Calculating Tank Wetted Area - Chemical Processing · Page 3 of 17 Horizontal Cylindrical Tank Equations Here are the specific equations for wetted surface areas of horizontal cylindrical

Page 1 of 17

Calculating Tank Wetted Area Saving time, increasing accuracy By Dan Jones, Ph.D., P.E.

alculating wetted area in a partially-filled horizontal or vertical cylindrical or elliptical tank can be complicated, depending on fluid height and the shape of the heads (ends) of a horizontal tank or the head (bottom) of a vertical tank. Exact equations are now available for several commonly-

encountered tank shapes. These equations can be used to make rapid and accurate wetted-area calculations. All equations are rigorous. All area equations give wetted areas in square units from tank dimensions in consistent linear units. All variables defining tank shapes required for tank wetted-area calculations are defined in the “Variables and Definitions” sidebar. Graphically, Figs. 1 and 2 show horizontal cylindrical tank parameters, Figs. 3 and 4 show vertical cylindrical tank parameters, and Fig. 5 shows some horizontal elliptical tank parameters . Elliptical horizontal tanks with hemiellipsoidal heads and vertical tanks with hemiellipsoidal bottoms are the only types of elliptical tanks considered. Horizontal Cylindrical Tanks Wetted area as a function of fluid height can be calculated for a horizontal cylindrical tank with either conical, hemispheroidal, guppy, spherical, or torispherical heads where the fluid height, h, is measured from the tank bottom to the fluid surface, see Figs. 1 and 2. A guppy head is a conical head of a horizontal tank where the apex of the conical head is level with the top of the cylindrical section of the tank as shown in Fig. 1. A torispherical head is an ASME-type (dished) head defined by a knuckle-radius parameter, k, and a dish-radius parameter, f, as shown in Fig. 2. A spheroidal head must be a hemispheroid; only a hemispheroid is valid – no “segment” of an spheroid will work as it will in the case of a spherical head where the head may be a spherical segment. For a spherical head, |a| £ R, where R is the radius of the cylindrical tank body. Where concave conical, hemispheroidal, guppy, spherical, or torispherical heads are considered, then |a| £ L/2. Both heads of a horizontal cylindrical tank must be identical for the equations to work; i.e., if one head is conical, the other must be conical with the same dimensions. However, the equations can be combined to deal with wetted-area calculations of horizontal tanks with heads of different shapes. For instance, if a horizontal cylindrical tank has a conical head on one end and a spherical head on the other end, calculate fluid wetted areas of two tanks, one with conical heads and the other with spherical heads, and average the results to get the desired wetted area. The heads of a horizontal tank may be flat (a = 0), convex (a > 0), or concave (a < 0). Wetted head areas are the same for convex or concave heads with the same dimensions, so the same formulas can be used for either convex or concave heads. The following variables must be within the ranges stated:

• |a| £ R for spherical heads • |a| £ L/2 for concave ends • 0 £ h £ 2R for all tanks • f > 0.5 for torispherical heads • 0 < k < 0.5 for torispherical heads • D > 0 • L ³ 0

C

Page 2 of 17

Variables and Definitions Sidebar (See Figs. 1-5) a is the distance a horizontal tank's heads extend beyond (a > 0) or into (a < 0) its cylindrical or elliptical

body section or the depth the bottom extends below the cylindrical or elliptical body section of a vertical tank. For a horizontal tank with flat heads or a vertical tank with a flat bottom a = 0.

b is the horizontal semiaxis of the elliptical cross section of a horizontal elliptical tank or the major

semiaxis of the elliptical cross section of a vertical elliptical tank. c is the vertical semiaxis of the elliptical cross section of a horizontal elliptical tank or the minor semiaxis

of the elliptical cross section of a vertical elliptical tank. C is the wetted cross-sectional length of the fluid in a horizontal tank's cylindrical section. D is the diameter of the cylindrical section of a horizontal or vertical tank. f is the dish-radius parameter for tanks with torispherical (dished) heads (ends or bottoms); fD is the dish

radius. h is the height of fluid in a tank measured from the lowest part of the tank to the fluid surface. k is the knuckle-radius parameter for tanks with torispherical heads (ends or bottoms); kD is the knuckle

radius. L is the length of the cylindrical section of a horizontal tank. r is the radius of a spherical head of a horizontal tank or a spherical bottom of a vertical tank. R is the radius of the cylindrical section of a horizontal or vertical tank. S is the wetted area, of fluid depth h, in a horizontal or vertical cylindrical tank.

