While single-phase pipe flow is almost always assumed to be one-dimensional, that is, homogeneous across the pipe's diameter, there can potentially be two-dimensional thermal effects in the laminar flow of high-viscosity oil. The heat that's initially imparted to the oil by heaters or by the pumping inefficiencies must conduct through the outer part of the oil before it can leave the pipe. If the oil has a strongly temperature-dependent viscosity, this can lead to significant viscosity variations across the pipe, and in turn to a flow regime that differs significantly from the usual Hagen-Poiseuille solution for laminar pipe flow. This article investigates the conditions necessary for the one-dimensional approximation to introduce significant errors in the frictional pressure gradient.
Flow in pipelines is generally assumed to be one-dimensional. That is, the pressure, temperature, and velocity are assumed to be uniform across the pipe. The one-dimensional approximation for the pressure is quite accurate under nearly all conditions, since any pressure gradient across the pipe will cause the fluid to move so as to balance the pressure. There is a small increase in pressure from the top of the pipe to the bottom to balance the gravitational head, and from the inside to the outside of a bend to balance the centrifugal force of the flow turning around the bend. For pipe lines these variations are small fractions of the pressure in the pipe. Actually, it is the head that is constant across the pipe. There are variations in the velocity and the temperature near the pipe wall, because the temperature of the environment is generally different from that of the fluid and the velocity of the wall is, of course, zero. For turbulent flow the eddy conductivity is so high that the constant temperature approximation is still valid, although not perfectly accurate. Heat transfer to the pipe wall can be calculated by a heat transfer coefficient or, usually, by calculating the conduction in the pipe wall and the surrounding environment. There is always a boundary layer near the pipe wall in which the velocity varies from the bulk fluid velocity to zero at the wall. For turbulent flow this boundary layer is very small compared to the pipe radius. The momentum transfer, which we usually refer to as the frictional resistance to the flow, can be calculated from the Darcy-Weisbach equation with a friction factor which depends on pipe roughness and the Reynolds number. For laminar flow, the boundary layer essentially fills the pipe. That is, the velocity varies all across the pipe. Steady-state isothermal laminar flow in a pipe, also known as Hagen-Poiseuille flow is one of the few fluid flow problems for which an exact solution can be found1. The velocity profile across the pipe is a parabola, with zero velocity at the wall and a centerline velocity equal to twice the
