Exercise: Identify sources of major losses and approximations

Exercise 1

Identify the major losses in this system. Justify assumptions and any correlations used. Describe methods to find the friction factor:

Image to explain different major losses within a piping system

Major losses in a system are losses that arise due to skin friction. Unlike Minor losses which are any losses that disturb the direction of the flow, major losses are often associated with the main elements of the system such as the tank and the pipes. Major losses can be calculated through the use of the Friction factor either graphically or through corresponding correlations such as the Colebrook-White Equation.

In this problem we draw our control volume around the top of the tanks 1 and 2 and connect them through the pipe. With this control volume we can see we have a total of 3 major losses, one for each tank and one for the pipes. Assuming steady state the mass balance becomes:

$m_{in}=m_{out}$

(1)

$m_{tank1}=m_{tank2}=m_{pipes}$

(2)

And since the mass flow rate is equal to the density of the fluid multiplied by its velocity and area:

$m=\rho VA$

(3)

where V is the velocity, ρ is the density, and A is the area.

$p_{tank1}{V}_{tank1}{A}_{tank1}=p_{tank2}{V}_{tank2}{A}_{tank2}=p_{pipes}{V}_{pipes}{A}_{pipes}$

(4)

Assuming the fluid density is constant then

$V_{tank1}{A}_{tank1}=V_{tank2}{A}_{tank2}=V_{pipes}{A}_{pipes}$

(5)

Since the area of both tank 1 and tank 2 are much larger than that of the pipes, and they must all be equal to each other, this means that the velocity of tanks 1 and 2 are very small in comparison to the velocity of the pipes. Because of this, it is often assumed for the velocities of the tanks to be negligible compared to the pipes when the difference in area between the pipes and the tanks are large. This means that the major losses are mainly occurring in the pipe walls. For this reason, we can neglect the major losses found in the tanks.

The major losses are found in the mechanical energy balance in the viscous loss term:

$H_{viscous}=H_{major}+H_{minor}$

(6)

where major and minor refer to major and minor losses. Since friction is a major loss, it is found in the major term as:

$f_D\left(\frac{Lpipes}{Dpipes}\right)\left(\frac{Vpipe{s}^2}{2g}\right)$

(7)

where $f_D$ is the friction factor, L is the length, D is the diameter, V is the velocity, and g is the gravitational constant. Since we neglect any friction from the tanks since their velocities are small compared to the pipe velocities we have:

$f_{Dpipes}\left(\frac{Lpipes}{Dpipes}\right)\left(\frac{Vpipe{s}^2}{2g}\right)$

(8)

The length of the pipes is found by adding up the total pipe lengths, and if the diameter of the pipes is the same, then the term is a constant. To find the friction factor, methods such as graphically through the use of a moody chart (which involves Reynolds number and the pipe roughness) or analytically through correlations such as the Colebrook-White equation are found here: Friction factor.