Page 3 of 17

Horizontal Cylindrical Tank Equations Here are the specific equations for wetted surface areas of horizontal cylindrical tanks with conical, hemispheroidal, guppy, spherical, and torispherical heads (use radian angular measure for all trigonometric functions, and D/2 = R > 0 for all equations): Conical heads.

÷øö

çèæ ---

-++= - 212

22

22 hRh)hR(RhRcosR

RRaCLS …………………..……………

ïî

ïíì

££ Dh

aall

0

Hemispheroidal heads.

( ) ( )ò ò-

-

-+--+-

+=

R

hR

R

RRaRaR

RCLS

22

0

222

42222224x

dydxyx

yx ………………………. ïî

ïíì

££ Dh

aall

0

special cases of hemispheroidal heads:

212 222 hRh)hR(RhRcosRCLS ---

-+= - ………………………………………….

îíì

££=

Dha00

22

22

22

22

aRR

aRRlnaR2

RaRLRS--

-+

-

p+p+p= ……………………….....……

îíì

=<<

RhRa0

( )DLRS +p= …………………………………………………………………..……. îíì

==RhRa

aRcos

Ra

RaRLRS 1

22

22 -

-

p+p+p= …………………………………..………..…

îíì

=>RhRa

22

22

22

22

aRR

aRRlnaR

RaR2LR2S--

-+

-

p+p+p= ……………………………..…….

îíì

=<<

DhRa0

( )DLR2S +p= ……………………………………………………………..…..…… îíì

==DhRa

aRcos

Ra

RaRLRS 1

22

22 222 -

-

p+p+p= …………………………………..… …….

îíì

=>DhRa

Page 4 of 17

Guppy heads.

ò ò-

-

-

÷÷ø

öççè

æ-

+÷÷

ø

ö

çç

è

æ

÷÷ø

öççè

æ÷øö

çèæ

--++=

Rh

R

R

)R(Ra

RRaCLS

22

0

222

12

14

x

dydxx

yx

y ……………….. ïî

ïíì

££ Dh

aall

0

Spherical heads.

………………….…….….. ïî

ïíì

££

£<

Dh

Ra

0

0

special cases for spherical heads:

212 222 hRh)hR(RhRcosRCLS ---

-+= - …………………………………...…….

îíì

££=

Dha00

Rh2CLS p+= …………………………….………………………….…………...…………. |a|=R ( )22 RaCLS +p+= ……………………………………………………………………….…. h=R ( )22 Ra2CLS +p+= …………………………………………………………..……. h=D

( ) ( )ò-

-

-+

-÷÷ø

öççè

æ ++=

R

hR

xaRa

Rasin

aRaCLS d

x

x2222

221

22

2

22

Page 5 of 17

Torispherical heads.

( )ïïïïïïï

î

ïïïïïïï

í

ì

£<-÷øö

çèæ -+p+p+p

£<a--

a++

a-£+

=

-=-

a

-

-

-

òò ò

ò ò

DhR.......................SkDasinkDRaDkaDfDL

Rh)sin(kD..........................DfcosfDcosfD)t,s(fCL

)sin(kDh...............................................................)t,s(fCL

S

hDh

sinfD

hR

a G

hkDh G

2121

2221

0 0

2

0 0

44

144

14

2

2

dxx

dydx

dydx

special cases for torispherical heads:

( ) ÷ø

öçè

æ-+p+p+

p= -

kDasinkDRakDaDfDLS 21

21 222

………………….…. h=R

( ) ÷ø

öçè

æ-+p+p+p= -

kDasinkDRaDk4aDf4DLS 21

21 …………………….. h=D

where in all above equations:

)tsyts)(t(

)t()ts()t,s(f22222222

2222222

21

xxx

yxxx

-+--+-

-+-++º

222222 2 )hR(tstsG ---+-+º xx

kDRs -º kDt º )kf(2k21sin 1

--

=a -

( )a-= cos1fDa1 a= coskDa2 21 aaa += factordishf = radiusdishfD = factorknucklek = radiusknucklekD = In all horizontal tank equations above, S is the wetted area of fluid in the tank in square units consistent with the linear units of tank dimension parameters, and C is the cross-sectional wetted length of fluid on the cylindrical body of the tank in linear units consistent with the linear units used for R and h. The equation for C is given by:

Dh...................................................................................................RhRcosDC ££÷øö

çèæ -

= - 01

Page 6 of 17

Figure 1. Parameters for Horizontal Cylindrical Tanks with Conical, Hemispheroidal, Guppy, or Spherical Heads. Cylindrical Tube Spherical head Hemispheroidal head r(sphere) D

Guppy R h

head Conical head

a(sphere) a(hemispheroid)

a L (cone; guppy) C Wetted cross-sectional length CROSS SECTION OF CYLINDRICAL TUBE

h 1. Both heads of a tank must be identical. Above diagram is for definition of parameters only. 2. Cylindrical tube of diameter D (D > 0), radius R (R > 0), and length L (L ³ 0). 3. For spherical head of radius r, r ³ R and |a| £ R. 4. For convex head other than spherical, 0 < a < ¥, for concave head a < 0. 5. L ³ 0 for a ³ 0, L ³ 2|a| for a < 0. 6. Spheroidal head must be a hemispheroid. 7. 0 £ h £ D.

Page 7 of 17

Figure 2. Parameters for Horizontal Cylindrical Tanks with Torispherical Heads. kD

R D a1 a2 a

fD h L kd(1-sin α) Horizontal Cylindrical Tank Examples The following examples can be used to verify correct application of all equations. For all horizontal tank examples the tank diameter is 72” and the cylindrical tank body is 120” long. In all cases find the wetted surface, S, in ft2,given the stated head type and dimensions and the fluid height, h, in the tank: Conical head: a = 48”, h = 24” S = 101.35826 ft2 a = 48”, h = 36” S = 141.37167 ft2 a = 48”, h = 48” S = 181.38508 ft2 Hemispheroidal head: a = 24”, h = 24” S = 102.59905 ft2 a = 24”, h = 36” S = 138.74815 ft2 a = 24”, h = 48” S = 174.89725 ft2 a = 36”, h = 24” S = 111.55668 ft2 a = 36”, h = 36” S = 150.79645 ft2 a = 36”, h = 48” S = 190.03622 ft2 a = 48”, h = 24” S = 121.09692 ft2 a = 48”, h = 36” S = 163.71486 ft2 a = 48”, h = 48” S = 206.33279 ft2

Page 8 of 17

Guppy head: a = 48”, h = 24” S = 94.24500 ft2 a = 48”, h = 36” S = 129.98330 ft2 a = 48”, h = 48” S = 167.06207 ft2 Spherical head: a = 24”, h = 24” S = 99.49977 ft2 a = 24”, h = 36” S = 135.08848 ft2 a = 24”, h = 48” S = 170.67720 ft2 a = 36”, h = 24” S = 111.55668 ft2 a = 36”, h = 36” S = 150.79645 ft2 a = 36”, h = 48” S = 190.03622 ft2 Torispherical head: f = 1, k = 0.06, h = 2.28” S = 22.74924 ft2 f = 1, k = 0.06, h = 24” S = 94.29092 ft2 f = 1, k = 0.06, h = 36” S = 127.74876 ft2 f = 1, k = 0.06, h = 48” S = 161.20660 ft2 f = 1, k = 0.06, h = 69.72” S = 232.74828 ft2

For this torispherical head, a = 12.19231”. When h = 2.28”, the head wetted area is only in the toroidal section of each head, i.e., h ≤ kD(1-sin α).

f = 0.9, k = 0.1, h = 3” S = 26.82339 ft2 f = 0.9, k = 0.1, h = 24” S = 96.18257 ft2 f = 0.9, k = 0.1, h = 36” S = 130.22802 ft2 f = 0.9, k = 0.1, h = 48” S = 164.27347 ft2 f = 0.9, k = 0.1, h = 69” S = 233.63265 ft2

For this torispherical head, a = 14.91694”. When h = 3”, the head wetted area is only in the toroidal section of each head, i.e., h ≤ kD(1-sin α).

