Engineering Script

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Description	<h2 style="margin: 0px; position: relative; font-variant-numeric: normal; font-variant-east-asian: normal; font-variant-alternates: normal; font-kerning: auto; font-optical-sizing: auto; font-feature-settings: normal; font-variation-settings: normal; font-variant-position: normal; font-weight: bold; font-stretch: normal; font-size: 14px; line-height: normal; font-family: "Trebuchet MS", Trebuchet, sans-serif; color: rgb(0, 0, 0);"><b><span style="font-family: Consolas;">Bernoulli Equation (SI unit)</span></b></h2><h2 style="margin: 0px; position: relative; font-variant-numeric: normal; font-variant-east-asian: normal; font-variant-alternates: normal; font-kerning: auto; font-optical-sizing: auto; font-feature-settings: normal; font-variation-settings: normal; font-variant-position: normal; font-weight: bold; font-stretch: normal; font-size: 14px; line-height: normal; font-family: "Trebuchet MS", Trebuchet, sans-serif; color: rgb(0, 0, 0);"><b><span style="font-family: Consolas;"><br></span></b></h2><h2 style="margin: 0px; position: relative; font-variant-numeric: normal; font-variant-east-asian: normal; font-variant-alternates: normal; font-kerning: auto; font-optical-sizing: auto; font-feature-settings: normal; font-variation-settings: normal; font-variant-position: normal; font-weight: bold; font-stretch: normal; font-size: 14px; line-height: normal; font-family: "Trebuchet MS", Trebuchet, sans-serif; color: rgb(0, 0, 0);"><p><span style="font-family: Consolas;">Bernoulli's law is an important principle used in fluid mechanics, describing the relationship between the velocity and pressure of a fluid. According to this law, as speed increases, pressure decreases, and as speed decreases, pressure increases. This principle is expressed by the following formula:</span></p><span style="font-family: Consolas;"><span style="color: rgb(128, 1, 128);"><b>Energy basis   : P1/ρ1 + u1^2/2 + z1 = P2/ρ2 + u2^2/2 + z2 [Pa, kg/m3, m/s, m]<br>Pressure basis : P1 + ρ1u1^2/2 + ρ1gz1 = P1 + ρ2u2^2/2 + ρ2gz2 [Pa, kg/m3, m/s, m]<br>Head basis     : P1/(ρ1g) + u1^2/2g + z1 = P2/(ρ2g) + u2^2/2g + z2 [Pa, kg/m3, m/s, m]<br></b></span> <br>Where,<br>ρ = fluid density (kg/m3)<br>g = acceleration due to gravity (9.8 m/s)<br>P1 = pressure at elevation-1 (Pa)<br>u1 = velocity at elevation-1 (m/s)<br>h1 = height of elevation-1 (m)<br>P2 = pressure at elevation-2 (Pa)<br>u2 = velocity at elevation-2 (m/s)<br>h2 = height at elevation-2 (m)</span></h2><h2 style="margin: 0px; position: relative; font-variant-numeric: normal; font-variant-east-asian: normal; font-variant-alternates: normal; font-kerning: auto; font-optical-sizing: auto; font-feature-settings: normal; font-variation-settings: normal; font-variant-position: normal; font-weight: bold; font-stretch: normal; font-size: 14px; line-height: normal; font-family: "Trebuchet MS", Trebuchet, sans-serif; color: rgb(0, 0, 0);"><span style="font-family: Consolas;"><br></span></h2><h2 style="margin: 0px; position: relative; font-variant-numeric: normal; font-variant-east-asian: normal; font-variant-alternates: normal; font-kerning: auto; font-optical-sizing: auto; font-feature-settings: normal; font-variation-settings: normal; font-variant-position: normal; font-weight: bold; font-stretch: normal; font-size: 14px; line-height: normal; font-family: "Trebuchet MS", Trebuchet, sans-serif; color: rgb(0, 0, 0);"><span style="font-family: Consolas;">Example of Bernoulli