Chapter Equations

Equations 1-2 Importance of Dimensions and Units 𝐹 = 𝑚𝑎 1-1 𝑊 = 𝑚𝑔 1-2 1J = 1N∙m 1-3 1-5 Density and Specific Gravity 𝜌= 𝑚 1-4 𝑉 𝑉 1 𝑣=𝑚=𝜌 𝑆𝐺 = 𝜌 𝜌 𝐻2 𝑂 𝛾𝑠 = 𝜌𝑔 1-5 1-6 1-7 1-8 Temperature and the Zeroth Law of Thermodynamics 𝑇 = 𝑎 + 𝑏𝑃 1-8 𝑇(K) = 𝑇(℃) + 273.15 1-9 𝑇(𝑅) = 𝑇(℉) + 459.67 1-10 𝑇(𝑅) = 1.8 ∙ 𝑇 (𝐾) 1-11 𝑇(℉) = 1.8 ∙ 𝑇(℃) + 32 1-12 ∆𝑇 (𝐾) = ∆𝑇(℃) 1-13 ∆𝑇 (𝑅) = ∆𝑇(℉) 1-14 1-9 Pressure 𝑃𝑔𝑎𝑔𝑒 = 𝑃𝑎𝑏𝑠 − 𝑃𝑎𝑡𝑚 1-15 𝑃𝑣𝑎𝑐 = 𝑃𝑎𝑡𝑚 − 𝑃𝑎𝑏𝑠 1-16 ∆𝑃 = 𝑃2 − 𝑃1 = −𝜌𝑔∆𝑧 = −𝛾𝑠 ∆𝑧 1-17 𝑃𝑏𝑒𝑙𝑜𝑤 = 𝑃𝑎𝑏𝑜𝑣𝑒 + 𝜌𝑔|∆𝑧| = 𝑃𝑎𝑏𝑜𝑣𝑒 + 𝛾𝑠 |∆𝑧| 1-18 𝑃 = 𝑃𝑎𝑡𝑚 + 𝜌𝑔ℎ or 𝑃𝑔𝑎𝑔𝑒 = 𝜌𝑔ℎ 1-19 𝑑𝑃 𝑑𝑧 = −𝜌𝑔 1-20 2 ∆𝑃 = 𝑃2 − 𝑃1 = − ∫1 𝜌𝑔 𝑑𝑧 𝑃1 = 𝑃2 → 𝐹1 𝐴1 = 𝐹2 𝐴2 → 𝐹2 𝐹1 = 1-21 𝐴2 1-22 𝐴1 1-10 Pressure Measurement Devices 𝑃𝑎𝑡𝑚 = 𝜌𝑔ℎ 1-23 𝑃2 = 𝑃𝑎𝑡𝑚 + 𝜌𝑔ℎ 1-24 𝑃1 + 𝜌1 𝑔(𝑎 + ℎ ) − 𝜌2 𝑔ℎ − 𝜌1 𝑔𝑎 = 𝑃2 1-25 𝑃1 − 𝑃2 = (𝜌2 − 𝜌1 )𝑔ℎ 1-26 2-2 Forms of Energy 𝑒= 𝐸 2-1 𝑚 𝐾𝐸 = 𝑚 𝑘𝑒 = 𝑉2 2-2 2 𝑉2 2-3 2 𝑃𝐸 = 𝑚𝑔𝑧 2-4 𝑝𝑒 = 𝑔𝑧 2-5 𝐸 = 𝑈 + 𝐾𝐸 + 𝑃𝐸 = 𝑈 + 𝑚 𝑒 = 𝑢 + 𝑘𝑒 + 𝑝𝑒 = 𝑢 + 𝑉2 2 𝑉2 2 + 𝑚𝑔𝑧 2-6 + 𝑔𝑧 2-7 𝑚̇ = 𝜌𝑉̇ = 𝜌𝐴𝑐 𝑉𝑎𝑣𝑔 2-8 𝐸̇ = 𝑚̇ 𝑒 2-9 𝑃 𝑒𝑚𝑒𝑐ℎ = 𝜌 + 𝑉2 2 + 𝑔𝑧 2-10 2 𝑃 𝑉 𝐸̇𝑚𝑒𝑐ℎ = 𝑚̇ 𝑒𝑚𝑒𝑐ℎ = 𝑚̇ (𝜌 + 2 + 𝑔𝑧) ∆𝑒𝑚𝑒𝑐ℎ = 𝑃2 −𝑃1 𝜌 + 𝑉22−𝑉12 2 2-11 + 𝑔(𝑧2 − 𝑧1 ) 2 2-12 2 𝑃 −𝑃 𝑉 −𝑉 ∆𝐸̇𝑚𝑒𝑐ℎ = 𝑚̇ ∆𝑒𝑚𝑒𝑐ℎ = 𝑚̇ ( 2 1 + 2 1 + 𝑔(𝑧2 − 𝑧1 )) 𝜌 2 2-13 2-3 Energy Transfer by Heat 𝑞= 𝑄 𝑚 2-14 𝑡 𝑄 = ∫𝑡 2 𝑄̇ 𝑑𝑡 2-15 𝑄 = 𝑄̇∆𝑡 2-16 1 2-4 Energy Transfer by Work 𝑊 𝑤=𝑚 2-17 𝑊̇𝑒 = 𝐕𝐼 2-18 2 𝑊𝑒 = ∫1 𝐕𝐼 𝑑𝑡 2-19 𝑊𝑒 = 𝐕𝐼∆𝑡 2-20 2-5 Mechanical Forms of Work 𝑊 = 𝐹𝑠 2-21 2 𝑊 = ∫1 𝐹 𝑑𝑠 2-22 𝑇 = 𝐹𝑟 → 𝐹 = 𝑇 2-23 𝑟 𝑠 = (2𝜋𝑟)𝑛 2-24 𝑇 𝑊𝑠ℎ = 𝐹𝑠 = ( ) (2𝜋𝑟𝑛) = 2𝜋𝑛𝑇 2-25 𝑊̇𝑠ℎ = 2𝜋𝑛̇ 𝑇 2-26 𝛿𝑊𝑠𝑝𝑟𝑖𝑛𝑔 = 𝐹 𝑑𝑥 2-27 𝐹 = 𝑘𝑥 2-28 𝑟 1 𝑊𝑠𝑝𝑟𝑖𝑛𝑔 = 2 𝑘(𝑥22 − 𝑥12 ) 2 2 𝑊𝑒𝑙𝑎𝑠𝑡𝑖𝑐 = ∫1 𝐹 𝑑𝑥 = ∫1 𝜎𝑛 𝐴 𝑑𝑥 2 𝑊𝑠𝑢𝑟𝑓𝑎𝑐𝑒 = ∫1 𝜎𝑠 𝑑𝐴 2-29 2-30 2-31 2-6 The First Law of Thermodynamics ∆𝐸𝑠𝑦𝑠𝑡𝑒𝑚 = 𝐸𝑓𝑖𝑛𝑎𝑙 − 𝐸𝑖𝑛𝑖𝑡𝑖𝑎𝑙 = 𝐸2 − 𝐸1 2-32 ∆𝐸 = ∆𝑈 + ∆𝐾𝐸 + ∆𝑃𝐸 2-33 𝐸𝑖𝑛 − 𝐸𝑜𝑢𝑡 = (𝑄𝑖𝑛 − 𝑄𝑜𝑢𝑡 ) + (𝑊𝑖𝑛 − 𝑊𝑜𝑢𝑡 ) + (𝐸𝑚𝑎𝑠𝑠,𝑖𝑛 − 𝐸𝑚𝑎𝑠𝑠,𝑜𝑢𝑡 ) = ∆𝐸𝑠𝑦𝑠𝑡𝑒𝑚 2-34 𝐸𝑖𝑛 − 𝐸𝑜𝑢𝑡 = ∆𝐸𝑠𝑦𝑠𝑡𝑒𝑚 2-35 𝑑𝐸𝑠𝑦𝑠𝑡𝑒𝑚 𝐸̇𝑖𝑛 − 𝐸̇𝑜𝑢𝑡 = 𝑑𝑡 2-36 𝑑𝐸 𝑄 = 𝑄̇∆𝑡, 𝑊 = 𝑊̇ ∆𝑡, 𝑎𝑛𝑑 ∆𝐸 = ( 𝑑𝑡 ) ∆𝑡 2-37 𝑒𝑖𝑛 − 𝑒𝑜𝑢𝑡 = ∆𝑒𝑠𝑦𝑠𝑡𝑒𝑚 2-38 𝛿𝐸𝑖𝑛 − 𝛿𝐸𝑜𝑢𝑡 = 𝑑𝐸𝑠𝑦𝑠𝑡𝑒𝑚 or 𝛿𝑒𝑖𝑛 − 𝛿𝑒𝑜𝑢𝑡 = 𝑑𝑒𝑠𝑦𝑠𝑡𝑒𝑚 2-39 𝑊𝑛𝑒𝑡,𝑜𝑢𝑡 = 𝑄𝑛𝑒𝑡,𝑖𝑛 𝑜𝑟 𝑊̇𝑛𝑒𝑡,𝑜𝑢𝑡 = 𝑄̇𝑛𝑒𝑡,𝑖𝑛 2-40 2-7 Energy Conversion Efficiencies 𝐷𝑒𝑠𝑖𝑟𝑒𝑑 𝑜𝑢𝑡𝑝𝑢𝑡 𝐸𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑐𝑦 = 𝑅𝑒𝑞𝑢𝑖𝑟𝑒𝑑 𝑖𝑛𝑝𝑢𝑡 𝜂𝑐𝑜𝑚𝑏.𝑒𝑞𝑢𝑖𝑝. = 𝑄𝑢𝑠𝑒𝑓𝑢𝑙 𝐻𝑉 = 2-41 𝑈𝑠𝑒𝑓𝑢𝑙 ℎ𝑒𝑎𝑡 𝑑𝑒𝑙𝑖𝑣𝑒𝑟𝑒𝑑 𝑏𝑦 𝑡ℎ𝑒 𝑐𝑜𝑚𝑏𝑢𝑠𝑡𝑖𝑜𝑛 𝑒𝑞𝑢𝑖𝑝𝑚𝑒𝑛𝑡 𝐻𝑒𝑎𝑡𝑖𝑛𝑔 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑓𝑢𝑒𝑙 𝑏𝑢𝑟𝑛𝑒𝑑 𝑊̇𝑛𝑒𝑡,𝑒𝑙𝑒𝑐𝑡𝑟𝑖𝑐 𝜂𝑜𝑣𝑒𝑟𝑎𝑙𝑙 = 𝜂𝑐𝑜𝑚𝑏.𝑒𝑞𝑢𝑖𝑝. 𝜂𝑡ℎ𝑒𝑟𝑚𝑎𝑙 𝜂𝑔𝑒𝑛𝑒𝑟𝑎𝑡𝑜𝑟 = 𝜂𝑚𝑒𝑐ℎ = 𝑀𝑒𝑐ℎ𝑎𝑛𝑖𝑐𝑎𝑙 𝑒𝑛𝑒𝑟𝑔𝑦 𝑜𝑢𝑡𝑝𝑢𝑡 𝜂𝑝𝑢𝑚𝑝 = 𝑀𝑒𝑐ℎ𝑎𝑛𝑖𝑐𝑎𝑙 𝑒𝑛𝑒𝑟𝑔𝑦 𝑖𝑛𝑝𝑢𝑡 𝐸𝑚𝑒𝑐ℎ,𝑜𝑢𝑡 𝐸𝑚𝑒𝑐ℎ,𝑖𝑛 =1− 𝑀𝑒𝑐ℎ𝑎𝑛𝑖𝑐𝑎𝑙 𝑒𝑛𝑒𝑟𝑔𝑦 𝑖𝑛𝑝𝑢𝑡 = 𝑀𝑒𝑐ℎ𝑎𝑛𝑖𝑐𝑎𝑙 𝑒𝑛𝑒𝑟𝑔𝑦 𝑑𝑒𝑐𝑟𝑒𝑎𝑠𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑓𝑙𝑢𝑖𝑑 𝑀𝑒𝑐ℎ𝑎𝑛𝑖𝑐𝑎𝑙 𝑝𝑜𝑤𝑒𝑟 𝑖𝑛𝑝𝑢𝑡 𝜂𝑔𝑒𝑛𝑒𝑟𝑎𝑡𝑜𝑟 = = 𝐸𝑙𝑒𝑐𝑡𝑟𝑖𝑐𝑎𝑙 𝑝𝑜𝑤𝑒𝑟 𝑜𝑢𝑡𝑝𝑢𝑡 𝑀𝑒𝑐ℎ𝑎𝑛𝑖𝑐𝑎𝑙 𝑝𝑜𝑤𝑒𝑟 𝑖𝑛𝑝𝑢𝑡 𝜂𝑝𝑢𝑚𝑝−𝑚𝑜𝑡𝑜𝑟 = 𝜂𝑝𝑢𝑚𝑝 𝜂𝑚𝑜𝑡𝑜𝑟 = 𝐸𝑚𝑒𝑐ℎ,𝑙𝑜𝑠𝑠 2-44 𝐸𝑚𝑒𝑐ℎ,𝑖𝑛 Δ𝐸̇𝑚𝑒𝑐ℎ,𝑓𝑙𝑢𝑖𝑑 𝑊̇𝑠ℎ𝑎𝑓𝑡,𝑖𝑛 = 𝑊̇𝑠ℎ𝑎𝑓𝑡,𝑜𝑢𝑡 𝑀𝑒𝑐ℎ𝑎𝑛𝑖𝑐𝑎𝑙 𝑒𝑛𝑒𝑟𝑔𝑦 𝑜𝑢𝑡𝑝𝑢𝑡 𝑀𝑒𝑐ℎ𝑎𝑛𝑖𝑐𝑎𝑙 𝑝𝑜𝑤𝑒𝑟 𝑜𝑢𝑡𝑝𝑢𝑡 2-43 𝐻𝐻𝑉×𝑚̇𝑓𝑢𝑒𝑙 𝑀𝑒𝑐ℎ𝑎𝑛𝑖𝑐𝑎𝑙 𝑒𝑛𝑒𝑟𝑔𝑦 𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑓𝑙𝑢𝑖𝑑 𝜂𝑡𝑢𝑟𝑏𝑖𝑛𝑒 = 𝜂𝑚𝑜𝑡𝑜𝑟 = = = |Δ𝐸̇ 𝑚𝑒𝑐ℎ,𝑓𝑙𝑢𝑖𝑑 | 𝑊̇𝑠ℎ𝑎𝑓𝑡,𝑜𝑢𝑡 𝑊̇𝑒𝑙𝑒𝑐𝑡,𝑖𝑛 = = 𝑊̇ 𝜂𝑡𝑢𝑟𝑏𝑖𝑛𝑒−𝑔𝑒𝑛 = 𝜂𝑡𝑢𝑟𝑏𝑖𝑛𝑒 𝜂𝑔𝑒𝑛𝑒𝑟𝑎𝑡𝑜𝑟 = 𝑊̇ 𝑒𝑙𝑒𝑐𝑡,𝑜𝑢𝑡 = |Δ𝐸̇ 𝑒𝑙𝑒𝑐𝑡,𝑜𝑢𝑡 𝑡𝑢𝑟𝑏𝑖𝑛𝑒,𝑒 = 𝑊̇𝑡𝑢𝑟𝑏𝑖𝑛𝑒 𝑊̇𝑡𝑢𝑟𝑏𝑖𝑛𝑒,𝑒 2-45 2-46 2-48 Δ𝐸̇𝑚𝑒𝑐ℎ,𝑓𝑙𝑢𝑖𝑑 𝑊̇𝑒𝑙𝑒𝑐𝑡,𝑖𝑛 𝑊̇ 𝑊̇𝑝𝑢𝑚𝑝,𝑢 𝑊̇𝑝𝑢𝑚𝑝 2-47 𝑊̇𝑒𝑙𝑒𝑐𝑡,𝑜𝑢𝑡 𝑊̇𝑠ℎ𝑎𝑓𝑡,𝑖𝑛 𝑊̇𝑝𝑢𝑚𝑝,𝑢 𝑊̇𝑒𝑙𝑒𝑐𝑡,𝑖𝑛 2-42 𝑚𝑒𝑐ℎ,𝑓𝑙𝑢𝑖𝑑 | 2-49 2-50 Mechanisms of Heat Transfer Δ𝑇 𝑄̇𝑐𝑜𝑛𝑑 = 𝑘𝐴 Δ𝑥 2-51 𝑑𝑇 𝑄̇𝑐𝑜𝑛𝑑 = −𝑘𝐴 2-52 𝑄̇𝑐𝑜𝑛𝑣 = ℎ𝐴(𝑇𝑠 − 𝑇𝑓 ) 2-53 𝑄̇𝑒𝑚𝑖𝑡,𝑚𝑎𝑥 = 𝜎𝐴𝑇𝑠4 2-54 𝑑𝑥 𝑄̇𝑒𝑚𝑖𝑡 = 𝜀𝜎𝐴𝑇𝑠4 2-55 𝑄̇𝑎𝑏𝑠 = 𝛼𝑄̇𝑖𝑛𝑐𝑖𝑑𝑒𝑛𝑡 2-56 4 ) 𝑄̇𝑟𝑎𝑑 = 𝜀𝜎𝐴(𝑇𝑠4 − 𝑇𝑠𝑢𝑟𝑟 2-57 3-5 Property Tables ℎ = 𝑢 + 𝑃𝑣 3-1 𝐻 = 𝑈 + 𝑃𝑉 3-2 𝑥= 𝑚𝑣𝑎𝑝𝑜𝑟 3-3 𝑚𝑡𝑜𝑡𝑎𝑙 𝑣𝑎𝑣𝑔 = 𝑣𝑓 + 𝑥𝑣𝑓𝑔 𝑥= 3-4 𝑣𝑎𝑣𝑔 −𝑣𝑓 3-5 𝑣𝑓𝑔 𝑢𝑎𝑣𝑔 = 𝑢𝑓 + 𝑥𝑢𝑓𝑔 3-6 ℎ𝑎𝑣𝑔 = ℎ𝑓 + 𝑥ℎ𝑓𝑔 3-7 𝑦 ≅ 𝑦𝑓@𝑇 3-8 ℎ ≅ ℎ𝑓@𝑇 + 𝑣𝑓@𝑇 (𝑃 − 𝑃𝑠𝑎𝑡@𝑇 ) 3-9 3-6 The Ideal-Gas Equation of State 𝑃𝑣 = 𝑅𝑇 3-10 kJ 8.31447 kmol∙K 8.31447 kPa∙m3 