Elementary Fluid Dynamics The Bernoulli Equation CEE 311

Elementary Fluid Dynamics: The Bernoulli Equation CEE 311

Streamlines Steady State

The Bernoulli Equation us first derive • Let the Bernoulli equation, which is one of the most well-known equations of motion in fluid mechanics, and yet is often misused. It is thus important to understand its limitations, and the assumptions made in the derivation. • The assumptions can be summarized as follows: ü Inviscid flow (ideal fluid, frictionless) ü Steady flow (unsteady Bernoulli will not be discussed in this course) ü Applied along a streamline ü Constant density (incompressible flow) ü No shaft work or heat transfer equation

Bernoulli Along a Streamline z (eqn 2. 2) Separate acceleration due to gravity. Coordinate system may be in any orientation! Component of g in s direction Note: No shear forces! Therefore flow must be frictionless. Steady state (no change in p wrt time) y x

i Along a Streamli ne Can we eliminate the partial derivative? chain rule Write acceleration as derivative wrt s 0 (n is constant along streamline) and

Integrate F =ma Along a Streamline Eliminate ds Now let’s integrate… But density is a function of ____. pressure If density is constant… Along a streamline

Bernoulli Equation • Assumptions needed for Bernoulli Equation äInviscid (frictionless) äSteady äConstant density (incompressible) äAlong a streamline Apply at two points along • Eliminate the constant in a streamline. the Bernoulli equation? ____________ Mechanical energy to thermal energy ________ Heat transfer, shaft work

Bernoulli Equation The Bernoulli Equation is a statement of the conservation of __________ Mechanical Energy p. e. k. e. Pressure head Piezometric head Elevation head Velocity head Total head

Example 1

Example 2

Example 2 -29. 9 kpa

Example 3

Example 3

The Energy Equation • The energy equation is more general than the Bernoulli equation, because it allows for (1) friction, (2) heat transfer, (3) shaft work, and (4) viscous work (another frictional effect). • Where Ws is the shaft work , h L , called the head loss, • In the absence of these two terms, the energy equation is identical to the Bernoulli equation • We must remember however that the

The Energy Equation

The Energy Equation

Example 1

Example 1 +

Example 2

Example 2 m/s

Example 2 p. L = 1000 x 9. 81 x (-6. 97) = -68. 4 k. N/m 2 (Ans. )

Example 3 Pump draws water from reservoir (A) and lifts it to a higher reservoir (B), as shown below, the head loss from A to the pump = 4 v 2/2 g and the head loss from the pump to B = 7 v 2/2 g. compute the pressure head the pump must deliver. The pressure head at the inlet of pump = -6 m. B 3 A 1 2

Example 3 3 1 2