Bernoulli Equation Analysis of Rapidly Expanding Pipe Flow As we have already

Bernoulli equation analysis of rapidly expanding pipe

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Bernoulli Equation Analysis of Rapidly-Expanding Pipe Flow As we have already noted, the assumptions we utilized in the preceding analysis of flow in a rapidly-expanding pipe via the control-volume momentum equation are the same as those used to derive Bernoulli’s equation. So it is worthwhile to repeat the analysis using the latter equation, again, in conjunction with the continuity equation. We will assume the pipe diameters are not extremely large so that even for dense liquids the hydrostatic terms in Bernoulli’s equation can be neglected (as the body-force terms were in the control-volume momentum equation). Then we can write Bernoulli’s equation between locations 1 and 3 as p 1 + ρ 2 u 2 1 = p 3 + ρ 2 u 2 3 . We rearrange this as p 1 p 3 = ρ 2 ( u 2 3 u 2 1 ) , and substitute the result relating u 3 to u 1 obtained earlier from the continuity equation to obtain p = ρ 2 u 2 1 bracketleftBigg parenleftbigg D 1 D 3 parenrightbigg 4 1 bracketrightBigg . (4.23) Comparison of this result from Bernoulli’s equation with that obtained from the control-volume momentum equation given in Eq. (4.22) shows quite significant differences, despite the fact that the same basic physical assumptions have been employed in both analyses. In particular, we can calculate the difference between these as p M p B = ρu 2 1 braceleftBiggbracketleftBigg parenleftbigg D 1 D 3 parenrightbigg 4 parenleftbigg D 1 D 3 parenrightbigg 2 bracketrightBigg 1 2 bracketleftBigg parenleftbigg D 1 D 3 parenrightbigg 4 1 bracketrightBiggbracerightBigg = 1 2 ρu 2 1 bracketleftBigg parenleftbigg D 1 D 3 parenrightbigg 4 2 parenleftbigg D 1 D 3 parenrightbigg 2 + 1 bracketrightBigg = 1 2 ρu 2 1 bracketleftBigg 1 parenleftbigg D 1 D 3 parenrightbigg 2 bracketrightBigg 2 , (4.24) where the subscripts B and M respectively denote “Bernoulli” and “Momentum” equation pressure changes. Equation (4.24) shows that the pressure change predicted by the momentum equation is always greater than that given by Bernoulli’s equation; but as D 3 D 1 ( i.e. , no pipe expansion), the two pressure differences coincide. We first comment that Eq. (4.24) shows that the pressure change predicted by the momentum equation is always greater than that predicted by Bernoulli’s equation, and second that experimental measurements quite strongly favor the control-volume momentum equation result— i.e. , Bernoulli’s
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122 CHAPTER 4. APPLICATIONS OF THE NAVIER–STOKES EQUATIONS equation does not give the correct pressure change for this flow, even though it should be applicable and even more, we actually were able to apply it along a streamline in the present case. So, what has gone wrong? Recall that we earlier noted the existence of prominent recirculation regions in the flow field immediately behind the expansion, as depicted in Fig. 4.8, and that these arise from viscous effects.
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