Engineering reference // heat transfer

Drawing No. EH–TH–019 // Thermal Engineering & HVAC

Heat Transfer Formula Sheet

A practical reference for the equations that connect temperature difference, thermal resistance, heat flux, surface convection, radiation and heat-exchanger performance. The sheet focuses on single-phase engineering heat transfer and clearly separates steady-state and transient assumptions.

Fast reference, with engineering context

Use the equations directly for screening calculations, then open the linked EngineerHub tools for input handling and unit conversion. Formula applicability and major limitations are stated beside each relation.

Start with Conduction Formula summary

Reference conventions

σ = 5.670374419×10⁻⁸Stefan–Boltzmann constant, W/(m²·K⁴)

q̇ > 0Use a stated sign convention for heat flow

T in KAbsolute temperature required for T⁴ radiation

Bi < 0.1Common lumped-capacitance screening criterion

Three modes of heat transfer

Conduction transfers energy through matter, convection couples a surface to moving fluid, and thermal radiation can transfer energy across a vacuum.

01 // Fourier conduction

For steady one-dimensional conduction through a plane wall with constant thermal conductivity, heat rate is proportional to area and temperature difference and inversely proportional to thickness.

Core relation

q̇ = kA(T₁ − T₂)/L

q″ = −k dT/dx

Symbol	Meaning	SI units	US customary
q̇	Heat-transfer rate	W	Btu/h
q″	Heat flux	W/m²	Btu/(h·ft²)
k	Thermal conductivity	W/(m·K)	Btu/(h·ft·°F)
A	Area normal to heat flow	m²	ft²
L	Conduction thickness	m	ft
T	Temperature	°C or K difference	°F or °R difference

Open Wall Heat Transfer Calculator Worked example ↓

Worked example — insulation slab

Given: 10 m² of insulation, k = 0.040 W/(m·K), thickness 0.10 m, and a 20 K temperature difference.

q̇ = kAΔT/L

q̇ = 0.040 × 10 × 20 / 0.10 = 80 W

Answer: q̇ = 80 W.

US check: ≈273 Btu/h.

Use temperature-dependent or apparent conductivity where the material/source requires it.

Fourier’s law is local and general; the compact slab form assumes one-dimensional, steady conduction and constant k.

For cylinders and spheres, resistance depends logarithmically or radially on geometry.
Thermal bridges and multidimensional edge effects can make a plane-wall model optimistic.

02 // Thermal resistance networks

Series thermal resistances can be summed exactly for one-dimensional steady heat flow when the same heat rate passes through each resistance.

Core relation

q̇ = ΔT / R_total

R_cond = L/(kA) ; R_conv = 1/(hA)

Symbol	Meaning	SI units	US customary
R	Thermal resistance	K/W	h·°F/Btu
h	Convection coefficient	W/(m²·K)	Btu/(h·ft²·°F)
k	Thermal conductivity	W/(m·K)	Btu/(h·ft·°F)
A	Reference area	m²	ft²
ΔT	Overall temperature difference	K	°F

Open Wall Heat Transfer Calculator Worked example ↓

Worked example — composite wall with films

Given: 10 m² wall; hᵢ = 8 W/(m²·K), 13 mm plaster with k = 0.16 W/(m·K), 100 mm insulation with k = 0.040 W/(m·K), hₒ = 25 W/(m²·K), and ΔT = 25 K.

R″ = 1/8 + 0.013/0.16 + 0.10/0.040 + 1/25 = 2.7463 m²·K/W

U = 1/R″ = 0.3641 W/(m²·K)

q̇ = UAΔT = 0.3641 × 10 × 25 = 91.0 W

Answer: U ≈ 0.364 W/(m²·K) and q̇ ≈ 91 W.

US check: ≈311 Btu/h.

Area-normalized resistance R″ is convenient for flat assemblies with one common area.

Thermal-contact resistance, fouling resistance and parallel heat paths should be included explicitly when important. In cylinders, each layer has a different area and should use the correct cylindrical resistance.

03 // Newton convection relation

Convection heat transfer between a surface and a fluid is modeled with a heat-transfer coefficient that contains the boundary-layer physics.

Core relation

q̇ = hA(T_s − T_∞)

q″ = h(T_s − T_∞)

Symbol	Meaning	SI units	US customary
h	Convection heat-transfer coefficient	W/(m²·K)	Btu/(h·ft²·°F)
A	Surface area	m²	ft²
Tₛ	Surface temperature	°C	°F
T∞	Bulk/free-stream fluid temperature	°C	°F
q̇	Heat rate	W	Btu/h

Open Heat Transfer Estimator Worked example ↓

Worked example — warm surface in moving air

Given: h = 25 W/(m²·K), A = 2.0 m², surface at 60°C, air at 30°C.

q̇ = 25 × 2.0 × (60 − 30)

q̇ = 1500 W

Answer: q̇ = 1.50 kW from the surface to the air.

