Heat Transfer formula solvers

Categories — 73 of them, 1787 solvers in all

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A B C D E F G H I J L M N O P R S T U V

Albedo and Reflected Radiation

G_r = \alpha\,G

Heat TransferSolar & Wind PowerShortwave radiation reflected by a surface, as its albedo times the incident irradiance. Whatever is not reflected is absorbed, so the complement (1 − α)G is the part that actually warms the ground — and it is reported alongside every answer here.

Biot Number

\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.

Brinkman Number

\mathrm{Br} = \frac{\mu \, v^{2}}{k \, \Delta T}

Heat TransferHeat generated by shearing the fluid against heat carried away by conduction. It is why a polymer melt leaves a die hotter than it entered, why a journal bearing runs warm on its own oil, and why viscous fluids in small passages cannot be treated as isothermal.

Capacity Rate Ratio (Cr)

C_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.

Clear-Sky Temperature (Berdahl-Martin)

T_{sky} = T_{air}\left[\,0.711 + 0.56\left(\tfrac{t_{dp}}{100}\right) + 0.73\left(\tfrac{t_{dp}}{100}\right)^{2}\right]^{1/4}

Air Quality & DispersionHeat TransferEffective radiating temperature of a cloudless sky, from air temperature and dew point, using the Berdahl-Martin (1984) clear-sky emissivity. This is why frost forms on a clear night and not a cloudy one: a clear sky radiates like a body tens of degrees colder than the air.

Combined Convection and Radiation Coefficient

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.

Conduction Through a Pipe Wall

\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.

Convection Film Resistance

R = \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.

Critical Radius of Insulation

r_{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.

Decrement Factor from Damping Depth

f = e^{-x/d}

Heat TransferThe fraction of the outdoor temperature swing that still exists at depth x into the material. It falls exponentially with thickness measured in damping depths, so the first few centimetres do most of the work and the last few do almost none. Not to be confused with the logarithmic decrement of a damped vibration, which is an unrelated quantity that happens to share a word.

Decrement Factor from Diffusivity and Period

f = \exp\!\left(-x\sqrt{\frac{\pi}{\alpha P}}\right)

Heat TransferThe same amplitude ratio as the decrement-factor page, written straight from the material's diffusivity and the length of the cycle so no damping depth has to be computed first. Handy when comparing two materials, or the same material against the daily and the annual swing.

Dittus-Boelter Correlation

\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.

Eckert Number

\mathrm{Ec} = \frac{v^{2}}{c_p \, \Delta T}

Heat TransferKinetic energy against enthalpy: whether the energy carried in a flow's motion is large enough, compared with the heat the temperature difference is already moving, that slowing the flow down heats it noticeably. The group that says when viscous heating has to be in the energy equation.

Effective R-Value with Framing (Parallel Path)

\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.

Effectiveness from NTU (Counterflow)

\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.

Fin Efficiency (Straight Fin)

\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

\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.

Fin Parameter mL (Straight Fin)

mL = 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.

Fouled Overall Coefficient

\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.

Fouling Factor on an Overall Coefficient

\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.

Fourier Number

\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.

Graetz Number

\mathrm{Gz} = \frac{D}{L} \, \mathrm{Re} \, \mathrm{Pr}

Heat TransferHow far into a duct the temperature profile is still developing. It compares the time a fluid spends in the tube with the time heat needs to diffuse from the wall to the centreline, and it decides whether an entry-length correlation or a fully developed one is the right tool.

Grashof Number

\mathrm{Gr} = \frac{g \, \beta \, \Delta T \, L^{3}}{\nu^{2}}

Heat TransferBuoyancy against viscosity: how hard the density difference a temperature difference creates pushes a fluid, compared with how hard the fluid's own stickiness resists. The natural-convection counterpart of the Reynolds number, and the starting point for every hot-wall-in-still-air calculation.

Heat Exchanger Duty (Q = U·A·F·LMTD)

\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.

Heat Exchanger Effectiveness (ε = Q/Qmax)

\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.

Heat Flow from Thermal Resistance

\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.

Heat Flux Through Insulation (q = ΔT/R)

q'' = \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.

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

\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.

Inside Swing Amplitude

A_i = f \, A_o

Heat TransferThe decrement factor turned into degrees. Multiply the outdoor swing by the fraction that survives the wall and you have the swing at the inside face — the number a room's occupant actually experiences, as opposed to a dimensionless ratio nobody can feel.

Jakob Number

\mathrm{Ja} = \frac{c_p \, \Delta T}{h_{fg}}

Heat TransferSensible heat against latent heat: how much of the energy crossing a boiling or condensing surface goes into changing the fluid's temperature rather than changing its phase. Small in almost every practical case, which is what makes phase change such an efficient way to move heat.

Lewis Number

\mathrm{Le} = \frac{\alpha}{D}

Heat TransferThermal diffusivity divided by mass diffusivity: whether heat or molecules spread faster through the same fluid. Equal to Schmidt divided by Prandtl, and the group that lets a wet-bulb thermometer, a cooling tower and a drying oven be analysed by one set of equations.

Log Mean Temperature Difference (Counterflow)

\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)

\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.

Lumped Capacitance Cooling Curve

T = 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.

Lumped Capacitance Time Constant

\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.

Maximum Possible Heat Transfer (Qmax)

\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.

Net Radiation Exchange Between Surfaces

\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.

