1

1 Nomenclature

Roman V velocity m.s−1

a acceleration m.s−2 w specific work energy J.kg−1

A area m2 W work energy Ja speed of sound m.s−1 x dryness fraction -D diameter m ∆z change of elevation mDh hydraulic diameter mf Fanning friction factor -F force N GreekF view factor - α absorptivity -g acceleration due to gravity m.s−2 β volume expansion coefficient -C heat capacity J.K−1 δ boundary layer thickness mcp const pressure specific heat capacity J.kg−1.K−1 ε wall roughness mcv const volume specific heat capacity J.kg−1.K−1 ε effectiveness -COP coefficient of performance - ε emisivity -e specific energy J.kg−1 η efficiency -E energy J γ ratio of specific heats (

cpcv

) -

h convective heat transfer coefficient W.m−2.K−1 µ dynamic viscosity Pa.sh specific enthalpy J.kg−1 ν kinematic viscosity m2.s−1

H enthalpy J ρ density kg.m−3

k loss coefficient - ρ reflectivity -k thermal conductivity W.m−1.K−1 σ Stefan-Boltzmann constant W.m−2.K−4

K bulk modulus Pa or N.m−2 τ shear streass N.m−2

L length or length scale m τ transmitivity -m mass kg T torque N.m−1

M molecular mass kg.mol−1 ω angular velocity s−1

n number of moles -n polytropic index - OtherN Avagadro’s number mol−1 X1,2,3..etc location or instantaneous value of X -p pressure Pa or N.m−2 x, y, z, r, θ spacial coordinates, radius and angle -P perimeter m ∆X finite change of X -q specific heat energy J.kg−1 δX infinitesimal change of X -

Q heat energy J X rate of X -

R radius m ⇀X a vector X -R thermal resistance K.W−1 X∗ critical value of X -R specific gas constant J.kg−1.K−1 X◦ stagnation value of X -

R universal gas constant (8.3145×103) J.mol−1.K−1 X◦ value of X at STP -s specific entropy J.kg−1.K−1 X average of X -S entropy J.K−1 X ′ modified value of X -t time s Xi inlet value of X -T temperature K Xe exit value of X -u specific internal energy J.kg−1 XH hot value of X -U internal energy J XC cold value of X -U overall heat transfer coefficient W.m−2.K−1 Xf value of X at saturated liquid -v specific volume m3.kg−1 Xg value of X at saturated vapour -V volume m3 Xfg change in X between Xf and Xg -

2 Material Properties

2.1 Viscosity variation with temperature

• Exponential model for liquids:

µ = µ0 × 10B

(T−C) (1)

where µ0, B and C are constants.

• For water µ0=2.414×105 Pa.s, B=247.8 K and C=140 K.

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• Poiseuille formula for dynamic viscosity:

µ = µ0

(1

1 +AT +BT 2

)(2)

where µ0, A and B are constants and T is the temperature in ◦C.

• For water, the value of µ0 is 0.00179 Pa.s, and the values of constants A and B are 0.033368 ◦C−1 and 0.000221 ◦C−2,respectively.

2.2 Material properties for air and water

Temp. Water Temp. Air at 1 atmµ(×10−3) ρ k cp µ(×10−6) ρ k cp

◦C Pa.s kg.m−3 W.m−1.K−1 kJ.kg−1.K−1 ◦C Pa.s kg.m−3 W.m−1.K−1 kJ.kg−1.K−1

10 1.31 1000 0.59 4.195 -150 8.60 2.79 0.012 1.02620 1 998 0.6 4.182 -100 11.8 1.98 0.016 1.00925 0.91 997 0.61 4.178 -50 14.6 1.53 0.020 1.00530 0.8 996 0.62 4.167 0 17.2 1.29 0.024 1.00540 0.65 992 0.63 4.175 20 18.2 1.21 0.026 1.00550 0.55 988 0.64 4.178 40 19.1 1.13 0.027 1.00560 0.47 983 0.65 4.181 60 20.2 1.07 0.029 1.00970 0.4 978 0.66 4.187 80 20.9 1.00 0.030 1.00980 0.36 971 0.67 4.194 100 21.8 0.95 0.031 1.00990 0.32 965 0.68 4.202 200 25.8 0.62 0.039 1.026100 0.28 958 0.68 4.211 400 32.7 0.52 0.052 1.068

3 Newton’s laws of motion

• Newton’s laws of motion

First Every object remains in a state of rest or in uniform motion in a straight line unless acted upon by a (nett) force.

