082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

2.4 Review of some common engineering functions and techniques 73
Thisiswhyitiscalledanon-idealvoltagesource.Engineerswouldprefertohave
a source that maintained a constant voltage no matter how much current was drawn
but itisnotpossible tobuild such a source.
Engineeringapplication2.6
Windpowerturbines
Windturbinesareanimportantsourceofelectricalpower.Themostcommontype,
andtheoneswhichareusuallyfoundinoffshoreinstallations,resembleadesktopfan
and are called horizontal axis turbines. The wind driving a turbine blade consists
ofmanymoleculesofair,eachhavingatinyamountofmass.Thismasspassingthe
bladeareaeachsecondcarrieskineticenergy,whichisthesourceofthewindpower.
The wind power,P, can be calculated using the formula
P = 1 2 Mv2
whereM is the total mass of air per second passing the blade in kg s −1 and v is the
velocity of the air inms −1 .
The mass per second can be calculated by considering the area swept out by the
blade,A, the density ofthe air, ρ, and the velocity:
M=ρAv
This equation can besubstituted inthe power equation
P = 1 2 (ρAv)v2 = 1 2 ρAv3 (2.8)
Theavailablewindpowerthereforeincreaseswiththecubeofthevelocity.Notethat
the power isacubic polynomial function of the independent variable, v.
At 20 ◦ C the air density is approximately 1.204 kg m −3 . Consider the case of an
offshoreturbinethathasasweptareaof6362m 2 andaratedwindspeedof15ms −1 .
The maximum theoretical power atthe rated speed istherefore
P = 1 2 ρAv3 = 1 2 ×1.204 ×6362 ×153 = 12.93 MW
Theactualratedpowerofthedeviceisapproximately3MWbecauseotherphysical
processesandlosseshavetobeaccountedfor,yetEquation(2.8)remainsoneofthe
mostfundamental inthe studyof wind power.
Manyexcellentcomputersoftwarepackagesexistforplottinggraphsandthese,aswell
as graphics calculators, may be used to solve polynomial equations. The real roots of
theequationP(x) = 0aregivenbythevaluesoftheinterceptsofthefunctiony =P(x)
and thexaxis, because on thexaxisyiszero.
Figure 2.20 shows a graph of y = P(x). The graph intersects the x axis at x = x 1
,
x = x 2
and x = x 3
, and so the equation P(x) = 0 has real roots x 1
, x 2
and x 3
, that is
P(x 1
) =P(x 2
) =P(x 3
) = 0.