In heat treating industry, cooling curves analysis is widely used to control stability of cooling intensity of liquid quenchants.  Unfortunately, cooling curves analysis doesn't provide enough data to be suitable for computer simulations and recipes calculations of real steel parts.

As known, alloy and high alloy steels are quenched in oils from austenitizing temperature to almost room temperature. . When quenching in oil, there is no need to interrupt quenching. In this case, cooling curves analysis is very suitable to control stability of oil as a quenchant.  However, with implementation of intensive quenching processes, one should pay a serious attention to critical heat flux densities. Otherwise one will be unable to develop recipes for intensive quenching (IQ) processes. Thus, a need in critical heat flux densities arises  with implementation of IQ processes.   

I  have not spent enough  time on the solution of the heat and mass transfer problems which involves solid and liquid phases concurrently such as the case of quenching of metal piece in oil etc.,  even though I have had enough experience when it comes to solid state alone!   Let me analyze the problem of quenching in a naive way by just concentrating  on the events  taking place at the interface  between liquid  oil and the solid metallic test piece. As  an experience  mathematician dealing with this time dependent boundary value problem,  I  have re cast the whole problem into three categories, namely 1) Dirichlet ' Temperature';  2) Newman ' Flux' and 3) Cauchy 'Mixed'.  During the continuous quenching  operation, the temperature as well as the flux  evaluated just  at the surface of the metallic substances changing with time simultaneously without having any simple connection between them unless one  relies oh the boundary layer theory,   and its heat transfer coefficient  h, which tells  us that  the Flux intensity is given by      J= h ( TSolid  -  TLiquid  ).  If  h  would be constant one needs only to know how the temperature of liquid  varies with time just next to boundary layer, where one has the greatest temperature gradient all the time compared to the rest of the composite media.  Note:  The maximum heat flux intensity for all times as Dear Nikolai mentioned above,   is that  the value of the flux intensity at the boundary just one turns on the quenching operation.

 Nikolai is perfectly right ! with this knowledge and  the final value of flux intensity _which may  be measured easily_  could be used to interpolate  J  versus time data knowing the total time for quenching, and  then plus the initial temperature distributions in liquid and solid phase are quite sufficient to solve the problem uniquely.  This of course very naive approach to the problem.  For the rigorous solution  one has to dealing with the convective term in the heat and mass transfer equations in liquid media plus heat leakage from the oil bath to its surrounding etc.  According to my reflection,  the most troublesome part  of  the complication  lies in the boundary layer theory itself, which may be only overcome by the extensive experimental feed backs in the measurements of  the heat transfer coefficient as a function of temperature, where the surface structure and roughness  may play significant role!!!  

Probably people are  taking another direct route where they constantly probing  the temperatures at the liquid side of the boundary at various spots , and then knowing heat conductivity of the quenching media they get the flux at the surface of the solid by extrapolations. If one collects that data would be quite to solve the heat transfer problem for solid object more accurately then above mentioned naive but more economical approach with reasonable engineering tolerance !!!

As always, thank you very much dear Tarik for interesting discussion. Critical heat flux densities qcr1 and qcr2 are connected with the transition  from nucleate boiling to film boiling and then back from film boiling again to nucleate boiling during quenching process. For transient boiling processes, it is better to  solve Neumann problem which can provide very accurate results and makes possibility to use easily critical heat flux densities.

As for your second comments, it is  interesting idea which is rather costly and will be difficult to apply when quenching complicated forms of steel components. I think, more promising is the sonar system which can perfectly control boiling processes since film boiling and transient nucleate boiling have their own songs which are very special.

Dear Nikolai,  sorry for the miss spelling it should be  NEUMANN , I made same mistake above but corrected now!    I think I could make my point very clearly. I was trying to show that how one can extract the time dependent Neumann boundary condition in practice during the numerical solution of this problem.  Since there is no heat source in this problem I could easily handle the numerical solution by Direct  Boundary element method, which I have used much more complicated problems than the quenching without boiling.

The Boiling case is very special where  boundary layer is no longer at the stationary state.  But  it rather shows stochastic  breakage at those spots, where bubbles are   pooping out.  That means this nucleationless stochastic process since sites are randomly selected similar to Stranski-Krastanow landing. If this so Neumann BC, which uses   flux vector (gradient of Green Function) all along the boundary of  the solid piece should randomized during the integration or summation procedure. Best Regards

I right now discovered following boiling paper. 

Dear Nicolai, I will check this boiling business while putting stochastic ingredients into the picture, which is numerically feasible. 

Thanks for the attached three interesting papers which I already uploaded. I consider our discussion as a very useful information both for us and readers.

Probably because the density of the material after different process of preparation, compared with the theoretical thermal conductivity of materials. This hampers the determination of thermal conductivity and qcr1 qcr2. Therefore they are considered to approximate to the theoretical and correction factors necessary. This of course is not very correct because it can lead to incorrect and distorted results.

Factors you mentioned  affect initial heat flux densities which take place during immersion of steel components into liquid. Critical heat flux densities are characteristics of a liquid. To design correctly quenching process, one should compare initial heat flux density  qi with the critical heat flux density qcr1. If  qi> qcr1, full film boiling stars. If qi< qcr1 , no film is observed. However, initial heat flux density cannot be determined from the  classic parabolic heat conductivity equation including boundary condition. Only hyperbolic heat conductivity equation with the appropriate boundary and initial condition allow correct calculation of initial heat flux density. This theory is in the process of its further development.  No enough data exist  to be provided to engineers.  Basically, I agree with you that many parameters should be carefully investigated and appropriate data collected.