Thermofluids Formula Sheet

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11 NomenclatureRoman V velocity m.s1a acceleration m.s2w specic work energy J.kg1A area m2W work energy Ja speed of sound m.s1x dryness fraction -D diameter m z change of elevation mDh hydraulic diameter mf Fanning friction factor -F force N GreekF view factor - absorptivity -g acceleration due to gravity m.s2 volume expansion coecient -C heat capacity J.K1 boundary layer thickness mcp const pressure specic heat capacity J.kg1.K1 wall roughness mcv const volume specic heat capacity J.kg1.K1 eectiveness -COP coecient of performance - emisivity -e specic energy J.kg1 eciency -E energy J ratio of specic heats (cpcv) -h convective heat transfer coecient W.m2.K1 dynamic viscosity Pa.sh specic enthalpy J.kg1 kinematic viscosity m2.s1H enthalpy J density kg.m3k loss coecient - reectivity -k thermal conductivity W.m1.K1 Stefan-Boltzmann constant W.m2.K4K bulk modulus Pa or N.m2 shear streass N.m2L length or length scale m transmitivity -m mass kg T torque N.m1M molecular mass kg.mol1 angular velocity s1n number of moles -n polytropic index - OtherN Avagadros number mol1X1,2,3..etc location or instantaneous value of X -p pressure Pa or N.m2x, y, z, r, spacial coordinates, radius and angle -P perimeter m X nite change of X -q specic heat energy J.kg1X innitesimal change of X -Q heat energy J X rate of X -R radius m X a vector X -R thermal resistance K.W1X critical value of X -R specic gas constant J.kg1.K1X stagnation value of X -R universal gas constant (8.3145103) J.mol1.K1X value of X at STP -s specic entropy J.kg1.K1X average of X -S entropy J.K1X

modied value of X -t time s Xi inlet value of X -T temperature K Xe exit value of X -u specic internal energy J.kg1XH hot value of X -U internal energy J XC cold value of X -U overall heat transfer coecient W.m2.K1Xf value of X at saturated liquid -v specic volume m3.kg1Xg value of X at saturated vapour -V volume m3Xfg change in X between Xf and Xg -2 Material Properties2.1 Viscosity variation with temperature Exponential model for liquids: = 010B(TC)(1)where 0, B and C are constants. For water 0=2.414105Pa.s, B=247.8 K and C=140 K.2 Poiseuille formula for dynamic viscosity: = 0_ 11 +AT +BT2_ (2)where 0, A and B are constants and T is the temperature in C. For water, the value of 0 is 0.00179 Pa.s, and the values of constants A and B are 0.033368 C1and 0.000221 C2,respectively.2.2 Material properties for air and waterTemp. Water Temp. Air at 1 atm(103) k cp (106) k cpC Pa.s kg.m3W.m1.K1kJ.kg1.K1 C Pa.s kg.m3W.m1.K1kJ.kg1.K110 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 1.0683 Newtons laws of motion Newtons laws of motionFirst 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 = maThird For every action there is an equal and opposite reaction. Equations of linear motionV2 = V1 +at (3)x2 = x1 +V1t + 12a (t)2(4)x2 = x1 + 12 (V2 +V1) t (5)V22 = V21 + 2a (x2x1) (6)4 Fluid Mechanics4.1 Fluid Statics Pascals lawdpdz = g (7) Force on a submerged planeyp = 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.34.2 Flow in pipes Continuity m = V A (9)d mdt = min mout (10) Hydraulic mean diameterDh = 4 AP (11) The Hagen-Poiseuille equationV = 14dpdx_R2r2_ (12)V = pR48L (13) Steady ow Energy Equation (SFEE)pin + V2in2 +gzin +wp = pout + V2out2 +gzout +wf +wt (14)where wf is the volumetric work lost due to friction, wp is the volumetric work supplied by a pump and wt is thevolumetric work generated by a turbine. Darcys Equation for losses in long pipeswf = 4f LDV22 (15) Fanning friction factorsf = 16Re laminar ow (16)f = (0.79ln (ReD) 1.64)2(17)1f = 1.8log10_6.9Re +_ 13.71

