# Laminar Flow Through a Pipe with Species Transport

In this application, AcuSolve is used to simulate the transport of a passive scalar species in a fully developed laminar flow. AcuSolve results are compared with analytical results as described in Kays and Crawford (1993). The close agreement of AcuSolve results with analytical results validates the ability of AcuSolve to model cases with species transport.

## Problem Description

The problem consists of species transport in a pipe 0.005 m in diameter and 0.1 m long, as shown in the following image, which is not drawn to scale. The species transport equation is bounded at the inlet with a value of 1.0, while the boundary condition at the wall is 0.0. These constraints impose a concentration gradient within the fluid. The concentration gradient causes the fraction of species 1 to decrease near the pipe wall due to diffusion. The reduced concentration of the species is transported downstream in the pipe via advection, and leads to a continued decrease in the concentration as flow propagates through the pipe.

## AcuSolve Results

The AcuSolve solution converged to a steady state and the results reflect the mean flow conditions with the species fraction changing along the length of the pipe. At the inlet, the concentration of species 1 is 100 percent. The fraction of species 1 decreases along the length of the pipe, developing a parabolic concentration profile due to the no-slip walls and the specified concentration at the wall.
Analytical values of the species fraction at different distances along the pipe are presented with the corresponding AcuSolve results in the following table. The analytical solution contains a series of five terms that are summed to yield the values shown in the table.
Table 1.
Distance along pipe (m) Analytical species 1 fraction AcuSolve species 1 fraction Percent deviation from analytical
0 1.0000* 1.0000 0.00
0.01 0.8225 0.8298 0.90
0.02 0.7308 0.7344 0.49
0.03 0.6593 0.6603 0.15
0.04 0.5992 0.5988 0.06
0.05 0.5469 0.5448 0.38
0.06 0.5006 0.4981 0.49
0.07 0.4589 0.4554 0.78
0.08 0.4212 0.4171 0.98
0.09 0.3869 0.3824 1.15
0.1 0.3555 0.3507 1.35

* The theoretical species fraction at the inlet is shown rather than the analytical. In order for the analytical solution to correspond exactly with the theoretical results, additional terms in the series summation would need to be evaluated.

## Summary

The AcuSolve solution compares well with analytical results for flow through a pipe with species transport. In this application, a fully developed laminar profile is achieved by enforcing periodic constraints on the pressure and velocity fields. The spatially developing species field is achieved by not enforcing periodicity and allowing the concentration to evolve along the full length of the model. The change in concentration along the pipe length is due to advection in the streamwise direction and diffusion of the species to the outer walls where the concentration is zero. The AcuSolve solution for fraction of species 1 is within 1.35 percent of analytical results.

## Simulation Settings for Laminar Flow Through a Pipe with Species Mixing

AcuConsole database file: <your working directory>\pipe_laminar_species\pipe_laminar_species.acs.

Global

• Problem Description
• Analysis type - Steady State
• Species equation - Advective Diffusive
• Num. species - 1
• Turbulence equation - Laminar
• Auto Solution Strategy
• Relaxation factor - 0.2
• Material Model
• Species1
• Density - 1.0 kg/m3
• Viscosity - 1.0e-5 kg/m-sec
• Diffusivity
• 1
• Diffusivity - 1.43e-5 kg/m-sec

Model

• Volumes
• Fluid
• Element Set
• Material model - Species1
• Surfaces
• Inflow
• Simple Boundary Condition - (disabled to allow for periodic conditions to be set)
• Nodal Boundary Conditions
• Species 1
• Type - Constant
• Constant value - 1.0
• Integrated Boundary Conditions
• Mass Flux
• Type - Constant
• Constant value - 1.96e-05 kg/sec
• Outflow
• Simple Boundary Condition - (disabled to allow for periodic conditions to be set)
• Wall
• Simple Boundary Condition
• Type - Wall
• Species 1 BC type - Value
• Species 1 - 0.0
• Periodics
• Periodic 1
• Individual Periodic BCs
• Velocity
• Type - Periodic
• Pressure
• Type - Single Unknown Offset

## References

W. M. Kays and M. E. Crawford. "Convective Heat and Mass Transfer", 3rd Edition. pp. 126-134. McGraw-Hill Book Co., Inc. New York. 1993.