Homework Project 9
Time Dependent Problems
Advanced Topics in Finite Elements
Summer 2015
John Connor
System Description
Time dependent problems are key to determining how a system reacts before reaching steady
state or in most cases over time because the system never reaches steady state. Time dependent
problems can be done for almost all types of analyses from mechanical loading to thermal fluids,
however a time dependent analysis can be very costly as far as computer resources. This is why
static analyses are the preferred type of analyses, but to account for time a static analysis cannot
be used.
In this project I will model two time dependent problems, the first is a Jominy end quench test
which is used to measure the hardenability of different materials and alloys. The second model
will evaluate the carbonization of a steel rod.
Governing Equations, Boundary Conditions and Input Data
For each of the analyses COMSOL multi-physics was used the Jominy test uses the heat transfer
in solids module while the carburization evaluation uses the coefficient PDE module to use a
theoretical approach to the analysis.
Jominy
The inputs and boundary conditions specific to the Jominy test are shown in Figure 1. The
inputs for the analysis are shown in Table 1. The design of the rod is based off a standard Jominy
test parameter and the 3 points placed into the rod will allow for the temperature to be measured
at those points over time.
Inward Heat Flux
HTC: 140
Temp: 293.15 [K]
Inward Heat Flux
HTC: 69.1
Temp: 293.15[K]
Axial Symmetry
Inward Heat Flux
HTC: 6661
Temp: 301.15[K]
Figure 1 – Jominy Test Overview
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Table 1 – Jominy Test Inputs
Name
Value [Unit]
Source
Radius
0.0125 [m]
Westmoreland
Height
0.1 [m]
Westmoreland
Point 1
0.002 [m]
Assumed
Point 2
0.012 [m]
Assumed
Point 3
0.022 [m]
Assumed
Initial Temperature
1293.15[K]
Assumed
Heat capacity at constant
475[J/(kg*K)]
COMSOL Defined
pressure
Thermal conductivity
44.5[W/(m*K)]
COMSOL Defined
Density
7850[kg/m^3]
COMSOL Defined
Carburization
The model created to evaluate the carburization of a steel bar is shown in Figure 2, the inputs for
the analysis are shown in Table 2.
Dirichlet Boundary
Condition: 1
Axial Symmetry
Figure 2 – Carburization Model Overview
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Table 2- Carburization Model Input Data
Name
Value [Unit]
Source
Radius
0.1 [m]
Assumed
Height
1 [m]
Assumed
Diffusion coefficient
{{5e-5, 0}, {0, 5eAssumed
5}}
Absorption coefficient
0
Assumed
Source term
0
Assumed
Mass coefficient
0
Assumed
Damping or mass coefficient
1
Assumed
Conservative flux convection
coefficient
{0, 0}
Assumed
Convection coefficient
{0, 0}
Assumed
Conservative flux source
{0, 0}
Assumed
The Galerkin finite element variation formulation used for the finite element analysis is:
∫ 𝑢̇ 𝑣𝑑𝑥 + ∫ 𝑢′ 𝑣 ′ 𝑑𝑥 = 0
Description of the Finite Element Model
The Jominy test and carbonization models were run several times with 2 mesh densities Figure 3
shows the “Normal” mesh density for the Jominy test (right) and the carburization model (left).
The Jominy test model ran for 600 seconds with one second incriminations creating 600 points
of values. The carburization model ran for 200 seconds with 20 second increments creating only
10 points of value.
Figure 3 – Mesh Densities of the 2 Models
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Results, Discussion and Conclusion
Figures 4 and 5 show the results of the Jominy test which clearly shows how the test specimen’s
temperature changes over time. The results depict that after about 400 seconds the test specimen
has essentially come down to the water temperature through the 4 points measured. At the end of
the analysis the entire test specimen only has approximately 16K difference in temperature
between the top and bottom of the specimen.
Figure 4 – Jominy Test Results Time=600 Surface: Temperature (K)
Figure 5 – Jominy Test Results Point Graph: Temperature (K)
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The carburization model results are shown in Figures 6 and 7 which show how the bar reacts
over time to the carburization process. It can be seen in Figure 7 that the carburization process
has already penetrated through the entire bar after just 200 seconds.
Figure 6 – Carburization Results at Time Zero
Figure 7 – Carburization Model at End of Analysis
References
1. Zienkiewicz, O. (2013). Finite Element Method: Its Basis and Fundamentals. Butterworth-Heinemann Ltd.
2. Westmoreland Mechanical Test http://www.wmtr.com/en.jominyend.html
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