Page 9 of 17

Vertical Cylindrical Tanks Wetted surface area in a vertical cylindrical tank with either a conical, hemispheroidal, spherical, or torispherical bottom can be calculated, where the fluid height, h, is measured from the center of the bottom of the tank to the surface of the fluid in the tank. See Figs. 3 and 4 for tank configurations and dimension parameters, which are also defined in the “Variables and Definitions” sidebar. A torispherical bottom is an ASME-type bottom defined by a knuckle-radius factor and a dish-radius factor as shown graphically in Fig. 4. The knuckle radius will then be kD and the dish radius will be fD. A spheroidal bottom must be a hemispheroid. To find the wetted area of a concave bottom vertical tank, the equations for a convex bottom vertical tank can be used with the following formula:

ïïî

ïïí

ì

³+p

<-=-+p

=

ah.....................................................................areabottomtotalDh

ah...........................hahforareabottomareabottomtotalDh

)concave(S

The following parameter ranges must be observed:

• a ³ 0 for all vertical tanks, a £ R for a spherical bottom • f > 0.5 for a torispherical bottom • 0 < k < 0.5 for a torispherical bottom • D > 0

Vertical Cylindrical Tank Equations Here are the specific equations for wetted surface areas in vertical cylindrical tanks with conical, spheroidal, spherical, and torispherical bottoms. Use radian angular measure for all trigonometric functions; a ≥ 0 and D > 0 for all equations: Conical bottom.

ïïî

ïïí

ì

>-p++p

£+p

=

ah.....................................................................................)ah(DRaR

ah..................................................................................................a

RaRhS

22

2

222

Page 10 of 17

Hemispheroidal bottom.

RhRS p+p= 22 …..………………………...……………………………………………………......... ïî

ïíì

>

=

0

0

h

a

( )îíì

<>

÷÷

ø

ö

çç

è

æ ---

-

p+

----p-p=

--

ahRa

....................a

RahasinaRcos

Ra

Ra

a)Ra()ha(aR)ha(

RS

2

2211

22

2

2

22242

RhS p= 2 ……………………………………...…………………………………………………........... îíì

£=ahRa

( ) ( ) ( ) ïî

ïíì

<

<<

--++--

÷øöç

èæ +-

-

p+

--+-p-p=

ah

Ra.....

aRhaaaRha

RaRaln

aR

Ra

a)aR()ha(aR)ha(

RS

0222422

22

22

2

2

22242

( )ahRaRcos

Ra

RaRS -p+-

p+p= - 21

22

22 ..............................................................................

îíì

³>ahRa

)ah(RRS -p+p= 22 2 ………………………...…………..………………………….……………… îíì

>=ahRa

ïî

ïíì

³

<<-p+

--

-+

-

p+p=

ah

Ra....................................................)ah(R

aRR

aRRlnaR

RaRS0

22 22

22

22

22

Page 11 of 17

Spherical bottom.

( ) ( )ïïî

ïïí

ì

>-p++p

£÷÷ø

öççè

æ +p

=

ah.............................................................................ahDRa

ah..........................................................................................aRah

S22

22

Torispherical bottom.

