Spring200310.450 ProcessDynamics,Operations,andControl

ProblemSets-8

1.Youhavean1800kgh-1streamofhexaneat100°C. Calculatethepressuredropforafully- openvalveofCv=17.

estimatingpressuredropthroughfullyopenvalve,givenmassflowandCv

massflow density

Cv

refdensity

1800kgh-1

580kgm-3

17gal.in.min-1.lbf-0.5

999kgm-3

0.5kgs-1

1.29151E-05 m4s-1N-0.5

calculatevolumetricflow

0.000862 m3s-1

calculatepressuredrop

2586.731 Pa

0.375076 psi

Ofcourse,yourealizethatthenumbersshowninthiscalculationsheetwouldberoundedtoa morerealisticprecisionforreportingresults:2590Pa.

2.Environmentaldisaster! TherewasaspillofBaD4StUF2fromatank;youarerequiredto estimatethemagnitudeoftherelease. From checkingtherecords,youfindthatthevalve wasopenat40%for17s,thatitisanequal-percentagevalvewhosecharacteristicsare representedbyarangeabilityof40,andCvis23. Thetankwasundernitrogenblanketat1.2

atmabsolute,thetemperaturewas45°C,liquidlevelwas80%,ambienttemperaturewas8°C. Liquidpropertiesaredensity926kgm-3, heat capacity 3270 kJ kg-1 K-1,surfacetension15 dynes cm-1.

nozzlesfor levelindicator

2.3

dimensions inm

4.8

1.11.6

Linehadbeenopened hereformaintenance, but not blanked off. Procedureviolation!

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Spring200310.450 ProcessDynamics,Operations,andControl

ProblemSets-8

drainingfromatankoveraninterval.Estimatethetotalspill.

first,assumethatthetransientissufficientlyshortthatthelevel

doesn'tchangemuch.CalculatethedrainratefromtheSSmechanical energybalance,andthenchecktotalamountdrainedwithchangeinlevel bymaterialbalance.

Furtherdifficultyinthatpipesizeisn'tgiven,sotheaccelerationcan'tbecalculated. Handlebyassumingvalvelosscoefismuchgreaterthanone.Theflowcalculatedwill

be conservatively high by some unknown amount.Then assume some reasonable pipe sizes, consistentwiththegivenvalueofCv,andcheckthevelocityforreasonablevalues.After pickingpipesize,cancheckthe'large-K'assumption.

conditionsatliquidsurface

elevation pressure

4.06m

1.2atm121590Pa

conditionsatdischargetoatmosphere

elevation pressure

1.6m

1atm101325Pa

fluidproperties

density

926kgm-3

valvedata

Cv23gal.in.min-1.lbf-0.51.74734E-05 m4s-1N-0.5

rangebility40

opening0.4

refdensity999kgm-3

otherdata

g9.807ms-2

duration17s

calculatetheratio(losscoef)/(pipeareasquared)

2548508643 m-4

calculatetheflowbasedentirelyonfrictionloss


6.492797821gpm

calculatevelocityfordifferentpipeareas

diameter(in)area(m2)velocity(ms-1) k

10.000506710.808330124140.8320751

20.0020268380.2020825312253.313202

30.0045603860.08981445811407.39809

1"pipeseemsmostreasonableforvelocityandgivenCv

andtheassumptionthatKdominatesissatisfiedinanycase.

computespillatthisflowrate

6.4477431 kg

0.006963 m3

14.21727349 lbm

1.839626049 gal

thiswillbeanegligiblechangeinliquidlevel

Theflowshouldvaryhardlyatallduringthespill,justifyingthesteady-stateapproximation.

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Spring200310.450 ProcessDynamics,Operations,andControl

ProblemSets-8

3.Youarecheckingoutaflowcontrolloopforinstability.Youfindthatthemaximumslope onaplotofflowfractionversusvalveopening(thoughtfullyleftforyoubyyour predecessor)is1.8. Ifthefullinstalledcapacityofthisvalveis35gpm,whatisthe maximumgainyouwillencounterinthevalvetransferfunction?

Thinkofthegainas sensitivityoftheoutputtotheinput. Forthevalve, wechangeflowbychangingpressureintheactuator,sothat

gaindF

dP

Werecognizethatactuatorpressureisrelatedtotheopening?.

gaindF d

d dP

NowtheinstalledcharacteristicisaplotofF/Ffullvs?. Themaximumslope(givenas1.8)isthus

Wealsoexpectthevalveopeningtobedirectlyproportionaltothe actuatorpressure

d 

10

dP = P= (1512)psi

Makingthesubstitutions,wefind

gaindFd1.8F

ddP

1

full (12)psi

(1.8)(35)gpm 5.25 gpm

= (12)psi= psi

Or,youcansimplyplugintoEquation(20.5.3)inthenotes…

Thisgainappliesattheopeningatwhichthecharacteristicshavethe maximumslope.

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