ENGR5003 must memorise

Everything you need in your head for the exam: formulas that are not on the formula sheet, and the definitions examiners ask for. Print it, or cover the right-hand side and test yourself.

Formulas not on the formula sheet

Needed in past-paper solutions

Linear interpolation: $\dfrac{y - y_1}{y_2 - y_1} = \dfrac{x - x_1}{x_2 - x_1}$	used in 6 questions, e.g. Lecture 8 examples - linear interpolation in the saturation table, Lecture 8 example - double interpolation in the superheated table, Old spec L5 worked example 2 - heating wet steam in a rigid tank
Mole balance with extent of reaction: $n_i = n_{i0} + \nu_i\varepsilon$ , $y_i = n_i/\sum n_i$	used in 3 questions, e.g. ENGR266 2021 Q3, ENGR266 2023 Q A2, ENGR266 2025 Section B Q4
Constant-pressure heat for a closed system: $Q = m\Delta h$	used in 2 questions, e.g. Old spec L5 worked example 1 - boiling water at constant pressure, Week 3 Q2
Carnot efficiency: $\eta = 1 - T_C/T_H$	used in 2 questions, e.g. Old spec L7 worked example - isochoric / isentropic / isothermal cycle, ENGR266 2022 Q1
Raoult's law (ideal liquid, ideal vapour): $y_iP = x_iP_i^{sat}$	used in 2 questions, e.g. ENGR263 2025 Q1(c), ENGR263 2019 Q A1(a)
Overall efficiency: $\eta_o = \eta_{th}\,\eta_b\,\eta_m$	used in 2 questions, e.g. ENGR217 2021 Q1, ENGR217 2022 Q A1
Power: $P = W_{net}\,N\,\eta_b\,\eta_m$ with $N = \dfrac{\mathrm{rpm}}{120}$ power strokes per second per cylinder for a 4-stroke	used in 2 questions, e.g. ENGR217 2021 Q1, ENGR217 2022 Q A1
Brake work: $W_b = \eta_m W_{net}$	used in 2 questions, e.g. ENGR217 2021 Q1, ENGR217 2022 Q A1
Redlich-Kwong: $P = \dfrac{RT}{V - b} - \dfrac{a}{T^{0.5}V(V + b)}$ , $a = \dfrac{0.42748R^2T_c^{2.5}}{P_c}$ , $b = \dfrac{0.08664RT_c}{P_c}$	used in 2 questions, e.g. ENGR266 2022 Q1, ENGR266 2023 Q A1
Pressure as force per unit area: $p = F/A$	used in 1 question, e.g. Lecture 2 exercise - piston static pressure
Static force balance on the piston: $pA = p_{atm}A + mg + F_{spring}$	used in 1 question, e.g. Lecture 2 exercise - piston static pressure
Shaft power: $\dot W = T\omega = 2\pi n T$ with $n$ in rev/s	used in 1 question, e.g. Lecture 2 exercise - drive-shaft power
Surface area of a sphere: $A = \pi D^2$	used in 1 question, e.g. Lecture 2 exercise - surface-tension work on a bubble
Exergy of heat $q$ supplied at mean temperature $T_{mH}$ : $e_{x,Q} = q\left(1 - \dfrac{T_0}{T_{mH}}\right) = q - T_0\Delta s$	used in 1 question, e.g. Lecture 3 Example 1 - energy vs exergy losses in a steam plant
Maximum (Carnot) work from heat: $w_{max} = q\left(1 - \dfrac{T_0}{T}\right)$	used in 1 question, e.g. Lecture 3 Example 1 - energy vs exergy losses in a steam plant
Exergy: the maximum useful work obtainable as a system comes into equilibrium with its surroundings	used in 1 question, e.g. Lecture 3 Example 2 - physical and chemical exergy of hydrogen
Exergy destroyed $= T_0 S_g$ , with $S_g \ge 0$ for any real process	used in 1 question, e.g. Lecture 3 Example 2 - physical and chemical exergy of hydrogen
