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

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1 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 Specific volume, v = V m = ρ Specific gravity, SG = Where r HO =000 kg/m 3 P gage = P Abs P atm m 3 kg Variation of pressure with depth: Apply between two pots the same fluid. Where below refers to pot at lower elevation and above at higher elevation P below = P above + ρg z The absolute and gage pressures a liquid open to the atmosphere at a depth of h from the free surface are: etic Energy: Where m=mass, V=velocity Potential Energy: P = P atm + ρgh E = m V ρ ρ H O P vac = P atm P abs P gage = ρgh ke = V PE = mgz pe = gz Where m=mass, g=gravitational acceleration, z=elevation SPECIFIC HEAT RELATIONS FOR IDEAL GAS: c v = δu u = u u = c v T dt δt v c p = δh δt p h = h h = c p T dt Variation of spec. heats with T is smooth and may be approx. as lear over small T terval. Can replace specific heat with C avg, yieldg: Electrical Work: When N Coulombs of electrical charge move through a potential difference V W e = VN In the rate form, W e = VI = I R = V R Electrical work done durg a time terval t: Where W e is the electrical power and I is the current. W e = VI dt Or when V and I rema constant durg terval t: W e = VI t Mechanical Forms of Work: W = Fs W = Fds Work done by a constant force, F on a body displaced a distance s Boundary Work: W b = Pdv Constant P process: W b = P(V V ) Polytropic process: W b = P V P V n Shaft Work: W sh = π n T (kw) Where n is the number of revolutions per unit time Sprg Work: W sprg = k x x Where x and x are the itial and fal displacements of the sprg. Polytropic Ideal Gas process: Polytropic Isothermal Ideal Gas process: INTERNAL ENERGY, ENTHALPY & SPECIFIC HEATS OF SOLIDS & LIQUIDS For an compressible substance: c p = c v = c u = u u = c avg T T h = u + v P c avg T T + v P W b = mr(t T ) n W b = PV ln V V = mrt o ln V V (W) Durg actual exp/comp process of gases, P and V are related by PV n =C. Where n and C are constants therefore between states, ideal gas, closed C = P V = P V = mrt 0 V V = P P u u = c v,avg T T, h h = c p,avg (T T ) c p = c v + R kj kg, k c p c v

2 P-v Diagram T-v Diagram SUPERHEATED STATE Region to the right of sat vap le & at a T above T cr To determe if S.H. P<P sat at given T T> T sat at given P v>v g at given P or T u> u g at given P or T h> h g at given P or T s> s g at given P or T COMPRESSED LIQUID STATE Region to the left of the Sat Liq le In the absence of C.L. data treat a C.L. as a Sat liq at given T To determe if C.L. P>P sat at given T T< T sat at given P v< v f at given P or T u< u f at given P or T h< h f at given P or T s< s f at given P or T Quality, x = y f y f@t Where y is v, u, s or h Quality, x = 0 SATURATED LIQUID-VAPOR MIXTURE STATE Region under the dome To determe the proportion of liquid and vapor phases the mixture, fd quality, x x = m vapor m total, Quality, 0 < x < To fd v, h, u, s at state. Where y is v, u, s or h y = y f + x y g y f To determe if Sat Liq-vap Mixture. v f v v g u f u u g h f h h g s f s s g m total= mliq +m vap =m f +m g

3 POWER CYCLES: Expandg the equation: (m m ) =

HEAT ENGINES Thermal Efficiency of HE, th th = W net, Q = Q Q W net, = Q Q ENERGY BALANCE: CLOSED SYSTEM E sys = E E U + E + PE = E E Expandg both the left and right side of the equation: SIMPLIFY

Step : If Stationary then E= PE=0 Step 3: Determe if have Q or Q if Adiabatic Q=0 Step 4: Determe if have W b, W paddle, W electrical m u u + m V V + mg z z = (Q +W ) (Q + W ) HEAT PUMP:

a Refrigerator is to keep a cold space cold Q L / ENERGY BALANCE: OPEN SYSTEM, STEADY STATE E sys = E E Expandg the equation: Q + W + m(h + V MASS BALANCE: OPEN SYSTEM, STEADY STATE m sys = m Sce m

