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Thermodynamics & Heat Transfer — formula sheet

Engineering thermodynamics, steam plant & heat transfer · 70 formulas · metric edition 1

The full booklet — learning zone, problems, answer key
Gauge and Absolute Pressure
Pabs=Pgauge+PatmP_{abs} = P_{gauge} + P_{atm}
Sensible Heat (Q = mcΔT)
Q=mcΔTQ = m c \Delta T
Latent Heat
Q=mLQ = m L
Boyle's Law
P1V1=P2V2P_1 V_1 = P_2 V_2
Charles's Law
V1T1=V2T2\frac{V_1}{T_1} = \frac{V_2}{T_2}
Gay-Lussac's Law
P1T1=P2T2\frac{P_1}{T_1} = \frac{P_2}{T_2}
Combined Gas Law
P1V1T1=P2V2T2\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}
Ideal Gas Law
PV=nRTP V = n R T
Gas Density from Molar Mass
ρ=PMRT\rho = \frac{PM}{RT}
Compressibility Factor (Z = PV/nRT)
Z=PVnRTZ = \frac{P V}{n R T}
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
Antoine Equation (Vapour Pressure)
log⁡10P=A−BC+T\log_{10} P = A - \frac{B}{C + T}
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)
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})
Steam Quality from Enthalpy
x=h−hfhfgx = \frac{h - h_f}{h_{fg}}
Wet Steam Enthalpy from h_f and h_g
h=(1−x) hf+x hgh = (1 - x) \, h_f + x \, h_g
Degrees of Superheat
ΔTsh=T−Ts(p)\Delta T_{sh} = T - T_s(p)
Flash Steam Percentage
%F=hf1−hf2hfg2×100\%F = \frac{h_{f1} - h_{f2}}{h_{fg2}} \times 100
Flash Steam Mass Rate
m˙f=%F100 m˙c\dot{m}_f = \frac{\%F}{100} \, \dot{m}_c
Steam Turbine Specific Work
w=h1−h2w = h_1 - h_2
Steam Turbine Power Output
P=m˙ wP = \dot{m} \, w
Turbine Isentropic Efficiency
ηisen=h1−h2h1−h2s\eta_{isen} = \frac{h_1 - h_2}{h_1 - h_{2s}}
Napier's Steam Leak Rate
m˙=A P70\dot{m} = \frac{A \, P}{70}
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}
Steam Coil Condensate Load
m˙=Q˙hfg\dot m = \frac{\dot Q}{h_{fg}}
Thermal Efficiency
η=WQh\eta = \frac{W}{Q_h}
Carnot Efficiency
η=1−TcTh\eta = 1 - \frac{T_c}{T_h}
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}
Otto Cycle Efficiency (Compression Ratio)
η=1−1rγ−1\eta = 1 - \frac{1}{r^{\gamma - 1}}
Brayton Cycle Efficiency (Pressure Ratio)
η=1−1rp(γ−1)/γ\eta = 1 - \frac{1}{r_p^{\left(\gamma - 1\right)/\gamma}}
Heat Conduction Rate
P=kAΔTdP = \tfrac{k A \Delta T}{d}
R-Value of an Insulation Layer (R = L/k)
R=LkR = \frac{L}{k}
Thermal Resistance of a Plane Wall
R=LkAR = \frac{L}{k A}
Thermal Resistances in Series
Rtot=R1+R2+R3R_{tot} = R_1 + R_2 + R_3
Total R-Value of an Assembly
Rtot=R1+R2+R3R_{tot} = R_1 + R_2 + R_3
U-Factor from Total R-Value (U = 1/R)
U=1RtotU = \frac{1}{R_{tot}}
Overall U from Total Resistance
U=1RtotAU = \frac{1}{R_{tot} A}
Heat Flow from Thermal Resistance
Q˙=ΔTR\dot{Q} = \frac{\Delta T}{R}
Heat Loss Through an Assembly (Q = A·ΔT/R)
Q˙=A ΔTRtot\dot{Q} = \frac{A \, \Delta T}{R_{tot}}
Heat Flux Through Insulation (q = ΔT/R)
q′′=ΔTRq'' = \frac{\Delta T}{R}
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}}
Conduction Through a Pipe Wall
Q˙=2πkL ΔTln⁡(r2/r1)\dot{Q} = \frac{2 \pi k L \, \Delta T}{\ln(r_2 / r_1)}
Critical Radius of Insulation
rcr=khr_{cr} = \frac{k}{h}
Newton's Law of Cooling (Q = hAΔT)
Q˙=hA ΔT\dot{Q} = h A \, \Delta T
Convection Film Resistance
R=1hAR = \frac{1}{h A}
Reynolds Number
Re=ρvDμRe = \frac{\rho v D}{\mu}
Prandtl Number
Pr=μcpk\mathrm{Pr} = \frac{\mu c_p}{k}
Nusselt Number
Nu=hLk\mathrm{Nu} = \frac{h L}{k}
Grashof Number
Gr=g β ΔT L3ν2\mathrm{Gr} = \frac{g \, \beta \, \Delta T \, L^{3}}{\nu^{2}}
Rayleigh Number
Ra=Gr Pr\mathrm{Ra} = \mathrm{Gr} \, \mathrm{Pr}
Dittus-Boelter Correlation
Nu=0.023 Re0.8 Prn\mathrm{Nu} = 0.023 \, \mathrm{Re}^{0.8} \, \mathrm{Pr}^{n}
Stefan-Boltzmann Law
P=εσAT4P = \varepsilon \sigma A T^4
Net Radiation Exchange Between Surfaces
Q˙=εσA(T14−T24)\dot{Q} = \varepsilon \sigma A (T_1^4 - T_2^4)
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)
Overall Heat Transfer Coefficient (U)
1U=1hi+Lk+1ho\frac{1}{U} = \frac{1}{h_i} + \frac{L}{k} + \frac{1}{h_o}
Fouled Overall Coefficient
1Uf=1Uc+1hf\frac{1}{U_f} = \frac{1}{U_c} + \frac{1}{h_f}
Fouling Factor on an Overall Coefficient
1Uf=1Uc+Rf\frac{1}{U_f} = \frac{1}{U_c} + R_f
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)}
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)}
Heat Exchanger Duty (Q = U·A·F·LMTD)
Q˙=UAF ΔTlm\dot{Q} = U A F \, \Delta T_{lm}
Stream Duty from Mass Flow (Q = ṁcΔT)
Q˙=m˙ cp ΔT\dot{Q} = \dot{m} \, c_p \, \Delta T
Number of Transfer Units (NTU)
NTU=UAm˙ cp\mathrm{NTU} = \frac{U A}{\dot{m} \, c_p}
Capacity Rate Ratio (Cr)
Cr=m˙mincminm˙maxcmaxC_r = \frac{\dot{m}_{min} c_{min}}{\dot{m}_{max} c_{max}}
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})
Heat Exchanger Effectiveness (ε = Q/Qmax)
ε=Q˙Q˙max\varepsilon = \frac{\dot{Q}}{\dot{Q}_{max}}
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)}}
Biot Number
Bi=hLck\mathrm{Bi} = \frac{h L_c}{k}
Fourier Number
Fo=k tρ c L2\mathrm{Fo} = \frac{k \, t}{\rho \, c \, L^{2}}
Lumped Capacitance Time Constant
τ=ρVchA\tau = \frac{\rho V c}{h A}
Lumped Capacitance Cooling Curve
T=T∞+(T0−T∞)e−t/τT = T_\infty + (T_0 - T_\infty) e^{-t/\tau}