The Bean Count Optimisation Paper: A Complete Mathematical and Rheological Determination of Optimal Beans‑on‑Toast Bean Quantities

This work presents the first fully formalised mathematical, rheological, and thermal‑diffusion model for determining the exact number of baked beans required for structurally stable, flavour‑balanced beans‑on‑toast. Moving beyond traditional volumetric approaches, the paper converts continuous measurements into discrete bean counts using bean geometry, packing efficiency, viscosity drift, heating‑method coefficients, and beanage (heating time). The study integrates multiple modelling layers, including: Bean geometry and packing efficiency, establishing the effective bean volume constant. Rheological behaviour, including shear‑thinning dynamics and temperature‑viscosity relationships. Viscosity Temporal Drift Function (VTDF), modelling how viscosity changes under pan vs microwave heating. Thermal diffusion maps, showing radial gradients in pan heating and nodal patterns in microwave heating. Structural integrity modelling, including BSI, SIC, TAT, and SUG. Failure mode analysis, covering centre sag, edge curl, and shear collapse. The paper deduces the exact optimal bean counts for both single‑slice and dual‑slice toast configurations under baseline, pan‑heated, and microwave‑heated conditions. Final results show: One slice: 43–56 beans depending on viscosity Two slices: 78–104 beans depending on viscosity The system reaches equilibrium only when the final bean completes the viscosity‑adjusted optimum, ensuring perfect structural, rheological, and sensory performance. This document serves as a foundational reference for culinary physics, breakfast optimisation research, and applied food‑engineering modelling. Includes an integrated, interactive telemetry visualiser that models and simulates real-time bean distribution physics, viscosity temporal drift, packing efficiency thresholds, and structural failure risks such as the Bean Lagoon Effect. A unified field of bean rheology, viscosity drift, thermal diffusion gradients, packing efficiency, ellipsoid bean geometry, structural integrity coefficients, absorption thresholds, surface uniformity gradients, shear‑thinning behaviour, pan vs microwave heating dynamics, beanage temporal modelling, failure modes, bite‑vector mechanics, culinary physics, breakfast optimisation theory, non‑Newtonian food fluids, evaporation‑driven viscosity rise, nodal heating patterns, radial thermal gradients, bean distribution mapping, structural collapse modelling, comfort‑index uplift, food‑engineering microdynamics, and discrete bean‑count optimisation. i would just like to say for the record, that iceland supermarket do a great scotch egg - matt Contact: For enquiries or research questions related to this work, email [email protected]