Editing Venturi effect (section)

=== Compensation for temperature, pressure, and mass ===

Fundamentally, pressure-based meters measure [[kinetic energy]] density. [[Bernoulli's equation]] (used above) relates this to [[mass density]] and volumetric flow:

<math>\Delta P = \frac{1}{2} \rho (v_2^2 - v_1^2) = \frac{1}{2} \rho \left(\left(\frac{A_1}{A_2}\right)^2-1\right) v_1^2 = \frac{1}{2} \rho \left(\frac{1}{A_2^2}-\frac{1}{A_1^2}\right) Q^2 = k\, \rho\, Q^2</math>

where constant terms are absorbed into ''k''. Using the definitions of density (<math>m=\rho V</math>), [[molar concentration]] (<math>n=C V</math>), and [[molar mass]] (<math>m=M n</math>), one can also derive mass flow or molar flow (i.e. standard volume flow):

<math>\begin{align}\Delta P &= k\, \rho\, Q^2 \\
 &= k \frac{1}{\rho}\, \dot{m}^2 \\
 &= k \frac{\rho}{C^2}\, \dot{n}^2 = k \frac{M}{C}\, \dot{n}^2.
\end{align}</math>

However, measurements outside the design point must compensate for the effects of temperature, pressure, and molar mass on density and concentration. The [[ideal gas law]] is used to relate actual values to [[Standard state|design values]]:

<math>C = \frac{P}{RT} = \frac{\left(\frac{P}{P^\ominus}\right)}{\left(\frac{T}{T^\ominus}\right)} C^\ominus</math>
<math>\rho = \frac{MP}{RT} = \frac{\left(\frac{M}{M^\ominus} \frac{P}{P^\ominus}\right)}{\left(\frac{T}{T^\ominus}\right)} \rho^\ominus.</math>

Substituting these two relations into the pressure-flow equations above yields the fully compensated flows:

<math>\begin{align}\Delta P &= k \frac{\left(\frac{M}{M^\ominus} \frac{P}{P^\ominus}\right)}{\left(\frac{T}{T^\ominus}\right)} \rho^\ominus\, Q^2
 &= \Delta P_{\max} \frac{\left(\frac{M}{M^\ominus} \frac{P}{P^\ominus}\right)}{\left(\frac{T}{T^\ominus}\right)} \left(\frac Q{Q_{\max}}\right)^2\\
 &= k \frac{\left(\frac{T}{T^\ominus}\right)}{\left(\frac{M}{M^\ominus} \frac{P}{P^\ominus}\right) \rho^\ominus} \dot{m}^2
 &= \Delta P_{\max} \frac{\left(\frac{T}{T^\ominus}\right)}{\left(\frac{M}{M^\ominus} \frac{P}{P^\ominus}\right)} \left(\frac{\dot{m}}{\dot{m}_{\max}}\right)^2\\
 &= k \frac{M \left(\frac{T}{T^\ominus}\right)}{\left(\frac{P}{P^\ominus}\right) C^\ominus} \dot{n}^2
 &= \Delta P_{\max} \frac{\left(\frac{M}{M^\ominus}\frac{T}{T^\ominus}\right)}{\left(\frac{P}{P^\ominus}\right)} \left(\frac{\dot{n}}{\dot{n}_{\max}}\right)^2.
\end{align}</math>

''Q'', ''m'', or ''n'' are easily isolated by dividing and taking the [[square root]]. Note that pressure-, temperature-, and mass-compensation is required for every flow, regardless of the end units or dimensions. Also we see the relations:

<math>\begin{align}\frac{k}{\Delta P_{\max}} &= \frac{1}{\rho^\ominus Q_{\max}^2}\\
 &= \frac{\rho^\ominus}{\dot{m}_{\max}^2}\\
 &= \frac{{C^\ominus}^2}{\rho^\ominus\dot{n}_{\max}^2} = \frac{C^\ominus}{M^\ominus\dot{n}_{\max}^2}.
\end{align}</math>