MEE405 — Heat and Mass Transfer II: Lecture Note

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Course code: MEE 405 Course title: Heat and Mass Transfer II Course credit load: 3 Units Series 1: Convection Fundamentals and Governing Equations • Part 1.1: Introduction to Convection Heat Transfer o Sub Part 1.1.1: Concept of Convection o Sub Part 1.1.2: Newton's Law of Cooling o Sub Part 1.1.3: The Convective Heat Transfer Coefficient • Part 1.2: The Velocity and Thermal Boundary Layers o Sub Part 1.2.1: The Velocity Boundary Layer in Convection o Sub Part 1.2.2: The Thermal Boundary Layer o Sub Part 1.2.3: Relationship Between Velocity and Thermal Boundary Layers • Part 1.3: The Energy Equation of Convection o Sub Part 1.3.1: Derivation of the Energy Equation o Sub Part 1.3.2: Terms of the Convection Energy Equation o Sub Part 1.3.3: Simplifications of the Energy Equation • Part 1.4: Continuity and Momentum Equations in Convection Analysis o Sub Part 1.4.1: Role of the Continuity Equation in Convection Analysis o Sub Part 1.4.2: Role of the Momentum Equation in Convection Analysis o Sub Part 1.4.3: Coupling Between Momentum and Energy Equations • Part 1.5: Solving the Governing Equations of Convection o Sub Part 1.5.1: Boundary Conditions for Convection Problems o Sub Part 1.5.2: Analytical Solution Approaches o Sub Part 1.5.3: Numerical Solution Approaches

Introduction Heat transfer by convection, the transfer of heat between a solid surface and an adjacent moving fluid, was introduced only briefly earlier in your studies, and this opening Series of the course now examines convection in genuine depth, beginning with Newton's law of cooling and the convective heat transfer coefficient, before turning to the velocity and thermal boundary layers that govern convective behaviour, and finally to the energy, continuity, and momentum equations that together provide the complete mathematical foundation from which convection heat transfer analysis proceeds. The aim of this Series is to equip you with a thorough understanding of convection fundamentals, including Newton's law of cooling, the velocity and thermal boundary layers, and the energy, continuity, and momentum equations and their combined role in convection heat transfer analysis. We shall commence this Series with an introduction to convection heat transfer, before turning to the velocity and thermal boundary layers. We will then examine the energy equation of convection, followed by the continuity and momentum equations in convection analysis, and conclude this Series by considering how to solve the governing equations of convection. Series Learning Outcomes Upon completing this Series, you should be able to: • SLO 1.1: explain the concept of convection heat transfer, • SLO 1.2: apply Newton's law of cooling, • SLO 1.3: define the velocity and thermal boundary layers, • SLO 1.4: explain the relationship between the velocity and thermal boundary layers, • SLO 1.5: derive the energy equation of convection, • SLO 1.6: explain the terms of the convection energy equation, • SLO 1.7: explain the role of the continuity equation in convection analysis, • SLO 1.8: explain the role of the momentum equation in convection analysis, • SLO 1.9: describe the boundary conditions required for convection problems, • SLO 1.10: distinguish analytical and numerical solution approaches for convection problems,

1.1 Introduction to Convection Heat Transfer 1.1.1 Concept of Convection Convection is the transfer of heat between a solid surface and an adjacent moving fluid, combining conduction occurring at the immediate fluid-solid interface, where the fluid is momentarily at rest relative to the surface, with the bulk motion of the fluid itself, which carries this heat away from or toward the surface far more effectively than conduction through a stationary fluid alone ever could. Convection is classified as natural, or free, convection when the fluid motion arises purely from buoyancy differences created by the temperature variation itself, or as forced convection when that fluid motion is instead driven externally by a fan, pump, or similarly imposed flow, and this distinction matters considerably throughout this Series, since the two convection types are analysed using related but genuinely distinct approaches. Worked example: A radiator warming a room primarily through natural convection, as buoyancy drives warm air upward and away from its surface, differs fundamentally from the same radiator being cooled by a fan blowing air across it, which instead relies on forced convection. Application: Correctly identifying whether a given convective heat transfer situation is natural or forced convection is the essential first step before selecting the appropriate analytical approach for that situation. Fill in the blank: Convection combines conduction at the fluid-solid interface with the ________ motion of the fluid itself. 1.1.2 Newton's Law of Cooling Newton's law of cooling expresses the rate of convective heat transfer between a solid surface and an adjacent fluid as directly proportional to the surface area involved and to the temperature difference between the surface and the fluid far from that surface, with the constant of proportionality, the convective heat transfer coefficient, absorbing all of the considerable underlying complexity of the actual convective process, fluid properties, flow velocity, surface geometry, and flow regime, into this single, experimentally or analytically determined value. Because Newton's law of cooling is, strictly speaking, a definition of the convective heat transfer coefficient rather than a fundamental physical law derived from first principles, the genuine engineering and scientific challenge in convection heat transfer lies almost entirely in determining an accurate value for this coefficient for a given specific flow situation, a determination that occupies the great majority of the material examined throughout the remainder of this Series and the next.

