Investigation of Transient Heating/Cooling of an Everyday Object via Conduction Analysis

MIET1081 Heat Transfer

Sem 1, 2025

1 Introduction and Objectives

Heat transfer is a critical phenomenon in engineering and everyday life.  Objects around us constantly experience heating or cooling due to interactions with their environment. Understanding how these objects respond to thermal changes is essential for designing efficient systems, predicting material behavior, and optimizing energy usage.   In this assignment, you will analyze the transient thermal behavior of an object using three different methods:

1. Lumped System Analysis: A simplified approach that assumes uniform tempera– ture distribution within the object.  Solve for the temperature as a function of time of the object regardless of the Biot number.

2. One-Term Approximation Solution: An analytical method applicable to standard geometries (plane walls, cylinders, and spheres) that accounts for spatial temperature variations.   Approximate the  object’s geometry as either a plane wall, cylinder, or sphere.  Use the one–term approximation solution to analyze the temperature distribution within the object.

3. Computational Fluid Dynamics (CFD): A numerical method that provides a de– tailed temperature distribution by solving the governing equations of heat transfer. Perform a CFD simulation to model the transient heating or cooling of the object, and of the simplified geometry.  Compare the results with those obtained from the lumped system analysis and the one–term approximation solution.

By comparing the results from these methods, you will evaluate their accuracy, limitations, and applicability to real–world problems.

2 Assignment Overview

2.1    Object Selection

1. Select an object from your daily life that experiences heating or cooling. Examples include cooling of a hot drink, heating of cold drink from the fridge, warming of a chilled fruit, a metal spoon left in boiling pasta pot.  Be thoughtful and select something that interests you. Investigate the transient thermal behavior. of the object using the following methods:

2. Describe the object, its geometry, material properties, and the heating/cooling conditions. Identify relevant properties: approximate shape, dimensions, thermal conductivity (k), specific heat (cp), density (p).

3. Define the problem: State the problem clearly: What are you trying to determine? (e.g., How long does it take for the object to cool from 90。C to 40。C?), what was the ambient temperature T∞ and an estimated or measured convective heat transfer coefficient h (if possible).

4. Explain simplifying assumptions:  constant properties, negligible radiation, uni– form. external temperature, etc.

2.2 Observational or Experimental Setup

If you have a thermometer, and ruler, you can define the initial and final conditions.  If not, you can use estimated values for the following:

– Initial temperature: Measure or estimate the initial temperature of the object, Ti.

– Dimension measurements: Obtain length, diameter, thickness, or radius as needed.

– Potential real measurements:  If feasible, use a thermometer or thermocouple to record the object’s temperature over time.

3 Lumped Capacitance Analysis

– Compute the Biot number:

where Lc  is the characteristic length (e.g., half–thickness for a plane wall, radius for a sphere/cylinder).

– Determine the Bi < 1, number and discuss its relevance for transient analysis.

– Derive the governing equation for transient heat transfer and solve for the tempera– ture as a function of time.

– Plot the temperature vs. time curve and discuss the result

4 One-Term Analytical Solution

4.1 Selection of Standard Geometry

1. Choose the closest matching geometry for your real object: plane wall, long cylinder, or sphere.

2. Use the one–term approximation solution to solve for the temperature distribution within the object.

3. Define the characteristic length LC accordingly.

4. Use tables or charts from standard heat transfer references to find the first eigenvalue λ1 and constant A1 corresponding to your geometry and Bi.

Some useful equations:

– Fourier Number:

– Biot Number:

5 CFD Simulation

5.1    Overview

In this part, you will use a CFD tool (e.g., ANSYS Fluent, to numerically solve the transient conduction problem. You will perform. two simulations:

1. Idealized Geometry: Plane wall, long cylinder, or sphere (matching the geometry used in Method 2).

2. Actual Object Geometry: A 3D model of your chosen everyday object (e.g., a mug, fruit, metal part, etc.).

5.2    Model Setup for Idealized Geometry

– Create a 2D or 3D model of the standard geometry (plane, cylinder, or sphere).

– Define the same material properties (p, Cp, k) used in the analytical solutions.

– Apply a convective boundary condition with h and T∞ on the relevant surfaces.

– Use transient solver settings.  Choose a suitable time step (e.g., Δt) to capture the temperature evolution properly.

5.3    Model Setup for the Actual Object

– Create a 3D CAD model of your real object.   Approximate complex features if necessary, but aim to capture the major dimensions.

– Assign the same material properties as used previously.

– Apply boundary conditions that mimic the real–life scenario (e.g., natural or forced convection over the outer surface).

– Use the same transient setup to compare with the idealized geometry case.

5.4    Post-Processing and Validation

– Obtain temperature contours or temperature–time histories at key points (center, surface) for later comparisons of :

1. Idealized geometry CFD results with the one–term analytical solution.

2. Real–object geometry CFD results with the simpler models (lumped and one– term) to identify deviations.

6 Comparison and Discussion

1. Consolidate Results: Present temperature vs. time plots from the lumped approach, one–term analytical solution, and both CFD models.

2. Analyze Discrepancies:

– Address possible reasons for differences (Biot number magnitude, shape ap– proximations, boundary condition assumptions, numerical discretization).

– Comment on whether the real geometry exhibits more complex temperature gradients.

3. Draw Conclusions:

– State which method(s) is most accurate and under what conditions.

– Highlight major insights about transient conduction in your chosen object.

7 Final Report Guidelines

Your final deliverable should include:

– Introduction/Methodology: Description of the object, the problem setup, and ratio– nale for choosing the geometry in the analytical approach.

