Understanding how gas flow condition inside a pipe influences the response of a flow sensor

Question

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1. What the problem is really asking

A small MEMS mass‑flow sensor is glued on a PCB that sits in the centre of a 5 cm‑diameter, 40 cm‑long PVC pipe.
Air is drawn through the pipe by a fan placed at end A (the “suction” side). The PCB is about 4 cm from the opposite end B (the “inlet” side).

When the lower half of the pipe’s inlet (the region right at B) is covered with tape, the sensor works normally.
When that tape is removed (the inlet is completely open), the sensor stops giving any output.

Why does covering half of the inlet change the sensor’s response?
Is the effect only due to a change in the average air speed, or does something else (velocity profile, turbulence, pressure distribution, etc.) also play a role?

The answer must explain, step by step, the fluid‑dynamic reasons why the sensor sees a usable flow only when the lower half of the inlet is blocked.

2. Step‑by‑step analysis

2.1. Geometry and basic numbers

Quantity	Symbol	Value (typical)
Pipe inner diameter	(D)	5 cm = 0.050 m
Pipe cross‑sectional area	(A)	(\displaystyle A=\frac{\pi D^{2}}{4}=1.96\times10^{-3}\;{\rm m^{2}})
Length of pipe	(L)	0.40 m
Distance of sensor from inlet B	(x)	0.04 m (≈ 4 cm)
Air density (room temperature)	(\rho)	1.2 kg m⁻³
Dynamic viscosity of air	(\mu)	(1.8\times10^{-5}) Pa·s
Fan‑generated volume flow (order of magnitude)	(Q)	1 L s⁻¹ = (1.0\times10^{-3}) m³ s⁻¹ (typical small brushless fan)
Corresponding average velocity (unblocked pipe)	(v_{\text{avg}})	(\displaystyle v_{\text{avg}}=\frac{Q}{A}\approx0.51\;{\rm m\,s^{-1}})

Note: The exact flow rate is not given, but a typical tabletop fan produces a few hundred cm³ s⁻¹ to a few litres s⁻¹, so the numbers above are a reasonable starting point.

2.2. Flow regime in the pipe

The Reynolds number for pipe flow is

[ \mathrm{Re}= \frac{\rho v D}{\mu} ]

Using the average velocity above:

[ \mathrm{Re}_{\text{unblocked}}\; \approx\; \frac{(1.2)(0.51)(0.05)}{1.8\times10^{-5}} \; \approx\; 1.7\times10^{3} ]

This is laminar (Re < 2300).

If the effective flow area is halved (see §2.4), the velocity roughly doubles, giving

[ \mathrm{Re}_{\text{blocked}}\; \approx\; 3.4\times10^{3} ]

This pushes the flow into the transitional / low‑turbulent regime.

Thus the presence or absence of the tape changes not only the magnitude of the velocity but also the type of flow (laminar ↔ turbulent).

2.3. What the MEMS sensor actually measures

The sensor works on a thermal time‑of‑flight (TOF) principle:

A short heating pulse is applied to the upstream platinum resistor.
The heated air parcel is carried downstream by the bulk flow.
The downstream resistor detects the arriving temperature rise.
The delay (\Delta t) between heating and detection is converted to an average velocity

[ v_{\text{TOF}} = \frac{L_{\text{sensor}}}{\Delta t} ]

where (L_{\text{sensor}}) is the spacing (≈ 0.5 mm–1 mm) between the two suspended structures.

Key requirement: the heated parcel must be advected downstream faster than it can spread by molecular diffusion. Otherwise the temperature rise is smeared out and the downstream resistor sees almost no distinct pulse.

The characteristic convective‑to‑diffusive ratio is the Péclet number

[ \mathrm{Pe}= \frac{v L_{\text{sensor}}}{\alpha} ]

with (\alpha) the thermal diffusivity of air (≈ 2.2 × 10⁻⁵ m² s⁻¹).
For a detectable signal we typically need (\mathrm{Pe}\gtrsim 5).

