Mecânica de Fluidos Ambiental 2015/2016 Lecture 8 Bernoulli equation Mecânica de Fluidos Ambiental 2015/2016

Mecânica de Fluidos Ambiental 2015/2016 Bernoulli’s equation Let us consider a elemental fixed streamtube (“tubo de corrente”) control volume such as indicated in the figure and an ideal fluid (without viscosity) Using the mass and momentum conservation principles, obtain an equation relating the energy in two sections Mecânica de Fluidos Ambiental 2015/2016

Bernoulli’s Equation – mass conservation Being a streamtube there is only flow across the tops Mecânica de Fluidos Ambiental 2015/2016

Bernoulli’s equation assumptions Ideal fluid (no viscosity) Incompressible flow ( constant) Permanent flow (partial time derivative null) Along a streamline of flow: different streamlines may have different “Bernoulli constants” Mecânica de Fluidos Ambiental 2015/2016

Bernoulli Equation – mass balance 𝑑 𝑑𝑡 𝐶𝑉 𝜌𝑑𝑣 + 𝑚 𝑜𝑢𝑡 − 𝑚 𝑖𝑛 =0≈ 𝜕𝜌 𝜕𝑡 𝑑𝑣+𝑑 𝑚 𝑚 =𝜌𝐴𝑉 𝑎𝑛𝑑 𝑑𝑣=𝐴𝑑𝑠 𝑑 𝑚 =𝑑 𝜌𝐴𝑉 =− 𝜕𝜌 𝜕𝑡 𝐴𝑑𝑠 If A is very small dA is even smaller and we are on a streamline Mecânica de Fluidos Ambiental 2015/2016

Bernoulli’s Equation – momentum balance the linear momentum relation in the streamwise direction: 𝑑 𝐹 𝑠 = 𝑑 𝑑𝑡 − 𝐶𝑉 𝑉𝜌𝑑𝑣 + 𝑚 𝑉 𝑜𝑢𝑡 − 𝑚 𝑉 𝑖𝑛 ≈ 𝜕 𝜕𝑡 𝜌𝑉 𝐴𝑑𝑠+𝑑 𝑚 𝑉 Where Vs = V , because s is itselfe in the streamline direction Mecânica de Fluidos Ambiental 2015/2016

Bernoulli’s Equation – momentum balance If we neglect the shear force on the walls (frictionless flow), the forces are due to pressure and gravity. The streamwise gravity force is due to the weight component of the fluid within the control volume: 𝑑𝐹 𝑠,𝑔𝑟𝑎𝑣 =−𝑑𝑃 sin 𝜃=−𝛾𝐴𝑑𝑠 sin 𝜃=−𝛾𝐴𝑑𝑧 Mecânica de Fluidos Ambiental 2015/2016

Bernoulli’s Equation – momentum balance The pressure force is more easily visualized, in Figure b, by first subtracting a uniform value p from all surfaces. The pressure along the slanted side of the streamtube has a streamwise component that acts not on A itself but on the outer ring of area increase dA. The net pressure force is thus: 𝑑𝐹 𝑆,𝑝𝑟𝑒𝑠𝑠 = 1 2 𝑑𝑝 𝑑𝐴−𝑑𝑝 𝐴+𝑑𝐴 ≈−𝐴𝑑𝑝 Mecânica de Fluidos Ambiental 2015/2016

Mecânica de Fluidos Ambiental 2015/2016 Bernoulli’s Equation 𝑑 𝐹 𝑠 =𝑑 𝐹 𝑠,𝑔𝑟𝑎𝑣 +𝑑 𝐹 𝑠,𝑝𝑟𝑒𝑠𝑠 =−𝛾𝐴 𝑑𝑧−𝐴𝑑𝑝 𝑑 𝐹 𝑠 = 𝜕 𝜕𝑡 𝜌𝑉 𝐴𝑑𝑠+𝑑 𝑚 𝑉 = 𝜕𝜌 𝜕𝑡 𝑉𝐴𝑑𝑠+ 𝜕𝑉 𝜕𝑡 𝜌𝐴𝑑𝑠+ 𝑚 𝑑𝑉+𝑉𝑑 𝑚 −𝛾𝐴 𝑑𝑧−𝐴𝑑𝑝= 𝜕𝜌 𝜕𝑡 𝑉𝐴𝑑𝑠+ 𝜕𝑉 𝜕𝑡 𝜌𝐴𝑑𝑠+ 𝑚 𝑑𝑉+𝑉𝑑 𝑚 The first and last terms on the right cancel by virtue of the continuity relation and we get: −𝛾𝐴 𝑑𝑧−𝐴𝑑𝑝= 𝜕𝑉 𝜕𝑡 𝜌𝐴𝑑𝑠+ 𝑚 𝑑𝑉 Divide what remains by A and rearrange into the final desired relation: 𝜕𝑉 𝜕𝑡 𝑑𝑠+ 𝑑𝑝 𝜌 +𝑉𝑑𝑉+𝑔𝑑𝑧 The Bernoulli’s equation for unsteady frictionless flow along a streamline Mecânica de Fluidos Ambiental 2015/2016

