Lecture 2 Flow classifications and continuity Dr Tim Gough: t.gough@bradford.ac.uk
General information 1 No tutorial week 3 3 rd October 2013 this Thursday. Attempt tutorial based on examples from today s lecture. As per usual, any problems email t.gough@bradford.ac.uk On email while here: www.esrf.eu Laboratory classes these are still being allocated. For those allocated in group 2J1 (Civil Eng.) the week 4 laboratory is postponed (tutor in Belfast). Please watch your email.
General information 2 Is Blackboard working yet? Currently 137 students registered on e vision. If yes, good. If no, then continue to use http://skeddanblog.wordpress.com/ Please also continue to watch email for announcements. Questions?
Lecture 1 recap Elastic solids Newtonian fluids Dynamic viscosity Kinematic viscosity
Lecture 1 recap Bernoulli P s is static pressure = gh is dynamic pressure Reynolds number is the ratio of inertial to viscous forces in a flow where: is density in kg/m 3 V is velocity in m/s L is a length in m is viscosity in Pa.s
Lecture 1 recap For pipe flow only! If Re < 2000 the flow is laminar If 2000 < Re < 4000 the flow is transitional If Re > 4000 the flow is turbulent
Fluid Flow
Fluid flow The motion of a fluid is usually extremely complex. Studies of fluids at rest, or in equilibrium, are simplified by the absence of shear forces within the fluid. When a fluid flows over a surface or other boundary, whether at rest or in motion, the velocity of the fluid in contact with the boundary must be the same as that of the boundary and a velocity gradient is created at right angles to the boundary.
Fluid flow The resulting change of velocity from layer to layer of fluid flowing parallel to the boundary gives rise to shear stresses in the fluid. If an individual particle of fluid is coloured, or otherwise rendered visible, it will describe a pathline. A pathline is the trace showing the position of the particle at successive intervals of time from a given point. If instead of colouring an individual particle, the flow pattern is made visible by injecting a stream of dye (liquid) or smoke (gas) the result will be a streakline (or filament line) which gives an instantaneous picture of the positions of all particles that have passed through the point at which the dye (or smoke) was injected. Since the flow may change with time a pathline and a streakline need not be the same.
Fluid flow When choosing a dye or other tracer it is clearly very important to match the density and other physical properties as closely as possible to the carrier fluid (isokinetic). Streamlines are curves that are everywhere tangential to the velocity vector.
Streaklines
Streaklines Flow around an aerofoil very low velocity
Uniform and steady flows
Uniform and steady flows Conditions in a body of fluid can vary from point to point and, at any given point, can vary from one moment of time to the next. Flow is described as uniform if the velocity at a given instant is the same in magnitude and direction at every point in the fluid. If, at a given instant, the velocity changes from point to point, the flow is described as non uniform. A steady flow is one in which the velocity, pressure and cross section of the stream may vary from point to point but do not vary with time. If, at any given point, conditions do change with time, the flow is described as unsteady.
Four possible types of flow 1. Steady uniform flow Conditions do not change with position or time. The velocity and cross sectional area of the stream of fluid are the same at each cross section. Pipe flow could be laminar or turbulent
Four possible types of flow 2. Steady non uniform flow Conditions change from point to point but not with time. The velocity and cross sectional area of the stream may vary from crosssection to cross section but, for each cross section, they will not vary with time. Flow through tapering pipe
Four possible types of flow 3. Unsteady uniform flow At a given instant of time the velocity at every point is the same, but the velocity will change with time. Start up flow of molten LDPE
Four possible types of flow 4. Unsteady non uniform flow The cross sectional area and velocity vary from point to point and also change with time. Wave travelling along a channel
Viscous and inviscid flows
Viscous and inviscid flows When a real fluid flows past a boundary, the fluid immediately in contact with the boundary will have the same velocity as the boundary. Further away from the boundary (or wall) perpendicularly the effects of the boundary diminish. Eventually the effects of the boundary become negligible. Since the effects of the boundary are due to the viscosity of the fluid, the flow near to the boundary is known as a viscous flow. Well away from the boundary the flow can be treated as being inviscid (or non viscous). The region near to the boundary is known as the boundary layer about which more later.
Viscous and inviscid flows Absence of viscous forces allows the fluid to slip along the pipe wall, providing a uniform velocity profile. Velocity profiles for inviscid (non viscous) flow
Viscous and inviscid flows Inviscid (non viscous) flow past aerofoil
Viscous and inviscid flows With viscosity involved, if the velocity at the wall is zero (no slip) then we must have a velocity profile near the wall. Velocity profiles for viscous flow
Viscous flow over flat plate The flow away from the walls can be treated as inviscid. But near the walls the viscosity is important.
Viscous flow over flat plate The region where viscous effects dominate is called the boundary layer (about which more later).
