should we consider the vertical exchanger lug support as sliding or fixed ?

Your question is about the supporting and the anchoring of a Vertical Heat Exchanger.
In order to give a proper answer to your question, more information is required.
What is the situation here? is this "vertical exchanger" a Thermosyphon Reboiler hanging off the side of a vertical vessel? Or is it something else?
How big is this vertical exchanger (VE)?
What size are the lines connecting to this VE?
Where do these lines come from and go to?
What are the temperatures of these lines?
What else can you tell us that might influence our answer?

1) Vertical exchanger ( sulfur reactor preheater )
2) OD-1720mm & 9144mm tube length
3) inlet / outlet size 48", 150#
4) shell design temp : -3degC/413degC
5) tube design temp : -3degC/252degC
6) Inlet (bottom ) line : 48" , 150# , CS material ,design temp 316degC , coming from exchanger (sulfur condensor)
7) Outlet (top) line : 48" , 150# , CS material , design temp 240degC , going to reactor ( sulfur reactor )

These are all "Design" Temperatures and have value only for the selection of Pipe wall thickness and Flange rating selection, they have no value in Plant layout and piping layout or Stress analysis.

4) shell design temp : -3degC/413degC
5) tube design temp : -3degC/252degC
6) Inlet (bottom ) line : 48" , 150# , CS material ,design temp 316degC , coming from exchanger (sulfur condensor)
7) Outlet (top) line : 48" , 150# , CS material , design temp 240degC , going to reactor ( sulfur reactor )

Never use Design Temp. for thermal expansion calcs.

Now, having the information that you have given I have one more question.
Is the Vertical Exchanger direct connected to any other piece of equipment by straight pipe?
If your answer is yes, then the support of the Exchanger should be "sliding".