heat-transfer-conduction_engrzHeat transfer is the flow of thermal energy driven by thermal non – equilibrium (i.e. the effect of a non – uniform temperature field), commonly measured as a heat flux (vector), i.e. the heat flow per unit time (and usually unit normal area) at a control surface.
The aim here is to understand heat transfer modelling, but the actual goal of most heat transfer (modelling) problems is to find the temperature field and heat fluxes in a material domain, given a previous knowledge of the subject (the general PDE), and a set of particular constraints: boundary conditions (BC), initial conditions (IC), distribution of sources or sinks (loads), etc.
There are also many cases where the interest is just to know when the heat – transfer process finishes,  and in a few other cases the goal is not in the direct problem (given the PDE+BC+IC, find the T – field) but on the inverse problem: given the  T -field and some aspects of PDE+BC+IC, find some missing parameters (identification problem), e.g. finding the required dimensions or materials for a certain heat insulation or conduction goal.
Heat – transfer problems arise in many industrial and environmental process es, particularly in energy utilization, thermal processing, and thermal control. Energy cannot be created or destroyed, but so – common it is to use energy as synonymous of exergy, or the quality of energy, than it is commonly said that energy utilization is concerned with energy generation from primary sources (e.g. fossil fuels, solar), to end – user energy consumption (e.g. electricity and fuel consumption ), through all possible intermediate steps of energy valorisation, energy transportation, energy storage, and energy conversion processes. The purpose of thermal processing is to force a temperature change in the system that enables or disables some material transformation (e.g. food pas teurisation, cooking, steel tempering or annealing). The purpose of thermal control is to regulate within fixed established bounds, or to control in time within a certain margin, the temperature of a system to secure is correct functioning.
As a model problem, consider the thermal problem of heating a thin metallic rod by grasping it at one end
with our fingers for a while, until we withdraw our grip and let the rod cool down in air; we may want to predict the evolution of the temperature at one end, or the heat flow through it, or the rod conductivity needed to heat the opposite end to a given value. We may learn from this case study how difficult it is to model the heating by our fingers, the extent of finger contact, the thermal convection through the air, etc.
By the way, if this example seems irrelevant to engineering and science (nothing is irrelevant to science), consider its similarity with the heat gains and losses during any temperature measurement with a typical
‘long’ thermometer (from the old mercury-in-glass type. to the modern shrouded thermocouple probe). A more involved problem may be to find the temperature field and associated dimensional changes during machining or cutting a material, where the final dimensions depend on the time-history of the temperature field.
Everybody has been always exposed to heat transfer problems in normal life (putting on coats and avoiding winds in winter, wearing caps and looking for breezes in summer, adjusting cooking power, and so on), so that certain experience can be assumed. However, the aim of studying a discipline is to understand it in depth; e.g. to clearly distinguish thermal -conductivity effects from thermal – capacity effects, the relevance of thermal radiation near room temperatures, and to be able to make sound predictions.
Typical heat – transfer devices like heat exchangers, condensers, boilers, solar collectors, heaters, furnaces, and so on, must be considered in a heat – transfer course, but the emphasis must be on basic heat-transfer models, which are universal, and not on the myriad of details of past and present equipment.

Heat transfer theory is based on thermodynamics, physical transport phenomena, physical and chemical
energy dissipation phenomena, space – time modelling, additional mathematical modelling, and experimental tests.