In the three standard rate laws for heat transfer, which quantity describes the situation - geometry, fluid, velocity, and flow regime - rather than a substance, a surface, or a universal constant?
Choose an answer, then check it.
Engineering Fundamentals · Thermal and Energy Systems
Heat Transfer
On this page 9 sections
In 30 seconds
Heat moves three ways, and each way has its own rate law. Conduction follows Fourier's law and depends on a material property, thermal conductivity. Convection follows Newton's law of cooling and depends on a coefficient that is not a property at all. Radiation follows the Stefan-Boltzmann law and depends on the fourth power of absolute temperature. Writing each mechanism as a Thermal resistance The ratio of temperature difference to heat transfer rate for one element of a heat path, in K/W. For a plane wall it equals thickness divided by the product of conductivity and area. Full entry → lets you combine them the way you combine resistors.
Why this matters
Almost every engineered system has a thermal problem hiding in it: a battery pack that must not overheat, a wall that must not lose the heat you paid for, an engine that must dump megawatts into moving air. Heat transfer is the tool that turns those worries into numbers you can check. It is also where careless work is easy to spot. Using Celsius in a fourth-power law, treating a convection coefficient as if it were a material constant, or mixing U.S. and SI R-values produce answers that are wrong by large factors, not small ones. Learning the assumptions behind each rate law is what separates a usable estimate from a confident mistake.
The college version
Three mechanisms, one question
Every heat transfer problem asks the same question: at what rate does energy cross this boundary? Three mechanisms can carry it. Conduction moves energy through the interactions of adjacent atoms and molecules, so it needs matter and no motion. Convection moves energy by the bulk motion and mixing of a fluid, so it needs a fluid that can flow. Radiation moves energy as electromagnetic waves emitted by any surface above absolute zero, and it needs no medium at all, which is why a vacuum flask still loses heat and why a spacecraft can only reject waste heat by radiating it. Real situations run all three at once. A hot pipe conducts through its wall, drives convection into the surrounding air, and radiates to the walls of the room. Engineering practice is to model each path separately, then decide which ones you are allowed to ignore.
Conduction: Fourier's law and the plane wall
For steady one-dimensional conduction, Fourier's law states that the heat transfer rate is proportional to the area, to the temperature difference, and inversely proportional to the thickness: Q = kA(T_hot - T_cold)/L. The constant k is the thermal conductivity, in W/(mK). Unlike the convective coefficient you will meet next, k really is a material property, although it varies with temperature and, for porous materials, with density and moisture. NIST's certified reference material for insulation makes this concrete: SRM 1450d, a molded fibrous-glass board, carries a certified conductivity equation lambda(T) = 1.10489 x 10^-4 T W/(m*K), valid from 280 K to 340 K, with a one percent expanded uncertainty. Note what that equation says. Conductivity is a function of temperature, it is certified only over a stated range, and it belongs to one specific material lot. A handbook number without a temperature and a material description is an estimate, not a property. The plane-wall result assumes steady state, one-dimensional flow, constant k, and no internal heat generation. Drop any of those and the simple formula stops applying.
Thermal resistance and the circuit analogy
Rearranging Fourier's law gives Q = (delta T)/R_th, with R_th = L/(kA). That is Ohm's law with temperature difference as the voltage, heat rate as the current, and L/(kA) as the resistance. The analogy is not a metaphor; it is the same linear algebra, so the same combination rules hold. Layers stacked one behind the other are in series and their resistances add, which is why the R-values of a multi-layered installation are added. Paths that carry heat side by side are in parallel and their conductances add. Convection joins the same network: a surface film with coefficient h contributes a resistance 1/(hA). Once every path is a resistance, a wall becomes a circuit you can solve by inspection. The layer with the largest resistance takes the largest temperature drop and controls the total, and the smallest resistance is usually the one worth attacking. Parallel paths are where intuition fails most often. A steel stud or a concrete slab edge crossing an insulation layer is a low-resistance path in parallel with a high-resistance one, and because parallel conductances add, a small area of metal can dominate the assembly. Engineers call this a Thermal bridge A high-conductivity path, such as a metal stud or a slab edge, that runs in parallel with an insulated path and carries a disproportionate share of the heat because parallel conductances add. Full entry →.
