Application of Dimensional Analysis to Convection

The heat transfer coefficient, h, is the variable whose value is being sought. Consider a series of bodies of the same geometrical shape, e.g., a series of elliptical cylinders (see Fig. 1.13), but of different size placed in various fluids. It can be deduced, either by physical argument or by considering available experimental results, that if, in the case of gas flows, the velocity is low enough for compressibility effects to be ignored, h will depend on  

Series of bodies with same geometrical shape. 

To derive an expression for a measure of the magnitude of the buoyancy force, consider an elemental volume of the fluid as shown in Fig. 1.15. First, consider the forces acting on this control volume when the fluid is unheated and at rest. 

Since the fluid is at rest, the hydrostatic pressure forces must just balance the weight of the fluid. Hence, if dA is the horizontal cross-sectional area of the control volume and if po is the density of the fluid in this unheated state, then the force balance requires that  

Now consider the same control volume when the surface is at a different temperature from the fluid far away from it and the fluid is in motion. The fluid near the surface will also be at a different temperature from the fluid far away from the surface and its density will as a result also be different. Let the density of the fluid in the control volume in this case be p. The situation is shown in Fig. 1.16. 

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The continuity equation (8.9) and the energy equation (8.12) are identical to those for forced convective flow. The x- and y-momentum equations, i.e., Eqs. (8.10) and (8.11), differ, however, from those for forced convective flow due to the presence of the buoyancy terms. The way in which these terms are derived was discussed in Chapter 1 when considering the application of dimensional analysis to convective heat transfer. In these buoyancy terms, is the angle that the x-axis makes to the vertical as shown in Fig. 8.3. 

Using finite element techniqnes, a mathematical model was developed for the two-dimensional analysis of non-isothermal and transient flow and mixing of a generalised Newtonian fluid with an inert filler. The model could incorporate no-slip, partial-slip or perfect-slip wall conditions using a universally applicable numerical technique. The model was used to simulate the convection of carbon black with flowing rubber in the dispersive section of a tangential rotor (Banbury) mixer. The Carreau equation was used to model the rheological behaviour of the fluid in this example. 31 refs. 

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