Minor losses

Minor Losses

Minor Viscous Losses in pipe flow occur due to changes in geometry or due to the addition of a component. They can be a significant part in calculating the velocity, pressure, or head in piping systems. Minor losses are generally expressed in terms of a Loss coefficient loss $K_L$ and can be calculated for each individual component of a piping system such as:

• Pipe entrance or exit
• Sudden expansion or contraction
• Bends, elbows, tees and other fittings
• Gradual expansions or contractions.

It is important to note that major losses are not always greater than minor losses.

The minor loss of an individual pipe system component can be calculated using equation 1:

$H_{viscous,minor}=K_L\left(\frac{v^2}{2g}\right)$

(1)

The total minor viscous losses of a system will be the sum of all the individual minor losses as shown in equation 2.

$H_{viscous,minor}=\sum K_{L_i}\left(\frac{v_i^2}{2g}\right)$

(2)

Minor Viscous Losses and The Mechanical Energy Balance

The minor losses of a system are a part of the of the $H_{viscous}$ term in the mechanical energy balance. Start with the steady state mechanical energy balance shown in equation 3:

$\sum _{out}\dot{m}\left(\frac{P}{\rho }+\frac{v^2}{2\alpha }+gz\right)-\sum _{in}\dot{m}\left(\frac{P}{\rho }+\frac{v^2}{2\alpha }+gz\right)={\dot{W}}_{shaft}-{\dot{E}}_{viscous}$

(3)

We will first assume the system the has no ${\dot{W}}_{shaft}$

Then we will manipulate equation 3 to make it in terms of $H_{viscous}$ by divide both sides by $\dot{m}g$ as shown below:

$\sum _{out}\left(\frac{P}{\rho g}+\frac{v^2}{2\alpha g}+z\right)-\sum _{in}\left(\frac{P}{\rho g}+\frac{v^2}{2\alpha g}+z\right)=-\frac{\left({\dot{E}}_{viscous}\right)}{\dot{m}g}$

The right side of this equation can now be rewritten in terms of $H_{viscous}$ by using equations 4 and 5.

$\left(\frac{{\dot{E}}_{viscous}}{\dot{m}}\right)={\hat{E}}_{viscous}$

(4)

$\frac{{\hat{E}}_{viscous}}{g}=H_{Viscous}$

(5)

This will give the mechanical energy balance in terns of $H_{viscous}$ as shown in equation 6

$\sum _{out}\left(\frac{P}{\rho g}+\frac{v^2}{2\alpha g}+z\right)-\sum _{in}\left(\frac{P}{\rho g}+\frac{v^2}{2\alpha g}+z\right)=-H_{viscous}$

(6)

The minor losses of a system will be taken into account in the $H_{viscous}$ term which can be expanded by using equation 7.

$H_{viscous}=(H_{viscous,major}+H_{viscous,minor})$

(7)

Assuming $H_{viscous,major}$ losses are negligible. We can combine equations 6 and 7 as shown in equation 8.

$\sum _{out}\left(\frac{P}{\rho g}+\frac{v^2}{2\alpha g}+z\right)-\sum _{in}\left(\frac{P}{\rho g}+\frac{v^2}{2\alpha g}+z\right)=-H_{viscous,minor}$

(8)

Since $H_{viscous,minor}$ is equal to the $\sum K_{L_i}\left(\frac{v_i^2}{2g}\right)$ then we can rewrite this equation into its final general form as shown in equation 9

$\sum _{out}\left(\frac{P}{\rho g}+\frac{v^2}{2\alpha g}+z\right)-\sum _{in}\left(\frac{P}{\rho g}+\frac{v^2}{2\alpha g}+z\right)=-\sum K_{L_i}\left(\frac{v_i^2}{2g}\right)$

(9)

K, The Loss Coefficient

K is the sum of all of the loss coefficients in the length of pipe and varies with different flow conditions. Some factors that affect the value of K are:

• the exact geometry of the particular component
• the flow of the Reynolds Number (laminar, turbulent)
• proximity to other fittings

Minor Losses in Parallel Pipes

Figure One

Identifying Minor Losses in a system

Let’s consider a piping system that contains a variety of components, some of which are in series and others are in parallel. We can start by first identifying each component that adds minor viscous losses to the system. These components are identified in figure One and are listed below:

• Gate Valves: fully open and (1/4) closed
• Tees - both in line and branch flow
• 90 degree bends
• Tank - expansion and contraction
• Packed Bed - expansion and contraction

Figure Two