## Laminar differential pressure flow measurement

Most Alicat mass flow controllers and meters are multivariate instruments that use laminar differential pressure flow measurement technology to obtain a mass flow rate. Here we explain the operating principle behind these devices.

### Step 1: Using Poiseuille’s Equation to calculate volumetric flow

Alicat devices are built with a special flow passage called a laminar flow element (LFE) that converts all turbulent flow into laminar flow. The differential pressure drop across the LFE is measured, and the Poiseuille Equation can then be used to calculate a volumetric flow rate.

**Volumetric flow = (P _{1}-P_{2})πr^{4}/8ηL = K(ΔP)/η**

P_{1} = Static pressure at the inlet; P_{2} = Static pressure at the outlet; r = Hydraulic radius of the restriction; η = Absolute viscosity of the fluid; L = Length of the restriction

The Poiseuille equation shows the linear relationship between volumetric flow rate, differential pressure (ΔP), and absolute viscosity (η). In the simplified equation, *K* is a constant that encompasses the geometric factors of the LFE.

### Step 2: Using volumetric flow to calculate mass flow

Laminar DP mass flow instruments use the volumetric flow rate and a series of density correction factors to calculate a mass flow rate.

**1. Temperature density correction = T _{s}/T_{a}**

T_{a} = Absolute temperature at flow conditions; T_{s} = Absolute temperature at standard conditions (STP)

**2. Pressure density correction= P _{a}/P_{s}**

P_{a} = Absolute pressure at flow conditions; P_{s} = Absolute pressure at standard conditions (STP)

**3. Compressibility = Z _{s}/Z_{a}**

Z_{a} = Compressibility at flow conditions; Z_{s} = Compressibility at standard conditions (STP)

When combined, you can produce an equation that uses volumetric flow, temperature, pressure, and gas compressibility to calculate a mass flow rate.

**Mass flow = (Volumetric flow)****(T _{s}/T_{a})(P_{a}/P_{s})(Z_{s}/Z_{a})**

(T_{s}/T_{a})(P_{a}/P_{s})(Z_{s}/Z_{a}) = Density correction factors

In an Alicat mass flow device, a discrete absolute pressure sensor is placed in the laminar region of the flow stream. This information is sent to the microprocessor and is combined with the data from the discrete absolute temperature sensor to make the correct mass flow calculations.

**I accidentally had the wrong gas selected. What do I do?**

Alicat devices are equipped with pre-loaded tables containing gas properties for 98+ gases, enabling them to continuously calculate viscosity and compressibility.

When using the Poiseuille equation, differential pressure (ΔP) is proportional to (Volumetric flow)(Flow viscosity). Imagine you were flowing hydrogen, but had the instrument set to nitrogen which has ~2x the viscosity of hydrogen. Since ΔP is directly proportional to the viscosity at a given flow rate, you can easily calculate corrected volumetric and mass flow rates. In this case, to obtain the corrected volumetric and mass flow rates for the hydrogen, you would double the indicated values.

## What is standard temperature and pressure (STP)?

Performing the mass flow calculations requires reference to a set of standard temperature and pressure conditions (STP) as indicated by the variables T_{s} and P_{s}. STP is usually defined at sea level conditions, but no single standard exists for this convention. Examples of common STP reference conditions include:

- 0 °C and 1013 mbar
- 25 °C and 14.696 psia
- 0 °C and 760 torr (or mmHg)

## How do I calculate a true mass flow rate?

Although a “true mass flow” is expressed in grams or kilograms per unit of time, it is standard for a mass flow rate to be expressed as a standardized volumetric flow rate. Examples include SLM/SLPM (standard liters per minute), SCCM (standard cubic centimeters per minute), and SCFH (standard cubic feet per hour).

By knowing the STP setting of the device and the density of a particular gas at that STP, it is possible to determine the flow rate in g/min, kg/h, etc.

**True mass flow = (mass flow rate)(gas density at STP or NTP)**

Imagine you are flowing helium at 250 SCCM. The gas density is 0.166g/L and your device is set to STP conditions of 25 °C and 14.696 PSIA.

**True mass flow =**** (250 SCCM)(1 L/1000 CC)(0.1636 g/L) = 0.0409 g/min of Helium**

Alicat laminar differential pressure flow meters and mass flow controllers can display mass flow rates as true mass flow rates. Simply select the desired unit, and your Alicat will perform all the calculations.