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# Fenômentos de Transporte

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the entity balance to momentum is that we will generate a vector equation rather than a scalar equation. We will, therefore, utilize balance equations, not all of which are conservation equations, and conservation equations, all of which are balance equations. Conservation equations are a sub-set of balance equations. 1.2.2 Steady-state processes Another special case of the entity balance which is often of interest is for a steady-state process. By a steady-state process we mean one which does not change in time - if we look at the process initially, then look at it again after some time has elapsed, the process still looks the same. This means that we are concerned with the entity balance in the form of rates as expressed in Equation (1.2-3). Note that this does not imply that all rates are zero. It does imply that accumulation rate is zero, because the very nature of accumulation means that content of the system is changing in time. Input and output rates, however, can be different from zero so long as they are constant in time, as can generation rates - it is only necessary that these rates balance one another so that there is no accumulation. Chapter 1: Essentials 17 For a steady-state process, therefore, the entity balance is written as Input rate + Generation rate = Output rate or [ Generation rate = Output rate - Input rate1 (1.2.2-1) 1.3 The Continuum Assumption We will treat only models in which each fluid property (for example, the density, the temperature, the velocity, etc.) is assumed to have a unique value at a given point at a given time. Further, we assume these values to be continuous in space and time. Even though we know that fluids are composed of molecules, and therefore, within volumes approaching molecular size, density fluctuates (and temperature is hard even to define), in order to use the concepts of calculus and differential equations we model such properties as continuous. For example, we define the density at a point as the mass per unit volume; but, since a point has no volume, it is not clear what we mean. By our definition we might mean (1.3-1) where p is the density, and Am is the mass in an elemental volume AV. Consider applying this definition to a gas at uniform temperature and pressure. Assume for the moment that we can observe Am and AV instantaneously, that is, that we can "freeze" the molecular motion as we make any one observation. For a range of values of AV suppose we make many observations of Am/AV at various times but at the Same point. At large values of AV, because the total number of molecules in our volume would fluctuate very little, the density from successive observations would be constant for all practical purposes. As we approach smaller volumes, however, our observed value would no longer be unique, because a gas is not a continuum, but is made up of small regions of large density (the molecules) connected by regions of zero density (empty space). In fact, we would see a situation somewhat as depicted in Figure 1.3-1, where, as our volume shrinks, we begin to see fluctuation in the observed density even in a gas at constant temperature and pressure. 18 Chapter 1: Essentials 0.001 0.01 0.1 1 10 100 1000 relative volume Figure 1.3-1 Breakdown of continuum assumption For example, if our AV were of the same order of volume as an atomic nucleus, the observed density would vary from zero to the tremendous density of the nucleus itself, depending on whether or not a nucleus (or, perhaps, some extra-nuclear particle) were actually present in AV at the time of the observation. To circumvent such problems we make the arbitrary restriction that we will treat only problems where we may assume that the material behavior we wish to predict can be described adequately by a model based on a continuum assumption. In practice, this usually means that we do not attempt to model problems where distances of the order of the mean free paths of molecules are important. The continuum assumption does fail under some circumstances of interest. For example, the size of the interconnected pores of a prous medium may be of the order of the mean free patb of gas molecules flowing through the pores. In this situation, "slip flow" (where the velocity at the gas-solid interface is not zero) can result. Another example is in the upper atmosphere where the mean free path of the molecules is very large - in fact, it may become of the order of a hundred miles. In this situation, slip flow can occur at the surface of vehicles traversing the space. Chapter I : Essentials 19 1.4 Fluid Behavior The reader is encouraged to make use of the resources available in flow visualization, e.g., Japan, S , 0. M. E., Ed. (1988). Visualized flow: Fluid motion in basic and engineering situations revealed by flow visualization. Thermodynamics and Fluid Mechanics. New York, NY, Pergamon, Japan, T. V. S. o., Ed. (1992). Atlas of Visualization. Progress in Visualization. New York, NY, Pergamon Press, and Van Dyke, M. (1982). An Album of Fluid Motion. Stanford, CA, The Parabolic Press, which contain numerous photos, both color and black and white. There are also many films and video tapes available. There is nothing quite so convincing as seeing the actual phenomena that the basic models in fluid mechanics attempt to describe - for example, that the fluid really does stick to the wall. 1.4.1 Laminar and turbulent flow We normally classify continuous fluid motion in several different ways: for example, as inviscid vs. viscous; compressible vs. incompressible. Viscosity is that property of a fluid that makes honey and molasses "thicken" with a decrease in temperature. Inviscid flow is modeled by assuming that there is no viscous effect. What we sometimes call ideal fluids or ideal fluid flow is modeled by not including in the equations of motion the property called viscosity. Viscous flow is modeled including these viscous effects. The viscosity of a fluid can range over many orders of magnitude. For example, in units of pPa s, air has a viscosity of about 10-5; water, about 10-3; honey, about 10; and molten polymers, perhaps 104; a ratio of largest to smallest of 1,OOO,OOO,OOO. (This is by no means the total range of viscous behavior, merely a sample of commonly encountered values in engineering problems .) In viscous flow we can have two further sub-classifications. The first is luminur flow, a flow dominated by viscous forces. In steady laminar flow, the fluid "particles" march along smoothly in files as if on a railroad track. If one releases dye in the middle of a pipe in which fluid is flowing at steady-state, as shown in Figure 1.4.1-1, at low velocities the dye will trace out a straight line. At higher velocities the fluid moves chaotically as shown and quickly mixes across the entire pipe cross-section. The low velocity behavior is Zaminar and the higher velocity is turbulent. 20 Chapter I : Essentials Figure 1.4.1-1 Injection of dye in pipe flow1* laminar (top) to turbulent (bottom) Laminar flow is dominated by viscous forces and turbulent flow is dominated by inertial forces, In a plot of point velocity versus time we would see a constant velocity in steady laminar flow, but in "steady" turbulent flow we would see instead a velocity varying chaotically about the time-average velocity at the point. Later in the text we will discuss dimensionless numbers and their meaning. At this point we merely state that value of the Reynolds number, a dimensionless group, lets us differentiate between laminar and turbulent flow. The Reynolds number is composed of the density and the viscosity of the fluid flowing in the pipe, the pipe diameter, and the area-averaged velocity, and represents the