# Laminar Flow Through a Pipe with Constant Wall Temperature

In this application, AcuSolve is used to simulate the flow of mercury through a heated pipe. The AcuSolve results are compared with analytical results for pressure drop as described in White (1991), and with temperature changes as described in Incropera and DeWitt (1981). The close agreement of AcuSolve results with analytical results validates the ability of AcuSolve to model cases with flow and imposed temperature constraints.

## Problem Description

The problem consists of mercury flowing through a circular pipe that is 0.005 m in diameter and 0.1 m long, as shown in the following image, which is not drawn to scale. Mercury enters the pipe with a constant temperature of 300 K and with a fully developed velocity profile with an average velocity of 0.005 m/s. The pipe wall has a constant temperature of 500 K. These constraints impose a temperature gradient within the fluid. As the fluid moves through the pipe it is slowly heated by the wall, resulting in an increased centerline temperature. Pressure decreases along the pipe length due to the friction imposed by the viscous stresses near the pipe wall. A periodic boundary condition is imposed on the inlet and outlet to allow the pressure to change with a constant offset, while the velocity is fully periodic.

## AcuSolve Results

The AcuSolve solution converged to a steady state and the results reflect the mean flow conditions. In this application, a fully developed laminar profile is achieved by enforcing periodic constraints on the pressure and velocity fields. The spatially developing temperature field is achieved by allowing the fluid's temperature to evolve along the full length of the model. The change in temperature along the pipe length is due to advection in the streamwise direction and heat conduction from the outer walls where the temperature is greater than the initial flow temperature. The temperature rises along the length of the pipe and the pressure drops along the length of the pipe. At the inlet, the temperature of the mercury is 300 K and the pressure is 0.496 Pa. At the outlet centerline, the temperature of the mercury has risen to 500.0 K and the pressure has dropped to -0.4965 Pa.
The following table shows the comparison of AcuSolve results with the analytical solutions from White (1994). Comparisons were made for the pressure drop between the inlet and outlet and the temperature at the center of the outlet.
Table 1.
Analytical solution AcuSolve solution Percent deviation from analytical
Pressure drop (Pa) 0.991 0.9925 0.15
Centerline temperature of flow at outlet (K) 500.0 500.0 < 0.0001

## Summary

The AcuSolve solution compares well with analytical results for flow through a pipe with constant temperature boundary conditions. The AcuSolve solution for the centerline temperature is nearly identical with the analytical results and the pressure reduces as expected for the prescribed flow conditions. The results of this simulation validate the ability of AcuSolve to accurately predict the heating of fluid flowing through a pipe with constant wall temperature.

## Simulation Settings for Laminar Flow Through a Pipe With Constant Wall Temperature

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

Global

• Problem Description
• Analysis type - Steady State
• Temperature equation - Advective Diffusive
• Turbulence equation - Laminar
• Auto Solution Strategy
• Relaxation Factor - 0.2
• Material Model
• Mercury
• Density - 13579.0 kg/m3
• Specific Heat - 139.3 J/kg-K
• Viscosity - 0.001548 kg/m-sec
• Conductivity - 8.69 W/m-K

Model

• Volume
• Fluid
• Element Set
• Material model - Mercury
• Surfaces
• Inflow
• Simple Boundary Condition - (disabled to allow for periodic conditions to be set)
• Integrated Boundary Conditions
• Mass Flux
• Type - Constant
• Constant value - 1.33e-3 kg/sec
• Temperature
• Type - Constant
• Constant value - 300.0 K
• Outflow
• Simple Boundary Condition - (disabled to allow for periodic conditions to be set)
• Wall
• Simple Boundary Condition
• Type - Wall
• Temperature BC type - Value
• Temperature - 500.0 K
• Periodics
• Periodic 1
• Individual Periodic BCs
• Velocity
• Type - Periodic
• Pressure
• Type - Single Unknown Offset

## References

F. M. White. "Fluid Mechanics". 3rd Edition. McGraw-Hill Book Co., Inc. New York. 1994.

F. P. Incropera and D. P. DeWitt. "Fundamentals of Heat Transfer". John Wiley & Sons. New York. 1981.