Second F = m× aThird For every action there is an equal and opposite reaction.

• Equations of linear motion

V2 = V1 + a∆t (3)

x2 = x1 + V1∆t+1

2a (∆t)

2(4)

x2 = x1 +1

2(V2 + V1) ∆t (5)

V 22 = V 2

1 + 2a (x2 − x1) (6)

4 Fluid Mechanics

4.1 Fluid Statics

• Pascal’s lawdp

dz= ρg (7)

• Force on a submerged plane

yp =IGA.yG

+ yG (8)

where yp is the distance to the centre of pressure and yG is distance to the centre of gravity, measured along the surfaceof the plane. IG is the second moment of area about the centroid, A is the area of the submerged plane.

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4.2 Flow in pipes

• Continuity

m = ρV A (9)

dm

dt= min − mout (10)

• Hydraulic mean diameter

Dh =4×AP

(11)

• The Hagen-Poiseuille equation

V = − 1

4µ

dp

dx

(R2 − r2

)(12)

V = −π∆pR4

8µL(13)

• Steady flow Energy Equation (SFEE)

pin +ρV 2

in

2+ ρgzin + ρwp = pout +

ρV 2out

2+ ρgzout + ρwf + ρwt (14)

where ρwf is the volumetric work lost due to friction, ρwp is the volumetric work supplied by a pump and ρwt is thevolumetric work generated by a turbine.

• Darcy’s Equation for losses in long pipes

wf = 4fL

D

V 2

2(15)

• Fanning friction factors

f =16

Relaminar flow (16)

f = (0.79ln (ReD)− 1.64)−2

(17)

1√f

= −1.8log10

(6.9

Re+

(1

3.71

ε

D

)1.11)

(18)

• Typical pipe roughnesses given below:

Material Roughness (mm)Coarse concrete 0.25Smooth concrete 0.025Drawn tubing 0.0025Glass, Plastic, Perspex 0.0025Cast Iron 0.15Old Sewers 3.0Mortar lined steel 0.1Rusted steel 0.5Forged steel 0.025Old water mains 1.0

• Loss coefficient for piping network components

wf = kV 2

2(19)

where k is the loss coefficient,

• values of k are given in table below:

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Component kSharp Entry 0.5Rounded Entry 0.25Contraction (50% area) 0.24Contraction (50% diameter,based on V2) 0.35

Expansion (based on V2)(A2

A1− 1)

180o elbow 0.990o elbow 0.945o elbow 0.4Globe valve (open) 10Angle valve (open) 2Gate valve (open) 0.15Gate valve (25% closed) 0.25Gate valve (50% closed) 2.1Gate valve (75% closed) 17Angle valve (open) 2Swing check valve (open) 2Ball valve (open) 17Ball valve (33% closed) 5.5Ball valve (66% closed) 200Diaphragm valve (open) 2.3Diaphragm valve (50% closed) 4.3Diaphragm valve (75% closed) 21Water meter 7

• Moody Diagram:

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4.3 Conservation of linear momentum

• Force on fluid in control volumeF = moutVout − minVin (20)

4.4 Lift and drag

• Lift force

FL = CL1

2ρV 2A (21)

where CL is the coefficient of lift

• Drag force

FD = CD1

2ρV 2A (22)

where CD is the coefficient of drag

• Coefficients of skin friction drag for laminar flow over flat plate

CD =1.328√

Re

(Re < 105

)(23)

• Coefficients of skin friction drag for turbulent flow over flat plate

CD =0.074

Re0.2

(105 < Re < 107

)(24)

CD =0.455

(log(Re))2.58

(107 < Re < 109

)(25)

• Coefficients of form drag around a cylinder

CD =24

Re(Re < 1) (26)

4.5 Compressible flow

• Isothermal compressible flow in a constant cross section pipe, neglecting change in gravitational potential energy

p22 = p2

1 + 2RT

(m

A

)2

ln

(p2

p1

)− 4fL

DRT

(m

A

)2

(27)

• Speed of sound

a =

√K

ρ=√γRT (28)

• Change in velocity with area of nozzel∂V

V= −∂A

A

1

1−M2(29)

• SFEE for isentropic compressible flow1

2V 2

1 + cpT1 =1

2V 2

2 + cpT2 (30)

• Adiabatic, isentropic, compressible flow

T2

T1=

(p2

p1

)( γ−1γ ) T2

T1=

(v1

v2

)γ−1

(31)

• Stagnation conditions

T

T0=

1

1 +M2 γ−12

T = T0 +V 2

2cp(32)

p

p0=

1[1 +M2 γ−1

2

] γγ−1

(33)

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• Critical conditions

T∗T0

=2

γ + 1(34)

p∗p0

=

(2

γ + 1

) γγ−1

(35)

where T∗ and p∗ are the critical temperature and pressure, respectively.

