Fluid Flow

Laminar Pipe Flow

For steady flow in a pipe (whether laminar or turbulent), a momentum balance on the fluid gives the shear stress at any distance from the pipe centerline.

In Equation (1), Φ = P + ρ gz. The volumetric flowrate Q can be related to the local shear rate by doing an integration by parts of Equation (2).

Newtonian fluid. For a Newtonian fluid, τ _rx = µY _rx, which gives the following volumetric flowrate, known as the Hagen-Poiseuille equation.

It can be written in dimensionless form in Equation (4) with the two terms defined in Equations (5) and (6).

Power law. A fluid that follows the power law model obeys the relationship τ _rx = – µ(– Y _rx) ⁿ. This gives the following equation.

Equation (7) can be rearranged into the following dimensionless form.

Bingham plastic. In this case, there is a solid-like “plug flow” region from the pipe centerline (where τ _rx = 0) to the point where – τ _rx = τ ₀ (that is, at r = r ₀ = R x τ ₀/ τ _w). The result is a flow integral modified from Equation (2). For a Bingham plastic, – τ _rx = τ ₀ + µ _∞ ( – Y _rx). Using this expression and the modified flow integral, the Buckingham-Reiner Equation (10) is found.

The equivalent dimensionless form is given by Equations (11), (12) and (13).

Turbulent Pipe Flow

Since most turbulent flows cannot be analyzed from a purely theoretical perspective, data and generalized dimensionless correlations are used.

Newtonian fluid. The friction factor for a Newtonian fluid in turbulent flow is a function of both N _Re and the pipe relative roughness, ε /D, which can be read off the Moody diagram [ 5]. The turbulent part of the Moody diagram (for N _Re > 4,000) is accurately represented by the Colebrook equation (14).

When N _Re is very large, the friction factor depends only on ε /D. This condition is noted with f _T as the “fully turbulent” friction factor in Equation (15).

The Churchill Equation [ 2] represents the entire Moody diagram, from laminar, through transition flow, to fully turbulent flow. It is presented here as Equations (16), (17), and (18).

Power law. For a power-law fluid, the friction factor depends only upon Equation (9) and the flow index, as represented by Equations (19) – (25) [ 3].

The value of N _Re where transition from laminar to turbulent flow occurs ( N _Re,plc) is given by Equation (25).

Bingham plastic. For the Bingham plastic, f _T is solely a function of N _{Re ∞} and N _He, as represented by Equations (26) – (29).