1.1:PROBLEMDEFINITION
Find:Listthreecommonunitsforeachvariable:
a.Volume flowrate (Q),mass flowrate (m),andpressure (p).
b.Force,energy,power.
c.Viscosity,surfacetension.
PLAN
UseTableF.1to findcommonunits
SOLUTION
a.Volume flowrate,mass flowrate,andpressure.
• Volume flowrate, m3 / s, ft3 / s orcfs,cfmor ft3 / m.
• Mass flowrate.kg/s,lbm/s,slug/s.
• Pressure.Pa,bar,psior lbf / in2 .
b.Force,energy,power.
• Force,lbf,N,dyne.
• Energy,J,ft·lbf,Btu.
• Power.W,Btu/s,ft lbf/s.
c.Viscosity.
• Viscosity,Pa·s,kg/(m·s),poise.
1.2:PROBLEMDEFINITION
Situation:Thehydrostaticequationhasthreecommonforms:
Find:Foreachvariableintheseequations,listthename,symbol,andprimarydimensionsofeachvariable.
PLAN
LookupvariablesinTableA.6.Organizeresultsusingatable.
SOLUTION
NameSymbolPrimarydimensions pressure pM/LT 2 specificweight γM/L2 T 2 elevation zL
piezometricpressure pz M/LT 2
changeinpressure ∆pM/LT 2
changeinelevation ∆zL
1.3:PROBLEMDEFINITION
Situation: Fiveunitsarespecified.
Find:
Primarydimensionsforeachgivenunit:kWh,poise,slug,cfm,CSt.
PLAN
1.FindeachprimarydimensionbyusingTableF.1.
2.Organizeresultsusingatable.
SOLUTION
UnitAssociatedDimensionAssociatedPrimaryDimensions
1.4:PROBLEMDEFINITION
Situation: Thehydrostaticequationis p γ + z = C
p ispressure, γ isspecificweight, z iselevationand C isaconstant.
Find: Provethatthehydrostaticequationisdimensionallyhomogeneous.
PLAN
Showthateachtermhasthesameprimarydimensions.Thus,showthattheprimary dimensionsof p/γ equaltheprimarydimensionsof z .Findprimarydimensionsusing TableF.1.
SOLUTION
1.Primarydimensionsof p/γ :
2.Primarydimensionsof z : [z ]= L
3.Dimensionalhomogeneity.Sincetheprimarydimensionsofeachtermislength, theequationisdimensionallyhomogeneous.Notethattheconstant C intheequation willalsohavethesameprimarydimension.
1.5:PROBLEMDEFINITION
Situation: Fourtermsaregivenintheproblemstatement.
Find:Primarydimensionsofeachterm.
a) ρV 2 /σ (kineticpressure).
b) T (torque).
c) P (power).
d) ρV 2 L/σ (Webernumber).
SOLUTION
a.Kineticpressure:
b.Torque.
c.Power(fromTableF.1).
d.WeberNumber:
Thus,thisisadimensionlessgroup
1.6:PROBLEMDEFINITION
Situation:
Thepowerprovidedbyacentrifugalpumpisgivenby: P = mgh
Find:
Provethattheaboveequationisdimensionallyhomogenous.
PLAN
1.Lookupprimarydimensionsof P and m usingTableF.1.
2.Showthattheprimarydimensionsof P arethesameastheprimarydimensions of ˙ mgh
SOLUTION
1.Primarydimensions:
[P ]= M L2 T 3
[ ˙ m]= M T
[g ]= L T 2
[h]= L
2.Primarydimensionsof ˙ mgh:
[mgh]=[m][g ][h]= µ M T ¶µ L T 2 ¶ (L)= M · L2 T 3
Since [mgh]=[P ] , Thepowerequationisdimensionallyhomogenous.
1.7:PROBLEMDEFINITION
Situation: Twotermsarespecified.
a. Z ρV 2 dA
b. d dt ZV ρVdV .
Find: Primarydimensionsforeachterm.