Van der Waals: $p = \dfrac{RT}{V_m - b} - \dfrac{a}{V_m^2}$	used in 1 question, e.g. Lecture 7 example - nitrogen vessel by ideal gas, van der Waals and Redlich-Kwong
Redlich-Kwong: $p = \dfrac{RT}{V_m - b} - \dfrac{a}{T^{0.5}V_m(V_m + b)}$ , $a = \dfrac{0.42748R^2T_c^{2.5}}{p_c}$ , $b = \dfrac{0.08664RT_c}{p_c}$	used in 1 question, e.g. Lecture 7 example - nitrogen vessel by ideal gas, van der Waals and Redlich-Kwong
Compressibility factor: $Z = \dfrac{pv}{R_gT}$	used in 1 question, e.g. Lecture 7 example - nitrogen vessel by ideal gas, van der Waals and Redlich-Kwong
Heat released per kg of mixture: $q_{in} = \dfrac{CV}{AFR + 1}$	used in 1 question, e.g. Lecture 11 worked example - realistic petrol engine efficiency
Brake work = indicated work minus friction; $\eta_b = w_b/q_{in}$	used in 1 question, e.g. Lecture 11 worked example - realistic petrol engine efficiency
Volumetric efficiency: $\eta_v = \dfrac{\dot V}{V_sN}$	used in 1 question, e.g. Old spec L15 Question 2 - petrol engine power band
Power output: $P = \eta_{th}\eta_v\eta_mQ_{in}N$ with $Q_{in}$ the heat for a full swept-volume charge	used in 1 question, e.g. Old spec L15 Question 2 - petrol engine power band
Pump work for an incompressible liquid: $w_p = v_f(p_2 - p_1)$	used in 1 question, e.g. Lecture 13 worked example 1 - ideal Rankine cycle
Isentropic relation with relative pressure: $\dfrac{P_{r2}}{P_{r1}} = \dfrac{p_2}{p_1}$	used in 1 question, e.g. Lecture 13 worked example 2 - ideal Brayton cycle with air tables
Linear interpolation	used in 1 question, e.g. Lecture 13 worked example 2 - ideal Brayton cycle with air tables
Heat exchanger models: cocurrent $T_2' = T_4' = \tfrac{1}{2}(T_2 + T_4)$ ; countercurrent $T_2' = T_4$ , $T_4' = T_2$	used in 1 question, e.g. Lecture 14 example - jet engine with a regenerative heat exchanger
Ideal turbocharger: work on the intake air = heat taken from the exhaust	used in 1 question, e.g. Lecture 14 example - turbocharged Otto engine
For a plate fin of width $w$ and thickness $t$ : $P = 2(w + t) \approx 2w$ , $A_c = wt$ , so $b \approx \sqrt{2h/(kt)}$	used in 1 question, e.g. ENGR203 2018 Q B1
Series resistances add: $R = R_1 + R_2 + \dots$ ; parallel resistances: $\dfrac{1}{R} = \dfrac{1}{R_1} + \dfrac{1}{R_2} + \dots$	used in 1 question, e.g. ENGR203 2018 Q B3
Energy: $1\ \mathrm{kWh} = 3.6\ \mathrm{MJ}$	used in 1 question, e.g. ENGR203 2018 Q B3
Wire resistance per unit length $R' = \rho_e/A_c$ ; Joule heating $\dot Q = I^2R$ ; volumetric generation $q_v = I^2\rho_e/A_c^2$	used in 1 question, e.g. ENGR203 2017 Q B2
1-D conduction with uniform generation, both ends at $T_0$ : $T(x) = T_0 + \dfrac{q_v}{2k}x(L - x)$ , $T_{max} - T_0 = \dfrac{q_vL^2}{8k}$	used in 1 question, e.g. ENGR203 2017 Q B2
Dalton: $P = \sum_i p_i$ , $y_i = p_i/P$	used in 1 question, e.g. ENGR263 2025 Q1(c)
Bubble point of a binary at fixed $P$ : $x_AP_A^{sat} + (1 - x_A)P_B^{sat} = P$	used in 1 question, e.g. ENGR263 2025 Q1(c)
Isentropic efficiencies: $\eta_c = \dfrac{W_{c,s}}{W_{c,a}}$ , $\eta_e = \dfrac{W_{e,a}}{W_{e,s}}$	used in 1 question, e.g. ENGR217 2021 Q1
