Reference ID: MET-FD16 | Process Engineering Reference Sheets Calculation Guide
Introduction & Context
The thermal conductivity (\(k\)) of multi-component food and biological materials is a spatial transport property. Unlike thermodynamic properties that may be mass-averaged, thermal conductivity depends on the volumetric distribution of phases. Consequently, mixing rules—such as the Parallel, Series, or Maxwell models—must be calculated using volume fractions (\(\phi_i\)) rather than mass fractions (\(X_i\)).
In Process Engineering, ignoring the density differences between constituents (e.g., water at \(\approx 1000 \, \text{kg/m}^3\) vs. protein at \(\approx 1300 \, \text{kg/m}^3\)) leads to significant errors in predicting heat transfer coefficients and sterilization lethality (\(F_0\)). Standard practice, such as the Choi-Okos model, requires converting composition data to volume fractions to ensure accurate thermal process design for heat exchangers, retorts, and freezers.
Methodology & Formulas
Determine Component Properties
Identify the thermal conductivity (\(k_i\)) and density (\(\rho_i\)) for each macro-component (Water, Protein, Fat, Carbohydrate, Ash) at the target temperature.
Convert Mass Fractions to Volume Fractions
Given mass fractions \(X_i\), first calculate the bulk specific volume (\(V\)):
\[
V = \sum \frac{X_i}{\rho_i}
\]
Then, calculate the volume fraction (\(\phi_i\)) for each component:
\[
\phi_i = \frac{X_i / \rho_i}{V}
\]
Apply the Parallel Mixing Rule
The effective thermal conductivity \(k\) (representing the upper physical bound) is the volume-weighted average:
\[
k = \sum \phi_i k_i = \phi_{\text{w}}k_{\text{w}} + \phi_{\text{p}}k_{\text{p}} + \phi_{\text{f}}k_{\text{f}} + \phi_{\text{c}}k_{\text{c}} + \phi_{\text{a}}k_{\text{a}}
\]
Material Regime
Typical \(k\) Range (W m⁻¹ K⁻¹)
Applicability (Volumetric)
High-moisture liquids/gels
0.50 – 0.65
\(\phi_{\text{water}} > 0.70\)
Intermediate-moisture solids
0.30 – 0.50
\(0.40 < \phi_{\text{water}} < 0.70\)
Fat-rich or porous materials
0.10 – 0.30
\(\phi_{\text{fat}}\) or \(\phi_{\text{air}}\) dominant
Thermal conductivity is a spatial transport property; it describes how heat moves through a volume of space. Because different components (like air, fat, and protein) occupy vastly different volumes for the same mass, mass-weighting fails to represent the actual physical path of heat flow. Volume fractions correctly account for the geometric distribution of the phases.
The Choi-Okos model provides temperature-dependent mathematical correlations for the thermal conductivity and density of pure food components (water, protein, fat, etc.) between -40 °C and 150 °C. It should be used whenever the process involves significant temperature swings, such as during UHT sterilization or blast freezing, to ensure the \(k_i\) and \(\rho_i\) values used in the mixing rule are accurate for the local temperature.
The Parallel model assumes components are arranged in layers parallel to the heat flow, providing the maximum theoretical conductivity. The Series model assumes layers perpendicular to heat flow, providing the minimum. Most liquid foods follow the Parallel model, while highly structured or porous foods may require a Maxwell-Eucken or Krischer model to account for the continuous vs. dispersed phases.
Air has a very low thermal conductivity (\(\approx 0.026 \, \text{W m⁻¹ K⁻¹}\)) and very low density (\(\approx 1.2 \, \text{kg/m}^3\)). Even a tiny mass fraction of air results in a massive volume fraction (\(\phi_{\text{air}}\)), which drastically reduces the bulk thermal conductivity. In porous materials like bread or powders, the volume fraction of air is the dominant factor in the mixing rule.
Worked Example – Predicting Thermal Conductivity of a Re-formulated Sauce
A dairy plant is re-formulating a high-protein dessert sauce. To size the holding tube for an aseptic line at 35 °C, the engineering team must calculate the thermal conductivity using volume fractions to account for the high density of proteins and carbohydrates relative to water and fat.
1. Knowns (at 35 °C)
Mass fractions (\(X_i\)): Water=0.65, Protein=0.18, Fat=0.12, Carb=0.05
Thermal conductivities (\(k_i\), W m⁻¹ K⁻¹): \(k_w\)=0.600, \(k_p\)=0.200, \(k_f\)=0.180, \(k_c\)=0.250
Densities (\(\rho_i\), kg m⁻³): \(\rho_w\)=994, \(\rho_p\)=1320, \(\rho_f\)=920, \(\rho_c\)=1600
Sum the volume-weighted contributions:
\[ k = (0.6869 \times 0.600) + (0.1433 \times 0.200) + (0.1370 \times 0.180) + (0.0328 \times 0.250) \]
\[ k = 0.4121 + 0.0287 + 0.0247 + 0.0082 = 0.4737 \, \text{W m⁻¹ K⁻¹} \]
Final Answer
The predicted thermal conductivity is 0.474 W m⁻¹ K⁻¹.
Note: Using mass fractions would have yielded 0.461 W m⁻¹ K⁻¹, a 2.7% underestimation that would propagate into significant errors in heat transfer calculations.
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