# Energy Equation

Energy equation in nanoFluidX is implemented so that it accommodates for conduction and convection heat transfer with initial or Dirichlet boundary conditions.

The rate of temperature change of a specific material is given by:(1)
$\frac{dT}{dt}=\frac{1}{{c}_{p}\rho }\nabla \left(\text{κ}\nabla T\right)$
With $t$ being time, $T$ being the temperature, $k$ the coefficient of thermal conductivity, $\rho$ the density and ${c}_{p}$ the specific heat capacity of the material. In SPH form the above analytic expression becomes (2)
${\left(\frac{dT}{dt}\right)}_{i}=\frac{1}{{c}_{p,i}}\sum _{j}\frac{4{m}_{j}}{{\rho }_{i}{\rho }_{j}}\frac{{\kappa }_{i}{\kappa }_{j}}{{\kappa }_{i}+{\kappa }_{j}}{T}_{ij}\frac{\nabla {W}_{ij}}{|{r}_{ij}|}\cdot n$

Where, the indices $i$ and $j$ standing for so called owner and neighboring particles respectively, and the $\mathrm{ij}$ index is a difference between the respective variables of particle $i$ and particle $j$ . The $m$ stands for mass of the particle, the $\nabla W$ is the gradient of the kernel, $r$ is the position vector and $n$ is the unit coordinate vector.

If you want to use the energy transport option, you need to turn the feature on by setting the energy_transport flag in Simulation parameters to true. Once this step is completed, you can opt for desired output, which can be either temperature or rate of temperature change (flux). The code will not work if an output is required and the energy transport flag is turned off.

Finally, set up the phase parameters, which are initial temperature (temp_0) in [K], evolve temperature flag (evolve_temp), specific heat capacity of the material (heat_cap) in [J/kg/K] and heat conduction coefficient of the material (heat_cond) in [W/m/K].
Note: Regarding the evolve_temp flag, this parameter is valid only for WALL or MOVINGWALL phases. If the evolve_temp flag is set to false, or if it is not defined (default is false), the set initial temperature will remain constant throughout the simulation. If set to true, the initial temperature will evolve in time and the temperature of the WALL or MOVINGWALL will be influenced by any surrounding phase. For FLUID phases, the temperature always evolves.