Please understand that I know absolutely nothing about your process, but let me ask a couple of questions to help direct you in a pattern of thought as you seek financial justification for this conversion:
1. Is there a risk involved in running the backwash manually? I.e., human safety, mechanical, chemical, getting distracted during the process, ...
2. Are any hazards created by performing the backwash manually? I.e., accidentally missing or repeating a step, accidental chemical release, ...
3. Are there any efficiency gains that could be obtained through automation? I.e., operator efficiency (can we reduce an operator whose only job is to run this manual process, or can he multi-task better), reduced chemical/component usage, ...
4. Are there any quality improvements that could be obtained through automation?
5. When (not if) that Micro 84 finally lays down and dies, what risks will that pose? How long will it take you to get back up and running?
6. How much have you been spending in the recent past on repairs? Forecast for the future?
Financial justification for replacing an obsolete PLC is a very difficult task. Alot of the bean counters only see benefit if the project will save money or create income. Many times, they don't understand (though they say they do... but that's a different story) the need to make continued investment in automation. They allow mechanical components to be replaced as they wear out, but not control...
Right now, your system is running, albeit not as well as you would like. I don't know -- will this upgrade help you make your product faster, cheaper, better? Those questions above will help determine that. If it won't, then your sales job to management becomes more difficult - and you have to convince them, not of a cost improvement, but of a cost avoidance. (questions 5 and 6) There truly is a cost associated with obsolesence. Our jobs are to define that cost and come up with reasonable means of offsetting those costs with control system upgrades.
Hope this helps. Good luck!