Firstly, are you familiar with the effectiveness-NTU method for heat exchanger design and rating? This was originally developed almost 50 years ago by Kays and London at Stanford. Many udergraduate texts explain how to use the method - it's amazingly simple to use by hand as well as in a spreadsheet. In my opinion, it is vastly superior computationally to the conventional LMTD - F Factor approach, but still entirely rigorous theoretically.
This method can solve BOTH design and rating problems with NO TRIAL AND ERROR.
Basically, the effectiveness is defined as the limiting (i.e., larger) delta T divided by the maximum delta T, defined as Thot(in) - Tcold(in). Constant temperature for a condensing or boiling fluid on either side is also handled correctly.
Two additional parameters are required:
(1) U*A/Cmin, where Cmin is the lower of the W*Cp values for the two streams.
(2) The Cmin/Cmax ratio. (Obviously, the Cmax is the higher of the W*Cp values).
For each type of exchanger geometry, charts are available that plot UA/Cmin on the X axis, and Effectiveness on the Y-axis. For the third parameter (Cmin/Cmax), there is a family of curves for different ratios going from 0 to 1.
Thus, there are different charts for counterflow exchangers, 1-2 exchangers, cross-flow exchangers, etc. There is a simple generalized equation that extends the 1-2 exchanger equations for additional passes, e.g., 2-4, 2-6, 3-6, etc. Best of all, there are analytical equations available that were used to derive the charts.
In rating work, the effectiveness is desired for given values of the other two parameters. The equations can be programmed easily in a spreadsheet. Also, I strongly recommend you learn to use the "solver" capability in Excel. Alternatively, you could write a macro to invoke the solver automatically.
For design, the effectiveness and Cmin/Cmax ratios are known and U*A/Cmin is desired. Once you find UA/Cmin, compute U*A and then estimate the exchanger film coefficients, fouling factors, etc. to estimate the overall U. Then the required A is found. Ideally, you should now reconfirm the film coefficients for the actual tube and shell layout for the required exchanger area. This is generally done using software such as BJAC, etc.
I hope this helps.