Mathcad - 75 Mm of Concrete (25 Degrees)

CATHODIC PROTECTION OF PIPELINES

Lj 40 ft⋅:= Joint length

ta 75 mm⋅:= Original nominal anode thickness

tc 0.635 mm⋅:= Corrosion coat thickness (anode internal lining + steel pipe anti-corrosion coating)

gap 4 in⋅:= Anode half shell gap

ρ0.32

1.6

ohm⋅ m⋅:= Environmental resistivityExposed

Buried

Fig 6.8, Cls 6.8.3

n 10:= Number of array elements

k 1 n..:= counter for array

Sk k Lj⋅:= Anode spacing

PIPELINE CATHODIC PROTECTION SYSTEM DESIGN - Square ended and Tapered Anodes(DnV RP B401 1993)

Input data is defined for the exposed and buried case using 2 row column matrices. The exposed values are entered in row 0, whilst the buried values are entered in row 1, as below. Current density values are based on water depth and location.

Pipeline Section: Exposed to Seawater with a Concrete Coating Thickness of 3 in (75 mm)

Universal Declarations

Exposed 0≡ Buried 1≡ x 0 1..:= All input data should be checked, however, highlighted values are most criticalInput Data Summary

Y 30 yr⋅:= Design life in years

Ta 25:= Anode Surface Temperature ( this value must be in OC, without units)

Depth 95ft:= Water depth

CR "Arctic":= Climactic region as defined by Tables 6.3.1 and 6.3.2. Input choices are: "Tropical", "Sub-Tropical", "Temperate", and "Arctic".

D 48 in⋅:= OD of pipe

Z:\Jennifer H\Sakhalin\ 1 10/26/2004


ff 11 %=

Ea1.05−

1.0−

V⋅:= Design Closed Circuit PotentialExposed

Buried

Tbl 6.6.2

Ec 0.80− V⋅:= Design Protective PotentialExposed

Buried

Cls 7.8.2

∆V Ec Ea−:= Difference in potential between steel and anode

∆V0.25

0.2

V= Cls 7.8.2

Cls 6.3.7iburied 20mA m

2−⋅:= Buried current density (same for all conditions)

Initial Current Density

Fig & Tbl 6.6.1,Cls 6.6.3εDNV

2500

2000

A⋅ hr⋅ kg1−

⋅:= Electrochemical EfficiencyExposed

Buried

u 0.8:= Utilisation factor for bracelet anodes, anode mass Tbl 6.9.1

ρa 0.09754 lb⋅ in3−

⋅:= Anode density ρa 2699.9 kg m3−

⋅= Assumed input data

fi 2 %⋅:= Initial coating breakdown factor

fm 0.05 0.002Y

yr30−

⋅+:= Mean coating breakdown factor fm 5 %=

Cls 6.5.3ff 0.07 0.004

Y

yr20−

⋅+:= Final coating breakdown factor



Table 6.3.1

ii0

150 mA m2−

⋅

⋅ CR "Tropical"=if

170 mA m2−

⋅

⋅ CR "Sub-Tropical"=if

200 mA m2−

⋅

⋅ CR "Temperate"=if

250 mA m2−

⋅

⋅ otherwise

Depth 30m≤if

130 mA m2−

⋅


150 mA m2−

⋅


180 mA m2−

⋅


220mA m2−

⋅ otherwise

otherwise

:=

ii1

iburied:=

ii250

20

mA m2−

⋅=

Mean Current DensityTable 6.3.2

im0

70 mA m2−

⋅


80 mA m2−

⋅


100 mA m2−

⋅


120 mA m2−

⋅

⋅ otherwise

Depth 30m≤if

60 mA m2−

⋅


70 mA m2−

⋅


80 mA m2−

⋅


100mA m2−

⋅ otherwise

otherwise

:=

im1

iburied:=

im120

20

mA m2−

⋅=

Final Current DensityTable 6.3.1



ε1

εDNV1Ta 20≤if

εDNV127 Ta 20−( )⋅ A⋅ hr⋅ kg

1−⋅−

otherwise

:=ε0

εDNV0:=

Buried:Exposed:

Electrochemical Efficiency Derating (Fig. 6.6.1):

if170

20

mA m2−

⋅=if if Ta 25> if Ta 25−( ) itc⋅+, if, :=

im120

20

mA m2−

⋅=im if Ta 25> im Ta 25−( ) itc⋅+, im, :=

ii250

20

mA m2−

⋅=ii if Ta 25> ii Ta 25−( ) itc⋅+, ii, :=

Temperature Correction (Ref. Cls. 6.3.8)

