2 nd Law: m.s in + Q in /T in + S gen = ds/dt + m.s out + Q out /T out. S gen /ṁ = -(Q in /m)/t + Δs,

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1 P xercises + f 4 6 Jan + Jan 5. kg ice at 6 K kg ater at 9 K. Heat Q at = is supplie y the surrunings. Specific heat ice : c i = 4 kj/kgk; ater c = 48 kj/kgk melting heat Δ m H = 4 kj/kg n La: m.s in + Q in / in + S gen = S/t + m.s ut + Q ut / ut S gen /ṁ = -Q in /m/ + Δs Δex = -Q in /m/ + Δs Q in /m = Q/m = 7 6 c i + Δ melt h c. = 484 kj per kg ΔS/m = Δs = c i 7/6 + Δ melt h / 7 + c 9/7 = 6 kj/kgk using ΔG = = ΔH ΔS at phase transitins ΔS = ΔH/ Δex = -Q in /m/ + Δs = Δex = -Q in /m+ Δs = -484 kj = 4 kj per kg P xercises + f 4 6 Jan + Jan 5 a. exergy ex ice = ex = h -h - s -s = h -h - s -s h = h sli = h liq - Δ melt h = = kj/ml s = s sli = s liq - Δ melt h/ melt = 68/7 = -99 J/ml K ex ice = = 4877 J/ml. energy alance heat exchanger: ṁ NH h 4 -h = ṁ HO h -h emp. -5 Δ 4 Lcatin ṁ NH /ṁ HO = h -h / h 4 -h = h - h sli/δ vap h NH at = -5 C 5 C el C Δ vap h NH at = -5 C = 8 6 = 8 J/ml ṁ NH /ṁ HO = 5-687/8 = 4 ml/ml

2 P xercises + f 4 6 Jan + Jan 5 c. Ẇ lst = Δx lst = Ṡ genho + Ṡ gennh = [s - s + s -s 4 ṁ NH /ṁ HO ] J/ml ice = [s sli - s + s -s 4 4] = 9 [ ] = 6 W/s/ml ice he same result is fun via Ẇ lst = Ėx in Ėx ut an Ėx in /ṁ HO = ėx + ėx 4 ṁ NH /ṁ HO an Ėx ut /ṁ HO = ėx + ėx ṁ NH /ṁ HO ith ex i = h i - h - s i - s. fficiency η X = x ut / x in = x in W lst / x in = W lst /x in Ėx in /ṁ HO = = 75 W/ml ice Ėx ut /ṁ HO = = 8 W/ml ice Data fr NH : h = 948 J/ml s = 674 J/ml K gives 54 J/ml fr ex an 6 J/ml fr ex 4 η X = 8 / 75 = 79 his is the prcess efficiency. Fr the heat exergy transfer η X = x x /x -x 4 = 498/[4 54 6] = 57 P xercises + f 4 6 Jan + Jan 5. W min = Δx separatin = 79 x N + x O - x air at p x air = 79 x N + x O + Δ mix x = 79 x N + x O + Δ mix H - Δ mix S ieal gases: Δ mix H = n p r changes & Δ mix S = - Σx i x i. Δx separatin = = 5 J/ml as a result f inefficiencies in real prcesses it ill e larger in practice: see Chapter

3 P xercises + f 4 6 Jan + Jan 5 4. First la: W in = ΔH + Q ut ΔH = H H aiaatic Q ut =. Q ut Ẇ in = ΔḢ = ṅ c p Δ = 57 W ṅ = ml/s ṅ ṅ Secn la: Ṡ gen = ΔS n heat exchange steay state ieal gas: Δs = s - s = c p / - p /p Ṡ gen = 7 W/K = p =p W in p Fr minimal rk: Ṡ gen = ΔS = c p / = p /p gives reversile = 5 C > reversile ecause f irreversiilities. efficiency = Ẇ in reversile / Ẇ in = c p Δ reversile / c p Δ = Δ reversile / Δ = = 5-/6- = 75 Wrk rate lst = per lst = c p reversile = Ṡ gen = 8 W = Ẇ in - Ẇ in reversile fr the cmpressr Fr the prcess: η X = Ėx ut / Ėx in = Ėx in + Ẇ in - Ẇ lst / Ėx in + Ẇ in gives =.9 Ẇ in - Ẇ lst /Ẇ in ecause Ėx in P xercises + f 4 6 Jan + Jan 5 5. Fr cling = 8 K ex q cp cp = 6 kj fr m = kg. Fr cnensatin at temperature : ex vap H = 5 kj/kg Δex + Δex = 67 kj/kg in reality ~ x higher hen using a s-calle Line prcess see curse efrigeratin

