A New Technique for Characterizing Multi-Temperature Convection with Application in Building Energy Simulation

Glazing systems with attachments such as shades and insect screens are known as complex fenestration systems (CFS). The ASHRAE Window Attachment Tool (ASHWAT) for modeling heat transfer through a CFS are based on a general network of thermal resistors. In these models, convective heat transfer at the indoor side of a CFS with an indoor-mounted attachment is represented by a delta resistor-network. The heat transfer coefficients that characterize this network cannot be calculated using the conventional methods, i.e. based only on the knowledge of the total heat transfer rates at the temperature nodes. Currently, approximate relations based on known limits and experience are used in ASHWAT to estimate the indoor-side convection coefficients. The CFS problem is part of the broader class of multi-temperature convection, i.e. problems entailing exclusively isothermal and adiabatic boundary conditions. Driven by the desire to calculate the convection coefficients of the CFS problem with improved accuracy, this thesis is devoted to the study of multi-temperature convection. An extension of the Newton law of cooling is proposed to formulate the multi-temperature convection problem in terms of multiple driving temperature differences. Consequently, the problem is characterized by multiple paired heat transfer coefficients. A technique dubbed dQdT was developed to obtain the paired heat transfer coefficients. The dQdT technique is based on a baseline solution to the full set of governing equations and subsequent solutions to the linearized energy equation with perturbed boundary conditions. dQdT can be implemented in both analytical and numerical solutions. In addition to enabling the extended Newton formulation, the dQdT technique provides a basis for determining the applicability of the resistor-network model to a convection problem. The validity of dQdT was demonstrated in several ways. The extended Newton formulation and the dQdT technique were applied to a wide range of convection problems: forced and free convection, internal and external flows, laminar and turbulent flows, hydrodynamically developed and hydrodynamically developing flows, constant- and variable-property flows. The extended Newton formulation was shown to be advantageous compared to the traditional formulation; it leads to a presentation of the solution that is more consistent with the physics of the problem while revealing more detail about the thermal phenomenon. Moreover, using the dQdT results improved correlations for the classical problems of convection in heated annuli and vertical channels were developed. Finally, the dQdT technique was applied to calculate the heat transfer coefficients of the CFS problem. It was shown that while the current ASHWAT estimates are in good agreement with the dQdT results for a CFS entailing a roller blind, there is a potential for improving the ASHWAT estimates for a CFS entailing a venetian blind. Using dQdT, the convection coefficients of a wide range of CFS configurations can now be accurately calculated

Laminar Free Convection from a Pair of Horizontal Cylinders: A Three-Temperature Problem

The formulation of multi-temperature convection problems in terms of a single driving temperature difference and a single heat transfer coefficient does not properly reflect the physics of the problem. Paired heat transfer coefficients, which designate both the source and the sink of heat transfer, are a suitable alternative. A numerical technique, namely dQdT, is proposed to compute such paired heat transfer coefficients. The dQdT technique entails a numerical solution of the governing equations and consequent solutions of the energy equation with perturbed boundary conditions. In the present study, dQdT is applied to the three-temperature problem of steady-state laminar free convection from two horizontal cylinders, with equal diameters but different surface temperatures, aligned vertically at a center-to-center spacing of two diameters. A number of moderate Rayleigh numbers, 2×104<Ra<2×105, are considered. The baseline solutions are validated against experimental data from the literature. A grid convergence study is performed to assess the discretization error. The utility of dQdT in characterizing this three-temperature free convection problem, specifically in quantifying the interaction of the cylinders, is demonstrated. It has also been shown that the paired heat transfer coefficients obtained through dQdT provide detailed information about the local interaction of the cylinders, which are not available otherwise.Smart Net-Zero Energy Buildings Strategic Network (SNEBRN) || Natural Sciences and Engineering Research Council of Canada (NSERC) || University of Waterlo

