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ME 322 Unit 2 Exam Master Cheat Sheet (1 Page Front & Back)
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ME 322: THERMODYNAMICS • UNIT 2 EXAM CHEAT SHEET (PAGE 1 / 2)

Closed Systems, Boundary Work, First Law & Thermal-Fluid Properties (ME 322 Equation Sheet Base)
PAGE 1: FRONT
PRESSURE, TEMPERATURE, AND DENSITY (EXACT FROM EQUATION SHEET) Units, Constants & Conversions
FORCE, PRESSURE & CRITICAL UNITS P & Units
  • • \(F = P \cdot A\)
  • • \(P_a = P_g + P_{\text{atm}}\) • \(P_{\text{vac}} = P_{\text{atm}} - P_a\)
  • • \(P_{\text{atm}} = 101.33\text{ kPa} = 14.70\text{ psia}\)
  • • \(g = 9.81\text{ m/s}^2 = 32.2\text{ ft/s}^2\)
  • • Units: \(1\text{ bar} = 100\text{ kPa} = 0.1\text{ MPa} = 10^5\text{ Pa} = 14.504\text{ psi}\)
  • • \(1\text{ psi} = 144\text{ lbf/ft}^2\) • \(1\text{ kPa} = 1\text{ kN/m}^2 = 1000\text{ N/m}^2\)
  • • \(1\text{ Btu} = 778.17\text{ ft}\cdot\text{lbf} = 1.055\text{ kJ}\) • \(1\text{ kJ} = 1\text{ kPa}\cdot\text{m}^3\)
TEMPERATURE SCALES & IDEAL GAS EQUATION T & Gas Law
  • • \(T_K = T_{^\circ\text{C}} + 273\) • \(T_R = T_{^\circ\text{F}} + 460\)
  • • \(T_R = \frac{9}{5} T_K\) • \(T_{^\circ\text{F}} = \frac{9}{5} T_{^\circ\text{C}} + 32\) • \(T_{\text{std.atm}} = 293\text{ K or } 530\text{ R}\)
  • • \(P v = R T\) • \(P V = m R T\) • \(P = \rho R T\)
  • • \(\rho = 1 / v\) • \(R = R_u / M\)
  • • \(R_u = 8314.4\text{ J/kmol}\cdot\text{K} = 1.9859\text{ Btu/lbmol}\cdot\text{R} = 1545.0\text{ ft}\cdot\text{lbf/lbmol}\cdot\text{R}\)
  • • Note: In ideal gas formulas, \(T\) must always be in absolute units (\(\text{K}\) or \(\text{R}\))!
PROPERTY DEFINITIONS & QUALITY
  • • \(x = \frac{m_{\text{vap}}}{m_{\text{tot}}}\) • \(x = \frac{f - f_f}{f_{fg}}\) • \(f = f_f + x f_{fg}\)
  • • \(f_{fg} = f_g - f_f\) • \(f \in \{v, u, h, s\}\)
  • • \(U = m u\) • \(\dot{U} = \dot{m} u\) • \(H = m h\) • \(\dot{H} = \dot{m} h\)
  • • \(W = m w\) • \(\dot{W} = \dot{m} w = \frac{W}{\Delta t}\) • \(Q = m q\) • \(\dot{Q} = \dot{m} q = \frac{Q}{\Delta t}\)
  • • \(V = m v\) • \(h = u + P v\)
  • • \(C_p = C_v + R\) • \(k = \frac{C_p}{C_v}\) • \(C = C_v = C_p\) (incompressible)
  • • \(Q_{\text{solid}} = m h_{if}\) • \(Q_{\text{evap}} = m h_{fg}\)
SPECIFIC HEATS, PROCESSES & PHASE RULES
  • • \(du = C_v dT\) • \(dh = C_p dT\)
  • • \(u_2 - u_1 = C_{v@T\text{avg}} (T_2 - T_1)\) • \(h_2 - h_1 = C_{p@T\text{avg}} (T_2 - T_1)\)
  • • \(P v^n = \text{const}\) • isentropic: \(n = k\)
  • Phase Identification Guide:
  • • Given \((P, T)\): \(T < T_{\text{sat}}(P)\) → Compressed Liq (\(v \approx v_f(T), u \approx u_f(T), h \approx h_f(T)\))
  • • \(T = T_{\text{sat}}(P)\) → Saturated Mixture (find \(x\)) • \(T > T_{\text{sat}}(P)\) → Superheated
