Volumen total de superadobe Al calcular el volumen de superadobe se obtienen las longitudes necesarias de saco y alambre, y el volumen de material de drenaje.
Para el cálculo del volumen de superadobe, se calcula la longitud de cada hilada —longitud de la circunferencia que describe el tubo— en el punto medio del saco —por similitud con el cálculo del volumen de un toro— y se multiplica por el área de la sección del saco lleno y compactado . En cada hilada, la longitud de la circunferencia es función del radio del domo a la altura del saco.
Por debajo de la línea de surgencia el radio es constante:
r
C
=
r
+
3
2
s
w
— para el volumen de C
{\displaystyle r_{C}={\color {Green}{r}+{3 \over 2}\color {Green}{s_{w}}}{\text{ — para el volumen de C}}}
r
D
,
E
=
r
+
1
2
s
w
— para los volúmenes de D y E
{\displaystyle r_{D,E}={\color {Green}{r}+{1 \over 2}\color {Green}{s_{w}}}{\text{ — para los volúmenes de D y E}}}
Por encima de la línea de surgencia, el radio
r
n
{\displaystyle r_{n}}
l
{\displaystyle l}
h
n
{\displaystyle h_{n}}
n
−
1
2
{\displaystyle n-{1 \over 2}}
s
h
{\displaystyle s_{h}}
l
2
=
h
n
2
+
l
n
2
{\displaystyle \color {Green}{l}^{2}=h_{n}^{2}+l_{n}^{2}}
l
n
=
l
−
r
+
r
n
{\displaystyle l_{n}=\color {Green}{l}-\color {Green}{r}+r_{n}}
h
n
=
(
n
−
1
2
)
s
h
{\displaystyle h_{n}=\left(n-{1 \over 2}\right)\color {Green}{s_{h}}}
l
2
=
[
(
n
−
1
2
)
s
h
]
2
+
(
l
−
r
+
r
n
)
2
{\displaystyle \color {Green}{l}^{2}=\left[\left(n-{1 \over 2}\right)\color {Green}{s_{h}}\right]^{2}+\left(\color {Green}{l}-\color {Green}{r}+r_{n}\right)^{2}}
l
−
r
+
r
n
=
l
2
−
[
(
n
−
1
2
)
s
h
]
2
{\displaystyle \color {Green}{l}-\color {Green}{r}+r_{n}={\sqrt {\color {Green}{l}^{2}-\left[\left(n-{1 \over 2}\right)\color {Green}{s_{h}}\right]^{2}}}}
r
n
=
r
−
l
+
l
2
−
[
(
n
−
1
2
)
s
h
]
2
{\displaystyle r_{n}=\color {Green}{r}-\color {Green}{l}+{\sqrt {\color {Green}{l}^{2}-\left[\left(n-{1 \over 2}\right)\color {Green}{s_{h}}\right]^{2}}}}
Añadiendo la mitad de la anchura del saco lleno, el radio resultante para el cálculo de los volúmenes del muro del domo por encima de la línea de surgencia (volúmenes en A) es:
r
n
(
A
)
=
r
−
l
+
1
2
s
w
+
l
2
−
[
(
n
−
1
2
)
s
h
]
2
{\displaystyle r_{n(A)}=\color {Green}{r}-\color {Green}{l}+{1 \over 2}\color {Green}{s_{w}}+{\sqrt {\color {Green}{l}^{2}-\left[\left(n-{1 \over 2}\right)\color {Green}{s_{h}}\right]^{2}}}}
Los volúmenes de B se calculan añadiendo a la fórmula anterior la anchura del saco lleno:
r
n
(
B
)
=
r
−
l
+
3
2
s
w
+
l
2
−
[
(
n
−
1
2
)
s
h
]
2
{\displaystyle r_{n(B)}=\color {Green}{r}-\color {Green}{l}+{3 \over 2}\color {Green}{s_{w}}+{\sqrt {\color {Green}{l}^{2}-\left[\left(n-{1 \over 2}\right)\color {Green}{s_{h}}\right]^{2}}}}
Área de la sección del saco Literatura y datos empíricos proporcionan información sobre la anchura y altura final del saco una vez lleno y compactado, pero no del área de la sección del saco: la forma de la sección condiciona el área, y esta el volumen resultante.
