page_title: CFD in buildings
section_title: Discretization of the governing differential equations for the transient heat transfer analysis
hierarchical_section_title: CFD in buildings / CFD approach for heat transfer analysis in buildings / Discretization of the governing differential equations for the transient heat transfer analysis
caption_attribution_description: English: FDM formulation of time dependent problems involve discrete points in time scale as well as in space.
context_page_description: CFD stands for computational fluid dynamics. As per this technique, the governing differential equations of a flow system or thermal system are known in the form of Navier–Stokes equations, thermal energy equation and species equation with an appropriate equation of state. In the past few years, CFD has been playing an increasingly important role in building design, following its continuing development for over a quarter of a century. The information provided by CFD can be used to analyse the impact of building exhausts to the environment, to predict smoke and fire risks in buildings, to quantify indoor environment quality, and to design natural ventilation systems.
context_section_description: Transient thermal analysis is more important than the steady thermal analysis, as this analysis include the variable ambient condition with time. In transient heat conduction, the temperature changes with time as well as position. The finite difference solution of transient heat conduction requires discretization in time in addition to space, as shown in figure-6.
  The nodal points and volume elements for the transient FDM formulation of 1-D conduction in a plane wall exist as shown in the figure-7.
  For this case the FDM explicit solution for equation (1) will be as follows,
  The above equation can be solved explicitly for the temperature  to give
  where,
  and 
  here, represents the cell Fourier no, represents thermal diffusivity,    represents specific heat at constant pressure,  represents time step, represents space step.
  Above equation is valid for all interior nodes and to find the relation for first and last node, apply boundary conditions (as applicable) as discussed in steady state heat transfer. For a convective & radiation boundary if solar radiation data  \, in ()  is available and absorptivity-transmissivity constant K is known, the relation for temperature is obtained as follows;
  Note: the thermal analysis for the roof and floor of a building can be done in same way, as discussed for walls.\\
caption: Figure-6: FDM fotirmulation of time dependent problem involves discrete points in time as well as in space
exif: {}