US check: ≈5118 Btu/h.

Treat h as a correlation/result, not a universal material constant.

The coefficient h depends on geometry, velocity, fluid properties, orientation, natural/forced convection and sometimes phase change. Use a suitable correlation or validated equipment value.

04 // Thermal radiation to large surroundings

For a gray surface exchanging radiation with a large isothermal enclosure, net radiation depends on emissivity and the fourth power of absolute temperature.

Core relation

q̇_rad = εσA(T_s⁴ − T_sur⁴)

σ = 5.670374419×10⁻⁸ W/(m²·K⁴)

Symbol	Meaning	SI units	US customary
ε	Hemispherical emissivity	dimensionless	dimensionless
σ	Stefan–Boltzmann constant	W/(m²·K⁴)	Btu/(h·ft²·°R⁴)
A	Radiating area	m²	ft²
Tₛ	Absolute surface temperature	K	°R
Tsur	Absolute surrounding temperature	K	°R

Open Heat Transfer Estimator Worked example ↓

Worked example — hot surface to room surroundings

Given: ε = 0.90, A = 1.0 m², surface = 100°C (373.15 K), surroundings = 20°C (293.15 K).

q̇ = 0.90σ(373.15⁴ − 293.15⁴)

q̇ ≈ 612.5 W

Answer: q̇ ≈ 613 W net radiative loss.

US check: ≈2090 Btu/h.

Use absolute temperature in the T⁴ terms.

The compact expression assumes a small gray surface in large surroundings. Exchange between finite surfaces requires view factors and surface-to-surface radiation analysis.

05 // Overall heat-transfer coefficient

An overall coefficient collapses a series of convection, conduction, fouling and other resistances onto a stated reference area.

Core relation

q̇ = UAΔT

1/(UA) = ΣR

Symbol	Meaning	SI units	US customary
U	Overall heat-transfer coefficient	W/(m²·K)	Btu/(h·ft²·°F)
A	Area on which U is defined	m²	ft²
R	Individual thermal resistance	K/W	h·°F/Btu
ΔT	Driving temperature difference	K	°F
UA	Overall conductance	W/K	Btu/(h·°F)

Open Wall Heat Transfer Calculator Worked example ↓

Worked example — overall U from the composite-wall example

Given: R″ = 2.7463 m²·K/W, A = 10 m², ΔT = 25 K.

U = 1/R″ = 0.3641 W/(m²·K)

q̇ = 0.3641 × 10 × 25 = 91.0 W

Answer: U ≈ 0.364 W/(m²·K); q̇ ≈ 91 W.

US check: U ≈0.0641 Btu/(h·ft²·°F).

Always state the area basis when U is used for cylindrical or unequal-area systems.

U is not a pure material property. It includes the particular construction, films, fouling and geometry used in the resistance network.

06 // Log-mean temperature difference

For a heat exchanger with nearly constant U and no phase/heat losses to the surroundings, LMTD represents the correct mean temperature driving force.

Core relation

ΔT_lm = (ΔT₁ − ΔT₂) / ln(ΔT₁/ΔT₂)

q̇ = UA F ΔT_lm

Symbol	Meaning	SI units	US customary
ΔT₁, ΔT₂	Terminal temperature differences	K	°F
F	LMTD correction factor	dimensionless	dimensionless
U	Overall heat-transfer coefficient	W/(m²·K)	Btu/(h·ft²·°F)
A	Heat-transfer area	m²	ft²
q̇	Heat duty	W	Btu/h

Open Heat Exchanger Calculator Worked example ↓

Worked example — counterflow exchanger

Given: Hot stream 150→90°C, cold stream 30→70°C; U = 350 W/(m²·K), A = 12 m², F = 1.

ΔT₁ = 150 − 70 = 80 K; ΔT₂ = 90 − 30 = 60 K

ΔTlm = (80−60)/ln(80/60) = 69.52 K

q̇ = 350 × 12 × 69.52 = 292 kW

Answer: ΔTlm ≈ 69.5 K and q̇ ≈ 292 kW.

US check: ≈0.996 million Btu/h.

Use the correct terminal differences for parallel or counterflow arrangement.

Multipass shell-and-tube and crossflow exchangers may need an LMTD correction factor or an ε–NTU method. If ΔT₁≈ΔT₂, the limiting LMTD is that common temperature difference.

07 // Lumped-capacitance transient

If internal temperature gradients are small, a body can be treated as spatially uniform while it heats or cools by convection.