La forma real del saco lleno y compactado tiene los laterales aproximadamente en forma de segmento circular. Por facilidad de cálculo se puede considerar que los laterales son semicírculos, aunque el área de la sección es menor y por lo tanto también el volumen calculado, lo cual puede suponer un defecto de material. Considerando la sección del saco como un rectángulo en el que quedarían inscritas las dos secciones anteriores, se facilitan los cálculos y se añade a los mismos un exceso de material que conviene tener en cuenta como parte de la merma de material. Además, de acuerdo con los datos anteriores , se puede considerar que la altura final del saco es aproximadamente el 27,50 % de la anchura del saco vacío:
L
v
— anchura del saco vacío
{\displaystyle \color {Green}{L_{v}}{\text{— anchura del saco vacío}}}
s
h
≈
0
,
275
×
L
v
— altura del saco lleno y compactado
{\displaystyle s_{h}\approx 0,275\times \color {Green}{L_{v}}{\text{— altura del saco lleno y compactado}}}
s
w
≈
0
,
725
×
L
v
— anchura del saco lleno y compactado
{\displaystyle s_{w}\approx 0,725\times \color {Green}{L_{v}}{\text{— anchura del saco lleno y compactado}}}
A
s
a
c
o
=
s
h
×
s
w
=
0
,
275
×
L
v
×
0
,
725
×
L
v
≈
0
,
2
×
L
v
2
{\displaystyle A_{saco}=s_{h}\times s_{w}=0,275\times \color {Green}{L_{v}}\times 0,725\times \color {Green}{L_{v}}\approx 0,2\times \color {Green}{L_{v}}^{2}}
V
A
{\displaystyle V_{A}}
N
— número de hiladas por encima de la línea de surgencia
{\displaystyle N{\text{ — número de hiladas por encima de la línea de surgencia}}}
l
2
=
h
2
+
(
l
−
r
)
2
{\displaystyle \color {Green}{l}^{2}=h^{2}+(\color {Green}{l}-\color {Green}{r})^{2}}
h
=
l
2
−
(
l
−
r
)
2
{\displaystyle h={\sqrt {\color {Green}{l}^{2}-\left(\color {Green}{l}-\color {Green}{r}\right)^{2}}}}
N
=
h
s
h
{\displaystyle N={\dfrac {h}{\color {Green}{s_{h}}}}}
N
=
l
2
−
(
l
−
r
)
2
s
h
{\displaystyle N={\dfrac {\sqrt {\color {Green}{l}^{2}-\left(\color {Green}{l}-\color {Green}{r}\right)^{2}}}{\color {Green}{s_{h}}}}}
V
A
(
n
)
=
A
s
a
c
o
×
2
π
r
n
(
A
)
=
A
s
a
c
o
×
2
π
(
r
−
l
+
1
2
s
w
+
l
2
−
[
(
n
−
1
2
)
s
h
]
2
)
{\displaystyle {\begin{aligned}V_{A}(n)&=A_{saco}\times 2\pi r_{n(A)}\\&=A_{saco}\times 2\pi \left(\color {Green}{r}-\color {Green}{l}+{\frac {1}{2}}\color {Green}{s_{w}}+{\sqrt {\color {Green}{l}^{2}-\left[\left(n-{\frac {1}{2}}\right)\color {Green}{s_{h}}\right]^{2}}}\right)\end{aligned}}}
V
A
=
A
s
a
c
o
×
2
π
∑
n
=
1
N
(
r
−
l
+
1
2
s
w
+
l
2
−
[
(
n
−
1
2
)
s
h
]
2
)
=
A
s
a
c
o
×
2
π
[
N
(
r
−
l
+
1
2
s
w
)
+
∑
n
=
1
N
l
2
−
[
(
n
−
1
2
)
s
h
]
2
]