PLAN
1.To findprimarydimensionsforterma,usetheideathatanintegralisdefined usingasum.
2.To findprimarydimensionsfortermb,usetheideathataderivativeisdefined usingaratio.
SOLUTION
Terma:
Termb:
Problem1.8
Nosolutionprovided.
1.9:PROBLEMDEFINITION
Applythegridmethod.
Situation: Densityofidealgasisgivenby:
= p RT
p =35 psi, R =1716ft- lbf / slug◦ R.
T =100 ◦ F=560 ◦ R.
Find: Calculatedensity(inlbm/ft3 )
PLAN
Followtheprocessgiveninthetext.LookupconversionratiosinTableF.1.
SOLUTION
(note:unitcancellationsnotshown).
1.10:PROBLEMDEFINITION
Applythegridmethod.
Situation: Windishittingawindowofbuilding.
∆p = ρV 2 2
ρ =1 2kg / m3 ,V =60 mph.
Find:
a.Expresstheanswerinpascals.
b.Expresstheanswerinpoundsforcepersquareinch(psi).
c.Expresstheanswerininchesofwatercolumn(inchH2 0).
PLAN
Followtheprocessforthegridmethodgiveninthetext.Lookupconversionratios inTableF.1.
SOLUTION
a) Pascals.
b) Poundspersquareinch.
c) Inchesofwatercolumn
p =432Pa
p =432Pa
p =0.0626 psi
=432Pa
1.11:PROBLEMDEFINITION
Applythegridmethod.
Situation:
Forceisgivenby F = ma.
a) m =10kg , a =10m/ s2
b) m =10lb, a =10ft/ s2
c) m =10slug , a =10ft/ s2 .
Find: Calculateforce.
PLAN
Followtheprocessforthegridmethodgiveninthetext.Lookupconversionratios inTableF.1.
SOLUTION
a)
Forceinnewtonsfor m =10kg and a =10m/ s2
F = ma =(10kg) ³10 m s2 ´ µ N s2 kg · m ¶
F =100N
b)
Forceinlbffor m =10 lbmand a =10ft/ s2 . F = ma =(10 lbm) µ10 ft s2 ¶µ lbf s2 32.2 lbm · ft ¶ F =3 11lbf
c)
Forceinnewtonsfor m =10 slugandaccelerationis a =10ft/ s2 F = ma =(10slug) µ10 ft s2
lbf s2 slug · ft
F =445N
4 448N lbf ¶
1.12:PROBLEMDEFINITION
Applythegridmethod.
Situation: Acyclististravellingalongaroad.
P = FV.
V =24mi/ h, F =5lbf
Find:
a)Findpowerinwatts.
b)Findtheenergyinfoodcaloriestoridefor1hour.
PLAN
Followtheprocessforthegridmethodgiveninthetext.Lookupconversionratios inTableF.1.
SOLUTION
a) Power P = FV =(5lbf) µ 4.448N lbf ¶ (24 mph) µ 1.0m/ s 2 237 mph
=239W
W · s N m ¶
b) Energy ∆E = P ∆t = µ 239J s ¶ (1h) µ 3600s h ¶µ 1 0 calorie(nutritional) 4187J ¶ ∆E =205 calories
1.13:PROBLEMDEFINITION
Applythegridmethod.
Situation: Apumpoperatesforoneyear. P =20hp
Thepumpoperatesfor 20 hours/day
Electricitycosts $0.10/kWh.
Find: Thecost(U.S.dollars)ofoperatingthepumpforoneyear.
PLAN
1.Findenergyconsumedusing E = Pt,where P ispowerand t istime.
2.Findcostusing C = E × ($0.1/kWh).
SOLUTION
1.EnergyConsumed E = Pt =(20hp) µ W 1.341 × 10 3 hp
20h d
365d y ¶ =1. 09 × 108 W · h µ kWh 1000W h ¶ E =1 09 × 105 kWh
2.Cost C = E ($0.1/kWh) = ¡1. 09 × 108 kWh¢ µ $0 10 kWh ¶ C =$10, 900