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ECUACIONES PARA EL CALCULO DE VARIABLES EN PROTECCION CATODICA, DE ACUERDO A PRACTICAS RECOMENDADAS

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P1: FCE
CE003-Disclaimer CE003-Peabody December 11, 2000 18:18 Char Count= 0
Disclaimer
Neither NACE International, its officers, directors, nor members thereof accept any
responsibility for the use of the equations discussed herein. The information is advisory
only and the use of the equations is solely at the risk of the user.
EXCEL VERSION OF THESE EQUATIONS
The equations presented on the following pages are available for use interactively on
this CD. For each set of equations there is an Excel spreadsheet on the CD. If you have
Excel or a programthat can open Excel spreadsheets, you may open the spreadsheet and
use the equations interactively by doing the following:
1. Navigate to the CD drive on your computer
2. Double click the directory called Equation
3. You will see two spreadsheets. To use the U.S. Standard version of the spreadsheet,
double click us stand.xls. To use the metric version double click metric.xls.
You may continue to browse through the PDF files on the CD while also accessing
the spreadsheets.
1
Dwight's Equation for Single Vertical Anode Resistance to Earth - inches
ρ = Soil resistivity in ohm-cm ρ = 10,000 ohm-cm
L =Rod length in inches L = 84.0 inches
d =Rod diameter in inches d = 8.00 inches
R
V
=
Resistance of vertical rod in ohms
R
V
= 25.5 ohms
Dwight's Equation for Single Vertical Anode Resistance to Earth - feet
ρ = Resistivity of backfill material (or earth) in ohm-cm ρ = 10,000 ohm-cm
L =Length of anode in feet L = 7.0 feet
d =Diameter of anode in feet d = 0.667 feet
R
V
=
Resistance of one vertical anode to earth in ohms
R
V
= 25.5 ohms
Dwight's Equation for Multiple Vertical Anodes in Parallel - feet
ρ = Soil resistivity in ohm-cm ρ = 10,000 ohm-cm
N =Number of anodes in parallel N = 10 each
L =Length of anode in feet L = 7.0 feet
d =Diameter of anode in feet d = 0.667 feet
S =Anode spacing in feet S = 10.0 feet
R =Resistance of vertical anodes in parallel to earth in ohms R = 4.51 ohms
NACE
Companion to the Peabody Book
October 26, 2000
Revision 1.1






− = 1
8 00521 . 0
d
L
ln
L
R
V
ρ






+ − = N ln
S
L
d
L
ln
NL
R
MV
656 . 0
2
1
8 00521 . 0 ρ






− = 1
8 0624 . 0
d
L
ln
L
R
V
ρ
NOTICE
For these equations to be valid the soil must be homogeneous and L >>d. With low resistance backfill, as
coke, d is taken as the diameter of the coke and L as the length of the backfill provided it does not extend
too much greater than the length of the anode (about 1.5 times the coke diameter).
Modified Dwight's Equation for Multiple Anodes Installed Horizontally
R
H
= Resistance, in ohms, of horizontal anode to earth
ρ = Resistivity, in ohm-cm, of backfill material (or earth)
L =Length of anode in feet
S =Twice depth of anode in feet
d =Diameter of anode in feet
ρ = 10,000 ohm-cm
L = 7.0 feet
S = 12.0 feet
d = 0.667 feet
R
H
= 22.49 ohms
NACE
Companion to the Peabody Book
October 26, 2000
Revision 1.1









+
− +
+ +
= 1
4 4 00521 . 0
2 2 2 2 2
L
L S
L
S
dS
L S L L
ln
L
R
H
ρ
NOTICE
For these equations to be valid the soil must be homogeneous and L >>d. With low resistance backfill, as
coke, d is taken as the diameter of the coke and L as the length of the backfill provided it does not extend
too much greater than the length of the anode (about 1.5 times the coke diameter).
Galvanic Anode Life
Weight = Weight = 240 lbs
Efficiency = Efficiency = 0.90
Utilization Factor = Utilization Factor = 0.85
Current = Current = 2.0 A
Mg Life = Zn Life = 3.9 yrs
Solution
Potential³
(Cu-CuSO4)
-1.1 V
-1.4 to -1.6 V
-1.7 to -1.8 V
Efficiency²
90%
50%
50%
Anode¹
Zinc
Std. Mg
Hi-Pot Mg
500
500
370
Consumption
Rate²
(kg/amp-hr)
23.7
17.4
17.4
Output²
(amp-hr/lb)
5.9 yrs
Mag Anode Zinc Anode
240 lbs
0.50
0.85
2.0 A
NACE
Companion to the Peabody Book
October 26, 2000
Revision 1.1
Amps. in Current
Factor n Utilizatio Efficiency Lbs in Wt. Anode
Life Years Magnesium
× × ×
=
116 . 0
Amps. in Current
Factor n Utilizatio Efficiency Lbs in Wt. Anode
Life Years Zinc
× × ×
=
0424 . 0
1. Anodes installed in suitable chemical backfill.
2. Current efficiency with current density. The shown efficiency, and the resulting consumption rate, are
at approximately 30 milliamps/ft² of anode surface. Efficiencies are higher at higher current densities and
lower at lower current densities.
3. The potentials are solution potentials. When calculating driving potentials, the difference between the
protected structure and the anode, allow for anode polarization. Anode polarization is also influenced by
current density at the anode surface. For magnesium polarization allow for 0.1 V anodic polarization.
Zinc in a proper backfill is not usually subject to significant anodic polarization and the solution potential
may be used.
Rectifier Total Circuit Resistance
R
Gbed
=Ground bed resistance (ohms) R
Gbed
=
3.08 ohms
R
C
=Cable resistance (ohms) R
C
=
0.097 ohms
R
S
=Pipeline/structure to earth resistance (ohms) R
S
=
7.53 ohms

