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Finding a function to describe the temperature of this idealised 2D rod is a boundary value problem with Dirichlet boundary conditions.Any solution function will both solve the heat equation, and fulfill the boundary conditions of a temperature of 0 K on the left boundary and a temperature of 273.15 K on the right boundary.
The problem of heat transfer in the presence of liquid flowing around the body was first formulated and solved as a coupled problem by Theodore L. Perelman in 1961, [1] who also coined the term conjugate problem of heat transfer. Later T. L. Perelman, in collaboration with A.V. Luikov, [2] developed this approach further.
In the most general case, it's easy to see that at least 6 more equations are required, possibly more if there are internal degrees of freedom (such as temperature) which may vary throughout spacetime. In practice, it is usually possible to simplify the problem by replacing the full set of equations of state with a simple approximation.
The temperature approaches a linear function because that is the stable solution of the equation: wherever temperature has a nonzero second spatial derivative, the time derivative is nonzero as well. The heat equation implies that peaks ( local maxima ) of u {\displaystyle u} will be gradually eroded down, while depressions ( local minima ...
Heating and cooling the material affects both the temperature and the thermodynamic free energy or Gibbs energy. Simulated annealing can be used for very hard computational optimization problems where exact algorithms fail; even though it usually achieves an approximate solution to the global minimum, it could be enough for many practical problems.
When stated in terms of temperature differences, Newton's law (with several further simplifying assumptions, such as a low Biot number and a temperature-independent heat capacity) results in a simple differential equation expressing temperature-difference as a function of time. The solution to that equation describes an exponential decrease of ...
Temperature is a physical ... be too coarse-grained as they average out the microscopic thermal information based on the scale of the representative sample ...
The temperature regime may be heating, cooling at a rate of temperature change that can include stepwise temperature changes, linear rate of change, temperature modulation with a set frequency and amplitude, free (uncontrolled) heating or cooling, or maintaining a constant increase in temperature.