Cls 6.3.8Temperature correction (for anode temp. > 25degC)itc 1 mA⋅ m2−

⋅:=

if170

20

mA m2−

⋅=

if1

iburied:=

if0

90 mA m2−

⋅


110 mA m2−

⋅


130 mA m2−

⋅


170 mA m2−

⋅

⋅ otherwise

Depth 30m≤if

80 mA m2−

⋅


90 mA m2−

⋅


110 mA m2−

⋅


130mA m2−

⋅ otherwise

otherwise

:=



Final Anode Thickness

Lmk x,

Y Imk x,

⋅

εx u⋅ ρa⋅ π ta⋅ D 2 tc⋅+ ta+( )⋅ 2 ta⋅ gap⋅− ⋅:=Substitute Ic into M and solve for Lm

Volume = Lm.[π.ta(D+2tc+ta) - 2tagap]

M = Volume x ρa

Cls 7.4.2Imk x,

πD Sk( )⋅ fm imx

⋅:=Current demand, Ic = Ac.fc.ic

Cls 7.7Required anode mass, M = [Ic(average).y]/u.εx

The following equations were used to obtain the mean anode length, Lmk,x

Mean Anode Length

Lik x,

0.315 ρx⋅ Iik x,

2

∆Vx( )2 π D 2 tc⋅+ 2 ta⋅+( )⋅ 2 gap⋅− ⋅

:=

Equate Current Demand, Ic and Current Output, Ia and solve for Li

Iik x,

πD Sk( )⋅ fi iix

⋅:=

The total current demand from coated pipe and uncoated J-lay collar is given as :

Cls 7.4.2Current demand, Ic = Ac.fc.ic

Cls 7.8.2Anode Current Output, Ia= ∆Vx/Ra

A = Exposed anode surface area, = Li [π(D+2ta+2tc)-2gap]

Tbl 6.7.1Anode resisitiviy, Ra0.315 ρ⋅

A:= The following equations were used to

obtain the initial anode length, Lik,x

Initial Anode Length

Square Ended Bracelet Anode

ε2500

1865

A hr⋅ kg1−

⋅=



Lf x( )

χk Lfk x,

←

k 0 n..∈for

χ

:=mf x( )

χk mfk x,

←

k 0 n..∈for

χ

:=Lrequired x( )

χk Lrequiredk x,

←

k 0 n..∈for

χ

:=

mrequired x( )

χk mrequiredk x,

←

k 0 n..∈for

χ

:=Lm x( )

χk Lmk x,

←

k 0 n..∈for

χ

:=Li x( )

χk Lik x,

←

k 0 n..∈for

χ

:=

mfk x,

ρa Lrequiredk x,

⋅ tf⋅ π D 2 tc⋅+ tf+( )⋅ 2 gap⋅− ⋅:=

mrequiredk x,

ρa Lrequiredk x,

⋅ ta⋅ π D 2 tc⋅+ ta+( )⋅ 2 gap⋅− ⋅:=

Lrequiredk x,

max Lik x,

Lmk x,

, Lfk x,

,

:=

Anode Mass

Lfk x,

0.315 ρx⋅ Ifk x,

⋅

2

∆Vx( )2 π D 2 tc⋅+ 2 tf⋅+( )⋅ 2 gap⋅− ⋅

:=Ifk x,

πD Sk( )⋅ ff ifx

⋅:=

Note : Use similar method as initial anode length, but substitute with tf from above

Final Anode Length

. . . final anode thicknesstf

ta21.011 %=tf 0.62 in=tf Find tf Vfinal,( )0:=

Vfinalπ

4D 2 tc⋅+ 2 tf⋅+( )2 D 2 tc⋅+( )2−

⋅ 2tf gap⋅−=

Vfinal 1 u−( ) Vinital⋅=Given

Vfinal 1 u−( ) Vinital⋅:=tf 0.20ta:=Guess

Vinitalπ

4D 2 tc⋅+ 2 ta⋅+( )2 D 2 tc⋅+( )2−

⋅ 2 ta⋅ gap⋅−:=



The following results tables can be applied to square-ended anode and tapered anode calculations, respectively. Mass is based on a square ended anode.

