Reference ID: MET-90A9 | Process Engineering Reference Sheets Calculation Guide
Introduction & Context
Thermal diffusivity, denoted by the Greek letter α, quantifies how quickly a temperature disturbance propagates through a material. In process engineering it is the key transport property that couples conductive heat flux to transient temperature fields. High α implies rapid thermal equilibration; low α indicates that internal temperature gradients persist. Typical applications include:
Heat-exchanger fouling analysis and thermal shock studies
Food, polymer and pharmaceutical processing where quality is temperature-history dependent
Non-Fourier conduction checks for micro- and nano-scale devices
Methodology & Formulas
The governing relation is derived from Fourier’s law and the energy balance for a rigid, isotropic medium with constant properties. Combining the conductive heat flux
\[ \vec{q} = -k\,\nabla T \]
with the transient energy equation
\[ \rho\,C_p\frac{\partial T}{\partial t} = \nabla\cdot(k\,\nabla T) \]
yields the classical diffusion equation
\[ \frac{\partial T}{\partial t} = \alpha\,\nabla^2 T \quad\text{where}\quad \alpha = \frac{k}{\rho\,C_p}. \]
Step-by-step procedure
Obtain thermal conductivity k, density ρ and specific heat capacity Cp at the relevant temperature. Units must be SI: W m−1 °C−1, kg m−3 and J kg−1 °C−1 respectively.
Insert the three properties into the algebraic formula above; no temperature conversion is required because k, ρ and Cp are already evaluated at the local temperature.
The resulting α has units of m2 s−1.
Regime
Characteristic condition
Implication for α
Laminar internal flow
\( Re < 2300 \)
Conduction dominates radial diffusion; accurate α essential for Graetz-type solutions.
Semi-infinite medium model valid; surface heat flux proportional to α½.
Quasi-steady conduction
\( Fo > 0.3 \)
Internal gradients decay exponentially with time constant \( \tau \approx L^2/(\pi^2\alpha) \).
Thermal diffusivity (α) measures how fast temperature changes propagate through a material. It is defined as α = k ⁄ (ρ·Cp), where k is thermal conductivity, ρ is density, and Cp is specific heat capacity. In process engineering it determines:
How quickly reactors, heat exchangers, or molds reach steady-state
Cycle times for heating and cooling steps
Risk of thermal shock or uneven curing in composite materials
Accurate α values let you size equipment correctly and avoid costly over-design.
The laser-flash (LFA) method is the plant-floor favorite because it needs only a small disk-shaped sample (~12 mm Ø × 3 mm) and yields α directly from the rise-time of the rear-face temperature. Key steps:
Coat sample with graphite to ensure uniform absorption
Run three shots at 25 °C, 100 °C, and 200 °C to bracket service range
Check repeatability within ±2 %; if larger, reprepare surface
Results traceable to ASTM E1461 can be obtained within two hours, fast enough for troubleshooting campaigns.
Thermal effusivity (e) governs surface temperature rise when two bodies touch; it is related to diffusivity by e = √(k·ρ·Cp) = k ⁄ √α. Rearranging gives α = (k ⁄ e)². Use effusivity when designing quench or contact-heating operations; use diffusivity when analyzing transient conduction inside a slab.
Extruded plastics, rolled metals, and fiber composites have direction-dependent α. Measure separately:
In-plane (α_x, α_y) using square samples cut at 0° and 90° to flow
Through-thickness (α_z) using the same laser-flash rig but with the beam on the edge
Report the tensor and use the appropriate component in your 2-D or 3-D simulation; neglecting anisotropy can over-predict heating times by 20–40 %.
Worked Example: Estimating Thermal Diffusivity of Tomato Purée in a Pasteuriser
A food-processing plant pasteurises tomato purée at 75 °C. To predict how quickly temperature fronts move through the product, engineers need the thermal diffusivity, \( \alpha \). Using handbook data for density, specific-heat capacity and thermal conductivity at this temperature, calculate \( \alpha \).
Knowns
Thermal conductivity, \( k \) = 0.625 W m\(^{-1}\) K\(^{-1}\)