Mecânica de Fluidos Ambiental 2015/2016 Bernoulli Equation 𝜕𝑉 𝜕𝑡 𝑑𝑠+ 𝑑𝑝 𝜌 +𝑉𝑑𝑉+𝑔𝑑𝑧 It is in differential form and can be integrated between any two points 1 and 2 on the streamline: 1 2 𝜕𝑉 𝜕𝑡 𝑑𝑠+ 1 2 𝑑𝑝 𝜌 + 1 2 𝑉 2 2 − 𝑉 1 2 +g 𝑧 2 − 𝑧 1 =0 To evaluate the two remaining integrals, one must estimate the unsteady effect 𝜕𝑉 𝜕𝑡 and the variation of density with pressure. At this time we consider only steady ( 𝜕𝑉 𝜕𝑡 =0) incompressible (constant-density) flow, and equation becomes: Mecânica de Fluidos Ambiental 2015/2016

Hydraulic and Energy Grade Lines A useful visual interpretation of Bernoulli’s equation is to sketch two grade lines of a flow. The energy grade line (“linha de energia”) shows the height of the total Bernoulli constant h0=z+p/+V2(2g). The hydraulic grade line (“linha piezométrica”) shows the height corresponding to elevation and pressure head z+p/ ,that is, the energy grade line minus the velocity head V2/(2g) Mecânica de Fluidos Ambiental 2015/2016

Hydraulic and Energy Grade Lines Figure illustrates the EGL and HGL for frictionless flow at sections 1 and 2 of a duct. The piezometer tubes measure the static pressure head z+p/ and thus outline the HGL. The pitot stagnation-velocity tubes measure the total head h0=z+p/+V2(2g), which corresponds to the EGL. In this particular case the EGL is constant, and the HGL rises due to a drop in velocity. HGL – linha piezométrica EGL – linha de energia Mecânica de Fluidos Ambiental 2015/2016

Mecânica de Fluidos Ambiental 2015/2016 Considerations The Mechanical Energy remains constant along a streamline in steady, incompressible, frictionless flow. Pressure is a form of energy: is the energy (work) necessary for moving a unit of volume from a region with null pressure into a region of pressure P. Inside pipes (pressurised flows) pressure is usually the main form of energy. In liquids the potential energy can be very important. Inside pipes, discharging liquids pressure and kinetic energy are usually the important forms of energy. In external flows pressure and kinetic energy are usually the most important forms of energy and determine the shape of the flow around a body. Mecânica de Fluidos Ambiental 2015/2016

Mecânica de Fluidos Ambiental 2015/2016 Exercise In a domestic water pipe the pressure is typically 6 kg/cm2. If the velocity is 1m/s, how much does the kinetic energy account for the total energy? If whole the pressure energy was transformed into kinetic energy, how much would be the velocity? Where do you expect the energy to be dissipated? Is the Bernoulli applicable in this flow? Mecânica de Fluidos Ambiental 2015/2016

Mecânica de Fluidos Ambiental 2015/2016 Computing the pressure and the kinetic energy: Mecânica de Fluidos Ambiental 2015/2016

Mecânica de Fluidos Ambiental 2015/2016 Chaminé Considere uma chaminé que escoa um gás cuja massa volúmica é 1.1 kgm-3 relacione a velocidade à saída com a altura da chaminé e com a massa volúmica do ar exterior. A equação de Bernoulli só é aplicável se as propriedades do fluido forem uniformes e por isso pode ser aplicada no interior da chaminé ou no exterior, mas não para relacionar pontos do interior com pontos do exterior. A diferença de pressões entre a entrada e a saída da chaminé é determinada pelas condições exteriores: Mecânica de Fluidos Ambiental 2015/2016

Mecânica de Fluidos Ambiental 2015/2016