AE1104 Physics 1. List of equations. Made by: E. Bruins Slot

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1 i AE04 Physics List of equations Made by: E. Bruins Slot

2 Chapter Introduction and basic concepts Newton s second law Weight F = M a (N) W = m g J = N m (N) Density Specific volume ρ = m V m 3 v = V m = ρ Specific weight γ s = ρ g N m 3 Kelvin to Celcius T (K) = T ( C) T (K) = T ( C) Rankine to Fahrenheit T (R) = T ( F) T (R) = T ( F) (R) =.8T (K) T ( F) =.8T ( C) + 32 Pa = N m 2 bar = 0 5 Pa = 0. MPa = 00 kpa Absolute, gage and vacuum pressure P gage = P abs P atm P vac = P atm P abs

3 The pressure at depth h from the free surface is P = P atm + ρgh or P gage = ρgh Relation for the variation of pressure with elevation dp dz = ρg P = P 2 P = 2 ρgdz The atmospheric pressure is measured by a barometer and is given by P atm = ρgh

4 Chapter 2 Energy, Energy Transfer and General Energy Analysis The total energy of a system on a unit mass basis e = E m kj Kinetic Energy Kinetic Energy on a unit mass basis KE = m V 2 2 ke = V 2 2 kj Potentional Energy PE = mgz Potentional Energy on a unit mass basis pe = gz kj Total Energy of a system E = U + KE + PE = U + m V mgz Total Energy of a system on a unit mass basis Mass flow rate e = u + ke + pe = u + V gz ṁ = ρ V = ρa c V avg s Energy flow rate Ė = ṁe ( kj s or ) kw 3

5 Mechanical Energy of a flowing fluid on a unit mass basis e mech = P ρ + V gz Mechanical Energy of a flowing fluid expressed in rate form ( P Ė mech = ṁe mech = ṁ ρ + V 2 ) 2 + gz Mechanical Energy change of a fluid during incompressible flow And e mech = P 2 P ρ + V 2 2 V g(z 2 z ) ) Ė mech = ṁe mech = ṁ( P2 P ρ + V 2 2 V g(z 2 z ) kj (kw) Heat transfer per unit mass of a system q = Q m kj Amount of heat transfer during a process When Q remains constant Q = t 2 t Qdt Q = Q t Work done per unit mass of a system w = W m kj The total volume change during a process between states and 2 The total work done during process -2 2 dv = V 2 V = V 2 δw = W 2 (Not W) Electrical work (where N is the amount of coulombs and V is a potentional difference) W e = V N

6 Electrical work expressed in rate form (Electrical Power) Ẇ e = V I (W) Electrical work done during time interval t Work done by a constant force Work done by a not constant force Torque 2 W e = V Idt W = Fs 2 W = Fds T = Fr F = T r This force acts through a distance s, which is related to the radius r by s = (2πr)n Shaft Work Power transmitted through the shaft T W sh = Fs = (2πrn) = 2πnT r Ẇ sh = 2πṅT (kw) Spring Work Total spring work δw spring = Fdx W spring = 2 k(x2 2 x 2 ) Work associated with the expansion or contraction of a solid bar 2 2 W elastic = Fdx = σ n Adx Work associated with the stretching of a film (also called surface tension work) 2 W sur f ace = σ s Adx

7 Energy balance E in E out = E system The change in the total energy of a system during a process (in the absence of electric, magnetic and surface tension effects) Where E = U + KE + PE U = m(u 2 u ) KE = 2 m(v 2 2 V 2) PE = mg(z 2 z ) Energy balance more explicitly E in E out = (Q in Q out ) + (W in W out ) + (E mass,in E mass,out ) = E system Energy balance for any system undergoing any kind of process Or in the rate form E in E out = E system Ė in Ė out = de system dt (kw) For constant rates, the total quantities during a time interval t are related to the quantities per unit time as Energy balance on a unit mass basis Q = Q t W = Ẇ t E = de dt t e in e out = e system kj Energy balance in differential form The Energy balance for a cycle δe in δe out = de system or δe in δe out = de system W net,out = Q net,in or Ẇ net,out = Q net,in Performance or efficiency Combustion efficiency Per f ormance = Desiredout put Requiredinput η combustion = Q HV = Amount of heat released during combustion Heating value of the fuel burned

