**A restrictive element in the main channel defines the relationship between differential pressure (ΔP) and gas flow:**

F = ƒ (ΔP)

**Figure 1. **Bypass configuration

The gas flow F is usually measured as mass flow Ṁ [mass per time]. It is possible to derive volumetric flow Q [volume per time] from mass flow, if required.

The volumetric flow is equivalent to the mass flow over gas density:

Q = Ṁ / ρ;

From the Ideal Gas Law, the gas density can be inferred as:

ρ = (MP) / (RT)

Definitions:

ΔP - Pressure drop on a flow-restrictive element

Ṁ - Mass flow

Q - Volumetric flow

ρ - Gas density

M - Molar mass

P - Pressure

R - Gas constant

T - Absolute temperature

### Standard Volumetric Flow Qs

Standard volumetric flow is a volumetric flow defined at “standard” pressure (P_{std}) and “standard” temperature (T_{std}). Different manufacturers refer to different standards (for example, T_{std} = 70 °F or 21.1 °C, P_{std} = 14.7 psia or 101.3 kPa).

“Standard cubic centimeters per minute [SCCM]” or “standard liters per minute [slm]” are commonly used units for standard volumetric flow.

For a given gas, volumetric flow at non-standard pressure (P) and non-standard temperature (T) can be found as:

Q = Qs (Ps/P) (T/Ts)

### Laminar and Orifice-Like Flow Restrictive Elements

Preferably, a pressure drop on an orifice increases quadraticly with the flow and a pressure drop on a laminar restrictive element increases linearly with the flow, as shown in Figure 2.

**Figure 2. **Characteristics for laminar (green) and orifice-like (blue) flow restrictive element.

Although the production cost of a laminar restrictive element is higher, it has two benefits in comparison to an orifice-like restrictor:

- increased sensitivity around zero flow
- wider flow measurement range (ΔF2 > ΔF1)

A flow restrictive element is in fact a combination of the two restrictors illustrated above; either the quadratic or linear pressure-from-flow characteristic dominates.

### Barometric Correction

For any thermo-anemometer type differential pressure sensor, such as the LDE/LME/LMI, output signal Vout is proportional to gas density ρ. This is the reason why barometric correction is needed for ΔP measurements.

From Poiseuille's equation, pressure drop on a laminar restrictor ΔP is proportional to mass flow Ṁ and also inversely proportional to gas density ρ:

ΔP ~ [µL/D^{4}] • Ṁ • 1/ρ |
**(2)** |

From (1) and (2)

V_{out} ~ [µL/D^{4}] • Ṁ |
**(3)** |

From Bernoulli’s equation, pressure drop on an orifice-like restrictor ΔP is proportional to mass flow in power of two Ṁ^{2} and also inversely proportional to gas density ρ:

ΔP ~ [1/D^{4}] • Ṁ^{2} • 1/ρ |
**(4)** |

From (1) and (4)

V_{out} ~ [1/D^{4}] • Ṁ^{2} |
**(5)** |

From (3) and (5) follows that the LDE/LME/LMI range of sensors intrinsically do not require barometric correction for mass flow measurements.

Definitions:

ΔP - Pressure drop on a flow-restrictive element

Ṁ^{2} - Mass flow

ρ - Gas density

µ - Gas viscosity

L - Length of a flow-restrictive element

D - Inner diameter of a flow-restrictive element

### Temperature Compensation

An embedded temperature sensor is provided in the LDE/LME/LMI families. Based on the application, the LDE/LME/LMI sensor range can be fully temperature compensated at the factory for differential pressure or for mass flow.

### Bypass Flow

The pressure/flow characteristic of a main channel restrictor is often defined without factoring bypass flow and bypass flow variation from one sample to another. As a result, a smaller flow in the bypass leads to better bypass/main channel split ratio and thus higher accuracy. A sensor’s pneumatic impedance Zp [pressure per flow] defines the amount of flow in the bypass channel. When the impedance is higher, the bypass flow will be lower.

The pneumatic impedance of the LDE/LME/LMI sensors ranges from 10,000s to 100,000s (Pa • s) / (ml).

For instance, if the pneumatic impedance Zp of a 250 Pa LDE sensor is 25,000 (Pa • s) / (ml), the bypass flow at nominal pressure F_{250} can be found as:

F_{250} = ΔP / Zp = 250 Pa / 25,000 (Pa • s) / (ml) = 0.01 mL/s.

## LDE/LME/LMI Features Suitable for Flow Metering

- There is no need for barometric or temperature compensation for mass flow application
- The highest-in-class pneumatic impedance ensures the highest bypass/main channel split ratio and the highest immunity to contamination
- The user can read out an embedded temperature sensor to correct temperatures in volumetric flow application (see section Standard Volumetric Flow Qs)
- Linearized sensor output makes it easy for expanding flow and pressure dynamic range by “cascading” the LDE/LME/LMI families. For instance, it is possible to read out a 50 Pa sensor in parallel with a 500 Pa sensor almost without data irregularities when transitioning from one sensor to another.

## Application Example – 30 SCCM Flowmeter

### Implementation

Shown below is the flowmeter sample meant for air flow measurements up to 30 SCCM at a pressure drop of 250 Pa or less. There are two main parts in the flowmeter – the laminar restrictor and the 250 Pa LME sensor. The restrictor is used as a capillary-like pipe integrated in the LME’s adapter. The adapter offers flexibility and convenience for coupling the flowmeter to a flow source.

**Figure 3a. **3D printed laminar flow restrictor inside transparent adapter sample for the LME sensor.

**Figure 3b. **LME sensor, adapter sample, and assembly.

### Evaluation Result

The plot below shows the evaluation result. The flowmeter’s output, whether digital or analog, increases linearly with the flow. The pressure drop goes no more than 250 Pa at 30 SCCM, and the split ratio between main and bypass flow ranges from 0.0001 to 0.0005.

**Figure 4. **Output vs. flow of LME pressure sensors, t=500 ms (10,000 samples, dt=50 µs).

### Expected Performance

Using the LDE/LME/LMI sensor series in bypass configuration, a SCCM level flowmeter is predicted to deliver better stability, accuracy, and power consumption when compared to other competitor products.

Device |
Flow range [SCCM] |
Vs [V] |
Ws [mW] |
Accuracy [%FS] |
Offset stability [%FS] |
Comp. temp. range [°C] |
Pressure drop [Pa] |

Competitor's high accuracy flowmeters |
50, 100, 200, 400, 750 |
3.3 / 5 |
40 / 65 |
±7 to ±15 |
±0.06 per 1000h |
0 ... 50 |
±25 to ±125 |

LDE/LME/LMI based mass flow meters |
50, 100, 200, 400, 750 |
3.3 / 5 |
12 / 35 |
±1.5 to ±3 |
±0.05 per year |
0 ... 70 |
±25 to ±500 |

This information has been sourced, reviewed and adapted from materials provided by First Sensor AG.

For more information on this source, please visit First Sensor AG.