# Mathematical model

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A search have been done in order to provide more understanding of the sand erosion in chokes, also a change to the flow field geometry to decrease the erosion.

The equations of continuous flow used in this experiment listed in the “Mathematical Model“ which it has been taken from Navier-stokes equations, also verifying the discrete particle hard sphere model by describing the inter particle collision and the fluid drag force which it is the clutching activity between fluid and particle, also the physical problem and simulation conditions explicated by using the certain dimensions for the inlet, outlet, geometry and velocity.

A conclusion was reached in the end of this experiment that the movement of solid particles can be simulated in the hard sphere model, also the impact angle, velocity, and location can be determined individually in this model, in the other hand the erosion can be decreased by using 4 ribs with height 3 mm width 5 mm and spacing 5 mm.

### According To “R.J.K. Wood, B.G. Mellor, M.L. Binfield”

A study have been done on thermally sprayed wc-co-cr with impingement rig of sand/water and the erosion rate results of sprayed are the same of the sintered tungsten carbide-cobalt-chrome in low energy impacts and the sintered tungsten-carbide performance for the high impacts was 4 times the sprayed.

The erodent used in this experiment was quartz sand from 1100 HV10 hardness and 2663 kg m-3 density and material is stainless steel with tungsten-carbide sprayed on it and coated with w-co-cr-c.

The methodology of this experiment was 101 reservoir circulated with piping of 19 mm diameter, the slurry goes through silver steel nozzle and the angle of impingement can be fixed between 30 and 90 degrees, the slurry is looping with a use of mono-pump with different speeds and the flow can measured by heinrichs electromagnetic flow meter.

### According To “Robert J.K. Wood”

The materials in oil and gas industry has been always a big issue for choke valves, pumps and pipes in this article the possibilities of choosing a material for choke valve have been discussed, since the oil production contains a mixture of oil, gas, water, and solid particles (sand) with high velocity and pressure, erosion is a great factor and it could make a serious damage to the material and that decreases the life time of the choke valve and the angle of the solid particles impact is between (0-40) degrees cutting can occur when the shear stress of the impact exceed the shear strength of the target, in the other hand the erosion can be more stronger in higher angles between (30-90) degrees as the stress exceed the yield strength of the material which it leads to micro-cracking and coalescence which it produce surface fragmentation.

### According “Hazim Al-Attar”

A steady have been done on three producing wells in the middle east to predict a new plotting technique to clarify the behaviour of sub-critical flow through well heads chokes with different choke sizes between 24/64 in and 128/64 in.

Through this study a linear relationship was observed with negative slope when a plot of log-log have been done on a graph paper the reason is the pressure differential of across the choke itself, also the instability of the downstream pressure which affect the upstream pressure.

### According To” E. J. Wentzel C. Allen”

This paper present an experiment for testing the erosion corrosion of tungsten-carbide with using a slurry erosion rig with test solution circulated by centrifugal pump and for dropping the pressure of the fluid an ejector was used and the angle of impact was 75 degree and the solution component's was 1N H2SO4 with sea water.

The material used in this experiment was divided into two groups tungsten-carbide based cermets is the first one and metal alloys with no tungsten-carbide particles the second one, the first one also divided into two groups 10 wt% binder cermets and 6 wt% binder cermets.

And the conclusion reached after doing this experiment that the WC based cermets erosion can be better by alloying and the lowest erosion rate was on co-cr grade with 10 wt%.

According to “E.A.M. Hussain, M.J. Robinson”

This paper present a study of sand erosion on 2205 stainless steel (UNS S31803) by using a jet impingement apparatus , a linear relation was found between the increases if the density and the kinetic energy of the sand particle.

Three cylindrical electrodes used in this experiment, the dimensions of the middle was 5 mm diameter and it was in rode shape, the two in the both ends were cylinder shape with internal diameter of 13 mm, 23 mm and 2 mm thickness each one was attached with electrical connection.

See water was used in the flow which it was nearly 31 of artificially sea water with 8.5 ms-1 caused by impeller driven pump with 5 mm nozzle, also sand particles with a size of 250 300 μm.

A conclusion was reached in the end of the experiment, the erosion on stainless steel depends on the kinetic energy of the sand and the hydrodynamic conditions, also the biggest erosion rate cased by sand particles directed to the object with angle of 90 degree.

### According To “M. Atkinson, E.V. Stepanov, D.P. Goulet, S.V. Sherikar, J. Hunter”

An experiment have been done for testing sand erosion under a high pressure with 3D flow channels by using CFD, the experiment was in 40 bar of nitrogen pressure with silica sand and the flow channels were made if organic glass (PMMA), the glass shape have been chosen by the tester to find a close resemblance to stainless steel.

The article divided to three parts the first interdiction and the second “ scaling erosion in reduced flow channels “ where the writer explain how the sand erosion can be calculated by some equations and the main equation is

P = K ρ in V2 in / 2 where p = is the pressure drop across the channel, ρ in is the density of the fluid and V in is the velocity and k is a constant number defined for each feature.

In part three the writer is clarifying the levels of the experiment and how it have been done, in the end pictures were listed to show the result of the experiment.

### According to “R.J.K. Wood“

This article shows studies of the performance of ceramic, metallic, and polymeric with diamond coatings on steel under sand velocities between 10-30 m/s and size of 60-235 μm and angles 30-90 degrees.

the writer listed the materials and coatings tested in two tables and described the dimensions of the pipes, coatings and coupons, also exposing an equation for the unit volume less per impact

W = V1 π d3 / 6 Qv t Cv

Where V1 = target volume loss

d = particle diameter

t = test duration

Cv = particle volume

The results of the experiment were shown in tables, figures and digits and in the end a conclusion have been reached based on the results which is the erosion performance of the coating mentioned in the first was determined relative to uncoated carbon steel and the diamond coating with CVD is the best and DLC is the worst.

### According to “A.P. Harsha, Deepak Kumar Bhaskar”

The purpose of this paper is to study the erosion behaviour of some kind of materials ferrous and non ferrous to understand sand erosion of normal and oblique impact angles between (15-90) degrees and (24-52) m/s velocities.

The writer plotted a table of metals selected for this experiment and specifications, physical and chemical properties of these metals.

The methodology of this study was done by using air jet erosion test rig and velocity of the particles was controlled by using rotating disc method and the particles were silica sand with a density equals 2600 kg/m2 and hardness of 880.

The results gained from this study after plotting each metal against the cumulative weight of impinging particles that the erosion rate ( E ) was increasing when the cumulative weight of impinging particles increased and in the end it goes in a steady statement the reason is the first particle made a micro cutting and removed a little of the material, in the end the writer comes with a conclusion that the erosion peak is about 60 degree impinging angle with 52 m/s velocity.

### According To “ Thananchai Leephakpreeda ”

An experiment has been done with using conventional flow meters for feed back in order to determine the flow rate without flow meters.

A picture of the experiment was listed in the “ Experiment “ section and also an explanation of the experiment procedure and the of every part, in the other hand the equation that been used in this experiment were listed in “Background principle” and clarifying each equation for calculating the flow rate in the control valve, the major equation for the calculating the flow rate is

Q = n Cv P Y ( x / v r z )1/2

Q = flow rate

n = numerical multiplier

Cv = valve coefficient

P = upstream pressure

Y = expansion factor

v = specific gravity

T = upstream temperature

z = compressibility factor

and after calculating the flow rate a plot was drawn showing the relationship between Q and P, in the end a conclusion was reached that the neural computing for evaluating the flow rate can be used without a flow meter.