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Thermodynamics & Heat Transfer: formula sheet

Engineering thermodynamics, steam plant & heat transfer · 71 formulas · metric edition 1

The full study guide: modules, problems, answer key
Module 1 · Properties of State
Thermodynamics & Heat Transfer-1 Gauge and Absolute Pressure
Pabs=Pgauge+PatmP_{abs} = P_{gauge} + P_{atm}
Absolute pressure is the gauge reading plus the surrounding atmospheric pressure.
Thermodynamics & Heat Transfer-2 Sensible Heat (Q = mcΔT)
Q=mcΔTQ = m c \Delta T
Heat needed to change a mass's temperature: mass times specific heat times the temperature change.
Thermodynamics & Heat Transfer-3 Latent Heat
Q=mLQ = m L
Heat absorbed or released when a mass changes phase at constant temperature.
Thermodynamics & Heat Transfer-4 Boyle's Law
P1V1=P2V2P_1 V_1 = P_2 V_2
At constant temperature, pressure times volume stays constant for a fixed amount of gas.
Thermodynamics & Heat Transfer-5 Charles's Law
V1T1=V2T2\frac{V_1}{T_1} = \frac{V_2}{T_2}
At constant pressure, gas volume is directly proportional to absolute temperature.
Thermodynamics & Heat Transfer-6 Gay-Lussac's Law
P1T1=P2T2\frac{P_1}{T_1} = \frac{P_2}{T_2}
At constant volume, gas pressure is directly proportional to absolute temperature.
Thermodynamics & Heat Transfer-7 Combined Gas Law
P1V1T1=P2V2T2\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}
For a fixed amount of gas, pressure times volume over absolute temperature stays constant between…
Thermodynamics & Heat Transfer-8 Ideal Gas Law
PV=nRTP V = n R T
Relates pressure, volume, amount, and temperature of a gas.
Thermodynamics & Heat Transfer-9 Gas Density from Molar Mass
ρ=PMRT\rho = \frac{PM}{RT}
Gives an ideal gas's density from its molar mass, pressure, and absolute temperature using R =…
Thermodynamics & Heat Transfer-10 Compressibility Factor (Z = PV/nRT)
Z=PVnRTZ = \frac{P V}{n R T}
How far a real gas departs from ideal behaviour, as a single multiplier on the ideal gas law.
Thermodynamics & Heat Transfer-11 Van der Waals Equation of State
(P+an2V2)(V−nb)=nRT\left(P + \frac{a n^{2}}{V^{2}}\right)\left(V - n b\right) = n R T
The first equation of state to describe a real gas, adding a term for molecular attraction (a) and…
Thermodynamics & Heat Transfer-12 Antoine Equation (Vapour Pressure)
log⁡10P=A−BC+T\log_{10} P = A - \frac{B}{C + T}
The three-constant fit that every handbook uses for pure-component vapour pressure.