R-value, and the unit trap that ruins answers
In building practice the per-unit-area thermal resistance L/k is called the R-value Thermal resistance expressed per unit area, used to rate insulation. Layer R-values add. In U.S. practice it is quoted in hft^2degF/Btu; in SI it is quoted in m^2*K/W, and the two numbers differ by a factor of 5.678263. Full entry →, and the reciprocal of the total assembly resistance is the U-factor. The U.S. Federal Trade Commission regulates R-value claims on home insulation under 16 CFR part 460. That rule requires advertised R-values to come from listed ASTM steady-state test methods, run on the insulation alone at a mean temperature of 75 degrees Fahrenheit with a temperature difference of 50 plus or minus 10 degrees Fahrenheit, and it requires the label to state that R means resistance to heat flow. Because the rule is written in U.S. customary units, an American R-value carries units of hft^2degF/Btu, while an SI R-value carries m^2K/W. They are not the same number. Using the NIST conversion factor, 1 Btu(IT)/(hft^2degF) equals 5.678263 W/(m^2K), so R_SI = R_US x 0.1761102. A wall with an SI resistance of 2.96 m^2*K/W is an R-16.8 wall on an American label. Someone who reads a European datasheet quoting R = 3 and assumes it means R-3 in U.S. terms has understated the insulation by a factor of about 5.7. This is not a hypothetical error; it is one of the most common unit mistakes in thermal work, and the defense is simple: never write an R-value without its units.
Convection: Newton's law of cooling and the honest status of h
Convection is written in a deliberately similar form, Q = hA(T_surface - T_fluid), where h is the convective heat transfer coefficient in W/(m^2*K). The similarity is misleading. Thermal conductivity is a property you can look up for a material; h is not. The DOE fundamentals handbook is blunt about it: no single property of the medium describes convection, and h depends on fluid velocity, fluid viscosity, heat flux, surface roughness, and the type of flow. It is determined empirically, from measurements or from correlations fitted to measurements, for a specific geometry and a specific flow. That is why every published table of h gives ranges rather than values, and why two ranges for what sounds like the same situation can differ by a factor of several. Two regimes are worth naming. In natural convection the fluid moves only because heating changes its density, so the flow is weak and h is small. In forced convection a fan or pump drives the flow, and h rises accordingly; turbulent flow gives a higher h than laminar flow because the near-wall stagnant film is thinner. When you see an h in a calculation, the right question is not what fluid is it, but what geometry, what velocity, and who measured it.
Radiation: the fourth power of absolute temperature
A surface at absolute temperature T emits thermal radiation at a rate Q = esigmaAT^4, where sigma is the Stefan-Boltzmann constant and e is the emissivity, a dimensionless number between 0 and 1 that measures how close the surface comes to an ideal black body. NIST's CODATA 2022 value of sigma is 5.670374419 x 10^-8 Wm^-2K^-4, exact in the revised SI. Textbooks often round it to 5.67 x 10^-8, which is the same number to three figures. What matters for exchange with the environment is the net rate, Q_net = esigmaA(T_surface^4 - T_surroundings^4). Both temperatures must be absolute. This is the single most common blunder in the whole subject: putting Celsius into a fourth power. Because the fourth power is so steep, radiation is easy to neglect at room temperature and impossible to neglect once things get hot. Doubling the absolute temperature multiplies emission by sixteen. Emissivity is the other lever, and it is a surface property rather than a bulk one, which is why a radiant barrier is a thin foil rather than a thick blanket: DOE's insulation fact sheet states that a radiant barrier must have an emittance of 0.1 or less and a reflectance of 0.9 or more, against about 0.9 for the ordinary wood and sheathing surfaces ORNL modeled in attics.