Equation </span><b><span style="font-family: Consolas;">(SI unit)</span></b></h2><h2 style="margin: 0px; position: relative; font-variant-numeric: normal; font-variant-east-asian: normal; font-variant-alternates: normal; font-kerning: auto; font-optical-sizing: auto; font-feature-settings: normal; font-variation-settings: normal; font-variant-position: normal; font-weight: bold; font-stretch: normal; font-size: 14px; line-height: normal; font-family: "Trebuchet MS", Trebuchet, sans-serif; color: rgb(0, 0, 0);"><b><span style="font-family: Consolas;"><br></span></b></h2><h2 style="margin: 0px; position: relative; font-variant-numeric: normal; font-variant-east-asian: normal; font-variant-alternates: normal; font-kerning: auto; font-optical-sizing: auto; font-feature-settings: normal; font-variation-settings: normal; font-variant-position: normal; font-stretch: normal; line-height: normal;"><p style="color: rgb(0, 0, 0); font-family: "Trebuchet MS", Trebuchet, sans-serif; font-size: 14px; font-weight: bold;"><span style="font-family: Consolas;">Water at 25 degC enters a pipe at the velocity of 1 m/s and a pressure of 101.3 kPa. The pipe has a constant diameter and friction loss is negligible. A change in the pipe’s elevation changes the downstream pressure in the pipe to 2.1 atm. Most nearly, what is the elevation change?</span></p><p style="color: rgb(0, 0, 0); font-family: "Trebuchet MS", Trebuchet, sans-serif; font-size: 14px; font-weight: bold;">P1 / (ρ1 * g) + u1^2 / 2g + z1 = P2 / (ρ2 * g) + u2^2 / 2g + z2<br></p></h2><h2 style="margin: 0px; position: relative; font-variant-numeric: normal; font-variant-east-asian: normal; font-variant-alternates: normal; font-kerning: auto; font-optical-sizing: auto; font-feature-settings: normal; font-variation-settings: normal; font-variant-position: normal; font-stretch: normal; line-height: normal;"><div style="color: rgb(0, 0, 0); font-family: "Trebuchet MS", Trebuchet, sans-serif; font-size: 14px; font-weight: bold;"><div>u1 = u2, ρ1 = ρ2</div><div>P1 / (ρ1 * g) + z1 = P2 / (ρ2 * g) + z2</div><div>101.3 kPa = 101,300 Pa</div><div>2.1 atm = 212,730 Pa</div><div>z2 - z1 = (P1 - P2) / ρ * g = (101,300 Pa - 212,730 Pa)/(997 kg/m3 * 9.81 m/sec) = -11.4 m</div><div><b style="font-family: Consolas; color: inherit;"><br></b></div><div><b style="font-family: Consolas; color: inherit;">Bernoulli Equation (Imperial unit)</b><br></div></div><div style="color: rgb(0, 0, 0); font-family: "Trebuchet MS", Trebuchet, sans-serif; font-size: 14px; font-weight: bold;"><br></div><div style="color: rgb(0, 0, 0); font-family: "Trebuchet MS", Trebuchet, sans-serif; font-size: 14px; font-weight: bold;"><b><span style="color: rgb(128, 1, 128); font-family: Consolas;">Energy basis   : P1gc/ρ1 + u1^2/2 + gz1 = P2gc/ρ2 + u2^2/2 + gz2 [lbf/ft2, lbm/ft3, ft/s, ft, 32.2 lbm-ft/lbf-sec2, 32.2 ft/sec2]</span></b><br></div></h2><h2 style="margin: 0px; position: relative; font-variant-numeric: normal; font-variant-east-asian: normal; font-variant-alternates: normal; font-kerning: auto; font-optical-sizing: auto; font-feature-settings: normal; font-variation-settings: normal; font-variant-position: normal; font-weight: bold; font-stretch: normal; font-size: 14px; line-height: normal; font-family: "Trebuchet MS", Trebuchet, sans-serif; color: rgb(0, 0, 0);"><b><span style="color: rgb(128, 1, 128); font-family: Consolas;">Pressure basis : P1 + ρ1u1^2/2gc + ρ1gz1/gc = P1 + ρ2u2^2/2gc + ρ2gz2/gc [lbf/ft2, lbm/ft3, ft/s, ft, 32.2 lbm-ft/lbf-sec2, 32.2 ft/sec2]<br>Head