A Note on Disk Drag Dynamics

0.53 MB·English

by  Neil J. Gunther

A Note on Disk Drag Dynamics
NeilJ.Gunther
PerformanceDynamicsCompany,CastroValley,CA94552
[email protected]
2 February1,2012
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Abstract

n
Theelectricalpowerconsumedbytypicalmagneticharddiskdrives(HDD)notonlyincreases
a
J linearlywiththenumberofspindlesbut,moresignificantly,itincreasesasveryfastpower-lawsof
speed(RPM)anddiameter. Sincethetheoreticalbasisforthisrelationshipisneitherwell-known
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norreadilyaccessibleintheliterature,weshowhowtheseexponentsarisefromaerodynamicdisk
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draganddiscusstheirimportforgreenstoragecapacityplanning.

]
F
P 1 Introduction
.
s
c Thesemi-empiricalrelationship[1,2,3]
[

1 Power∝ Platters× RPM2.8 × Diameter4.6, (1)
v
2 hasbeenusedtomotivate“greener”magneticHDDdesigns[4]. Ofthethreevariablesthatcanused
0
tomatchagivenpowerconstraint,diameterandRPM(revolutionsperminute)havethegreatestimpact
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duetothehighdegreeoftheirrespectivepositiveexponents(Fig.1). Itisnoteworthythat(1)isnota
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. functionofcapacity(e.g.,GB);thetypicalmetricusedforstoragecapacityplanning.
1
0
fx
2 20
Lnf
1 20.0
: !" 15.0 x5 x3 x2
v 15 x5 x3 x2 !"
i 10.0
X
7.0
r Out[45]= 10 Out[42]= 5.0 x
a
3.0
5 x
2.0
1.5
x 1.0 Lnx
0 1 2 3 4 5 1.0 1.5 2.0 3.0 5.0
(a)Linearaxes (b)Log-logaxes
!"
Figure1: Afastfunctionfamily: f(x) = xp withpowers p =1,2,3,5.
TheHDDsusedinlaptopsaredesignedtooperateintheenergy-limitedenvironmentofabattery-
powered system. Consequently, those disks tend to be smaller, slower, and have fewer platters.
Server HDDs, on the other hand, tend to follow the converse trend. Since the RPM can be varied
dynamically(DRPM)[4],diskdrivemanufacturersnowofferthiscapabilityinsomemodels[5].
2 Calculating Power
ForthepurposesofcomparingthepowerconsumptionoftwoHDDmodels(say,aandb)1 itismore
convenienttoexpress(1)intheratioform
P (cid:18)N (cid:19)(cid:18)Ω (cid:19)2.8(cid:18)D (cid:19)4.6
b = b b b . (2)
P N Ω D
a a a a
whichavoidsthenecessityofotherwisedeterminingaconstantofproportionality.
Inthissimplernotation,P isthepowerconsumed(Watts),Nisthenumberofplattersperspindle,
Ω is the angular speed (RPM), and D is the platter diameter (not the external form factor) usually
expressed in inches in U.S. vendor data sheets. Another convenience of using (2) is that the ratio of
twoHDDparametersthatarecommon(e.g.,thesamediameter)simplifiestounitvalue.
Example1(RPMVariation). TheparametersinthefollowingtableareforHDDswithasingleplatterN =
a
N =1andthesamediameterD = D =2.6inches[4].
b a b
Singleplatter2.6inchHDD
P P Ω Ω P∗
a b a b b
0.91 1.13 15,098 16,263 1.121
2 35.55 19,972 55,819 35.550
35.55 499.73 55,819 143,470 499.782
Hence,onlytheratiooftheangularspeedscontributesin(2)totheestimatedpowerP∗ inthelastcolumn.
b
Because the exponents in (1) are relatively large, the functions of RPM and diameter are highly
nonlinearandthatmeans:
• PotentiallylargeenergysavingscanbeachievedevenwithinalimitedselectionofHDDmodels,
especiallywhentakeninaggregateacrossSANs,NAS,JBoDs,orRAIDstorageconfigurations.
• The key disk parameters are available from vendor data sheets, although some caveats still
apply[6].
Example2(DiameterVariation). ThedatausedinthisexamplearetakenfromTables3.3–3.5in1.Theeffect
ofreducingHDDdiameter(d)isshowninFig.2. AthighestRPM,boththe d = 2.6inchandd = 2.1inch
HDDsarelocatedhigheronthepowerhillthanthed =1.6inchHDD.
3 Theoretical Justification
We now turn to providing a theoretical justification for the semi-empirical relationship in (1). As a
startingpoint,weassume:
1. Theempiricalexponentscanbeassociatedwithintegers(roundedup)
2. Planerrotationoftheplatterimposesaxialsymmetry
3. Rotationalfrictionispresentsincetheplatterisnotinvacuo
Athinrotatingplatterimpliesthatinertiallinearrelationships,likethekineticenergyalone,can-
notproduce(1). Rather,rotationalquantities,likemomentsofinertia,aremoreimportant.
