Bernoulli’s principle for a perfect fluid without compressibility and viscosity is expressed by Eq. (1).<\/p>\r\n\r\n\r\n\r\n
$$\\frac{v_{2}^{2}-v_{1}^{2}}{2}+g(z_{2}-z_{1})+\\frac{P_{2}-P_{1}}{\u03c1}=0\u30fb\u30fb\u30fb(1)$$<\/p>\r\n
The first term represents kinetic energy, the second term represents potential energy, the third term represents pressure, and the energy of the fluid is the sum of these terms.<\/p>\r\n
It also means that kinetic energy and pressure can be converted into potential energy. This has very important industrial implications.<\/p>\r\n
Because it shows that you can lift a fluid to a higher level if you pump it to a higher pressure.<\/p>\r\n
Due to the location constraints, plants are often structured to be long and vertical structures.<\/p>\r\n
Equipment such as heat exchangers, distillation columns, and agitation vessels may be installed high up in the structure, so it is necessary to transport the fluid to them.<\/p>\r\n
Thanks to Bernoulli’s principle<\/span>, engineers can calculate how much energy and pump lift is needed to lift a fluid to a given height.<\/p>\r\n\r\n\r\n\r\n\r\n\r\n We take the energy balance of the flow in the flow pipe as shown in the above figure.<\/p>\r\n Assuming that the fluid is a perfect fluid, the energy loss associated with the flow can be neglected.<\/p>\r\n Here, we consider each energy (kinetic energy, potential energy, and pressure) at the inlet and outlet of the flow pipe.<\/p>\r\n In the physics of rigid bodies, each energy is Eq. (2)-(4).<\/p>\r\n $$Kinetic\\ Energy=\\frac{1}{2}mv^{2}\u30fb\u30fb\u30fb(2)$$<\/p>\r\n \r\n\r\n<\/p>\r\n $$Potential\\ Energy=mgz\u30fb\u30fb\u30fb(3)$$<\/p>\r\n \r\n\r\n<\/p>\r\n $$Work\\ by\\ pressure (force\\ \u30fb\\ distance)=PA\u30fbx\u30fb\u30fb\u30fb(4)$$<\/p>\r\n where m<\/em> is the mass of a substance, v<\/em> is the velocity of a substance, g<\/em> is the gravitational acceleration, z<\/em> is location of a substance, P<\/em> is pressure, A<\/em> is Area where the force acts, x<\/em> is travel distance.<\/span><\/p>\r\n The units of each equation are kg\u30fbm2\/s2 = N\u30fbm = J (Joule).<\/p>\r\n Basically, if we apply Eq. (2) to (4) to the fluid and take the energy balance at the inlet and outlet, we get the form of Bernoulli’s principle.<\/p>\r\n First of all, the mass of the fluid is expressed by the following equation.<\/p>\r\n $$\u03c1Fdt$$<\/p>\r\n where \u03c1<\/em> is the fluid density, F<\/em> is the volume flow rate , and dt<\/em> is the fluid flow time in a very short time.<\/p>\r\n The volume flow rate F<\/em> can be expressed as the cross-sectional area of the section A<\/em> and the flow velocity v<\/em>.<\/p>\r\n $$F=Av$$<\/p>\r\n The distance x<\/em> traveled by the fluid is given by the flow velocity v<\/em> and the time dt<\/em> of fluid flow.<\/p>\r\n $$x=vdt$$<\/p>\r\n Based on the above relationship, if the energy at the inlet of the flow pipe and the energy at the outlet are conserved and equal, then Eq. (5) holds.<\/p>\r\n $$\\begin{align}&\\frac{1}{2}\u03c1F_{1}dtv_{1}^{2}+\u03c1F_{1}dt\u30fbg\u30fbz_{1}+P_{1}A_{1}\u30fbv_{1}dt\\\\&=\\frac{1}{2}\u03c1F_{2}dtv_{2}^{2}+\u03c1F_{2}dt\u30fbg\u30fbz_{2}+P_{2}A_{2}\u30fbv_{2}dt\u30fb\u30fb\u30fb(5)\\end{align}$$<\/p>\r\n The left side at Eq.(5) shows the energy at the inlet of the flow pipe and the right side shows the energy at the outlet of the flow pipe.<\/p>\r\n Assuming that the fluid is always flowing continuously, Eq. (6) holds because the volume flow rate F<\/em> is equal at all locations.<\/p>\r\n $$F_{1}=F_{2}=v_{1}A_{1}=v_{2}A_{2}\u30fb\u30fb\u30fb(6)$$<\/p>\r\n Converting the volume flow rate F<\/em> in Eq. (5) to v1<\/sub>A1<\/sub><\/em> in Eq. (6) and dividing both sides of Eq. (5) by \u03c1v1<\/sub>A1<\/sub>dt<\/em>, Eq. (7) is formed.<\/p>\r\n $$\\begin{align}&\\frac{1}{2}v_{1}^{2}+gz_{1}+\\frac{P_{1}}{\u03c1}\\\\&=\\frac{1}{2}v_{2}^{2}+gz_{2}+\\frac{P_{2}}{\u03c1}\u30fb\u30fb\u30fb(7)\\end{align}$$<\/p>\r\n If we put the form right side – left side = 0, we get Eq. (1), and Bernoulli’s principle can be derived.