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4/25/2016 Volume and Wetted Area of Partially Filled Horizontal Vessels – Neutrium
https://neutrium.net/equipment/volumeandwettedareaofpartiallyfilledvessels/ 1/12
NeutriumDONATECONTACTPODCASTARTICLES
VOLUME AND WETTED AREA OF PARTIALLY FILLED HORIZONTAL VESSELS
SUMMARY
The calculation of a horizontal vessels wetted area and volume is
required for engineering tasks such fire studies and the determination
of level alarms and control set points. However the calculation of these
parameters is complicated by the geometry of the vessel, particularly
the heads. This article details formulae for calculating the wetted area
and volume of these vessels for various types of curved ends including:
hemispherical, torispherical, semi-ellipsoidal and bumped ends.
1. DEFINITIONS
: Wetted Area
: Inside Diameter of Vessel
: Outside Diameter of Vessel
: Liquid level above vessel bottom
: Length of vessel, tan-line to tan-line
: Straight Flange
: Inside Vessel Radius
f
A
Di
Do
h
L
Lf
R
4/25/2016 Volume and Wetted Area of Partially Filled Horizontal Vessels – Neutrium
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: Inside crown radius
: Inside knuckle radius
: Vessel Wall Thickness
: Partially Filled Liquid Volume
: Total Volume of head or vessel
: Inside Dish Depth
: Eccentricity of elliptical heads
2. INTRODUCTION
The calculation of the liquid volume or wetted area of a partially filled
horizontal vessel is best performed in parts, by calculating the value
for the cylindrical section of the vessel and the heads of the vessel and
then adding the areas or volumes together. Below we present the
wetted area and partially filled volume for each type of head and the
cylindrical section.
Rc
Rk
t
Vp
Vt
z
ε
4/25/2016 Volume and Wetted Area of Partially Filled Horizontal Vessels – Neutrium
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The partially filled volume is primarily used for the calculation of tank
filling times and the setting of control set points, alarm levels and
system trip points.
The wetted area is the area of contact between the liquid and the wall
of the tank. This is primary used in fire studies of process and storage
vessels to determine the emergency venting capacity required to
protect the vessel.
The volume and wetted area of partially filled vertical vessels is
covered separately.
3. HEMISPHERICAL HEADS - HORIZONTAL VESSEL
4/25/2016 Volume and Wetted Area of Partially Filled Horizontal Vessels – Neutrium
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3. HEMISPHERICAL HEADS - HORIZONTAL VESSEL
Hemispherical heads have a depth which is half their diameter. They
have the highest design pressures out of all the head types and as such
are typically the most expensive head type. The formula for calculating
the wetted area and volume of one head are presented as follows.
3.1 Wetted Area
3.2 Volume
4. SEMI-ELLIPSOIDAL OR ELLIPTICAL HEADS - HORIZONTAL VESSEL
A = πhDi
2
= π (3R − h)Vp16
h2
= (3 − 2 )Vp D3i
π
12( )h
Di
2 ( )h
Di
3
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The semi-ellipsoidal heads are shallower than the hemispherical heads
and deeper than the torispherical heads and therefore have design
pressures and expense lying between these two designs.
The most common variant of semi-ellipsoidal head is the 2:1 elliptical
head which has a depth equal to 1/4 of the vessel diameter. The
formula for calculating the wetted area and volume for one 2:1 semi-
elliptical head are presented as follows.
4.1 Wetted Area
For a 2:1 semi-ellipsoidal head ε is equal to 0.866, for other geometries
the formula below may be used to calculate ε.
Aw
B
ε
= ( − 0.5) B + 1 + lnπD2
i
8
⎛⎝⎜
h
Di
14ε
⎛⎝⎜
4ε ( − 0.5) + BhDi
2 − 3√
⎞⎠⎟
⎞⎠⎟
= 1 + 12( − 0.5)h
Di
2− −−−−−−−−−−−−−−−√= 1 −
4z2
D2i
− −−−−−−√
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The wetted area calculated using this method does not include the
straight flange of the head. The length of the straight flange must be
included in the calculation of the wetted area of the cylindrical
section.
4.2 Volume
Where,
for ASME 2:1 Elliptical heads:
for DIN 28013 Semi ellipsoidal heads:
The volume calculated does not include the straight flange of the head,
only the curved section. The straight flange length must be included in
the calculation of the volume of the cylindrical section.
= C (3 − 2 )Vp D3i
π
12( )h
Di
2 ( )h
Di
3
C = 1/2
C = 0.49951 + 0.10462 + 2.3227t
Do( )t
Do
2
4/25/2016 Volume and Wetted Area of Partially Filled Horizontal Vessels – Neutrium
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5. TORISPHERICAL HEADS - HORIZONTAL VESSEL
Torispherical heads are the most economical and therefore is the most
common head type used for process vessels. Torispherical heads are
shallower and typically have lower design pressures than semi-
elliptical heads. The formula for the calculation of the wetted area and
volume of one partially filled torispherical head is presented as
follows.