Combining modes, fins, and the limits of adding insulation
When several mechanisms act in series it is standard practice to collapse them into one overall heat transfer coefficient U, defined by Q = UA(delta T_overall), where U bundles the convective film on each side with the conduction through the solid between them. U is the reciprocal of the total resistance per unit area, so it is bounded by the worst link in the chain: no amount of improvement elsewhere can beat the largest resistance you cannot remove. Fins exploit the other side of that logic. When the surface convection resistance dominates, adding area helps, so a fin is a piece of high-conductivity material that carries heat out of the base and presents more surface to the fluid. Fins only pay when the surface resistance is the bottleneck, which is why heat sinks are finned on the air side, where the surface resistance is largest. The same reasoning explains why adding insulation shows diminishing returns. Heat flow goes as 1/R_total, not as R_total, so the first layer removes a large fraction of the loss and each subsequent layer removes a fraction of what is left. Doubling the insulation never halves the loss once the other resistances matter, and it eventually buys very little. Installation quality can undo the arithmetic entirely: DOE guidance notes that compressed insulation does not deliver its rated R-value.
When things are still changing: lumped capacitance
Everything above assumes steady state. During warm-up, cool-down, or a thermal transient, the temperature inside a body varies with both position and time, and the governing equation is a partial differential equation. There is one shortcut worth knowing at this level. If a body conducts heat internally much faster than its surface exchanges heat with the fluid, its interior stays nearly uniform and you can treat it as a single lump at one temperature that decays exponentially toward the fluid temperature. The test is the Biot number, Bi = h*L_c/k, where L_c is a characteristic length such as volume divided by surface area. Bi compares internal conduction resistance to external convection resistance. The general criterion is Bi < 0.1; NASA thermal analysts apply exactly this test and report where it fails. A thin copper plate in air passes easily. A sheet of gypsum board in the same air does not, because gypsum conducts poorly enough that its own interior lags. When Bi exceeds 0.1, the lumped model understates internal temperature differences, and you need a spatial solution.
What this lesson is and is not
This is educational material. It is not engineering design guidance. Every property value here is a typical or certified-reference value at a stated condition and date, not a design value, and none of it should be used to size insulation, specify a thermal system, or justify a construction detail. Building envelope and thermal system design in the United States is governed by adopted energy codes. Under the Energy Conservation and Production Act, DOE must determine whether each revised edition of ANSI/ASHRAE/IES Standard 90.1 or the International Energy Conservation Code improves energy efficiency, after which states review, update, or certify their own codes. Those codes, and the standards they reference, set the required U-factors, R-values, and test methods. Real thermal design is done by a licensed engineer working to the governing code, using tested assembly values and the correlations appropriate to the actual geometry and flow.
Eli explains
The same idea, in plain words
Explain it like I’m 10
Heat always travels from hot to cold, but it has three different ways of getting there, and each way has its own speed limit. It can hand energy from atom to atom through solid stuff, which is conduction. It can ride along with moving air or water, which is convection. Or it can beam across empty space as invisible light, which is radiation. The clever trick engineers use is to think of every one of these paths as a section of pipe that heat has to squeeze through. A thick blanket is a narrow pipe. A metal bolt is a wide pipe. Line the pipes up end to end and the narrowest one sets the flow. Put them side by side and heat takes the widest one, which is why a single metal bolt through a thick blanket can leak more heat than you would ever guess. Radiation is the odd one out. It gets far stronger as things get hotter, and it does not care whether there is anything in between.
Picture it like this
Think of a wall as a stack of coffee filters and heat as water poured on top. Each layer slows the flow by its own amount, so the total delay is the sum of the layers, and the tightest filter is what really holds the water back. Punch a straw through the stack and almost everything pours through the straw, no matter how good the filters are.
Where the picture stops working
The analogy covers series and parallel resistances well, and nothing else. Water pouring through filters is a one-way flow of stuff; heat is not a substance, and both surfaces in a radiation problem are emitting toward each other at once, with only the difference showing up as net transfer. Filters also have a fixed tightness, whereas a convection film's resistance changes the moment the air starts moving. And no filter behaves like radiation, which strengthens with the fourth power of absolute temperature and crosses a vacuum where there is no filter at all.
