Solving Vessel Equations: A Better Way

Irregularly shaped vessels present challenges for determining liquid volumes. New tools can help

Sasha Gurke Knovel Corp.

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 accurately carrying out this calculation.

In my own experience as a chemical 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 equations are readily available for common vessel shapes. But what if you are using a vessel that is a vertical cylinder 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 liquids within these irregular shapes to calculate the volume properly.

Vessel-calculation challenges

Let’s begin with an example scenario. Suppose a chemical engineer works at a pharmaceutical facility that produces 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 concentration, 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 calculating the volume of this irregularly shaped vessel based on the liquid level.
In this scenario, suppose that the vessel in question is a vertical cylindrical vessel comprised of a conical bottom and elliptical top. The elliptical portion of the vessel is partially filled with liquid, while the cylindrical and conical portions are fully filled (Figure 1). What should be the approach to calculating the portion that is partially filled?

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

To calculate the total volume, you need to combine the different equations — one for each of these three basic shapes of the vessel: the conical bottom, the elliptical top and the vertical cylinder.
At this point, two complexities arise. Engineers are forced to search through databases and manuals for the equations that are appropriate for the irregular parts of the tank, and then calculate the volume using some kind of calculation software. While many engineers favor Microsoft Excel as their calculation software of choice, keep in mind that the program was not specifically 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 reliable equations and quick calculations.
In a similar scenario (depicted in Figure 1), the author and colleagues first either found and verified, or derived equations, in some instances using integrals, for each shape involved. Glancing through a reliable engineering book, such as Perry’s Chemical Engineers’ Handbook, revealed nothing useful for this problem.
We had better luck conducting Internet-based research, but it was not until poring through many search results that we came across the following article by Dan Jones — “Calculating Tank Volume” (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 calculations proved to be a problem in itself. First, the equations had to be assembled 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 information 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 everything from scratch — which we did 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 laptops. 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 vessel, 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 embedded 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 programming 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 engineering calculation tools available such as PTC’s (Needham, Mass.; www.ptc.com) Mathcad, which has automatic unit conversion and can check equations for mathematical errors. However, a browser version of Mathcad is not available, and that limits its usefulness 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 calculation tool that is both powerful and distributed free of charge — SMath Studio (en.smath.info/forum/yaf_topics12_Download-SMath-Studio.aspx).
SMath has a browser version called SMath Live. While it is functionally 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 following features:
• The ability to handle numeric and symbolic calculations
• Capabilities for 2-D and 3-D graphs
• Software versions designed for different platforms and operating systems
• Partial support of Mathcad files (*.xmcd)
• The ability to use mathematical units (either built-in or user defined)
• Multi-language worksheets
• Multi-language interface (28 languages)
• The capacity to use programming functions directly on the worksheet
• Infrastructure to support third-party plug-ins
• An auto-complete feature with description of all supported entries
• The ability to use the tool in collaboration (via server)
• Equation snippets

Improved volume calculations

The tools chemical engineers have at their disposal are critical for maintaining high levels of productivity. Ideally, 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 resided on company servers are now located on a third party’s remote servers and are accessed via the Web.
Cloud computing assures today’s engineers 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.

FIGURE 2. A Web-based equation library can help in vessel calculations

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

Currently in the early stages of development, there is an engineering cloud-based productivity tool (Figure 2) comprising of SMath Live integrated with a searchable and browsable library of common engineering equations, including those for partially filled shapes, that could help you calculate liquid volume as a function 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 engineering workflow as an early-stage design tool. The stages of a typical engineering 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 partially filled shape. The same approach could be used for calculating the volume 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 vertical cylindrical vessel with conical bottom and elliptical top. This example was assembled from calculations for three basic shapes: cone bottom, elliptical top and vertical cylinder. Each calculation contains limiting conditions and validation routines, as well as graphic representation of a shape. These conditions and validation routines are easily adoptable for the vessel shown in the example.

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 engineers might face

A prototype of this cloud-based calculation tool is now underway. We believe that the future of engineering 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

Author

Sasha Gurke is engineering technical fellow at Knovel Corp. (240 West 37th Street, New York, NY 10018; Email: sgurke@knovel.com; Phone: 617-803-8344 ). A chemist and chemical engineer, Gurke has more than 30 years of experience in the technical information field. He co-founded Knovel in 1999 and as senior vice president, he was actively involved in product development and management. 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 Abstracts Service/American Chemical Society in product development and editorial positions. His industrial experience includes working as a chemist at water treatment and paint manufacturing plants. Gurke holds a master’s degree in chemical technology from St. Petersburg State University of Technology and Design.