4.6 Water Hammer

• Pressure drop due to water hammer.∆p = ρV a (36)

• Augmented bulk modulus (K′) for non-rigid pipes.

1

K′=

1

K+D

tEgiving a =

√K′ρ

(37)

where D is the internal diameter of the pipe, t is the thickness of the pipe wall and E is the Young’s modulus.

σθ =pD

2t(38)

where σθ is the hoop stress, t is the thickness of the pipe wall and E is the Young’s modulus.

5 Heat Transfer

5.1 Thermal expansion

• Linear expansion

∆L = L1β

3∆T (39)

• Area expansion

∆A = A12β

3∆T (40)

• Volumetric expansion∆V = V1β∆T (41)

where β3 is the coefficient of linear expansion, sometimes referred to as α in other texts.

5.2 1D heat transfer

• Conduction

Q = kA

L∆T (42)

• Conduction in thick walled cylinder

Q = 2πkLT1 − T2

ln(R2

R1

) (43)

• ConvectionQ = hA∆T (44)

where h can be found using the Nusselt number, given in equation 99.

• Resistor analogy for composite surfaces

Q =∆T

R1 +R2 +R3 + . . .+Rn(45)

7

Rcond,planar =L

kA(46)

Rcond,clyind =ln(R2

R1

)2πkL

(47)

Rconvect =1

hA(48)

(49)

• RadiationQ = σεA

(T 4

1 − T 42

)(50)

where σ is the Stefan-Boltzmann constant, of 5.67051 ×10−8 W.m−2.K−4

5.3 Radiation heat transfer view factors

• Radiation equation with view factorsQij = AiFijσεi

(T 4i − T 4

j

)(51)

• Reciprocity RelationAiFij = AjFji (52)

5.4 Forced convection

• For an isothermal flat plate

Nu = 0.032Re12

LPr13 (53)

valid for ReL < 105 and 0.6 < Pr < 60.

Nu = 0.0296Re45

LPr13 (54)

valid for 108 > ReL > 105 and 0.6 < Pr < 60.

Nu = 0.037Re45

LPr13 (55)

valid for ReL > 108 and 0.6 < Pr < 60.

• For an isothermal horizontal cylinder

Nu = CRemDPr13 (56)

C = 0.193,m = 0.618 for 4000 < Re < 40000 and C = 0.027,m = 0.805 for 40000 < Re < 400000.

• Dittus-Boelter equation for forced convection in pipes.

Nu = 0.023Re45

DPrn (57)

where n = 0.4 for heating fluid and n = 0.3 for cooling fluid. Valid for Re ≥ 10000 and 0.7 ≤ Pr ≥ 160

• Log mean temperature difference

∆TLM =∆T1 −∆T2

ln∆T1

∆T2

(58)

• Exit temperature (Te) for constant wall temperature pipe

Te = Tw + (Ti − Tw) exp

(−hAmcp

)(59)

where Ti is the inlet temperature and tempw is the wall temperature.

8

5.5 Heat exchanger design

• Capacity rates (m× cp)

Cmin = min(CH , CC) Cmax = max(CH , CC) C∗ =CminCmax

(60)

• Correction factor Fq = FUA∆TLM (61)

where F is a function of P and R:

P =Ttube,o − Ttube,iTshell,i − Ttube,i

R =CtubeCshell

=Tshell,i − Tshell,oTtube,o − Ttube,i

(62)

• Effectiveness-NTUQmax = Cmin (TH,i − TC,i) (63)

ε =Qact

Qmax(64)

5.6 Natural convection

• Free convection at a vertical wall (Churchill and Chu)

NuL = 0.68 +0.67Ra

1/4L[

1 + ( 0.492Pr )9/16

]4/9 (65)

for RaL ≤ 109.