5.1 Wetted Area
We can approximate the partially filled surface area of the
torispherical head using the formula for elliptical heads. This
approximation will over estimate the surface area because a
torispherical head is flatter than a ellipsoidal head. This assumption is
conservative for pool fire relieving calculations.
⎛ ⎛ ( ) ⎞⎞
4/25/2016 Volume and Wetted Area of Partially Filled Horizontal Vessels – Neutrium
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The wetted area calculated using this method does not include the
straight flange of the head. The length of the straight flange must be
included in the calculation of the wetted area of the cylindrical
section.
5.2 Volume
Where,
for ASME Torispherical heads:
for DIN 28011 Torispherical heads:
Aw
B
ε
= ( − 0.5) B + 1 + lnπD2
i
8
⎛⎝⎜
h
Di
14ε
⎛⎝⎜
4ε ( − 0.5) + BhDi
2 − 3√
⎞⎠⎟
⎞⎠⎟
= 1 + 12( − 0.5)h
Di
2− −−−−−−−−−−−−−−−√= 1 −
4z2
D2i
− −−−−−−√
= C (3 − 2 )Vp D3i
π
12( )h
Di
2 ( )h
Di
3
C = 0.30939 + 1.7197 − 0.16116 + 0.98997− 0.06Rk Do
Di
t
Do( )t
Do
2
C = 0.37802 + 0.05073 + 1.3762t
Do( )t
Do
2
4/25/2016 Volume and Wetted Area of Partially Filled Horizontal Vessels – Neutrium
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top
The volume calculated does not include the straight flange of the head,
only the curved section. The straight flange length must be included in
the calculation of the volume of the cylindrical section.
6. BUMPED HEADS - HORIZONTAL VESSEL
Bumped heads have the lowest cost but also the lowest design
pressures, unlike torispherical or ellipsoidal heads they have no
knuckle. They are typically used in atmospheric tanks, such as
horizontal liquid fuel storage tanks or road tankers.
Here we present formulae for calculated the wetted area and volume
for an arbitrary liquid level height in a single Bumped head.
6.1 Wetted Area
We can approximate the partially filled surface area of the bumped
head using the formula for elliptical heads. This approximation will
over estimate the surface area, which is conservative for pool fire
relieving calculations.
⎛ ⎛ ( ) ⎞⎞
4/25/2016 Volume and Wetted Area of Partially Filled Horizontal Vessels – Neutrium
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6.2 Volume
The partially filled volume equation is an approximation, but will give
a reasonable accuracy for vessel volume calculations.
7. CYLINDRICAL SECTION - HORIZONTAL VESSEL
Here we present formulae for calculated the wetted area and volume
for an arbitrary liquid level height in the cylindrical section of a
horizontal drum.
Aw
B
ε
= ( − 0.5) B + 1 + lnπD2
i
8
⎛⎝⎜
h
Di
14ε
⎛⎝⎜
4ε ( − 0.5) + BhDi
2 − 3√
⎞⎠⎟
⎞⎠⎟
= 1 + 12( − 0.5)h
Di
2− −−−−−−−−−−−−−−−√= 1 −
4z2
D2i
− −−−−−−√
= π (3 − z)Vt13
z2 Rc
= (1 − )Vp3Vt
4( )h
R
2h
3R
4/25/2016 Volume and Wetted Area of Partially Filled Horizontal Vessels – Neutrium
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7.1 Wetted Area
7.2 Volume
Where the vessel has torispherical or ellipsoidal heads the straight
flange length of the head should be included in the cylindrical section
length when calculating the volume or surface area.
8. REFERENCES
= 2LRco ( )Ap s−1 R − h
R
= L co (1 − 2 )Ap Di s−1 h
Di
= L( co ( ) − (R − H) )Vp R2 s−1 R − h
R2Rh − h2− −−−−−−−
√
= L ( co (1 − 2 ) − ( − ) )Vp D2i
14
s−1 h
Di
12
h
Di−
h
Di( )h
Di
2− −−−−−−−−−−√
4/25/2016 Volume and Wetted Area of Partially Filled Horizontal Vessels – Neutrium
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Article Created: March 1, 2013
ARTICLE TAGS
1. B Wiencke, 2009, Computing the partial volume of pressure vessels
2. R Doane, 2007, Accurate Wetted Areas for Partially Filled Vessels
3. E Ludwing, 1997, Applied Process Design for Chemical and
Petrochemical Plants (Volume 2)
4. E Weisstein, 2013, Cylindrical Segment. From MathWorld
Bumped Cylindrical Dished Hemispherical Horizontal Drum
Liquid Level Partially Filled Torospherical Vessel Vessel Head
Volume Wetted Area
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