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Heat transfer models

Build from energy conservation to conduction, convection, radiation, transient cooling and heat exchangers, with twenty worked examples.

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Matter pathway: atom → solid → liquid → gas → plasma. Quantum mechanics and quantum field theory provide foundations across the pathway; they are not additional phases. This is a connected modeling route, not a universal heating curve. Actual phases depend on pressure, composition, and kinetics.

1. Conduction and energy balance

Definitions & inputs. k conductivity,ρ density,cp heat capacity,T temperature,qdot volumetric generation.

  1. Fourier’s law sends heat down the temperature gradient.

    q′′=−k∇T\mathbf q''=-k\nabla T

  2. Balance energy storage against conduction and generation.

    ρcp∂tT=k∇2T+q˙\rho c_p\partial_tT=k\nabla^2T+\dot q

  3. Integrate the steady one-dimensional no-generation equation across a slab.

    Rwall=L/(kA),Q=ΔT/RwallR_{wall}=L/(kA),\quad Q=\Delta T/R_{wall}

Interpretation. Composite walls add series resistances when the same heat rate crosses each layer.

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2. Surface exchange

Definitions & inputs. h convection coefficient,A area,ε emissivity,σ Stefan–Boltzmann constant.

  1. Define convection through the surface-to-fluid temperature difference.

    Qc=hA(Ts−T∞)Q_c=hA(T_s-T_\infty)

  2. Subtract absorbed enclosure irradiation from emitted radiation.

    Qr=ϵσA(Ts4−Tsur4)Q_r=\epsilon\sigma A(T_s^4-T_{sur}^4)

  3. Factor the fourth-power difference to obtain a temperature-dependent linearized coefficient.

    hr=ϵσ(Ts+Tsur)(Ts2+Tsur2)h_r=\epsilon\sigma(T_s+T_{sur})(T_s^2+T_{sur}^2)

Interpretation. View factors and interacting finite surfaces require a radiation network.

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3. Transient cooling

Definitions & inputs. Lc=V/A,α=k/(ρcp),Bi Biot number,θ=T−T∞.

  1. Compare internal conduction resistance with surface resistance and define thermal time.

    Bi=hLc/k,τ=ρVcp/(hA)Bi=hL_c/k,\quad\tau=\rho Vc_p/(hA)

  2. Apply an energy balance to the uniform body.

    dθ/dt=−θ/τd\theta/dt=-\theta/\tau

  3. Integrate from the initial uniform temperature.

    θ/θ0=e−t/τ\theta/\theta_0=e^{-t/\tau}

Interpretation. If Bi is large, solve spatial conduction with appropriate geometry and boundary conditions.

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4. Fins and exchangers

Definitions & inputs. m fin parameter,k conductivity,P perimeter,Ac cross section; Cdot heat-capacity rate.

  1. Solve the fin conduction-convection equation with the insulated-tip boundary condition.

    m=hP/(kAc),ηf=tanh⁡(mL)/(mL)m=\sqrt{hP/(kA_c)},\quad\eta_f=\tanh(mL)/(mL)

  2. Integrate the stream energy balance.

    Q=m˙cp(To−Ti)Q=\dot m c_p(T_o-T_i)

  3. Integration along an ideal parallel or counterflow exchanger gives the log-mean driving difference.

    Q=UAΔTlm,ΔTlm=(ΔT1−ΔT2)/ln⁡(ΔT1/ΔT2)Q=UA\Delta T_{lm},\quad\Delta T_{lm}=(\Delta T_1-\Delta T_2)/\ln(\Delta T_1/\Delta T_2)

Interpretation. Multipass geometry and phase changes need the corresponding exchanger formulation.

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Graphical worked example

Lumped body: T0=100°C, ambient 20°C, τ=900 s; small Biot number required. X axis: Cooling time (s). Y axis: Body temperature (°C).
Lumped body: T0=100°C, ambient 20°C, τ=900 s; small Biot number required. Related worked calculation · Download SVG · Plot data

Twenty worked examples

Open a problem to see its defined inputs, assumptions, equation, numerical substitution, result, and interpretation. Values are illustrative analytical exercises.

Example 01. Plane-wall heat flow

Definitions & inputs. k=.5 W/(m K),A=2 m²,L=.1 m,ΔT=20 K.

  1. Choose the governing model and isolate the requested quantity.

    Q=kAΔT/LQ=kA\Delta T/L

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    Q=0.5(2)(20)/0.1Q=0.5(2)(20)/0.1

  3. Evaluate the expression; the result uses the units shown.

    Result=200 W\mathrm{Result}=200\ {\rm W}

Interpretation. Steady one-dimensional conduction excludes edge losses.

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Example 02. Wall thermal resistance

Definitions & inputs. k=.5 W/(m K),A=2 m²,L=.1 m.

  1. Choose the governing model and isolate the requested quantity.

    R=L/(kA)R=L/(kA)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    R=0.1/[0.5(2)]R=0.1/[0.5(2)]

  3. Evaluate the expression; the result uses the units shown.

    Result=0.1 K W−1\mathrm{Result}=0.1\ {\rm K\,W}^{-1}

Interpretation. Multiply resistance by heat rate to obtain temperature drop.

