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Fluid Mechanics - Law of Continuity

Any fluid moving through a pipe obeys the Law of Continuity, which states that the product of average velocity (v), pipe cross-sectional area (A), and fluid density (ρ) for a given flow stream must remain constant:
 

Fluid continuity is an expression of a more fundamental law of physics: the Conservation of Mass. If we assign appropriate units of measurement to the variables in the continuity equation, we see that the units cancel in such a way that only units of mass per unit time remain:

 

This means we may define the product ρAv as an expression of mass flow rate, or W:

 

In order for the product  to differ between any two points in a pipe, mass would have to mysteriously appear and disappear. So long as the flow is continuous (not pulsing), and the pipe does not leak, it is impossible to have different rates of mass flow at different points along the flow path without violating the Law of Mass Conservation. The continuity principle for fluid through a pipe is analogous to the principle of current being the same everywhere in a series circuit, and for equivalently the same reason1.

We refer to a flowing fluid as incompressible if its density does not substantially change2. For this limiting case, the continuity equation simplifies to the following form:



Examining this equation in light of dimensional analysis, we see that the product Av is also an expression of flow rate:


 

Cubic meters per second is an expression of volumetric flow rate, often symbolized by the variable Q:


 

The practical implication of this principle is that fluid velocity is inversely proportional to the cross-sectional area of a pipe. That is, fluid slows down when the pipe’s diameter expands, and visa-versa. We see this principle easily in nature: deep rivers run slow, while rapids are relatively shallow (and/or narrow).

For example, consider a pipe with an inside diameter of 8 inches (2/3 of a foot), passing a liquid flow of 5 cubic feet per minute. The average velocity (v) of this fluid may be calculated as follows:

 

 

Solving for A in units of square feet:

 

 

Now, solving for average velocity v:

 
 

1In an electric circuit, the conservation law necessitating equal current at all points in a series circuit is the Law of Charge Conservation.

2Although not grammatically correct, this is a common use of the word in discussions of fluid dynamics. By definition, something that is “incompressible” cannot be compressed, but that is not how we are using the term here. We commonly use the term “incompressible” to refer to either a moving liquid (in which case the actual compressibility of the liquid is inconsequential) or a gas/vapor that does not happen to undergo substantial compression or expansion as it flows through a pipe. In other words, an “incompressible” flow is a moving fluid whose ρ does not substantially change, whether by actual impossibility or by circumstance.


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