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## Mass Flow Calculations

If we wish to calculate mass flow instead of volumetric flow, the equation does not change much. The relationship between volume (V ) and mass (m) for a sample of fluid is its mass density (ρ):

Similarly, the relationship between a volumetric flow rate (Q) and a mass flow rate (W) is also the fluid’s mass density (ρ):

Solving for W in this equation leads us to a product of volumetric flow rate and mass density:

A quick dimensional analysis check using common metric units confirms this fact. A mass flow rate in kilograms per second will be obtained by multiplying a mass density in kilograms per cubic meter by a volumetric flow rate in cubic meters per second:

Therefore, all we have to do to turn our general volumetric flow equation into a mass flow equation is multiply both sides by fluid density (ρ):

It is generally considered “inelegant” to show the same variable more than once in an equation if it is not necessary, so let’s try to consolidate the two densities (ρ) using algebra. First, we may write ρ as the product of two square-roots:

Next, we will break up the last radical into a quotient of two separate square roots:

Now we see how one of the square-rooted ρ terms cancels out the one in the denominator of the fraction:

It is also considered “inelegant” to have multiple radicands in an equation where one will suffice, so we will re-write our equation for aesthetic improvement1:

As with the volumetric flow equation, all we need in order to arrive at a suitable k value for any particular flow element is a set of values taken from that real element in service, expressed in whatever units of measurement we desire.

For example, if we had a venturi tube generating a differential pressure of 2.30 kilo-Pascals (kPa) at a mass flow rate of 500 kilograms per minute of naphtha (a petroleum product having a density of 0.665 kilograms per liter), we could solve for the k value of this venturi tube as such:

Now that we know a value of 404 for k will yield kilograms per minute of liquid flow through this venturi tube given pressure in kPa and density in kilograms per liter, we may readily predict the mass flow rate through this tube for any other pressure drop and fluid density we might happen to encounter. The value of 404 for k relates the disparate units of measurement for us:

As with volumetric flow calculations, the calculated value for k neatly accounts for any set of measurement units we may arbitrarily choose. The key is first knowing the proportional relationship between flow rate, pressure drop, and density. Once we combine that proportionality with a specific set of data experimentally gathered from a particular flow element, we have a true equation properly relating all the variables together in our chosen units of measurement.

If we happened to measure 6.1 kPa of differential pressure across this same venturi tube as it flowed sea water (density = 1.03 kilograms per liter), we could calculate the mass flow rate quite using the same equation (with the k factor of 404):

1This re-write is solidly grounded in the rules of algebra. We know that √a√b = √ab, which is what allows us to do the re-write.

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written by Mery@CalcGeek.com, November 19, 2014