This is a complex system. You want to model it and do experiments on it over a range of parameters. Here are my starting suggestions:

In the same way that you cannot solve a simple physics problem correctly without the right free-body diagram, you cannot model a complex heat transfer system completely without the correct picture of the energy balance for the control volume of interest. As for the experiments, accurate predictions can only be done using well-calibrated experiments. Think ... Why do people disagree about climate change? One set of people disagree because they do not believe the experimental data are measured correctly, and the other set do not believe the models are developed properly for the given system picture.

Otherwise, I honestly have to wonder whether you should invite someone with experience to collaborate with you directly rather than trying to piece together a successful program of study from random answers in this forum. I apologize when I misread your postings so far. However, the system you have posted is complex, your questions are broad, and your characterization of the system is vague. In this case, can you appreciate why I am left to think as I do?

Syed, I agree with Jeffrey, this is a very complex problem that you need to break down into manageable pieces and having a knowledgeable partner/mentor would be probably be quite helpful to you.

 A great deal about heat transfer is well understood. Quite often heat transfer can be broken down into regions dominated by either radiation, conduction, or convection.

In a typically furnace firebox the heat transfer from the flame to the furnace tube is dominated by radiation from flame to tube, flame to furnace wall and furnace wall to tube and tube to tube.  Calculating the correct flame temperature and furnace tube profiles is often a difficult iterative task .

Once the tube temperature profile is known conduction takes over as the primary mechanism to get the heat to transfer from the tube wall to the material flowing inside the tube a steady state situation will occur so that the radiant heat from the flame to the tube wall will equal the heat transferred across the tube to the process material flowing inside and the tube temperatures will remain constant until conditions are changed.

The hot flue gas will leave the radiant section of the furnace will usually run across process tubes to heat some process fluid or generate steam via conduction in the furnace convection section.

The goal of most process furnaces is to achieve a steady state temperature profile and  heat transfer to desired products with a minimum amount  of  control (opening and closing valves controlling fuel, air or process flows.

Your problem is a little different than the steady state problem.  You are asking how to adjust conditions to heat or cool a process fluid at a constant rate.  This is usually  achieved is achieved with a process control scheme with a feedback loop.   To increase furnace effluent product at a given rate fuel firing is gradually increase and temperature rise is measured.  If the rate of increase is too high the fuel gas controller will close the fuel valve if it is too slow it will open it.

Cooling the furnace effluent at a given rate can be done in a similar fashion be gradually reducing the firing and changing coolant rates.

Theoretically all of the heat transfer rates are calculable and the process control feedback loop is not necessary.  But there are always a few unknown variable that make that a very dangerous approach. In practice  It is much safer to calculate a variable setting check if the desired result is achieved and adjust as necessary.

I am not sure if I have answered you question but I hope this response helps to move the discussion forward.

If a cylindrical body is sliced in a slab placed in an isolated system (having heat source), the temperature distribution (heating) can easily be calculated through boundary conditions (initial furnace values at given period of time alone were reasonable, so no flame measurement was employed). As described, I already broke down my object in specific mode of heat transfer for closed systems with reasonable result accuracy. Thus cooling/heating rate in an isolated system is not an issue.

When I move the object into open system cooling rate  should be calculated by transient heat transfer equation keeping other parameters constant (lump analysis). Sorting high Biot number values out could ease these calculations.

Sounds to me as though you have the study and strategy outlined nicely. I cannot see what else I can contribute specifically to the overwhelming scope of this effort. Any recommendations I might make at this point are likely to be so general that I would insult your intelligence to repeat them here. Alternatively, I will need more of my time than I can afford just trying to recognize a well-focused, specific question that I can truly answer rather than to weed through a continual repeat of a broadly presented thesis proposal.

In closing, I recommend that you engage someone in a consultant or advisory capacity to help you focus your thoughts on what you really should be asking here. This continues otherwise to be a sincere but misplaced request to help design a successful thesis effort rather than a directed question that can be answered in a directed manner.

Best regards and success in your thesis efforts.