Using the average velocity for the unblocked pipe:

[ \mathrm{Pe}_{\text{unblocked}} = \frac{0.51\;(1\times10^{-3})}{2.2\times10^{-5}} \approx 23 ]

That looks large enough, but two hidden factors reduce the effective velocity that reaches the downstream sensor:

Velocity profile – near the centre of a laminar pipe the velocity is highest; close to the wall it goes to zero. If the sensor sits near the centre, it sees the centre velocity; if it is slightly off‑centre it can see a much smaller local speed.
Recirculation / dead‑zone – at low Re and in a short pipe the flow may not be fully developed; a large portion of the inlet cross‑section can form a stagnation zone that swirls back toward the inlet. In that zone the net axial velocity is near zero, so the thermal pulse never reaches the downstream resistor.

Consequently, even though the average (\mathrm{Pe}) is > 1, the local Péclet number at the sensor location can be far below the detection threshold when the inlet is completely open.

2.4. Effect of covering the lower half of the inlet

2.4.1. Simple continuity argument

When the lower half of the inlet is taped, the effective flow area at the entrance becomes roughly

[ A_{\text{eff}} \approx \frac{1}{2}A = 9.8\times10^{-4}\;{\rm m^{2}} ]

Because the fan’s pressure head is essentially unchanged (the fan is a constant‑pressure source for a small pipe), the volumetric flow (Q) remains almost the same, so the average velocity at the inlet doubles:

[ v_{\text{inlet}} \approx 2\,v_{\text{avg}} \;\approx\; 1.0\;{\rm m\,s^{-1}} ]

The Reynolds number now rises to ~3 × 10³, i.e. the flow becomes transitional/turbulent. Turbulent flow mixes the core and wall regions, flattening the velocity profile and removing the low‑speed “dead‑zone” that existed in the laminar case.

2.4.2. Formation of a vena contracta

A sudden reduction in cross‑section (the half‑pipe blockage) creates a vena contracta just downstream of the blockage. The streamlines converge toward the centre of the open half, producing a jet‑like region with a velocity higher than the simple continuity estimate (often 1.2–1.5 × the average). The jet is directed upward (because the bottom half is blocked) and passes right over the PCB that is centred in the pipe.

Result: the sensor experiences a locally higher velocity (perhaps 1.5 m s⁻¹) and a well‑defined direction (upward). The Péclet number at the sensor jumps to

[ \mathrm{Pe}_{\text{blocked}} \approx \frac{1.5\;(1\times10^{-3})}{2.2\times10^{-5}} \approx 68 ]

comfortably above the detection limit, and the thermal pulse arrives at the downstream resistor as a clear, sharp peak → the sensor outputs a normal flow reading.

2.4.3. Pressure‑gradient effect

Blocking the lower half also shifts the pressure distribution across the pipe cross‑section. The suction side (fan) creates a nearly uniform pressure drop along the pipe; the blocked side, however, sees a higher static pressure because the flow cannot pass there. This pressure gradient pushes the bulk flow toward the open side, reinforcing the upward jet. The jet stays stable over the 4 cm distance between the inlet and the sensor.

2.5. Why the sensor fails when the inlet is fully open

Low local velocity – In a fully open, smooth 5 cm pipe at the modest fan pressure, the centreline velocity may be only ~0.5 m s⁻¹, while the region where the sensor sits (a few mm off the centre due to PCB thickness) can be < 0.2 m s⁻¹. The local Péclet number falls below ~10, giving a very weak, broadened thermal pulse that the electronics cannot resolve.
Laminar parabolic profile – The velocity at the pipe wall is zero. The sensor’s suspended structures extend a few hundred microns into the flow; part of each structure sits in a region of near‑zero speed, diluting the effective advection speed.
Developing flow and recirculation – The pipe is short (L/D = 8). For laminar entry flow the developing length is roughly

[ L_{\text{dev}}\approx 0.05\,\mathrm{Re}\,D \approx 0.05\,(1700)\,(0.05) \approx 4.3\;{\rm m} ]

Because the pipe is far shorter than this, the flow at the sensor location is not fully developed; a large portion of the inlet flow still forms a plug that slowly spreads to the walls. The resulting large shear layer near the walls can generate a weak reverse‑flow region that cancels the downstream temperature signal.

Diffusion dominates – With such low axial convection, the heated air parcel spreads radially by molecular diffusion before it can travel the ≈ 1 mm distance to the downstream resistor. The downstream resistor therefore measures only the ambient temperature rise (≈ 0 °C) and the TOF algorithm returns “zero flow”.

In short: the sensor needs a minimum convective transport speed and a stable, uni‑directional flow at its exact location. The unblocked pipe fails to provide either

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