R
T
=Total circuit resistance (ohms)
R
T
=
10.71 ohms
Deep Anode Ground Bed Resistance
ρ = Effective soil resistivity (ohm-cm) ρ = 10,000 ohm-cm
L =Anode length (feet) L = 40 feet
d =Anode diameter (feet) d = 0.667 feet
R
H
=
Resistance to earth of a vertical single anode (ohms)
R
H
= 6.73 ohms
Rectifier Efficiency
K =Meter constant DC Amps = 1.00 A
N =#of revolutions of disk DC Volts = 2.00 V
T =Time in seconds K = 0.005
N = 4
T = 10 sec
Efficiency = 27.78%
NACE
Companion to the Peabody Book
October 26, 2000
Revision 1.1
S C Gbed T
R R R R + + =
T
KN
Power Input AC
3600
=
100 × =
Power Input AC
Power Output DC
Efficiency %
Amps DC x Volts DC Power Output DC =






− = 1
8 00521 . 0
d
L
ln
L
R
V
ρ
Impressed Current - # of Anodes Required
Wt = Weight per anode (pounds) Wt = 60 lbs
CR =Consumption rate (lbs./amp-year) CR = 0.75 lbs/A-yr
DL =Desired life (years) DL = 20 yrs
Current =Current required (amps) Current = 15.00 A
UF =Utilization factor UF = 0.60
#anodes = 7 each
# of Anodes Required Based on Current Discharge
* from manufacturer data
MD =Maximum discharge per anode (amps) MD = 2.50 A
Current =Current required (amps) Current = 15.00 A
#anodes = 6 each
Cable Resistance
R
CABLE
=Resistance per 1000 feet (Mft) R
CABLE
= 0.254 ohms/Mft
L
CABLE
=Length in feet (sum of positive and negative cables) L
CABLE
= 384 feet
R
C
=Cable resistance R
C
= 0.097 ohms
NACE
Companion to the Peabody Book
October 26, 2000
Revision 1.1
*
MD
Current
Anodes of Number =
1000
CABLE CABLE
C
L R
R
×
=
Anode per Weight Factor n Utilizatio
Required Current Life Desired Rate n Consumptio
Anodes of Number
×
× ×
=
Dwight's Equation for Single Vertical Anode Resistance to Earth - millimeters
ρ = Soil resistivity in ohm-cm ρ = 10,000 ohm-cm
L =Rod length in mm L = 2134 mm
d =Rod diameter in mm d = 203 mm
R
V
=
Resistance of vertical rod in ohms
R
V
= 25.6 ohms
Dwight's Equation for Single Vertical Anode Resistance to Earth - meters
ρ = Resistivity of backfill material (or earth) in ohm-cm ρ = 10,000 ohm-cm
L =Length of anode in meters L = 2.13 m
d =Diameter of anode in meters d = 0.203 m
R
V
=
Resistance of one vertical anode to earth in ohms
R
V
= 25.6 ohms
Dwight's Equation for Multiple Vertical Anodes in Parallel - meters
ρ = Soil resistivity in ohm-cm ρ = 10,000 ohm-cm
N =Number of anodes in parallel N = 10 each
L =Length of anode in meters L = 2.13 m
d =Diameter of anode in meters d = 0.203 m
S =Anode spacing in meters S = 3.0 m
R =Resistance of vertical anodes in parallel to earth in ohms R = 4.52 ohms
NACE
Companion to the Peabody Book
October 26, 2000
Revision 1.1M






− = 1
8 59 . 1
d
L
ln
L
R
V
ρ






− = 1
8 00159 . 0
d
L
ln
L
R
V
ρ






+ − = N ln
S
L
d
L
ln
NL
R 656 . 0
2
1
8 00159 . 0 ρ
NOTICE
For these equations to be valid the soil must be homogeneous and L >>d. With low resistance backfill, as
coke, d is taken as the diameter of the coke and L as the length of the backfill provided it does not extend
too much greater than the length of the anode (about 1.5 times the coke diameter).
Modified Dwight's Equation for Multiple Anodes Installed Horizontally
R
H
= Resistance, in ohms, of horizontal anode to earth
ρ = Resistivity, in ohm-cm, of backfill material (or earth)
L =Length of anode in meters
S =Twice depth of anode in meters
d =Diameter of anode in meters
ρ = 10,000 ohm-cm
L = 2.13 m
S = 3.7 m
d = 0.203 m
R
H
= 22.52 ohms
NACE
Companion to the Peabody Book
October 26, 2000
Revision 1.1M









