Editing Thermal conduction (section)

==Transient thermal conduction==
{{main|Heat equation}}

===Interface heat transfer===
{{unreferenced section|date=May 2013}}
The heat transfer at an interface is considered a transient heat flow. To analyze this problem, the [[Biot number]] is important to understand how the system behaves. The Biot number is determined by:
<math display="block"> \text{Bi} = \frac{hL}{k} </math>
The heat transfer coefficient <math>h</math>, is introduced in this formula, and is measured in <math display="block"> \mathrm{\frac{J}{m^{2} s K}} </math>. If the system has a Biot number of less than 0.1, the material behaves according to Newtonian cooling, i.e. with negligible temperature gradient within the body.<ref>{{cite book |last1=III |first1=H. Palmour |last2=Spriggs |first2=R. M. |last3=Uskokovic |first3=D. P. |title=Science of Sintering: New Directions for Materials Processing and Microstructural Control |date=11 November 2013 |publisher=Springer Science & Business Media |isbn=978-1-4899-0933-6 |page=164 |url=https://books.google.com/books?id=tUX2BwAAQBAJ |language=en}}</ref> If the Biot number is greater than 0.1, the system behaves as a series solution. however, there is a noticeable temperature gradient within the material, and a series solution is required to describe the temperature profile. The cooling equation given is:
<math display="block">q = -h \, \Delta T, </math>
This leads to the dimensionless form of the temperature profile as a function of time:
<math display="block"> \frac{T-T_f}{T_i - T_f} = \exp \left ( \frac{-hAt}{\rho C_p V} \right ). </math>
This equation shows that the temperature decreases exponentially over time, with the rate governed by the properties of the material and the heat transfer coefficient.<ref name="Biot Number Significance">{{cite web |last1=Aggarwal |first1=Nikita |title=Biot Number Calculator - Significance and Calculations - ChemEnggCalc |url=https://chemenggcalc.com/biot-number-calculator-and-significance/ |website=ChemEnggCalc |access-date=11 November 2024 |language=en |date=19 July 2024}}</ref>
The [[heat transfer coefficient]], {{math|''h''}}, is measured in <math> \mathrm{\frac{W}{m^2 K}} </math>, and represents the transfer of heat at an interface between two materials. This value is different at every interface and is an important concept in understanding heat flow at an interface.

The series solution can be analyzed with a [[nomogram]]. A nomogram has a relative temperature as the {{math|''y''}} coordinate and the Fourier number, which is calculated by
<math display="block">\text{Fo}= \frac{\alpha t}{L^2}. </math>

The Biot number increases as the Fourier number decreases. There are five steps to determine a temperature profile in terms of time.
# Calculate the Biot number
# Determine which relative depth matters, either ''x'' or ''L''.
# Convert time to the Fourier number.
# Convert <math>T_i</math> to relative temperature with the boundary conditions.
# Compared required to point to trace specified Biot number on the nomogram.