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ÓBUDA UNIVERSITY. Heat and Flow Technology II. Use only inside. Dr. Ferenc Szlivka Professor. Bernoulli’s equation applications Chapter 6. Fluid energy per unit mass. Restrictions on Bernoulli ’s equation. Restrictions are the following : - steady flow
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ÓBUDA UNIVERSITY Heat and Flow Technology II. Use only inside Dr. Ferenc Szlivka Professor Dr. Szlivka: Heat and Flow Technology II_6
Bernoulli’s equation applications Chapter 6. Dr. Szlivka: Fluid Mechanics 6.
Restrictions on Bernoulli’s equation Restrictions are the following: - steady flow - irrotational flow or the two point are on the same streamline - the acting force has a potencial function „U” (irrotational force field, rot g =0) - the density is constant. t is valid only for incompressible fluids since the density of the fluid is assumed to be the same at the two sections of interest - there can be no friction and no energy transfer into the fluid There can be no mechanical devices between the two sections of interest which would add or remove energy from the system, since the equation states that the total energy in the fluid is constant.
PROCEDURE FOR APPLYING BERNOULLI'S EQUATION • 1. Decide which items are known and what is to be found. • 2. Choose a proper coordinate system for writing the potential function of U. • 3. Decide which two sections in the system will be used when writing Bernoulli's equation. One section is chosen for which much data is known. The second is usually the section at which something is to be calculated. • 4. Write Bernoulli's equation for the two selected sections in the system. • 5. Solve the equation algebraically for the desired term. • 6. Substitute known quantities and calculate the result. Dr. Szlivka: Fluid Mechanics 6.
Outflow from a tank, Torrichelli‘s theorem data: Questions: a./ Calculate the velocity if the pressure is b./ Calculate the velocity if the pressure is The tank is open to the atmosphere.
Solution a./ Torrichelli’s theorem b./
Siphon Questions: a./ How big is the out coming flow velocity ? b./ Draw the pressure distribution along the piston!
Solution Between "O" and "F" points writing the Bernoulli’s equation
Solutions A2 A1 Between "O" and „A1" points Between "O" és „A2" points Between "O" és „B" points Between "O" és „C" points Between "O" és „D" points Between "O" és „E" points
Pressure distribution at different „H” Between "O" and „A1" points Between "O" and „A2" points Between "O" and „B" points Between "O" and „C" points Between "O" and „D" points Between "O" and „E" points
Pressure distribution at different „H” r g Hmax Pv =4 kPa
data: Venturi tube Question: Calculate the volume flow rate flowing through the Venturi tube! Dr. Szlivka: Fluid Mechanics 6.
Solution: Calculate the pressure difference between the point 1 and 2 showed by the „U” tube. Use de hydrostatic equation!
Solution: Apply the Bernoulli equation in the flowing fluid between the points 1 and 2.
Solution: Continuity equation for sections 1and 2.
Solution: The volume flow rate is proportional with the square root of pressure difference showed by the „U” tube.
The figure shows a vortex on the water surface. The nucleus rotates like a rigid body with an constant angular velocity. Questions: a./ Calculate the water surface function! b./ How big is the circulation around a circle ? c./ Calculate the depth of the center!
Solution: Apply the Bernoulli equation in the swirl in a standing coordinate system between the points "1'' and "2". The point 2 is in infinite where r→∞.
Solution: In the nucleus in a rotating coordinate system between the points „3'‘ (the deepest point) and „4".
Solution: On the border of the nucleus and swirl the „z” coordinates are the same in the both expressions. On the border of the nucleus and swirl the velocities are the same in the both expressions.
Solution: Calculation of Circulation b./ Integrate along the circumstances of the circle rn c./ The coordinate of the deepest point (z0).
Hurricane Substitute the „z” with term.
Type of Fans (Blowers, Ventilator) Radial flow fan semi axial flow fan Axial flow fan
Working principe of the fans Total pressure, static pressure , dynamic pressure On the left hand side is the „ " the dynamic pressure, The "p" is the static pressure, "pt", is the total pressure, the sum of the two mentioned terms.
The working Principe of the fans Usefull power Total pressure difference
Euler’s turbine equationt The "v" is the absolute, "w„ is the relative and "u„ is the tangential velocity. Write the Bernoulli’s equation between the point "1„ at the incoming section and point "2" at the outgoing section in a rotating coordinate system.
Euler’s turbine equationt The circulation calculation around the outside circumference. The circulation calculation around the inside circumference. The circulation generated by the impeller
Euler’s turbine equationt using for pumps 2 1 The total ideal pressure head
Radial flow fan ideal characteristic curves Backward curved vane radial flow fan ideal characteristic curve Radial vane radial flow fan Ideal characteristic curve Forward curved vane radial flow fan Ideal characteristic curve
Backward curved vane radial flow fan effective characteristic curve
Forward curved vane radial flow fan effective characteristic curve
Simple pump, rotating „S” tube The "S" tube is rotating around a vertical axes with a constant angular velocity, "w". The tube works like a simple pump. Data: Questions: a./ How big is the volume flow rate coming out from the tube during the rotation? b./ How big is the power for rotating the tube, in ideal case?
Solution: From the standing coordinate system the flow is unsteady. In the rotating coordinate system the flow is steady, but rotational. Respect to the Kelvin’s vortex low we can use the Bernoulli’s equation in the rotating system. 2 Using the equation between the points 0 and 2, to consider the centrifugal potential function. 0 velocity potential velocity potential 0 2
Solution b/. 2 We neglect the friction forces and calculate only with the kinetic and potential energy change in the standing system. The energy change of a unit mass: 0
Unsteady flow calculation for an outflow from a tank data: 1 2
Unsteady outflow from a tank Questions: a./Calculate the steady velocity! b./ Calculate the velocity and the cancellation functions in the pipe! c./ How big is the time when the velocity approaches the steady velocity? d./ Draw the pressure distribution along the pipe in different time! értéknél!
Unsteady outflow from a tank a./ Steady solution: 1 2
Unsteady outflow from a tank b./ Unsteady solution: Unsteady Bernoulli’s equation 1 2 The first term is the integral of the local acceleration in a given instant along the points "1" és "2".
Unsteady outflow from a tank b./ The continuity equation for unsteady flow when density is constant 1 2 The „a” is the local acceleration in the point "1" and "2„. When the cross section area is constant the acceleration must be also constant in a given instant. The acceleration is constant along the pipe in a given instant.
Unsteady outflow from a tank b./ Unsteady solution: Unsteady Bernoulli’s equation 1 2
Unsteady outflow from a tank The solution of equation 1 Define: 2