Fuel economy: distance per litre $=$ speed $\times$ time to burn 1 litre	used in 1 question, e.g. ENGR217 2022 Q A1
Mass flow (continuity): $\dot m = \rho A C = \dfrac{AC}{v}$ , with $pv = R_gT$	used in 1 question, e.g. ENGR217 2022 Q A2
Brake power: $P_b = P\,\eta_v\,\eta_m$	used in 1 question, e.g. ENGR217 2023 Q A1
Mass left in the tank: $\dfrac{m_2}{m_1} = \left(\dfrac{p_2}{p_1}\right)^{1/\gamma}$	used in 1 question, e.g. ENGR217 2023 Q A2
Continuity: $\dot m = \rho AC = \dfrac{pAC}{R_gT}$	used in 1 question, e.g. ENGR217 2023 Q A2
Cut-off ratio $r_c = v_3/v_2$ , compression ratio $r = v_1/v_2$	used in 1 question, e.g. ENGR217 2025 Q1
Carnot efficiency $\eta_C = 1 - \dfrac{T_L}{T_H}$ (temperatures in K)	used in 1 question, e.g. ENGR217 2025 Q1
Mean effective pressure $\mathrm{MEP} = \dfrac{w_{net}}{v_{max} - v_{min}} = \dfrac{w_{net}}{v_1 - v_2}$	used in 1 question, e.g. ENGR217 2025 Q1
Rigid tank charged from a main: $m_2u_2 - m_1u_1 = h_{in}(m_2 - m_1)$	used in 1 question, e.g. ENGR217 2025 Q2
Free frictionless piston: equal pressures on both sides at every instant	used in 1 question, e.g. ENGR217 2025 resit Q1
Entropy change of a solid (incompressible): $\Delta S = mc\ln\dfrac{T_2}{T_1}$	used in 1 question, e.g. ENGR217 2025 resit Q2
Entropy change of surroundings at constant $T_0$ : $\Delta S_{env} = \dfrac{Q_{env}}{T_0}$	used in 1 question, e.g. ENGR217 2025 resit Q2
Fin parameter $m = \sqrt{\dfrac{hp}{kA_c}}$ ; for a pin fin $p = \pi D$ , $A_c = \pi D^2/4$ , so $m = \sqrt{4h/(kD)}$	used in 1 question, e.g. ENGR272 week 6 mock Q2
Heat flux from wall generation (insulated outside): $\dot q_s\,\pi D_i L = \dot e_{gen}\,\dfrac{\pi}{4}(D_o^2 - D_i^2)L$	used in 1 question, e.g. ENGR272 week 6 mock Q3
Overall efficiency $\eta_o = \eta_{th}\eta_b\eta_m$ ; power $P = W_{net}N\eta_b\eta_m$ , $N = \mathrm{rpm}/120$ ; $W_b = \eta_mW_{net}$	used in 1 question, e.g. ENGR217 2024 Q B1
Compressibility factor: $Z = \dfrac{PV}{RT}$ , reduced properties $T_r = T/T_c$ , $P_r = P/P_c$	used in 1 question, e.g. ENGR266 2021 Q1
[EXTERNAL] Pitzer second-virial correlation: $Z = 1 + \hat B \dfrac{P_r}{T_r}$ , $\hat B = B^0 + \omega B^1$ , $B^0 = 0.083 - \dfrac{0.422}{T_r^{1.6}}$ , $B^1 = 0.139 - \dfrac{0.172}{T_r^{4.2}}$	used in 1 question, e.g. ENGR266 2021 Q1
[EXTERNAL] Lee-Kessler corresponding states: $Z = Z^0 + \omega Z^1$ with $Z^0, Z^1$ read from the Lee-Kessler tables	used in 1 question, e.g. ENGR266 2021 Q1
Refrigerator COP: $COP_R = \dfrac{q_L}{w_{in}} = \dfrac{h_1 - h_4}{h_2 - h_1}$	used in 1 question, e.g. ENGR266 2021 Q2
Compressor isentropic efficiency: $\eta_c = \dfrac{h_{2s} - h_1}{h_2 - h_1}$	used in 1 question, e.g. ENGR266 2021 Q2
Linear interpolation in the R134a tables	used in 1 question, e.g. ENGR266 2021 Q2
[EXTERNAL] Modified Raoult's law: $y_i P = x_i \gamma_i P_i^{sat}$ , so $P = x_1\gamma_1 P_1^{sat} + x_2\gamma_2 P_2^{sat}$	used in 1 question, e.g. ENGR266 2021 Q3
