Abstract
This chapter develops a methodology for modeling of heat transfer dynamics in a building atrium via Koopman mode decomposition (KMD). We address the phenomenon of heat transfer due to movement of air inside an atrium, where the air slowly moves over the distance between individual rooms. The heat transfer is modeled as a heat advection equation with a vector-valued velocity coefficient and a nonlinear input term from heating, ventilation, and air-conditioning (HVAC) operations, which corresponds to a control variable. KMD is applied to the equation so that the heat transfer and HVAC operation are characterized in terms of frequencies and wavenumber vectors of Koopman modes (KMs). The velocity coefficient is then identified with a KM governing the heat transfer. The effectiveness of the modeling is demonstrated with measurement data on temperature field and HVAC operation of a practically used building. A possibility of how to use this modeling for control of heat transfer dynamics is discussed at the end of this chapter.
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Notes
- 1.
Although \(\mathbf {V}_\text {eff}\) and \(D_\text {eff}\) are constant in the typical averaging and homogenization frameworks, we now treat them as space dependent functions in order to represent global trend of \(\mathbf {v}(\mathbf {r},t)\) over the domain \(V_\text {c}\). The space- (and time-)dependency of \(D_\text {eff}\) and \(\mathbf {V}_\text {eff}\) is investigated in the literature: see, e.g., [27].
- 2.
KMs {10,11} also have large magnitudes of \(\Vert \mathbf {A}_m\Vert \) and \(\Vert \mathbf {B}_m\Vert \). However, they can be neglected because the original data contain no oscillatory component associated with the period \(T_{10}=6.490\) h. Indeed, we applied the discrete Fourier transform (DFT) to the original data \(\{T(\mathbf {r}_p,n\varDelta t) | n=0,1,\ldots , N\}\) for every position \(\mathbf {r}_p\), and confirmed that the values of the period of dominant components derived by DFT are different from \(T_{10}\): see [25].
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Acknowledgements
The authors appreciate Mr. Chosei Kaseda, Mr. Junnya Nishiguchi, and Mr. Kei Koga (Azbil Corporation) for the provision of the photographs and measurement data, and insightful discussion. This work was partly supported by JST, CREST #JPMJCR15K3.
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Kono, Y., Susuki, Y., Hikihara, T. (2020). Modeling of Advective Heat Transfer in a Practical Building Atrium via Koopman Mode Decomposition. In: Mauroy, A., Mezić, I., Susuki, Y. (eds) The Koopman Operator in Systems and Control. Lecture Notes in Control and Information Sciences, vol 484. Springer, Cham. https://doi.org/10.1007/978-3-030-35713-9_18
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