Worked example: Calculating the convective heat loss from a heated pipe surface exposed to flowing air applies Newton's law of cooling directly, once an appropriate convective heat transfer coefficient for that specific pipe and flow condition has been determined. Application: Recognising that Newton's law of cooling is fundamentally a definition, with essentially all the analytical difficulty concentrated in determining the coefficient itself, correctly frames the genuine scope of convection heat transfer analysis. Fill in the blank: Newton's law of cooling expresses convective heat transfer rate as proportional to surface area and to ________ difference. 1.1.3 The Convective Heat Transfer Coefficient The convective heat transfer coefficient depends on a considerable number of factors specific to the flow situation being analysed, including the fluid's own thermal and physical properties, thermal conductivity, viscosity, density, and specific heat, the flow's velocity and regime, laminar or turbulent, and the surface's own geometry and orientation, meaning this coefficient is never a fixed, universal property of a fluid alone but always a property of the complete flow situation under consideration. Determining a reliable value for the convective heat transfer coefficient for a specific situation is achieved through one of several complementary approaches, direct experimental measurement, analytical solution of the underlying governing equations for sufficiently simple geometries, or the empirical correlations examined in the next Series, correlations themselves ultimately derived from careful experimental measurement across a systematically varied range of flow conditions. Worked example: The convective heat transfer coefficient for air flowing slowly across a flat plate differs considerably from the coefficient for the same air flowing rapidly across that same plate, confirming that the coefficient depends on flow condition rather than being a fixed fluid property. Application: Recognising that the convective heat transfer coefficient is a property of the entire flow situation, not of the fluid alone, is essential before attempting to look up or calculate an appropriate value for any specific convection problem. Fill in the blank: The convective heat transfer coefficient is a property of the complete ________ situation, not of the fluid alone.

Figure 1.1: Convective heat transfer from a solid surface to a moving fluid. Pilot questions 1.1 1. Define convection heat transfer. (Linked to SLO 1.1) 2. Distinguish natural convection from forced convection. (Linked to SLO 1.1) 3. State Newton's law of cooling. (Linked to SLO 1.2) 4. Explain why Newton's law of cooling is considered a definition rather than a fundamental law. (Linked to SLO 1.2) 5. List the factors that influence the convective heat transfer coefficient. (Linked to SLO 1.2) 1.2 The Velocity and Thermal Boundary Layers 1.2.1 The Velocity Boundary Layer in Convection The velocity boundary layer is the thin region of fluid adjacent to a solid surface within which velocity changes rapidly from zero at the surface, satisfying the no-slip condition, up to very nearly the free-stream velocity found further from the surface, exactly the same boundary layer concept already examined in earlier fluid mechanics studies, now revisited specifically for its central role in governing convective heat transfer.

The velocity boundary layer's thickness and character, laminar or turbulent, directly influence how effectively heat can be transferred between the surface and the fluid, since a thinner boundary layer generally permits a steeper temperature gradient near the surface, and therefore a higher convective heat transfer coefficient, than a thicker boundary layer of otherwise similar flow condition would allow. Worked example: A surface exposed to a high-velocity flow generally develops a thinner velocity boundary layer than the same surface exposed to a low-velocity flow, correspondingly producing a higher convective heat transfer coefficient at the higher velocity. Application: Recognising that a thinner velocity

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