Thermodynamics & Heat Transfer-13 Clausius–Clapeyron Equation (Two-Point Form)
ln⁡ ⁣(P2P1)=−ΔHvapR(1T2−1T1)\ln\!\left(\frac{P_2}{P_1}\right) = -\frac{\Delta H_{vap}}{R}\left(\frac{1}{T_2} - \frac{1}{T_1}\right)
Relates two points on a liquid's vapour-pressure curve to its molar enthalpy of vaporisation,…
Module 2 · Steam, Turbines & Power Cycles
Thermodynamics & Heat Transfer-14 Saturation Temperature and Pressure of Steam
Tsat=Ts(psat)psat=ps(Tsat)T_{sat} = T_s(p_{sat}) \qquad p_{sat} = p_s(T_{sat})
Saturation temperature from pressure and saturation pressure from temperature, on the IAPWS-IF97…
Thermodynamics & Heat Transfer-15 Steam Quality from Enthalpy
x=h−hfhfgx = \frac{h - h_f}{h_{fg}}
Dryness fraction of wet steam from its measured enthalpy, the saturated liquid enthalpy and the…
Thermodynamics & Heat Transfer-16 Wet Steam Enthalpy from h_f and h_g
h=(1−x) hf+x hgh = (1 - x) \, h_f + x \, h_g
Enthalpy of a wet steam mixture as the mass-weighted blend of saturated water and dry saturated…
Thermodynamics & Heat Transfer-17 Degrees of Superheat
ΔTsh=T−Ts(p)\Delta T_{sh} = T - T_s(p)
How many degrees a steam temperature sits above saturation at its own pressure, with the saturation…
Thermodynamics & Heat Transfer-18 Superheated Steam Enthalpy h(p, T)
h=R T τ(γτ∘+γτr),τ=540 KTh = R\,T\,\tau\left(\gamma^{\circ}_{\tau} + \gamma^{r}_{\tau}\right), \quad \tau = \frac{540\ \mathrm{K}}{T}
Specific enthalpy of superheated steam from its pressure and temperature, straight off IAPWS-IF97…
Thermodynamics & Heat Transfer-19 Flash Steam Percentage
%F=hf1−hf2hfg2×100\%F = \frac{h_{f1} - h_{f2}}{h_{fg2}} \times 100
Percentage of hot condensate that flashes to steam when let down to a lower pressure, from the…
Thermodynamics & Heat Transfer-20 Flash Steam Mass Rate
m˙f=%F100 m˙c\dot{m}_f = \frac{\%F}{100} \, \dot{m}_c
Flash steam a condensate stream releases when it is let down to a lower pressure, from the flash…
Thermodynamics & Heat Transfer-21 Steam Turbine Specific Work
w=h1−h2w = h_1 - h_2
Work a steam turbine takes out of every kilogram of steam: the enthalpy at the stop valve less the…
Thermodynamics & Heat Transfer-22 Steam Turbine Power Output
P=m˙ wP = \dot{m} \, w
Shaft power a steam turbine develops from its steam rate and the specific work taken out of each…
Thermodynamics & Heat Transfer-23 Turbine Isentropic Efficiency
ηisen=h1−h2h1−h2s\eta_{isen} = \frac{h_1 - h_2}{h_1 - h_{2s}}
Isentropic efficiency of a steam turbine: the actual enthalpy drop divided by the ideal drop a…
Thermodynamics & Heat Transfer-24 Napier's Steam Leak Rate
m˙=A P70\dot{m} = \frac{A \, P}{70}
Steam lost through a hole, a blowing trap or a lifted relief valve by Napier's rule: pounds per…
Thermodynamics & Heat Transfer-25 Desuperheater Water Injection Rate
m˙w=m˙1 h1−h2h2−hw\dot{m}_w = \dot{m}_1 \, \frac{h_1 - h_2}{h_2 - h_w}
Spray water a desuperheater needs to bring superheated steam down to a target enthalpy, straight…
Thermodynamics & Heat Transfer-26 Steam Coil Condensate Load
m˙=Q˙hfg\dot m = \frac{\dot Q}{h_{fg}}
Steam a coil condenses per unit time from its heat duty and the latent heat at the coil pressure:…
Thermodynamics & Heat Transfer-27 Thermal Efficiency
η=WQh\eta = \frac{W}{Q_h}
Fraction of heat input that a heat engine converts into useful work.
Thermodynamics & Heat Transfer-28 Carnot Efficiency
η=1−TcTh\eta = 1 - \frac{T_c}{T_h}
The maximum possible efficiency of any heat engine operating between two absolute temperatures.
Thermodynamics & Heat Transfer-29 Rankine Cycle Thermal Efficiency
η=(h1−h2)−(h4−h3)h1−h4\eta = \frac{\left(h_1 - h_2\right) - \left(h_4 - h_3\right)}{h_1 - h_4}
Net work over heat added for a steam power cycle, taken straight off the four state-point…
Thermodynamics & Heat Transfer-30 Otto Cycle Efficiency (Compression Ratio)
η=1−1rγ−1\eta = 1 - \frac{1}{r^{\gamma - 1}}
Air-standard efficiency of a spark-ignition engine from its compression ratio alone.
