= Pneumotachograph
= Pneumotachometer
{synonym}
A <pneumotachograph> measures and records gas <volumetric flow rate> from the <pressure difference> produced across a resistive flow head. In a mesh-type implementation, two <pressure> taps sample opposite sides of a fine screen, and tubing connects them to a differential <pressure sensor>. The flow head need not contain electronics; the <pressure> transducer and amplifier convert its response to a recorded signal. The manufacturer manual documents this arrangement and limits proportionality to an appropriate flow range.<Front-end Signal Conditioners Owner’s Guide>[Chapter 10, printed p. 149, Using the Spirometer, Figure 10–6]{ref}
For an original quasisteady linear model, let signed volume flow rate be $Q$, head <hydraulic resistance> $R_f$ for the specified gas and conditions, and <pressure>-drop convention $\Delta p=p_{\rm upstream\ tap}-p_{\rm downstream\ tap}$ for a fixed positive direction. Then
$$
\Delta p=R_fQ,\qquad e=e_0+G\Delta p,
\qquad \boxed{Q=\frac{e-e_0}{GR_f}.}
$$
Here $G$ is the <pressure>-to-output gain and $e_0$ the zero-flow offset. Reversing flow reverses the modeled <pressure> difference. The <pressure> taps retain fixed labels when direction changes; renaming them upstream and downstream on each reversal would discard the sign.
A bundle of $n$ identical long circular capillaries gives an illustrative viscous model,
$$
Q=\frac{n\pi r^4}{8\mu\ell}\Delta p,
$$
under the assumptions of fully developed <laminar flow>, negligible compressibility and negligible entrance losses. This <Poiseuille flow> calculation explains one way to obtain an approximately linear <pressure>/flow relation. It is not an exact geometry model for every mesh head. Changes in gas <viscosity>, contamination, leakage or nonlinear inertial losses can change the calibration.
For a numerical signal-chain example, choose $R_f=50\,\mathrm{Pa/(L/s)}$ and $G=0.020\,\mathrm{V/Pa}$. A flow of $0.40\,\mathrm{L/s}$ produces $\Delta p=20\,\mathrm{Pa}$ and an offset-corrected output of $0.40\,\mathrm V$. If this flow lasts two seconds, the transported volume is $0.80\,\mathrm L$. The head also adds a real <pressure> drop to the gas path; a sensitive measurement can therefore alter the system being measured.
<Zero-offset drift of an integrated flow measurement> follows directly from time integration. For inferred flow $\widehat Q=Q+b$ with constant offset $b$,
$$
\widehat{\Delta V}=\int_{t_1}^{t_2}\widehat Q\,dt
=\Delta V+b(t_2-t_1).
$$
An offset of $0.010\,\mathrm{L/s}$ accumulates $0.30\,\mathrm L$ of volume error in thirty seconds. Zeroing the no-flow signal addresses this additive contribution; it does not establish the flow scale. The manual treats zeroing and known-volume calibration separately.<Front-end Signal Conditioners Owner’s Guide>[printed pp. 151–153, Calibrating the Flow Head and Reducing Drift]{ref}
A proposed bench calibration connects a known-displacement piston or calibration syringe, records the output and verifies that its integral recovers the displaced gas volume under the stated conditions. Repeat several piston speeds and both directions. Agreement for one integral does not prove the instantaneous waveform is accurate: <pressure>-tap tubing, amplifier filters and sensor dynamics can distort fast changes while preserving an overall area approximately.
For a measured integral $J=\int(e-e_0)\,dt$ and known displaced volume $V_s$, the linear conversion factor is $K=V_s/J$, so $Q=K(e-e_0)$. State the volume’s <pressure> and temperature conditions, and test the intended rate range before extending that factor. The manufacturer’s known-volume syringe procedure supplies this physical calibration principle.<Front-end Signal Conditioners Owner’s Guide>[printed pp. 151–152, Injecting a known volume and integrating]{ref} Calibration establishes the response of the complete gas-flow measurement chain under its tested conditions.
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