1ah......................................................................................................fDh2S £p=

aha.........................kDhasin

kDasin)kDR(ahkDDafS £<÷÷

ø

öççè

æ÷ø

öçè

æ ---+-p+p= --

1121

11 22

( ) ah.................................ahRkDasin)kDR(akDDafS ³-p+÷

ø

öçè

æ-+p+p= - 222 21

21

( ) ( ) a=a-=-

-=a - cosDkacos1Dfa

kf2k21sin 21

1

factordishf = radiusdishfD = 21 aaa +=

factorknucklek = radiusknucklekD =

Page 12 of 17

Figure 3. Parameters for Vertical Cylindrical Tanks with Conical, Hemispheroidal, or Spherical Bottoms. D

h a spherical bottom hemispheroidal bottom Conical bottom Figure 4. Parameters for Vertical Cylindrical Tanks with Torispherical Bottoms. D R a fD

h kD

a2 a1

Page 13 of 17

Vertical Cylindrical Tank Examples The following examples can be used to verify correct application of all equations. For all examples, the tank diameter is 72” and the bottoms are convex or concave, as noted. In all cases find the wetted surface, S, in ft2, given the stated head type and dimensions and the fluid height, h, in the tank: Convex Bottom (ft2) Concave Bottom (ft2) Conical head: a = 36”, h = 36” S = 39.98595 S = 96.53461 a = 48”, h = 24” S = 11.78097 S = 73.04203 a = 48”, h = 36” S = 26.50719 S = 100.72731 a = 48”, h = 48” S = 47.12389 S = 122.52211 a = 48”, h = 60” S = 65.97345 S = 141.37167 a = 48”, h = 72” S = 84.82300 S = 160.22123 Hemispheroidal head: a = 24”, h = 12” S = 24.71061 S = 38.63931 a = 24”, h = 24” S = 44.50037 S = 82.19948 a = 24”, h = 48” S = 82.19948 S = 119.89859 a = 36”, h = 18” S = 28.27433 S = 56.54867 a = 36”, h = 36” S = 56.54867 S = 113.09734 a = 36”, h = 60” S = 94.24778 S = 150.79645 a = 48”, h = 24” S = 32.46693 S = 74.69926 a = 48”, h = 48” S = 69.46708 S = 144.86530 a = 48”, h = 72” S = 107.16619 S = 182.56441 Spherical head: a = 24”, h = 12” S = 20.42035 S = 39.26991 a = 24”, h = 24” S = 40.84070 S = 78.53982 a = 24”, h = 48” S = 78.53982 S = 116.23893 a = 36”, h = 18” S = 28.27433 S = 56.54867 a = 36”, h = 36” S = 56.54867 S = 113.09734 a = 36”, h = 60” S = 94.24778 S = 150.79645 Torispherical head: f = 1, k = 0.06, h = 2” S = 6.28319 S = 6.39290 f = 1, k = 0.06, h = 6” S = 18.84956 S = 23.47205 f = 1, k = 0.06, h = 12” S = 33.19882 S = 51.74638 f = 1, k = 0.06, h = 12.19231” S = 33.50098 S = 52.65262 f = 1, k = 0.06, h = 36.19231” S = 71.20010 S = 90.35173

For this torispherical head, a = 12.19231”, and when h = 2”, the head wetted area is only in the spherical part of the convex head and only in the torioidal part of the concave head.

f = 0.9, k = 0.1, h = 6” S = 16.96460 S = 20.22281 f = 0.9, k = 0.1, h = 8” S = 22.61947 S = 28.98943

f = 0.9, k = 0.1, h = 12” S = 31.28983 S = 46.58235 f = 0.9, k = 0.1, h = 14.91694” S = 35.98024 S = 59.41171 f = 0.9, k = 0.1, h = 38.91694” S = 73.67936 S = 97.11083

For this torispherical head, a = 14.91694”, and when h = 6”, the head wetted area is only in the spherical part of the convex head and only in the toroidal part of the concave head.

