Query

Your question can be interpreted in two ways:

1. You want to know if you can operate the vessel at the designed MAWP as per the manufacturers data report (U-1) - this is easy to answer, see below.
2. You want to increase the operating pressure above what the vessel was originally designed for - this is a little more difficult, see below.

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1. If the original thickness was 15 mm, which includes a 3 mm corrosion allowance - this means that the required thickness is 12 mm. Since you have an actual thickness of 13 mm, which is above the required - there is no need to perform calculations - you are able to operate the vessel up to a maximum of design pressure as per the manufacturer's data report (U-1)
2. If you are looking to increase the operating/design pressure of the vessel above what it was originally designed for:
1. Talk with your Authorized Inspector/insurer/jurisdictional representative.
2. You will have to perform a rerate of the vessel, which could include:
1. NBIC NB-23
2. API 510
3. API 572
4. API 579 / ASME FFS-1

But to specifically answer your question:

BE VERY CAREFUL in backcalculating required thickness. There may be more to it then just the thickness required due to internal pressure as you have implied. The shell also must be designed for any external loadings, which are not accounted for in the simple pressure equation. The thickness used in the equation depends on if you are attempting to re-rate the vessel (e.g. if you had a large corrosion allowance and where changing the service of the vessel and wanted to operate at a higher pressure, you could theoretically calculate using a thickness of 13 mm with no corrosion allowance). But, in general, the formulas in ASME VIII are in the corroded condition, that is they do not include the corrosion allowance, so it would appear that what you are attempting to do (which may not even be required as per #1 above) you would use t=10 mm & the joint efficiency would be as per the original design.

Don't do that.

Get a report on the vessel instead from the local Engineer/Surveyor from the company that supplies burst/collapse indemnity insurance cover on the equipment. Its details, and recent insurance reports in its condition, should already be in the General Register for insurance inspections at the facility, which register is always the first point of reference.

After all, it will have had recent hydraulic tests witnessed by that individual, won't it? WON'T IT?