{\displaystyle {\begin{aligned}V_{A}&=A_{saco}\times 2\pi \sum _{n=1}^{N}\left(\color {Green}{r}-\color {Green}{l}+{\frac {1}{2}}\color {Green}{s_{w}}+{\sqrt {\color {Green}{l}^{2}-\left[\left(n-{\frac {1}{2}}\right)\color {Green}{s_{h}}\right]^{2}}}\right)\\&=A_{saco}\times 2\pi \left[N\left(\color {Green}{r}-\color {Green}{l}+{\frac {1}{2}}\color {Green}{s_{w}}\right)+\sum _{n=1}^{N}{\sqrt {\color {Green}{l}^{2}-\left[\left(n-{\frac {1}{2}}\right)\color {Green}{s_{h}}\right]^{2}}}\right]\end{aligned}}}
V
B
{\displaystyle V_{B}}
h
c
— altura del contrafuerte por encima de la línea de surgencia (m)
{\displaystyle \color {Green}{h_{c}}{\text{ — altura del contrafuerte por encima de la línea de surgencia (m)}}}
C
— número de hiladas del contrafuerte por encima de la línea de surgencia
{\displaystyle C{\text{ — número de hiladas del contrafuerte por encima de la línea de surgencia}}}
C
=
h
c
s
h
{\displaystyle C={\dfrac {\color {Green}{h_{c}}}{\color {Green}{s_{h}}}}}
V
B
(
n
)
=
A
s
a
c
o
×
2
π
r
n
(
B
)
=
A
s
a
c
o
×
2
π
(
r
−
l
+
3
2
s
w
+
l
2
−
[
(
n
−
1
2
)
s
h
]
2
)
{\displaystyle {\begin{aligned}V_{B}(n)&=A_{saco}\times 2\pi r_{n(B)}\\&=A_{saco}\times 2\pi \left(\color {Green}{r}-\color {Green}{l}+{\frac {3}{2}}\color {Green}{s_{w}}+{\sqrt {\color {Green}{l}^{2}-\left[\left(n-{\frac {1}{2}}\right)\color {Green}{s_{h}}\right]^{2}}}\right)\end{aligned}}}
V
B
=
A
s
a
c
o
×
2
π
∑
n
=
1
C
(
r
−
l
+
3
2
s
w
+
l
2
−
[
(
n
−
1
2
)
s
h
]
2
)
=
A
s
a
c
o
×
2
π
[
C
(
r
−
l
+
3
2
s
w
)
+
∑
n
=
1
C
l
2
−
[
(
n
−
1
2
)
s
h
]
2
]
{\displaystyle {\begin{aligned}V_{B}&=A_{saco}\times 2\pi \sum _{n=1}^{C}\left(\color {Green}{r}-\color {Green}{l}+{\frac {3}{2}}\color {Green}{s_{w}}+{\sqrt {\color {Green}{l}^{2}-\left[\left(n-{\frac {1}{2}}\right)\color {Green}{s_{h}}\right]^{2}}}\right)\\&=A_{saco}\times 2\pi \left[C\left(\color {Green}{r}-\color {Green}{l}+{\frac {3}{2}}\color {Green}{s_{w}}\right)+\sum _{n=1}^{C}{\sqrt {\color {Green}{l}^{2}-\left[\left(n-{\frac {1}{2}}\right)\color {Green}{s_{h}}\right]^{2}}}\right]\end{aligned}}}
V
C
{\displaystyle V_{C}}
n
C
— número de hiladas hasta la línea de surgencia del contrafuerte
{\displaystyle \color {Green}{n_{C}}{\text{ — número de hiladas hasta la línea de surgencia del contrafuerte}}}
V
C
=
n
c
×
A
s
a
c
o
×
(
2
π
r
c
)
=
n
c
×
A
s
a
c
o
×
2
π
(
r
+
3
2
s
w
)
{\displaystyle {\begin{aligned}V_{C}&=\color {Green}{n_{c}}\times A_{saco}\times (2\pi r_{c})\\&=\color {Green}{n_{c}}\times A_{saco}\times 2\pi \left(\color {Green}{r}+{\frac {3}{2}}\color {Green}{s_{w}}\right)\end{aligned}}}
V
D
{\displaystyle V_{D}}