  • • Given \((P, y)\) or \((T, y)\): \(y < y_f\) → Comp. Liq | \(y_f \le y \le y_g\) → Mixture | \(y > y_g\) → Superheated
  • • Incompressible Liquid: \(\Delta u = c \Delta T\) • \(\Delta h = c \Delta T + v \Delta P\) • Pump: \(w = v \Delta P\)
Process Path Governing Condition / Law Boundary Work Equation \(W_{\text{out}} = \int_1^2 P dV\) [Area under P-V Curve]
General Integral \(W = \int_1^2 F dx\) \(W_{\text{out}} = \int_1^2 P dV\) (Area under process curve on P-V diagram)
Isometric \(V = \text{const} \implies dV = 0\) \(W_{\text{out}} = 0\) (No boundary work in rigid tank / isometric process)
Isobaric \(P = \text{const}\) \(W_{\text{out}} = P(V_2 - V_1) = m P(v_2 - v_1)\)
Isothermal \(T = \text{const}\) (Ideal Gas: \(PV = mRT\)) \(W_{\text{out}} = P_1 V_1 \ln\left(\frac{V_2}{V_1}\right) = m R T \ln\left(\frac{V_2}{V_1}\right) = P_1 V_1 \ln\left(\frac{P_1}{P_2}\right)\)
Polytropic \(P v^n = \text{const} \quad (n \ne 1)\) \(W_{\text{out}} = \frac{P_2 V_2 - P_1 V_1}{1 - n} = \frac{m R (T_2 - T_1)}{1 - n}\) • Exponent: \(n = \frac{\ln(P_2 / P_1)}{\ln(V_1 / V_2)}\)
Spring Piston \(P(V) = P_1 + \frac{k_s}{A^2}(V - V_1)\)
(Linear P-V line)		 \(W_{\text{out}} = \text{Trapezoid Area} = \left(\frac{P_1 + P_2}{2}\right)(V_2 - V_1) = P_1 \Delta V + \frac{1}{2} k_s x^2\)
Spring stored work: \(W_{\text{spring}} = \frac{1}{2} k_s x^2 = \frac{1}{2} k_s \left(\frac{\Delta V}{A}\right)^2\) • \(W_{\text{atm}} = P_{\text{atm}} \Delta V\)
Shaft & Electric Shaft: \(\dot{W}_{\text{sh}} = 2\pi \dot{n} \mathcal{T} = \omega \mathcal{T}\)
Electric: \(\dot{W}_e = V I = I^2 R\)		 Drivetrain balance: \(\dot{W}_{\text{turb}} = \dot{W}_{\text{comp}} + \dot{W}_{\text{gen}}\)
Electric work over time: \(W_e = V \cdot I \cdot \Delta t\) \([\text{Watts} = \text{J/s}]\)
1ST LAW FOR CLOSED SYSTEMS
  • • \(E_{\text{in}} - E_{\text{out}} = \Delta E_{\text{sys}}\) • \(Q_{\text{in}} - W_{\text{out}} = m(u_2 - u_1)_{\text{sys}}\)
  • • Rate form: \(\dot{Q}_{\text{in}} - \dot{W}_{\text{out}} = \frac{dE_{\text{sys}}}{dt} \approx m \frac{du}{dt}\)
  • • Sign Convention: \(Q_{\text{in}} > 0\), \(Q_{\text{out}} < 0\) | \(W_{\text{out}} > 0\) (expansion), \(W_{\text{in}} < 0\) (compression)
  • • Thermodynamic Cycle: \(\Delta U_{\text{cycle}} = 0 \implies \mathbf{W_{\text{net,out}} = Q_{\text{net,in}} = Q_{\text{in}} - Q_{\text{out}}}\)
CONSTANT-PRESSURE HEATING RULE (\(P = \text{const}\))
  • • \(Q_{\text{in}} - W_{b,\text{out}} = \Delta U = (U_2 - U_1) + P(V_2 - V_1)\)
  • • \(Q_{\text{in}} = (U_2 + P V_2) - (U_1 + P V_1) = H_2 - H_1\)
  • • \(Q_{\text{in}} = \Delta H = m(h_2 - h_1) = m C_p (T_2 - T_1)\)
  •   CRITICAL RULE: For closed constant-P process with boundary work only, Heat Added = Enthalpy Change!