Net Radiative Cooling to the Sky

q_{net} = \varepsilon \sigma \left(T_s^{4} - T_{sky}^{4}\right)

Heat TransferThermodynamicsNet longwave heat lost per square metre from a surface facing the open sky, as the difference of fourth powers between the surface and the effective sky temperature. On a clear night this runs to 60-100 W/m², which is what pulls a surface below air temperature.

Newton's Law of Cooling (Q = hAΔT)

\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.

Number of Transfer Units (NTU)

\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.

Nusselt Number

\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.

Overall Heat Transfer Coefficient (U)

\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.

Overall U from Total Resistance

U = \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.

Periodic Temperature Profile

T(x,t) = T_m + A \, e^{-x/d} \sin\!\left(\frac{2\pi t}{P} - \frac{x}{d}\right)

Heat TransferThe full harmonic solution: the temperature at depth x and time t when the surface swings sinusoidally about a mean. The exponential is the decrement factor and the term subtracted inside the sine is the phase lag, so this one line contains both of the previous pages — and it is what you use when you want a temperature rather than a ratio.

Planetary Effective Temperature

T_e = \left(\frac{S\,(1 - \alpha)}{4\sigma}\right)^{1/4}

Astronomy & GravitationPhysicsHeat TransferThe temperature a planet would sit at if it were a bare blackbody sphere: absorbed sunlight balanced against thermal radiation, so the whole of a world's climate reduces to two numbers — the sunlight reaching it and the fraction it throws straight back.

Prandtl Number

\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.

R-Value of a Snow Wall

R = \frac{L}{0.138 - 1.01\,\rho^{*} + 3.233\,\rho^{*2}}

Snow & IceHeat TransferThermodynamicsArea-specific thermal resistance of a snow wall, straight from its thickness and its density, with Sturm's conductivity fit folded in so you never have to look k up. It answers the question a shelter builder actually asks — how much wall is worth building — and nothing whatsoever about whether the shelter is safe.

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

R = \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.

Rayleigh Number

\mathrm{Ra} = \mathrm{Gr} \, \mathrm{Pr}

Heat TransferGrashof multiplied by Prandtl, and the number that actually decides whether a still fluid starts to move. Natural-convection correlations turn out to depend on this product rather than on either factor alone, which is a finding about the physics and not merely a definition.

Stanton Number for Heat Transfer

\mathrm{St} = \frac{h}{\rho \, v \, c_p}

Heat TransferThe convection coefficient measured against the heat the stream is carrying past: what fraction of the oncoming flow's thermal capacity is actually delivered to the wall. The thermal twin of the mass-transfer Stanton number, and the group the Colburn j-factor is written in.

Stefan Frost Penetration Depth

X = \sqrt{\frac{2 k_f \, n \, \Delta T \, t}{L}}

Heat TransferJosef Stefan's 1891 result for how deep a freezing front reaches into wet ground. All the heat removed goes into freezing the pore water, so the depth grows as the square root of the accumulated freezing index — twice the cold season for only 1.4 times the depth. The classic first estimate for a frost line, a footing depth, or the cover a buried water line needs.

Stream Duty from Mass Flow (Q = ṁcΔT)

\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.

Surface Temperature Depression Under a Clear Sky

T_s = T_{air} - \frac{q_{net}}{h_c}

Heat TransferAir Quality & DispersionHow far a surface settles below air temperature when it loses net radiation to a clear sky and gains heat back only by convection. This is why frost appears on a windscreen while the porch thermometer reads a few degrees above freezing.

Thermal Conductivity of Snow (Sturm 1997)

k_{\mathrm{eff}} = 0.138 - 1.01\,\rho^{*} + 3.233\,\rho^{*2}

Snow & IceHeat TransferThermodynamicsEffective thermal conductivity of seasonal snow from its density alone, using the quadratic fit Sturm and colleagues published in 1997 from 488 measurements. This is the equation behind the fact that a snow shelter works: at 300 kg/m³ snow conducts about the same heat as softwood, and the air trapped between the grains is doing almost all of it.

Thermal Damping Depth

d = \sqrt{\frac{\alpha P}{\pi}}

Heat TransferThe depth at which a repeating temperature swing has faded to 1/e — about 37% — of its size at the surface. It is the natural ruler for every periodic heat problem: a wall thin compared with it barely delays anything, and a wall several times thicker than it has effectively swallowed the cycle.

Thermal Diffusivity

\alpha = \frac{k}{\rho \, c}

Heat TransferHow fast a temperature change travels through a material, as distinct from how much heat the material lets through. Conductivity divided by volumetric heat capacity: the numerator carries heat forward, the denominator soaks it up along the way. Every periodic and transient result on this page is built on it.

Thermal Resistance of a Plane Wall

R = \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.

Thermal Resistances in Series

R_{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.

Thermal Time Lag

\varphi = \frac{x}{2}\sqrt{\frac{P}{\pi \alpha}}

Heat TransferHow many hours late the peak of a repeating temperature swing arrives at depth x. This is the number behind every thick-wall building on earth: the afternoon heat that lands on the outside face does not reach the room until the evening, and by then the outside has cooled and takes it back.

Thermal Time Lag from Damping Depth

\varphi = \frac{x P}{2 \pi d}

Heat TransferThe same time lag, expressed through the damping depth instead of the diffusivity. In this form the structure of the answer is plain: the delay is the number of damping depths travelled, divided by 2π, times the period. One damping depth costs one radian of phase, and one radian of a 24-hour cycle is 3.82 hours.

Total R-Value of an Assembly

R_{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 = \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.

View Factor Reciprocity

A_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.