n
D
— número de hiladas hasta la línea de surgencia
{\displaystyle \color {Green}{n_{D}}{\text{ — número de hiladas hasta la línea de surgencia}}}
V
D
=
n
D
×
A
s
a
c
o
×
(
2
π
r
D
)
=
n
D
×
A
s
a
c
o
×
2
π
(
r
+
1
2
s
w
)
{\displaystyle {\begin{aligned}V_{D}&=\color {Green}{n_{D}}\times A_{saco}\times (2\pi r_{D})\\&=\color {Green}{n_{D}}\times A_{saco}\times 2\pi \left(\color {Green}{r}+{\frac {1}{2}}\color {Green}{s_{w}}\right)\end{aligned}}}
V
E
{\displaystyle V_{E}}
n
E
— número de hiladas en los cimientos
{\displaystyle \color {Green}{n_{E}}{\text{ — número de hiladas en los cimientos}}}
V
E
=
n
E
×
A
s
a
c
o
×
(
2
π
r
E
)
=
n
E
×
A
s
a
c
o
×
2
π
(
r
+
1
2
s
w
)
{\displaystyle {\begin{aligned}V_{E}&=\color {Green}{n_{E}}\times A_{saco}\times (2\pi r_{E})\\&=\color {Green}{n_{E}}\times A_{saco}\times 2\pi \left(\color {Green}{r}+{\frac {1}{2}}\color {Green}{s_{w}}\right)\end{aligned}}}
Volumen total
V
=
∑
x
=
A
E
V
x
=
V
A
+
V
B
+
V
C
+
V
C
+
V
E
=
A
s
a
c
o
×
2
π
[
N
(
r
−
l
+
1
2
s
w
)
+
∑
n
=
1
N
l
2
−
[
(
n
−
1
2
)
s
h
]
2
]
+
A
s
a
c
o
×
2
π
[
C
(
r
−
l
3
2
s
w
)
+
∑
n
=
1
C
l
2
−
[
(
n
−
1
2
)
s
h
]
2
]
+
A
s
a
c
o
×
2
π
×
n
C
(
r
+
3
2
s
w
)
+
A
s
a
c
o
×
2
π
×
n
D
(
r
+
1
2
s
w
)
+
A
s
a
c
o
×
2
π
×
n
E
(
r
+
1
2
s
w
)
=
A
s
a
c
o
×
2
π
[
(
N
+
C
)
(
r
−
l
+
1
2
s
w
)
+
C
s
w
+
∑
n
=
1
N
l
2
−
[
(
n
−
1
2
)
s
h
]
2
+
∑
n
=
1
C
l
2
−
[
(
n
−
1
2
)
s
h
]
2
+
(
n
C
+
n
E
+
n
D
)
(
r
+
1
2
s
w
)
+
n
C
s
w
]
=
A
s
a
c
o
×
2
π
[
(
N
+
C
)
(
r
−
l
+
1
2
s
w
)
+
(
n
C
+
n
E
+
n
D
)
(
r
+
1
2
s
w
)
+
(
C
+
n
C
)
s
w
+
∑
n
=
1
N
l
2
−
[
(
n
−
1
2
)
s
h
]
2
+
∑
n
=
1
C
l
2
−
[
(
n
−
1
2
)
s
h
]
2
]
=
A
s
a
c
o
×
2
π
[
(
N
+
C
+
n
C
+
n
D
+
n
E
)
(
r
+
1
2
s
w
)
−
l
(
N
+
C
)
+
(
C
+
n
C
)
s
w
+
∑
n
=
1
N
l
2
−
[
(
n
−
1
2
)
s
h
]
2
+
∑
n
=
1
C
l
2
−
[
(
n
−
1
2
)
s
h
]
2
]
=
A
s
a
c
o
×
2
π
[
(
N
+
C
+
n
C
+
n
D
+
n
E
)
(
r
+
1
2
s
w
)
−
l
(
N
+
C
)
+
(
C
+
n
C
)
s
w
+
2
∑
n
=
1
C
l
2
−
[
(
n
−
1
2
)
s
h
]
2
+
∑
n
=
C
+
1
N
l
2
−
[
(
n
−
1
2
)
s
h
]
2
]
=
0
,
2
×
L
v
2
×
2
π
[
(
N
+
C
+
n
C
+
n
D
+
n
E
)
(
r
+
1
2
×
0
,
725
L
v
)
−
l
(
N
+
C
)
+
(
C
+
n
C
)
×
0
,
725
L
v
+
2
∑
n
=
1
C
l
2
−
[
(
n
−
1
2
)
×
0
,
275
L
v
]
2
+
∑
n
=
C
+
1
N
l
2
−
[
(
n
−
1
2
)
×
0
,
275
L
v
]
2
]