8 Mechanical efficiency Pump efficiency Turbine efficiency Motor efficiency Generator efficiency η mech = E mech,out E mech,in η pump = Ė mech, f luid Ẇ sha ft,in η turbine = Ẇsha ft,out Ė mech, f luid = = E mech,loss E mech,in η motor = Ẇsha ft,out Ẇ elect,in η generator = Ẇelect,out Ẇ sha ft,in Combined efficiency of a pump-motor combination = Ẇpump,u Ẇ pump Ẇturbine Ẇ turbine,e η pump motor = η pump η motor = Ẇpump,u Ẇ elect,in = Ė mech, f luid Ẇ elect,in Combined efficiency of a turbine-generator combination η turbine generator = η turbine η generator = Ẇelect,out Ẇ turb,in = Ẇelect,out Ė mech, f luid Rate of heat conduction Q cond = k t A T x In the limiting case of x 0 (Fouriers law) Rate of heat transfer by convection Maximum rate of radiation Radiation emitted by a real surface Q cond = k t A dt dx Q conv = ha(t s T f ) Q emit,max = σat 4 s (W) (W) Q emit = εσat 4 s (W) Rate at which a surface absorbs radiation Q abs = α Q incident (W)

9 Net rate of radiation heat transfer Q rad = εσa(t 4 s T 4 s ) (W)

10 Chapter 3 Properties of Pure Substances The quality x as the ratio of the mass of vapor to the total mass of the mixture (for saturated mixtures only) Where x = m vapor m total m total = m liquid + m vapor = m f + m g The total volume in a tank containing a saturated liquid-vapor mixture is V = V f +V g V = mv m t v avg = m f v f + m g v g Dividing my m t yields m f = m t m g m t v avg = (m t m g )v f + m g v g Since x = m g /m t. This relation can also be expressed as v avg = ( x)v f + xv g m 3 v avg = v f + xv f g Where v f g = v g v g. Solving for quality we obtain x = v avg v f v f g The analysis given above can be repeated for internal energy and enthalpy with the following results u avg = u f + xu kj f g ) h avg = h f + xh f g All the results are of the same format, and they can be summarized in a single equation as Where y is v, u or h. y avg = y f + xy f g ( kj 9

11 Ideal-gas Equation of State The gasconstant R is determinded from R = R u M T P = R Pv = RT v kj kpa m3 or K K Where R u is the universal gas constant The mass of a system m = MN () The ideal-gas Equation of State can be written in several different forms V = mv PV = mrt mr = (MN)R = NR u PV = NR u T V = N v P v R u T The properties of an ideal gas at two different states are related to each other by Compressibility factor P V T = P 2V 2 T 2 or Z = Pv RT Pv = ZRT Gases behave differently at a given temperature and pressure, but they behave very much the same at temperatures and pressure normalized with respect to their critical temperatures and pressures. The normalization is done as P R = P P cr and T R = T T cr Pseudeo-reduced specific volume v actual v R = RT cr /P cr Van der Waals Equation of State (P + av 2 ) (v b) = RT The determination of the two constants appearing in this equation is based on the observation that the critical isotherm on a P v diagram has a horizontal inflection point of the cricital point. Thus, the first and the second derivatives of P with respect to v at the critical point must be zero. That is P = 0 and v T =T cr =const ( 2 ) P v 2 = 0 T =T cr =const

12 By performing the differentiations and eliminating v cr, the constants a and b are determined to be Beattie-Bridgeman Equation of State a = 27R2 T 2 cr 64P cr and b = RT cr 8P cr where P = R ut v 2 A = A 0 a v ( c vt 3 ) ( v + B) A v 2 and B = B 0 b v Benedict-Webb-Rubin Equation of State P = R ( ut v + B 0 R u T A 0 C 0 T 2 Virial Equation of State ) v 2 + br ut a v 3 + aα v 6 + c ( v 3 T 2 + γ v )e γ v 2 2 Vapor Pressure P = RT v + a(t ) v 2 + b(t ) v 3 + c(t ) v 4 + d(t ) v 5 + P atm = P a + P v