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 coecient for piping network componentswf = kV22 (19)where k is the loss coecient, values of k are given in table below:4Component kSharp Entry 0.5Rounded Entry 0.25Contraction (50% area) 0.24Contraction (50% diameter,based on V2) 0.35Expansion (based on V2)_A2A11_180oelbow 0.990oelbow 0.945oelbow 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:54.3 Conservation of linear momentum Force on uid in control volumeF = moutVout minVin (20)4.4 Lift and drag Lift forceFL = CL12V2A (21)where CL is the coecient of lift Drag forceFD = CD12V2A (22)where CD is the coecient of drag Coecients of skin friction drag for laminar ow over at plateCD = 1.328Re_Re < 105_ (23) Coecients of skin friction drag for turbulent ow over at plateCD = 0.074Re0.2_105< Re < 107_ (24)CD = 0.455(log(Re))2.58_107< Re < 109_ (25) Coecients of form drag around a cylinderCD = 24Re (Re < 1) (26)4.5 Compressible ow Isothermal compressible ow in a constant cross section pipe, neglecting change in gravitational potential energyp22 = p21 + 2RT_ mA_2ln_p2p1_ 4fLD RT_ mA_2(27) Speed of sounda =K =_RT (28) Change in velocity with area of nozzelVV = AA11 M2 (29) SFEE for isentropic compressible ow12V21 +cpT1 = 12V22 +cpT2 (30) Adiabatic, isentropic, compressible owT2T1=_p2p1_(1 )T2T1=_v1v2_1(31) Stagnation conditionsTT0= 11 +M2 12T = T0 + V22cp(32)pp0= 1_1 +M2 12 1(33)6 Critical conditionsTT0= 2 + 1 (34)pp0=_ 2 + 1_ 1(35)where T and p are the critical temperature and pressure, respectively.4.6 Water Hammer Pressure drop due to water hammer.p = V a (36) Augmented bulk modulus (K) for non-rigid pipes.1K = 1K + DtE giving a =K (37)where D is the internal diameter of the pipe, t is the thickness of the pipe wall and E is the Youngs modulus. = pD2t (38)where is the hoop stress, t is the thickness of the pipe wall and E is the Youngs modulus.5 Heat Transfer5.1 Thermal expansion Linear expansionL = L13T (39) Area expansionA = A123 T (40) Volumetric expansionV = V1T (41)where 3 is the coecient of linear expansion, sometimes referred to as in other texts.5.2 1D heat transfer ConductionQ = kALT (42) Conduction in thick walled cylinderQ = 2kL T1T2ln_R2R1_ (43) ConvectionQ = hAT (44)where h can be found using the Nusselt number, given in equation 99. Resistor analogy for composite surfacesQ = TR1 +R2 +R3 +. . . +Rn(45)7Rcond,planar = LkA (46)Rcond,clyind =ln_R2R1_2kL (47)Rconvect = 1hA (48)(49) RadiationQ = A_T41 T42_ (50)where is the Stefan-Boltzmann constant, of 5.67051 108W.m2.K45.3 Radiation heat transfer view factors Radiation equation with view factorsQij = AiFiji_T4i T4j_ (51) Reciprocity RelationAiFij = AjFji (52)5.4 Forced convection For an isothermal at plateNu = 0.032Re12LPr13(53)valid for ReL < 105and 0.6 < Pr < 60.Nu = 0.0296Re45LPr13(54)valid for 108> ReL > 105and 0.6 < Pr < 60.Nu = 0.037Re45LPr13(55)valid for ReL > 108and 0.6 < Pr < 60. For an isothermal horizontal cylinderNu = 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 = 0.023Re45DPrn(57)where n = 0.4 for heating uid and n = 0.3 for cooling uid. Valid for Re 10000 and 0.7 Pr 160 Log mean temperature dierenceTLM = T1T2lnT1T2(58) Exit temperature (Te) for constant wall temperature pipeTe = Tw + (TiTw) exp_hA mcp_ (59)where Ti is the inlet temperature and tempw is the wall temperature.85.5 Heat excha