• For the horizontal surface with top surface of a hot object in a colder environment or bottom surface of a cold object ina hotter environment.

NuL = 0.54Ra14

L 104 < RaL < 107 (66)

NuL = 0.15Ra13

L 107 < RaL < 1011 (67)

(68)

• For the horizontal surface with the bottom surface of a hot object in a colder environment or top surface of a cold objectin a hotter environment.

NuL = 0.27Ra14

L 105 < RaL < 1010 (69)

5.7 Combustion

• Molar masses of atoms and heats of combustion of fuels

Chemical M (g.mol−1) Fuel LCV (MJ.kg−1) HCV (MJ.kg−1)H 1 Petrol 44.4 47.3He 4 Diesel - 44.8C 12 Ethanol (C2H5OH) - 29.7N 14 Methane (CH4) 50 55.5O 16 Ethane (C2H6) 47.8 51.9

• Volume occupied by one mole, from the ideal gas equation

V

n=RT

p(70)

At 1 bar and 25 degrees C, the recommended value is 24.5 litres per mole.

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6 Thermodynamics

6.1 Laws of Thermodynamics

Zeroth If two thermodynamic systems are each in thermal equilibrium with a third, then they are in thermal equilibriumwith each other.

First Energy can neither be created nor destroyed. It can only change from one form to another.

Second The entropy of an isolated system not in equilibrium will tend to increase over time, approaching a maximum valueat equilibrium.

Third As temperature approaches absolute zero, the entropy of a system approaches a constant minimum.

6.2 Method for Solving Thermodynamics Problems

Draw a simple pictureto represent the system

Decide a naming systemand label the picture

Draw a control volume

Decide if control volume is open or closed,steady or transient

Write down any relaventequations

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6.3 Conventions for Conservation of Energy

System Surroundings

1

23 n

• In finite formE1 + E2 + E3 + ...+ En = ∆E (71)

• In rate form

E1 + E2 + E3 + ...+ En =dE

dt(72)

6.4 Thermodynamic relations

• Ratio of specific heats:

γ =cpcv

(73)

• Enthaply

H = U + pV (74)

h = u+ pv (75)

dh = cpdT (76)

• Internal energy

du = cvdT (77)

• Gas constant

R = cp − cv R =R

M(78)

where R = 8.314 kJ.kmol−1K−1

• Polytropic processespV n = constant (79)

where n is the polytropic constant.

Process nIsochoric (const volume) n = infIsothermal (const temperature) n = 1Isobaric (const pressure) n = 0Isentropic (const entropy) n = γ

• Entropy

q = sdT (80)

ds =dq

T(81)

s2 − s2 = cvlnT2

T1+Rln

v2

v1= cpln

T2

T1+Rln

p2

p1(82)

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• Isentropic expansion and compression

T2

T1=

(p2

p1

)( γ−1γ )

=

(v1

v2

)(γ−1)p2

p1=

(T2

T1

)( γγ−1 )

=

(v1

v2

)(γ)v2

v1=

(T1

T2

)( 11−γ )

=

(p1

p2

)( 1γ )

(83)

The value of the ratio of specific heats, γ, can usually be considered to be 1.4 for air.

6.5 Gas Laws

• Boyle’s law for a constant temperature (isothermal) process

pV = const p1V1 = p2V2 (84)

• Charles’ law for a constant pressure (isobaric) process

V

T= const

V1

T1=V2

T2(85)

• Gay-Lussac’s law for constant volume (isochoric) process

p

T= const

p1

T1=p2

T2(86)

• Ideal gas equation:

pV = nRT (87)

pV = mRT (88)

pv = RT (89)

p = ρRT (90)

• Thermodynamic properties of common gasses at STP.

Gas cp (kJ.kg−1.K−1) cv (kJ.kg−1.K−1) γ R (kJ.kg−1.K−1)Air 1.005 0.718 1.40 0.287Carbon dioxide 0.884 0.655 1.289 0.189Hydrogen 14.32 10.16 1.41 4.12Methane 2.22 1.70 1.30 0.518Natural Gas 2.34 1.85 1.27 0.5Nitrogen 1.04 0.743 1.40 0.297Oxygen 0.919 0.659 1.40 0.260

6.6 Thermodynamic devices

• Work done

W12 =

∫ 2

1

pdV (91)

• Heat engine thermal efficiency:

ηth =w

˙qH(92)

where qH is the heat energy from the hot source.