Resistor-Network Formulation of Multitemperature Forced-Convection Problems

Please note that this file contains the final draft version of this technical paper. Minor differences will be found between this version and the final version printed by the publisher. The reader should contact the publisher if the final version, as printed, is preferred. Copyright © 2016 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. Copies of this paper may be made for personal and internal use, on condition that the copier pay the per-copy fee to the Copyright Clearance Center (CCC). All requests for copying and permission to reprint should be submitted to CCC at www.copyright.com; employ the ISSN 0887-8722 (print) or 1533-6808 (online) to initiate your request. Foroushani, S., Naylor, D., & Wright, J. L. (2016). Resistor-Network Formulation of Multitemperature Forced-Convection Problems. Journal of Thermophysics and Heat Transfer, 1–8. https://doi.org/10.2514/1.T4993Many convection heat transfer problems involve more than two isothermal heat sources/sinks. A network of multiple convective resistors connecting temperature nodes representing the isothermal sources (walls, inlet flows, etc.) can be used to represent this class of problem. However, the convective resistances that characterize this network cannot generally be evaluated using energy balances resulting from a single solution to the energy equation. A technique based on solutions of the energy equation with perturbed boundary conditions is developed to overcome this difficulty. The resulting technique is verified by comparison with energy-balance results previously obtained for a special symmetric case. The technique is also applied to a superposition solution for hydrodynamically developed laminar flow in an annulus and to numerical solutions of simultaneously developing flow in an asymmetrically heated annulus under both laminar and turbulent flow conditions. This work is part of an ongoing research project on the resistor-network modeling and characterization of multitemperature convection problems.Smart Net-Zero Energy Buildings Strategic Research Network (SNEBRN) || Natural Sciences and Engineering Research Council of Canada (NSERC) || University of Waterlo

Convective Heat Transfer In Hydrodynamically-Developed Laminar Flow In Asymmetrically-Heated Annuli: A Three-Temperature Problem

Heat transfer in hydrodynamically-developed flow in asymmetrically-heated channels and annuli has been studied extensively. This study is an extension of earlier work where heat transfer in an asymmetrically-heated parallel-plate channel was examined in a resistor-network framework. It was shown that the formulation of the problem in terms of a delta thermal-resistor network has several advantages. A delta network can also be used to represent heat transfer in asymmetrically-heated annuli. Nevertheless, the evaluation of the three paired convective resistances that characterize the network is not straightforward. In the present paper, a new technique based on solutions of the energy equation with perturbed boundary conditions is proposed. The proposed technique is first verified by comparison with the results previously obtained for the parallel-plate channel problem. A superposition solution to the energy equation is obtained for hydrodynamically-developed laminar flow in an asymmetrically-heated annulus. The developed technique is then applied to the annulus problem to obtain the corresponding resistances. Results are validated by examining limiting cases.Smart Net-Zero Energy Buildings Strategic Research Network (SNEBRN) || Natural Sciences and Engineering Research Council of Canada (NSERC) || University of Waterlo

Asymmetric Graetz Problem: The Analytical Solution Revisited

Copyright © 2016 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. Copies of this paper may be made for personal and internal use, on condition that the copier pay the per-copy fee to the Copyright Clearance Center (CCC). All requests for copying and permission to reprint should be submitted to CCC at www.copyright.com; employ the ISSN 0887-8722 (print) or 1533-6808 (online) to initiate your request. Foroushani, S. S. M., Wright, J. L., & Naylor, D. (2017). Asymmetric Graetz Problem: The Analytical Solution Revisited. Journal of Thermophysics and Heat Transfer, 31(1), 237–242. https://doi.org/10.2514/1.T4944Forced-convective heat transfer from the walls of an asymmetrically heated channel to the fluid passing through in a laminar, hydrodynamically developed flow is known as the asymmetric Graetz problem. Several analytical and numerical solutions for this problem have been published, and many variations and extensions have been investigated. Recently, there has been a renewed interest in this problem due to its applications in emerging areas such as microchannels and fuel cells. In the present work, the asymmetric Graetz problem is examined in a resistor-network framework. The formulation of the problem in terms of three convective resistances forming a delta network leads to temperature-independent Nusselt numbers that are free of the singularities found in previous results. The proposed approach also offers new information regarding the split of heat transfer between the channel walls and the flow. This work is part of an ongoing project on resistor-network modeling and characterization of multitemperature convection problems.Smart Net-Zero Energy Buildings Strategic Research Network (SNEBRN) || Natural Sciences and Engineering Research Council of Canada (NSERC) || University of Waterlo