Thermodynamics & Heat Transfer-31 Brayton Cycle Efficiency (Pressure Ratio)
η=1−1rp(γ−1)/γ\eta = 1 - \frac{1}{r_p^{\left(\gamma - 1\right)/\gamma}}
Air-standard efficiency of a gas turbine from compressor pressure ratio alone.
Module 3 · Conduction & the Building Envelope
Thermodynamics & Heat Transfer-32 Heat Conduction Rate
P=kAΔTdP = \tfrac{k A \Delta T}{d}
Steady-state heat flow through a slab by Fourier's law of conduction.
Thermodynamics & Heat Transfer-33 R-Value of an Insulation Layer (R = L/k)
R=LkR = \frac{L}{k}
Area-specific thermal resistance of one layer, from its thickness and thermal conductivity: the RSI…
Thermodynamics & Heat Transfer-34 Thermal Resistance of a Plane Wall
R=LkAR = \frac{L}{k A}
Conduction resistance of a flat slab in kelvin per watt, from its thickness, thermal conductivity…
Thermodynamics & Heat Transfer-35 Thermal Resistances in Series
Rtot=R1+R2+R3R_{tot} = R_1 + R_2 + R_3
Total resistance of a composite wall, where the same heat crosses each layer in turn so the layer…
Thermodynamics & Heat Transfer-36 Total R-Value of an Assembly
Rtot=R1+R2+R3R_{tot} = R_1 + R_2 + R_3
Total R-value of a wall, roof or floor built up from three layers in series, where the same heat…
Thermodynamics & Heat Transfer-37 U-Factor from Total R-Value (U = 1/R)
U=1RtotU = \frac{1}{R_{tot}}
Converts an assembly's total R-value into the U-factor used by energy codes and window labels, and…
Thermodynamics & Heat Transfer-38 Overall U from Total Resistance
U=1RtotAU = \frac{1}{R_{tot} A}
Converts an assembly's total resistance in kelvin per watt into the overall coefficient U quoted on…
Thermodynamics & Heat Transfer-39 Heat Flow from Thermal Resistance
Q˙=ΔTR\dot{Q} = \frac{\Delta T}{R}
Ohm's law for heat: the flow through an assembly equals the temperature difference across it…
Thermodynamics & Heat Transfer-40 Heat Loss Through an Assembly (Q = A·ΔT/R)
Q˙=A ΔTRtot\dot{Q} = \frac{A \, \Delta T}{R_{tot}}
Steady heat loss through a wall, roof or floor from its area, the inside-to-outside temperature…
Thermodynamics & Heat Transfer-41 Heat Flux Through Insulation (q = ΔT/R)
q′′=ΔTRq'' = \frac{\Delta T}{R}
Heat flow per unit area through an insulated assembly, straight from the temperature difference and…
Thermodynamics & Heat Transfer-42 Effective R-Value with Framing (Parallel Path)
1Reff=ffrRfr+1−ffrRcav\frac{1}{R_{eff}} = \frac{f_{fr}}{R_{fr}} + \frac{1 - f_{fr}}{R_{cav}}
Whole-wall R-value once the studs are counted, area-weighting the framing and cavity paths as…
Thermodynamics & Heat Transfer-43 Conduction Through a Pipe Wall
Q˙=2πkL ΔTln⁡(r2/r1)\dot{Q} = \frac{2 \pi k L \, \Delta T}{\ln(r_2 / r_1)}
Radial conduction through a cylindrical pipe or insulation layer, where the area grows outward so…
Thermodynamics & Heat Transfer-44 Critical Radius of Insulation
rcr=khr_{cr} = \frac{k}{h}
The outer radius below which adding insulation to a small cylinder increases heat loss, because…
Module 4 · Convection, Radiation & Heat Exchangers
Thermodynamics & Heat Transfer-45 Newton's Law of Cooling (Q = hAΔT)
Q˙=hA ΔT\dot{Q} = h A \, \Delta T
Convective heat rate from a surface, set by the film coefficient, the wetted area and the…
Thermodynamics & Heat Transfer-46 Convection Film Resistance
R=1hAR = \frac{1}{h A}
Thermal resistance of a boundary-layer film in kelvin per watt, the reciprocal of the film…
Thermodynamics & Heat Transfer-47 Reynolds Number
Re=ρvDμRe = \frac{\rho v D}{\mu}
The dimensionless ratio of inertial to viscous forces that decides laminar versus turbulent flow.
