equation

P1: FCECE003-Disclaimer CE003-Peabody December 11, 2000 18:18 Char Count= 0

Disclaimer

Neither NACE International, its officers, directors, nor members thereof accept anyresponsibility for the use of the equations discussed herein. The information is advisoryonly 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 onthis CD. For each set of equations there is an Excel spreadsheet on the CD. If you haveExcel or a program that can open Excel spreadsheets, you may open the spreadsheet anduse the equations interactively by doing the following:

1. Navigate to the CD drive on your computer2. Double click the directory called Equation3. 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 accessingthe spreadsheets.

1

Dwight's Equation for Single Vertical Anode Resistance to Earth - inches

= Soil resistivity in ohm-cm = 10,000 ohm-cmL = Rod length in inches L = 84.0 inchesd = 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

= Resistivity of backfill material (or earth) in ohm-cm = 10,000 ohm-cmL = Length of anode in feet L = 7.0 feetd = 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

= Soil resistivity in ohm-cm = 10,000 ohm-cmN = Number of anodes in parallel N = 10 eachL = Length of anode in feet L = 7.0 feetd = Diameter of anode in feet d = 0.667 feetS = Anode spacing in feet S = 10.0 feetR = Resistance of vertical anodes in parallel to earth in ohms R = 4.51 ohms

NACECompanion to the Peabody Book

October 26, 2000Revision 1.1

= 1800521.0

dLln

LRV

+= Nln

SL

dLln

NLRMV 656.0

21800521.0

= 180624.0

dLln

LRV

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 = Resistivity, in ohm-cm, of backfill material (or earth)L = Length of anode in feet

S = Twice depth of anode in feetd = Diameter of anode in feet

= 10,000 ohm-cmL = 7.0 feetS = 12.0 feetd = 0.667 feet

R H = 22.49 ohms



++

++= 1

4400521.0 22222

LLS

LS

dSLSLL

lnL

RH

NOTICE



Galvanic Anode Life

Weight = Weight = 240 lbsEfficiency = Efficiency = 0.90

Utilization Factor = Utilization Factor = 0.85Current = Current = 2.0 A

Mg Life = Zn Life = 3.9 yrs

Solution Potential

(Cu-CuSO4)-1.1 V

-1.4 to -1.6 V

-1.7 to -1.8 V

Efficiency90%

50%

50%

AnodeZinc

Std. Mg

Hi-Pot Mg

500

500

370

Consumption Rate

(kg/amp-hr)23.7

17.4

17.4

Output(amp-hr/lb)

5.9 yrs

Mag Anode Zinc Anode240 lbs

0.500.85

2.0 A



Amps. in CurrentFactor nUtilizatioEfficiencyLbs in Wt. Anode

Life Years Magnesium

=

116.0

Amps. in CurrentFactor nUtilizatioEfficiencyLbs in Wt. AnodeLife 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/ft 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 ohmsR C = Cable resistance (ohms) R C = 0.097 ohmsR 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

= Effective soil resistivity (ohm-cm) = 10,000 ohm-cmL = Anode length (feet) L = 40 feetd = 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 AN = # of revolutions of disk DC Volts = 2.00 VT = Time in seconds K = 0.005

N = 4T = 10 sec

Efficiency = 27.78%



SCGbedT RRRR ++=

TKNPower Input AC 3600=

100=Power Input ACPower Output DCEfficiency %

Amps DC x Volts DCPower Output DC =

= 1800521.0

dLln

LRV

Impressed Current - # of Anodes Required

Wt = Weight per anode (pounds) Wt = 60 lbsCR = Consumption rate (lbs./amp-year) CR = 0.75 lbs/A-yrDL = Desired life (years) DL = 20 yrs

Current = Current required (amps) Current = 15.00 AUF = 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 ACurrent = Current required (amps) Current = 15.00 A

# anodes = 6 each

Cable Resistance

R CABLE = Resistance per 1000 feet (Mft) R CABLE = 0.254 ohms/MftL CABLE = Length in feet (sum of positive and negative cables) L CABLE = 384 feet