4 P xercises + f 4 6 Jan + Jan 5 6. ransmittivity τ = K K K K m m he raiatin functin tales r equatin give fr a lacky:. µm 58 K = 74 µmk f. = 5.5 µm 58 K = 45 µmk f.5 = 9664 τ =.9 f.5 - f. = 84 P xercises + f 4 6 Jan + Jan 5 7.a Fr the ttal emissivity f the rickall: = 5 K. λ = 75 µmk f - λ = ; λ = 5 µmk f - λ = 64 ε = = 6 tal emissive per = ε = 6 σ 5 4 = 6 W/m. f f f f

5 P xercises + f 4 6 Jan + Jan 5 7 Fr iffuse irraiatin O a iffuse surface: α λ = ε λ α = ε = f f f f N: λ c = µmk f - λ = 7 ; λ c = µmk f - λ = 986 α c = = 95 P xercises + f 4 6 Jan + Jan 5 8. = 7 K ε = A = A = 5 m² = A = 77 K ε = 5 = K 8 A A Vie factr frm iagrams: F - = 85 = F - F - = -F - = 75 = F - here may e a resistance thus = J A ut A is very large 7 78 F A F A

6 P xercises + f 4 6 Jan + Jan 5 8 At nes J an J : J J J J J J J J = σ 4 = 489 kw/m ; = σ 4 = 4 kw/m ; = σ 4 = 46 kw/m J = 47 kw/m ; J = 55 kw/m Heat lst y surface : Heat lst y surface : q q J J kw 59 kw Heat receive y rm = surface : J J J J q q q 7 kw; ith J P xercises + f 4 6 Jan + Jan 5 xam March 9: Questin C C ṁ A = 5 kg/s utb C ṁ B = kg/s c pa = c pb = c p a. Heat fl Q A = ṁ A c pa Δ A = ṁ B c pb Δ B = Q B = Q = 8 kw B ut = Bin Q/ ṁ B c pb = 975 C. x Q mc p x ut in ut Q mc p mc in p ut in ut in tal exergy change Δx = Δx A + Δx B x m c A pa uta uta ina m Bc ina pb utb utb

7 P xercises + f 4 6 Jan + Jan 5 xam March 9: Questin x m c A pa uta uta ina m Bc ina pb utb utb he terms ith ut in cancel ut ecause f heat alance ṁ A c pa Δ A = ṁ B c pb Δ B Δ x m c A pa uta ina m Bc pb utb = = -64 kw lsses are ΔĖx = 64 kw c. Lsses are -Δ Ėx/Q = 64 kw / 8 kw = 96 %; efficiency = -96 = 84 % P xercises + f 4 6 Jan + Jan 5 xam March 9: Questin Data MgCO CaCO MgCO CaCO ΔG f kj/ml a. Δex = Δ = D C M = ΔG f D +.. ΔG f C +. ΔG f M +. = ΔG f D - ΔG f C - ΔG f M = 464 kj/ml. mixing: Δex = = - = -7 kj/ml c. Overall Δex > s nt spntaneus ut sufficiently high pressure ill.

8 P xercises + f 4 6 Jan + Jan 5 xam March 9: Questin a. Vie factrs: L /W = /6 = 65 ; L /W = 5 iagram curse material F - = - = /A F - = /A F - = 5 m Q AF 6-7 kw A F. his gives surface resistances A A ttal... 8m A AF A 9 A A 9 A 44 5 Q - = σ 4-4 / ttal = 7kW anser frm A * frm a / frm = 7 kw*5/8 = 6 kw P xercises + f 4 6 Jan + Jan 5 xam 8 May : questin a. Max per utput W max = J - º J S = = 77 W/m is 9% f 77 W/m With 6 m gives 445 MW.. 77 W/m 6 m = 477 MW cmpare t MWe means /4.77 % = % Pssile ul e 9% see a. effectiveness = /9 = % c. Heat frm iler t surrunings: 5 σ = 5 MW emains fr electricity generatin 445 MW 5 = 4 MW; Output is MWe efficiency = /4 = 9%. Irreversiility = º S gen = W max W real = 445 = 45 MW S gen = 45 MW/ K = 5 kw/k Lsses: cllectr 477 MW 445 MW = MW = 67% iler raiatin 5 MW = % Per generatin 4 MW = 5% tal lsses 79% MW electricity frm 477MW slar irraiatin = %