new earth potential equation & application ii by rogelio de ias casas

New Earth Potential Equations and Applications

II

Rogelio de las Casas

Introduction Cathodic Protection System Design

Determine ideal distance between groundbed and protected structure

The optimization of the groundbed location improve the CP system efficiency

Predict the optimum distance of groundbed from foreign line structures

Calculate the amount of interference current on foreign structures.

Potential to remote earth of any amount or anode configuration can be determine.

Linear current density on pipelines in function of groundbed distance from pipeline.

Use of New Potential Equations in Cathodic Protection System Design Permit determine Remote Earth distance:

Point where the anode’s potential influence on the earth surface is not longer observed.

Can be used for Groundbeds Installed Either Close To or Remote From the Structure Close groundbeds concentrate current on a small area of the

structure Remote groundbeds protect a larger area of the structure with

less concentrated current Optimal placement distance from the structure is calculated

for remote and distribute groundbeds. Permit determine Influence on Foreign Structures;

(Interference by stray current)

Horizontal Anodes in a Line Perpendicular to Structure, with anodes perpendicular to Structure as well.

The integral is:

The potential equation for the action of one anode over any point in the earth will be:

Groundbed Schematic diagram

Horizontal Anodes in a Line Perpendicular to Structure, with anodes perpendicular to Structure as well.

1

0222

222

lnN

p ξLSpytξ)LSp(yx

ξSpytξ)Sp(yxο):Uremot(x,y

The potential equation for groundbed with N anodes is then:

The potential equation, for the case of potential in the surface of the earth (z=0 m) is:

LNIdis

2

:

Where:

S: Distance between consecutive anodes (meters)N: Total number of anodesL: Anode length including

coke backfill (meters)

t: Depth of the anode (meters)Idis: Current applied at groundbed (Amperes): Soil resistivity at the groundbed location

(ohm-m) Distance of first anode from pipeline

Individually Located Horizontal Anodes in Parallel Distribution with Respect to the Structure

The integral is:

Groundbed Schematic diagram

The potential equation in any point in the earth for single anodes is:

Individually Located Horizontal Anodes in Parallel

Distribution with Respect to the Structure

1

0222

222

)(ln

N

p Lxtξ)Sp(yLx

xtξ)Sp(yxο):Uremot(x,y

LNIdis

2

:

The potential equation in the case of multiple anodes in a groundbed configuration is:

The potential equation, for the case of potential in the surface of the earth (z=0 m) is:

S: Distance between consecutive anodes (meters)

N: Total number of anodesL: Anode length including


t: Depth of the top of the anode (meters)Idis: Current applied at groundbed (Amperes): Soil resistivity at the groundbed location


Where

Vertical Anodes in a Groundbed Perpendicular to a Structure

The potential for single vertical electrode, acting in the earth is:

The integral is:

Groundbed configuration diagram

Vertical Anodes in a Groundbed Perpendicular to a

Structure

1

0222

222 )(ln

N

p tξ)Sp(yxt

Ltξ)Sp(yxLtο):Uremot(x,y

The potential equation in the case of multiple anodes in a groundbed configuration is:

The potential at the surface of the earth (z=0 m) will be:

LNIdis

2

:

S: Distance between consecutive anodes (meters)

N: Total number of anodesL: Anode length including


t: Depth of the top of the anode (meters)Idis: Current applied at groundbed (Amperes): Soil resistivity at the groundbed location


Horizontal Distributed Anodes inside stations.

Where Xw and Yw are the coordinates of the distributed anodes.

This is the total potential distributionon the plant:

Compressor station, refinery, electrical substation, etc.

Equations’ Application

Equations Provide: Optimal groundbed location Graphical representation of

anodes combined potential applied to structure

Visual representation of cathodic protection efficiency

Identification of impact that groundbed location has on foreign structures

Graph of Combined Potentials on a pipeline provided by a groundbed with multiple individually anodes in parallel distribution with respect to the pipe. Example of Equation 2.

Example of a horizontal groundbed that is not remote from pipe

Example of the use of the new equations, in the case of perpendicular to pipe horizontal anode

installation, equation 1.

Remote earth example calculation for horizontal groundbed, anodes perpendicular to pipe.

To determine the remote earth distance to install the groundbed from the pipe, the following subprogram can be used.

Freshour remote distance determination.

Distance from last anode (ft):

Potential meas. (volt)

5 0.06610 0.13415 0.21220 0.28840 0.50760 0.67880 0.796

100 0.898120 0.985140 1.052160 1.12180 1.179200 1.225220 1.275240 1.3260 1.34280 1.36300 1.375320 1.4340 1.42360 1.433380 1.439400 1.44 Percentage of potential change between 600 ft and 480 ft: 1.929474420 1.45 Remote distance: 600 ft440 1.478 Potential to remote earth at 332' : 0.083 Volt460 1.492 Percentage of potential change between remote earth (600 ft from the last anode) 480 1.503 and minimum remote distance calculated during design calculations (332'): 5.42600 1.532 The actual distance selected for groundbed was 460 ft,

So we can regard that the groundbed is remote from the pipeline.Potential to remote earth 460' from last anode: 0.040 VoltPercentage of potential change between remote earth (600 ft from the last anode) and actual distance between first anode and pipeline, 460 ft: 2.610966


0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0 100 200 300 400 500 600 700

Distance from last anode (ft)

Pote

ntia

l bet

wee

n tw

o re

fere

nce

cells

(V)

Potential mea

Alexander Rd.

