Lecture 4 : Fluid Mechanics and Transport (II) Flashcards
1. Poiseuille's equation ; 2. Pressure drop in a pipe: reynold's number, flow, then calculate pressure drop; 3. Moody chart: friction factor as a function of relative roughness and Reynold's number; 4. Bernoulli's equation and application
What is the equation for velocity of fluid at point r, as a function of ΔP?
u(r) = (ΔP/4μL).(R^2-r^2)
where R is the radial distance (aka radius, from centre of pipe to the top of the pipe) and ΔP is the pressure drop as fluid flows from point A to B
What does Poiseuille’s equation describe?
Poiseulle’s equation describes that the volumetric flow rate of liquid across a pipe cross section is proportional to pressure difference.
What is the Poiseuille’s equation?
v̇ = (π.R^4.ΔP)/8μL
where R is the radial distance
For laminar conditions, what is the relationship between ū (mean velocity) and u_max (maximum velocity)?
ū /u_max = 0.5
*given in equation list
For turbulent flow, what is the equation relating u(r) (velocity at point r) and u_max (maximum velocity)?
given in equation list
u(r) = u_max.(1-(r/R))^1/7
where r = distance of interest from centerline of pipe, R = radial distance (centerline all the way to wall of pipe, aka radius of the pipe)
For turbulent conditions, what is the relationship between ū (mean velocity) and u_max (maximum velocity)?
ū /u_max = 0.82
For laminar flow, what is meant by the friction factor, f?
It is the ratio between shear stress at the wall, σ, to the kinetic energy of fluid per unit volume
Shear stress : opposing force ; kinetic energy : forward energy (due to forward velocity)
What is the equation for kinetic energy?
Kinetic energy = 0.5mv^2,
m = mass ; v = velocity
What is the equation for friction factor, which can be applied to all situations (i.e. whether laminar or turbulent flow)
f = ΔPD / (2L𝜌ū^2)
What is the equation for pressure drop in fully developed laminar flow?
ΔP = 32μūL / (D^2)
What is the equation to represent the relationship between the Fanning friction factor, f, and the Reynold’s number under laminar conditions?
f = 16/Re
given in eqn list
What is the equation to represent the relationship between the Fanning friction factor, f, and the Reynold’s number under turbulent flow?
refer to notes (too messy to type)
1/sqrt(f) = -3.6log((6.9/Re)+(εD/3.7)^1.1)
