CIVL2131 Formulas, Units, Constants and Variables

$$Q=UA$$

Discharge or Volume Flow Rate (m³/s) = Velocity x

$$F=ma$$

Force = mass x acceleration

$$F=\rho Q\Delta U$$

Force = Density (mass/volume) x Flow Rate (m³/s) x Velocity (m/s)

Kinetic Head = Velocity² / 2 * gravity

The height a column of fluid would rise if all it’s kinetic energy were converted into potential energy → Represents the energy a fluid possesses due to its motion.

Pressure Head = Pressure / {rho} * gravity

It’s the pressure that must be overcome to move fluid from one place to another. It is the static head (vertical distance the fluid must be pushed against gravity) + friction head (resistance caused by fluid rubbing against insides of pipes, valves and fittings).

Total Head = Pressure Head + Kinetic Head + Potential Head

$$\tau$$

Shear Stress

The force per unit area exerted on the fluid by the boundary and vice versa is the shear stress.

$$\mu$$

Dynamic Viscosity, (Ns/m²)

A measure of how easily a fluid flows.

$$F/A={mu}\cdot du/dy$$

Newton’s Law of Viscosity

Fluids that don’t obey this are Non-Newtonian fluids, which change their viscosity depending on amount of force or pressure applied. They act like a fluid at rest and solid when struck or squeezed.

$$\sigma$$

Surface Tension (N/m)

Arises from the elasticity of the surface and causes capillary rise in tubes. (You then have to measure the meniscus (bottom of the round bit))

Wetted Length

Equating weight of column of fluid to surface tension force acting on wetted length

Flow Variations

Describes flow variation with time and space

Temporal variation:

• Steady Flows - Velocity and Depth constant with time

• Unsteady Flows - Velocity and Depth vary with time

Spatial variation:

• Uniform - Flow properties constant in flow direction

• Non-Uniform - Flow properties vary in flow direction

Governing Principles

• Continuity

• Momentum

• Energy

Continuity

Conservation of Mass

$${\rho}Q_1={\rho}Q_2$$

or for incompressible fluids / no change in density:

$$Q_{1} = U_{1}A_{1} = U_{2}A_{2} = Q_{2}$$

Momentum

Conservation of Momentum (Newton’s 2^nd Law)

Rate of change of momentum = sum of forces

Momentum flux (or rate at which momentum passes through a cross-section) is the mass flow rate x velocity

$$\rho QU=\left(\rho UA\right)U$$

• The rate of change between cross-sections is $\rho Q\Delta U$

This requires a resultant force F in the direction of motion.
Momentum equation

$$\sum F=\rho Q\Delta U=\rho Q\left(U_2-U_1\right)$$

Energy

Total head (or head), H

Sum of three forms of energy
The energy of a flow is the sum of potential energy, kinetic energy and pressure energy and is conserved if there are no energy losses.

$p_1+\frac12\rho U_1^2+\rho gz_1=const$ ← Energy equation {E}

• Considers all energy inputs, losses and changes

$H=\frac{p_1}{\rho g}+\frac{U_1^2}{2g}+z_1=const$ ← Bernoulli equation {B}

• Simplified version of energy equation that only applies to ideal, frictionless flows where no energy is added or removed