Worked example
Take one square meter of wall, from the inside out: an inside air film, 12.7 mm of gypsum board, 90 mm of fibrous-glass insulation, and a 1.0 mm copper cladding sheet with an outside air film. Property values, all typical or certified-reference values at a stated date and not design values: gypsum k = 0.258 W/(mK) (NIST measurement of U.S. Type X board, paper on, room temperature, 2008); insulation k = 1.10489e-4 x 297.15 = 0.03283 W/(mK) (NIST SRM 1450d certified equation at 297.15 K, inside its 280-340 K range); copper k = 240 Btu/(hftdegF) x 1.730735 = 415.4 W/(mK) (DOE handbook, 1992, converted with the NIST SP 811 factor). The film coefficients are assumed for illustration, h_inside = 8 and h_outside = 25 W/(m^2K); they are not sourced values, because h is not a property.
Resistances per square meter: inside film 1/8 = 0.1250; gypsum 0.0127/0.258 = 0.0492; insulation 0.090/0.03283 = 2.7412; copper 0.0010/415.4 = 0.0000024; outside film 1/25 = 0.0400 m^2K/W. In series they add to R_total = 2.9555 m^2K/W, so U = 1/R_total = 0.3384 W/(m^2*K). With inside air at 21 degC and outside air at -4 degC, q = U x 25 = 8.46 W/m^2. The insulation carries 92.7 percent of the resistance and 23.2 K of the 25 K drop; the copper carries 0.00008 percent and drops nothing measurable. In American units the same wall is R-16.8 (2.9555 / 0.1761102), a reminder that the SI and U.S. numbers for one wall differ by 5.68.
Now the parallel case. Let a solid copper bridge replace the insulation over 10 percent of the area. The bridged path has R = 0.2144 m^2K/W, so U = 4.663 there. Area-weighted, U_assembly = 0.9 x 0.3384 + 0.1 x 4.663 = 0.7708 W/(m^2K) and q = 19.27 W/m^2. Ten percent of the area more than doubled the loss, a 128 percent increase. That is a thermal bridge.
Radiation, same square meter: a surface at 60 degC facing surroundings at 25 degC, emissivity 0.9 (the value ORNL used for ordinary wood and sheathing surfaces). Q_net = 0.9 x 5.670374419e-8 x (333.15^4 - 298.15^4) = 225.39 W. Now make the classic mistake and put Celsius into the fourth power: 0.9 x 5.670374419e-8 x (60^4 - 25^4) = 0.64 W. That is 0.28 percent of the right answer, wrong by a factor of 351. Not a rounding error, a different physical claim. Fitting a foil radiant barrier instead, at emissivity 0.05, gives 12.52 W, a 94 percent reduction, with no change in thickness. And to see why radiation takes over when things get hot, write it as an effective coefficient h_r = esigma(T_s^2 + T_sur^2)(T_s + T_sur): at 30 degC h_r is 5.55, at 200 degC it is 12.3, and at 800 degC it is 86.8 W/(m^2*K), by which point radiation is carrying an order of magnitude more than a natural-convection film would.
Key takeaway
Conduction, convection, and radiation each have a rate law, and each rate law has assumptions: k is a material property, h is not, and radiation depends on the fourth power of absolute temperature. Turn every path into a thermal resistance and the whole problem becomes a circuit you can solve and check.
Quick check
3 questions here, of 5 in this lesson’s practice set. Answers stay hidden until you check.
A datasheet from a European supplier lists an insulation board at R = 3. A U.S. contractor reads it as equivalent to R-3 on an American insulation label. What has gone wrong?
Choose an answer, then check it.
A 1 m^2 surface at 60 degC faces surroundings at 25 degC with emissivity 0.90. Using sigma = 5.670374419e-8 Wm^-2K^-4, what is the net radiative heat transfer rate, and what happens if Celsius values are used in the fourth power instead of kelvin?
