# How to solve a multi-step mass balance

I have a problem, where I'm trying to solve the mass balance of a multi-step process with limited information available.

What I have is a material supply of known flow and composition. This supply is split into multiple streams (in my current case 7). For these streams, I do not know the individual flows, I only know they are not all equal.

Each of these streams is supplied to a separation process where they are separated into a product and a waste stream. I know the composition of each of those product and waste streams, bot no flows at all.

As such, the system looks like this:

From the compositions, I can easily calculate the ratio of product and waste stream for each separation process (i.e. the values of x1 ~ x7 above).

My problem is that I can't figure out how to calculate each of the process flows 1 through 7, or whether or not this is even possible with the information available.

I know that if I can set up a system of linear equations, where I have the same number of independant equations as unknowns, then it can easily be solved. However, I seem to only have 1 equation with 7 unknowns (sum of all process flows = total supply flow).

I would appreciate any help here.

• you can't get there from here Commented Jan 8 at 13:56
• Thanks @Tiger Guy, after several failed attempts I came to the same conclusion. In the end, I had to gather some more data and then I could create a solver scenario in Excel, which performs some iterative guess work until the mass balance is correct. Commented Jan 9 at 17:06
• If there is equal division of Feed or Feed split is specified...then it is is easy to perform calculations using system of linear equations Commented Feb 21 at 14:19