Conservation Equations in Fluid Flow
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1 Conservation Equations in Fluid Flow Q. Choose the correct answer (i) Mathematical statement of Renolds transport theorem is given b dn (a). da d C t dn (b) da. d C t dn (c) d. da C t dn (d) d. da C t where N is an etensive propert, is the propert per unit mass, is the velocit vector and da is the elemental area vector on the control surface. [Ans.(c)] (ii) A fish tank is being carried on a car moving with constant-horizontal acceleration. The level of water will (a) remain unchanged (b) rise on the front side of the tank onl (c) rise on the front side of the tank and fall on the back side (d) rise on the back side of the tank and fall on the front side [Ans.(d)] Q. A tank as shown in the figure below has a nozzle of eit diameter at a depth H below the free surface. At the side opposite to that of nozzle, another nozzle of diameter is attached to the tank at a depth H. Neglecting the frictional effects, find the diameter in terms of so that the net horizontal force on the tank is zero. H H Control olume F A fied control volume as shown b the dashed line in the above figure is considered for the analsis.
2 For nozzle, appling Bernoulli s equation between a point on the free surface and nozzle eit along a streamline, we have g H g g g or gh and mass flow rate m A gh For nozzle, appling Bernoulli s equation between a point on the free surface and nozzle eit along a streamline, we have g H g g g or gh gh and mass flow rate m A gh Let F be the horizontal force in the positive direction of on the fluid mass in the control volume inscribing the tank as shown in the figure. Appling the momentum theorem for the control volume, we get F m m gh gh gh gh gh It is given that the net horizontal force acting on the tank is zero, and therefore, we obtain gh from which we get
3 Q. Eample 5. A 5 reducing pipe-bend in a horizontal plane (shown in the figure below) has an inlet diameter of 6 mm and outlet diameter of mm. The pressure at the inlet is kpa gauge and rate of flow of water through the bend is.5 m /s. Neglecting friction, calculate the net resultant horizontal force eerted b the water on the bend. Assume uniform conditions with straight and parallel streamlines at inlet and outlet and the fluid to be frictionless. pa pa 5 A fied control volume as shown b the dashed line in the above figure is considered for the analsis. The inlet velocit is Q.5.5 m/s A.6 The outlet velocit is Q.5 6. m/s A. Appling Bernoulli s equation along a streamline connecting sections and, we have p p g g g g.5 6. or p or, p. Pa. kpa Appling the momentum theorem to the control volume, we get pa pacos 5F Qcos 5 pasin 5F Qsin 5 and where F and F are the forces in the and directions eerted b the bend on water in the control volume. Substituting the respective values in the above two equations, we get
4 .. sin 5F.56.sin 5 which give, F.6 kn.6.. cos 5 F.5 6.cos 5.5 F 7.96 kn Therefore, the resultant force on the water is F kn The resultant force makes an angle of tan with the negative direction of -ais. According to Newton s third law, the force eerted b the water on the bend is equal and opposite to the force F. Q. An open rectangular tank of length L m, height h m and wih (perpendicular to the plane of the figure below) m is initiall half-filled with water. The tank suddenl accelerates along the horizontal direction with an acceleration =.5g (where, g is the acceleration due to gravit). Will an water spill out of the tank? g h h L Let us consider the case that the tank accelerates with a maimum acceleration a along the horizontal direction without spilling the water. Since the volume of the water in the tank remains unchanged, the free surface takes the shape as shown in the figure below. g h h a,ma L
5 From the principle of relative equilibrium, one can write h a,ma tan L g a,ma or. g or a,ma.g This is the maimum acceleration that can be given without spilling the water. Since the given acceleration is higher than this value, the water will spill out of the tank. 5
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