Thermodynamics & Heat Transfer-48 Prandtl Number
Pr=μcpk\mathrm{Pr} = \frac{\mu c_p}{k}
Fluid property group comparing how fast momentum diffuses with how fast heat does, setting the…
Thermodynamics & Heat Transfer-49 Nusselt Number
Nu=hLk\mathrm{Nu} = \frac{h L}{k}
Dimensionless convection coefficient: the ratio of convective transfer at a surface to pure…
Thermodynamics & Heat Transfer-50 Grashof Number
Gr=g β ΔT L3ν2\mathrm{Gr} = \frac{g \, \beta \, \Delta T \, L^{3}}{\nu^{2}}
Buoyancy against viscosity: how hard the density difference a temperature difference creates pushes…
Thermodynamics & Heat Transfer-51 Rayleigh Number
Ra=Gr Pr\mathrm{Ra} = \mathrm{Gr} \, \mathrm{Pr}
Grashof multiplied by Prandtl, and the number that actually decides whether a still fluid starts to…
Thermodynamics & Heat Transfer-52 Dittus-Boelter Correlation
Nu=0.023 Re0.8 Prn\mathrm{Nu} = 0.023 \, \mathrm{Re}^{0.8} \, \mathrm{Pr}^{n}
Turbulent tube-flow Nusselt number, valid for Re above 10,000, Pr from 0.6 to 160 and L/D over 10,…
Thermodynamics & Heat Transfer-53 Stefan-Boltzmann Law
P=εσAT4P = \varepsilon \sigma A T^4
Power radiated by a hot surface, using the Stefan–Boltzmann constant σ = 5.670374419×10⁻⁸ W/(m²·K⁴).
Thermodynamics & Heat Transfer-54 Net Radiation Exchange Between Surfaces
Q˙=εσA(T14−T24)\dot{Q} = \varepsilon \sigma A (T_1^4 - T_2^4)
Net radiant heat from a grey surface to large surroundings, using the Stefan-Boltzmann constant and…
Thermodynamics & Heat Transfer-55 Combined Convection and Radiation Coefficient
ht=hc+εσ(Ts+Tsur)(Ts2+Tsur2)h_t = h_c + \varepsilon \sigma (T_s + T_{sur})(T_s^2 + T_{sur}^2)
Total surface coefficient adding a linearised radiation term to the convective film, so one h…
Thermodynamics & Heat Transfer-56 Overall Heat Transfer Coefficient (U)
1U=1hi+Lk+1ho\frac{1}{U} = \frac{1}{h_i} + \frac{L}{k} + \frac{1}{h_o}
Overall coefficient U for a plane wall with fluid on both sides, adding the inside film, the wall…
Thermodynamics & Heat Transfer-57 Fouled Overall Coefficient
1Uf=1Uc+1hf\frac{1}{U_f} = \frac{1}{U_c} + \frac{1}{h_f}
Adds a fouling deposit as one more resistance in series, reducing the clean overall coefficient to…
Thermodynamics & Heat Transfer-58 Fouling Factor on an Overall Coefficient
1Uf=1Uc+Rf\frac{1}{U_f} = \frac{1}{U_c} + R_f
Service-condition U of a heat exchanger, adding the TEMA fouling factor as an extra area-specific…
Thermodynamics & Heat Transfer-59 Log Mean Temperature Difference (Counterflow)
ΔTlm=ΔT1−ΔT2ln⁡(ΔT1/ΔT2)\Delta T_{lm} = \frac{\Delta T_1 - \Delta T_2}{\ln(\Delta T_1 / \Delta T_2)}
Effective driving temperature difference in a counterflow exchanger, from the terminal differences…
