# Physics-Based Modeling

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## Physics-Based Modeling - overview

Below we describe on-going work on the use of physics-based models for data center energy management applications within MMT. Such applications combine the use of equations of mathematical physics, coupled with real-time sensor measurements, and their numerical solution in an effort to devise methods suitable for operational and real-time usage.

Heat transfer modeling

Thermal, or cooling, zones

### Air Flow and Heat Transfer Modeling

Current state-of-the-art numerical modeling approaches in data centers rely on the use of computational fluid dynamics (CFD) models based on the Navier-Stokes equations for air flow coupled with an equation for heat transfer. While this approach has been successfully used and is the most accurate modeling technology, especially for planning purposes, it comes with long calculation times, making it unsuitable for performing (near) real-time simulations. In addition, it is often not clear whether the quality of the modeling input data (or the lack of it) warrants such detailed or time-consuming computations.

Hence one of the focus areas in our group is centered on the development of simpler and, computationally speaking, faster CFD models for air and heat transfer in data centers, leveraging the fact that the availability of real-time measurement data could be “traded” against the complexity of the partial differential equations (PDEs) comprising the model. Real-time sensor measurements, which provide information about current (i.e., realistic) conditions in a data center, are used to define the required input information, mainly in the form of boundary data for the boundary value problems defining the model.

Results from our numerical simulations indicate that, even with a limited amount of sensor data supplied as input, the proposed air and heat transfer model for data centers is capable of producing numerical temperature distributions that capture well the behavior observed in experimental data collected from high resolution scans of temperature in a data center. This suggests the suitability of the model for operational and real-time usage as part of a data center energy management system.

**References**

- V. L�pez and H. F. Hamann.
*Heat Transfer Modeling in Data Centers*. Submitted. January 2011. - V. L�pez and H. F. Hamann. Measurements-based modeling for data centers. In
*Proceedings of ITherm 2010*, June 2010. - M. M. Toulouse, G. Doljac, V. P. Carey, and C. Bash. Exploration of a potential-flow-based compact model of air-flow transport in data centers. In
*Proceedings of the ASME International Mechanical Engineering Congress & Exposition*, November 2009. - J. W. VanGilder and S. K. Shrivastava. Capture index: an air-flow based rack cooling performance metric.
*ASHRAE Transactions*, 113(1):126-136, 2007.

### Thermal / Cooling Zones

The concept of *thermal*, or *cooling*, *zones* has been proposed in recent years as a means for providing operational information regarding which physical areas (or zones) in a data center are being supplied by the different air conditioning units (ACUs), in order to gain insights as to optimal use of cooling equipment [1, 2]. We note that the term “thermal zone” is being used generically to denote a region within a data center that is being influenced or impacted by the air being supplied by a particular ACU. As a concept, though, it is not uniquely defined. For instance, in [2, 3] thermal zones are defined via correlations computed using temperature data, whereas the study in [1] approximates thermal zones from an air velocity field (without employing temperature data) by explicitly tracing streamlines of the velocity field.

Currently in MMT, the latter definition is adopted. Rather than explicitly tracing the streamlines, though, the problem of computing thermal zones is being reformulated as a boundary value problem (BVP) for convective transport [4]. Use of such BVP to model thermal zones in MMT is designed so that zone boundaries correspond to sharply defined fronts of the solution of the BVP. The thermal zones boundaries then correspond to points for which the gradient of the solution of the BVP is non-zero. At the interior of each zone, one has that the gradient of the solution is (numerically) zero (the solution is constant within a thermal zone). This leads to identification of the zones by a simple post-processing of the numerical solution of the BVP, making the procedure convenient to apply, especially in three-dimensional domains. This methodology for modeling thermal zones is indeed conceptually simple, yet that is one of its advantages, at least in applications where visualization by means of streamlines is not necessarily optimal, as in MMT [1].

**References**

- H. F. Hamann , V. L�pez and A. Stepanchuk. Thermal zones for more efficient data center energy management. In
*Proceedings of ITherm 2010*, June 2010.

- C. J. Bonilla, E. Ferrer, and C. Bash. Thermal zone mapping: Data center design and assessment automated visualization tool for thermal metric analysis.
*IMAPS ATW*, October 2007.

- H. Li and H. F. Hamann. A statistical approach to thermal zone mapping. To appear in
*Proceedings of InterPACK 2011*, July 2011.

- V. L�pez and H. F. Hamann. A numerical technique for the approximation of thermal zones. To appear in
*Proceedings of InterPACK 2011*, July 2011.