ME 322 Thermodynamics • Unit 2 Exam Reference (Page 1) Turn over for Open Flow Control Volumes & Steady-Flow Devices →

ME 322: THERMODYNAMICS • UNIT 2 EXAM CHEAT SHEET (PAGE 2 / 2)

Conservation of Mass, 1st Law Open Flow (SFEE) & All 6 Steady-Flow Devices (ME 322 Equation Sheet Base)
PAGE 2: BACK
CONSERVATION OF MASS & ENERGY TERMS
  • • \(\dot{m} = \rho V A = \frac{V A}{v}\) • \(\dot{V} = V A\)
  • • \(\sum_{\text{in}} \dot{m} = \sum_{\text{out}} \dot{m} + \frac{dm}{dt}\)
  • • \(\sum_{\text{in}} m = \sum_{\text{out}} m + [m_2 - m_1]_{\text{CV}}\)
  • • Single Stream Steady: \(\dot{m}_1 = \dot{m}_2 \implies \mathbf{\frac{A_1 V_1}{v_1} = \frac{A_2 V_2}{v_2}}\)
  • • \(ke = \frac{1}{2} V^2\) \([\text{kJ/kg: } \frac{V^2}{2000}]\) • \(KE = \frac{1}{2} m V^2\) • \(\dot{KE} = \frac{1}{2} \dot{m} V^2\)
  • • \(pe = g z\) \([\text{kJ/kg: } \frac{gz}{1000}]\) • \(PE = m g z\) • \(\dot{PE} = \dot{m} g z\)
  • • \(\dot{W}_{\text{pump}} = \dot{m} v (P_{\text{out}} - P_{\text{in}}) = \dot{m} \frac{P_{\text{out}} - P_{\text{in}}}{\rho}\)
1ST LAW OPEN FLOW & USUF RATE FORMS
  • • \(E_{\text{in}} - E_{\text{out}} = \Delta E_{\text{sys}}\)
  • • \(\dot{Q}_{\text{in}} - \dot{W}_{\text{out}} = \dot{m}(h_{\text{out}} - h_{\text{in}})_{\text{CV}}\) (Single stream steady)
  • • \(Q_{\text{in}} - W_{\text{out}} + \sum_{\text{in}} m_i (h_i + ke_i + pe_i) - \sum_{\text{out}} m_o (h_o + ke_o + pe_o) =\)
      \([m_2(u_2 + ke_2 + pe_2) - m_1(u_1 + ke_1 + pe_1)]_{\text{sys}}\) (Unsteady USUF)
  • • \(\dot{Q}_{\text{in}} - \dot{W}_{\text{out}} + \sum_{\text{in}} \dot{m}_i (h_i + ke_i + pe_i) - \sum_{\text{out}} \dot{m}_o (h_o + ke_o + pe_o) =\)
      \(\frac{d}{dt}[m(u + ke + pe)]_{\text{CV}}\) (General rate form)
  • • English KE Conversion: \(\text{ke} = \frac{V^2}{2 \times 32.174 \times 778.17} = \mathbf{\frac{V^2}{50{,}074}}\text{ [Btu/lbm]}\)
Device & Schematic Key Assumptions Property Trends Governing Energy & Sizing Equations
Nozzle & Diffuser
Nozzle: \(A\downarrow, V\uparrow\)
Diffuser: \(A\uparrow, V\downarrow\) 		\(\dot{W} = 0\)