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Example 03. Series wall flow

Definitions & inputs. R1=.1,R2=.2 K/W,total ΔT=30 K.

  1. Choose the governing model and isolate the requested quantity.

    Q=ΔT/(R1+R2)Q=\Delta T/(R_1+R_2)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    Q=30/0.3Q=30/0.3

  3. Evaluate the expression; the result uses the units shown.

    Result=100 W\mathrm{Result}=100\ {\rm W}

Interpretation. The same heat rate passes through both layers.

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Example 04. Convection heat flow

Definitions & inputs. h=10 W/(m² K),A=2 m²,Ts−T∞=30 K.

  1. Choose the governing model and isolate the requested quantity.

    Q=hAΔTQ=hA\Delta T

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    Q=10(2)(30)Q=10(2)(30)

  3. Evaluate the expression; the result uses the units shown.

    Result=600 W\mathrm{Result}=600\ {\rm W}

Interpretation. The convection coefficient is a prescribed input.

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Example 05. Convection resistance

Definitions & inputs. h=20 W/(m² K),A=.5 m².

  1. Choose the governing model and isolate the requested quantity.

    R=1/(hA)R=1/(hA)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    R=1/[20(0.5)]R=1/[20(0.5)]

  3. Evaluate the expression; the result uses the units shown.

    Result=0.1 K W−1\mathrm{Result}=0.1\ {\rm K\,W}^{-1}

Interpretation. Fouling and contact resistances are additional elements.

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Example 06. Net thermal radiation

Definitions & inputs. ε=.8,A=1 m²,Ts=400 K,Tsur=300 K,σ=5.670374419×10⁻⁸.

  1. Choose the governing model and isolate the requested quantity.

    Q=ϵσA(Ts4−Tsur4)Q=\epsilon\sigma A(T_s^4-T_{sur}^4)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    Q=0.8(5.670374419×10−8)(4004−3004)Q=0.8(5.670374419\times10^{-8})(400^4-300^4)

  3. Evaluate the expression; the result uses the units shown.

    Result=793.8524 W\mathrm{Result}=793.8524\ {\rm W}

Interpretation. Large enclosure, view factor one, and gray diffuse behavior are assumed.

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Example 07. Linearized radiation coefficient

Definitions & inputs. Same ε=.8,Ts=400 K,Tsur=300 K.

  1. Choose the governing model and isolate the requested quantity.

    hr=ϵσ(Ts+Tsur)(Ts2+Tsur2)h_r=\epsilon\sigma(T_s+T_{sur})(T_s^2+T_{sur}^2)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    hr=0.8(5.670374419×10−8)(700)(250000)h_r=0.8(5.670374419\times10^{-8})(700)(250000)

  3. Evaluate the expression; the result uses the units shown.

    Result=7.938524 W m−2K−1\mathrm{Result}=7.938524\ {\rm W\,m}^{-2}{\rm K}^{-1}

Interpretation. Multiplying by 100 K reproduces the net radiation heat flux.

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Example 08. Sensible heating energy

Definitions & inputs. m=2 kg,cp=900 J/(kg K),ΔT=50 K.

  1. Choose the governing model and isolate the requested quantity.

    E=mcpΔTE=mc_p\Delta T

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    E=2(900)(50)E=2(900)(50)

  3. Evaluate the expression; the result uses the units shown.

    Result=90000 J\mathrm{Result}=90000\ {\rm J}

Interpretation. No phase change or environmental loss is included.

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Example 09. Heating time

Definitions & inputs. Required energy 90 kJ,net constant heater power 300 W.

  1. Choose the governing model and isolate the requested quantity.

    t=E/Pt=E/P

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    t=90000/300t=90000/300

  3. Evaluate the expression; the result uses the units shown.

    Result=300 s\mathrm{Result}=300\ {\rm s}

Interpretation. Net power already subtracts heat losses.

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Example 10. Thermal diffusivity

Definitions & inputs. k=200 W/(m K),ρ=2700 kg/m³,cp=900 J/(kg K).

  1. Choose the governing model and isolate the requested quantity.

    α=k/(ρcp)\alpha=k/(\rho c_p)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    α=200/[2700(900)]\alpha=200/[2700(900)]

  3. Evaluate the expression; the result uses the units shown.

    Result=8.230453×10−5 m2s−1\mathrm{Result}=8.230453\times10^{-5}\ {\rm m}^2{\rm s}^{-1}

Interpretation. Diffusivity sets the rate of temperature spreading.

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Example 11. Biot number

Definitions & inputs. h=10 W/(m² K),Lc=.01 m,k=200 W/(m K).

  1. Choose the governing model and isolate the requested quantity.

    Bi=hLc/kBi=hL_c/k

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    Bi=10(0.01)/200Bi=10(0.01)/200

  3. Evaluate the expression; the result uses the units shown.

    Result=0.0005 \mathrm{Result}=0.0005\ {}

Interpretation. The small value supports a lumped-temperature approximation.

