Doing the Math – Roast Vessel Dimensions

2010 March 8
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Well it has finally come to this! Doing the math. I HATE math. As a visual thinker, it is not my strong suit even if I did well in college math courses. Math does have its place , though. I’ve been wondering for quite a while what the size of the roasting vessel will need to be for a capacity of 15kg (33lbs) of green coffee per batch. Solid geometry is the easiest way to figure it out.

Part 1 – Density of Coffee

When coffee is roasted, the density of the bean decreases substantially. First of all, many components vaporize (in the case of silver skin, separate) and go out the exhaust stack. Secondly, as the water in the cells turns to steam, the cells inflate and the whole bean grows in size much like popcorn. The weight goes down and the volume goes up, therefore density is reduced and the space need to contain the beans increases. The darker one roasts the beans, the lower the density becomes. That’s why French Roast makes the bag look much fuller or is packaged in a 12oz. bag when Full City Roast fits fine in a 16oz. bag.

We are really concerned, then, with the least dense coffee to determine the roasting vessel size. I don’t plan on roasting beyond a French Roast, so I decided to measure the density of a good French Roast as a benchmark. I weighed 500ml of Mr. Espresso French Roast and found that it weighs 158gr net. Note that it probably lost 18% of  the original green weigh for this degree of roast. If you do the math, the volume that 1kg of this French Roast requires to contain it is 3,200cc. This means that 15kg of the French Roasts requires 48,000cc or around 3,000 cubic inches of space to contain the beans.

Part 2 – Sizing of a Bowl

There are many shapes of bowls that can contain the beans. I have focused for the moment on truly spherical bowls. You could have a shallow, wide bowl. You could have a bowl that is essentially half of a sphere (the deepest one possible). I suspect that the half sphere is not ideal in that airflow and agitation might be problematic. Excessively shallow bowls would have to be very wide, which is impractical. Some balance of depth and diameter will have to be arrived at.  I did three tables of calculations for spheres. One at 50,000cc volume (solid with beans), 60,000cc which allows for 20% agitation and head space, and 75,000cc which allows for 50% agitation space and head space. The calculations are in the table below.

Assuming the the 75,000cc model is a good one, we can see that the bowl size varies from 8″ deep and 52″ in diameter to a 1/2 sphere of 26″ diameter and 13″ deep. A 10″ deep bowl is about 3′ in diameter. All of these seem to be within the range of size that would be appropriate in a shop setting.

Part 3- Sizing of a Drum

The drum is a fairly simple geometry, yet there are the two variables: length and diameter. For any given volume you could have a short, large diameter drum. You could have a drum that is essentially a long tube. Drums that are too short or too long will have problems with even heating, agitation and ability to quickly evacuate the drum at the end of the roasting process. Some balance of length and diameter will have to be arrived at here, as well.  I did three tables of calculations for drums. One at 50,000cc volume (solid with beans), 60,000cc which allows for 20% agitation and head space, and 75,000cc which allows for 50% agitation space and head space. The calculations are in the table below.

I have marked in pink the values that do not make sense. The drum is either too short or too long. If 75,000cc is a good value, then drums in the range of 22″ long and 16″ diameter to 34″ long and 13″ in diameter seem appropriate. Again, these are sizes that would fit will in a shop environment.

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4 Responses
  1. Kevin Briggs permalink
    March 22, 2010

    Joe,
    Thank you for this posting about the roaster and coffee-fill volumes.

    Some people consider the amount of coffee within a roaster volume as important design and process control parameters.

    I am currently thinking that the roaster-volume and the coffee-volume are important but I am uncertain about the effects and interactions. My thinking process is “stuck” on the “air requirements for the roasting process” where the air-flow-rate around the beans may be very important; other parameters interact with this parameter. The roaster-volume, coffee-volume and air-flow-paths all contribute to the rate of air over the coffee beans.

    I have created a brief analytical analysis of the cylinder configuration. I have documents that describes my analysis (PDF and XML/MathML – no artwork, I am not good at it and I don’t have the tools to complete in a brief time). I have a spreadsheet summarizing my analysis.

    I can Email these documents if you are interested in them.

    I am currently working on the sphere analysis. I have a “back of envelope analysis” done but I have not prepared any documents.

    • March 22, 2010

      Kevin, please do email them to me. I am VERY interested.

      • Kevin Briggs permalink
        March 24, 2010

        Thank you for your interest! There are not many people interested in the information I have.

        Briefly, I have tried to calculate the “height” of the coffee-volume within the roaster-volume. I assumed the coffee is “like a fluid”; filling the lower elevations first. I have used “dimension-less terms” so that many configurations may be evaluated; e.g., the ratio coffee-volume to roaster-volume, the ratio “height” to cylinder-radius, etc.

        I apologize for not knowing your Email address. I have not located a method to attach files through this interface. When I have the Email address, or become aware of how to attach files through the interface, then I will send the information I have to you.

        I have my “first version” cylinder analysis completed; analysis and spreadsheet. I have the spreadsheet in an Excel format. I created the analysis with XHTML/MathML but I did a “quick conversion” to PDF for distribution. I am assuming you would prefer the PDF file; XHTML/MathML file can be viewed on browsers that have MathML built-in, I am only aware of Amaya and Firefox.

        I am more than half-way through the analysis document for a sphere. I have not created the spreadsheet.

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