External Forced Convection

4.1 Introduction to Laminar Boundary Layers 4.1.1 Introduction Chapters 1 through 3 consider conduction heat transfer in a stationary medium. Energy transport within the material of interest occurs entirely by conduction and is governed by Fourier’s law. Convection is considered only as a boundary condition for the relatively simple ordinary or partial differential equations that govern conduction problems. Convection is the transfer of energy in a moving medium, most often a liquid or gas ﬂowing through a duct or over an object. The transfer of energy in a ﬂowing ﬂuid is not only due to conduction (i.e., the interactions between micro-scale energy carriers) but also due to the enthalpy carried by the macro-scale ﬂow. Enthalpy is the sum of the internal energy of the ﬂuid and the product of its pressure and volume. The pressure-volume product is related to the work required to move the ﬂuid across a boundary. You were likely introduced to this term in a thermodynamics course in the context of an energy balance on a system that includes ﬂow across its boundary. The additional terms in the energy balance related to the ﬂuid ﬂow complicate convection problems substantially and link the heat transfer problem with an underlying ﬂuid dynamics problem. The complete solution to many convection problems therefore requires sophisticated computational ﬂuid dynamic (CFD) tools that are beyond the scope of this book. The presentation of convection heat transfer that is provided in this book looks at convection processes at a conceptual level in order to build insight. In addition, the capabilities and tools that are required to solve typical convection heat transfer problems are presented. As engineers, we are most often interested in the interaction between a ﬂuid and a surface; speciﬁcally the transport of momentum and energy between the surface and the ﬂuid. The transport of momentum is related to the force exerted on the surface and it is usually represented in terms of a drag force or a shear stress. The transport of energy is expressed in terms of the heat transfer coefﬁcient. These are the engineering quantities of interest and they are governed by the behavior of boundary layers, the thin layer of ﬂuid that is adjacent to the surface and they are affected by its presence. We will attempt to obtain physical intuition regarding the behavior of boundary layers and understand how the transport of momentum and energy are related. We will explore the equations that govern these transport processes and see how they can be simpliﬁed and non-dimensionalized. We will look at exact solutions to these simpliﬁed equations, where they exist, and develop some tools that provide approximate solutions. Most importantly, we will examine the correlations that are enabled by the non-dimensionalized equations and understand their proper use and the limits of their applicability. The convection heat transfer correlations included in this book are also built into EES, which simpliﬁes their application. However, it is important that any

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External Forced Convection

solution be checked against physical intuition and understood at a deeper level than just “this is what the correlation predicts.”

4.1.2 The Laminar Boundary Layer

Figure 4-1(a) and (b) illustrate, qualitatively, the laminar ﬂow of a cold ﬂuid over a heated plate that is at a uniform temperature (Ts ). The ﬂow approaching the plate (i.e., at x < 0) has a uniform velocity (u∞ ) in the x-direction, no velocity in the y-direction (v∞ = 0), and a uniform temperature (T∞ ). The quantities u∞ and T∞ are referred to as the free-stream velocity and temperature, respectively. The difference between laminar and turbulent ﬂow will be discussed in more detail in Section 4.5. For now, it is sufﬁcient to understand that the laminar ﬂow is steady, provided that the free-stream velocity and temperature do not change with time. An instrument placed in...