{\displaystyle {\begin{aligned}V&=\sum _{x=A}^{E}V_{x}=V_{A}+V_{B}+V_{C}+V_{C}+V_{E}\\&=A_{saco}\times 2\pi \left[N\left(\color {Green}{r}-\color {Green}{l}+{\frac {1}{2}}\color {Green}{s_{w}}\right)+\sum _{n=1}^{N}{\sqrt {\color {Green}{l}^{2}-\left[\left(n-{\frac {1}{2}}\right)\color {Green}{s_{h}}\right]^{2}}}\right]\\&+A_{saco}\times 2\pi \left[C\left(\color {Green}{r}-\color {Green}{l}{\frac {3}{2}}\color {Green}{s_{w}}\right)+\sum _{n=1}^{C}{\sqrt {\color {Green}{l}^{2}-\left[\left(n-{\frac {1}{2}}\right)\color {Green}{s_{h}}\right]^{2}}}\right]\\&+A_{saco}\times 2\pi \times \color {Green}{n_{C}}\left(\color {Green}{r}+{\frac {3}{2}}\color {Green}{s_{w}}\right)\\&+A_{saco}\times 2\pi \times \color {Green}{n_{D}}\left(\color {Green}{r}+{\frac {1}{2}}\color {Green}{s_{w}}\right)\\&+A_{saco}\times 2\pi \times \color {Green}{n_{E}}\left(\color {Green}{r}+{\frac {1}{2}}\color {Green}{s_{w}}\right)\\&=A_{saco}\times 2\pi \left[\left(N+C\right)\left(\color {Green}{r}-\color {Green}{l}+{\frac {1}{2}}\color {Green}{s_{w}}\right)+C\color {Green}{s_{w}}+\sum _{n=1}^{N}{\sqrt {\color {Green}{l}^{2}-\left[\left(n-{\frac {1}{2}}\right)\color {Green}{s_{h}}\right]^{2}}}+\sum _{n=1}^{C}{\sqrt {\color {Green}{l}^{2}-\left[\left(n-{\frac {1}{2}}\right)\color {Green}{s_{h}}\right]^{2}}}+\left(\color {Green}{n_{C}}+\color {Green}{n_{E}}+\color {Green}{n_{D}}\right)\left(\color {Green}{r}+{\frac {1}{2}}\color {Green}{s_{w}}\right)+\color {Green}{n_{C}}\color {Green}{s_{w}}\right]\\&=A_{saco}\times 2\pi \left[\left(N+C\right)\left(\color {Green}{r}-\color {Green}{l}+{\frac {1}{2}}\color {Green}{s_{w}}\right)+\left(\color {Green}{n_{C}}+\color {Green}{n_{E}}+\color {Green}{n_{D}}\right)\left(\color {Green}{r}+{\frac {1}{2}}\color {Green}{s_{w}}\right)+\left(C+\color {Green}{n_{C}}\right)\color {Green}{s_{w}}+\sum _{n=1}^{N}{\sqrt {\color {Green}{l}^{2}-\left[\left(n-{\frac {1}{2}}\right)\color {Green}{s_{h}}\right]^{2}}}+\sum _{n=1}^{C}{\sqrt {\color {Green}{l}^{2}-\left[\left(n-{\frac {1}{2}}\right)\color {Green}{s_{h}}\right]^{2}}}\right]\\&=A_{saco}\times 2\pi \left[\left(N+C+\color {Green}{n_{C}}+\color {Green}{n_{D}}+\color {Green}{n_{E}}\right)\left(\color {Green}{r}+{\frac {1}{2}}\color {Green}{s_{w}}\right)-\color {Green}{l}\left(N+C\right)+\left(C+\color {Green}{n_{C}}\right)\color {Green}{s_{w}}+\sum _{n=1}^{N}{\sqrt {\color {Green}{l}^{2}-\left[\left(n-{\frac {1}{2}}\right)\color {Green}{s_{h}}\right]^{2}}}+\sum _{n=1}^{C}{\sqrt {\color {Green}{l}^{2}-\left[\left(n-{\frac {1}{2}}\right)\color {Green}{s_{h}}\right]^{2}}}\right]\\&=A_{saco}\times 2\pi \left[\left(N+C+\color {Green}{n_{C}}+\color {Green}{n_{D}}+\color {Green}{n_{E}}\right)\left(\color {Green}{r}+{\frac {1}{2}}\color {Green}{s_{w}}\right)-\color {Green}{l}\left(N+C\right)+\left(C+\color {Green}{n_{C}}\right)\color {Green}{s_{w}}+2\sum _{n=1}^{C}{\sqrt {\color {Green}{l}^{2}-\left[\left(n-{\frac {1}{2}}\right)\color {Green}{s_{h}}\right]^{2}}}+\sum _{n=C+1}^{N}{\sqrt {\color {Green}{l}^{2}-\left[\left(n-{\frac {1}{2}}\right)\color {Green}{s_{h}}\right]^{2}}}\right]\\&=0,2\times \color {Green}{L_{v}}^{2}\times 2\pi \left[\left(N+C+\color {Green}{n_{C}}+\color {Green}{n_{D}}+\color {Green}{n_{E}}\right)\left(\color {Green}{r}+{\frac {1}{2}}\times 0,725\color {Green}{L_{v}}\right)-\color {Green}{l}\left(N+C\right)+\left(C+\color {Green}{n_{C}}\right)\times 0,725\color {Green}{L_{v}}+2\sum _{n=1}^{C}{\sqrt {\color {Green}{l}^{2}-\left[\left(n-{\frac {1}{2}}\right)\times 0,275\color {Green}{L_{v}}\right]^{2}}}+\sum _{n=C+1}^{N}{\sqrt {\color {Green}{l}^{2}-\left[\left(n-{\frac {1}{2}}\right)\times 0,275\color {Green}{L_{v}}\right]^{2}}}\right]\end{aligned}}}
![{\displaystyle \color {Green}{l}^{2}=\left[\left(n-{1 \over 2}\right)\color {Green}{s_{h}}\right]^{2}+\left(\color {Green}{l}-\color {Green}{r}+r_{n}\right)^{2}}](https://es.wikipedia.org/api/rest_v1/media/math/render/svg/6c9490e05670c41021fe36da68d636de2b25d5a1)
![{\displaystyle \color {Green}{l}-\color {Green}{r}+r_{n}={\sqrt {\color {Green}{l}^{2}-\left[\left(n-{1 \over 2}\right)\color {Green}{s_{h}}\right]^{2}}}}](https://es.wikipedia.org/api/rest_v1/media/math/render/svg/d2c79f9d56a44b96d60d5eed11f6135a8450ccd2)
![{\displaystyle r_{n}=\color {Green}{r}-\color {Green}{l}+{\sqrt {\color {Green}{l}^{2}-\left[\left(n-{1 \over 2}\right)\color {Green}{s_{h}}\right]^{2}}}}](https://es.wikipedia.org/api/rest_v1/media/math/render/svg/dbc87af13446f7d6f461eb7523256e0be5dd3ccc)
![r_{n(A)} = \color{Green}{r} - \color{Green}{l} + {1 \over 2}\color{Green}{s_w} + \sqrt{\color{Green}{l}^2 - \left[\left(n - {1 \over 2} \right)\color{Green}{s_h} \right]^2}](https://es.wikipedia.org/api/rest_v1/media/math/render/svg/49ae68cfb834b686089baf27472b4805d2a6c60d)
![r_{n(B)} = \color{Green}{r} - \color{Green}{l} + {3 \over 2}\color{Green}{s_w} + \sqrt{\color{Green}{l}^2 - \left[\left(n - {1 \over 2} \right)\color{Green}{s_h} \right]^2}](https://es.wikipedia.org/api/rest_v1/media/math/render/svg/7e274f3f544683cf55588b51c633568df8ee6c52)
![\begin{align}
V_A(n) & = A_{saco} \times 2 \pi r_{n(A)} \\
& = A_{saco} \times 2 \pi \left(\color{Green}{r} - \color{Green}{l} + \frac{1}{2}\color{Green}{s_w} + \sqrt{\color{Green}{l}^2-\left[\left(n - \frac{1}{2} \right)\color{Green}{s_h} \right]^2} \right)
\end{align}](https://es.wikipedia.org/api/rest_v1/media/math/render/svg/86a954d8fbb8483dc7d189df28f69b113b6f5324)
![\begin{align}
V_A & = A_{saco} \times 2 \pi \sum_{n=1}^N \left(\color{Green}{r} - \color{Green}{l} + \frac{1}{2}\color{Green}{s_w} + \sqrt{\color{Green}{l}^2-\left[\left(n - \frac{1}{2} \right)\color{Green}{s_h} \right]^2} \right) \\
