MAWP = maximum allowable working pressure. It is a code-defined term that depends on the strength of the material, the efficiency of weld joints, etc.
Design pressure can mean some other things, though often the terms are considered synonymous.
For example, in designing a refrigeration system, the design pressure of the system might be 180 psi. Various codes might require pressure vessels on the high-pressure side of the system to be rated for 250 psi. This would be a different design pressure, to be met by the vessel manufacturer. In other words, vessel design pressure might differ from system design pressure.
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