kmol∙K bar∙m3 0.0831447 𝑅𝑢 = kmol∙K Btu 1.98588 lbmol∙R 10.7316 { 1545.37 3-11 psia∙ft3 lbmol∙R ft∙lbf lbmol∙R 𝑚 = 𝑀𝑁 3-12 𝑉 = 𝑚𝑣 → 𝑃𝑉 = 𝑚𝑅𝑇 3-13 𝑚𝑅 = (𝑀𝑁)𝑅 = 𝑁𝑅𝑢 → 𝑃𝑉 = 𝑁𝑅𝑢 𝑇 3-14 𝑉 = 𝑁𝑣 → 𝑃𝑣 = 𝑅𝑢 𝑇 3-15 𝑃1 𝑉1 𝑇1 = 𝑃2 𝑉2 𝑇2 3-16 3-7 Compressibility Factor – A Measure of Deviation from Ideal-Gas Behavior 𝑍= 𝑃𝑣 3-17 𝑅𝑇 𝑃𝑣 = 𝑍𝑅𝑇 𝑍= 3-18 𝑣𝑎𝑐𝑡𝑢𝑎𝑙 3-19 𝑣𝑖𝑑𝑒𝑎𝑙 𝑃𝑅 = 𝑃 and 𝑇𝑅 = 𝑃𝑐𝑟 𝑇 3-20 𝑇𝑐𝑟 𝑣 𝑣𝑅 = 𝑅𝑇𝑎𝑐𝑡𝑢𝑎𝑙 ⁄𝑃 𝑐𝑟 3-21 𝑐𝑟 3-8 Other Equations of State (𝑃 + 𝑎= 𝑃= 𝑎 𝑣2 ) (𝑣 − 𝑏) = 𝑅𝑇 2 27𝑅 2𝑇𝑐𝑟 64𝑃𝑐𝑟 𝑅𝑢 𝑇 𝑣 2 𝑅𝑇𝑐𝑟 and 𝑏 = (1 − 𝑐 𝑣𝑇3 3-22 3-23 8𝑃𝑐𝑟 ) (𝑣 + 𝐵) − 𝐴 𝑣 3-24 2 𝑎 𝑏 𝐴 = 𝐴0 (1 − 𝑣 ) and 𝐵 = 𝐵0 (1 − 𝑣 ) 𝑃= 𝑃= 𝑅𝑢 𝑇 𝑣 𝑅𝑇 𝑣 𝐶 + (𝐵0 𝑅𝑢 𝑇 − 𝐴0 − 𝑇02 ) + 𝑎(𝑇) 𝑣2 + 𝑏(𝑇) 𝑣3 + 𝑐(𝑇) 𝑣4 + 1 𝑣 + 2 𝑑(𝑇) 𝑣5 3-25 𝑏𝑅𝑢 𝑇−𝑎 𝑣 3 + 𝑎𝛼 𝑣 6 + 𝑐 3 𝑣 𝑇2 (1 + 𝛾 𝑣 2 ) 𝑒 −𝛾 ⁄𝑣 2 +⋯ 3-26 3-27 4-1 Moving Boundary Work 𝛿𝑊𝑏 = 𝐹 𝑑𝑠 = 𝑃𝐴 𝑑𝑠 = 𝑃 𝑑𝑉 4-1 2 𝑊𝑏 = ∫1 𝑃 𝑑𝑉 4-2 2 2 𝐴𝑟𝑒𝑎 = 𝐴 = ∫1 𝑑𝐴 = ∫1 𝑃 𝑑𝑉 4-3 2 𝑊𝑏 = ∫1 𝑃𝑖 𝑑𝑉 4-4 2 𝑊𝑏 = 𝑊𝑓𝑟𝑖𝑐𝑡𝑖𝑜𝑛 + 𝑊𝑎𝑡𝑚 + 𝑊𝑐𝑟𝑎𝑛𝑘 = ∫1 (𝐹𝑓𝑟𝑖𝑐𝑡𝑖𝑜𝑛 + 𝑃𝑎𝑡𝑚 𝐴 + 𝐹𝑐𝑟𝑎𝑛𝑘 ) 𝑑𝑥 2 2 𝑊𝑏 = ∫1 𝑃 𝑑𝑉 = 𝑃0 ∫1 𝑑𝑉 = 𝑃0 (𝑉2 − 𝑉1 ) 2 2𝐶 𝑊𝑏 = ∫1 𝑃 𝑑𝑉 = ∫1 𝑃 = 𝐶𝑉 −𝑛 𝑉 2 𝑑𝑉 𝑑𝑉 = 𝐶 ∫1 𝑉 4-5 4-6 𝑉 𝑉 = 𝐶 ln 𝑉2 = 𝑃1 𝑉1 ln 𝑉2 1 1 4-7 4-8 2 2 𝑊𝑏 = ∫1 𝑃 𝑑𝑉 = ∫1 𝐶𝑉 −𝑛 𝑑𝑉 = 𝐶 𝑊𝑏 = 𝑚𝑅(𝑇2 −𝑇1 ) 1−𝑛 𝑉2−𝑛+1 −𝑉2−𝑛+1 −𝑛+1 = 𝑃2 𝑉2−𝑃1 𝑉1 1−𝑛 𝑛≠1 4-9 4-10 4-2 Energy Balance for Closed Systems 𝐸𝑖𝑛 − 𝐸𝑜𝑢𝑡 = Δ𝐸𝑠𝑦𝑠𝑡𝑒𝑚 4-11 𝑑𝐸𝑠𝑦𝑠𝑡𝑒𝑚 𝐸̇𝑖𝑛 − 𝐸̇𝑜𝑢𝑡 = 𝑑𝑡 4-12 𝑑𝐸 𝑄 = 𝑄̇Δ𝑡, 𝑊 = 𝑊̇ Δ𝑡, and Δ𝐸 = ( ) Δ𝑡 4-13 𝑒𝑖𝑛 − 𝑒𝑜𝑢𝑡 = Δ𝑒𝑠𝑦𝑠𝑡𝑒𝑚 4-14 𝛿𝐸𝑖𝑛 − 𝛿𝐸𝑜𝑢𝑡 = 𝑑𝐸𝑠𝑦𝑠𝑡𝑒𝑚 or 𝛿𝑒𝑖𝑛 − 𝛿𝑒𝑜𝑢𝑡 = 𝑑𝑒𝑠𝑦𝑠𝑡𝑒𝑚 4-15 𝑊𝑛𝑒𝑡,𝑜𝑢𝑡 = 𝑄𝑛𝑒𝑡,𝑖𝑛 or 𝑊̇𝑛𝑒𝑡,𝑜𝑢𝑡 = 𝑄̇𝑛𝑒𝑡,𝑖𝑛 4-16 𝑄𝑛𝑒𝑡,𝑖𝑛 − 𝑊𝑛𝑒𝑡,𝑜𝑢𝑡 = Δ𝐸𝑠𝑦𝑠𝑡𝑒𝑚 or 𝑄 − 𝑊 = Δ𝐸 4-17 𝑄 − 𝑊𝑜𝑡ℎ𝑒𝑟 = 𝐻2 − 𝐻1 4-18 𝑑𝑡 4-3 Specific Heats 𝜕𝑢 𝑐𝑣 = (𝜕𝑇 ) 4-19 𝑣 𝜕𝑢 𝑐𝑝 = (𝜕𝑇 ) 4-20 𝑝 4-4 Internal Energy, Enthalpy, and Specific Heats of Ideal Gases 𝑢 = 𝑢 (𝑇) 4-21 ℎ = ℎ(𝑇) 4-22 𝑑𝑢 = 𝑐𝑣 (𝑇) 𝑑𝑇 4-23 𝑑ℎ = 𝑐𝑝 (𝑇) 𝑑𝑇 4-24 2 Δ𝑢 = 𝑢2 − 𝑢1 = ∫1 𝑐𝑣 (𝑇) 𝑑𝑇 2 4-25 Δℎ = ℎ2 − ℎ1 = ∫1 𝑐𝑝 (𝑇) 𝑑𝑇 4-26 𝑢2 − 𝑢1 = 𝑐𝑣,𝑎𝑣𝑔 (𝑇2 − 𝑇1 ) 4-27 ℎ2 − ℎ1 = 𝑐𝑝,𝑎𝑣𝑔 (𝑇2 − 𝑇1 ) 4-28 𝑐𝑝 = 𝑐𝑣 + 𝑅 4-29 𝑐𝑝 = 𝑐𝑣 + 𝑅𝑢 𝑘= 4-30 𝑐𝑝 4-31 𝑐𝑣 4-5 Internal Energy, Enthalpy, and Specific Heats of Solids and Liquids 𝑐𝑝 = 𝑐𝑣 = 𝑐 4-32 𝑑𝑢 = 𝑐𝑣 𝑑𝑇 = 𝑐 (𝑇) 𝑑𝑇 4-33 2 Δ𝑢 = 𝑢2 − 𝑢1 = ∫1 𝑐 (𝑇) 𝑑𝑇 4-34 Δ𝑢 ≅ 𝑐𝑎𝑣𝑔 (𝑇2 − 𝑇1 ) 4-35 𝑑ℎ = 𝑑𝑢 + 𝑣 𝑑𝑃 + 𝑃 𝑑𝑣 = 𝑑𝑢 + 𝑣 𝑑𝑃 4-36 Δℎ = Δ𝑢 + 𝑣Δ𝑃 ≅ 𝑐𝑎𝑣𝑔 Δ𝑇 + 𝑣Δ𝑃 4-37 ℎ@𝑃,𝑇 ≅ ℎ𝑓@𝑇 + 𝑣𝑓@𝑇 (𝑃 − 𝑃𝑠𝑎𝑡@𝑇 ) 4-38 Thermodynamic Aspects of Biological Systems 𝑊(kg) 𝐵𝑀𝐼 = 𝐻 2(m2) 4-39 5-1 Conservation of Mass 𝐸 = 𝑚𝑐 2 5-1 𝛿𝑚̇ = 𝜌𝑉𝑛 𝑑𝐴𝑐 5-2 𝑚̇ = ∫𝐴 𝛿𝑚̇ = ∫𝐴 𝜌𝑉𝑛 𝑑𝐴𝑐 5-3 1 ∫ 𝑉 𝑑𝐴𝑐 𝐴𝑐 𝐴𝑐 𝑛 5-4 𝑐 𝑉𝑎𝑣𝑔 = 𝑐 𝑚̇ = 𝜌𝑉𝑎𝑣𝑔 𝐴𝑐 5-5 𝑉̇ = ∫𝐴 𝑉𝑛 𝑑𝐴𝑐 = 𝑉𝑎𝑣𝑔 𝐴𝑐 = 𝑉𝐴𝑐 5-6 𝑉̇ 𝑚̇ = 𝜌𝑉̇ = 𝑣 5-7 𝑚𝑖𝑛 − 𝑚𝑜𝑢𝑡 = Δ𝑚𝐶𝑉 5-8 𝑐 𝑚̇𝑖𝑛 − 𝑚̇𝑜𝑢𝑡 = 𝑑𝑚𝐶𝑉 𝑑𝑡 𝑚𝐶𝑉 = ∫𝐶𝑉 𝜌 𝑑𝑉 𝑑𝑚𝐶𝑉 𝑑𝑡 𝑑 = 𝑑𝑡 ∫𝐶𝑉 𝜌 𝑑𝑉 5-9 5-10 5-11 ⃗ ∙ 𝑛⃗ 𝑉𝑛 = 𝑉 cos 𝜃 = 𝑉 5-12 ⃗ ∙ 𝑛⃗) 𝑑𝐴 𝛿𝑚̇ = 𝜌𝑉𝑛 𝑑𝐴 = 𝜌(𝑉 cos 𝜃) 𝑑𝐴 = 𝜌(𝑉 5-13 ⃗ ∙ 𝑛⃗) 𝑑𝐴 𝑚̇𝑛𝑒𝑡 = ∫𝐶𝑆 𝛿𝑚̇ = ∫𝐶𝑆 𝜌𝑉𝑛 𝑑𝐴 = ∫𝐶𝑆 𝜌(𝑉 5-14 𝑑 ∫ 𝜌 𝑑𝑉 𝑑𝑡 𝐶𝑉 ⃗ ∙ 𝑛⃗ ) 𝑑𝐴 = 0 + ∫𝐶𝑆 𝜌(𝑉 5-15 𝑑 ∫ 𝜌 𝑑𝑉 𝑑𝑡 𝐶𝑉 + ∑𝑜𝑢𝑡 𝜌|𝑉𝑛 |𝐴 − ∑𝑖𝑛 𝜌|𝑉𝑛 |𝐴 = 0 5-16 𝑑 ∫ 𝜌 𝑑𝑉 𝑑𝑡 𝐶𝑉 + ∑𝑖𝑛 𝑚̇ − ∑𝑜𝑢𝑡 𝑚̇ = 0 or 𝑑𝑚𝐶𝑉 𝑑𝑡 = ∑𝑖𝑛 𝑚̇ − ∑𝑜𝑢𝑡 𝑚̇ 5-17 ∑𝑖𝑛 𝑚̇ = ∑𝑜𝑢𝑡 𝑚̇ 5-18 𝑚̇1 = 𝑚̇2 → 𝜌1 𝑉1 𝐴1 = 𝜌2 𝑉2 𝐴2 5-19 ∑𝑖𝑛 𝑉̇ = ∑𝑜𝑢𝑡 𝑉̇ 5-20 𝑉̇1 = 𝑉̇2 → 𝑉1 𝐴1 = 𝑉2 𝐴2 5-21 5-2 Flow Work and The Energy of a Flowing Fluid 𝐹 = 𝑃𝐴 5-22 𝑊𝑓𝑙𝑜𝑤 = 𝐹𝐿 = 𝑃𝐴𝐿 = 𝑃𝑉 5-23 𝑤𝑓𝑙𝑜𝑤 = 𝑃𝑣 5-24 𝑒 = 𝑢 + 𝑘𝑒 + 𝑝𝑒 = 𝑢 + 𝑉2 2 + 𝑔𝑧 5-25 𝜃 = 𝑃𝑣 + 𝑒 = 𝑃𝑣 + (𝑢 + 𝑘𝑒 + 𝑝𝑒) 𝜃 = ℎ + 𝑘𝑒 + 𝑝𝑒 = ℎ + 𝑉2 5-26 + 𝑔𝑧 5-27 + 𝑔𝑧) 5-28 𝑉 𝐸̇𝑚𝑎𝑠𝑠 = 𝑚̇ 𝜃 = 𝑚̇ (ℎ + 2 + 𝑔𝑧) 5-29 𝐸𝑚𝑎𝑠𝑠 = 𝑚𝜃 = 𝑚 (ℎ + 2 𝑉2 2 2 𝐸𝑚𝑎𝑠𝑠,𝑖𝑛 = ∫𝑚 𝜃𝑖 𝛿𝑚𝑖 = ∫𝑚 (ℎ𝑖 + 𝑖 𝑖 𝑉𝑖2 2 + 𝑔𝑧𝑖 ) 𝜎𝑚𝑖 5-30 5-3 Energy Analysis of Steady-Flow Systems ∑𝑖𝑛 𝑚̇ = ∑𝑜𝑢𝑡 𝑚̇ 5-31 𝑚̇1 = 𝑚̇2 → 𝜌1 𝑉1 𝐴1 = 𝜌2 𝑉2 𝐴2 5-32 𝑑𝐸𝑠𝑦𝑠𝑡𝑒𝑚 𝐸̇𝑖𝑛 − 𝐸̇𝑜𝑢𝑡 = 𝑑𝑡 = 0 5-33 𝐸̇𝑖𝑛 = 𝐸̇𝑜𝑢𝑡 5-34 𝑄̇𝑖𝑛 + 𝑊̇𝑖𝑛 + ∑𝑖𝑛 𝑚̇ 𝜃 = 𝑄̇𝑜𝑢𝑡 + 𝑊̇𝑜𝑢𝑡 + ∑𝑜𝑢𝑡 𝑚̇ 𝜃 5-35 2 𝑉 𝑄̇𝑖𝑛 + 𝑊̇𝑖𝑛 + ∑𝑖𝑛 𝑚̇ (ℎ + 2 + 𝑔𝑧) 𝑝𝑒𝑟 𝑖𝑛𝑙𝑒𝑡 2 𝑉 𝑄̇ − 𝑊̇ = ∑𝑜𝑢𝑡 𝑚̇ (ℎ + 2 + 𝑔𝑧) 2 𝑝𝑒𝑟 𝑒𝑥𝑖𝑡 = 𝑄̇𝑜𝑢𝑡 + 𝑊̇𝑜𝑢𝑡 + ∑𝑜𝑢𝑡 𝑚̇ (ℎ + − ∑𝑖𝑛 𝑚̇ (ℎ + 𝑉2 2 + 𝑔𝑧) 𝑉2 2 + 𝑔𝑧) 𝑝𝑒𝑟 𝑖𝑛𝑙𝑒𝑡 2 𝑉 −𝑉 𝑄̇ − 𝑊̇ = 𝑚̇ [ℎ2 − ℎ1 + 2 1 + 𝑔(𝑧2 − 𝑧1 )] 2 𝑞 − 𝑤 = ℎ2 − ℎ1 + 𝑉22−𝑉12 2 + 𝑔(𝑧2 − 𝑧1 ) 𝑞 − 𝑤 = ℎ2 − ℎ1 𝑝𝑒𝑟 𝑒𝑥𝑖𝑡 5-36 5-37 5-38 5-39 5-40 5-4 Some Steady-Flow Engineering Devices ℎ2 ≅ ℎ1 5-41 5-5 Energy Analysis of Unsteady-Flow Processes 𝑚𝑖𝑛 − 𝑚𝑜𝑢𝑡 = Δ𝑚𝑠𝑦𝑠𝑡𝑒𝑚 5-42 𝑚𝑖 − 𝑚𝑒 = (𝑚2 − 𝑚1 )𝐶𝑉 5-43 𝐸𝑖𝑛 − 𝐸𝑜𝑢𝑡 = Δ𝐸𝑠𝑦𝑠𝑡𝑒𝑚 5-44 (𝑄𝑖𝑛 + 𝑊𝑖𝑛 + ∑𝑖𝑛 𝑚𝜃) − (𝑄𝑜𝑢𝑡 + 𝑊𝑜𝑢𝑡 + ∑𝑜𝑢𝑡 𝑚𝜃) = (𝑚2 𝑒2 − 𝑚1 𝑒1 )𝑠𝑦𝑠𝑡𝑒𝑚 5-45 𝑄 − 𝑊 = ∑𝑜𝑢𝑡 𝑚ℎ − ∑𝑖𝑛 𝑚ℎ + (𝑚2 𝑢2 − 𝑚1 𝑢1 )𝑠𝑦𝑠𝑡𝑒𝑚 5-46 General Energy Equation 𝑑𝐸𝑠𝑦𝑠 𝑄̇ − 𝑊̇ = or 𝑄̇ − 𝑊̇ = 𝑑𝑡 𝑒 = 𝑢 + 𝑘𝑒 + 𝑝𝑒 = 𝑢 + 𝑉2 2 𝑑 ∫ 𝜌𝑒 𝑑𝑉 𝑑𝑡 𝑠𝑦𝑠 + 𝑔𝑧 5-47 5-48 𝑊𝑡𝑜𝑡𝑎𝑙 = 𝑊𝑠ℎ𝑖𝑓𝑡 + 𝑊𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 + 𝑊𝑣𝑖𝑠𝑐𝑜𝑢𝑠 + 𝑊𝑜𝑡ℎ𝑒𝑟 5-49 ⃗ ∙ 𝑛⃗) 𝛿𝑊̇𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 = 𝑃 𝑑𝐴 𝑉𝑛 = 𝑃 𝑑𝐴 (𝑉 5-50 ⃗ ∙ 𝑛⃗) 𝑑𝐴 = ∫ 𝑃 𝜌(𝑉 ⃗ ∙ 𝑛⃗) 𝑑𝐴 𝑊̇𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒,𝑜𝑢𝑡 = ∫𝐴 𝑃(𝑉 𝐴𝜌 5-51 𝑃 ⃗ ∙ 𝑛⃗) 𝑑𝐴 𝑊̇𝑛𝑒𝑡,𝑜𝑢𝑡 = 𝑊̇𝑠ℎ𝑎𝑓𝑡,𝑛𝑒𝑡 𝑜𝑢𝑡 + 𝑊̇𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒,𝑛𝑒𝑡 𝑜𝑢𝑡 = 𝑊̇𝑠ℎ𝑎𝑓𝑡,𝑛𝑒𝑡 𝑜𝑢𝑡 + ∫𝐴 𝜌 𝜌(𝑉 5-52 𝑑𝐸𝑠𝑦𝑠 𝑄̇𝑛𝑒𝑡,𝑖𝑛 − 𝑊̇𝑠ℎ𝑎𝑓𝑡,𝑛𝑒𝑡 𝑜𝑢𝑡 − 𝑊̇𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒,𝑛𝑒𝑡 𝑜𝑢𝑡 = 𝑑𝑡 5-53 𝑑𝐸𝑠𝑦𝑠 𝑑𝑡 𝑑 ⃗ ∙ 𝑛⃗) 𝑑𝐴 = 𝑑𝑡 ∫𝐶𝑉 𝑒𝜌 𝑑𝑉 + ∫𝐶𝑆 𝑒𝜌(𝑉 5-54 𝑑 ⃗⃗⃗𝑟 ∙ 𝑛⃗) 𝑑𝐴 𝑄̇𝑛𝑒𝑡,𝑖𝑛 − 𝑊̇𝑠ℎ𝑎𝑓𝑡,𝑛𝑒𝑡 𝑜𝑢𝑡 − 𝑊̇𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒,𝑛𝑒𝑡 𝑜𝑢𝑡 = 𝑑𝑡 ∫𝐶𝑉 𝑒𝜌 𝑑𝑉 + ∫𝐶𝑆 𝑒𝜌(𝑉 5-55 𝑑 𝑃 ⃗⃗⃗𝑟 ∙ 𝑛⃗) 𝑑𝐴 𝑄̇𝑛𝑒𝑡,𝑖𝑛 − 𝑊̇𝑠ℎ𝑎𝑓𝑡,𝑛𝑒𝑡 𝑜𝑢𝑡 = 𝑑𝑡 ∫𝐶𝑉 𝑒𝜌 𝑑𝑉 + ∫𝐶𝑆 (𝜌 + 𝑒) 𝜌(𝑉 5-56 𝑑 𝑃 𝑃 𝑄̇𝑛𝑒𝑡,𝑖𝑛 − 𝑊̇𝑠ℎ𝑎𝑓𝑡,𝑛𝑒𝑡 𝑜𝑢𝑡 = 𝑑𝑡 ∫𝐶𝑉 𝑒𝜌 𝑑𝑉 + ∑𝑜𝑢𝑡 𝑚̇ (𝜌 + 𝑒) − ∑𝑖𝑛 𝑚̇ (𝜌 + 𝑒) 5-57 2 2 𝑑 𝑃 𝑉 𝑃 𝑉 𝑄̇𝑛𝑒𝑡,𝑖𝑛 − 𝑊̇𝑠ℎ𝑎𝑓𝑡,𝑛𝑒𝑡 𝑜𝑢𝑡 = 𝑑𝑡 ∫𝐶𝑉 𝑒𝜌 𝑑𝑉 + ∑𝑜𝑢𝑡 𝑚̇ (𝜌 + 𝑢 + 2 + 𝑔𝑧) − ∑𝑖𝑛 𝑚̇ (𝜌 + 𝑢 + 2 + 𝑔𝑧) 5-58 2 2 𝑑 𝑉 𝑉 𝑄̇𝑛𝑒𝑡,𝑖𝑛 − 𝑊̇𝑠ℎ𝑎𝑓𝑡,𝑛𝑒𝑡 𝑜𝑢𝑡 = 𝑑𝑡 ∫𝐶𝑉 𝑒𝜌 𝑑𝑉 + ∑𝑜𝑢𝑡 𝑚̇ (ℎ + 2 + 𝑔𝑧) − ∑𝑖𝑛 𝑚̇ (ℎ + 2 + 𝑔𝑧) 5-59 6-3 Heat Engines 𝑊𝑛𝑒𝑡,𝑜𝑢𝑡 = 𝑊𝑜𝑢𝑡 − 𝑊𝑖𝑛 6-1 𝑊𝑛𝑒𝑡,𝑜𝑢𝑡 = 𝑄𝑖𝑛 − 𝑄𝑜𝑢𝑡 6-2 𝑇ℎ𝑒𝑟𝑚𝑎𝑙 𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑦 = 𝜂𝑡ℎ = 𝑇𝑜𝑡𝑎𝑙 ℎ𝑒𝑎𝑡 𝑖𝑛𝑝𝑢𝑡 𝑊𝑛𝑒𝑡,𝑜𝑢𝑡 𝑄𝑜𝑢𝑡 6-5 