Pickup EMF from a passing slug

What voltage does a magnetised slug induce as it passes through the pickup coil?

The formula

ε = \frac{N \cdot μ_{0} \cdot M \cdot A \cdot κ \cdot v}{ℓe}

ε: Peak EMF, V
N: Pickup turns
M: Slug magnetisation, A/m
A: Bore cross-section, π D² ÷ 4, m²
κ: Fill fraction
v: Flow speed, m/s
ℓe: Edge length: the greater of the coil length and the edge width, m

LaTeX

ε = \frac{N \cdot \mu_0 \cdot M \cdot A \cdot κ \cdot v}{ℓe}

Work it out

N Pickup turns

Total turns on the pickup coil. The EMF is proportional to it; so is the coil's resistance, which the loaded-output page charges for.

D Tube inner diameter

The magnetised cross-section — the bore the medium fills.

ℓc Coil length

The pickup winding's length along the tube. A coil cannot see an edge sharper than itself.

M Slug magnetisation

A/m

What the medium carries as it arrives — the remanence page's figure downstream, or the solenoid page's inside the drive field. Per unit volume of medium, not of particle.

v Flow speed

in/s

The medium's speed through the coil. The estate holding is 50 in/s for the slurry.

w Edge width

How far the flow has smeared the slug's front by the time it reaches the coil — the laminar-smearing page's figure. Zero for a perfectly sharp edge.

ℓs Slug length

The length of magnetised medium — flow speed times the drive pulse's on-time.

or see a worked example with typical numbers

Method

The bore's area is πD²/4, and the flux through it when the slug fills it is μ₀·M·A — Φ per turn.
The linkage rises over the edge length ℓ_e, which is the coil's length or the smeared edge's width, whichever is greater: the coil cannot resolve an edge sharper than itself, and the medium cannot supply one sharper than the flow has left it. The crossing time is ℓ_e over the flow speed.
If the slug is shorter than ℓ_e it starts leaving before it has finished entering, and only reaches the fraction ℓ_s/ℓ_e of the full flux — the fill fraction κ, capped at 1.
The peak EMF is N times the flux rise, κ·Φ, over the crossing time: N·μ₀·M·A·κ·v/ℓ_e. The leaving edge gives the same magnitude with the opposite sign — the pulse is bipolar.
The sharp-edge ceiling is the same expression with the edge width taken as zero and the slug as long as the coil: N·μ₀·M·A·v/ℓ_c. The page states the actual figure against it.

Assumptions

The linkage rises linearly over the edge length — a ramp. The real rise for a square front through a finite coil is a ramp; for a smeared front it is a smoother S, whose peak slope is somewhat higher than the mean slope used here. The figure is a fair mean, within a factor of about 1.5 of the true peak either way.
The slug's magnetisation is uniform across the bore and along its length, and it is what the input says — this page does not know whether the medium kept it; the medium-relaxation and remanence pages do.
The coil is a thin winding at the bore: the flux linked is the flux in the tube. A winding standing well off the tube links the same flux (it is the tube's flux that matters) but has a larger area of nothing round it, which does not change the EMF.
Open-circuit EMF, before the coil's own resistance and inductance and the load take their share. The loaded-output page does that.
Nothing here is Meyer's: it is Faraday's law, −N dΦ/dt, put to a slug in a tube. The estate material claims an output from the pickup coils and gives no figure for it; this is what the physics allows from the stated speed, and a reader can put in the magnetisation they believe in.