[EXTERNAL] Margules equations: $\ln\gamma_1 = x_2^2\left[A_{12} + 2(A_{21} - A_{12})x_1\right]$ , $\ln\gamma_2 = x_1^2\left[A_{21} + 2(A_{12} - A_{21})x_2\right]$	used in 1 question, e.g. ENGR266 2021 Q3
van der Waals: $P = \dfrac{RT}{V - b} - \dfrac{a}{V^2}$ , $a = \dfrac{27R^2T_c^2}{64P_c}$ , $b = \dfrac{RT_c}{8P_c}$	used in 1 question, e.g. ENGR266 2022 Q1
Compressibility factor: $Z = PV/RT$	used in 1 question, e.g. ENGR266 2022 Q1
Refrigerator COP: $COP_R = \dfrac{h_1 - h_4}{w_{in}}$	used in 1 question, e.g. ENGR266 2022 Q2
Refrigerant mass flow: $\dot m = \dfrac{\dot Q_L}{h_1 - h_4}$	used in 1 question, e.g. ENGR266 2022 Q2
Carnot refrigerator COP: $COP = \dfrac{T_L}{T_H - T_L}$	used in 1 question, e.g. ENGR266 2022 Q2
Energy balance for a refrigeration cycle: $Q_H = Q_L + W_{in}$ , $COP_R = Q_L/W_{in}$	used in 1 question, e.g. ENGR266 2023 Q A1
[EXTERNAL] Excess property: $V^E = V - \sum_i x_i V_i$	used in 1 question, e.g. ENGR266 2023 Q A2
[EXTERNAL] Modified Raoult's law: $y_i P = x_i\gamma_i P_i^{sat}$ , so $P = \sum_i x_i\gamma_i P_i^{sat}$	used in 1 question, e.g. ENGR266 2023 Q A2
Vapour-compression COP (flowchart numbering): $COP = \dfrac{h_2 - h_1}{h_3 - h_2}$	used in 1 question, e.g. ENGR266 2024 Q B1
Isentropic compressor: $s_3 = s_2$	used in 1 question, e.g. ENGR266 2024 Q B1
Molar from specific volume: $V = v\,M$	used in 1 question, e.g. ENGR266 2024 Q B1
Reduced properties: $T_r = T/T_c$ , $P_r = P/P_c$	used in 1 question, e.g. ENGR266 2025 Section B Q1
[EXTERNAL] Lee-Kessler generalised correlation: $Z = Z^0 + \omega Z^1$ , $Z^0$ and $Z^1$ from the Lee-Kessler tables at $(T_r, P_r)$	used in 1 question, e.g. ENGR266 2025 Section B Q1
Molar volume from tables: $V = vM$	used in 1 question, e.g. ENGR266 2025 Section B Q1
Carnot efficiency: $\eta = 1 - \dfrac{T_C}{T_H}$	used in 1 question, e.g. ENGR266 2025 Section B Q2
Steam property values $h_f$ , $h_g$ , $h_{fg}$ at the given pressures and temperatures	used in 1 question, e.g. ENGR217 2017 Q B2
Isentropic process: $s_2 = s_1$	used in 1 question, e.g. ENGR217 2018 Q A1
Steam property values at 20 bar and 1 bar	used in 1 question, e.g. ENGR217 2018 Q A1
Continuity: $\dot m = \rho A C$	used in 1 question, e.g. ENGR217 2018 Q A2
Linear interpolation in the steam tables	used in 1 question, e.g. ENGR217 2019 Q A1
Steam property values (saturated temperature and pressure tables, superheated table)	used in 1 question, e.g. ENGR217 2019 Q A1
Isentropic efficiencies: $\eta_C = \dfrac{T_{2s} - T_1}{T_{2a} - T_1}$ , $\eta_T = \dfrac{T_3 - T_{4a}}{T_3 - T_{4s}}$	used in 1 question, e.g. ENGR217 2019 Q A3
Redlich-Kwong: $P = \dfrac{RT}{V - b} - \dfrac{a}{T^{1/2} V (V + b)}$ , $a = 0.42748\dfrac{R^2 T_c^{2.5}}{P_c}$ , $b = 0.08664\dfrac{R T_c}{P_c}$	used in 1 question, e.g. ENGR266 2018 Q A1
Equivalent generic-cubic form: $a(T) = 0.42748\,\alpha(T_r)\dfrac{R^2T_c^2}{P_c}$ with $\alpha = T_r^{-1/2}$	used in 1 question, e.g. ENGR266 2018 Q A1
R134a saturated $v_f$ , $v_g$ at the given temperature	used in 1 question, e.g. ENGR266 2018 Q A1
Carnot refrigerator: $COP_R = \dfrac{Q_L}{W_{in}} = \dfrac{T_L}{T_H - T_L}$	used in 1 question, e.g. ENGR266 2018 Q A2