Core relation

(T − T∞)/(Tᵢ − T∞) = exp[−hAt/(ρVc_p)]

Bi = hL_c/k ; L_c = V/A

Symbol	Meaning	SI units	US customary
t	Elapsed time	s	s or min
ρ	Solid density	kg/m³	lbm/ft³
V	Solid volume	m³	ft³
cₚ	Solid specific heat	J/(kg·K)	Btu/(lbm·°F)
h	Convection coefficient	W/(m²·K)	Btu/(h·ft²·°F)
Bi	Biot number	dimensionless	dimensionless

Open Heat Transfer Estimator Worked example ↓

Worked example — cooling a small steel part

Given: ρ = 7800 kg/m³, V = 1.0×10⁻⁴ m³, cₚ = 470 J/(kg·K), h = 15 W/(m²·K), A = 0.060 m², k = 45 W/(m·K), Tᵢ = 100°C, T∞ = 20°C, t = 300 s.

τ = ρVcₚ/(hA) = 407.3 s

T = 20 + 80 exp(−300/407.3) = 58.3°C

Bi = h(V/A)/k = 5.6×10⁻⁴

Answer: T ≈ 58.3°C after 300 s; Bi is far below 0.1.

US check: ≈136.9°F.

Bi < 0.1 is a common screening condition for lumped analysis.

If Bi is not small, internal conduction resistance matters and a spatial transient solution (e.g., Heisler charts, analytical series or numerical model) is required.

08 // Heat-exchanger effectiveness

Effectiveness compares actual exchanger duty with the maximum thermodynamically possible duty based on the smaller heat-capacity rate.

Core relation

ε = q̇ / q̇_max

q̇_max = C_min(T_h,in − T_c,in)

Symbol	Meaning	SI units	US customary
ε	Heat-exchanger effectiveness	dimensionless	dimensionless
C	Heat-capacity rate = ṁcₚ	W/K	Btu/(h·°F)
Cmin	Smaller heat-capacity rate	W/K	Btu/(h·°F)
q̇	Actual heat duty	W	Btu/h

Open Heat Exchanger Calculator Worked example ↓

Worked example — water-to-water exchanger

Given: Hot side ṁ = 2 kg/s, cₚ = 4.2 kJ/(kg·K), 90→60°C; cold-side capacity rate is larger; cold inlet = 20°C.

Cmin = 2×4200 = 8400 W/K

q̇ = 8400(90−60) = 252 kW

q̇max = 8400(90−20) = 588 kW

ε = 252/588 = 0.429

Answer: ε ≈ 0.429.

Effectiveness alone does not identify pressure drop, fouling or economic optimum.

The ε–NTU method is especially useful when outlet temperatures are unknown. The ε(NTU,C_r) relation depends on flow arrangement.

09 // Quick formula summary

Compact print reference. Use the detailed sections above for definitions and limitations.

Topic	Equation	Purpose	Tool
Conduction	q̇ = kAΔT/L	Steady wall conduction	Wall heat transfer
Thermal resistance	q̇=ΔT/ΣR	Layered systems	Wall heat transfer
Convection	q̇=hAΔT	Surface-fluid heat transfer	Estimator
Radiation	q̇=εσA(Tₛ⁴−Tsur⁴)	Gray surface to enclosure	Estimator
Overall U	q̇=UAΔT	Collapse resistances	Wall heat transfer
LMTD	q̇=UAFΔTlm	Heat exchanger duty	Heat exchanger
Lumped transient	θ/θᵢ=e^(−hAt/ρVcₚ)	Small-Bi transient	Estimator
Effectiveness	ε=q/qmax	Exchanger performance	Heat exchanger

10 // Assumptions & limitations

Fundamental equations are only useful when their assumptions match the actual problem.

11 // Technical references

ASHRAE 2025 is the primary current engineering reference. The DOE handbook is an archived fundamentals/training source, not a current design code.

Heat Transfer Formula Sheet

Fast reference, with engineering context

Reference conventions

Three modes of heat transfer

01 // Fourier conduction

Use temperature-dependent or apparent conductivity where the material/source requires it.

02 // Thermal resistance networks

Area-normalized resistance R″ is convenient for flat assemblies with one common area.

03 // Newton convection relation

Treat h as a correlation/result, not a universal material constant.

04 // Thermal radiation to large surroundings

Use absolute temperature in the T⁴ terms.

05 // Overall heat-transfer coefficient

Always state the area basis when U is used for cylindrical or unequal-area systems.

06 // Log-mean temperature difference

Use the correct terminal differences for parallel or counterflow arrangement.

07 // Lumped-capacitance transient

Bi < 0.1 is a common screening condition for lumped analysis.

08 // Heat-exchanger effectiveness

Effectiveness alone does not identify pressure drop, fouling or economic optimum.

09 // Quick formula summary

10 // Assumptions & limitations

11 // Technical references

12 // Related EngineerHub tools

Related resources