\(\dot{Q} \approx 0\) (adiabatic)
\(\Delta\text{pe} \approx 0\) 		Nozzle: \(V\uparrow\uparrow, P\downarrow, T\downarrow, h\downarrow\)
Diffuser: \(V\downarrow\downarrow, P\uparrow, T\uparrow, h\uparrow\) 		\(h_1 + \frac{V_1^2}{2000} = h_2 + \frac{V_2^2}{2000}\)
Exit Velocity: \(V_2 = \sqrt{V_1^2 + 2000(h_1 - h_2)}\) (If \(V_1 \approx 0 \implies V_2 = \sqrt{2000(h_1 - h_2)}\))
Exit Area: \(A_2 = \frac{\dot{m} v_2}{V_2}\) • If heat loss: \(\frac{\dot{Q}}{\dot{m}} = (h_2 - h_1) + \frac{V_2^2 - V_1^2}{2000}\)
Turbine
Expanding casing
Produces shaft work 		\(\dot{W}_{\text{out}} > 0\)
\(\dot{Q} \approx 0\) (or heat loss)
\(\Delta\text{ke} \approx 0, \Delta\text{pe} \approx 0\) 		\(P\downarrow\downarrow, T\downarrow\downarrow, h\downarrow\downarrow\)
Fluid expands and turns shaft blades. 		\(\dot{W}_{\text{turb,out}} = \dot{m}(h_1 - h_2)\) • Ideal Gas: \(\dot{W}_{\text{turb}} = \dot{m} c_p (T_1 - T_2)\)
With heat loss: \(\dot{W}_{\text{turb}} = \dot{m}(h_1 - h_2) - \dot{Q}_{\text{loss}}\)
Specific power: \(w_{\text{out}} = h_1 - h_2\) [kJ/kg]
Compressor & Pump
Compressor: Gas
Pump: Liquid 		\(\dot{W}_{\text{in}} > 0\)
\(\dot{Q} \approx 0\) (adiabatic)
\(\Delta\text{ke} \approx 0, \Delta\text{pe} \approx 0\) 		\(P\uparrow\uparrow, T\uparrow, h\uparrow\)
Work is consumed to pressurize fluid. 		Compressor (Gas): \(\dot{W}_{\text{comp,in}} = \dot{m}(h_2 - h_1) = \dot{m} c_p (T_2 - T_1)\)
Pump (Liquid, \(v \approx \text{const}\)): \(\dot{W}_{\text{pump,in}} = \dot{m} v (P_2 - P_1) = \dot{m} \frac{\Delta P}{\rho}\)
With \(\Delta\text{ke}\): \(\dot{W}_{\text{pump}} = \dot{m}\left[v(P_2 - P_1) + \frac{V_2^2 - V_1^2}{2000}\right]\)
Throttling Valve
Expansion valve,
orifice, porous plug 		\(\dot{W} = 0\)
\(\dot{Q} \approx 0\)
\(\Delta\text{ke} \approx 0, \Delta\text{pe} \approx 0\) 		ISENTHALPIC: \(h_1 = h_2\)
\(P\downarrow\downarrow\) (large drop)
Ideal gas: \(T_1 = T_2\) 		\(h_2 = h_1\) (Constant Enthalpy!)
• Ideal Gas: Since \(h = c_p T \implies \mathbf{T_2 = T_1}\) (No temp change!)