• Carnot thermal efficiency:

ηth,car = 1− TLTH

(93)

• COP of a heat pump and refrigerator:

COPHP =qH

Win

COPR =qL

Win

(94)

where qL is the heat energy from the cold source.

12

• Carnot COP:

COPHP =1

1− TLTH

COPR =1

THTL− 1

(95)

• Isentropic efficiencies, compressors and turbines

ηC =hout,s − hinhout − hin

ηT =hin − houthin − hout,s

(96)

• Spark ignition and compression ignition engine thermal efficiency.

SPARK : ηth = 1−[

1

rγ−1v

]COMPRESSION : ηth = 1−

[1

rγ−1v

]rγc − 1

γ (rc − 1)(97)

where rc is the cut off ration and rv is the compression ratio

7 Dimensionless Numbers

• Reynolds number

Re =ρUD

µ(98)

• Nusselt number

Nu =hL

k(99)

• Mach number

M =V

a(100)

• Prandtl numberPr =

cpµ

k(101)

• Grashof number

Gr =gβρ2 (TS − Tf )L3

µ2(102)

where β is 1T for an ideal gas.

• Rayleigh numberRa = Gr× Pr (103)

• Biot number

Bi =hL

k(104)

8 Approximate Values for Quantities

8.1 Mass

10 g a pen100 g mobile phone500 g bottle of drink1 kg house brick1.5 kg bag of sugar70 kg person1000 kg mass of small car1500 kg mass of family saloon

8.2 Length

8 cm finger20 cm width A4 sheet30 cm length A4 sheet2 m door height30 m Owen building

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8.3 Volume

5 ml tea spoon25 ml shot of whisky330 ml can of coke568 ml a pint of beer750 ml bottle of wine2 l bottle of coke2,500,000 l Olympic swimming pool

8.4 Velocity

1.5 m/s walking3 m/s running10 m/s sprinter13 m/s residential speed limit30 m/s motorway speed limit45 m/s British train150 m/s good train330 m/s speed of sound in air

8.5 Density

1.2 kg/m3 air700 kg/m3 petrol800 kg/m3 oil1,000 kg/m3 water2,700 kg/m3 aluminium7,800 kg/m3 steel13,500 kg/m3 mercury

8.6 Viscosity @ 20◦C

1×10−5 Pa.s CO2/Ar/He/N2

2×10−5 Pa.s air0.000 6 Pa.s Petrol0.001 Pa.s water0.2 Pa.s oil2 Pa.s honey

8.7 Power

10 W energy saving lightbulb100 W electric light bulb (incandescent)2.5 kW domestic kettle100 kW Car engine (135 bhp)5 MW Big wind turbine4,000 MW Drax power station output40,000 MW UK power consumption

8.8 Conductivity of solids

0.004 W/mK Cork0.033 W/mK Styrofoam0.04 W/mK Fibreglass0.04 W/mK General insulation0.05 W/mK Paper0.055 W/mK Wood (balsa)0.12 W/mK Wood (pine)0.14 W/mK Leather0.15 W/mK Wood (oak)0.17 W/mK Plasterboard0.2 W/mK PVC0.5 W/mK HD Polyethene0.75 W/mK Asphalt

8.9 Further conductivity of solids

0.004 W/mK Cork0.033 W/mK Styrofoam0.04 W/mK Fibreglass0.04 W/mK General insulation0.05 W/mK Paper0.055 W/mK Wood (balsa)0.12 W/mK Wood (pine)0.14 W/mK Leather0.15 W/mK Wood (oak)0.17 W/mK Plasterboard0.2 W/mK PVC0.5 W/mK HD Polyethene0.75 W/mK Asphalt1 W/mK Bricks1.05 W/mK Glass16 W/mK Stainless Steel35 W/mK Lead55 W/mK Carbon Steel109 W/mK Brass250 W/mK Aluminium400 W/mK Copper

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9 Space for your notes

15

10 Credits

Compiled and edited by Dr Andrew Garrard (a.garrard@shu.ac.uk)Cover illustration by Jack Good (www.jackgood.co.uk)

All material within this book is held under copyright c©2011 Andrew Garrard, Sheffield Hallam Univesity. Permission isgranted to copy and freely distribute provided credit is given to the original author.

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