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ES3C9 FLUID MECHANICS FOR MECHANICAL ENGINEERS Example Class 3 (1) Reservoir pumping A necked-down section in a pipe flow, called a Venturi, develops a low throat pressure which can aspirate fluid upward from a reservoir, as in figure 2. Using Bernoulli’s equation with no losses, derive an expression for the velocity V1 which is just sufficient to bring reservoir fluid into the throat. D2 D1 V1 V2 Water P2=Pa h Pa Water Fig. 2: Water will begin to aspirate into the throat when pa p1 gh . Hence: Volume flow: Bernoulli D12 4 V1 D22 V2 V1 V2 D2 / D1 V1 V2 2 with D2 / D1 2 4 1 1 z 0 : p1 V12 patm V22 patm p1 V12 V22 1V22 2 2 2 2 Solve if pa p1 gh V2 Similarly, V1,min V2,min 2 gh 1 2 gh 1 D1 / D2 (2) Non-dimensional analysis The Morton number Mo, used to correlate bubble-dynamics studies, is a dimensionless combination of acceleration of gravity g, viscosity , density , and tension coefficient Y. If Mo is proportional to g, find its form. The relevant dimensions are the following: g LT 2 , ML1T 1 , ML3 , MT 2 To have g in numerator, we need the combination: Mo g a b c a b c L M M M 0 0 0 2 3 2 M LT T LT L T M a b c 0 L 1 a 3b 0 T 2 a 2c 0 a b c 0 1 2a 3c 0 2 a 2c 0 Solve for a = 4, b= - 1, c= - 3, or: Mo g 4 (3) The Titanic (length 269m, maximum width 28m, and height 30.5m) sank on April 14, 1912, hitting an iceberg and sinking 160 minutes later. Recently, a sonar study of the bow of the Titanic on the ocean floor has revealed that the holes caused by the iceberg are much smaller than originally thought. Until this study, it was assumed that a large, 100m long gash was ripped in the Titanic’s side, but now the sonar reveals that the area of the hole was only 1.4m2 (the size of a typical door!). The hole of the Titanic was approximately 6.1m below sea level at the start of the sinking. Was the hole large enough to sink the titanic in 160 minutes? Fig. 1: Governing Equations: By selecting the entire ship as an open system, the conservation of mass is given by: d d V .n dAr 0 dt 1 In this problem, however, only part of the selected volume is filled with fluid. Therefore, at any instant t, the density is defined only over the volume f t that the fluid occupies. Therefore, the unsteady term reduces to d d d d f dt dt f For a streamline originating at the free surface (0) and terminating at the hole (1), the Bernoulli equation is expressed as V02 V12 P0 gz0 P1 gz1 2 2 where P, V, and z are respectively the pressure, velocity, and elevation from an arbitrary reference level, of the points. Basic Assumptions: The problem can be simplified by assuming inviscid, incompressible flow, with constant uniform velocity at the hole. We also represent the Titanic as an empty rectangular box – a rather conservative assumption. The water inside the ship is assumed opened to the atmosphere. At the hole, we have the following conditions: V u1i and n i . Substituting the appropriate terms in the conservation of mass, we get d d f u1dAr 0 dt f hole Since sinking is defined as the point when the Titanic is fully filled, the upper limit of the integration is set to f t 229726 m3 , where t is the total volume of the d f u1 Ah , where dt Ah is the area of the hole. By integration, the final result then is f u1 Ahtsink . The velocity of the water at the hole can be evaluated using the Bernoulli equation by selecting an arbitrary point at the free surface where P0 Patm , V0 0 , and z h0 , and a point at the hole where P1 Patm , V1 0 , z 0 and (the reference elevation is chosen at the level of the hole). Therefore, the water velocity at the hole is u1 2 gh0 10.94 m s-1 . The analysis then predicts that under the assumptions and conditions stated, the Titanic. For uniform conditions at the hole, and constant density Titanic would have sunk in tsink t 15000s 4.2h Ah 2 gh0 This is higher than the actual sinking time. A more detail analysis of the same problem must consider the fact that while the Titanic sunk both the depth of the hole and the level of the water inside the ship increased. The analysis must also consider that the Titanic was not empty but partially filled with people, luggage ... Nevertheless, since the predicted 4.2 hours is not very far off from 2.7 hours, the analysis suggests that the Titanic could have sunk because of a hole that was much smaller than originally thought. u Ah