Boundary layers over a flat plate
Discharge and mean velocity
Discharge and mean velocity The total quantity of fluid flowing in time past any particular crosssection of a stream is called the discharge or flow at that section. It can be expressed either in terms of: Mass flowrate (kg/s) or Volumetric flowrate, Q (m 3 /s) Clearly we can move between the two flowrates, for an incompressible fluid using the simple relation:
Discharge and mean velocity In an ideal fluid, where there is no friction, the velocity, V, would be the same at every point across the cross section. If the cross section has an area A then we can say that the volume passing would be V x A, i.e. V r r R r V a) Laminar flow b) Turbulent flow In a real fluid however the velocity at the wall is the same as that of the wall (see the no slip condition later)
Discharge and mean velocity V r r R r V a) Laminar flow b) Turbulent flow If V is the velocity at any radius r, the flow Q through an annular element of radius r and thickness r will be: Hence: 2 In many problems we simply assume a constant velocity equal to the mean velocity to give: /
Continuity of flow
Continuity of flow Mass of fluid entering Control volume Mass of fluid leaving Except in nuclear processes, mass is neither created nor destroyed. Thus the principle of conservation of mass can be applied to a flowing fluid. i.e. Mass of fluid entering per unit time = Mass of fluid leaving per unit time + Increase of mass of fluid in control volume per unit time
Continuity of flow For steady flow we can write: Mass of fluid entering per unit time = Mass of fluid leaving per unit time For a streamtube (no fluid crosses boundary): 1 2 Area = A 1 Velocity = V 1 Density = 1 Area = A 2 Velocity = V 2 Density = 2
Continuity of flow This is the equation of continuity for the flow of a compressible fluid through a streamtube. For flow of a real fluid through a pipe or conduit we can use the mean velocity, again: Or for an incompressible fluid where 1 = 2 this reduces to:
Continuity of flow example 1 Branched pipes
Continuity of flow example 1 Water flows from point A to point B through a pipe of 50 mm diameter. At B the pipe expands to a diameter of 75 mm until point C. At C the pipe splits into branches CD and CE. Branch E has a diameter of 30 mm. The mean velocity of the flow in BC is 2 m/s and in CD is 1.5 m/s and the flowrate in CD is twice that in CE. Assuming no frictional losses, calculate: a) The flowrates through AB, BC, CD and CE b) The velocity of the flows in AB and CE and c) The diameter of the pipe for branch CD.
Continuity of flow example 1 4 4 4 50 10 4 75 10 4 30 10 4 1.96 10 4.42 10 7.07 10
Continuity of flow example 1 1.96 10 4.42 10 7.07 10 2 4.42 10. / By continuity,. / And, 2 3 8.84 10 /
Continuity of flow example 1 1.96 10 4.42 10 7.07 10 8.84 10 / 8.84 10 / 3 8.84 10 / so And, Now, 8.84 10 2 2 2.95 10. / so 3. / 2.95 10. / 7.07 10
Continuity of flow example 1 Similarly, And, so so 8.84 10. / 1.96 10 5.90 10 1.5 3.93 10 4 4 4 3.93 10 5 10.
Continuity of flow example A B Q 2 =? = 2 m/s d 2 = 75 mm C Q 3 = 2Q 4 =? = 1.5 m/s d 3 =? D Q 1 =? =? d 1 = 50 mm Q 4 = 0.5Q 3 =? =? d 4 = 30 mm E Q 1 = Q 2 = 8.84 x 10 3 m 3 /s Q 4 = 2.95 x 10 3 m 3 /s Q 3 = 5.9 x 10 3 m 3 /s V 1 = 4.51 m/s V 4 = 4.17 m/s D 3 = 7.07 cm
Continuity of flow example 2 Porous walls
Continuity of flow example 2 Water flowing through an 8 cm diameter pipe enters a porous section which allows a radial velocity through the wall surfaces for a distance of 1.2 metres. If the entrance average velocity is 12 m/s, find the exit average velocity if: a) is 15 cm/s out of the pipe walls and b) is 10 cm/s into the pipe. flow 1.2 m
Continuity of flow example 2 12 / 8 0.08 4 By continuity, 5.03 10 so Rearranging:
Continuity of flow example 2 12 / 8 a) =15 cm/s = 0.15 m/s 5.03 10 2 2 0.04 1.2 0.302 12 5.03 10 0.15 0.302 5.03 10.... /
Continuity of flow example 2 12 / 8 b) = 10 cm/s = 0.10 m/s 5.03 10 2 2 0.04 1.2 0.302 12 5.03 10 0.10 0.302 5.03 10. /
Continuity of flow example 3 Surge tank
Continuity of flow example 3 A surge tank may be attached to a pressurised pipe flow in order to accommodate sudden changes in pressure. It can either absorb sudden rises in pressure or quickly provide extra fluid in case of a drop in pressure. Used in all branches of engineering. Often found on racing cars undergoing high levels of lateral acceleration to ensure that the inlet to the fuel pump is never starved of fuel. Surge tank
Continuity of flow example 3 The pipe flow fills a cylindrical surge tank as shown here. At time t = 0, the water depth in the tank is 30 cm. Estimate the time required to fill the remainder of the tank. flow. / d = 75 cm d = 12 cm 1 m. / Firstly calculate pipe areas, 0.12 4 1.13 10 0.75 4 4.42 10
Continuity of flow example 3 1.13 10 4.42 10 2.5 / 1.9 / By continuity, So, Rearrange to find,
Continuity of flow example 3 1.13 10 4.42 10 2.5 / 1.9 / 2.5 1.13 10 1.9 1.13 10 4.42 10 0.02825 0.02147 0.442 0.01534 / So, 0.01534 0.442 6.78 10 /
Continuity of flow example 3 6.78 10 / To fill remainder of tank the water has to rise from 30 cm to 1 metre. 0.75 4 1 0.3 0.3093 So time to fill remainder of tank,...
Continuity of flow example 4 Jet engine
Continuity of flow example 4 Pratt and Whitney J52 turbojet engine
Continuity of flow example 4 At cruise conditions, air flows into a jet engine at a steady rate of 30 kg/s. Fuel enters the engine at a steady rate of 0.3 kg/s. The average velocity of the exhaust gases is 500 m/s relative to the engine. If the engine exhaust effective cross sectional area is 0.3 m 2 estimate the density of the exhaust gases in kg/m 3. Inlet Exit Airflow Combustion A e Thrust 30 / 0.3 / 500 / 0.3
Continuity of flow example 4 30 / 0.3 / 500 / 0.3 By continuity, 30 0.3 30.3 / And, Rearrange,... /
The End