Distance from last anode (ft):


2 0.044100 0.968200 1.147300 1.226400 1.312500 1.351545 1.36

Percentage of potential change between 545 ft and 500 ft 2.972561Remote distance: 450 ftPotential to remote earth measured at 362': 0.068 VoltPercentage of potential change between remote earth (545 ft from the last anode) and minimum remote distance calculated during design calculations (362 ft): 5.00Actual distance selected for groundbed was 465 ft,So we can regard that the groundbed is remote from the pipeline:Potential to remote earth measured at 465 ft from last anode: 0.007 VoltPercentage of potential change between remote earth (600 ft from the last anode) and the actual distance between first anode and pipeline 460 ft: 0.5


00.20.40.60.8

11.21.41.6

0 100 200 300 400 500 600

Distance from last anode (ft)

Pote

ntia

l bet

wee

n tw

o re

fere

nce

cells

(V)


Calculation example in congested area.

In the case of congested area, when we need to keep the local groundbed at the same energetic distance from the structures to be protected; we can calculate the optimum distance to keep the groundbed delivering the same protection level to different structures around the groundbed location.

Example for the case of gas distribution station.

A five anode one trench horizontal groundbed is designed to be installed between a 30” gas line and a 30” water line, with a 16” gas line crossing the above lines perpendicularly, at the south side of the station.

The distance between the two 30” lines is 35.052 m (115 ft). The potential at the lines with the horizontal anode parallel to the two 30” lines, located in the middle point between the two lines is calculated as follow:


Using the equation for horizontal groundbed with a single anode (single bed), without the remote component included in the equation, for the case of anodes parallel to pipe:

Design data:


Potential at parallel lines: This is the potential to remote earth that the two

lines will see from the horizontal anode.

Example for the case of gas distribution station

Now if we want to keep the same influence over the 16” line at the south, the distance from the groundbed to this pipe can be determined using another Mathcad subprogram with the following conditions:


So 40 ft is the distance between the south end of the groundbed and the 16”, to keep the Same influence over this line than over the other parallel lines.


In the case of the two parallel pipelines, with another line crossing these two lines, we can determine accurately the distances in each direction to keep the same influence over all the buried structures.


Also in the case of congested areas you can determine the current density at the structure surface.

Now, if we calculate the gradient of the potential, we can obtain the current density at the structure surface as follow:


This is the equation for the potential at the two parallel 30” pipelines

These are the coordinates of the points where we want to Calculate the current density from the groundbed location.

This is the equation to calculate the potential gradient at the point x, y determined above.

And this is the equation to calculate the current density at the structure surface due to the horizontal groundbed; at the closest point between the groundbed and the two 30” lines.


Now, the current density for the 16” line that is perpendicular to the two 30” line, the calculation is as follow:

Potential equation

Potential gradient equation

Coordinates where the current density is needed:

Current density equation:

Current density value

Interference current calculation

For the case when the groundbed is affecting a foreign line, the amount of current getting to the foreign line can be calculated.

Potential due to a vertical semi-deep anode:

The potential gradient can be calculated as follow:


Potential gradient calculation using Mathcad.

Parametric equation for a pipeline located at the x axis.

Tangent vector to the pipeline, parallel to x axis

Tangent vector to the pipeline, parallel to y axis


The remote earth distance from the semi-deep well calculated as follow:


The length of pipe that is receiving current from the semi-deep well is calculated as follow:

These are the integration limits on the pipe to calculate the total current getting in to it due to the semi-deep well.


This is the normal vector to the pipeline centered on x axis.

This is the current density due to the Semi-deep anode well

Magnitude of the normal vector

Total of interference current picked up by the foreign line.

Potential profile of distributed anodes on Compressor station

The potential profile in the compressor station due to the presence of all the distributed anodes.

Where Xw and Yw provide the North-South and East-West anode coordinateson the Compressor station, and z is the anode’s depth. This is the equation for horizontal anodes, the same can be developed for vertical anodes as well.

The entire potential distribution on the station is:

It provides the potential to remote earth in each point inside the station as a function of each and all anodes.

Potential profile of distributed anodes on Compressor station

And the potential profile in the station is:

First graph shows the anodes spatial distribution and potentialprofile

Second graph includes the potential profileAnd total anodes influence in the station.

Potential, current density and linear current density on Gas wells

Total potential due to deep well and Gas well in ground.

Potential profile in the surface of the ground.

Potential profile in the well axis

For the case that the deep well is just 5 meters (16.4 ft) from the Gas well


Well casing parametric equation: Well casing graph:

Current density profile in the well. Linear current density on the well surface


Potential profile in the ground. Potential profile in the well access.

Current densityProfile on Gaswell

Linear Current density in Well surface

For the case that the deep well is 100 meters (328 ft) from the Gas well

Conclusion

New Equations Facilitate Optimum Groundbed Placement Increased efficiency of Cathodic Protection System Ensure entire structure protected, either for locate, distributed or remote

groundbeds. Prevent interference with foreign structures

Cathodic Protection System Area of Influence is Accurately Identified Equations provide graphical representation of potential influence on the

soil surface. Equations identify cumulative effect of multiple anode systems In the case of distributed anode system, the appropriate location of the

anodes can be determine and the current density at the structure surface accurately calculated.

The amount of interference current can be predicted with accuracy. The current density on the structure surface can be calculated.

Questions?

new earth potential equation & application ii by rogelio de ias casas

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