Results Summary Table - Square Ended Anode - Exposed Case φ Exposed:=

Li Lm Lf Lrequired mrequired mf mrequired Lrequired

joints ft in in in in lb lb kg cm1 40.0 0.1 1.9 1.3 1.9 81.2 16.2 36.922 80.0 0.3 3.7 5.2 5.2 229.5 45.9 104.313 120.0 0.8 5.6 11.8 11.8 516.3 103.3 234.704 160.0 1.4 7.4 20.9 20.9 917.9 183.6 417.245 200.0 2.1 9.3 32.7 32.7 1434.3 286.9 651.946 240.0 3.1 11.1 47.1 47.1 2065.3 413.1 938.79 117.777 280.0 4.2 13.0 64.1 64.1 2811.1 562.2 1277.79 160.298 320.0 5.4 14.8 83.7 83.7 3671.7 734.3 1668.96 209.369 360.0 6.9 16.7 106.0 106.0 4647.0 929.4 2112.27 264.97

10 400.0 8.5 18.5 130.9 130.9 5737.0 1147.4 2607.74 327.13

Anode Spacing

S

Lj

S

ft

Li φ( )

in

Lm φ( )

in

Lf φ( )

in

Lrequired φ( )

in

mrequired φ( )

lb

mf φ( )

lb

Results Summary Table - Square Ended Anode - Buried Case φ Buried:=

Li Lm Lf Lrequired mrequired mf mrequired Lrequired

joints ft in in in in lb lb kg cm1 40.0 0.0 0.4 0.7 0.7 31.0 6.2 14.099 1.7686412 80.0 0.1 0.8 2.8 2.8 124.1 24.8 56.396 7.0745633 120.0 0.2 1.2 6.4 6.4 279.2 55.8 126.891 15.917774 160.0 0.3 1.7 11.3 11.3 496.3 99.3 225.583 28.298255 200.0 0.5 2.1 17.7 17.7 775.4 155.1 352.474 44.216026 240.0 0.8 2.5 25.5 25.5 1116.6 223.3 507.563 63.671067 280.0 1.0 2.9 34.7 34.7 1519.9 304.0 690.849 86.663398 320.0 1.4 3.3 45.3 45.3 1985.1 397.0 902.334 113.1939 360.0 1.7 3.7 57.3 57.3 2512.4 502.5 1142.016 143.2599

Anode Spacing

S

Lj

S

ft

Li φ( )

in

Lm φ( )

in

Lf φ( )

in

Lrequired φ( )

in

mrequired φ( )

lb

mf φ( )

lb



Tapered anode length can be estimated as equal to the calculated length plus the thickness of the anode, assuming the following geometry:

Lcalc

Ltapered

Lcalc

Ltapered

In this case, the taper angle was assumed to be 45 degrees. This simplifies the length estimate.

Results Summary Table - Tapered Anode - Exposed Case φ Exposed:=

Li Lm Lf Lrequired mrequired mf

joints ft in in in in lb lb1 40.0 3.0 4.8 1.9 4.8 81.2 16.22 80.0 3.3 6.7 5.9 8.2 229.5 45.93 120.0 3.7 8.5 12.4 14.7 516.3 103.34 160.0 4.3 10.4 21.6 23.9 917.9 183.65 200.0 5.1 12.2 33.3 35.7 1434.3 286.96 240.0 6.0 14.1 47.7 50.1 2065.3 413.17 280.0 7.1 15.9 64.7 67.1 2811.1 562.28 320.0 8.4 17.8 84.4 86.7 3671.7 734.39 360.0 9.8 19.6 106.6 108.9 4647.0 929.4

10 400.0 11.5 21.5 131.5 133.8 5737.0 1147.4

Anode Spacing

S

Lj

S

ft

Li φ( ) ta+

in

Lm φ( ) ta+

in

Lf φ( ) tf+

in

Lrequired φ( ) ta+

in

mrequired φ( )

lb

mf φ( )

lb

Results Summary Table - Tapered Anode - Buried Case φ Buried:=

Li Lm Lf Lrequired mrequired mf

joints ft in in in in lb lb1 40.0 3.0 3.4 3.7 1.3 31.0 6.22 80.0 3.0 3.8 5.8 3.5 124.1 24.83 120.0 3.1 4.2 9.3 7.0 279.2 55.84 160.0 3.3 4.6 14.3 11.9 496.3 99.35 200.0 3.5 5.0 20.6 18.3 775.4 155.16 240.0 3.7 5.4 28.4 26.1 1116.6 223.37 280.0 4.0 5.8 37.6 35.3 1519.9 304.08 320.0 4.3 6.3 48.2 45.9 1985.1 397.09 360.0 4.7 6.7 60.3 57.9 2512.4 502.5

10 400.0 5.1 7.1 73.7 71.4 3101.8 620.4

Anode Spacing

S

Lj

S

ft

Li φ( ) ta+

in

Lm φ( ) ta+

in

Lf φ( ) ta+

in

Lrequired φ( ) tf+

in

mrequired φ( )

lb

mf φ( )

lb



required

cm4.63

13.0929.4452.3481.78

117.77160.29209.36264.97327.13

required

cm1.7686417.07456315.9177728.2982544.2160263.6710686.66339113.193

143.2599

1in 0.025 m=


Mathcad - 75 Mm of Concrete (25 Degrees)

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