𝑄𝑖𝑛 𝑊𝑛𝑒𝑡,𝑜𝑢𝑡 𝑄𝐻 6-3 6-4 𝑄𝑖𝑛 𝜂𝑡ℎ = 1 − 𝜂𝑡ℎ = 𝑁𝑒𝑡 𝑤𝑜𝑟𝑘 𝑜𝑢𝑡𝑝𝑢𝑡 𝑄 or 𝜂𝑡ℎ = 1 − 𝑄 𝐿 𝐻 6-6 6-4 Refrigerators and Heat Pumps 𝐶𝑂𝑃𝑅 = 𝐷𝑒𝑠𝑖𝑟𝑒𝑑 𝑜𝑢𝑡𝑝𝑢𝑡 𝑅𝑒𝑞𝑢𝑖𝑟𝑒𝑑 𝑖𝑛𝑝𝑢𝑡 = 𝑄𝐿 𝑊𝑛𝑒𝑡,𝑖𝑛 𝑊𝑛𝑒𝑡,𝑖𝑛 = 𝑄𝐻 − 𝑄𝐿 𝐶𝑂𝑃𝑅 = 𝑄 𝑄𝐿 𝐻 −𝑄𝐿 𝐶𝑂𝑃𝐻𝑃 = 6-8 1 ⁄ 𝐻 𝑄𝐿 −1 =𝑄 𝐷𝑒𝑠𝑖𝑟𝑒𝑑 𝑜𝑢𝑡𝑝𝑢𝑡 𝑅𝑒𝑞𝑖𝑟𝑒𝑑 𝑖𝑛𝑝𝑢𝑡 𝐶𝑂𝑃𝐻𝑃 = 𝑄 𝑄𝐻 𝐻 −𝑄𝐿 6-7 6-9 =𝑊 𝑄𝐻 𝑛𝑒𝑡,𝑖𝑛 1 = 1−𝑄 𝐿 ⁄𝑄𝐻 𝐶𝑂𝑃𝐻𝑃 = 𝐶𝑂𝑃𝑅 + 1 6-10 6-11 6-12 6-9 The Thermodynamic Temperature Scale 𝑄𝐻 𝑄𝐿 = 𝑓(𝑇𝐻 , 𝑇𝐿 ) 6-13 𝑄1 𝑄2 𝑄𝐻 𝑄𝐿 ( 𝜙(𝑇 ) = 𝑓(𝑇1 , 𝑇3 ) = 𝜙(𝑇1 ) 6-14 3 = 𝑄𝐻 𝜙(𝑇𝐻 ) 𝜙(𝑇𝐿 ) ) 𝑄𝐿 𝑟𝑒𝑣 = 6-15 𝑇𝐻 6-16 𝑇𝐿 𝑇(℃) = 𝑇 (𝐾) − 273.15 6-17 6-10 The Carnot Heat Engine 𝜂𝑡ℎ,𝑟𝑒𝑣 = 1 − 𝑇𝐿 6-18 𝑇𝐻 < 𝜂𝑡ℎ,𝑟𝑒𝑣 irreversable heat engine 𝜂𝑡ℎ { = 𝜂𝑡ℎ,𝑟𝑒𝑣 reversable heat engine > 𝜂𝑡ℎ,𝑟𝑒𝑣 impossible heat engine 6-19 6-11 The Carnot Refrigerator and Heat Pump 𝐶𝑂𝑃𝑅,𝑟𝑒𝑣 = 1 6-20 𝑇𝐻 ⁄𝑇𝐿 −1 𝐶𝑂𝑃𝐻𝑃,𝑟𝑒𝑣 = 1 6-21 1−𝑇𝐿 ⁄𝑇𝐻 < 𝐶𝑂𝑃𝑅,𝑟𝑒𝑣 irreversable refrigerator 𝐶𝑂𝑃𝑅 { = 𝐶𝑂𝑃𝑅,𝑟𝑒𝑣 reversable refrigerator > 𝐶𝑂𝑃𝑅,𝑟𝑒𝑣 impossible refrigerator 6-22 7-1 Entropy ∮ 𝛿𝑄 𝑇 ≤0 7-1 𝛿𝑄 ∮( 𝑇 ) 𝑖𝑛𝑡 𝑟𝑒𝑣 =0 7-2 ∮ 𝑑𝑉 = 0 7-3 𝛿𝑄 𝑑𝑆 = ( 𝑇 ) 7-4 𝑖𝑛𝑡 𝑟𝑒𝑣 2 𝛿𝑄 Δ𝑆 = 𝑆2 − 𝑆1 = ∫1 ( ) 𝑇 𝑖𝑛𝑡 𝑟𝑒𝑣 𝑄 Δ𝑆 = 𝑇 7-5 7-6 0 7-2 The Increase of Entropy Principle 2 𝛿𝑄 𝑆2 − 𝑆1 ≥ ∫1 𝑇 7-7 𝑑𝑆 ≥ 𝛿𝑄 7-8 𝑇 2 𝛿𝑄 Δ𝑆𝑠𝑦𝑠 = 𝑆2 − 𝑆1 = ∫1 𝑇 + 𝑆𝑔𝑒𝑛 7-9 Δ𝑆𝑖𝑠𝑜𝑙𝑎𝑡𝑒𝑑 ≥ 0 7-10 𝑆𝑔𝑒𝑛 = Δ𝑆𝑡𝑜𝑡𝑎𝑙 = Δ𝑆𝑠𝑦𝑠 + Δ𝑆𝑠𝑢𝑟𝑟 ≥ 0 7-11 7-3 Entropy Change of Pure Substances Δ𝑆 = 𝑚Δ𝑠 = 𝑚(𝑠2 − 𝑠1 ) 7-12 7-4 Isentropic Processes Δ𝑠 = 0 or 𝑠2 = 𝑠1 7-13 7-5 Property Diagrams Involving Entropy 𝛿𝑄𝑖𝑛𝑡 𝑟𝑒𝑣 = 𝑇 𝑑𝑆 2 7-14 𝑄𝑖𝑛𝑡 𝑟𝑒𝑣 = ∫1 𝑇 𝑑𝑆 7-15 𝛿𝑞𝑖𝑛𝑡 𝑟𝑒𝑣 = 𝑇 𝑑𝑠 7-16 2 𝑞𝑖𝑛𝑡 𝑟𝑒𝑣 = ∫1 𝑇 𝑑𝑠 7-17 𝑄𝑖𝑛𝑡 𝑟𝑒𝑣 = 𝑇0 Δ𝑆 7-18 𝑞𝑖𝑛𝑡 𝑟𝑒𝑣 = 𝑇0 Δ𝑠 7-19 7-6 What is Entropy? 𝑆 = 𝑘 ln 𝑊 7-20a 𝑆 = −𝑘 ∑ 𝑝𝑖 log 𝑝𝑖 7-20b 7-7 The 𝑻 𝒅𝒔 Relations 𝛿𝑄𝑖𝑛𝑡 𝑟𝑒𝑣 − 𝛿𝑊𝑖𝑛𝑡 𝑟𝑒𝑣,𝑜𝑢𝑡 = 𝑑𝑈 7-21 𝑇 𝑑𝑆 = 𝑑𝑈 + 𝑃 𝑑𝑉 7-22 𝑇 𝑑𝑠 = 𝑑𝑢 + 𝑃 𝑑𝑣 7-23 ℎ = 𝑢 + 𝑃𝑣 → 𝑑ℎ = 𝑑𝑢 + 𝑃 𝑑𝑣 + 𝑣 𝑑𝑃 } 𝑇 𝑑𝑠 = 𝑑ℎ − 𝑣 𝑑𝑃 𝑇 𝑑𝑠 = 𝑑𝑢 + 𝑃 𝑑𝑣 𝑑𝑠 = 𝑑𝑠 = 𝑑𝑢 𝑇 𝑑ℎ 𝑇 + − 𝑃 𝑑𝑣 𝑇 𝑣 𝑑𝑃 𝑇 7-24 7-25 7-26 7-8 Entropy Change of Liquids and Solids 𝑑𝑠 = 𝑑𝑢 = 𝑇 𝑐 𝑑𝑇 