basis     : P1gc/(ρ1g) + u1^2/2g + z1 = P2gc/(ρ2g) + u2^2/2g + z2 [lbf/ft2, lbm/ft3, ft/s, ft, 32.2 lbm-ft/lbf-sec2, 32.2 ft/sec2]</span></b><div><span style="font-family: Consolas;"><br></span></div><div><span style="color: rgb(68, 68, 68); font-family: Consolas; font-size: 15px; font-weight: 400;">Where,<br>ρ = fluid density (lbm/ft3)<br>g = acceleration due to gravity (32.2 ft/sec2)<br>P1 = pressure at elevation-1 (lbf/ft2, = psia * 144 in2/ft2)<br>u1 = velocity at elevation-1 (ft/s)<br>h1 = height of elevation-1 (ft)<br>P2 = pressure at elevation-2 (lbf/ft2, = psia * 144 in2/ft2)<br>u2 = velocity at elevation-2 (ft/s)<br>h2 = height at elevation-2 (ft)</span></div><div><span style="color: rgb(68, 68, 68); font-family: Consolas; font-size: 15px; font-weight: 400;"><br></span></div></h2><h2 style="margin: 0px; position: relative; font-variant-numeric: normal; font-variant-east-asian: normal; font-variant-alternates: normal; font-kerning: auto; font-optical-sizing: auto; font-feature-settings: normal; font-variation-settings: normal; font-variant-position: normal; font-weight: bold; font-stretch: normal; font-size: 14px; line-height: normal; font-family: "Trebuchet MS", Trebuchet, sans-serif; color: rgb(0, 0, 0);"><span style="font-family: Consolas;">Example of Bernoulli Equation </span>(Imperial unit)</h2><h2 style="margin: 0px; position: relative; font-variant-numeric: normal; font-variant-east-asian: normal; font-variant-alternates: normal; font-kerning: auto; font-optical-sizing: auto; font-feature-settings: normal; font-variation-settings: normal; font-variant-position: normal; font-weight: bold; font-stretch: normal; font-size: 14px; line-height: normal; font-family: "Trebuchet MS", Trebuchet, sans-serif; color: rgb(0, 0, 0);"><div style="color: rgb(68, 68, 68); font-family: Consolas; font-size: 15px; font-weight: 400;"><p>The pump supplies water at a flow rate of 4.0 ft3/sec from an open reservoir through a horizontal 8 inch pipe. the head loss from the reservoir to the suction of the pump is 2 ft-lbf/lbm. and the discharge is at 90 psi in to 6 inch pipe (5.761 inch). The pump has an efficiency of 65%. The head that must be delivered by the pump to water is?</p></div><span style="color: rgb(68, 68, 68); font-family: Consolas; font-size: 15px; font-weight: 400;">P1gc/(ρ1g) + u1^2/2g + z1 + hpump = P2gc/(ρ2g) + u2^2/2g + z2 + hf<br>ρ1 = ρ2 = 62.4 lbm/ft3<br>u1 = 0<br>z1 = 0<br>z2 = 0<br>hf = 0<br>P2 - P1 = 90 psi = 90 lbf/in2 = 12,960 lbf/ft2<br>6 inch pipe diameter = 3.14(5.761 inch (1 ft / 12 inch)/2)^2 = 0.181 ft2<br>u2 = 4.0 ft3/sec / 0.181 ft2 = 22.1 ft/sec<br>gc = 32.2 lbm-ft/lbf-sec2, <br>g = 32.2 ft/sec2<br>hpump = (P2-P1)gc/(ρg) + u2^2/2g = 12,960 lbf/ft2 / 62.4 lbm/ft3 + 22.1^2 / (2 * 32.2) = 215 ft</span></h2><h2 style="margin: 0px; position: relative; font-variant-numeric: normal; font-variant-east-asian: normal; font-variant-alternates: normal; font-kerning: auto; font-optical-sizing: auto; font-feature-settings: normal; font-variation-settings: normal; font-variant-position: normal; font-weight: bold; font-stretch: normal; font-size: 14px; line-height: normal; font-family: "Trebuchet MS", Trebuchet, sans-serif; color: rgb(0, 0, 0);"><span style="color: rgb(68, 68, 68); font-family: Consolas; font-size: 15px; font-weight: 400;"><br></span><blockquote style="color: rgb(68, 68, 68); font-family: Consolas; font-size: 15px; font-weight: 400; border: none; margin-bottom: 0px; margin-left: 40px; padding: 0px;"><div></div></blockquote></h2><p></p>