Calculating the volume of a liquid in a vessel of a complex shape is a common task for chemical engineers. However, there are

several difficulties associated with ac-curately carrying out this calculation.

In my own experience as a chemi-cal engineer, I have become familiar with the complexities of calculations related to determining the volume of a liquid contained in a vessel with an irregular shape.

Precise volume-determination equa-tions are readily available for common vessel shapes. But what if you are using a vessel that is a vertical cyl-inder with a hemispherical top and bottom? Or, what if you are working with a horizontal elliptical vessel with concave heads? No matter the type of vessel you are working with, chemical engineers need to account for the liq-uids within these irregular shapes to calculate the volume properly.

Vessel-calculation challengesLet’s begin with an example scenario. Suppose a chemical engineer works at a pharmaceutical facility that pro-duces cough syrup. In that capacity, the engineer may have to prepare a solution in a 1,000-gal vessel or tank with an irregular shape. To prepare the proper concentration of cough syrup, he or she may need to add 50 pounds of an active pharmaceutical ingredient into sugar syrup.

Before adding anything to this liquid base to prepare the proper concentra-

tion, the exact fluid volume must be known. One option is to measure the volume using a meter pump, but this method will not produce an accurate result. The alternative is to verify the exact amount of liquid needed by cal-culating the volume of this irregularly shaped vessel based on the liquid level.

In this scenario, suppose that the vessel in question is a vertical cylindri-cal vessel comprised of a conical bottom and elliptical top. The elliptical portion of the vessel is partially filled with liq-uid, while the cylindrical and conical portions are fully filled (Figure 1). What should be the approach to calculating the portion that is partially filled?

To calculate the total volume, you need to combine the different equa-tions — one for each of these three basic shapes of the vessel: the conical bottom, the elliptical top and the verti-cal cylinder.

At this point, two complexities arise. Engineers are forced to search through databases and manuals for the equa-tions that are appropriate for the ir-regular parts of the tank, and then cal-culate the volume using some kind of calculation software. While many en-gineers favor Microsoft Excel as their calculation software of choice, keep in mind that the program was not spe-cifically designed for entering complex equations. As a result, this process for calculating the volume of a particular vessel can be a time-consuming and inefficient process. Engineers cannot afford to waste time — they need reli-able equations and quick calculations.

In a similar scenario (depicted in Figure 1), the author and colleagues first either found and verified, or de-rived equations, in some instances using integrals, for each shape in-volved. Glancing through a reliable en-

Feature Report

30 ChemiCal engineering www.Che.Com June 2013

Cover Story

Solving Vessel Equations: A Better Way

V R h

where RD

c c π 2

2

:=

:=

•

t

h

b

D

H

t

h

b

D

H

Volume of vertical cylindrical shell:

Shell diameter D := 50 in.

Volume of conical bottom:

Bottom height b := 30 in.

Volume of elliptical top:

Conditions:

Top height t := 10 in.

Shell height H := 150 in.Liquid level h := 150 in.

VR

b hb

π3 b

:= • •

2

3 VR

t ht

π3

3t2– ht2:= • •

2

t2

hc := if (H+b≥h) (h>b)

V := if (h≤b) Vb

else if h ≤ (H + b) Vc + Vb else Vc + Vb + Vt

h – b else 0

< ht := if (h>(H+b)) h – H – b else 0

hb := if (h<b) h else b

Figure 1. Calculating the volume of a liquid in an irregularly shaped vessel in-volves combining equations for the various portions of the vessel, such as a cylinder portion, a conical portion and an elliptical portion, in this case

Sasha GurkeKnovel Corp.

Irregularly shaped vessels present challenges

for determining liquid volumes.

New tools can help

gineering book, such as Perry’s Chem-ical Engineers’ Handbook, revealed nothing useful for this problem.

We had better luck conducting In-ternet-based research, but it was not until poring through many search results that we came across the fol-lowing article by Dan Jones — “Cal-culating Tank Volume” (http://www.webcalc.com.br/blog/Tank_Volume.pdf). Also see (Chem. Eng., Sept. 2011, pp 55–63).

Using the equations provided in Jones’ article for practical calcula-tions proved to be a problem in itself. First, the equations had to be as-sembled in a sensible way to account for all the shapes containing liquid in the vessel. In addition, depending on the level of the liquid you are working with, there are different equations that have to be used.