Resistor-Network Formulation of Multi-Temperature Free Convection Problems

Copyright © 2016 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. Copies of this paper may be made for personal and internal use, on condition that the copier pay the per-copy fee to the Copyright Clearance Center (CCC). All requests for copying and permission to reprint should be submitted to CCC at www.copyright.com; employ the ISSN 0887-8722 (print) or 1533-6808 (online) to initiate your request. Foroushani, S., Wright, J. L., & Naylor, D. (2017). Resistor-Network Formulation of Multitemperature Free-Convection Problems. Journal of Thermophysics and Heat Transfer, 1–6. https://doi.org/10.2514/1.T5024In recent work, the resistor-network formulation of forced-convection problems and a technique (dQdT) for evaluating the paired convective resistances that characterize the network were presented. This technique entails solutions of the energy equation with perturbed boundary conditions. In the present paper, the dQdT technique is extended to free convection. The analytical solution to the classical two-temperature problem of free convection at an isothermal vertical flat plate is used to verify the technique. Then, dQdT is applied to the three-temperature problem of free convection in an asymmetrically heated vertical channel based on numerical solutions of the energy equation. Sample results are presented and known limits are discussed to demonstrate the validity of the results. This paper is part of a series on the resistor-network formulation of convection problems.Smart Net-Zero Energy Buildings Strategic Research Network || Natural Sciences and Engineering Research Council of Canada || University of Waterlo

Forced Convection From a Pair Of Spheres: A Three-Temperature Problem

Interaction of spherical particles in a fluid flow is important in combustion, chemical processes and air pollution. In this study, a recently developed technique for characterizing multi-temperature convective heat transfer is used to investigate convection from a pair of spheres with different surface temperatures. The technique entails numerical solutions of the full set of governing equations and subsequent solutions of the energy equation with perturbed boundary conditions. Steady-state heat transfer in intermediate Reynolds number flows over spheres in side-by-side and tandem arrangements in a water tunnel is studied. The results are expressed in terms of local and total Nusselt numbers. The variation of sphere-sphere and sphere-fluid Nusselt numbers with flow conditions is examined.Smart Net-Zero Energy Buildings Strategic Network (SNEBRN) || Natural Sciences and Engineering Research Council of Canada (NSERC) || University of Waterlo

Sensitivity of the Solar Heat Gain Coefficient of Complex Fenestration Systems to the Indoor Convection Coefficient

Accurate estimation of the solar gain of a fenestration system is important in analyzing the energy performance of buildings. Recently, models were developed for complex fenestration systems – glazing systems with attachments such as venetian blinds and insect screens. These models use a three-node network for modeling heat transfer at the indoor-side of a glazing system. Empirical expressions based on observation and known limits were originally proposed for the corresponding convection coefficients. To address any ambiguity or error associated with these expressions, a research project is underway to develop techniques for evaluating these convection coefficients more accurately. The purpose of the current paper is to quantify the sensitivity of the U-value and solar heat gain coefficient of complex fenestration systems to the indoor-side convection coefficients. Configurations comprised of low-e glazings, roller blinds, venetian blinds, drapes and insect screens are examined in design summer and winter conditions using the window analysis software VISION5. Results show that the presence of an indoor-mounted attachment can significantly change the solar heat gain coefficient of a fenestration system. Nevertheless, the solar heat gain coefficient and the overall heat transfer coefficient are not sensitive to the indoor convection coefficient

Assessing Convective Heat Transfer Coefficients Associated with Indoor Shading Attachments Using a New Technique Based on Computational Fluid Dynamics