– Theoretical Analyses:

– Lumped Capacitance derivation and result.

– One–term approximation for the chosen standard geometry.

– CFD Simulation:

– Description of the simplified geometry setup (plane wall, cylinder, sphere).

– Description of the actual object geometry setup.

– Boundary conditions, mesh, solver settings, and convergence checks.

– Simulation results (plots, temperature contours, etc.).

– Comparison and Discussion:

– Side–by–side comparisons of Lumped, One–term Analytical, and both CFD re– sults.

– Discussion of any discrepancies or observations.

– Conclusion and Future Work:

– Summary of key findings.

– Suggestions for improving the model (e.g. what are your thoughts on radiation effects, h estimation, advanced material models).

– References:

– Cite textbooks, journal papers, and software manuals as needed.

– Appendix of student work:

– All students must complete the lumped system analysis, and one–term approx– imation calculations. This part of the work is to help compare and discuss ideas among the group. Please scan and submit each student’s calculations.

– Group meeting detailing minutes and actions (minimum of three meetings)

8 Grading Rubric

9 Detailed Rubric Explanations

Object Selection & Setup (10%)

– Technical accuracy:  Object choice is highly appropriate and clearly relevant to transient conduction.

– Completeness: The setup description is comprehensive, including precise geometry, all key dimensions, material properties (p, cp, k) with values, initial and ambient temperatures, and a well–estimated or measured convective heat transfer coefficient

(h).   The problem is clearly defined (e.g. exact cooling/heating objective stated) and all simplifying assumptions (constant properties, negligible radiation, etc.) are explicitly listed and justified.

– Depth:  Demonstrates thorough understanding by rationalizing choices (why the object and conditions were chosen) and considering broader context.

– Presentation: Information is very well–organized and clearly written, making this section easy to follow with no errors or omissions.

Lumped Analysis (20%):

– Technical accuracy: Performs a correct lumped capacitance analysis with no errors. Derives the governing transient heat balance equation and its solution accurately, applies it correctly to the object, and calculates the Biot number, discussing its relevance.

– Completeness: All steps are shown: from formula derivation to solving for temper– ature over time, including proper units and constants. A clear temperature–vs.–time plot is presented, correctly labeled, and discussed.

– Depth: Goes beyond the basics by evaluating the validity of the lumped assumption (e.g. when the Biot number is small enough) and interpreting results (time constant, approach to ambient temperature).

– Presentation:  Very clearly presented with neat equations, well–defined variables, and a logical explanation of the method.

Analytical (One-Term) Method (20%):

– Technical accuracy: Successfully applies the one–term analytical solution, choosing the correct standard geometry (plane wall, long cylinder, or sphere).   Correctly identifies characteristic length (Lc), Biot number, first eigenvalue (λ1), and coefficient (A1). Uses them properly to compute the transient temperature distribution.

– Completeness:  Every step is included:  dimensionless parameters (Fourier, Biot), retrieving or computing λ1  and A1  from references, and presenting a temperature– vs.–time plot or center–temperature curve.

– Depth: Explains the implications of the one–term approximation, why higher terms are neglected, and how the geometry choice affects accuracy.

– Presentation:  Extremely clear, well–organized derivation and application of the solution, with each variable and assumption fully documented.

CFD Simulation (25%):

– Technical accuracy: Executes a comprehensive transient CFD simulation (for both idealized and actual geometry, if applicable) with appropriate mesh, boundary con– ditions (convective cooling, correct h and T∞ ), solver settings, and time–stepping.

– Completeness:  Describes the model setup thoroughly (geometry, mesh, material properties, solver parameters, convergence criteria) and presents clear results (tem– perature contours, temperature–vs.–time plots at key points).

– Depth:  Analyzes the simulation outcomes in detail, comparing idealized vs. real geometry, and discussing any numerical issues. Explains differences from theoretical models (lumped and one–term) with strong reasoning.

– Presentation: Well–structured and easy to follow, with labeled figures and tables. All relevant parameters and choices are justified clearly.

Comparison & Discussion (15%):

– Technical accuracy:   Provides an insightful and accurate comparison of all ap– proaches (Lumped, One–Term, and CFD). Correctly interprets and quantifies simi– larities or differences.

– Completeness:  Discusses each pair of methods (Lumped vs. Analytical, Lumped vs. CFD, Analytical vs. CFD) and, if done, compares idealized vs. actual geometry. References numerical data or plots to illustrate findings clearly.

– Depth:  Explores the reasons behind any discrepancies (e.g. assumptions, geome– try approximations, numerical factors), commenting on accuracy, limitations, and conditions in which each method is valid.

– Presentation:  Organized into a cohesive narrative showing how results align or differ.  Uses tables/figures effectively to support the arguments and draws logical conclusions from the data.

Report Quality and Appendix of Student Work (10%):

– Organization & clarity:  Exceptionally well–organized, with sections flowing logi–

cally (Introduction, Methods, Results, Discussion, Conclusion). Headings/subheadings are effectively used.  Clear, concise, and professional.  Few or no typographical or

grammatical errors.  Language is appropriate, and technical terms are used accu– rately.  Figures/tables are neatly labeled, referenced in text, and clearly annotated. Any references are properly cited in a consistent style. The layout adheres to guide– lines and appears polished.

– Student Contributions and Meetings:  All student work has been submitted to demonstrate learning of the lumped system analysis, and one–term approximation calculations.   Group meeting  detailing minutes and actions (minimum of three meetings)