COMSOL 6.4 - An Integro-Partial Differential Equation

An Integro-Partial Differential Equation¹

This example contains an analysis of conductive and radiative heat transfer in a hollow pipe, where the ends are held at two different temperatures. To solve this integro-partial differential equation, the model makes use of the destination operator and a nonlocal integration coupling.

Model Definition

This example investigates how to solve the integro-partial differential equation

(1)

where L is the pipe length, Di and Do are respectively the inner and outer diameters of the pipe, ρ is the density, Cp is the heat capacity, κ is the thermal conductivity, σ is Stefan’s constant (the Stefan-Boltzmann constant), ε is the emissivity, and k(x, x') is the kernel corresponding to the radiation view factor. This equation arises in the physical description of 1D heat conduction and radiation along a pipe. Figure 1 shows the model geometry.

Before setting up the model, make the following assumptions:

•

Inside the tube, neglect convection and consider only radiation and conduction.

•

Assume blackbody radiation with ε = 1.

•

Model heat transfer only in the x direction (assume θ symmetry).

•

The pipe’s outer wall is perfectly insulated so that no heat escapes to the outside world by either radiation or conduction.

The definition of the kernel k(x, x') is

where ξ = |x − x'|/ Di as explained in Ref. 1.

Also consider the following boundary conditions and initial condition:

Figure 1: Model geometry.

Results and Discussion

The temperature distribution along the length of the pipe at t = 3600 s appears in Figure 2. The straight line is the solution for the radiation-free model obtained by setting the emissivity to zero:

Figure 2: Temperature distribution along the pipe at t = 3600 s with radiation (ε = 1) and without radiation (ε = 0).

Comparison with the Full 3D Radiation Model

To illustrate the validity of the 1D model, you can set up the entire stationary 3D model using the Heat Transfer Module. Its Heat Transfer interface handles surface-to-surface radiation boundary conditions, making it easy to verify the results. Figure 3 shows the temperature on the 3D cylinder’s surface, while Figure 4 compares the temperature distributions along the axial direction for the 1D and 3D models. Clearly the results are in excellent agreement.

Figure 3: 3D temperature distribution in the pipe.

Figure 4: The temperature distributions for the 1D model and the 3D model.

Notes About the COMSOL Implementation

To model the equation, use the Heat Transfer interface and include the radiation effects in the source term, Q, using a nonlocal integration coupling.

To enter convolution integrals of the type needed here, use the dest operator, which instructs COMSOL Multiphysics to evaluate the expression on which it operates on the destination points instead of the source points. In the expression k(x, x'), x' is the variable to integrate over, whereas the model does not integrate over x. To specify that x should remain a coordinate variable that can take on values from the entire domain, write it as dest(x) inside the nonlocal integration coupling.

Reference

1. R. Siegel and J. Howell, Thermal Radiation Heat Transfer, 4th ed., Taylor & Francis Group, New York, 2001.

COMSOL_Multiphysics/Equation_Based/integro_partial

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

In the tree, select > .

Click .

Click

In the tree, select > .

Click

Global Definitions

Parameters 1

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
epsilon	1	1	Emissivity
T_cold	300[K]	300 K	Temperature, cold end
DT_max	1200[K]	1200 K	Maximum temperature difference
T_init	T_cold	300 K	Initial temperature
D_i	1[in]	0.0254 m	Inner diameter
D_o	1.1*D_i	0.02794 m	Outer diameter
L	0.2[m]	0.2 m	Length

Geometry 1

Interval 1 (i1)

In the window, under right-click and choose .

In the window for , locate the section.

In the table, enter the following settings:


Coordinates (m)
0
L

Click

Definitions

Define variables for the radiation terms on the left-hand side of Equation 1. For this purpose, you need a nonlocal integration coupling.

Integration 1 (intop1)

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Variables 1

In the toolbar, click

In the window for , locate the section.

From the list, choose .

Locate the section. In the table, enter the following settings:


Name	Expression	Unit	Description
xi	abs(dest(x)-x)/D_i
k	1-(2xi^3+3xi)/(2*(xi^2+1)^1.5)		Integral kernel
Q_source	4/(D_o^2-D_i^2)epsilonsigma_constintop1(kT^4)	W/m³	Heat source
Q_loss	-4D_i/(D_o^2-D_i^2)epsilonsigma_constT^4	W/m³	Heat loss

Materials

Material 1 (mat1)

In the window, under right-click and choose .

By default, the first material applies for all domains. COMSOL Multiphysics indicates any undefined material parameters required by the physics interfaces defined on those domains.

In the window for , locate the section.

In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Thermal conductivity	k_iso ; kii = k_iso, kij = 0	13	W/(m·K)	Basic
Density	rho	8700	kg/m³	Basic
Heat capacity at constant pressure	Cp	300	J/(kg·K)	Basic