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 validated against two independent experimental datasets. For corrugated flexible ducts (Dai et al., 2021), it achieved a precise prediction error of +4.5%, whereas the Colebrook–White equation underestimated the pressure drop by −14.8%. In gas–liquid–solid three-phase flows (Al-Hadhrami et al., 2014), the phenomenological framework proved highly adaptable beyond conventional HVAC domains. For standard galvanized steel ducts in HVAC applications (Cbase = 0.042), the model yields consistently stable, conservative margins (ranging from +21% to +27%) relative to the Colebrook–White equation.

Operating with a constant-time computational complexity of O(1), the Balbi Equation eliminates iterative processes, offering a physically motivated and computationally efficient alternative 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)