ENG581: Applied Heat and Mass Transfer

Prac 1: Transient Conduction

Theory

Transient conduction prediction via the finite difference method and semi-infinite solid method.

Setup for the simulation:

• 3 different metal strips (Aluminium, Gold and Copper) of 10 cm in length, of unit area cross section with one end immersed in water at 98°C, the other end and sides insulated.
• 4 thermocouples placed along each metal strip 25mm apart.
• 1 thermocouple to measure water temperature.
• 1 data logger to record the temperature at the thermocouple.

Assume the conductivity values for the metals.

Students are encouraged to refer to Sections 5.7 to 5.10 in the prescribed textbook for a comprehensive understanding of the concepts needed to address this prac. A thorough analysis is expected, and students should also consult additional sources and include appropriate references in their assignment.

Tasks:

Predict the transient heat conduction of the three metals and predict the temperature at
the 4 thermocouple points for the first 200 seconds.

Using Excel/Matlab, implement one of the following equations:

a. Use initial conditions of 26°C and dt = 1sec or dx = 0.1cm
Assume suitable boundary conditions as needed.

Finite difference method:

1. Create a plot of the temperature at the 4 thermocouple locations for the first 200 sec when the
hot water is brought in contact with the setup.
2. Discuss the relevant stability criterion and how your choice of delta t and delta x fits with the
relevant stability criterion
3. Using your spreadsheet, demonstrate what happens if the stability criterion is not followed.
4. What other limitations are there for the explicit method? Do some research and demonstrate
these limitations using your spreadsheet results.

Challenge to the finite difference method:

• Consider another method to find the same results:
  • MATLAB's PDE solver (PDEPE), OR
  • attempt to implement an implicit method. Please provide the relevant references.
• Compare these results to the results for the first method.

Semi-infinite solid method:

1. Implement the semi-infinite solid formulae for the experimental conditions
2. Create a plot of the temperature at the 4 thermocouple locations for the first 200 sec when the
hot water is brought in contact with the setup.
3. Discuss how you evaluated the ERF function and the validity of that choice.

Method comparison:

1. Compare the results obtained from each of the methods you used (at least the finite-difference
method and the semi-infinite solid method)
2. Which do you think is the more accurate?
3. What factors will affect the difference between these two methods?
4. Do you have any control of these factors?

Data analysis

Plot the sample observation experimental results and the theoretical results on one plot. Discuss
potential reasons for any differences between the experimental results and the theoretical results
(consider validity of boundary conditions, initial conditions, material properties).

Please present a professional analysis i.e. error, variance, etc. Use this information in your discussionand analysis. Also, discuss how the thermal diffusivity of Aluminum, Gold, and Copper affects the temperature distribution along the strip. Provide a comparison and discuss the significance of thermal diffusivity in transient heat conduction.

Transient Heat Conduction Practical Assessment Summary

Purpose:

• Predict and analyse transient heat conduction along metal strips using numerical and analytical methods.
• Develop skills in finite difference methods (explicit/implicit), semi-infinite solid analysis, and computational simulations (Excel/MATLAB).
• Compare theoretical predictions with experimental observations and interpret thermal behaviour.

Key Assessment Requirements:

1. Experimental Setup & Theory:

• Three metal strips: Aluminium, Gold, Copper (10 cm, unit cross-section)
• One end immersed in water at 98°C; other end insulated
• Four thermocouples along each strip (25 mm apart)
• One thermocouple measures water temperature; data logged over 200 seconds
• Thermal conductivity values assumed from references

2. Finite Difference Method (FDM – Explicit):

• Predict temperature at the four thermocouples using initial conditions of 26°C, dt = 1 s, dx = 0.1 cm
• Plot temperature vs time for 200 seconds
• Discuss stability criterion and effects of violating it
• Identify limitations of the explicit FDM

3. Alternative Method:

• Implement PDE solver (MATLAB PDEPE) or an implicit FDM method
• Compare results with explicit FDM

4. Semi-Infinite Solid Method:

• Apply analytical formula for transient conduction
• Plot temperature vs time
• Discuss evaluation of the ERF function and validity of approach

5. Method Comparison & Data Analysis:

• Compare results from FDM and semi-infinite methods
• Discuss which method is more accurate and factors influencing differences
• Plot experimental vs theoretical data and analyse deviations
• Discuss error, variance, and influence of thermal diffusivity on temperature distribution
• Compare Aluminium, Gold, and Copper and explain significance of thermal diffusivity

Step-by-Step Mentor Guidance

1. Understanding the Task:
   • Mentor explained transient conduction theory, thermal diffusivity, and relevance of the experiment.

   • Clarified objectives of comparing multiple methods and interpreting results.

2. Finite Difference Method Implementation:
   • Guided student in setting up Excel/MATLAB simulations with specified initial and boundary conditions.
   • Explained how to calculate stability criteria and evaluate delta t and delta x values.
   • Reviewed plotting techniques and interpretation of time-dependent temperature profiles.
3. Exploring Limitations & Alternative Methods:
   • Demonstrated how violating stability criteria affects results.
   • Introduced MATLAB PDEPE and implicit FDM methods as alternatives.

   • Encouraged referencing sources for method implementation and limitations.

4. Semi-Infinite Solid Analysis:
   • Guided student in applying semi-infinite solid equations and using the error function (ERF)

   • Discussed assumptions, validity, and practical implications of analytical method.

5. Method Comparison and Data Analysis:
   • Mentor instructed on overlaying theoretical and experimental results in plots.
   • Discussed factors affecting deviations (boundary conditions, thermal properties, measurement errors).

   • Encouraged discussion of thermal diffusivity differences among metals and their significance in transient conduction.

6. Professional Reporting:
   • Ensured student included error analysis, variance, clear plots, and detailed discussion.
   • Checked references and adherence to professional scientific writing standards.

Outcome Achieved

• Successfully predicted temperature distributions for Aluminium, Gold, and Copper using multiple methods.
• Plotted and compared explicit FDM, alternative methods, semi-infinite solutions, and experimental data.
• Demonstrated understanding of stability criteria, thermal diffusivity, and limitations of numerical methods.
• Produced a professional report with clear plots, error analysis, variance, and thorough discussion.

Learning Objectives Covered

1. Transient Conduction Analysis: Applied finite difference and semi-infinite solid methods to predict heat transfer.

2. Computational Simulation Skills: Used Excel/MATLAB for modelling transient conduction and data visualisation.

3. Method Comparison & Critical Evaluation: Compared numerical and analytical methods, evaluating accuracy and limitations.

4. Data Interpretation & Reporting: Analysed experimental vs theoretical results, discussed errors, and prepared professional documentation.

5. Understanding Material Properties: Evaluated effects of thermal diffusivity of Aluminium, Gold, and Copper on transient heat conduction.

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