Rotation also implies that there is aerodynamic friction due to the platter spinning. This is also
assumed to be greater than friction in the spindle bearings. Since the HDD platter resides in a sta-
tionaryhousing,weassumethereisnotranslationaldragproportionalthecross-sectionalareaofthe
platter;astherewouldbeforafan-bladeorpropellerpullingair.
1Suchasthekindofcalculationsrequiredfordatacenterandstoragecapacityplanning.
2
Figure2: SpecificHDDconfigurations(dots)plottedonthepowersurfacedefinedby(1). Thedesign
trendistomovetowardthebottomcorneroftheHDDpowersurface.
3.1 Pressure
The Bernoulliequationtellsusthatthepressureatsomepointinafluid(e.g.,airwithdensity ρ)is
givenby:
1
P = p+ρgh+ ρv2.
ρ
2
For the spinning disk, we are only interested in the dynamic contribution of the third term. The
externalandhydrostaticcontributionscanbeignored.
Sincepressureisforceperunitarea(F/A),theaerodynamicfrictionalforcewillbe
1
F = ρv2C A, (3)
ρ d
2
whereC ,thedragcoefficientisanadditionalfudgefactorthatcoversamultitudeofsinsastohowthe
d
fluiddragactuallyoccurs2.
Thedragforcein(3)istobeunderstoodasbeingtangential,ratherthancentripetal. Thevelocity
v at any point on the platter surface has magnitude relative to the air in the tangential direction of
thediskspin. Similarly,thearea Aisonthesurfaceoftheplatter(notitsedge)wheretheairliterally
drags. TheroughnessoftheplattersurfaceiscapturedinC .
d
3.2 Power
Initsmechanicalform,poweristherateofdoingwork(W)orexpendingenergy.
dW
P = . (4)
dt
ForanHDD,however,it’sprobablymoreconvenienttomeasuretheenergylossduetoaerodynamic
frictionintermsofthecurrentdrawn(I)atvoltage(V):
dq
P = IV = V. (5)
dt
Ifwecountthechargesqaselectrons,thendqVismeasuredinelectron-volts(eV),whichisameasure
ofenergy3. Hence,(4)and(5)aredimensionallyequivalent.
2Factorssuchas:shape,roughness,viscosity,compressibility,boundary-layerseparation,etc.
3KeVinananalogTV.
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Moreover, the work performed in (4) is by virtue of the friction force in (3) acting along an ele-
mental(tangential)arclengthds,i.e.,dW = F ds. Hence,thepower
ρ
ds
P = F = F v, (6)
ρ ρ
dt
canbeexpresseddirectlyintermsofthedragforceandthetangentialvelocity.
dA dr
dΘ
r
Out[97]=
Figure3: DifferentialareaelementdAdefinedonadiskplatter
However, each elemental area (dA) on the platter moves with a different tangential speed and
thereforeexperiencesadifferentforce. Thus,wehavetointegrateallpossiblepatchesoverthewhole
platter. Applying(3),thepowerintegralcanbewrittenas:
(cid:90)
P ∝ ρv3dA, (7)
ignoringanyproportionalityconstant.
FromFig.3thesectorialareaontheplattercanbewrittenastheproductofthearcdelta(ds)and
radialdelta(dr):
dA = ds.dr = (rdθ)dr. (8)
Similarly,thetangentialvelocityvectorvisrelatedtothe(axial)angularvelocityΩbyv = Ω×rbut,
sinceallthesevectorsareorthogonalonthediskwecanusethescalarform
v = |Ω|r. (9)
Substituting(8)and(9)into(7)producesthetwo-dimensionalintegral
(cid:90) R (cid:90) 2π (cid:90) R (cid:90) 2π
P ∝ ρ dr rdθ r3Ω3 ≡ ρΩ3 drr4 dθ,
0 0 0 0
whichyields
2π
P ∝ ρΩ3R5. (10)
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WithD =2R,(10)isidenticalto(1)uptoconstantsofproportionality.
4 Summary
Thetheoreticalcontributionsto(1)canbesummarizeinthefollowingsimplesteps:
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1. FactorthepowerP asΩ3R3×R2
2. R2comesfromthetotalareaoftheplatter
3. ΩRistheangularspeedvofairatanypointonthespinningplatter
4. Whythecube: v3 ≡ Ω3R3?
5. Afactorv2comesfromtheBernoullipressure(F/A)in(3)
6. AnothervfactorcomesfromthedefinitionofpowerP = v×F/Ain(6)
Afactorof2shouldalsoincludedsincethedragwilloccuronboththeupperandlowersideof
theplatter,butwearenotkeepingtrackofconstantshere.Therecouldalsobeinter-platterturbulence
buttheReynold’snumberislikelyalreadysubstantialduetoheadmovement.
It has also been shown recently that electrical power consumption is proportional to the third
powerofthelogicalblocknumber[7]. ThisislikelyrelatedtoΩ3via(9).
TheprecedingdiscussionconcerningHDDaerodynamicpowerconsumptionwillbecomemoot
asSSDsbecomeincreasinglycost-effectiveandreliable[8,9,10].
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