# Simulation of the running in process in external gear pumps and experimental verification

## Abstract

Before marketing external gear pumps are subjected to a running in process to increase their efficiency. However, this is one of the most time-consuming tasks of the entire manufacturing process. Therefore, a mathematical model for optimizing the running in process can be a useful tool for time-to-market reduction. In particular, in this paper a model for the analysis of the dynamic behaviour of external gear pumps, developed by the authors in previous works, is modified and used for simulating the running in process. The modified model is presented and validated via experimental data. A good correlation between simulation and test results guarantees the effectiveness of the model in determining the amount and the distribution of the removed material during the running in process. A meaningful reduction (16%) of the global running in time has been achieved with the introduction of a modified running in process drawn from simulation results.

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## Abbreviations

b b :

Bearing block width (see Fig. 1b).

b k :

Face width of gear k.

B oil :

Oil bulk modulus.

C r :

Radial clearance in journal bearings.

f bxk :

Bearing reaction applied to gear k in direction X.

f byk :

Bearing reaction applied to gear k in direction Y.

f mgm :

Mean meshing force over one pitch.

f pxk :

Pressure force applied to gear k in direction X.

f pyk :

Pressure force applied to gear k in direction Y.

f pxkm :

Mean value over one pitch of pressure force applied to gear k in direction X.

f pykm :

Mean value over one pitch of pressure force applied to gear k in direction Y.

h b :

Radial clearance between casing and bearing blocks.

h rn :

Radial clearance defined as the difference between the internal radius of the casing and the outside radius of the gear (nominal).

h wk :

Radial thickness of removed material for gear k.

M m :

Motor driving torque.

M pk :

Pressure torque applied to gear k.

M pkm :

Mean value over one pitch of pressure torque applied to gear k.

p :

Pressure in the generic control volume.

p i :

Pressure in isolated tooth space i.

p in :

Pressure in the inlet volume.

p out :

Pressure in the outlet volume.

p t :

Pressure in the trapped volume.

R b :

r bk :

Base radius of gear k.

r ext :

Outside radius of gears.

r k :

Radius of the contact point for gear k (see Fig. 5, the expressions are given in [10]).

V :

Volume of the generic control volume.

V i :

Volume of the isolated tooth space i.

W kj :

Volume of removed material by gear k during step j.

W Tj :

Volume of removed material by both gears during step j.

(x k ,y k ):

Coordinates of the centre of gear k in reference frame O k X k Y k .

z k :

Number of teeth of gear k.

α w :

Pressure angle in working conditions.

ΔQ :

Difference between the volumetric flow rate, coming into a control volume and coming out.

θ :

Angular displacement.

θ k :

Angular displacement of gear k.

θ p :

Angular pitch.

μ :

Lubricant dynamic viscosity.

ρ :

Oil density.

ψ k :

Angle coordinate along the casing housing for gear k (Fig. 1).

ψ ink :

Angle limiting the isolated tooth spaces at the inlet side for gear k.

ψ outk :

Angle limiting the isolated tooth spaces at the outlet side for gear k.

ω :

Mean angular speed of gears in steady-state operational conditions.

i :

Denotes isolated spaces between teeth.

j :

Denotes running in steps.

k=1,2:

Denotes gears.

| k :

Applied to gear k.

n, m :

Number of isolated tooth spaces for gear 1 and 2, respectively.

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Correspondence to Emiliano Mucchi.

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Mucchi, E., D’Elia, G. & Dalpiaz, G. Simulation of the running in process in external gear pumps and experimental verification. Meccanica 47, 621–637 (2012). https://doi.org/10.1007/s11012-011-9470-9