Heat and Materials with Application
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MAT6669/MAT6679: Heat and Materials with Application
MATLAB modelling problem sheet
Please create separate MATLAB scripts for each sub-question where appropriate, using the following
naming convention: “Question1a.m”, “Question1b.m”, “Question1c.m”, etc. If you create your own
functions, include them within the submission. Note some scripts will be similar to previous scripts,
with minor modifications to answer the next question.
Include a single word document for presenting derivations and figures.
Zip the files together for the assignment submission. 10 marks are available for documentation and
presentation of figures generated.
The assignment questions are designed to increase in difficulty. You may wish to solve the first few
parts of the question and then revisit the last questions once you are experienced in solving heat
transfer and diffusion problems using finite difference, and coding in MATLAB.
Some questions are optional. Any marks awarded for these questions are bonus marks, however
your final mark is limited to a maximum of 100%.
Table 1: Question spice level
You must complete this question if you wish to pass this assignment. If you
succeeded in fixing the derivation and bugs in the provided code, you
should be able to adapt these codes to solve this problem.
This question is mildly difficult, requiring you to extend the derivation and
modify the fixed code to solve. You must succeed in these questions to
achieve merit within the assignments.
This question will challenge you and must be successfully answered to
achieve distinction.
MAT6669/MAT6679: Heat and Materials with Application
Magnus Anderson
2
Problem 1: Homogenization times
The chemical segregation of Rhenium in a cast and extruded Nickel superalloy has been measured
using wavelength dispersive spectroscopy (WDX). The diffusivity of Rhenium in the FCC austenite
phase has a diffusivity coefficient of 7.868× 10−6m2/s, and an activation energy of 287.3 kJ/mol. A
solid solution heat treatment is required to homogenize the dendritic segregation of Rhenium so
that the difference between the maximum and minimum concentrations are less than 0.15 at%.
a) How long is needed to homogenize the chemical segregation using the diffusion distance
formula for a 3D problem and a grain size of 12µm and a temperature of 1150°C?
(2 marks)
A more accurate calculation of the homogenization process is required. This is to be achieved by
modelling diffusion of the concentration field measured at the grain boundary. The concentration
field is approximated by the following Gaussian waveform:
(, = ) = 0 +
(4)1/2
(
−2
4
) 1
= (4)
1/2 2
Where 0 is 8%, is 5%, is the distance from the grain boundary and is 100s. The concentration
profile is presented in Figure 1, showing different options of how to model the concentration profile
at the grain boundary interface.
a)
b)
Figure 1: a) The initial concentration field of Nb considering plus or minus half a grain. b) The initial concentration gradient
accounting for symmetry in the concentration profile.
b) Write down a finite difference scheme for solving the diffusion equation in one dimension,
using Ficks’ first law of diffusion. Choose a representation of the concentration profile shown
in Figure 1 and specify suitability boundary conditions. (4 marks)
c) Implement the model and calculate the concentration field after 3h at 1150°C. Create a
figure that compares your calculation to the analytical solution. (5 marks)
d) How long is needed to reduce homogenize the segregation so that the difference between
the maximum and minimum concentration is less than 0.15%? (3 marks)
e) The diffusivity of Rhenium in FCC nickel has the following composition dependence.
MAT6669/MAT6679: Heat and Materials with Application
Magnus Anderson
3
= 0exp (
0
) 3
0 = 7.8682 × 10
−6 × (1 + 9.9286 × 10−4) 4
0 = 287776.2 × (1 − 0.1537924) 5
Where 0 is the activation energy (J/mol) and 0 is the diffusivity coefficient (m
2/s). is
the gas constant, and is the absolute temperature. Note the composition in Equations 4
and 5 is in atomic fraction, not atomic percent. Present the finite difference scheme that
includes a composition dependent diffusivity. (4 marks)
f) Quantify the impact of the composition dependent diffusivity upon predicting the required
homogenization time. (4 marks)
Problem 2: 1D steady state temperature profiles in a pipe
The partial differential equation describing heat transfer in the radial direction of a cylinder is given
in Equation 6.
