Continuity Equation with Example calculation

by Martin Schlobach | Last updated: Jan 21, 2021 | 0 comments

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In this article, I introduce the Continuity Equation, explain why it is important to know it and show the context of the continuity equation with an example calculation. Since this article refers to the article on volumetric flow rate and flow velocity, I recommend you to read it as well.

Table of Contents

1 Why is it important to know the continuity equation?
2 Continuity equation explained
3 Example calculation Continuity equation
- 3.1 1. example calculation – flow velocity for different pipe cross-sectional areas
4 Conclusion

Why is it important to know the continuity equation?

In a heating or cooling system, the heated or cooled water must be transported from the source (heating system or chiller) through pipes to a transmitter (radiator or cooling unit). Here it is important to ensure that the flow velocity is not too high, otherwise there will be flow noise and excessive pressure losses. Both are characteristics that can lead to increased energy consumption and are unintentional.

For this reason, it is important to observe guide values for flow velocities when calculating the pipe network, because the continuity equation describes the change in flow velocity with a changing pipe cross section and constant volumetric flow rate.

Continuity equation explained

The continuity equation explains a special feature of the behaviour of the volumetric flow rate $\dot V$ and the flow velocity $\omega$ when the pipe diameter changes. The change in cross-section $A$ of a pipe from cross-section A1 to cross-section A2 says that the volume rate of flow remains constant, but the flow velocity is increased the smaller the pipe cross-section becomes. The prerequisite for this is that no liquid is fed in or discharged.

In the presentation of the continuity equation I have illustrated the issue once again. The following applies to the Continuity equation:

${\dot V_1=\dot V_2$

$A_1\cdot \omega_1=A_2\cdot \omega_2}$

$\frac{A_1}{A2}=\frac{\omega_2}{\omega_1}$

Example calculation Continuity equation

In the following example I would like to demonstrate the connection with a small calculation. We take the calculated values from the example calculation – Calculate Volumetric Flow Rate for Radiators. There we have already calculated the pipe cross-sectional area for DN 10 and DN 15 and given a volumetric flow rate of $\dot V$ =33.55l/h.

1. example calculation – flow velocity for different pipe cross-sectional areas

Task

When changing the pipe cross-sectional area from DN 15 to DN 10, the flow velocity in a pipe changes while the volumetric flow rate remains constant. Calculate the flow velocities for DN 15 and DN 10.

Given

volumetric flow rate $\dot V$ = 33,55 l/h = 0,00000932 m³/s
Pipe cross-sectional area $A$ for DN 10 = 0,000123 m²
Pipe cross-sectional area $A$ for DN 15 = 0,0002 m²

Wanted

Flow velocity $\omega$ for DN 15
Flow velocity $\omega$ for DN 10

Solution

Flow velocity $\omega$ for DN 15

$\ \omega=\frac{\dot V}{A} \Longrightarrow \omega=\frac{0,00000932 m^3/s}{0,0002 m^2}=\underline{\underline{0,046 m/s}}$

The flow velocity $\omega$ DN 15 is therefore 0.046 m/s.

Flow velocity $\omega$ for DN 10

$\ \omega=\frac{\dot V}{A} \Longrightarrow \omega=\frac{0,00000932 m^3/s}{0,000123 m^2}=\underline{\underline{0,076 m/s}}$

The flow velocity $\omega$ DN 10 is therefore 0.076 m/s.

Conclusion

As the example calculation shows once again, the flow velocity changes when the cross-section of a pipe changes at constant volumetric flow rate. The flow velocity increases the smaller the pipe cross section becomes. (At the same volume – i.e. no liquid is added or removed).

In the following I have collected some reference values for flow speeds. Clues of the flow velocities in district heating pipes are the following [medium: water] – source Recknagel:

DN 50: 1,0 m/s
DN 100: 1,4 m/s
DN 150: 1,6 m/s
DN 200: 2,1 m/s
DN 300: 2,5 m/s
≥ DN 500: 3,0 m/s

In the heating system, for example, the following guide values for flow velocities apply – source SBZ Monteur:

Main distribution pipes: 0.3 m/s to 1.0 m/s
Radiator connection pipes: 0.5 m/s to 0.8 m/s

I hope I was able to give you an understanding of the Continuity equation with this article. If you have any questions, suggestions or criticism, I look forward to your comments.

Greetings, Martin

Related links and sources:
SBZ Monteur
Recknagel – Paperback for heating and air-conditioning technology
Europa Berufsschule – Kontinuitätsgesetz
Wikipedia – Volumetric Flow Rate
Wikipedia – Flow Velocity

About Me

Hi, my name is Martin and I’m a passionate engineer in the field of buildings technology. Here you can read who I am and why I write this blog.

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