Harmonic Growth—The Harmonic Number Applied to ROI, Business Scaling, Nth Plant Economics, Micro Combined Heat and Power (mCHP), Biorefineries, and Biochar Production

• Published in: Mathematical methods in the applied sciences
• Main Author: Jasper, Micah N.
• Format: Journal Article
• Language: English
• Published: 10.08.2025
• ISSN: 0170-4214; 1099-1476
• DOI: 10.1002/mma.11156

Scaling is a topic of utmost importance in business. When to scale and how has been the subject of countless books and seminars. This paper looks at modular businesses that have income and costs directly proportional to the number of units. This work assumes that all of the plants or units have the same capital cost, operating costs, and revenue and that the company reinvests all net profit into purchasing more units. Furthermore, this model ignores depreciation, inflation, fluctuating market prices, and market saturation. With these assumptions, the question of when the company can afford to scale and purchase the next unit is directly related to the payback period and the harmonic number. This work describes a novel growth function called the harmonic growth function based on harmonic numbers. This function is related to the standard ROI. The harmonic growth function is explained and applied to a micro combined heat and power (mCHP) station, a biomass company expanding the number of biorefineries, and biochar retort kilns for biochar production.