Kinematics of Fluid Flow Kinematics is the science which deals with study of motion of liquids without considering the forces causing the motion. Rate of Flow Quantity of fluid passing through any section (area) per unit time Where: A Cross-sectional area (the area of the surface at right angle to the velocity vector) V Mean or average velocity over the entire sectional area Q Volume flow rate (cfs ft 3 /sec, or m 3 /sec) m o Mass flow rate (slug/sec, or kg/sec) ρ Density of the flow (slug/ft 3, or kg/m 3 ) G Weight flow rate (lb/sec, or kn/sec) γ Specific weight (lb/ft 3, or kn/m 3 ) Volumetric Flow Rate (Q): The volume of water collected (passing through a cross -sectional area) in a certain time. Mass Flow Rate (m o ): Units: SI = kg/s BG = slug/sec 1
Weight Flow Rate (G): Units: SI = kn/s BG = lb/sec Average Velocity (V) Q = VA Classification of Fluid Flow: 2
Average Velocity and Discharge: At time t = 0, the fluid in the pipe was at section 1, and after time Δt, fluid moves to another section of the pipe at a distance = Δx. Volume of the fluid = A Δx Where v is the average velocity along the flow path. The flow in a pipe has a parabolic velocity distribution as shown in a circular pipe of a radius R. To calculate the average velocity in term of maximum velocity, taking r as the radial distance to any local velocity v. Where: Q Flow rate at a section. v The velocity component normal to flow. 3
Average velocity is half of the max. Velocity. 4
Stream line: Is the imaginary line drawn through a flow field in such that the tangent to the line at any point on the line indicates the direction of velocity vector at that point. Stream tube: The concept of stream line can be extended further to form stream surface and stream tube. Stream tube like boundary bounded by a number of stream lines. Conservation of mass- Continuity equation Continuity equation is a mathematical expression for the principle of conservation of mass flow. V= velocity ρ= mass density A= cross-sectional area Mass rate of flow at sec.1= ρ1 V1 A1 (kg./s) Mass rate of flow at sec.2= ρ2 V2 A2 (kg./s) As neither mass is created nor destroyed in the stream tube ρ1 V1 A1= ρ2 V2 A2= m Here m is the mass flow rate in (kg./s), for incompressible, steady flow ρ1= ρ2 V1 A1= V2 A2 = Q Q1= Q2 The discharge Q is called the volumetric flow rate (m 3 /s). Q = V A Weight flow rate. 5
Example: 6
Conservation of Energy- Bernoulli s Equation: In driving Bernoulli s equation, we will assume: 1- Viscous (friction) effects are negligible (Ideal fluid). 2- The flow is steady (constant with respect to time) 3- The equation applies along a streamline. 4- The fluid is incompressible. 5- No energy is added or removed from the fluid. Bernoulli s equation states that the sum of pressure head, kinetic energy and potential energy per unit mass is constant along a streamline. In most cases in closed pipes, all streamlines can be assumed to have the same energy level. (Bernoulli s equation) Where: H= total head of fluid flow. = pressure energy or pressure head. = kinetic energy. = potential energy per unit mass. 7
This is known as Bernoulli s equation, which is an expression of conservation of energy. If the fluid is static, the velocities are zero, and Bernoulli s equation reduces to: Energy line (E.L.): Energy line is a graphical representation of the energy at each section with respect to a chosen datum. Hydraulic Grade Line (H.G.L.): The HGL lies below the energy line by an amount of velocity head at that section. The two lines are parallel for all sections of equal cross- sectional area. 8
Applications of Bernoulli s Equation CASE 1: Flow through orifice. Applying Bernoulli to Points 1, 2 in the fluid: But, p 1 = p 2 = atmospheric pressure = 0 gage pressure. v 1 =0 (Still water surface in a large tank with small outlet pipe) Where h = z 1 z 2 i.e. exit velocity is proportional to the fluid depth. CASE 2: Venturi meter for measuring flow rate. In Bernoulli s Equation if z 1 = z 2, Then: i.e. where the velocity is high, the pressure is low and vise versa. 9
CASE 3: Measuring velocity using Pitot tube: (z 1 = z 2, and V 2 = 0) Where, v 1 = Flow velocity p 1 = Static pressure = p p 2 = Stagnation pressure = p s Case 4: siphon 11