Background 1. Energy conservation equation If there is no friction

2. If there is frictional loss, then In many cases Background

Q. Where are all frictional loss can occur ? in pipe, in valves, joints etc First focus on pipe friction In pipe, Can we relate the friction to other properties ? Background

Example for general case: At the normal operating condition given following data 0 gauge pressure Example What should be the pressure at inlet ?

Solution : taking pressure balance Example (continued) For pipe, Force balance Hence we can find total pressure drop

We have said nothing about fluid flow properties However, Normally we do not know the Usually they depend on flow properties and fluid properties Flow properties Empirical

In general we want to find  f is a measure of frictional loss higher f implies higher friction This is Fanning-Friction factor f f Friction Factor: Definition

So we write Friction factor This is for pipe with circular cross section

Here f is function of other parameters For laminar flow, don’t worry about f, just use For turbulent flow, Is it possible to get expression for shear ? Friction factor: Turbulent Flow Using log profile

Because original equation Equation for Friction Factor

This is equivalent of laminar flow equation relating f and Re (for turbulent flow in a smooth pipe) Equation for Friction Factor

Friction Factor: Laminar Flow For laminar flow

Use of f is for finding effective shear stress and corresponding “head loss” or “ pressure drop” In the original problem, instead of saying “normal operating condition” we say Pressure drop using Friction Factor Laminar or turbulent?

For turbulent flow We can solve for f, once you know f, we can get shear Pressure drop using Friction Factor Once you know shear, we can get pressure drop If flow is laminar, ( i.e. Re < 2300 ), we use

And original equation becomes, In above equation the value of f can be substitute from laminar and turbulent equation Laminar flow – straight forward Turbulent flow – iterative or we can use graph Pressure drop using Friction Factor 0 gauge pressure

Determination of Q or D Given a pipe (system) with known D and a specified flow rate (Q ~ V), we can calculate the pressure needed i.e. is the pumping requirement We have a pump: Given that we have a pipe (of dia D), what is flow rate that we can get? OR We have a pump: Given that we need certain flow rate, of what size pipe should we use?

Determination of Q or D We have a pump: Given that we have a pipe (of dia D), what is flow rate that we can get? To find Q i.e. To find average velocity (since we know D) Two methods: (i) Assume a friction factor value and iterate (ii) plot Re vs (Re 2 f) Method (i) Assume a value for friction factor Calculate V av from the formula relating  P and f Calculate Re Using the graph of f vs Re (or solving equation), re-estimate f; repeat

Determination of Q or D Method (ii) From the plot of f vs Re, plot Re vs (Re 2 f) From the known parameters, calculate Re 2 f From the plot of Re vs (Re 2 f), determine Re Calculate V av

We take original example, assume we know p, and need to find V and Q Let us say Iteration 1: assume f = gives V = 1.73m/s, Re = 3.5x10 5, f = Iteration 2: take f = gives V = 1.15m/s, Re = 2.1x10 5, f = Iteration 3: take f = gives V = 1.04 m/s, Re = 2.07x10 5, f =

If flow is laminar, you can actually solve the equation

If you are given pressure drop and Q, we need to find D

Iteration 1: Assume f = 0.01 Iteration 2: take f = and follow the same procedure Solving this approximately (how?), we get