Balbi Equation - Fundamental Concepts and Principles

A New Explicit, Non-Iterative and Unified Model for Distributed Pressure Drop Calculation in Ducts

Abstract

This paper presents the Balbi Equation, a phenomenological and semi-empirical formulation for estimating distributed pressure drop in circular ducts. Rather than replacing established methods, the proposed approach offers an alternative physical interpretation of flow resistance while remaining consistent with classical fluid mechanics and established engineering practice.

The model combines analytical development with experimental calibration through five fundamental relationships: (1) the viscous wavelength, λ = (ν / vₘ) · kBalbi, representing an effective characteristic length associated with viscous momentum diffusion; (2) a unified exponential velocity profile, u(r) = vₘₐₓ · (1 − e^(-((R − r)/λ)^α)), applicable to laminar, transitional, and turbulent regimes; (3) a continuous flow-regime function, α(Re) = 1 + 1 / (1 + (Re / 2800)⁴), providing a smooth transition between flow regimes; (4) a pressure-drop expression, ΔP / L = (2μvₘₐₓα) / (Rλ) · (1 / 1000), derived directly from the proposed velocity profile; and (5) a calibration relationship, kBalbi = Cbase Re^0.25 (ε / Dh)^0.1, which relates the characteristic viscous scale to wall roughness and flow conditions.

The formulation was evaluated against two independent experimental datasets. For corrugated flexible ducts (Dai et al., 2021), it achieved a mean prediction error of +4.5%, whereas the Colebrook–White equation underestimated the measured pressure drop by −14.8%. In gas–liquid–solid three-phase flows (Al-Hadhrami et al., 2014), the phenomenological framework was also evaluated beyond the conventional HVAC application domain. For standard galvanized steel ducts in HVAC applications (Cbase = 0.042), the model yields pressure-drop predictions approximately 21% to 27% higher than those obtained with the Colebrook–White equation under the conditions investigated.

Operating with constant-time computational complexity, O(1), the Balbi Equation avoids iterative processes, offering a physically motivated and computationally efficient formulation for spreadsheet and software implementation in engineering design.

Fundamental equations:

λ = (ν / vₘ) · kBalbi

kBalbi = Cbase Re^0.25 (ε / Dh)^0.1

u(r) = vₘₐₓ · (1 − e^(-((R − r)/λ)^α))

α(Re) = 1 + 1 / (1 + (Re / 2800)⁴)

ΔP / L = (2μvₘₐₓα) / (Rλ) · (1 / 1000)