<\/p>\r\n $$\\frac{v_{2}^{2}-v_{1}^{2}}{2}+g(z_{2}-z_{1})+\\frac{P_{2}-P_{1}}{\u03c1}=0\u30fb\u30fb\u30fb(1)$$<\/p>\r\n By the way, if you divide both sides of Eq. (7) by the gravitational acceleration g<\/em>, the unit system of the equation becomes m<\/em> (meter).<\/p>\r\n $$\\frac{v_{1}^{2}}{2g}+z_{1}+\\frac{P_{1}}{\u03c1g}=\\frac{v_{2}^{2}}{2g}+z_{2}+\\frac{P_{2}}{\u03c1g}$$<\/p>\r\n In this case, the total energy hydraulic head H<\/em>\u00a0is expressed by Eq. (8).<\/p>\r\n $$H=\\frac{v^{2}}{2g}+z+\\frac{P}{\u03c1g}=constant\u30fb\u30fb\u30fb(8)$$<\/p>\r\n Eq. (8) is the base equation when calculating the pump head.<\/p>\r\n Each term in the equation is referred to as<\/p>\r\n $$Velocity\\ Head\uff1a\\frac{v^{2}}{2g}$$<\/p>\r\n \r\n\r\n<\/p>\r\n $$Positional\\ Head\uff1az$$<\/p>\r\n \r\n\r\n<\/p>\r\n $$Pressure\\ Head\uff1a\\frac{P}{\u03c1g}$$<\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n Eq. (1) and (8) are valid for perfect fluids, which are fluids in which viscosity does not work.<\/p>\r\n However, because of the viscous nature of real world fluids, energy loss due to friction, etc. occurs.<\/p>\r\n So, in order to apply Eq. (8) to a real fluid, we add two terms to Eq. (8) to represent the energy loss.<\/p>\r\n $$H=\\frac{v^{2}}{2g}+z+\\frac{P}{\u03c1g}+h_{f}+h_{l}\u30fb\u30fb\u30fb(9)$$<\/p>\r\n Eq. (9) is used in practice to calculate the pump head.<\/p>\r\n The hf<\/sub><\/em> is called the frictional loss hydraulic head and represents the energy loss due to friction on the pipe wall.<\/p>\r\n The hl<\/sub><\/em> is called the shape loss hydraulic head, and it represents the energy loss that occurs at points where the flow path changes significantly, such as at a sudden expansion or contraction of the pipe or at a bend.<\/p>\r\n Therefore, when calculating the head of the pump, it is necessary to determine the head by adding the energy loss according to Eq. (9).<\/p>","protected":false},"excerpt":{"rendered":" The energy conservation law for fluids is called Bernoulli’s principle in particular. Thanks to Bernoulli’s principle, engineers can calculate how much energy and pump lift is needed to lift a fluid to a given height.<\/p>\n","protected":false},"author":1,"featured_media":71,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[7],"tags":[],"class_list":["post-61","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-fluid-dynamics"],"_links":{"self":[{"href":"https:\/\/chemical-engineering-review.com\/en\/wp-json\/wp\/v2\/posts\/61","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/chemical-engineering-review.com\/en\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/chemical-engineering-review.com\/en\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/chemical-engineering-review.com\/en\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/chemical-engineering-review.com\/en\/wp-json\/wp\/v2\/comments?post=61"}],"version-history":[{"count":13,"href":"https:\/\/chemical-engineering-review.com\/en\/wp-json\/wp\/v2\/posts\/61\/revisions"}],"predecessor-version":[{"id":77,"href":"https:\/\/chemical-engineering-review.com\/en\/wp-json\/wp\/v2\/posts\/61\/revisions\/77"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/chemical-engineering-review.com\/en\/wp-json\/wp\/v2\/media\/71"}],"wp:attachment":[{"href":"https:\/\/chemical-engineering-review.com\/en\/wp-json\/wp\/v2\/media?parent=61"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/chemical-engineering-review.com\/en\/wp-json\/wp\/v2\/categories?post=61"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/chemical-engineering-review.com\/en\/wp-json\/wp\/v2\/tags?post=61"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}Derivation of Bernoulli’s Principle<\/h2>\r\n
![\"\"](\"https:\/\/chemical-engineering-review.com\/en\/wp-content\/uploads\/sites\/2\/2022\/01\/bernoullis-principle1.png\")
<\/p>\r\n<\/p>\r\nExtension to real viscous fluids<\/h2>\r\n