Thermodynamic Phase Transitions and Heat Transfer Physics in Porous Matrices during Roasting

Coffee roasting is a complex phase transition and thermodynamic process that fundamentally reorganizes the physicochemical structure within a green coffee bean. During the roasting process, the semi-crystalline cellulose and hemicellulose matrices—the primary components of the bean—undergo unique physical state changes depending on variations in temperature and moisture content. To understand this, quantifying the glass transition temperature (T_g) phase diagram dynamics and the associated heat transfer and energy balance is essential.

1. Glass Transition (T

_g) Phase Diagram Dynamics and State Transitions

Green coffee beans exist in a 'Glassy State' at room temperature, characterized by high hardness and brittleness. As roasting begins and thermal energy is supplied, the internal amorphous polymer chains gain thermal activation energy and transition into a flexible, easily deformable 'Rubbery State'. The critical temperature at which this transition occurs is defined as the glass transition temperature (T_g). T_g is not a fixed constant but is strongly dependent on the moisture content (w), which acts as a plasticizer within the bean. According to T_g curves following the Flory-Huggins theory or the Gordon-Taylor equation, higher moisture content (approx. 10–12%) allows water molecules to increase the free volume between polymer chains, lowering the T_g to near room temperature (40–60°C). As roasting progresses and moisture evaporates due to dehydration, the plasticizing effect decreases, causing the T_g to rise rapidly to 150–180°C or higher. When the bean temperature crosses the T_g curve and transitions into the rubbery state, the internal viscoelasticity increases sharply, enabling plastic deformation of cell walls and porous expansion driven by internal vapor pressure. Conversely, during the quenching process in the latter stages of roasting, the temperature drops rapidly, freezing the bean (Vitrification) back into a hard and brittle glassy state, resulting in a porous matrix structure (Porosity, ε ≈ 0.3–0.5) with high brittleness.

2. Heat Transfer Equations in Porous Coffee Beds (Conduction, Convection, Radiation)

The heat flux (q) transferred to the beans in a drum roaster is described by the simultaneous, parallel combination of three mechanisms: conduction, convection, and radiation. Each heat transfer mode is quantified by the following governing equations:

First, conduction, which occurs through contact between the roaster drum walls and the beans, follows Fourier's Law:

$ q_{\text{cond}} = -k \cdot A \cdot ( abla T) $

Here, k is the effective thermal conductivity of the bean cell walls and lignocellulosic structure (W/m·K), A is the contact cross-sectional area, and ∇T is the spatial temperature gradient. This is a key factor in determining internal temperature polarization during the initial drying stage.

Second, convection between the hot air circulated by the blower and the bean surface is defined by Newton's Law of Cooling:

$ q_{\text{conv}} = h \cdot A \cdot (T_{\text{gas}} - T_s) $

Here, h is the effective convective heat transfer coefficient (W/m²·K), which is derived from the Nusselt number (Nu), a function of the fluid's Reynolds number (Re) and Prandtl number (Pr). T_g is the temperature of the hot air, and T_s is the bean surface temperature. Modern hot-air roasters maximize this convective heat to induce uniform heat transfer into the interior of the beans.

Third, radiation emitted from the internal drum walls and burner heat sources is based on the Stefan-Boltzmann Law:

$ q_{\text{rad}} = \epsilon \cdot \sigma \cdot A \cdot (T_{\text{surr}}^4 - T_s^4) $

Here, ε is the emissivity of the bean surface, σ is the Stefan-Boltzmann constant (5.67 × 10^-8 W/m²·K⁴), and T_surr is the surrounding radiant temperature inside the roaster. The contribution of radiative heat increases rapidly in the mid-to-late stages of roasting in high-temperature zones.

3. Thermodynamics of Moisture Evaporation and Energy Balance

The transient energy balance within the bean during the roasting process is expressed by the first law of thermodynamics as follows:

$ m \cdot C_p \cdot \frac{dT_s}{dt} = q_{\text{cond}} + q_{\text{conv}} + q_{\text{rad}} - Q_{\text{evap}} + Q_{\text{rxn}} $

Here, m is the mass of the bean, C_p is the specific heat of the bean as a function of temperature and moisture content, and dT_s/dt is the rate of temperature change (Rate of Rise, ROR). The latent heat energy balance of moisture evaporation (Q_evap), which is the largest endothermic term in the energy balance, is given by:

$ Q_{\text{evap}} = \frac{dm_w}{dt} \cdot \Delta H_{\text{vap}} $

(dm_w/dt) is the moisture evaporation rate per unit time, and ΔH_vap is the effective enthalpy of vaporization required for moisture to vaporize against the capillary pressure of the micro-porous bean structure. In the early stage of roasting (drying phase), the moisture evaporation rate is very high, consuming most of the supplied heat as Q_evap, which causes the ROR to rise gradually. However, after the first crack, when moisture content falls below 3–4%, Q_evap decreases sharply, making ROR control extremely difficult. Furthermore, as the enthalpy of the thermal decomposition reaction (Q_rxn) of organic matter occurring around 200°C transitions from endothermic to exothermic (Q_rxn > 0), a thermal runaway phenomenon in the beans may occur. Therefore, precise thermodynamic balancing through external hot-air supply and exhaust damper adjustment is required.