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Example 12. Lumped thermal time

Definitions & inputs. m=1 kg,cp=900 J/(kg K),h=10 W/(m² K),A=.1 m².

  1. Choose the governing model and isolate the requested quantity.

    τ=mcp/(hA)\tau=mc_p/(hA)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    τ=900/[10(0.1)]\tau=900/[10(0.1)]

  3. Evaluate the expression; the result uses the units shown.

    Result=900 s\mathrm{Result}=900\ {\rm s}

Interpretation. Verify a small Biot number separately.

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Example 13. Temperature after one time constant

Definitions & inputs. T0=100°C,T∞=20°C,t=τ.

  1. Choose the governing model and isolate the requested quantity.

    T=T∞+(T0−T∞)e−1T=T_\infty+(T_0-T_\infty)e^{-1}

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    T=20+80e−1T=20+80e^{-1}

  3. Evaluate the expression; the result uses the units shown.

    Result=49.43036 ∘C\mathrm{Result}=49.43036\ {}^\circ{\rm C}

Interpretation. Temperature differences may be in Celsius for this nonradiative relation.

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Example 14. Cooling to ten-percent excess

Definitions & inputs. τ=900 s,θ/θ0=.1.

  1. Choose the governing model and isolate the requested quantity.

    t=−τln⁡(θ/θ0)t=-\tau\ln(\theta/\theta_0)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    t=−900ln⁡0.1t=-900\ln0.1

  3. Evaluate the expression; the result uses the units shown.

    Result=2072.327 s\mathrm{Result}=2072.327\ {\rm s}

Interpretation. Ambient temperature is held constant.

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Example 15. Fourier number

Definitions & inputs. α=10⁻⁵ m²/s,t=100 s,L=.1 m.

  1. Choose the governing model and isolate the requested quantity.

    Fo=αt/L2Fo=\alpha t/L^2

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    Fo=10−5(100)/(0.1)2Fo=10^{-5}(100)/(0.1)^2

  3. Evaluate the expression; the result uses the units shown.

    Result=0.1 \mathrm{Result}=0.1\ {}

Interpretation. Length convention must match the geometry-specific transient solution.

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Example 16. Cylindrical conduction

Definitions & inputs. k=1 W/(m K),L=1 m,r1=.01 m,r2=.02 m,ΔT=20 K.

  1. Choose the governing model and isolate the requested quantity.

    Q=2πkLΔT/ln⁡(r2/r1)Q=2\pi kL\Delta T/\ln(r_2/r_1)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    Q=40π/ln⁡2Q=40\pi/\ln2

  3. Evaluate the expression; the result uses the units shown.

    Result=181.2944 W\mathrm{Result}=181.2944\ {\rm W}

Interpretation. Steady radial flow has area increasing with radius.

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Example 17. Fin efficiency

Definitions & inputs. Straight insulated-tip fin with mL=1.

  1. Choose the governing model and isolate the requested quantity.

    ηf=tanh⁡(mL)/(mL)\eta_f=\tanh(mL)/(mL)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    ηf=(e2−1)/(e2+1)\eta_f=(e^2-1)/(e^2+1)

  3. Evaluate the expression; the result uses the units shown.

    Result=0.7615942 \mathrm{Result}=0.7615942\ {}

Interpretation. Fin efficiency compares actual heat loss with a fin uniformly at base temperature.

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Example 18. Stream heating

Definitions & inputs. Mass flow .1 kg/s,cp=4200 J/(kg K),rise 10 K.

  1. Choose the governing model and isolate the requested quantity.

    Q=m˙cpΔTQ=\dot m c_p\Delta T

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    Q=0.1(4200)(10)Q=0.1(4200)(10)

  3. Evaluate the expression; the result uses the units shown.

    Result=4200 W\mathrm{Result}=4200\ {\rm W}

Interpretation. The heat-capacity approximation is constant over the temperature range.

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Example 19. Log-mean temperature difference

Definitions & inputs. End driving differences 40 K and 20 K.

  1. Choose the governing model and isolate the requested quantity.

    ΔTlm=(40−20)/ln⁡(40/20)\Delta T_{lm}=(40-20)/\ln(40/20)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    ΔTlm=20/ln⁡2\Delta T_{lm}=20/\ln2

  3. Evaluate the expression; the result uses the units shown.

    Result=28.8539 K\mathrm{Result}=28.8539\ {\rm K}

Interpretation. Both differences use consistent hot-minus-cold terminal temperatures.

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Example 20. Latent melting energy

Definitions & inputs. m=.5 kg,latent heat 334 kJ/kg; already at melting point.

  1. Choose the governing model and isolate the requested quantity.

    E=mLfE=mL_f

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    E=0.5(334)E=0.5(334)

  3. Evaluate the expression; the result uses the units shown.

    Result=167 kJ\mathrm{Result}=167\ {\rm kJ}

Interpretation. Sensible preheating and postheating are separate terms.

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Symbols and units

Each derivation and problem defines its own symbols and inputs. Symbols may be reused with different meanings in other subjects. Keep units consistent, retain sufficient precision during calculation, and apply the stated validity limits.