& = A_{saco} \times 2 \pi \left[ N \left(\color{Green}{r} - \color{Green}{l} + \frac{1}{2}\color{Green}{s_w} \right) + \sum_{n=1}^N \sqrt{\color{Green}{l}^2-\left[\left(n - \frac{1}{2} \right)\color{Green}{s_h} \right]^2} \right]
\end{align}](https://es.wikipedia.org/api/rest_v1/media/math/render/svg/6bce79740517e32aeadec354ded1d84dc61bb1b3)
![\begin{align}
V_B(n) & = A_{saco} \times 2 \pi r_{n(B)} \\
& = A_{saco} \times 2 \pi \left(\color{Green}{r} - \color{Green}{l} + \frac{3}{2}\color{Green}{s_w} + \sqrt{\color{Green}{l}^2-\left[\left(n - \frac{1}{2} \right)\color{Green}{s_h} \right]^2} \right)
\end{align}](https://es.wikipedia.org/api/rest_v1/media/math/render/svg/642142bfbf64afa94eb5f3d3ea0e6877d1831ff1)
![\begin{align}
V_B & = A_{saco} \times 2 \pi \sum_{n=1}^C \left(\color{Green}{r} - \color{Green}{l} + \frac{3}{2}\color{Green}{s_w} + \sqrt{\color{Green}{l}^2-\left[\left(n - \frac{1}{2} \right)\color{Green}{s_h} \right]^2} \right) \\
& = A_{saco} \times 2 \pi \left[ C \left(\color{Green}{r} - \color{Green}{l} + \frac{3}{2}\color{Green}{s_w} \right) + \sum_{n=1}^C \sqrt{\color{Green}{l}^2-\left[\left(n - \frac{1}{2} \right)\color{Green}{s_h} \right]^2} \right]
\end{align}](https://es.wikipedia.org/api/rest_v1/media/math/render/svg/8d5079d82031620d26b6fd2b971e097a03471f51)
![\begin{align}
V & = \sum_{x=A}^{E} V_x = V_A + V_B + V_C + V_C + V_E
\\
& = A_{saco} \times 2 \pi \left[ N \left(\color{Green}{r} - \color{Green}{l} + \frac{1}{2}\color{Green}{s_w} \right) + \sum_{n=1}^N \sqrt{\color{Green}{l}^2-\left[\left(n - \frac{1}{2} \right) \color{Green}{s_h} \right]^2} \right]
\\
& + A_{saco} \times 2 \pi \left[ C \left(\color{Green}{r} - \color{Green}{l} \frac{3}{2}\color{Green}{s_w} \right) + \sum_{n=1}^C \sqrt{\color{Green}{l}^2-\left[\left(n - \frac{1}{2} \right)\color{Green}{s_h} \right]^2} \right]
\\
& + A_{saco} \times 2 \pi \times \color{Green}{n_C} \left(\color{Green}{r} + \frac{3}{2} \color{Green}{s_w} \right)
\\
& + A_{saco} \times 2 \pi \times \color{Green}{n_D} \left(\color{Green}{r} + \frac{1}{2} \color{Green}{s_w} \right)
\\
& + A_{saco} \times 2 \pi \times \color{Green}{n_E} \left(\color{Green}{r} + \frac{1}{2} \color{Green}{s_w} \right) \\
& = A_{saco} \times 2 \pi
\left[
\left(N + C \right)
\left(\color{Green}{r} -\color{Green}{l} + \frac{1}{2} \color{Green}{s_w} \right)
+ C \color{Green}{s_w}
+ \sum_{n=1}^N \sqrt{
\color{Green}{l}^2 -
\left[
\left(n - \frac{1}{2} \right) \color{Green}{s_h}
\right]^2
}