13 Chapter 4 Energy Analysis of Closed Systems Boundary work δw b = Fds = PAds = Pdv The total boundary work 2 W b = PdV The total area under the process curve Area = A = da = pdv Pressure for a polytropic process P = CV n Work done during a polytropic process 2 W b = 2 pdv = CV n dv = C V 2 n+ V n+ = P 2V 2 P V n + n Since C = P V n = P 2V2 n. For an ideal gas (PV = mrt ), this equation can also be written as W b = mr(t 2 T ) n n For the special case of n = the boundary work becomes 2 W b = 2 pdv = Energy balance for a closed system undergoing a cycle CV dv = PV ln ( V2 W net,out = Q net,in or Ẇ net,out = Q net,in (for a cycle) V ) 2

14 Various forms of the first-law relation for closed systems General Stationary systems Per unit mass Differential form Q W = E Q W = U q w = e δq δw = de Specific heat at constant volume (= the change in internal energy with temperature at constant volume) δu kj kj c v = δt v C or K Specific heat at constant pressure (= the change in enthalpy with temperature at constant pressure) c P = ( δh δt ) P kj kj C or K Using the definition of enthalpy and the equation of state of an ideal gas, we have } h = u + PV h = u + RT Pv = RT The differential changes in the internal energy and enthalpy of an ideal gas and du = c v (T )dt dh = c P (T ) the change in internal energy or enthalpy for an ideal gas during a process from state to state 2 is determined by integreting these equations and u = u 2 u = 2 c v (T )dt h = h 2 h = 2 c P (T )dt kj kj A special relationship between C P and C v for ideal gasses can be obtained by differentiating the relation h = r + RT, which yields dh = du + RdT Replacing dh by c P dt and du by c v dt and dividing the resulting expression by dt, we obtain kj c P = c v + R K Specific heat ratio k = c P c v

15 For incompressible substances (solids and liquids) c P = C v = c The change in internal energy of incompressible substances between states and states 2 is obtained by 2 u = u 2 u = c(t )dt kj = c avg (T 2 T ) The change in enthalpy of incompressible substances between states and states 2 is obtained by kj h = u + v P

16 Chapter 5 Mass and Energy Analysis of Control Volumes Conservation of mass principle Mass flow rate Volume flow rate The total energy of a flowing fluid m in m out = m system and ṁ in ṁ oud = dm system dt ṁ = ρva V = VA = ṁ ρ θ = h + ke + pe = h + V gz The general mass and energy balances for any system undergoing any process can be expressed as E in E }{{ out } = E system }{{} Net energy transfer by heat,work, and mass Changes in internal, kinetic, potential, etc., energies It can also be expressed in the rate form as E system Ė in Ė }{{ out = }}{{ dt } Rate of net energy transfer by heat,work, and mass Rate of changes in internal, kinetic, potential, etc., energies Conservation of mass and energy equations for steady-flow processes ṁ = ṁ in out Q Ẇ = ṁ (h + V 2 ) out 2 + gz ṁ (h + V 2 ) in 2 + gz }{{}}{{} for each exit for each inlet For single-stream (one-inlet-one-exit) sytems they simplify to ṁ = ṁ 2 v V A = v 2 V 2 A 2 [ ] Q Ẇ = ṁ h 2 h + V 2 2 V g(z 2 z ) 5

17 When kinetic and potential energy changes associated with the control volume and the fluid streams are negligible, the mass and energy balance relations for a uniform-flow system are expressed as m in m out = m system Q W = mh mh + (m 2 u 2 m u ) system out in

18 Chapter 6 The Second Law of Thermodynamics The thermal efficiency of a heat engine η th = W net,out Q H = Q L Q H Coefficient of performance COP R = COP HP = Q L W net,in = Q H QL Q H W net,in = Q L Q H Thermodynamic temperature scale related to the heat transfers between a reversible device and the high- and low-temperature reservoirs QH = T H Q L T H Thermal efficiency of a Carnot heat engine, as well as all other reversible heat engines rev η th,rev = T L T H Coefficient of performance of reversible refrigerators and heat pumps and COP R,rev = T H TL COP HP,rev = T L T H 7

19 Chapter 7 Entropy Definition of entropy ds = dq T int,rev For the special case of an internally reversible, isothermal process, this gives Increase of entropy principle S = Q T 0 S gen 0 The entropy change and isentropic relations for a process with pure substances Any process: s = s 2 s Isentropic process: s 2 = s The entropy change and isentropic relations for a process with incompressible substances Any process: s 2 s = c avg ln T 2 T Isentropic process: T 2 = T The entropy change and isentropic relations for a process with ideal gases with constant specific heats (approximate treatment) Any process: s 2 s = c v,avg ln T 2 T + Rln v 2 v s 2 s = c P,avg ln T 2 T + Rln P 2 P Isentropic process: T2 T T2 T P2 P s=constant = ( v v 2 ) k k = P k s=constant P 2 s=constant = ( v v 2 ) k 8