Another common frustration for chemical engineers is that the data found online must be validated as well. Often with research conducted online, the reliability and validity of the the in-formation found is not clear or defined. Checking the validity of the equations that are found can be complicated. To be sure that an equation is validated, engineers may need to recalculate ev-erything from scratch — which we did

ChemiCal engineering www.Che.Com June 2013 31

The Tafel equation is an equation in electrochemicalkinetics relating the rate of an electrochemical reaction tothe overpotential. The Tafel equation was first deducedexperimentally and was later shown to have a theoreticaljustification. The equation is named after Swiss chemistJulius Tafel (1862-1918)

Tafel equation

GENERAL Tafel equation

Tafel Equation

+ –x ÷

≤ ≥

010101

if

Ωα

for linewhile

Arithmetic

Matrices

Boolean

Functions

Plot

Programming

Symbols

Plot

∆V is the overpotential, V

A is the so-called “Tafel slope”, V

i is the current denisity, A/m2 and

i0 is the so called “exchange current density”

Calc 1

NEW UPLOAD DOWNLOAD

Tafel diagram n vs log i

2

1η

log i0

log i

01•10–6 1•10–5 1•10–4 0.001

+

xx Calc 2x Calc 3x

∆V Ai

i= ×

ln0

fx

contributed by Jane Chemist

references Publication One, Publication Two is Longer

citations Electrochemical Methods, Fundementals and Applications

Figure 3. Cloud-based calculation tools can improve engineering workflow

Time

Concept development

High-levelrequirements

Detailedrequirements

High-leveldesign

Detaileddesign

Scale-up, inte-gration and testing

Implementation

Subystemverification

Acceptance

Operation andmaintenance

Part/unit replacementand

process improvements

Initial design stages: sizing, conceptualizing,

material selection

Equation library(corporate/Wiki)

Databases(materials/components)

Life engineering calculation tools (SMath-based)

Optional PLM integration layervia API

Codes andstandardsFigure 2. A Web-based equation li-

brary can help in vessel calculations

in this case — and of course, it ended up costing us even more time.

Once we found and validated the equations and vessel dimensions, the next step was to use a calculation tool that is easy to integrate with the data. We first turned to Microsoft Excel, probably the most-often-used tool in many engineers’ toolboxes. For most calculations and analyses, an Excel spreadsheet would suffice. Several generations of engineers now have grown up using Excel — it’s a common, inexpensive software that is readily available on most desktops and lap-tops. In addition to its familiarity, it’s relatively easy to input large amounts of data into Excel.

However, in this case of calculating volume in an irregularly shaped ves-sel, it was not. It became clear that it would become an exceedingly time-consuming process just to enter the equations and variables.

Other reasons why Excel was not the right calculation tool in this case is that programming is required with external data. Second, all calculations must be performed in a consistent system of units with conversion factors embed-ded in equations. This is because Excel does not automatically understand the units of measurement and does not support calculations in different unit

systems unless additional program-ming is introduced. Third, we planned to create a browser-based application, but the Internet version of Excel has exhibited performance issues and is not highly rated by many users.

Beyond Excel, there are engineer-ing calculation tools available such as PTC’s (Needham, Mass.; www.ptc.com) Mathcad, which has automatic unit conversion and can check equa-tions for mathematical errors. How-ever, a browser version of Mathcad is not available, and that limits its use-fulness in cloud-based applications.

SMath To calculate the volume of a liquid in a vessel of a complex shape, a task that should take only minutes, we tested a tool that is readily available online and that could integrate our data. We found an engineering desktop calcu-lation tool that is both powerful and distributed free of charge — SMath Studio (http://en.smath.info/forum/yaf_topics12_Download-SMath-Stu-dio.aspx).

SMath has a browser version called SMath Live. While it is function-ally similar to the desktop version, it needs further development. SMath, developed specifically for engineering calculations, is now used by thousands

of engineers and engineering students around the world.

This tool consists of a powerful math engine core, user-friendly worksheet-based graphical user interface (GUI) and plug-ins — some of which are open source software — that connect the core with GUI. SMath has the fol-lowing features:•The ability to handle numeric and

symbolic calculations•Capabilitiesfor2-Dand3-Dgraphs•Softwareversionsdesignedfordiffer-

ent platforms and operating systems•Partial support of Mathcad files

(*.xmcd)•Theabilitytousemathematicalunits

(either built-in or user defined)•Multi-languageworksheets•Multi-language interface (28 lan-

guages)•The capacity to use programming

functions directly on the worksheet•Infrastructure to support third-

party plug-ins•An auto-complete feature with de-

scription of all supported entries•Theabilitytousethetoolincollabo-

ration (via server)•Equationsnippets

Improved volume calculationsThe tools chemical engineers have at their disposal are critical for main-