Page 14 of 17

Horizontal Elliptical Tanks with Hemiellipsoidal Heads A horizontal elliptical tank is a tank with a horizontal body which has an elliptical cross-section. The only case considered is one with hemiellipsoidal heads (which includes flat heads if a=0). Figure 5 shows the only configurations considered for horizontal elliptical tanks – the elliptical axes are parallel or perpendicular to level ground. The horizontal axis has length 2b and the vertical axis has length 2c. For a concave head |a| ≤ L/2. The wetted surface areas of convex and concave horizontal elliptical tanks are identical, other parameters being the same. Figure 5. Parameters for Horizontal Elliptical Tanks. 2c 2b 2c 2b h

Page 15 of 17

Horizontal Elliptical Tank Equations The wetted surface area of a horizontal elliptical tank with hemiellipsoidal heads with b > 0 and c > 0 is given by:

chc..............................................................................................................SSS

ch..............................................................................................................SSS

HB

HB

22

1

£<+=

£+=

where:

( ) ( ) cha

bccbcba

bccbbcaS

cha

.............................................................hchc

)hc(bchccosbcS

chc.....................................................................Sbb

cLS

chbb

cLS

c

hc

ccb

H

H

B

b

B

hchcb

B

200

14

200

222

214

12

22

2

0

2222222

222

2222222

222

21

1

0

224

22

2

2

0

224

22

1

££³

--+

--+º

££=

--

--

º

£<--

+º

£-

+º

ò ò

ò

ò

-

-

-

-

.........dxdyyx

yyx

x

dxx

x

..........................................................................dxx

x

y

Horizontal Elliptical Tank Examples The following examples will verify correct application of the equations for wetted surface areas, S, of horizontal elliptical tanks with hemiellipsoidal heads. All tank dimensions are in inches, the wetted surface areas are in ft2, and the elliptical body length for each example is 120 inches. a″ b″ c″ h″ S ft2 S ft2 (flat head) 12 36 24 12 68.62583 64.66981 24 103.76827 98.17675 36 138.91071 131.68370 12 24 36 12 43.85382 41.69564 36 103.76827 98.17675 60 163.68272 154.65787 42 36 24 12 85.63406 64.66981 24 129.40913 98.17675 36 173.18419 131.68370 42 24 36 12 53.20415 41.69564 36 129.40913 98.17675 60 205.61411 154.65787

Page 16 of 17

Vertical Elliptical Tanks with Hemiellipsoidal Bottoms A vertical elliptical tank is a tank with a vertical body which has an elliptical cross-section. The only case considered is one with a hemiellipsoidal bottom (which includes a flat bottom if a=0). If the major semiaxis of the elliptical cross-section is ′b′, then the minor semiaxis is ′c′ and vice versa. Only convex bottoms can be calculated with the equations, concave bottoms can be handled using the same technique described in the section on vertical cylindrical tanks. For all vertical tanks a ≥ 0. Vertical Elliptical Tank Equations The wetted surface area of a vertical elliptical tank with a hemiellipsoidal bottom is given by:

( )

ïïïïïï

î

ïïïïïï

í

ì

>=

-++p

>>

-+-+

££>

=

ò

ò

00

14

014

00

0

224

22

0

224

22

2

1

h,c,ba

.................................................................................bb

chbc

ahc,b,a

...........................................................................bb

cahS

ahc,b,a

........................................................................................................................S

S

b

b

dxx

x

dxx

x

where:

( ) ( )

( ) ( ) dxdyyxy

yxx

dxdyyx

yyx

x

y

y

ò ò

ò ò-

- --

--+

--+º

--+

--+º

cc

cb

hahac ahchac

acb

bccbcba

bccbbcaS

bccbcba

bccbbcaS

0 0

2222222

222

2222222

222

2

2

0

2

0

2222222

222

2222222

222

1

22

2 22222

14

14

Vertical Elliptical Tank Examples The following examples of vertical elliptical tanks with hemiellipsoidal bottoms will serve to verify correct application of the equations. Linear dimensions in inches, areas in ft2: a″ b″ c″ h″ S ft2 S ft2 (flat bottom) S ft2 (concave bottom) 18 30 24 12 17.68779 29.88880 29.52672 18 24.93188 36.97921 46.20313 60 74.56480 86.61213 95.83605

Page 17 of 17

Dan Jones is a retired senior process chemist from Stockhausen Louisiana, LLC, Garyville, LA. Contact him at [email protected] or (225)276-5465.

DEJ 4/11/17 File: TANKAREA2.DOC