Ultimately, design standards are going to be set by your organization and driven by end customer requirements. In general, ASME is followed for legal compliance and code stamping, as it pertains to pressure vessels in general. It’s the broadest set of guidelines, as it can be applied to many different applications, such as boilers, chillers, heat exchangers, etc.
TEMA on the other hand, is a more practical manufacturing guideline for the design and fabrication of shell and tube exchangers. It addresses things like nomenclature, tolerances, and tube sheet design. While there is overlap with ASME code, TEMA has a more specialized application and is followed by both manufacturers and end users.
Lastly, API guidelines reference both standards above, while providing additional guidance for end users in the oil and gas industry.
Overall, industry standards are in place to ensure quality, consistency, and public safety. Although many of these guidelines are not legally required, manufacturers are adopting them to stay competitive in a growing marketplace. Understanding what these organizations are and the standards they set can help benefit your organization too.