MME 273b FLUID MECHANICS – QUIZ 1
The Quiz is LIMITED OPEN BOOK. A single 8.5” x 11” double-sided crib sheet is allowed (no photocopies). Calculators are allowed. State any assumptions clearly and draw diagrams where required.
Background information
The diagram below shows part of a liquid process plant. Pipe A runs horizontally (out of the page) and carries oil of specific gravity SG = 0.70. The gauge pressure at the centre of Pipe A (marked by a + sign on the diagram) is 48,000 N/m2. A manometer system containing different fluids is connected to the pipe and this system is itself linked to a series of tanks. The cube-shaped Tank G contains mercury (SG = 13.60) and has dimensions of 2 m x 2 m x 2 m. There is also a small part of the system that contains air (at a pressure PC). The Tank D is separated from the adjacent tank by a rectangular gate, EF, that is 2 m high and 1 m wide (out of the page). The gate is hinged along its top edge at E (out of the page) and the hinge may be assumed to be frictionless. The gate itself may be considered to be weightless. The atmospheric pressure is 101,300 N/m2. The density of water may be taken as 1,000 kg/m3.
Questions
(a) What is value of the absolute air pressure, PC ? [Answer = 128,895 Pa]
(b) What is the hydrostatic force acting downwards on the square surface BB' ? [2.542 MN]
(c) What depth, h, of oil (SG = 0.70) is required in Tank D to ensure that the gate EF remains closed and hanging vertically as, shown in the figure ? [8.389 m]
(d) The technicians running the plant carry out a simple test each day to check the dynamic viscosity m of the oil in Tank D. They coat a smooth flat plate with the oil so that the oil layer has a thickness d. They tilt the plate so that it is at an angle of q to the horizontal. Then, they take a rectangular block of mass M, which has a flat base of bearing area A, and place this block on the oil. The time t taken for the block to slide a distance s down the plate (measured parallel to the plate) is recorded. The measurement period is chosen to be when the block is not accelerating or decelerating. Derive an expression for the viscosity of the oil in terms of the above parameters. You may assume that the oil itself is quite viscous, and so does not run down the plate, and that the oil layer thickness remains constant. [m = m g sinq d t / (A s)]
(1) The above figure is not drawn to scale.
(2) The 2nd moment of area, about the centroid, of a rectangle of height a and width b is Ixx = b a3 / 12