Heat Transfer With Pulsatile Flow in a Tube
A Thesis Submitted in Partial Fulfilment of the Requirements for the Degree of Doctor of Philosophy
AI TAKEAWAY
Abstract
The partial differential equations governing heat transfer with pulsatile flow in a tube, which serves as a model of a simple heat exchange device are solved in this thesis. The governing equations are the Navier Stokes equations and the convection diffusion equation. Two boundary value problems are solved. In the first case a Neumann boundary condition is specified which represents constant heat flux at the wall of the tube with fluid entering a thermal region at a constant temperature. A regular perturbation expansion is used to obtain higher order harmonics downstream for the temperature field. The main assumption is that the temperature field becomes fully developed downstream as the velocity field becomes fully developed downstream. The perturbation parameter is the ratio of pressure gradient amplitudes of unsteady flow to that of steady flow. Using a Green’s function the first order solution is obtained. As a measure of heat transfer enhancement, a bulk temperature is formulated for the convective process involved and a change in unsteady Nusselt number to that of steady Nusselt number is evaluated analytically. In the second part of the thesis the more difficult problem of heat transfer in pulsatile flow with constant wall temperature is considered. Although a complete solution is not possible as in the first part, it is possible to use a combination of the Generalized Integral Transform Techniques and Laplace transforms to solve this problem upstream and downstream in the thermal region of the tube. The approximate solution indicates that a plane wave propogates down the tube and the phase of the wave defines critical values in frequency and time along the tube. Use of Dirac’s distribution makes it possible to define a bulk temperature and a change in unsteady bulk temperature from that of steady bulk temperature is presented. By means of classical analysis an inequality involving the two quantities is presented. As a result a measure of heat transfer in pulsatile flow compared to that of steady flow is presented.
About the Authors
Table of Contents
Interactive Book
Chapter 1: Introduction
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Chapter 2: Motivation
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Chapter 3: Previous Work
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Chapter 4: Present Work
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Chapter 5: Governing Equations
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Chapter 6: Basic Assumptions
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Chapter 7: Governing System of Equations
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Chapter 8: Preliminary Steady State Problems
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Chapter 9: Heat Transfer With Slug Flow in a Tube
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Chapter 10: Method of Solution
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Chapter 11: The Graetz Problem
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Chapter 12: Asymptotic Method to Solve Graetz Problem
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Chapter 13: Heat Transfer with Pulsatile Flow and Constant Heat Flux
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Chapter 14: Overview
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Chapter 15: Governing Equations
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Chapter 16: Regular Perturbation Method
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Chapter 17: Steady Temperature
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Chapter 18: Green's Functions
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Chapter 19: Oscillatory Temperature
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Chapter 20: Uniqueness of Solution
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Chapter 21: Riemann Surfaces
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Chapter 22: A Surface for log z
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Chapter 23: Verification of Boundary Conditions
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Chapter 24: Zero Frequency
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Chapter 25: Higher Order Perturbation Terms - Convergence Criteria
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Chapter 26: Results and Discussion
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Chapter 27: Conclusions
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Chapter 28: Heat Transfer with Pulsatile Flow and Constant Temperature
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Chapter 29: The Generalized Integral Transform Technique
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Chapter 30: Heat Transfer in Pulsatile Flow with Constant Wall Temperature
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Chapter 31: Distributions
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Chapter 32: Method of Solution
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Chapter 33: Results and Discussion
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Chapter 34: Concluding Remarks
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Chapter 35: Appendix A: Maple Code for the Graetz Problem
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Chapter 36: Appendix B: Maple Code for Constant Heat Flux Problem
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Chapter 37: Appendix C: Axial Gradient of Temperature Downstream
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Chapter 38: Appendix D: Picard-Lindelöf Theorem
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Chapter 39: Proof
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Chapter 40: Appendix E: Properties of the Dirac Delta Function
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Academic Disclosures
Ethical Approval
Not applicable for this article.
Conflict of Interest
The authors declare no conflict of interest.
Data Availability
Not applicable for this article.
Acknowledgement
I would like to express my thanks and appreciation to my supervisor Dr. M. Zamir for his guidance and direction over the past few years. I also appreciate helpful discussions with Dr. Stan Deakin. I also would like to thank National Sciences and Engineering Research Council of Canada for their financial support.
Dedication
To My Father and Mother
AI Use Disclosure
No generative AI was used for analysis or results.
Trial Registration
Null — Not applicable for mathematical/engineering research
Funding Organization
NSERC Canada mentioned explicitly in acknowledgements