Concentric Laminar Annular Pressure Loss Calculator

Calculate exact Newtonian concentric-annulus friction loss and the radial velocity profile, with geometry and conservative laminar-range checks.

Results

Calculate to view linked results. Default values are examples, not recommended operating inputs.

Learn, compare and practice

Worked explanations, controlled studies and printable exercises connected to this calculation.

When a laminar annular model stops being appropriate

Calculation sequence

Enter the outer bore ID and inner pipe OD for the same uniform interval, not two pipe IDs. Rates are actual liquid volume per second at the modeled condition. Use a measured Newtonian dynamic viscosity and density at that condition. The inner-to-outer diameter ratio must be between 0.01 and 0.999. This intentionally excludes both a vanished inner wall and an unresolved near-contact gap.
The annular conductance retains both radii and the logarithmic geometry term. It is not a circular-pipe Poiseuille formula with hydraulic diameter substituted. The software evaluates the conductance and velocity brackets with decimal arithmetic to avoid subtractive cancellation in narrow gaps; the independent regression fixtures use 80-digit arithmetic and cross-sectional velocity integration.
Divide rate by the annular area to obtain mean velocity and use D minus d for the diagnostic hydraulic diameter. This workflow rejects Re_H above 1,000, a conservative software envelope rather than a universal annular transition criterion. The stated fully developed interval must already satisfy the entrance-flow assumptions; entry length and local losses are not automatically inferred or added.
Read the friction-only pressure gradient and its product with interval length. The chart samples the radial axial-velocity distribution and enforces zero velocity at both stationary walls. The maximum is evaluated at the analytical interior stationary point rather than chosen from the chart grid. The reported Darcy factor is derived from the exact gradient; it is not simply 64/Re_H.

Worked check

The defaults describe a synthetic 0.2-m bore around a 0.1-m stationary pipe, Q = 0.001 m3/s, mu = 0.1 Pa.s, rho = 1,200 kg/m3 and L = 1,000 m. Mean velocity is about 0.0424413 m/s and Re_H is about 50.9296. Friction loss is about 20,212.7 Pa. Doubling length doubles loss without changing the velocity profile; doubling rate doubles the viscous gradient within the supported laminar envelope.

Synthetic examples for reproducibility, not measured field results.

Assumptions and limitations

Single incompressible Newtonian liquid, stationary concentric circular walls and fully developed laminar axial flow only. No eccentricity, pipe rotation, roughness-driven turbulence, washouts, yield stress, shear thinning, cuttings, gas or wall slip is modeled. Do not substitute a Herschel-Bulkley apparent viscosity and call this a non-Newtonian annular solution.
The pressure output excludes hydrostatic elevation change, entrance/exit losses, fittings and changing geometry. Add appropriate separate components only with consistent signs and fluid properties. It is not total standpipe pressure, an ECD prediction or evidence of adequate hole cleaning.
Positive inputs and the supported radius-ratio/Reynolds envelopes are enforced. Density affects the Reynolds and Darcy diagnostics, but not the exact viscous gradient at fixed rate and viscosity. Numerical agreement with the analytical solution does not verify a field fluid's Newtonian behavior or installation geometry.

Implemented equations

Exact concentric-annulus conductance

\begin{aligned}C_a&=R_o^4-R_i^4\\&\quad-\frac{(R_o^2-R_i^2)^2}{\ln(R_o/R_i)}\\G_p&=\frac{8\mu Q}{\pi C_a}\\\Delta p_f&=G_pL\end{aligned}

Velocity and hydraulic-diameter diagnostics

\begin{aligned}u(r)&=\frac{G_p}{4\mu}\left[R_o^2-r^2\right.\\&\left.\quad-\frac{R_o^2-R_i^2}{\ln(R_o/R_i)}\ln\frac{R_o}{r}\right]\\D_H&=2(R_o-R_i)\\Re_H&=\frac{\rho\bar uD_H}{\mu}\\f_D&=\frac{2D_HG_p}{\rho\bar u^2}\end{aligned}

Sources and maintenance

Published by PetroCalcHub Team. Implementation and source checks: 2026-10-21. This does not claim independent expert review.

Report a correction

Related tools and references

Yield-power-law concentric annular solution Annular average velocity Drilling hydraulics guide Measured Herschel-Bulkley fitting (not this Newtonian model)