+ \sum_{n=1}^C \sqrt{
\color{Green}{l}^2 -
\left[
\left(n - \frac{1}{2} \right) \color{Green}{s_h}
\right]^2
}
+ \left(\color{Green}{n_C} + \color{Green}{n_E} + \color{Green}{n_D}\right) \left(\color{Green}{r} + \frac{1}{2} \color{Green}{s_w} \right)
+ \color{Green}{n_C} \color{Green}{s_w}
\right] \\
& = A_{saco} \times 2 \pi
\left[
\left(N + C \right)
\left(\color{Green}{r} -\color{Green}{l} + \frac{1}{2} \color{Green}{s_w} \right)
+ \left(\color{Green}{n_C} + \color{Green}{n_E} + \color{Green}{n_D}\right) \left(\color{Green}{r} + \frac{1}{2} \color{Green}{s_w} \right)
+ \left(C + \color{Green}{n_C} \right) \color{Green}{s_w}
+ \sum_{n=1}^N \sqrt{
\color{Green}{l}^2 -
\left[
\left(n - \frac{1}{2} \right) \color{Green}{s_h}
\right]^2
}
+ \sum_{n=1}^C \sqrt{
\color{Green}{l}^2 -
\left[
\left(n - \frac{1}{2} \right) \color{Green}{s_h}
\right]^2
}
\right] \\
& = A_{saco} \times 2 \pi
\left[
\left(N + C + \color{Green}{n_C} + \color{Green}{n_D} + \color{Green}{n_E} \right)
\left(\color{Green}{r} + \frac{1}{2} \color{Green}{s_w} \right)
- \color{Green}{l} \left(N + C \right)
+ \left(C + \color{Green}{n_C} \right) \color{Green}{s_w}
+ \sum_{n=1}^N \sqrt{
\color{Green}{l}^2 -
\left[
\left(n - \frac{1}{2} \right) \color{Green}{s_h}
\right]^2
}
+ \sum_{n=1}^C \sqrt{
\color{Green}{l}^2 -
\left[
\left(n - \frac{1}{2} \right) \color{Green}{s_h}
\right]^2
}
\right] \\
& = A_{saco} \times 2 \pi
\left[
\left(N + C + \color{Green}{n_C} + \color{Green}{n_D} + \color{Green}{n_E} \right)
\left(\color{Green}{r} + \frac{1}{2} \color{Green}{s_w} \right)
- \color{Green}{l} \left(N + C \right)
+ \left(C + \color{Green}{n_C} \right) \color{Green}{s_w}
+ 2 \sum_{n=1}^C \sqrt{
\color{Green}{l}^2 -
\left[
\left(n - \frac{1}{2} \right) \color{Green}{s_h}
\right]^2
}
+ \sum_{n=C+1}^N \sqrt{
\color{Green}{l}^2 -
\left[
\left(n - \frac{1}{2} \right) \color{Green}{s_h}
\right]^2
}
\right] \\
& = 0,2 \times \color{Green}{L_v}^2 \times 2 \pi
\left[
\left(N + C + \color{Green}{n_C} + \color{Green}{n_D} + \color{Green}{n_E} \right)
\left(\color{Green}{r} + \frac{1}{2} \times 0,725 \color{Green}{L_v} \right)
- \color{Green}{l} \left(N + C \right)
+ \left(C + \color{Green}{n_C} \right) \times 0,725 \color{Green}{L_v}
+ 2 \sum_{n=1}^C \sqrt{
\color{Green}{l}^2 -
\left[
\left(n - \frac{1}{2} \right) \times 0,275 \color{Green}{L_v}
\right]^2
}
+ \sum_{n=C+1}^N \sqrt{
\color{Green}{l}^2 -
\left[
\left(n - \frac{1}{2} \right) \times 0,275 \color{Green}{L_v}
\right]^2
}
\right]
\end{align}](https://es.wikipedia.org/api/rest_v1/media/math/render/svg/7712ed19cbf282a041b3e62223c485f1173dffc8)