Reading a P-x-y diagram: bubble point from the bubble (liquid, $x_1$ ) curve, the coexisting vapour from the dew ( $y_1$ ) curve at the same $P$ (tie line)	used in 1 question, e.g. ENGR266 2018 Q A3
Lever rule on a T-x-y diagram: liquid fraction $\dfrac{L}{F} = \dfrac{y_1 - z_1}{y_1 - x_1}$ , vapour fraction $\dfrac{V}{F} = \dfrac{z_1 - x_1}{y_1 - x_1}$ (from the component balance $z_1 = (L/F)x_1 + (V/F)y_1$ )	used in 1 question, e.g. ENGR266 2019 Q A3
Slope of a constant-property line on T-s: $(\partial T/\partial s)_p = T/c_p$ , $(\partial T/\partial s)_v = T/c_v$	used in 1 question, e.g. Week 1 Q1
Heat exchanger: heat gained by the compressed air = heat lost by the exhaust	used in 1 question, e.g. Week 2 Q2
Indicated work = net work - pumping work; $MEP = \dfrac{W_{indicated}}{V_s}$	used in 1 question, e.g. Week 4 Q2
Volumetric (breathing) efficiency: $\eta_v = \dfrac{\dot V_a}{V_sN}$	used in 1 question, e.g. Week 4 Q3
Mechanical efficiency: $\eta_m = \dfrac{W_b}{W_{net}}$	used in 1 question, e.g. Week 4 Q3
Overall efficiency: $\eta_o = \eta_{th}\eta_v\eta_m$	used in 1 question, e.g. Week 4 Q3
Fuel energy: $Q_{in} = m_f \times CV$	used in 1 question, e.g. Week 4 Q3
Isentropic efficiencies: $\eta_C = \dfrac{T_{2s} - T_1}{T_2 - T_1}$ , $\eta_T = \dfrac{T_3 - T_4}{T_3 - T_{4s}}$	used in 1 question, e.g. Week 5 Q1
Ideal (countercurrent) regenerator: $T_2' = T_4$	used in 1 question, e.g. Week 5 Q1
Pump work for an incompressible liquid: $w_P = v_f(p_2 - p_1)$	used in 1 question, e.g. Week 5 Q2
Isentropic efficiencies: $\eta_P = \dfrac{h_{2s} - h_1}{h_2 - h_1}$ , $\eta_T = \dfrac{h_3 - h_4}{h_3 - h_{4s}}$	used in 1 question, e.g. Week 5 Q2
Surface-tension work: $\delta W = -\sigma_s\,dA$ (work done by the system), i.e. work $\sigma_s\,\Delta A$ must be done on a film to create area $\Delta A$	used in 1 question, e.g. Old spec Tutorial 1 Problem 1
Circle area $A = \pi r^2$	used in 1 question, e.g. Old spec Tutorial 1 Problem 1
Absolute pressure = gauge pressure + atmospheric pressure; temperatures in kelvin	used in 1 question, e.g. Old spec Tutorial 1 Problem 2
Work done by the atmosphere on the outside of the piston $W_{atm} = p_0\,\|\Delta V\|$	used in 1 question, e.g. Old spec Tutorial 1 Problem 3
Phase test: compare the given $u$ (or $T$ ) with $u_f$ , $u_g$ (or $T_{sat}$ ) at the given pressure or temperature	used in 1 question, e.g. Old spec Tutorial 1 Problem 6
Compressed liquid: $u \approx u_f(T)$	used in 1 question, e.g. Old spec Tutorial 1 Problem 6
Piston equilibrium: $p = p_{atm} + \dfrac{mg}{A}$	used in 1 question, e.g. Old spec Tutorial 1 Problem 8
Linear interpolation in the saturation (pressure) table	used in 1 question, e.g. Old spec Tutorial 1 Problem 8
Linear interpolation in the saturation tables	used in 1 question, e.g. Old spec Tutorial 1 Problem 9
Mass of each phase: $m_f = V_f/v_f$ , $m_g = V_g/v_g$	used in 1 question, e.g. Old spec Tutorial 1 Problem 10
Superheated steam table and linear interpolation (in $T$ and in $p$ )	used in 1 question, e.g. Old spec Tutorial 1 Problem 10