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Heat Transfer formula solvers

Newton's Law of Cooling (Q = hAΔT)

Q˙=hA ΔT\dot{Q} = h A \, \Delta T

Heat TransferThermodynamicsHVAC & HydronicsConvective heat rate from a surface, set by the film coefficient, the wetted area and the surface-to-fluid temperature difference.

Overall Heat Transfer Coefficient (U)

1U=1hi+Lk+1ho\frac{1}{U} = \frac{1}{h_i} + \frac{L}{k} + \frac{1}{h_o}

Heat TransferThermodynamicsHVAC & HydronicsOverall coefficient U for a plane wall with fluid on both sides, adding the inside film, the wall and the outside film as resistances in series.

Thermal Resistance of a Plane Wall

R=LkAR = \frac{L}{k A}

Heat TransferThermodynamicsConduction resistance of a flat slab in kelvin per watt, from its thickness, thermal conductivity and the area heat crosses.

Convection Film Resistance

R=1hAR = \frac{1}{h A}

Heat TransferThermodynamicsHVAC & HydronicsThermal resistance of a boundary-layer film in kelvin per watt, the reciprocal of the film coefficient times the wetted surface area.

Thermal Resistances in Series

Rtot=R1+R2+R3R_{tot} = R_1 + R_2 + R_3

Heat TransferThermodynamicsTotal resistance of a composite wall, where the same heat crosses each layer in turn so the layer resistances simply add.

Heat Flow from Thermal Resistance

Q˙=ΔTR\dot{Q} = \frac{\Delta T}{R}

Heat TransferThermodynamicsHVAC & HydronicsOhm's law for heat: the flow through an assembly equals the temperature difference across it divided by its total thermal resistance.

Overall U from Total Resistance

U=1RtotAU = \frac{1}{R_{tot} A}

Heat TransferThermodynamicsHVAC & HydronicsConverts an assembly's total resistance in kelvin per watt into the overall coefficient U quoted on exchanger and envelope datasheets.

Conduction Through a Pipe Wall

Q˙=2πkL ΔTln⁡(r2/r1)\dot{Q} = \frac{2 \pi k L \, \Delta T}{\ln(r_2 / r_1)}

Heat TransferThermodynamicsHVAC & HydronicsRadial conduction through a cylindrical pipe or insulation layer, where the area grows outward so the resistance follows a logarithm.

Critical Radius of Insulation

rcr=khr_{cr} = \frac{k}{h}

Heat TransferThermodynamicsThe outer radius below which adding insulation to a small cylinder increases heat loss, because added surface beats added resistance.

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)}

Heat TransferThermodynamicsHVAC & HydronicsEffective driving temperature difference in a counterflow exchanger, from the terminal differences at the hot and cold ends of the shell.

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 TransferThermodynamicsHVAC & HydronicsEffective driving temperature difference when both streams enter at the same end, pairing the two inlets and the two outlets.

Heat Exchanger Duty (Q = U·A·F·LMTD)

Q˙=UAF ΔTlm\dot{Q} = U A F \, \Delta T_{lm}

Heat TransferThermodynamicsHVAC & HydronicsThe LMTD design equation with the correction factor F, which derates the counterflow driving force for shell-and-tube or crossflow arrangements.

Stream Duty from Mass Flow (Q = ṁcΔT)

Q˙=m˙ cp ΔT\dot{Q} = \dot{m} \, c_p \, \Delta T

Heat TransferThermodynamicsHVAC & HydronicsHeat picked up or given off by one exchanger stream, from its mass flow, specific heat and the temperature change across the unit.

Number of Transfer Units (NTU)

NTU=UAm˙ cp\mathrm{NTU} = \frac{U A}{\dot{m} \, c_p}

Heat TransferThermodynamicsDimensionless size of an exchanger: its conductance UA divided by the heat capacity rate of the minimum stream, ṁ times its specific heat.

Capacity Rate Ratio (Cr)

Cr=m˙mincminm˙maxcmaxC_r = \frac{\dot{m}_{min} c_{min}}{\dot{m}_{max} c_{max}}

Heat TransferThermodynamicsRatio of the smaller to the larger stream heat capacity rate ṁcₚ, the second dimensionless group the effectiveness-NTU method needs.

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)}}

Heat TransferThermodynamicsCounterflow effectiveness from the two dimensionless groups NTU and Cr, valid for any Cr from 0 to 1 with the balanced case handled as a limit.

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 TransferThermodynamicsThe thermodynamic ceiling on exchanger duty: the minimum capacity rate multiplied by the full inlet-to-inlet temperature difference.

Heat Exchanger Effectiveness (ε = Q/Qmax)

ε=Q˙Q˙max\varepsilon = \frac{\dot{Q}}{\dot{Q}_{max}}

Heat TransferThermodynamicsHVAC & HydronicsEffectiveness as the ratio of actual duty to the thermodynamic maximum, the performance figure that needs no outlet temperatures to interpret.

Fouled Overall Coefficient

1Uf=1Uc+1hf\frac{1}{U_f} = \frac{1}{U_c} + \frac{1}{h_f}

Heat TransferThermodynamicsHVAC & HydronicsAdds a fouling deposit as one more resistance in series, reducing the clean overall coefficient to the fouled value used for design margin.

Fin Parameter mL (Straight Fin)

mL=L2hktmL = L \sqrt{\frac{2h}{k t}}

Heat TransferThermodynamicsThe dimensionless group governing straight-fin performance, combining fin length, thickness, material conductivity and the surface film coefficient.