−
1
(
) = 0 6
If the inner and outer temperature are fixed and the thermal conductivity does not change with
temperature, the analytical solution for steady state conditions where
→ 0 is given by
() =
1 ln (
2
) + 2 ln (
1
)
ln (
2
1
)
7
Where is the position along the radius of interest, 1 is the inner diameter and 2 is the outer
diameter. 1 and 2 are the temperatures at the inner and outer diameters of the cylinder. Please
refer to Dr Travis’ Lecture 5 for details about the derivation of this equation.
Model the 1D radial temperature profile within a stainless-steel pipe where the temperature at the
inner radius is 300°C and at the outer radius it is 180°C. Assume a density of 7850kg/m3, a thermal
conductivity of 17 W/m/K, and a specific heat of 0.444 kJ/kg/K. The inner radius is 12.5cm and the
pipe is 3cm thick.
a) Derive the finite difference scheme with and without temperature dependent
thermophysical properties. (7.5 marks)
b) Create a labelled figure which compares the analytical and numerical solutions to this
problem. (7.5 marks)
c) The thermophysical properties of stainless steel are now considered temperature
dependent. The temperature dependence of the thermal conductivity and specific heat are
given in equations 8 and 9, given in units of W/m/K and J/kg/K, respectively. Create a
labelled figure that compares the impact of the temperature dependent properties with the
analytical solution. (Hint: an iterative approach is needed to approximate the temperature
dependent thermal conductivity) (7.5 marks)
() = 14.52 + 0.128 (°) 8
MAT6669/MAT6679: Heat and Materials with Application
Magnus Anderson
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() = 414.7 + 0.155(°) 9
d) What heat fluxes are required on the inner and outer radii to maintain this temperature
profile with the original temperature independent thermal conductivity, and with the
temperature dependent thermal conductivity? (7.5 marks)
Problem 3: Quench simulation
A series of experiments have been performed to characterize the water quench process of cylindrical
pressure vessels. The quench tank has adjustable agitation. The experiments have measured the
cooling rate of the cylinder at different locations across the radial direction for different agitation
settings. The quench agitation can be set to a maximum setting of 10. The cylinders are placed
vertically above the agitation propellers, achieving faster flow rates of water through the internal
diameter of the workpiece in comparison to the outer surface. The component is heated to 900°C
prior to the quench.
The quenched components are made out of the steel A508-3. The cylindrical geometry has an
internal diameter of 2.5m and thickness of 0.225m. The thermocouple data can be downloaded
from Blackboard for three experiments at different agitation levels. The location of the
thermocouples is illustrated in Figure 2. The thermocouples are labeled “Ac1”, “Ac2”, and “Ac3”.
Figure 2: The geometry of the cylinder and location of the three thermocouples positioned close to the inner, middle, and
outer diameter of the component.
The thermophysical properties of A508-3 are given in Equations 10 to 12. The emissivity is
approximately 0.316. Note the temperature dependence of the thermal conductivity changes at
1066K and the temperature dependence is given in units of Kelvin.
() = 7939.8 + 0.32912 () 10
≥ 1066 () = 14.6 + 0. 01(K)
< 1066 () = 50.3 − 0.02(K)
11
MAT6669/MAT6679: Heat and Materials with Application
Magnus Anderson
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() = 472.2 + 0.25 () 12
a) Derive the finite difference schemes for a 1D heat transfer for the outer diameter, inner
diameter, and for conduction inside the component. The partial differential equation for
heat transfer through conduction considering cylindrical geometry is given in Equation 6.
Consider radiative and convective heat losses, in addition to temperature dependent
thermophysical properties. (6 marks)
b) Determine the heat transfer coefficients for the outer and inner diameter that best
describes the datasets. (8 marks)
c) The heat treatments need to be designed to obtain predominately bainitic microstructures
and reduce the amount of ferrite formed. To avoid excessive ferrite formation, a cooling rate
greater than 0.28K/s is needed within the temperature window of 710°C to 570°C. What
depth within the inner diameter meets this criteria for each condition? (7.5 marks)
d) Determine the relationship between the calibrated heat transfer coefficients and the
agitation of the quench tank. (4 marks)