 103 N

Now wish to find the maximum bending moment in the X_X plane. Using the formula:

Bending Moment at any point= The sum of forces to the left of that point X distance from left to the point

Mrunner 0 J=Nm

Ma Forcejet L3 Ma 55.481 JMbelt Forcejet L3 L2 L1 Ra L2 L1 Mbelt 103.54 JMb Td L1 Ra L2 Forcejet L2 L3 Mb 2.842 10 14 JAlmost Zero

Y_Y Plane Radial Forcesg 9.807

m

s2

mrunner 7.5 kg

Weightrunner mrunnerg Weightrunner 73.55 N

Have two choices here,

1. To model the weight of the runner as a Uniformly Distributed Load (UDL) would be more representative of the real situation. The UDL could be modelled as a point load equal to the (UDL X distance of it's influence), acting at the midpoint of the UDL. The magnitude of this would be less than senario 2.

2. Model the UDL caused by runner self weight as a point load of magnitude, equal to the weight of runner, acting at the midpoint of the supposed UDL. An overestimation of the runner self weight influence but simpler to do and will provide conservative estimate.

The decision was taken to go with senario 2. As it is quicker and will give a conservative answer.

Moments about B

Rav

Weightrunner L2 L3 L2

Rav 108.854N

Assuming the belt tension is completely horizontal as assumed in the micro-hydro design manual.

Now wish to find the maximum bending moment in Y_Y plane. Using the formula:

Bending Moment at any point= The sum of forces to the left of that point X distance from left to the point

Mrunnerv 0

Mav WeightrunnerL3 Mav 8.826 J

Mbeltv Weightrunner L3 L2 L1 Rav L2 L1 Mbeltv 4.413 J

Mbv Weightrunner L2 L3 Rav L2 Mbv 0 JThus the maximum bending moment is at Belt centre with a magnitude of M.belt=0.1 KNm in the X_X plane.

The maximum moment created by the belt tension and jet force in the x-x plane is approximatley 10 times the maximum moment created in the Y-Y plane by the self weight of the runner. Remember this is with an overestimation of the effect of the runner on the system, Senario 2, from above. In reality the effect of the runner would be even less.

Now calculate the necessary shaft diameter. Several formula are available ofr this purpose. We will use a historical approach based on a code established by the ASME (American Society of Mechanical Engineers) in 1927. This method is based on the maximum shear stress theory and will tend to give a conservative result.

Structural Steel A36 , Hibbler, Mechanics of Materials, gives values for yield and ultimate tensile strength of respectively.

Yield Strength Sy 250N

mm2

Ultimate tensile Strength Sut 400N

mm2

Permissible shear stress formulae

tpa 0.30Sy tpb 0.18Sut

tpa 7.5 107 Pa tpb 7.2 107 Pa

Need to take the lower value, a permissible shear stress of tpb= 72MPa.

Need to change tpb into N/mm2 to be consistent with the units of yield and ultimate tensile

strength and give a shaft diameter in mm. 1Mpa= 1 N/mm2.

tpbnew 72N

mm2

Note this unit change is done manually mathcad has not been set up to do this. So if i change the material properties of the steel i will have to go back and manually adjust the permissible stress value into the corect units.

For belt drives, the shaft rotating and the load is considered steady. Thus, shock factor, Cm=1.5 and fatigue factor Ct=1

Cm 1.5 Ct 1

tbelt Mbelt Where M.belt = t.belt

d5.1

tpb

Cm Ma 2 Ct tbelt 2 0.5

1

3

d 0.021m

Might not be able to get a shaft exactly 21 mm. So dactual_shaft is set up to be able to change shaft diameter for deflection calculations and allow input of stock size shaft that will actually be used. Allows mathcad sheet to adjusted to increase shaft diameter if deflection critria below is not satisfied.

dactualshaft 0.026 m

Deflection of Shafts Weightrunner 73.55 N Forcejet 462.341N

Modulus of elasticity of shaft material 200GPa Td 2.1 103 N

Moment of inertia of the shaft E 200 109 Pa

I dactualshaft

464

I 2.243 10 8 m4

X_X Plane

Formulae for calculating the deflections pg 233 Harvey.A, Micro-Hydro design manual.

Forcejet 462.341N Td 2.1 103 N

Deflection at C due to belt onlyDeflection at d due to belt only

dcbelt

Td L23

48 E I ddbelt

Td L3 L2216 E I

Deflection at d due to force jetDeflection at C due to force jet

dcforcejet

Forcejet L3 L22

16 E I ddforcejet

Forcejet L32 L2 L3

3 E I

Acceptable allowable deflection is 0.0005 times the distance between bearings.

dallowable 0.0005 0.25 m dallowable 1.25 10 4 m 0.125mm

Determine the deflection d at the turbine runner

d caused by force_jet only

ddforcejet 1.83 10 4 m

d caused by T only

ddbelt 2.195 104 m

Deflection form both Force_jet and belt tension at point d

dtotald ddforcejet ddbelt dtotald 3.644 10 5 m Ok

Determine the deflection c at the turbine runner

c caused by force_jet only

dcforcejet 4.831 105 m

c caused by T only

dcbelt 1.524 10 4 m

Deflection form both Force_jet and belt tension at point c

dtotalc dcforcejet dcbelt dtotalc 1.041 10 4 m Ok

Deflection at C is more than D this makes sense because the force of jet is less than that of belt tension.

Can recduce deflection by increasing the shaft diameter

Both deflections are under the maximum allowable value.

Therefore the final values for shaft diameter and lenght are 26 mm dia shaft or above of length 44 mm.

dactualshaft 0.026 m

ltotal 0.44 m