fluidsCh1 - Chapter 1 Motivation This chapter adapted from...

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Chapter 1 Motivation This chapter, adapted from Tritton’s book, presents the need for more ad- vanced tools. For internal ±ows, the canonical case is the ±ow in a circular pipe. The student should be familiar with the laminar (Poiseuille) parabolic velocity pro²le, with the Moody diagram for pressure loss in a pipe, and with the modi²ed Bernoulli equation including friction and minor losses. So we understand laminar friction and pressure drop, but how about entrance losses? Secondary ±ows at the entrance and exit? Flows in curved pipes? Pulsatile ±ow? At the undergraduate level, external ±ows are studied in relation with drag in uniform ±ows (usually involving empirical dimensionless curves of drag coe³cients, and a description of ±ow separation and the drag crisis), ±at plate boundary layers, and possibly some simple potential ±ows such as the ±ow around a cylinder. In this chapter, we revisit these topics to motivate the introduction of the more advanced concepts required to understand e.g. ±ow separation. We will end up with many questions marks, many unresolved di³culties to be addressed later in the course. 1.1 Internal Flow 1.1.1 A simple problem There are a number of practical design issues related to the pipe ±ow issuing from a head tank (Fig. 1.1). There are also some added features (e.g. about entrance pressure drop) that can be explained on the basis of undergraduate 17
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18 CHAPTER 1. MOTIVATION Figure 1.1: A simple pipe fow set-up tools. And an engineer should not overlook these approaches: simple ±rst! But there are also a number o² ²acts that show the need ²or more advanced concepts, more sophisticated tools (and more advanced mathematics). On the practical side, it would be important to maintain constant fow rate. This can be achieved in several ways: use a constant-displacement pump, insulated ²rom the pipe by a ba³ed plenum; or use an overfow pipe, so the supply rate into the head tank needs no monitoring; etc. You might want to discuss alternative designs. By running this experiment over a wide range o² Reynolds numbers, and making sure the pipe is long enough so entrance and exit losses can be neglected (how long is that?), one can collect the data represented on the Moody diagram (Fig. 1.2), here shown ²or a smooth pipe. The analysis was carried out in the previous chapter. Re²er to the previous chapter ²or the basic analysis, using Bernoulli’s equation and control volume in the pipe. For automated calculations, one
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1.1. INTERNAL FLOW 19 10 2 10 3 10 4 10 5 10 6 10 7 10 8 10 -3 10 -2 10 -1 10 0 Reynolds number Friction factor Figure 1.2: Sketch of Moody diagram for pipe friction can make use e.g. of Colebrook’s formula for the friction factor f 1 f = - 2 .log 10 ( e 0 3 . 7 + 2 . 51 Re D f ) , (1.1) where e 0 is the pipe’s relative roughness (Fig. 1.2). We now turn to some of the details.
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This note was uploaded on 10/16/2010 for the course L.C.SMITH MAE643 taught by Professor Xx during the Spring '10 term at Syracuse.

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fluidsCh1 - Chapter 1 Motivation This chapter adapted from...

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