5: The discharge pressure (P) of a gear pump (Fig. 3) is a function of flow rate (Q), gear diameter (D), fluid viscosity (u) and gear angular speed (w). P = f (Q, D, μ, wo). Use the pi theorem to rewrite this function in terms of dimensionless parameters.
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- 1. Start from the following equations from class notes (treat as given): The shear stress in pipe flow T = -μ- dr du 16 [1-(-)²), AP D² 16μL The fluid velocity in the pipe u = Derive clearly the following results: (a) The shear stress on the wall of the pipe, Tw. (b) The average velocity in the pipe flow, ū. (c) The Darcy friction factor, defined by f where L is the length of the pipe. 8 tw pū² (d) Express the result of (c) in terms of the Reynolds number, Re. (e) Find an expression for the head loss due to shear stress at the wall, AH₁, in terms of f,u, D and L. (f) What are the main assumptions in the fluid flow, in order to analyse the flow in this way?7. An oil (sp.gr 0.9) is flowing through a 1.2m diameter pipe at a rate of 2.5 m³/s. The kinematic viscosity of oil is 3 X106 m²/s. In order to model this flow, water is used to flow through a 120mm diameter pipe having kinematic viscosity of 0.012X104 m²/s. Find the model discharge and velocity.Find the pressure drop between locations a and b for flow in a pipe using the following data: Working fluid: Water Pipe: Stainless steel, Schedule No. 120 Nominal pipe size (cm): 10 Flow rate (m3 /min): 2.271 Temperature (C): 38 Length (m): 45.72 Solve for the following 3 cases: a) Za = Zb =33.5 m, b) Za = Zb + 15 m, and c) Za = Zb + 46 m.
- (b) A pitot-static probe is use to determine the flow velocity by measuring the differential pressure. The pitot formula to obtain the flow velocity is, 2(P- P,) V = where, V is the velocity, P is pressure and pis fluid density. i) The pressure difference sensor use in the system is electronic types and the output of the device is measured in voltage. The output of the pressure device is 3.5 V and the linear relationship between the device and the pressure difference is 10 kPa/V. If the measured fluid is water at 20°C, determine the pressure difference inside the system the water velocity. ii) The Pitot-static tube is also commonly use in aircraft. An aircraft flying at 3000 m above sea level when the differential pressure reading clocked 3 kPa. Determine the speed of the aircraft.Problem 1: The discharge pressure (P) of a centrifugal pump shown below is a function of flow rate (Q), impeller diameter (D), fluid density (p), and impeller angular speed (12). P = f (Q. D, p. 92). Use the Buckingham pi technique to rewrite this function in terms of dimensionless parameters, 1 g (n₂). P= P(Q,D, Dimensions 2) N= 5 Q. P1. Filtration Consider this data: Filtrate (ml) Axis Title 9 8 7 6 10 3 2 1 0 ● 0 5 Filtrate Volume (ml) 10 ● Filtrate Volume (ml) 15 20 time (min) 25 30 35 What is the coefficient of fluid loss CL? What is the spurt loss Vsp? time (min) PANASON 1 2 4 7 10 15 20 30 Filtrate Volume (ml) 1.4 3 3.4 3.6 4 3.9 4.8 5 6 01 6.5 8
- Dimensional Analysis and Hydraulic Similitude (fluid mechanics) 1. Water at 60F at 12 ft/s in a 6-in. pipe. (a) For dynamic similarity, determine the velocity of medium fuel oil at 90F flowing in a 12-in. pipe. (b) Determine the diameter of the pipe that should be used if a medium lubricating oil at 70oF if flowing at a velocity of 50 ft/s. find the: a. velocity : ____________________ fps b. diameter : ____________________ in.Problem 5 s): The discharge pressure (P) of a screw pump (Fig. 5) is a function of flow rate (Q), screw diameter (D), fluid viscosity (u) and screw angular speed (w). P = f (Q, D, μ, w). Use the pi theorem to rewrite this function in terms of dimensionless parameters, ₁ g (T₂). Choose Q, D, and u as repeating variables. Screw Fig. 5: Screw pumpMeasurements at a certain point of a pipe have been done where the following parameters were recorded: Fluid of density = 887 kg/m3, Fluid velocity = 4 m/s, Pressure= 11.3 KN/m2 If the total energy per unit weight at this point = 32 m, then the potential energy is: O Zero m 10 m 20 m 30 m &
- An incompressible fluid oscillates harmonically (V = Vosinut, where Vis the velocity) with a frequency of 9 rad/s in a 6-in.-diameter pipe. A 1/5 scale model is to be used to determine the pressure difference per unit length, Ap, (at any instant) along the pipe. Assume that Api= f(D, Vo, w, t, u, p) where D is the pipe diameter, w the frequency, t the time, the u fluid viscosity, and p the fluid density. If the same fluid is used in the model and the prototype, at what frequency should the model operate? Wm i rad/s1. In Buckingham π theorem, if n is the number of variables and m is the number of basic dimensions, the number of independent dimensionless parameters would be,A. (m – n) B. (n – m) C. (m + n) D. (n/m)In the field of air pollution control, one often needs to sample the quality of a moving airstream. In such measurements a sampling probe is aligned with the flow as sketched in Fig. A suction pump draws air through the probe at volume flow rate V· as sketched. For accurate sampling, the air speed through the probe should be the same as that of the airstream (isokinetic sampling). However, if the applied suction is too large, as sketched in Fig, the air speed through the probe is greater than that of the airstream (super iso kinetic sampling). For simplicity consider a two-dimensional case in which the sampling probe height is h = 4.58 mm and its width is W = 39.5 mm. The values of the stream function corresponding to the lower and upper dividing streamlines are ?l = 0.093 m2/s and ?u = 0.150 m2/s, respectively. Calculate the volume flow rate through the probe (in units of m3/s) and the average speed of the air sucked through the probe.