# Laminar Flow Through a Pipe With Imposed Heat Flux

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 heat flux.

## Problem Description

The problem consists of mercury flowing through a heated, 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. The pipe has an imposed heat flux of 2000 W/m2. The laminar velocity field is modeled as periodic to achieve a fully developed velocity profile. As the fluid moves through the pipe it is slowly heated by the heat flux on 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.

## AcuSolve Results

The AcuSolve solution converged to a steady state and the results reflect the mean flow conditions. The greatest pressure is at the inlet, with pressure decreasing as a function of distance away from the inlet. The temperature of the mercury in the pipe increases as a function of distance from the inlet due to the impact of the heat flux on the pipe walls.
The analytical solution for the pressure drop, between the inlet and outlet, as well as the analytical solution for the temperature of the mercury at the center of the outlet are presented with the corresponding AcuSolve results in the following table.
 Analytical solution AcuSolve solution Percent deviation from analytical Pressure drop (Pa) 0.991 0.994 0.303 Centerline temperature of flow at outlet (K) 316.917 316.269 0.204

## Summary

The AcuSolve solution compares well with analytical results for flow through a pipe with an imposed heat flux. In this application, the pressure drop in the fluid region is driven by the viscous stresses near the pipe wall. The temperature in the fluid region is influenced by the imposed heat flux on the walls. The values of pressure and temperature that are predicted by AcuSolve are within 0.3 percent of the analytical solution.

## Simulation Settings for Laminar Flow Through a Pipe With Imposed Heat Flux

AcuConsole database file: <your working directory>\pipe_laminar_heat\pipe_laminar_heat.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
• Viscosity - 0.001548 kg/m-sec

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 - Flux
• Heat flux - 2000.0 W/m2
• Periodics
• Periodic 1
• Individual Periodic BCs
• Velocity
• Type - Periodic
• Pressure
• Type - Single Unknown Offset
• Temperature
• Type - Single Unknown Offset

## References

F. M. White. "Viscous Fluid Flow". Section 3-2.1. McGraw-Hill Book Co., Inc.. New York. 1991.

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