4 HEAT ENGINES Thermal Efficiency of HE, th th = W net, Q = Q Q W net, = Q Q ENERGY BALANCE: CLOSED SYSTEM E sys = E E U + E + PE = E E Expandg both the left and right side of the equation: SIMPLIFY ENERGY BALANCE FOR CLOSED SYSTEM Step : Defe system of terest and simplify E-bal. Step : If Stationary then E= PE=0 Step 3: Determe if have Q or Q if Adiabatic Q=0 Step 4: Determe if have W b, W paddle, W electrical m u u + m V V + mg z z = (Q +W ) (Q + W ) HEAT PUMP: REFRIGERATOR: The objective of a HP is to keep a warm space warm Coefficient of Performance for HP, COP HP COP HP = COP HP = Desired put Required put = Q L = COP HP = COP R + W net, The objective of a Refrigerator is to keep a cold space cold Q L / ENERGY BALANCE: OPEN SYSTEM, STEADY STATE E sys = E E Expandg the equation: Q + W + m(h + V MASS BALANCE: OPEN SYSTEM, STEADY STATE m sys = m Sce m sys =0 for SS 0 = m m m Sce E sys = 0 for SS 0 = E E + gz) = Q + W + m(h + V mass flow rate, m = ρva For Steady, compressible flow: V = V ENERGY BALANCE: OPEN SYSTEM, UNSTEADY-FLOW + gz) Volumetric flow rate, V = VA = m/ρ CARNOT REFRIGERATOR COP R,REV = T L /T H CARNOT REFIGERATION CYCLE COP R = COP R = Desired put Required put = Q l W net, Q l = Q L Q L / E sys = E E U + E + PE = E E Expandg the equation: U + E + PE = Q + W + MASS BALANCE: OPEN SYSTEM, UNSTEADY-FLOW m(h + V + gz) (Q + W + m(h + V + gz)) m sys = (m m ) = th < th,rev Irreversible HE = th,rev Reversible HE CARNOT HEAT ENGINE: th,rev = T L T H CARNOT HEAT PUMP: COP HP,REV = T L /T H Q L REV = T L T H > th,rev Impossible HE CARNOT HE CYCLE

5 ENTROPY, S: ds = δq T INT REV FIND THE CHANGE IN ENTROPY: S = S S = INCREASE OF ENTROPY PRINCIPLE: S SYS = S S = δq T INTERNALLY REVERSIBLE ISOTHERMAL HEAT TRANSFER : S = Q T O S gen = 0 > 0 Irreversible Reversible < 0 Impossible ISENTROPIC PROCESS: INT REV δq T + S gen A ternally reversible, adiabatic process S = 0 or S = S ENTROPY CHANGE OF LIQUIDS AND SOLIDS: S S = dt c(t) T c avgln T T Where c avg is the average specific heat of the substance over the given temperature terval SPECIAL CASE: ISENTROPIC LIQUIDS & SOLIDS S S = c avg ln T T = 0 T = T ISENTROPIC EFFICIENCIES OF STEADY-FLOW DEVICES TURBINE T = Actual Turbe Work Isentropic Turbe Work = w a h h a w s h h s COMPRESSOR Isentropic Compressor Work T = Actual Compressor Work ENTROPY CHANGE OF IDEAL GAS: CONSTANT SPECIFIC HEAT (Approximate Analysis: for when T is small < 300 ) S S = c v,avg ln T T + R ln v v S S = c p,avg ln T T R ln P P ISENTROPIC PROCESS OF IDEAL GAS: CONSTANT SPECIFIC HEAT T T S=CONST = v v T T S=CONST = P P P P S=CONST = v v k (k )/ GENERAL ENTROPY BALANCE S sys = S S + S gen k Mechanisms of Entropy Transfer = Q and m = w s w a h s h h a h R/c v = k ENTROPY BALANCE: OPEN SYSTEM S sys = S S = + m T i s i m e s e + S gen ENTROPY BALANCE: OPEN SYSTEM-STEADY FLOW 0 = + m T i s i m e s e + S gen NOZZLE Actual E a nozzle exit N = Isentropic E at nozzle exit = V a = h h a V S h h s PUMP P = w s w a = v(p P ) h a h ENTROPY CHANGE OF IDEAL GAS: VARIABLE SPECIFIC HEAT (Exact Analysis: for when T is large & specific heats vary non-learly w/ T range) S S = S S R ln P P ISENTROPIC PROCESS OF IDEAL GAS: VARIABLE SPECIFIC HEAT P P S=CONST = P r P r v v S=CONST = v r v r ENTROPY BALANCE: CLOSED SYSTEM S sys = S S = + S T gen ENTROPY BALANCE: ADIABATIC CLOSED SYSTEM S sys = S gen ENTROPY BALANCE: ADIABATIC CLOSED SYSTEM AND SURROUNDINGS S gen = S sur = Q surr T surr S = S system + S surroundgs

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

AE1104 Physics 1. List of equations. Made by: E. Bruins Slot i AE04 Physics List of equations Made by: E. Bruins Slot 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 =