Choose an answer, then check it.
Study tools & related lessonsYou’ll learn to · Common mistakes · Easily confused · Key vocabulary · Related
You’ll learn to
- Distinguish conduction, convection, and radiation by the mechanism that moves the energy and by the medium each requires.
- Apply Fourier's law, Newton's law of cooling, and the Stefan-Boltzmann law, stating the assumption each one rests on.
- Analyze a composite wall by combining thermal resistances in series and in parallel, and report the result as an overall heat transfer coefficient.
- Explain why the convective heat transfer coefficient is not a material property and why quoted values are ranges.
- Evaluate whether a transient problem satisfies the lumped-capacitance criterion using the Biot number.
- Convert between U.S. customary and SI R-values and identify the errors caused by confusing them.
Common mistakes
Putting Celsius or Fahrenheit temperatures into the Stefan-Boltzmann law.
The fourth power only works on absolute temperature. In the worked example, using 60 and 25 degrees Celsius instead of 333.15 K and 298.15 K gives 0.64 W instead of 225.39 W, an answer 351 times too small. Convert to kelvin before raising to the fourth power, every time.
Looking up h in a table the way you would look up thermal conductivity.
h is not a material property. It depends on geometry, fluid, velocity, surface roughness, and whether the flow is laminar or turbulent, and it is obtained empirically. Published values are ranges tied to specific configurations, so an h borrowed from a different geometry is an assumption, not data, and should be labeled as one.
Comparing an SI R-value with a U.S. R-value as if they were the same quantity.
R in m^2K/W and R in hft^2*degF/Btu differ by a factor of 5.678263. An SI R of 2.96 is a U.S. R-16.8. Insulation labeled under the FTC rule in the United States is in customary units, so always write the units with the number and convert with the NIST factor before comparing.
Averaging conductivities, or averaging R-values, across a wall that has studs or fasteners in it.
Series layers add resistances; parallel paths add conductances. Those are different operations and give different answers. In the worked example a copper bridge over 10 percent of the area raised the assembly U from 0.3384 to 0.7708 W/(m^2*K), a 128 percent increase in heat loss that no averaging of R-values would predict.
Assuming a lumped, single-temperature model is fine for any small object during a transient.
Check the Biot number, Bi = hL_c/k, and use the lumped model only when Bi < 0.1. With h = 8 W/(m^2K), a 6 mm copper plate gives Bi = 1.2 x 10^-4 and is safely lumped, while 12.7 mm gypsum board gives Bi = 0.39 and is not; treating the gypsum as uniform hides real internal temperature differences.
Easily confused
Conduction vs. Convection
Conduction transports energy through interactions between adjacent molecules with no bulk motion and is governed by a genuine material property, k. Convection transports energy by bulk fluid motion and is governed by h, which describes a situation rather than a substance.
Natural convection vs. Forced convection
Natural convection is driven only by buoyancy from temperature-induced density differences, so the flow is weak and h is small. Forced convection is driven by a pump or fan, giving faster flow, a thinner near-wall film, and a substantially larger h.
Thermal conductivity, k vs. Convective coefficient, h
Both appear in similar-looking rate laws, but k is a property of a material measurable in a standardized apparatus, while h is an empirical coefficient valid only for the geometry, fluid, and flow it was measured in.
Resistances in series vs. Resistances in parallel
Layers stacked through the thickness add their resistances, so the largest resistance dominates. Paths running side by side add their conductances, so the smallest resistance dominates, which is why a small metal bridge can control an insulated assembly.
U.S. customary R-value vs. SI R-value
They measure the same physical quantity in different units: hft^2degF/Btu versus m^2*K/W. R_SI = R_US x 0.1761102, so the same wall is R-16.8 in American units and 2.96 in SI.
Radiation at room temperature vs. Radiation at high temperature
Because emission goes as absolute temperature to the fourth power, radiation is comparable to a natural-convection film near room temperature but dominates once surfaces are hot: the effective radiation coefficient in the worked example rises from 5.55 W/(m^2K) at 30 degC to 86.8 W/(m^2K) at 800 degC.
