Mathcad - 75 Mm of Concrete (25 Degrees)
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Transcript of Mathcad - 75 Mm of Concrete (25 Degrees)
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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
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CATHODIC PROTECTION OF PIPELINES
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
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CATHODIC PROTECTION OF PIPELINES
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−
⋅
⋅ CR "Tropical"=if
150 mA m2−
⋅
⋅ CR "Sub-Tropical"=if
180 mA m2−
⋅
⋅ CR "Temperate"=if
220mA m2−
⋅ otherwise
otherwise
:=
ii1
iburied:=
ii250
20
mA m2−
⋅=
Mean Current DensityTable 6.3.2
im0
70 mA m2−
⋅
⋅ CR "Tropical"=if
80 mA m2−
⋅
⋅ CR "Sub-Tropical"=if
100 mA m2−
⋅
⋅ CR "Temperate"=if
120 mA m2−
⋅
⋅ otherwise
Depth 30m≤if
60 mA m2−
⋅
⋅ CR "Tropical"=if
70 mA m2−
⋅
⋅ CR "Sub-Tropical"=if
80 mA m2−
⋅
⋅ CR "Temperate"=if
100mA m2−
⋅ otherwise
otherwise
:=
im1
iburied:=
im120
20
mA m2−
⋅=
Final Current DensityTable 6.3.1
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CATHODIC PROTECTION OF PIPELINES
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−
⋅
⋅ CR "Tropical"=if
110 mA m2−
⋅
⋅ CR "Sub-Tropical"=if
130 mA m2−
⋅
⋅ CR "Temperate"=if
170 mA m2−
⋅
⋅ otherwise
Depth 30m≤if
80 mA m2−
⋅
⋅ CR "Tropical"=if
90 mA m2−
⋅
⋅ CR "Sub-Tropical"=if
110 mA m2−
⋅
⋅ CR "Temperate"=if
130mA m2−
⋅ otherwise
otherwise
:=
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CATHODIC PROTECTION OF PIPELINES
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−
⋅=
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CATHODIC PROTECTION OF PIPELINES
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⋅−:=
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CATHODIC PROTECTION OF PIPELINES
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
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CATHODIC PROTECTION OF PIPELINES
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
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CATHODIC PROTECTION OF PIPELINES
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=
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