20 The entropy change and isentropic relations for a process with ideal gases with variable specific heats (exact treatment) Any process: s 2 s = s 2 s Rln P 2 P Steady flow work for a reversible process For incompressible substances it simplifies to Isentropic process: s 2 = s + Rln P 2 P ( P2 ) P v2 v s=constant = P r2 P r s=constant = v r2 v r 2 w rev = vdp ke pe w rev = v(p 2 P ) ke pe The reversible work inputs to a compressor compressing an ideal gas from T, P to P 2 in an isentropic, polytropic or isothermal manner [ Isentropic: w comp,in = kr(t 2 T ) k = krt k ] P2 k k P [ Polytropic: w comp,in = nr(t 2 T ) n = nrt n ] P2 n n P Isothermal: w comp,in = RT ln P 2 P Isentropic or adiabatic efficiency for turbines, compressors and nozzles η T = η C = η N = Actual turbine work Isentropic turbine work = w a Isentropic compressor work Actual compressor work w s = h h 2a h h 2s = w s w a = h 2s h h 2a h Actual KE at nozzle exit Isentropic KE at nozzle exit = V 2a 2 = h h 2a V2s 2 h h 2s The entropy balance for any system undergoing any process can be expressed in the general form as or, in rate form as S in S }{{ out } + S gen }{{} = S system }{{} Net entropy transfer by heat and mass Entropy generation Change in entropy ds system Ṡ in Ṡ }{{ out + Ṡ } gen = }{{}}{{ dt } Rate of net entropy transfer by heat and mass Rate of entropy generation Rate of change in entropy

21 For a general stead-flow process it simplifies to Ṡ gen = ṁ e s e ṁ i s i Q k T k

22 Chapter 8 Exergy: A Measure of Work Potentional Not mandatory for the exam 2

23 Chapter 9 Gas Power Cycles Thermal efficiency of the Carnot cycle η th,carnot = T L T H In reciprocating engines, the compression ratio r and the mean effective pressure MEP are defined as r = V max V min MEP = = V BDC V T DC w net v max v min The thermal efficiency of the ideal Otto cycle (spark-ignition reciprocating engines) under cold-airstandard assumptions is η th,otto = r k The thermal efficiency of the ideal Diesel cycle (compression-ignition reciprocating engines) under cold-air-standard assumptions is η th,diesel = [ ] r k c r k k(r c ) The thermal efficiency of the ideal Brayton cycle (modern gas-turbine engines) under cold-airstandard assumptions is η th,brayton = r (k )/k p The deviation of the actual compressor and the turbine from the idealized isentropic ones can be accurately accounted for by utilizing their isentropic efficiencies, defined as and η C = w s w a = h 2s h h 2a h η T = w a w s = h 3 h 4a h 3 h 4S Where states and 3 are the inlet states, 2a and 4a are the actual exit states, and 2s and 4s are the isentropic exit states. Effectiveness (the extent to which a regenerator approaches an ideal regenerator) ε = q regen,act q regen,max 22

24 Under cold-air-standard assumptions, the thermal efficiency of an ideal Brayton cycle with regeneration becomes T η th,regen = (r P ) k k The net thrust devoloped by the ideal jet-propulsion cycle Propulsive power Propulsive efficiency T 3 F = ṁ(v exit V inlet ) Ẇ P = ṁ(v exit V inlet )V aircra ft η P = Propulsive power Energy input rate = ẆP Q in For an ideal cycle that involves heat transfer only with a source at T H and a sink at T L, the exergy destruction is ( qout x dest = T 0 q ) in T L T H

ENERGY TRANSFER BY WORK: Electrical Work: When N Coulombs of electrical charge move through a potential difference V

ENERGY TRANSFER BY WORK: Electrical Work: When N Coulombs of electrical charge move through a potential difference V Weight, W = mg Where m=mass, g=gravitational acceleration ENERGY TRANSFER BY WOR: Sign convention: Work done on a system = (+) Work done by a system = (-) Density, ρ = m V kg m 3 Where m=mass, V =Volume