Consider an exchanger with cold fluid t and hot fluid T. By definition t-in < T-in. Usually t-out is also < T-out. But in some cases, t-out > T-out. This is a temperature cross.
Example: water enters an exchanger at 85 degF and leaves at 95 degF. The heating medium is tempered water at 100 degF and it leaves at 90 degF. There is a temperature cross.
If this occurs in a perfect counter-current heat exchanger there is no problem. But what if there are two or more tube passes? The Rule of Thumb is that you cannot have a temperature cross in such an exchanger.
However, temperature crosses are found in multi-pass exchangers. It can happen because the shell fluid leaves (or enters) the exchanger in the middle of the tube bundle.
I created a simple Excel spreadsheet to illustrate the concept and compute a hypothetical temperature profile through the exchanger. You'll see that the section of the exchanger closest to the tube in/out nozzles does most of the work (corresponding to the two ends of the tube path). The workbook includes a macro to perform the calculations in the algorithm that is described.
Steve