How shall I report a critical error in a given paper? This is a follow-up of my previous question about this course task:


  Find an existing, published research paper in the field of Chemical Engineering and try to autonomously replicate its results (plots, data, equations et similia).


In my replication study, I found that there was a report error; that is, a mathematical model was wrongly written, but the results shown on the paper were related to its correct form.


  After my revision and replication trial, I found that using their same model, I do not obtain the paper results. There is no presence of replication error by me, and my TA pointed out flaws (of fundamental nature, for example F = mv instead of F = ma) in the model, that once corrected allowed me to obtain the results shown in the paper.


Now, I discovered a pure error in one of the basic assumptions of the model (namely, the molar mass of molecular oxygen set to 16 u, and not 32 u). This error in itself changes all the results of the paper, since it invalidates the reactor model and the subsequent optimization.

I have verified and consistently proved that the error exists, as I can reproduce the paper results including said error. What are my options now? I want someone to take notice and fix this paper, since I have proved it contains wrong results.
Talk to an experienced researcher in the field before getting too excited.  For instance, you could talk to the professor of your course, if they are an active researcher in this area.

The question of what constitutes a "critical error" is subjective and often hard for a student to evaluate.  For instance, perhaps the authors are just using conventions that you are not familiar with, or you have misunderstood what they are saying.  

It can also happen that although the text of the paper is literally incorrect, an experienced researcher can see immediately what was intended, and won't be confused by it.  These sorts of errors are typically not taken seriously and are not considered to warrant a formal correction to the paper, which requires a lot of work for the authors and editors.

It's good that you've discussed this with your TA, but your TA is likely a graduate student who doesn't yet have a lot of experience with research and publication in this area.  

I think it's pretty common for students to get very excited about finding errors of any kind in published work, and to feel that "the world needs to know".  I've had this experience myself.  But community standards may not be what you think they are, and if you try to raise a big fuss about something that the research community regards as trivial, it'll just waste time and make you look a bit silly.

If an expert agrees that the error really is critical, then they can also advise you on how to proceed.  Typically you would begin by contacting the authors, and they would talk to the journal editor about publishing a correction (or "corrigendum").  In the unlikely event that they don't acknowledge the error, you can also raise the issue with the journal editor directly.  If the discovery of the error is itself something of significant scientific interest, then you could think about writing a new paper about it.  But don't cross that bridge until you come to it.