Mr. Richard Lee of Plumlee International Consulting usually heat exchangers have two sets of test pressures per side, one for strength tests, and the other for “operating” or “leak” tests. The strength tests are set by the design code and if you have the original design data sheets for your equipment then the information should be shown on these. If you do not then you will have to do the calculations yourself, the exact method will depend upon which design code you use, the most common one being TEMA (which uses the ANSI/ASME pressure vessel code for reference in this area).
Most shell and tube exchangers are designed such that each side of the unit will withstand the full design pressure, with only atmospheric pressure on the other side. In order to save money, some larger units will have the tube-sheets especially designed to withstand only a much lower differential pressure (requiring both sides to be tested simultaneously). This important information should be shall quite clearly on the design sheets and on the vessel nameplate (assuming that either are available). If the only need is to check that a gasket has been properly installed then it can be permissible to perform a lower pressure test based on the operating pressure. The acceptability of this lower pressure test will often depend upon the consequences of a leak.