Question 1 This is a problem requiring Control Volume (CV) analysis. A horizontal belt moves with a speed U as shown in

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Question 1 This Is A Problem Requiring Control Volume Cv Analysis A Horizontal Belt Moves With A Speed U As Shown In 1 (103.82 KiB) Viewed 29 times

Question 1 This is a problem requiring Control Volume (CV) analysis. A horizontal belt moves with a speed U as shown in the figure below. The closed chamber of height b between the belt and a stationary base contains lubrication oil. The belt tends to drag the lubricant from the left to the right. A pressure difference is therefore externally applied such that there is no net transfer of oil from left to right. Belt τεί η) Oil Oil Oil Assume the following. 1. The flow in the chamber is steady. 2. The fluid is incompressible with a constant density p. 3. Pressure varies only in the I-direction. 4. Only the r-component of the velocity, Vz, is important. 5. Gravitational effects are negligible. 6. The width of the chamber in the z-direction perpendicular to the plane of the paper is of unit magnitude. Use the Eulerian CV shown by the dashed line in the figure above for your analysis. The pressures at Sections 1 and 2 are p, and P2, respectively. Under steady operating conditions, v, is a quadratic function of y of the following form: vr (9) = v % - MŚ (1-7), where M is a constant that depends on the applied difference P2 - P1. The goal of this design analysis is to firstly relate M to P2 – pı, find the pressure difference required to achieve the desired operating condition of zero net transfer of the lubricant from left to right, and the mechanical power required to keep the belt in motion under the desired operating condition. 3

[2] [4] (a) The given flow is expected to have horizontal streamlines. Can you verify this is indeed the case using the equation of a streamline? (b) Verify that given velocity field is incompressible. (c) The volumetric flow rate per unit width across any cross-section is Q = So vzdy. Using the given velocity profile, obtain an expression relating Q to U, M and b. Therefore, if Q = 0 at the desired operating condition, show that M, = 3U, where the subscript o denotes the desired operating condition. (d) What rate of accumulation of c-momentum do you expect in the control volume? Justify your answer. (e) The net rate of advective transfer of r-momentum into an Eulerian CV is [3] [5] $ Purv.nda. where the integral is over the entire control surface of the CV. Using this, show that the net rate of advective transfer of r-momentum into the CV shown in the figure is zero. (f) Show that the c-component of the total pressure force on the fluid CV per unit width in the 2-direction, Fpx = -(P2 - P.) b. (g) The only frictional tractions on the fluid CV with non-zero -components are due to the belt and the base. Use the full tensorial form of Newton's Law of Viscosity to show that these c-components are: [5] [5] aus = = Uz ду ly=b ; base = -1 ду ly=0 where u is the viscosity of the fluid. (h) Therefore, use the velocity profile given show that the r-component of the total frictional drag force on the CV per unit width is: [5] Ff. 1 2ML , = b b = (1) [5] = (i) Use the law of momentum conservation to express the constant M in terms of the applied pressure difference, Ap = P2 - Pi and any other constant parameters in the system. Therefore, what value of Ap should be applied to achieve the desired operating condition where M, = 3U for an application where the lubricant has a viscosity of 80 mPa s, the gap height is 2 mm, the belt length is 3 m, running at a speed of 1 m/s? () For the application given above, what is the power required to keep the belt of unit width (in the 2-direction) running at the desired operating condition? [2]