An integro-partial differential equation

This model comes courtesy of Daniel Smith and Ali Shajii of MKS Instruments, Wilmington, Massachusetts, USA.

Introduction

• The heat distribution in a hollow pipe, whose ends are held at two different temperatures, is studied.

• The outside surface is assumed to be thermally isolated and the inner surfaces have radiation boundary conditions. The role of convection in the heat transfer is taken to be negligible.

• A comparison is made with a purely conductive model in order to visualize the effect of the heat radiation.

• The temperature is assumed to be constant along the thickness of the pipe. Rotational symmetry is also assumed, allowing the pipe to be modeled in 1D.

• A 3D model is also shown as a comparison

0<x<L

Di

Do

Geometry

• The geometry of the pipe is shown below:

Geometry

• Since we have radial and rotational symmetry, we will only consider temperature variations along the x-axis, which is directed along the axis of the pipe.

• Therefore the 1D geometry consists of a line between x=-0.1 and x=0.1.

Domain equations

• The heat transfer equation is given by:

L

ii

i

i

ip D

xdxTxxk

DD

DT

DD

D

x

T

xt

TC

0

4

220

422

0

,44

• The kernel k = k(x,xf) is defined as:

2/32

3

12

321

where:iD

xx

Boundary and Initial Conditions

• The pipe is at a held constant temperature T=1500 at the left end.

• The pipe is held at a temperature T=300*tanh(t) at the right end.

• The initial condition: 3000, xT

Coupling Variables

• The radiation heat source at a point x is given by the radiation emanating from all other points in the pipe. This will result in the convolution integral in the source term.

• We use an integration coupling variable and the built-in COMSOL Multiphysics function dest in order to evaluate this convolution integral.

• The source domain and the destination domain is the entire 1D geometry.

Results

• Temperature distribution at time = 100 s

Results

• Temperature distribution with (blue line) and without radiation (red line)

Comparison with full 3D model

• Results from a stationary 3D Heat Transfer model using surface-to-surface radiation on the inside surface of the tube.

Comparison with full 3D model

• Temperature distribution along the z-axis in the 3D (blue) model compared to the distribution obtained from the 1D (red) model.

Concluding Remarks• The heat radiation lets the pipe cool down faster at the hotter end

and elevates the temperature close to the cooler end.

• Results indicate that the effect of the radiation on the temperature distribution is substantial near the hot end of the pipe.

• Results from the 1D model display very good agreement with the full 3D model of heat transfer through radiation and conduction in the pipe, thus verifying the validity of the 1D approximation.

• Use of 1D models save computational memory and time.