Modeling and Design of Net ZEBs as Integrated Energy Systems - Modeling, Design, and Optimization of Net-Zero Energy Buildings - page 84

rate to the rate of thermal energy storage) has to be less than 0.5. For
uniform conductivity and uniform one-dimensional grids
(2.44)
where is the grid spacing, is the conductivity, is the density, and is
the specific heat of the material.
See Section 2.1.5 for a one-dimensional example. For multidimensional
discretization, Eq. (2.44) can be rewritten as Eq. (2.45) , where is the
capacitance of the node, and is the conductance between the current
node and the adjacent node in direction . If the node receives heat flux
(e.g., solar radiation), the in the corresponding heat flux direction can
be increased to a larger equivalent conductance to account for this heat flux.
(2.45)
Implicit differencing approach is commonly used. The implicit formulation
assumes that the future values prevail throughout the current time step.
Generally, a system of equations can be solved simultaneously to obtain the
variable values at the next time step. Alternatively, their values can also be
initially guessed and then fine-tuned through iterations until overall energy
balance is achieved (Incropera and DeWitt, 2002). The advantage of the
implicit approach is that the limitation on the time step no longer exists.
However, long time steps are not appropriate for cases when the values
of the variables are changing rapidly. Furthermore, when controlled heat
sources are used, time steps cannot be longer than the control time interval.
The Crank-Nicolson approach lies between the explicit and implicit
approaches - it assumes linear variations of the values over a time step.
For the fluid flowing inside the BITES systems, Eq. (2.46) (Chen, Athienitis
andGalal,2013)givestheanalyticalsolutionforthelocaltemperatureofthe
fluid with the assumptions of constant boundary and large Peclet number
(Incropera and DeWitt, 2002; Patankar, 1980).