• Refrigerant/Steam: \(T\downarrow\downarrow\) to \(T_{\text{sat}}(P_2)\). Flashes into wet mixture: \(x_2 = \frac{h_1 - h_f}{h_{fg}}\) • \(u_2 = h_1 - P_2 v_2\)
Mixing Chamber
Open feedwater heater,
shower T-junction 		\(\dot{W} = 0\)
\(\dot{Q} \approx 0\) (insulated)
\(\Delta\text{ke} \approx 0, \Delta\text{pe} \approx 0\) 		\(P_1 = P_2 = P_3 = P\)
(Inlets & exit must be at same pressure!) 		Mass: \(\dot{m}_1 + \dot{m}_2 = \dot{m}_3\) • Energy: \(\dot{m}_1 h_1 + \dot{m}_2 h_2 = \dot{m}_3 h_3\)
Mass Ratio: \(\frac{\dot{m}_1}{\dot{m}_2} = \frac{h_3 - h_2}{h_1 - h_3}\) • (For ideal gas: \(\frac{T_3 - T_2}{T_1 - T_3}\))
Heat Exchanger
Shell-and-tube, radiator,
condenser, boiler 		\(\dot{W} = 0\)
\(\Delta\text{ke} \approx 0, \Delta\text{pe} \approx 0\)
Fluids DO NOT mix! 		Hot fluid: \(T\downarrow, P \approx \text{const}\)
Cold fluid: \(T\uparrow, P \approx \text{const}\) 		• Entire Exchanger (Insulated): \(\dot{m}_{\text{hot}}(h_{h,\text{in}} - h_{h,\text{out}}) = \dot{m}_{\text{cold}}(h_{c,\text{out}} - h_{c,\text{in}})\)
• Single Stream: \(\dot{Q} = \dot{m}_{\text{cold}}(h_{c,\text{out}} - h_{c,\text{in}}) = \dot{m}_{\text{hot}}(h_{h,\text{in}} - h_{h,\text{out}})\)
• Non-Insulated: \(\dot{Q}_{\text{hot}} = \dot{Q}_{\text{cold}} + \dot{Q}_{\text{loss}}\) (If 10% lost → \(\dot{Q}_{\text{cold}} = 0.90 \dot{Q}_{\text{hot}}\))
UNSTEADY UNIFORM-STATE UNIFORM-FLOW (USUF)
  • • Mass Balance: \(m_{\text{in}} - m_{\text{out}} = (m_2 - m_1)_{\text{CV}}\)
  • • Energy Balance: \(Q - W + \sum m_i \theta_i - \sum m_e \theta_e = (m_2 u_2 - m_1 u_1)_{\text{CV}}\)
  • • Evacuated Tank Filling (\(m_1=0, m_e=0, W=0, Q=0\)):
  •   \(u_2 = h_{\text{line}}\) • For Ideal Gas: \(c_v T_2 = c_p T_{\text{line}} \implies \mathbf{T_2 = k T_{\text{line}}}\)
  •   (Tank gas is ALWAYS hotter than supply line by factor \(k \approx 1.4\)!)
  • • Rigid Tank Discharging: \(V = \text{const} \implies W_b = 0\).
⚠️ TOP 8 EXAM TRAPS & COMMON MISTAKES
  • 1. Gage vs Abs: \(P_{\text{abs}} = P_{\text{gage}} + 14.7\text{ psi}\) (or \(+ 101.3\text{ kPa}\)). Never put gage into \(Pv=RT\)!
  • 2. Kelvin / Rankine: Every single \(T\) alone MUST be absolute. \(\Delta T\) is same in \(^\circ\text{C}\) or \(\text{K}\).
  • 3. Kinetic Energy: In nozzles/turbines, DIVIDE \(V^2\) by 2000 in SI, or 50,074 in English!
  • 4. Throttling: Ideal gas → \(T_1 = T_2\). Liquid/refrigerant → \(T_2 = T_{\text{sat}}(P_2) \ll T_1\).
  • 5. Isobaric Heating: Closed system with boundary work: \(Q = \Delta H = m(h_2 - h_1)\).
  • 6. Spring Piston: Pressure varies linearly! Boundary work is trapezoid: \(W_b = \frac{P_1+P_2}{2}\Delta V\).
  • 7. Pump Work: \(w = v(P_2 - P_1)\). Use \(v \approx v_f\) at inlet temp. In English, divide by \(5.404\) for Btu/lbm!
  • 8. Work Signs: Expansion = \(+W_{\text{out}}\). Compression / Pump = \(+W_{\text{in}}\). Turbine = \(+W_{\text{turb}}\).
ME 322 Thermodynamics • Unit 2 Exam Cheat Sheet (2 Pages Front & Back) Good luck on your exam! You've got this!