What are the benefits of a diaphragm pump
over a simplex pump?
Is there any specific design criteria that should be used/stated in an engineering specification for a diaphragm pump?
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I can't give a definite answer, since I hardly plan either type. So please read this answer as an incomplete list of questions you could ask the sales engineers of your pumps.
Both can be operated hydraulically or pneumatically (for example if you don't want to bring a motor into an ex-zone or there would be cooling problems). In the case of piston pumps you'd need extra air/oil pistons for that. The diaphragm can be operated directly by the fluid, so the pump can be simpler. So for a sump application, you could have a diaphragm pump in your sump powered by a compressor nearby.
I've seen small and huge piston pumps, but I've never seen huge diaphragm pumps (huge= flowrate >10m³/h). I assume some economy of scale here that makes large piston pumps cheaper than comparable diaphragm pumps (But I could be totally wrong).
Diaphragm pumps have no seals that touch the medium. This is an advantage since you probably need to service them less. This also means that they are better for aggressive media or applications where cleanliness is important.
With any pump, you need to know if there are any solids or abrasive stuff in your medium.
As for your question what specifications should be stated, ask your vendor(s) "What do you need to know to size and configure a pump?"