Thermodynamics & Heat Transfer-60 Log Mean Temperature Difference (Parallel Flow)
ΔTlm=ΔT1−ΔT2ln⁡(ΔT1/ΔT2)\Delta T_{lm} = \frac{\Delta T_1 - \Delta T_2}{\ln(\Delta T_1 / \Delta T_2)}
Effective driving temperature difference when both streams enter at the same end, pairing the two…
Thermodynamics & Heat Transfer-61 Heat Exchanger Duty (Q = U·A·F·LMTD)
Q˙=UAF ΔTlm\dot{Q} = U A F \, \Delta T_{lm}
The LMTD design equation with the correction factor F, which derates the counterflow driving force…
Thermodynamics & Heat Transfer-62 Stream Duty from Mass Flow (Q = ṁcΔT)
Q˙=m˙ cp ΔT\dot{Q} = \dot{m} \, c_p \, \Delta T
Heat picked up or given off by one exchanger stream, from its mass flow, specific heat and the…
Thermodynamics & Heat Transfer-63 Number of Transfer Units (NTU)
NTU=UAm˙ cp\mathrm{NTU} = \frac{U A}{\dot{m} \, c_p}
Dimensionless size of an exchanger: its conductance UA divided by the heat capacity rate of the…
Thermodynamics & Heat Transfer-64 Capacity Rate Ratio (Cr)
Cr=m˙mincminm˙maxcmaxC_r = \frac{\dot{m}_{min} c_{min}}{\dot{m}_{max} c_{max}}
Ratio of the smaller to the larger stream heat capacity rate ṁcₚ, the second dimensionless group…
Thermodynamics & Heat Transfer-65 Maximum Possible Heat Transfer (Qmax)
Q˙max=m˙mincmin(Th,in−Tc,in)\dot{Q}_{max} = \dot{m}_{min} c_{min} (T_{h,in} - T_{c,in})
The thermodynamic ceiling on exchanger duty: the minimum capacity rate multiplied by the full…
Thermodynamics & Heat Transfer-66 Heat Exchanger Effectiveness (ε = Q/Qmax)
ε=Q˙Q˙max\varepsilon = \frac{\dot{Q}}{\dot{Q}_{max}}
Effectiveness as the ratio of actual duty to the thermodynamic maximum, the performance figure that…
Thermodynamics & Heat Transfer-67 Effectiveness from NTU (Counterflow)
ε=1−e−NTU(1−Cr)1−Cr e−NTU(1−Cr)\varepsilon = \frac{1 - e^{-\mathrm{NTU}(1 - C_r)}}{1 - C_r \, e^{-\mathrm{NTU}(1 - C_r)}}
Counterflow effectiveness from the two dimensionless groups NTU and Cr, valid for any Cr from 0 to…
Thermodynamics & Heat Transfer-68 Biot Number
Bi=hLck\mathrm{Bi} = \frac{h L_c}{k}
Ratio of internal conduction resistance to surface convection resistance; below 0.1 a body may be…
Thermodynamics & Heat Transfer-69 Fourier Number
Fo=k tρ c L2\mathrm{Fo} = \frac{k \, t}{\rho \, c \, L^{2}}
Dimensionless time for transient conduction, written from conductivity, density and specific heat…
Thermodynamics & Heat Transfer-70 Lumped Capacitance Time Constant
τ=ρVchA\tau = \frac{\rho V c}{h A}
Thermal time constant of a body cooling at uniform temperature, its stored heat per kelvin divided…
Thermodynamics & Heat Transfer-71 Lumped Capacitance Cooling Curve
T=T∞+(T0−T∞)e−t/τT = T_\infty + (T_0 - T_\infty) e^{-t/\tau}
Exponential temperature history of a body at uniform temperature, and the time it needs to reach…