R C = Cable resistance R C = 0.097 ohms



*MD

CurrentAnodesofNumber =

1000CABLECABLE

CLRR

=

AnodeperWeightFactornUtilizatioRequiredCurrentLifeDesiredRatenConsumptioAnodesofNumber

=

Dwight's Equation for Single Vertical Anode Resistance to Earth - millimeters

= Soil resistivity in ohm-cm = 10,000 ohm-cmL = Rod length in mm L = 2134 mmd = 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

= Resistivity of backfill material (or earth) in ohm-cm = 10,000 ohm-cmL = Length of anode in meters L = 2.13 md = 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

= Soil resistivity in ohm-cm = 10,000 ohm-cmN = Number of anodes in parallel N = 10 eachL = Length of anode in meters L = 2.13 md = Diameter of anode in meters d = 0.203 mS = Anode spacing in meters S = 3.0 mR = Resistance of vertical anodes in parallel to earth in ohms R = 4.52 ohms


October 26, 2000Revision 1.1M

= 1859.1

dLln

LRV

= 1800159.0

dLln

LRV

+= Nln

SL

dLln

NLR 656.021800159.0

NOTICE



Modified Dwight's Equation for Multiple Anodes Installed Horizontally

R H = Resistance, in ohms, of horizontal anode to earth = Resistivity, in ohm-cm, of backfill material (or earth)L = Length of anode in meters

S = Twice depth of anode in metersd = Diameter of anode in meters

= 10,000 ohm-cmL = 2.13 mS = 3.7 md = 0.203 m

R H = 22.52 ohms



++

++= 1

4400159.0 22222

LLS

LS

dSLSLL

lnL

RH

NOTICE



Galvanic Anode Life

Weight = Weight = 109 kgEfficiency = Efficiency = 0.90

Utilization Factor = Utilization Factor = 0.85Current = Current = 2.0 A

Mg Life = Zn Life = 3.9 yrs5.9 yrs

Mag Anode Zinc Anode109 kg

0.500.85

2.0 A

1100

1100

815

Consumption Rate

(kg/amp-hr)10.8

7.9

7.9

Output(amp-hr/kg)Anode

Zinc

Std. Mg

Hi-Pot Mg

Efficiency90%

50%

50%

Solution Potential

(Cu-CuSO4)-1.1 V

-1.4 to -1.6 V

-1.7 to -1.8 V



AmpsinCurrentFactornUtilizatioEfficiencykginWtAnodeLifeYearsMagnesium = .256.0

AmpsinCurrentFactornUtilizatioEfficiencykginWtAnodeLifeYearsZinc = .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/ft 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 ohmsR C = Cable resistance (ohms) R C = 0.097 ohmsR 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

= Effective soil resistivity (ohm-cm) = 10,000 ohm-cmL = Anode length (meters) L = 12 md = 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 AN = # of revolutions of disk DC Volts = 2.00 VT = Time in seconds K = 0.005

N = 4T = 10 sec

Efficiency = 27.78%



SCGbedT RRRR ++=

TKNPower Input AC 3600=

100=Power Input ACPower Output DCEfficiency %

Amps DC x Volts DCPower Output DC =

= 1800521.0

dLln

LRV

Impressed Current - # of Anodes Required

Wt = Weight per anode (kg) Wt = 27.2 kgCR = Consumption rate (kg/amp-year) CR = 0.34 kg/A-yrDL = Desired life (years) DL = 20 yrs

Current = Current required (amps) Current = 15.00 AUF = 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 ACurrent = Current required (amps) Current = 15.00 A

# anodes = 6 each

Cable Resistance

R CABLE = Resistance per km R CABLE = 0.833 ohms/kmL CABLE = Length in meters (sum of positive and negative cables) L CABLE = 117 m

R C = Cable resistance R C = 0.097 ohms



AnodeperWeightFactornUtilizatioRequiredCurrentLifeDesiredRatenConsumptioAnodesofNumber

=

1000CABLECABLE

CLRR =

*MD

CurrentAnodesofNumber =

U.S. CustomaryDwight-Vertical AnodesDwight-Horizontal AnodesGalvanic AnodesRectifier TotalImpressed Current

MetricDwight-Vertical AnodesDwight-Horizontal AnodesGalvanic AnodesRectifier TotalImpressed Current

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