Fluids - Chapter Notes

Published on: Wed Jun 02 2010

The Density (rho) is mass per unit volume, the unit is kg/m3. [math]displaystyle rho = frac{m}{V}[/math] rho equals m over V [math] m = rho V[/math] m equals rho V Specific Gravity is ratio of the density of the substance to the density of water at 4C. Pressure is force per unit area, when the force is magnitude acting perpendicular to the surface area A. [math]displaystyle P = frac {F}{A}[/Math] Pressure is a scalar with the unit name Pascal, which is N/m2. Pressure due to liquid is , remember the funny looking p is “rho” aka DENSITY. As the pressure increases, density increases as well though so for cases with gas, pressure is [math] displaystyle frac{dP}{dy} = -rho g[/math]. Another way to express is equation is [math] displaystyle P_2 - P_1 = -int_{y_1}^{y_2} rho g dy [/Math] If you can ignore variations in density this can be expressed as: [math] P_2 - P_1 = - rho g(y_2 - y_1) [/math] For an open contain of liquid, you simply add the pressure from the atmosphere [Math] P = P_0 + rho gh[/math] Pressure head: Sometimes the h in Rho g h is called the pressure head. Atmospheric Pressure is 101.3 kPa for 14.7lb/in2. sometimes we use bars, which are 10,000 N/m2. Pressure gauges register gauge pressure. This does not include the atmospheric pressure, so we need to add it in, by adding the pressure of the atmosphere to the gauge. Pascal’s Principle states “If an external pressure is applied to a confined fluid, the pressure at every point within the fluid increases by that amount” [Math] P_{out} = P_{in}[/math] [math] displaystyle frac{F_{out}}{A_{out}} = frac{F_{In}}{A_{In}}[/math] [math] displaystyle frac{F_{out}}{F_{in}} = frac{A_{out}}{A_{In}}[/math] This ratio gives mechanical advantage, Fout/Fin. The buoyant force occur’s because pressure on the bottom surface of a submerged object is greater than the downward pressure on the top of the object. It is equal to the weight of fluid displaced by the object. [math]F_b = m'g = rho Vg[/math] where m’g is the weight of the body of fluid equal to the volume of the original submerged object. When an objects float Fb=mg in general. When an object is partially submerged. [math] displaystyle frac{V_{displaced}}{V_0} = frac {rho_0}{rho_{final}}[/math] Tools Hydrometer – measured specific gravity of a liquid Fluid Dynamics (Hydrodynamics) Streamline (laminar flow) Turbulent flow Mass flow rate equals change in mass over change in time. [math]rho_1A_1v_1 = rho_2A_2v_2[/math] This is the equation of continuity. It reads "initial Density times Area times Velocity equals final Density times Area times Velocity". If density is constant, The equation of continuity can be written as [math]A_1v_1 = A_2v_2[/math] Bernoulli’s Principle “Where the velocity of a fluid is high, the pressure is low, and where the velocity is low, the pressure is high” Bernoulli’s Equation: [math] displaystyle P_1 + frac{1}{2}rho v_1^2 + rho gy_1 = P_2 + frac{1}{2} rho v_2^2+ rho gy_2[/math] or in other words [math] displaystyle P + frac{1}{2}rho v^2 + rho gy[/math] is constant and for both, y is the height of the center of the tube above a fixed reference level. Liquid leaves a spigot with the same speed a freely falling object would attain if dropped from the same height. [math] v_1 = sqrt{2g(y_2 - y_1)}[/math] And if you want to use Bernoulli's principle in the case where there is no significant change in height: [math]displaystyle P_1 + frac{1}{2} rho v_1^2 = P_2 + frac{1}{2} rho v_2^2 [/math]