Key vocabulary
- Thermal conductivity (k)
- A material property, in W/(m*K), giving the heat transfer rate per unit area for a unit temperature gradient. It varies with temperature and, in porous materials, with density and moisture.
- Thermal resistance
- The ratio of temperature difference to heat transfer rate for one element of a heat path, in K/W. For a plane wall it equals thickness divided by the product of conductivity and area.
- R-value
- Thermal resistance expressed per unit area, used to rate insulation. Layer R-values add. In U.S. practice it is quoted in hft^2degF/Btu; in SI it is quoted in m^2*K/W, and the two numbers differ by a factor of 5.678263.
- U-factor / overall heat transfer coefficient
- The reciprocal of the total thermal resistance per unit area, in W/(m^2*K). It gives the heat transfer rate per unit area per degree of overall temperature difference across an assembly.
- Convective heat transfer coefficient (h)
- The proportionality constant in Q = hA(delta T), in W/(m^2*K). It is not a material property: it depends on geometry, fluid, velocity, surface condition, and flow regime, and is obtained from measurement or fitted correlation.
- Natural versus forced convection
- Natural convection is fluid motion driven only by density differences caused by heating; forced convection is fluid motion driven by an external device such as a pump or fan.
- Emissivity (e)
- A dimensionless surface property from 0 to 1 giving the fraction of black-body radiation a real surface emits at the same temperature. A perfect black body has an emissivity of 1.
- Stefan-Boltzmann constant (sigma)
- The constant relating black-body emissive power to the fourth power of absolute temperature, 5.670374419 x 10^-8 Wm^-2K^-4, exact in the revised SI.
- Thermal bridge
- A high-conductivity path, such as a metal stud or a slab edge, that runs in parallel with an insulated path and carries a disproportionate share of the heat because parallel conductances add.
- Biot number (Bi)
- The dimensionless ratio h*L_c/k comparing a body's internal conduction resistance to its surface convection resistance. Below about 0.1 the body can be treated as having a single uniform temperature.
Sources & references
- DOE Fundamentals Handbook: Thermodynamics, Heat Transfer, and Fluid Flow, Volume 2 of 3 (DOE-HDBK-1012/2-92) — U.S. Department of Energy
- University Physics Volume 2, 1.6 Mechanisms of Heat Transfer — OpenStax, Rice University
- CODATA Value: Stefan-Boltzmann constant — NIST Physical Measurement Laboratory (Fundamental Physical Constants)
- NIST Guide to the SI, Appendix B.9: Conversion Factors Listed by Kind of Quantity — National Institute of Standards and Technology (NIST Special Publication 811, 2008 edition)
- Retrospective Analysis of NIST Standard Reference Material 1450, Fibrous Glass Board, for Thermal Insulation Measurements — NIST Journal of Research, Volume 119 (2014)
- Measurement of Thermal Properties of Gypsum Board at Elevated Temperatures — NIST (Manzello, Park, Mizukami, Bentz), Proceedings of the Fifth International Conference on Structures in Fire
- 16 CFR Part 460 - Labeling and Advertising of Home Insulation (the FTC R-value Rule) — U.S. Federal Trade Commission / Office of the Federal Register
- Insulation Fact Sheet (DOE/CE-0180, 2008) — U.S. Department of Energy, Office of Energy Efficiency and Renewable Energy
- Analysis in Support of the Radiant Barrier Fact Sheet 2010 Update — Oak Ridge National Laboratory (Stovall et al.), Thermal Performance of the Exterior Envelopes of Whole Buildings XI
- On-Orbit Xenon Refueling Loading Times and Transient Analysis (TFAWS 2019, TFAWS19-ID-16) — NASA Glenn Research Center / Thermal and Fluids Analysis Workshop
- Building Energy Codes Program: Determinations — U.S. Department of Energy
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Researched 2026-08-19
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