Cover Story

32 ChemiCal engineering www.Che.Com June 2013

Figure 4. (A–H) Various standard shapes that can be combined include cylinders, cones, ellipsis and hemispherical. The diagrams and equations show some of the possible situations for volume measurement that engi-neers might face

V H–h+Rhh

Hπ

D2

3H:=

where R :=

22

A. Conical top:

H

h

D

Vb hπ H6 R

3 R–h:= • • • •2

B. Elliptical head:

D2

where R :=

h

H

D

R

C. Elliptical bottom:

V hπ3 H

D

R3 H– h:= • • • •

2

2

2

2where R :=

hH

D

R

Note: := means “is defined to be”

Hh

W

L

D. Horizontal elliptical vessel:

V L (h –b) (h –b)h –ba

b:= • • • •b – asin

π2

2 2 + b b

2 + b 2

H2

b :=W2

where a := Note: := means “is defined to be”

taining high levels of productivity. Ide-ally, engineers should use tools that are seamless, can save time, and avoid costly errors in the workflow. One way to accomplish this is through cloud computing, where software programs and data that have traditionally re-sided on company servers are now lo-cated on a third party’s remote servers and are accessed via the Web.

Cloud computing assures today’s en-gineers quick and easy access to data from anywhere on a variety of devices. It also allows engineers to easily share data with their peers across the globe. Fortunately, as technology continues to move into the cloud, engineers will have more effective and reliable tools to integrate data, such as equations with calculation software, into their design and workflow.

Currently in the early stages of de-velopment, there is an engineering cloud-based productivity tool (Figure 2) comprising of SMath Live inte-

E. Hemispherical head:

Vb hπ6 (1.5 • D–h):= • •

2

h

D

F. Hemispherical top:

h

D

V hπ3 (D+h):= • •

23

4D –

2D –

G. Horizontal cylinder:

D

R

L

h

V LR–h

:= • R • •acos – (R – h)2 2

R

D2

where R :=

2 • R •

h –h

H. Horizontal hemicylindtrical top:

R

L

h

V Lh

:= • R • •acos – h–22 2 2

R R – h0.5 •

π • R –

ChemiCal engineering www.Che.Com June 2013 33Circle 11 on p. 56 or go to adlinks.che.com/45773-11

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Cover Story

34 ChemiCal engineering www.Che.Com June 2013

grated with a searchable and brows-able library of common engineering equations, including those for par-tially filled shapes, that could help you calculate liquid volume as a func-tion of liquid level much faster than before. A chemical engineer could use this cloud-based product to find shapes and assemble them in any

reasonable combination to calculate the volume of liquid in any partially filled vessel. Such a product will be useful when integrated into engineer-ing workflow as an early-stage design tool. The stages of a typical engineer-ing workflow where this tool can be integrated can be seen in Figure 3.

This type of Web-based product

would enable users to find and select equations for various shapes and then assemble them like Lego blocks onto an SMath Live worksheet. If you are working with any unusually shaped shells, bottoms or heads, you can build any vessel from them using smaller pieces (Figure 4). You can continue to build up to more complex shapes and calculate the volume of the entire shape or the volume of liquid in par-tially filled shape. The same approach could be used for calculating the vol-ume of dry particulates, suspensions and so on.

Initial results are encouraging and can be seen in Figure 1, which shows an example of a calculation for a verti-cal cylindrical vessel with conical bot-tom and elliptical top. This example was assembled from calculations for three basic shapes: cone bottom, el-liptical top and vertical cylinder. Each calculation contains limiting condi-tions and validation routines, as well as graphic representation of a shape. These conditions and validation rou-tines are easily adoptable for the ves-sel shown in the example.

A prototype of this cloud-based calculation tool is now underway. We believe that the future of engineer-ing will be characterized by tools that integrate data and calculation software and are available in the cloud. Development and deployment of these sophisticated tools will be critical for maintaining high levels of engineering productivity in the chemical industry. ■

Edited by Scott Jenkins

AuthorSasha Gurke is engineer-ing technical fellow at Knovel Corp. (240 West 37th Street, NewYork,NY10018;Email:sgurke@knovel.com; Phone: 617-803-8344).Achemistandchemical engineer, Gurke has more than 30 years of experi-ence in the technical infor-mation field. He co-founded Knovel in 1999 and as senior vice president, he was actively

involved in product development and manage-ment. Knovel was acquired by Elsevier in 2012, and Gurke continues to play an important role in new product development and strategy. Prior to Knovel, he spent 15 years with Chemical Ab-stracts Service/American Chemical Society in product development and editorial positions. His industrial experience includes working as a chemist at water treatment and paint manufac-turing plants. Gurke holds a master’s degree in chemical technology from St. Petersburg State University of Technology and Design.

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