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Heat Transfer

Heat conduction through walls and insulation, overall heat transfer, heat exchanger sizing, radiation and heating-up processes.

Heat Loss of an Insulated Pipe

Cylindrical heat conduction + outside heat transfer

Pipe outside diameter d₁ [mm]

Insulation thickness s [mm]

Fluid temp. T_i [°C]

Ambient temp. T_a [°C]

λ insulation [W/mK]

α outside [W/m²K]

Pipe length L [m]

Heat loss per meter q̇–

Total heat loss Q̇–

Surface temperature–

Comparison: without insulation*–

*Without insulation, same α, radiation neglected. The actual loss of bare pipes is usually even higher.

Theory

Heat flow per meter of pipe through a cylindrical insulation layer with outside heat transfer:

q̇ = π · (T_i − T_a)ln(d₂/d₁)2·λ + 1α·d₂

d₂ = d₁ + 2·s: outside diameter of the insulation
λ: thermal conductivity (mineral wool ≈ 0.04 W/mK)
α: outside heat transfer coefficient (still air ≈ 8–12 W/m²K)
Inside heat transfer and the (steel) pipe wall are negligible.

The surface temperature follows from q̇ = α·π·d₂·(T_s−T_a).

Overall Heat Transfer of a Plane Wall (U-Value)

Up to 3 layers + surface heat transfer

α inside [W/m²K]

α outside [W/m²K]

Layer 1: s₁ [mm]

λ₁ [W/mK]

Layer 2: s₂ [mm] (0 = none)

λ₂ [W/mK]

Layer 3: s₃ [mm] (0 = none)

λ₃ [W/mK]

Area A [m²]

ΔT [K]

U-value–

Heat flow Q̇–

Theory

The overall heat transfer coefficient U combines all individual thermal resistances:

1U = 1αᵢ + Σ sⱼλⱼ + 1αₐ

Q̇ = U · A · ΔT

Typical α values:

still air (inside): 8 W/m²K
moving air (outside): 25 W/m²K
flowing water: 500–4,000 W/m²K
condensing steam: 5,000–10,000 W/m²K

Heat Exchanger (LMTD Method)

Required heat transfer area

Flow arrangement

Hot in T_h1 [°C]

Hot out T_h2 [°C]

Cold in T_c1 [°C]

Cold out T_c2 [°C]

Mass flow hot side ṁ [kg/h]

c_p hot side [kJ/kgK]

U-value [W/m²K] (water/water ≈ 800–1,500)

Heat duty Q̇–

ΔT_log–

Required area A–

Theory

Sizing via the log mean temperature difference (LMTD):

Q̇ = ṁ · c_p · (T_h1 − T_h2)

ΔT_log = ΔT₁ − ΔT₂ln(ΔT₁ / ΔT₂)

A = Q̇U · ΔT_log

Counterflow: ΔT₁ = T_h1−T_c2, ΔT₂ = T_h2−T_c1
Parallel flow: ΔT₁ = T_h1−T_c1, ΔT₂ = T_h2−T_c2
For the same heat duty, counterflow needs less area.

Thermal Radiation

Stefan-Boltzmann law

Surface temp. T₁ [°C]

Ambient temp. T₂ [°C]

Emissivity ε [-]

Area A [m²]

Radiant heat flux q̇–

Radiant power Q̇–

Theory

Net radiation from a body to its surroundings:

Q̇ = ε · σ · A · ( T₁⁴ − T₂⁴ )

σ = 5.67·10⁻⁸ W/m²K⁴ (temperatures in kelvin!). Emissivities:

black body: 1.0
oxidized steel: ≈ 0.8
painted surface: ≈ 0.9
polished aluminum: ≈ 0.05

At high temperatures, radiation dominates over convection.

Heating Up / Cooling Down

Amount of heat and heating power

Mass m [kg]

c [kJ/kgK] (water 4.19 · steel 0.47 · oil 1.9)

Start temp. T₁ [°C]

Target temp. T₂ [°C]

Heating time t [h]

Efficiency η [-]

Amount of heat Q–

Q–

Required heating power P–

Theory

Amount of heat needed to raise the temperature:

Q = m · c · (T₂ − T₁)

Required heating power for a desired heating time t:

P = Qt · η

η accounts for losses (insulation, transfer)
1 kWh = 3,600 kJ
Rule of thumb: heating 1 m³ of water by 1 K takes ≈ 1.16 kWh

Rough estimate for preliminary design. Does not replace a code-compliant design calculation.