## Qi r cos aQj

c Find also an expression for the force on the plane, and evaluate it for the case b = 10cm, t = 1cm, Q: = 10L.s-1, a = 60°.

2.3 Consider a steady laminar flow between two fixed parallel plates at y = 0 and y = D (cf. Figure 2.4). The fluid is pushed to the right (x increasing) by a constant pressure gradient 'dp/'dx(< 0).

a What are the forces acting on an element of fluid of length Ax and width Az, lying between y and y + Ay? Show that the balance of forces on this fluid element is given by d / du \ dp dy x 9y J 9x b By integrating the last expression, show that the velocity at a distance y above the lower plate is

The plates have width W ^ D, so that edge effects are negligible. Show that the volumetric flow rate between the plates is

2.4 Laminar flow in a pipe

A constant pressure gradient (dp/dx) between the ends pushes fluid straight down a circular pipe of diameter D = 2R. Let x be the distance along the pipe and r the distance from the axis.

a As in Problem 2.3, show that the balance of forces on an annulus of length Ax lying between r and r + Ar leads to the equation d ( du \ dp — I r — I = — r— dr dr dx

Hint: The shear force at r is (2wx)r(r), and r varies across the pipe.

b By integration, show that the velocity at a distance r from the axis is ur)-- 4p(|)<R2—r c Hence show that the volume of fluid flowing out of the pipe per unit time is dp

Q \dx d The mean velocity in the pipe is u = 0/A. By putting the results so far into (2.12) show that uf = I6v/D, thus verifying (2.15).

2.5 Having been informed of the results of Example 2.1, the accountant of the engineering firm concerned suggests that it would be cheaper to use narrow PVC pipe to carry the flow.

a Disillusion the accountant by calculating the theoretical friction head incurred in passing 0.1m3 s-1 of water through 200 m of PVC pipe of diameter 5 cm, and show that it exceeds the available head. b If gravity is the only force available to move the water, calculate an upper limit to the flow which could in fact be pushed through this pipe.

Hint: Take Hf = L (vertical pipe), estimate f and calculate the corresponding u.

2.6 A steel pipe of diameter D and length L is to carry a flow Q. Assuming that the pipe friction coefficient f varies only slowly with Reynolds number, show that the head loss due to friction is proportional to D-5 (for fixed L and Q).

### Bibliography

The following selection from the many books on fluid mechanics may prove useful. There are many other good books besides those listed. For work on turbomachin-ery, books written for engineers are usually more useful than those written for mathematicians, who too often ignore friction and forces. Since the basics of fluid dynamics have not changed, old textbooks can still be useful, especially if they use SI units (which many older books do not).

Batchelor, G.K. (1967) An Introduction to Fluid Dynamics, Cambridge University Press. {Classic text, reissued unchanged in 2000. A most precise statement of the foundations, see especially Chapter 3, with many examples. Repays careful reading, but perhaps unsuitable for beginners.} Francis, J.R. (1974, 4th edn) A Textbook of Fluid Mechanics, Edward Arnold, London. {Clear writing makes easy reading for beginners. More engineering detail than Kay and Nedderman.} Hughes, W.F., Brighton, J.A. and Winowich, N. (1999, 3rd edn) Theory and Problems of Fluid Dynamics, Schaum's Outline Series. {Multitude of worked examples.}

Kay, J.M. and Nedderman, R.M. (1985) Fluid Mechanics and Transfer Processes,

Cambridge University Press. {A concise and wide-ranging introduction.} Mott, R.L. (2000, 5th edn) Applied Fluid Mechanics, Prentice Hall, USA. {Widely used student text for beginners, with exceptionally clear explanations.} Tritton, D.J. (1988) Physical Fluid Dynamics, Oxford Science Publications, Oxford

U.P. {Careful mathematical formulation, related carefully to physical reality.} Webber, N. (1971) Fluid Mechanics for Civil Engineers, Chapman and Hall, London. {Delightfully simple but useful introduction for students with little knowledge of physics or engineering.}

Chapter 3

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