Vol. 337 No. 1 (2026)
DOI https://doi.org/10.18799/24131830/2026/1/5475
Stationary and non-stationary thermohydraulic processes of liquid in a pipeline with laminar flow
Relevance. Many branches of engineering require, to varying degrees, the calculation of pipeline systems, the analysis of fluid flow, and the investigation of heat transfer phenomena occurring during fluid motion near solid boundaries. Depending on the fluid viscosity and flow velocity, the flow regime may be laminar or turbulent, which significantly influences the temperature distribution along the pipeline. In oil and gas fields operating under harsh climatic conditions, electric heating systems are widely used for oil, gas, and water pipelines installed on the surface or for individual sections exposed at the surface in well clusters, process facilities, booster pumping stations, preliminary water and gas separation units, compressor stations, and oil collection points. The selection of the electric heating configuration and power is determined by thermal engineering calculations. However, existing engineering approaches are largely based on simplified empirical models and do not adequately account for the coupled interaction of hydrodynamic, thermal, and electrical processes, which leads to increased energy consumption and reduced reliability of fluid and gas transportation. This article presents a calculation of the velocity distribution in the cross-sectional and longitudinal directions of a pipe in order to determine the corresponding temperature distribution. Aim. To determine the fluid velocity distribution in the pipe cross-section and the temperature distribution along the pipe under laminar flow conditions based on the Navier–Stokes equations, the continuity equation, and the heat conduction equation. Methods. Analytical method is employed to solve the Navier–Stokes equations and the continuity equation. Result and conclusions. The authors have obtained the analytical expressions for the fluid velocity distribution in the pipe cross-section for both steady-state and transient laminar flow regimes. The paper demonstrates the effect of the flow regime on the heat transfer coefficient at the fluid–pipe wall interface. The authors determined the Nusselt eigenvalues and eigenfunctions, enabling the fluid temperature field to be represented as a series and the temperature distribution along the pipeline to be calculated. The results can be applied in thermal engineering calculations and in the design of electric heating systems for pipelines.
Keywords:
Heat and mass transfer, forced convection, fluid flow, convective heat and mass transfer, Nusselt functions, laminar flow, axial symmetry
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