Maqqadri, Cnu879, Rockdock, and I both have initially interpreted your writing as stating that your TEG fluid is brown in color. While that may be the case, what you are now describing is an external, brown coating that has tainted your equipment externally.
What you may be seeing is the result of the fog/mist that is vented out of your rich TEG flash drum – especially if you are running your absorber rather “warm” (100+ oF) and your unit is located in a humid site (like Indonesia). You didn’t supply the rest of the P&ID, so I have to assume you are venting the flash gas and not flaring it or burning it in your reboiler. If that is not the case, then you may have other venting taking place near by. Your photo is an excellent illustration of what is happening and I can detect what appears to be a foggy vent about half-way up the TEG Still tower.
I share your concern for the problem. It isn’t only aesthetics; if your unit looks dirty and bad, it is an indication that the maintenance may also be bad – and that isn’t good. All TEG units should reflect good, clean operation.