© 2015 ASHRAE (www.ashrae.org). Published in ASHRAE Conference Papers, Winter Conference, Chicago, IL. For personal use only. Additional reproduction, distribution, or transmission in either print or digital form is not permitted without ASHRAE's prior written permission.ASHRAE research project RP-1311 concluded with the creation of simulation models, the ASHWAT models, for glazing systems with shading attachments. Such assemblies are known as Complex Fenestration Systems (CFS). ASHWAT represents several key advances. An “equivalent layer” approach is used to track radiation at glazing layers, slat-type blinds, roller blinds, pleated drapes and insect screens. Longwave and off-normal solar optical properties are available for each of these shading layer types. A general thermal resistance framework allows for calculation of U-factor and Solar Heat Gain Coefficient (SHGC). Radiant flux, solar beam/diffuse or longwave, within the CFS multi-layer array is tracked by ASHWAT using an accounting system that is straightforward. To support this operation a large effort was devoted to the compilation of optical properties for shading layers - based on a significant level of measurement and analysis. It is also necessary to account for convective heat transfer between the various CFS layers. Methods to obtain convective heat transfer coefficients for glazing cavities are well established, even for a glazing cavity that includes a venetian blind. However, the convective heat transfer coefficients in the vicinity of a shading layer mounted next to a glazing system are not so readily obtained. Added complexity arises because each of three temperature nodes – indoor glazing surface, shading layer and room air – exchanges thermal energy with the other two nodes so three convective heat transfer coefficients are needed. No previous work provides an analysis of this three-resistor system. ASHWAT currently incorporates estimates of the three convective heat transfer coefficients, as functions of glass-to-shade spacing, based largely on known limiting cases and a limited amount of observation related to the behaviour of venetian blinds. Convective heat transfer coefficients can readily be evaluated in a two-resistor network. Quantification of the heat transfer between the nodes, for a given temperature difference, is sufficient. This is not the case for a three-node system. However, recent theoretical work shows that the three heat transfer coefficients can be obtained using a unique numerical perturbation procedure. Also, if this perturbation procedure is applied through Computational Fluid Dynamics (CFD) it is possible, using a specific sequence of steps, to escape the numerical error associated with small perturbations. The same approach cannot be applied to experimentation with natural convection. Theory is presented and sample results are compared to the ASHWAT estimates for a shading layer mounted adjacent to the indoor side of a window.Smart Net-Zero Energy Buildings Research Network (SNEBRN) || Natural Science and Engineering Research Council (NSERC) of Canada || University of Waterlo

Indoor-Side Convection Coefficients for Complex Fenestration Systems with Roller Blinds

© 2017 ASHRAE (www.ashrae.org). Published in ASHRAE Conference Papers, Winter Conference, Las Vegas, NV. For personal use only. Additional reproduction, distribution, or transmission in either print or digital form is not permitted without ASHRAE's prior written permission.In order to characterize the resistor network that describes convective heat transfer on the indoor side of a complex fenestration system, three convection coefficients are needed. Although methods to obtain convection coefficients in glazing cavities are well established, the convection coefficients in the vicinity of an indoor-mounted attachment are not so readily available. In principle, convection coefficients of a three-resistor network cannot be obtained based on numerical solutions or measurement only. The ASHWAT models for simulating windows with attachments currently provide estimates of the three convection coefficients as functions of glass-to-shade spacing, based largely on known limits. Recently, a numerical technique, dubbed dQdT, was developed for evaluating the heat transfer coefficients of multi-temperature convection problems. This technique entails numerical solutions of the full set of governing equations and subsequent solutions of the energy equation with perturbed boundary conditions. In earlier work, dQdT was applied to a fenestration system with an indoor-mounted roller blind. To keep the flow laminar and the numerical solutions simple, a relatively short window was considered. Preliminary results suggested that ASHWAT gives good estimates of the convection coefficients and accurately predicts the general trends. Nevertheless, ASHWAT overestimated the glass-to-shade heat transfer coefficient for intermediate spacing, while underpredicting glass-to-air and shade-to-air heat transfer coefficients for larger spacings. The effect of spacing also seemed to be underestimated by ASHWAT. The present study was undertaken to further examine the accuracy of the ASHWAT estimates for windows with realistic dimensions, various glazing-attachment spacings and taking into account the transition of the flow to turbulence. Excellent agreement between the dQdT results and the ASHWAT predictions was obtained under summer design conditions, confirming the validity and utility of the current ASHWAT correlations. A minor adjustment to improve the accuracy of the current ASHWAT estimates is suggested.Smart Net-Zero Energy Buildings strategic Research Network (SNEBRN) || Natural Science and Engineering Research Council (NSERC) of Canada || University of Waterlo