Taller de prensa n.º 4: El periódico escolar

In explicit FD approach, Fourier number ( ∙ ⁄ , the ratio of the heat conduction rate to the rate of thermal energy storage) needs to be less than 0.5 to stabilize the marching of the FD equations forward in time (Incropera and DeWitt 2002). Therefore in our case of one-dimensional heat transfer, the maximum allowable time step can be calculated as follow:

_ _(3.24)

where _ is the capacitance of the edge CV, and is the conductance

between the edge capacitance node and the adjacent inner capacitance node.

One of the advantages of using LPFD models is their less critical time step requirement in explicit formulation, extending from less than five minutes for a 1-cm thick layer to more than an hour for a 5-cm layer in one-dimensional model. However, large time step may not be adequate to obtain a sufficiently accurate solution reflecting practical operating situations. Effects of time step selection are discussed in the companion paper Part 2 (Section 3.3).

3. Conclusion

Modeling techniques for frequency response (FR) and lumped-parameter finite difference (LPFD) approaches are presented in this section (paper) for active building-integrated thermal energy storage (BITES) systems. The methodology is applied to ventilated concrete slabs (VCS) system. The techniques are applicable to other ventilated, electric, and hydronic BITES systems.

Network modeling techniques, such as heat source transformation with Thévenin theorem, heat flow division, and Y-diakoptic method are presented as means to develop transfer

function models in frequency domain. Thévenin transformation and heat flow division are equivalent in the treatment of the heat flow from the flowing air. Y-diakoptic method can split the BITES system into two parts at the internal heat source level, and hence facilitates the formulation and calculation. Discrete Fourier series (DFS) representation in complex frequency form are used to represent the boundary excitations, such as the surface

temperature variation, solar radiation and heat flux associated with flowing air. Since the heat transfer equations are also solved and represented in complex frequency domain, simple and efficient solutions can be readily obtained. Frequency domain results can be easily

transformed back to the time domain. The criteria for choosing the number of harmonics have also been discussed. Furthermore, equations for one-dimensional discretization and time step selection are discussed, taking into account convective and radiative heat transfer on the boundaries. A method used in simplified models for calculating the heat transfer between flowing air and ventilated BITES systems is developed. Application of these techniques is presented in the second part of the study (Section 3.3).

Section 3.3

Frequency domain and finite difference modeling

of ventilated concrete slabs and comparison with

field measurements: Part 2, application

Based on a paper in-press:

Chen, Y., A. Athienitis and K. Galal. 2013b. Frequency domain and finite difference modeling of ventilated concrete slabs and comparison with field measurements: Part 2, application. International Journal of Heat and Mass Transfer, in press.

Section Abstract

This section (paper) is the second of two that present techniques and guidelines for

frequency response (FR) and lumped-parameter finite difference (LPFD) approaches for the thermal modeling of building-integrated thermal energy storage (BITES) systems. To assist the thermal analysis and control of active BITES systems, development of FR and LPFD models are presented in this two-part study. Modeling methodology and techniques are presented in paper Part 1 (Section 3.2) using ventilated concrete slabs (VCS) for

demonstration. In this part, the methodology is applied to two types of VCS. The modeling results from different FR and explicit LPFD models with different time steps and

discretization schemes are presented. The results are compared to each other, and with field- measured data from a solar demonstration house with a VCS. Simulation results show that time step of half an hour for FR models results in less than 3% errors in thermal

performance. For LPFD models, discretization with a Biot number smaller than 0.5 can reduce errors to about 5%.

1. Introduction

In Section 3.2 of this chapter, the model development for FR and explicit LPFD models of ventilated BITES systems are presented, using VCS systems for demonstration. In this part, the techniques are applied to two common types of VCS systems (Fig. 3.12). VCS-b systems have air channels at the bottom of the slab (“b” stands for “bottom”) while VCS-c systems use their hollow cores as air channels (“c” stands for “center” or “core”). This cross section can represent either a slab-on-grade or an intermediate floor slab. The insulation layer can be replaced by a false ceiling and the air layer between the slab and the ceiling. Insulation is optional, but should be used if occupants would like to limit the heat to the opposite direction. The air flow in the channels does not interact directly with room air (i.e. a close- loop system). The FR and LPFD modeling results are compared with each other. The LPFD modeling results of the VCS-b system are also compared with field-measurement. The accuracies for corresponding choices of time step and discretization are quantified. The purpose of this part is to provide modeling guidelines for FR models and explicit LPFD approaches for ventilated BITES systems.

(a) VCS-b configuration (b) VCS-c configuration

Fig. 3.12: Schematics of two types of VCS

Concrete Insulation Room air node Soil node Air channel Concrete Insulation Room air node Soil node Air channel Air channel