7-27 𝑇 2 𝑠2 − 𝑠1 = ∫1 𝑐(𝑇) 𝑑𝑇 𝑇 ≅ 𝑐𝑎𝑣𝑔 ln 𝑇2 𝑇1 𝑇 𝑠2 − 𝑠1 = 𝑐𝑎𝑣𝑔 ln 𝑇2 = 0 → 𝑇2 = 𝑇1 1 7-28 7-29 7-9 The Entropy Change of Ideal Gases 𝑑𝑠 = 𝑐𝑣 𝑑𝑇 𝑇 +𝑅 𝑑𝑣 7-30 𝑣 2 𝑑𝑇 𝑠2 − 𝑠1 = ∫1 𝑐𝑣 (𝑇) 𝑇 2 𝑑𝑇 𝑠2 − 𝑠1 = ∫1 𝑐𝑝 (𝑇) 𝑇 + 𝑅 ln 𝑣2 𝑣1 𝑃 − 𝑅 ln 𝑃2 1 𝑇 𝑣 𝑠2 − 𝑠1 = 𝑐𝑣,𝑎𝑣𝑔 ln 𝑇2 + 𝑅 ln 𝑣2 1 1 𝑇 𝑃 𝑠2 − 𝑠1 = 𝑐𝑝,𝑎𝑣𝑔 ln 𝑇2 − 𝑅 ln 𝑃2 1 1 𝑇 𝑣 𝑠2 − 𝑠1 = 𝑐𝑣,𝑎𝑣𝑔 ln 𝑇2 + 𝑅 ln 𝑣2 1 1 𝑇 𝑃 𝑠2 − 𝑠1 = 𝑐𝑝,𝑎𝑣𝑔 ln 𝑇2 − 𝑅 ln 𝑃2 1 𝑇 𝑠° = ∫0 𝑐𝑝 (𝑇) 2 ∫1 𝑐𝑝 (𝑇) 𝑑𝑇 1 𝑑𝑇 𝑃 𝑃 𝑜 𝑠2 − 𝑠1 = 𝑠2 − 𝑠1 − 𝑅𝑢 ln 𝑃2 𝑇1 𝑣 = ln ( 1) 𝑅 ⁄𝑐𝑣 𝑣 𝑘−1 ( 2) = (𝑣1) ( 2) 𝑇 = (𝑃2 ) 𝑃 = ( 1) 𝑇1 𝑠=𝑐𝑜𝑛𝑠𝑡. 𝑇1 𝑠=𝑐𝑜𝑛𝑠𝑡. ( 2) 𝑃1 𝑠=𝑐𝑜𝑛𝑠𝑡. 7-36 7-41 𝑣2 𝑇 7-35 7-40 1 𝑇2 7-34 7-39 1 ln 7-33 7-38 𝑠2 − 𝑠1 = 𝑠2𝑜 − 𝑠1𝑜 − 𝑅 ln 𝑃2 𝑜 7-32 7-37 𝑇 = 𝑠2𝑜 − 𝑠1𝑜 𝑇 7-31 (ideal gas) 2 𝑃 (𝑘−1) ⁄𝑘 (ideal gas) 1 𝑣 7-42 7-43 𝑘 𝑣2 𝑇𝑣 𝑘−1 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 (ideal gas) 7-44 7-45 𝑇𝑃 1−𝑘 𝑘 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 7-46 𝑃𝑣 𝑘 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 7-47 𝑃 𝑠2𝑜 − 𝑠1𝑜 + 𝑅 ln 𝑃2 7-48 1 𝑃 𝑃 ( 2) = 𝑃𝑟2 𝑣 = 𝑣𝑟2 𝑃1 𝑠=𝑐𝑜𝑛𝑠𝑡. 7-49 𝑟1 𝑣 ( 2) 𝑣1 𝑠=𝑐𝑜𝑛𝑠𝑡. 7-50 𝑟1 7-10 Reversible Steady-Flow Work 2 𝑤𝑟𝑒𝑣 = − ∫1 𝑣 𝑑𝑃 − Δ𝑘𝑒 − Δ𝑝𝑒 7-51 2 𝑤𝑟𝑒𝑣 = − ∫1 𝑣 𝑑𝑃 7-52 2 𝑤𝑟𝑒𝑣,𝑖𝑛 = ∫1 𝑣 𝑑𝑃 + Δ𝑘𝑒 + Δ𝑝𝑒 7-53 𝑤𝑟𝑒𝑣 = −𝑣(𝑃2 − 𝑃1 ) − Δ𝑘𝑒 − Δ𝑝𝑒 7-54 𝑣(𝑃2 − 𝑃1 ) + V22 −V21 2 + g(z2 − z1 ) = 0 7-55 7-11 Minimizing the Compressor Work 2 𝑤𝑟𝑒𝑣,𝑖𝑛 = ∫1 𝑣 𝑑𝑃 𝑤𝑐𝑜𝑚𝑝,𝑖𝑛 = 𝑤𝑐𝑜𝑚𝑝,𝑖𝑛 = 7-56 𝑘𝑅(𝑇2 −𝑇1 ) 𝑘−1 𝑛𝑅(𝑇2 −𝑇1 ) 𝑛−1 = = 𝑘𝑅𝑇1 𝑘−1 𝑛𝑅𝑇1 𝑛−1 𝑃 [( 2 ) (𝑘−1)⁄𝑘 𝑃1 𝑃 [( 2 ) (𝑛−1)⁄𝑛 𝑃1 − 1] 7-57a − 1] 7-57b 𝑃 𝑤𝑐𝑜𝑚𝑝,𝑖𝑛 = 𝑅𝑇 ln 𝑃2 7-57c 1 𝑤𝑐𝑜𝑚𝑝,𝑖𝑛 = 𝑤𝑐𝑜𝑚𝑝 𝐼,𝑖𝑛 + 𝑤𝑐𝑜𝑚𝑝 𝐼𝐼,𝑖𝑛 = 𝑃𝑥 = (𝑃1 𝑃2 )1⁄2 or 𝑃𝑥 𝑃1 = 𝑛𝑅𝑇1 𝑛−1 𝑃 [( 𝑥 ) (𝑛−1) ⁄𝑛 𝑃1 𝑃2 − 1] + 𝑛𝑅𝑇1 𝑛−1 𝑃 [( 2 ) 𝑃𝑥 (𝑛−1)⁄𝑛 − 1] 7-58 7-59 𝑃𝑥 7-12 Isentropic Efficiencies of Steady-Flow Devices 𝐴𝑐𝑡𝑢𝑎𝑙 𝑡𝑢𝑟𝑏𝑖𝑛𝑒 𝑤𝑜𝑟𝑘 𝜂 𝑇 = 𝐼𝑠𝑒𝑛𝑡𝑟𝑜𝑝𝑖𝑐 𝑡𝑢𝑟𝑏𝑖𝑛𝑒 𝑤𝑜𝑟𝑘 = 𝜂𝑇 ≅ ℎ1−ℎ2𝑎 ℎ1−ℎ2𝑠 𝑤𝑎 𝑤𝑠 7-60 7-61 𝜂𝐶 = 𝐼𝑠𝑒𝑛𝑡𝑟𝑜𝑝𝑖𝑐 𝑐𝑜𝑚𝑝𝑟𝑒𝑠𝑠𝑜𝑟 𝑤𝑜𝑟𝑘 𝐴𝑐𝑡𝑢𝑎𝑙 𝑐𝑜𝑚𝑝𝑟𝑒𝑠𝑠𝑜𝑟 𝑤𝑜𝑟𝑘 𝑤 = 𝑤𝑠 7-62 𝑎 ℎ −ℎ 𝜂𝐶 ≅ ℎ 2𝑠 −ℎ1 2𝑎 7-63 1 𝑤 𝜂𝑃 = 𝑤𝑠 = 𝑣(𝑃2 −𝑃1 ) 7-64 ℎ2𝑎−ℎ1 𝑎 𝑤 𝜂𝐶 = 𝑤 𝑡 7-65 𝑎 𝐴𝑐𝑡𝑢𝑎𝑙 𝐾𝐸 𝑎𝑡 𝑛𝑜𝑧𝑧𝑙𝑒 𝑒𝑥𝑖𝑡 𝜂𝑁 = 𝐼𝑠𝑒𝑛𝑡𝑟𝑜𝑝𝑖𝑐 𝐾𝐸 𝑎𝑡 𝑛𝑜𝑧𝑧𝑙𝑒 𝑒𝑥𝑖𝑡 = 𝜂𝑁 ≅ 2 𝑉2𝑎 2 𝑉2𝑠 ℎ1−ℎ2𝑎 7-66 7-67 ℎ1−ℎ2𝑠 7-13 Entropy Balance 𝑆𝑖𝑛 − 𝑆𝑜𝑢𝑡 + 𝑆𝑔𝑒𝑛 = Δ𝑆𝑠𝑦𝑠𝑡𝑒𝑚 7-68 Δ𝑆𝑠𝑦𝑠𝑡𝑒𝑚 = 𝑆𝑓𝑖𝑛𝑎𝑙 − 𝑆𝑖𝑛𝑖𝑡𝑖𝑎𝑙 = 𝑆2 − 𝑆1 7-69 𝑆𝑠𝑦𝑠𝑡𝑒𝑚 = ∫ 𝑠 𝛿𝑚 = ∫𝑉 𝑠𝜌 𝑑𝑉 7-70 𝑆ℎ𝑒𝑎𝑡 = 𝑄 (T=constant) 𝑇 2 𝛿𝑄 𝑆ℎ𝑒𝑎𝑡 = ∫1 𝑇 7-71 𝑄 ≅ ∑ 𝑇𝑘 7-72 𝑘 𝑆𝑤𝑜𝑟𝑘 = 0 7-73 𝑆𝑚𝑎𝑠𝑠 = 𝑚𝑠 7-74 ̇ ̇ 𝑆𝑚𝑎𝑠𝑠 = ∫𝐴 𝑠𝜌𝑉𝑛 𝑑𝐴𝑐 and 𝑆𝑚𝑎𝑠𝑠 = ∫ 𝑠 𝛿𝑚 = ∫Δ𝑡 𝑆𝑚𝑎𝑠𝑠 𝑑𝑡 7-75 𝑆𝑖𝑛 − 𝑆𝑜𝑢𝑡 + 𝑆𝑔𝑒𝑛 = Δ𝑆𝑠𝑦𝑠𝑡𝑒𝑚 7-76 𝑑𝑆𝑠𝑦𝑠𝑡𝑒𝑚 ̇ − 𝑆𝑜𝑢𝑡 ̇ + 𝑆𝑔𝑒𝑛 ̇ 𝑆𝑖𝑛 = 𝑑𝑡 7-77 (𝑠𝑖𝑛 − 𝑠𝑜𝑢𝑡 ) + 𝑠𝑔𝑒𝑛 = Δ𝑠𝑠𝑦𝑠𝑡𝑒𝑚 7-78 𝑐 ∑ 𝑄𝑘 𝑇𝑘 + 𝑆𝑔𝑒𝑛 = Δ𝑆𝑠𝑦𝑠𝑡𝑒𝑚 = 𝑆2 − 𝑆1 7-79 𝑆𝑔𝑒𝑛 = Δ𝑆𝑎𝑑𝑖𝑎𝑏𝑎𝑡𝑖𝑐 𝑠𝑦𝑠𝑡𝑒𝑚 7-80 𝑆𝑔𝑒𝑛 = ∑ Δ𝑆 = Δ𝑆𝑠𝑦𝑠𝑡𝑒𝑚 + Δ𝑆𝑠𝑢𝑟𝑟𝑜𝑢𝑛𝑑𝑖𝑛𝑔𝑠 7-81 ∑ ∑ 𝑄𝑘 𝑇𝑘 𝑄̇𝑘 𝑇𝑘 + ∑ 𝑚𝑖 𝑠𝑖 − ∑ 𝑚𝑒 𝑠𝑒 + 𝑆𝑔𝑒𝑛 = (𝑆2 − 𝑆1 )𝐶𝑉 ̇ + ∑ 𝑚̇𝑖 𝑠𝑖 − ∑ 𝑚̇𝑒 𝑠𝑒 + 𝑆𝑔𝑒𝑛 = 𝑑𝑆𝐶𝑉 𝑑𝑡 7-82 7-83 ̇ 𝑄 ̇ 𝑆𝑔𝑒𝑛 = ∑ 𝑚̇𝑒 𝑠𝑒 − ∑ 𝑚̇𝑖 𝑠𝑖 − ∑ 𝑇𝑘 7-84 𝑘 ̇ 𝑄 ̇ 𝑆𝑔𝑒𝑛 = 𝑚̇ (𝑠𝑒 − 𝑠𝑖 ) − ∑ 𝑘 7-85 ̇ 𝑆𝑔𝑒𝑛 = 𝑚̇ (𝑠𝑒 − 𝑠𝑖 ) 7-86 𝑇𝑘 Reducing the Cost of Compressed Air 𝐸𝑛𝑒𝑟𝑔𝑦 𝑠𝑎𝑣𝑖𝑛𝑔𝑠 = (𝑃𝑜𝑤𝑒𝑟 𝑠𝑎𝑣𝑒𝑑)(𝑂𝑝𝑒𝑟𝑎𝑡𝑖𝑛𝑔 ℎ𝑜𝑢𝑟𝑠) 7-87 𝜂𝑚𝑜𝑡𝑜𝑟 𝐶𝑜𝑠𝑡 𝑠𝑎𝑣𝑖𝑛𝑔𝑠 = (𝐸𝑛𝑒𝑟𝑔𝑦 𝑠𝑎𝑣𝑖𝑛𝑔𝑠)(𝑈𝑛𝑖𝑡 𝑐𝑜𝑠𝑡 𝑜𝑓 𝑒𝑛𝑒𝑟𝑔𝑦) 𝑤𝑐𝑜𝑚𝑝,𝑖𝑛 = 𝑤𝑟𝑒𝑣𝑒𝑟𝑠𝑖𝑏𝑙𝑒 𝑐𝑜𝑚𝑝,𝑖𝑛 𝜂𝑐𝑜𝑚𝑝 2 =𝜂 𝑛𝑅𝑇1 𝑃 [( 2 ) (𝑛−1) 𝑃1 𝑐𝑜𝑚𝑝 𝑘 ⁄(𝑘−1) 𝑃 𝑙𝑖𝑛𝑒 𝑚̇𝑎𝑖𝑟 = 𝐶𝑑𝑖𝑠𝑐ℎ𝑎𝑟𝑔𝑒 (𝑘+1) 𝑅𝑇𝑙𝑖𝑛𝑒 √𝑘𝑅 ( (𝑛−1) ⁄𝑛 2 𝑘+1 7-88 − 1] 7-89 ) 𝑇𝑙𝑖𝑛𝑒 7-90 𝑃𝑜𝑤𝑒𝑟 𝑠𝑎𝑣𝑒𝑑 = 𝑃𝑜𝑤𝑒𝑟 𝑤𝑎𝑠𝑡𝑒𝑑 = 𝑚̇𝑎𝑖𝑟 𝑤𝑐𝑜𝑚𝑝,𝑖𝑛 7-91 ̇ 𝑊𝑐𝑜𝑚𝑝 𝑊̇𝑒𝑙𝑒𝑐𝑡𝑟𝑖𝑐 = 𝜂 7-92 𝑚𝑜𝑡𝑜𝑟 𝑊̇𝑒𝑙𝑒𝑐𝑡𝑟𝑖𝑐,𝑠𝑎𝑣𝑒𝑑 = 𝑊̇𝑒𝑙𝑒𝑐𝑡𝑟𝑖𝑐,𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 − 𝑊̇𝑒𝑙𝑒𝑐𝑡𝑟𝑖𝑐,𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 = 𝑊̇𝑐𝑜𝑚𝑝 ( (𝑅𝑎𝑡𝑒𝑑 𝑝𝑜𝑤𝑒𝑟)(𝐿𝑜𝑎𝑑 𝑓𝑎𝑐𝑡𝑜𝑟) ( 𝜂 1 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 −𝜂 1 𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 ) 𝐸𝑛𝑒𝑟𝑔𝑦 𝑠𝑎𝑣𝑖𝑛𝑔𝑠 = 𝑊̇𝑒𝑙𝑒𝑐𝑡𝑟𝑖𝑐,𝑠𝑎𝑣𝑒𝑑 × 𝐴𝑛𝑛𝑢𝑎𝑙 𝑜𝑝𝑒𝑟𝑎𝑡𝑖𝑛𝑔 ℎ𝑜𝑢𝑟𝑠 1 𝜂𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 − 1 𝜂𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 )= 7-93 7-94

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