+
− +
+ +
= 1
4 4 00159 . 0
2 2 2 2 2
L
L S
L
S
dS
L S L L
ln
L
R
H
ρ
NOTICE
For these equations to be valid the soil must be homogeneous and L >>d. With low resistance backfill, as
coke, d is taken as the diameter of the coke and L as the length of the backfill provided it does not extend
too much greater than the length of the anode (about 1.5 times the coke diameter).
Galvanic Anode Life
Weight = Weight = 109 kg
Efficiency = Efficiency = 0.90
Utilization Factor = Utilization Factor = 0.85
Current = Current = 2.0 A
Mg Life = Zn Life = 3.9 yrs 5.9 yrs
Mag Anode Zinc Anode
109 kg
0.50
0.85
2.0 A
1100
1100
815
Consumption
Rate²
(kg/amp-hr)
10.8
7.9
7.9
Output²
(amp-hr/kg) Anode¹
Zinc
Std. Mg
Hi-Pot Mg
Efficiency²
90%
50%
50%
Solution
Potential³
(Cu-CuSO4)
-1.1 V
-1.4 to -1.6 V
-1.7 to -1.8 V
NACE
Companion to the Peabody Book
October 26, 2000
Revision 1.1M
Amps in Current
Factor n Utilizatio Efficiency kg in Wt Anode
Life Years Magnesium
× × ×
=
. 256 . 0
Amps in Current
Factor n Utilizatio Efficiency kg in Wt Anode
Life Years Zinc
× × ×
=
. 0935 . 0
1. Anodes installed in suitable chemical backfill.
2. Current efficiency with current density. The shown efficiency, and the resulting consumption rate, are
at approximately 30 milliamps/ft² of anode surface. Efficiencies are higher at higher current densities and
lower at lower current densities.
3. The potentials are solution potentials. When calculating driving potentials, the difference between the
protected structure and the anode, allow for anode polarization. Anode polarization is also influenced by
current density at the anode surface. For magnesium polarization allow for 0.1 V anodic polarization.
Zinc in a proper backfill is not usually subject to significant anodic polarization and the solution potential
may be used.
Rectifier Total Circuit Resistance
R
Gbed
=Ground bed resistance (ohms) R
Gbed
=
3.08 ohms
R
C
=Cable resistance (ohms) R
C
=
0.097 ohms
R
S
=Pipeline/structure to earth resistance (ohms) R
S
=
7.53 ohms

R
T
=Total circuit resistance (ohms)
R
T
=
10.71 ohms
Deep Anode Ground Bed Resistance - meters
ρ = Effective soil resistivity (ohm-cm) ρ = 10,000 ohm-cm
L =Anode length (meters) L = 12 m
d =Anode diameter (meters) d = 0.203 m
R
H
=
Resistance to earth of a vertical single anode (ohms)
R
H
= 6.74 ohms
Rectifier Efficiency
K =Meter constant DC Amps = 1.00 A
N =#of revolutions of disk DC Volts = 2.00 V
T =Time in seconds K = 0.005
N = 4
T = 10 sec
Efficiency = 27.78%
NACE
Companion to the Peabody Book
October 26, 2000
Revision 1.1M
S C Gbed T
R R R R + + =
T
KN
Power Input AC
3600
=
100 × =
Power Input AC
Power Output DC
Efficiency %
Amps DC x Volts DC Power Output DC =






− = 1
8 00521 . 0
d
L
ln
L
R
V
ρ
Impressed Current - # of Anodes Required
Wt = Weight per anode (kg) Wt = 27.2 kg
CR =Consumption rate (kg/amp-year) CR = 0.34 kg/A-yr
DL =Desired life (years) DL = 20 yrs
Current =Current required (amps) Current = 15.00 A
UF =Utilization factor UF = 0.60
#anodes = 7 each
# of Anodes Required Based on Current Discharge
* from anode manufacturer data
MD =Maximum discharge per anode (amps) MD = 2.50 A
Current =Current required (amps) Current = 15.00 A
#anodes = 6 each
Cable Resistance
R
CABLE
=Resistance per km R
CABLE
= 0.833 ohms/km
L
CABLE
=Length in meters (sum of positive and negative cables) L
CABLE
= 117 m
R
C
=Cable resistance R
C
= 0.097 ohms
NACE
Companion to the Peabody Book
October 26, 2000
Revision 1.1M
Anode per Weight Factor n Utilizatio
Required Current Life Desired Rate n Consumptio
Anodes of Number
×
× ×
=
1000
CABLE CABLE
C
L R
R
×
=
*
MD
Current
Anodes of Number =

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