Heat transfer

Reference values#

Values reported or used in the sources. Models fit some values to data; they are not material constants or targets. Coefficients in °C⁻¹ and K⁻¹ are numerically equal, since they refer to temperature differences.

The physics in equations#

Convection at the surface. Single-bean models treat the heat flux into the bean as proportional to the gap between roasting temperature and surface temperature, scaled by a heat-transfer coefficient h:⁷ q = h (T_air − T_s). The coefficient is not a property of coffee. It depends on the properties of the roasting air³ and, in the correlation below, on air speed.⁵ Several coffee models estimate it with the Ranz–Marshall correlation, Nu = 2 + 0.6 Re^1/2 Pr^1/3, where the Nusselt number Nu is h made dimensionless with the equivalent bean diameter.^5,6,8 The Reynolds number Re grows with air speed, so faster air raises h.⁵ As air speed approaches zero, the formula gives Nu close to 2, its lowest value. This is why the 2025 drum study, with air moving at only 0.02 m/s from drum rotation, calculated a low convective coefficient.⁶

Radiation. A surface emits at a rate Q/t = σ e A T⁴, where e is the emissivity, A the area and T the absolute temperature; the net exchange with surroundings at T₂ is σ e A (T₂⁴ − T₁⁴).¹² The constant σ is 5.670 374 419 × 10⁻⁸ W m⁻² K⁻⁴.¹³ A worked example, Coffearium’s arithmetic from the law and not a roaster measurement: raising a wall from 300 °C (573 K) to 400 °C (673 K) raises its emission by a factor of (673/573)⁴ ≈ 1.90. For a bean at 150 °C, the net radiation it receives rises by about 2.3 times, with emissivity and geometry held constant. No source opened for this entry gives a measured emissivity for coffee beans or roaster walls.

Conduction inside. Within the bean, a 2023 study notes, conductive heat transfer is governed by porosity and density.³ In one 2025 model, conductivity falls with bean density, which itself falls as moisture leaves.⁵ In the 2017 Model 2, the pores hold liquid water (0.58 W/(m·K)), vapour (0.016) and carbon dioxide (0.015), next to a solid matrix at 0.037 W/(m·K).⁷

The Biot number. The Biot number sets the surface coefficient against the bean’s conductivity: Bi = h L / k, with L the characteristic length (volume over surface) and k the conductivity. The 2025 drum study reported Bi = 0.48–0.82, which its authors read as a relevant role of conductive resistance within the bean.⁶ The reported values follow from that formula with the study’s own h, L and k (Coffearium’s check: 60.20 × 8.8056 × 10⁻⁴ / 0.11 ≈ 0.48).⁶ The 2017 Model 1 used a Nusselt number of 1; the authors note that this number is sometimes called the heat-transfer Biot number.⁷ The 2025 study nevertheless used a lumped model with a single bean temperature.⁶

Heat-transfer coefficients reported#

Studies obtained their coefficients in different ways: fitted to measured bean temperatures, calculated from a correlation, or chosen to match earlier data. They are therefore not strictly comparable.^6,7,11

• Single bean in a gas-fired drum. The overall coefficient, fitted to measured temperatures, rose from 60.20 to 102.55 W m⁻² °C⁻¹ between 200 and 280 °C.⁶ The convective part, calculated, stayed at 15.91–17.43.⁶ The bean touched neither wall nor other beans, so conduction was excluded by design.⁶ The authors attribute the gap to radiation (see Contested).⁶
• Model sphere. A 2017 model of a bean as a sphere of 4 mm radius in air at 200 °C used 20 and 16 W/(m²·K), chosen to agree with earlier data.⁷ Its boundary condition included convection but no radiation.⁷
• Spouted bed. A 1997 study derived one overall coefficient of 12.023 W/(m²·K) from the time–temperature curve and described the heating as unsteady-state heat transfer.¹¹

In a spouted bed, a 2021 particle-tracking study expects a high coefficient in the spout, where beans meet the full air stream.⁴ Within the bed, it lists bean-to-bean contact conduction and bean-to-bean radiation among the mechanisms.⁴ As beans lighten, more of them are carried into the dilute zone where convection dominates, so the authors expect the global coefficient to rise as the roast proceeds.⁴

Models of the roasting bean#

Models of heat transfer in coffee differ in what they include and leave out.

• Fabbri et al. 2011 built a three-dimensional finite-element model of heat and moisture inside one bean, in simplified and 3D-scanned shapes, for a rotating cylinder roaster under natural convection.⁹ Predicted bean-centre temperatures matched thermocouples in a drum prototype with an RMSE of 5.97 °C.⁹
• Alonso-Torres et al. 2013 used computational fluid dynamics for a semi-ellipsoid bean in hot air at 200–250 °C and 2 m/s, with h from the Ranz–Marshall correlation.⁸ The authors report good agreement with measured temperature and moisture, and note that accuracy fell slightly at higher roasting temperatures.⁸
• Fadai et al. 2017 compared a model adapted from earlier work with a new first-principles model that tracks water and vapour in the pores.⁷ Model 1 becomes nearly uniform in temperature and moisture after the first minute; the authors judge Model 2’s slower drying front more realistic.⁷
• Bustos-Vanegas et al. 2025 used a lumped model with one bean temperature, grouping convection and radiation into one coefficient, and added evaporation and reaction heat.⁶ Evaporation had a greater effect on the predicted temperature than reaction heat.⁶
• Al-Shemmeri et al. 2025 simulated a whole batch in a pilot-scale (0.5 kg) spouted-bed roaster. It neglected metal-to-bean conduction and bean-to-bean conduction and radiation, and it cannot resolve gradients inside the bean.⁵

No model opened for this entry follows a full drum batch with wall contact, hot air and radiation together.

Measuring heat transfer#

Researchers measure bean temperature with fine thermocouples placed in the bean. A 2005 fluid-bed study drilled 0.3 mm holes into beans for a 0.25 mm thermocouple.¹⁰ The 2000 thesis took care that the tip sat in the bean tissue, not in the central folding gap.² In the 2005 study, the bean core never reached the set air temperature; the authors attributed this to radiation and called it typical of small equipment with a large air-to-bean ratio.¹⁰

Production roasters do not measure bean temperature directly. Their probe reads a mix of bean and air temperature, specific to each machine.^1,4 Coefficients in research are fitted to measured bean temperatures or calculated from correlations.^6,11

Risks and defects#

• Surface over-roasted, core under-roasted. In extreme fast roasting, high heat transfer at the surface can give “burnt” and “greenish” notes together, according to the textbook chapter.¹
• Scorched flat sides. A drum turning too fast presses beans against the hot wall, over-develops the outside and, at worst, scorches the flats.¹ The thesis names mixing to prevent scorching as a main task of a roaster.²
• Misreading the probe. The probe reads bean and air together, so a change in airflow or drum speed can move the reading without the same change in the beans.^1,4
• Uneven heating. Beans must keep moving for homogeneous heat transfer; any movement also stresses the beans mechanically.¹

Open questions#

• Shares in a loaded production roaster. No study opened for this entry measured conduction, convection and radiation separately in a full batch.^1,6
• Emissivity of beans and walls. No measured value was found, so the radiation share cannot be calculated from first principles.
• Gradients at tips and edges. Models use spheres, semi-ellipsoids or a scanned bean, and none opened for this entry reports temperatures at tips or edges.^7,8,9
• Heat paths and flavour. The thesis recommends a fairly high conductive share and low air-to-bean ratios, linking low ratios to better cup quality and less oxygen contact², but no sensory study opened for this entry varied the heat path alone at equal roast degree and time.

Key studies#