Fin Efficiency (Straight Fin)

ηf=tanh⁡(mL)mL\eta_f = \frac{\tanh(mL)}{mL}

Heat TransferThermodynamicsEfficiency of a straight fin with an adiabatic tip, comparing its real duty with the duty it would give if it were all at base temperature.

Fin Heat Transfer Rate

Q˙f=ηf hAf ΔTb\dot{Q}_f = \eta_f \, h A_f \, \Delta T_b

Heat TransferThermodynamicsHVAC & HydronicsDuty of a fin or finned surface: the ideal convective rate over the whole fin area, derated by the fin efficiency.

Biot Number

Bi=hLck\mathrm{Bi} = \frac{h L_c}{k}

Heat TransferThermodynamicsRatio of internal conduction resistance to surface convection resistance; below 0.1 a body may be treated as having one uniform temperature.

Fourier Number

Fo=k tρ c L2\mathrm{Fo} = \frac{k \, t}{\rho \, c \, L^{2}}

Heat TransferThermodynamicsDimensionless time for transient conduction, written from conductivity, density and specific heat so no diffusivity input is needed.

Lumped Capacitance Time Constant

τ=ρVchA\tau = \frac{\rho V c}{h A}

Heat TransferThermodynamicsThermal time constant of a body cooling at uniform temperature, its stored heat per kelvin divided by the surface conductance hA.

Lumped Capacitance Cooling Curve

T=T∞+(T0−T∞)e−t/τT = T_\infty + (T_0 - T_\infty) e^{-t/\tau}

Heat TransferThermodynamicsExponential temperature history of a body at uniform temperature, and the time it needs to reach any temperature between start and ambient.

Prandtl Number

Pr=μcpk\mathrm{Pr} = \frac{\mu c_p}{k}

Heat TransferFluid MechanicsThermodynamicsFluid property group comparing how fast momentum diffuses with how fast heat does, setting the relative thickness of the two boundary layers.

Nusselt Number

Nu=hLk\mathrm{Nu} = \frac{h L}{k}

Heat TransferFluid MechanicsThermodynamicsDimensionless convection coefficient: the ratio of convective transfer at a surface to pure conduction through the same fluid layer.

Dittus-Boelter Correlation

Nu=0.023 Re0.8 Prn\mathrm{Nu} = 0.023 \, \mathrm{Re}^{0.8} \, \mathrm{Pr}^{n}

Heat TransferFluid MechanicsTurbulent tube-flow Nusselt number, valid for Re above 10,000, Pr from 0.6 to 160 and L/D over 10, with n = 0.4 heating and 0.3 cooling.

Net Radiation Exchange Between Surfaces

Q˙=εσA(T14−T24)\dot{Q} = \varepsilon \sigma A (T_1^4 - T_2^4)

Heat TransferThermodynamicsNet radiant heat from a grey surface to large surroundings, using the Stefan-Boltzmann constant and the difference of fourth-power temperatures.

View Factor Reciprocity

A1F1→2=A2F2→1A_1 F_{1 \to 2} = A_2 F_{2 \to 1}

Heat TransferThermodynamicsReciprocity relation for radiation view factors, which lets you recover the unknown factor between two surfaces from the known one and their areas.

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)

Heat TransferThermodynamicsHVAC & HydronicsTotal surface coefficient adding a linearised radiation term to the convective film, so one h covers both mechanisms over a modest ΔT.

R-Value of an Insulation Layer (R = L/k)

R=LkR = \frac{L}{k}

Heat TransferHVAC & HydronicsThermodynamicsArea-specific thermal resistance of one layer, from its thickness and thermal conductivity — the RSI or R-value quoted on every insulation label.

Total R-Value of an Assembly

Rtot=R1+R2+R3R_{tot} = R_1 + R_2 + R_3

Heat TransferHVAC & HydronicsThermodynamicsTotal R-value of a wall, roof or floor built up from three layers in series, where the same heat crosses each layer so the R-values simply add.

U-Factor from Total R-Value (U = 1/R)

U=1RtotU = \frac{1}{R_{tot}}

Heat TransferHVAC & HydronicsThermodynamicsConverts an assembly's total R-value into the U-factor used by energy codes and window labels, and back — the two are simple reciprocals.

Heat Loss Through an Assembly (Q = A·ΔT/R)

Q˙=A ΔTRtot\dot{Q} = \frac{A \, \Delta T}{R_{tot}}

Heat TransferHVAC & HydronicsThermodynamicsSteady heat loss through a wall, roof or floor from its area, the inside-to-outside temperature difference and the assembly's total R-value.

Heat Flux Through Insulation (q = ΔT/R)

q′′=ΔTRq'' = \frac{\Delta T}{R}

Heat TransferHVAC & HydronicsThermodynamicsHeat flow per unit area through an insulated assembly, straight from the temperature difference and the R-value, with no area needed.

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}}

Heat TransferHVAC & HydronicsThermodynamicsWhole-wall R-value once the studs are counted, area-weighting the framing and cavity paths as parallel conductances rather than averaging their R-values.

Fouling Factor on an Overall Coefficient

1Uf=1Uc+Rf\frac{1}{U_f} = \frac{1}{U_c} + R_f

Heat TransferThermodynamicsHVAC & HydronicsService-condition U of a heat exchanger, adding the TEMA fouling factor as an extra area-specific resistance on top of the clean coefficient.