scholarly journals An Analytical Model for Transient Heat Transfer with a Time-Dependent Boundary in Solar- and Waste-Heat-Assisted Geothermal Borehole Systems: From Single to Multiple Boreholes

2021 ◽  
Vol 11 (21) ◽  
pp. 10338
Author(s):  
Mohammed A. Hefni ◽  
Minghan Xu ◽  
Ferri Hassani ◽  
Seyed Ali Ghoreishi-Madiseh ◽  
Haitham M. Ahmed ◽  
...  
Keyword(s):  
Heat Flux ◽  
Energy Storage ◽  
Analytical Model ◽  
Thermal Performance ◽  
Closed Loop ◽  
Waste Heat ◽  
Analytical Framework ◽  
Thermal Design ◽  
Time Dependent ◽  
Borehole Wall

With the increasing engineering applications of geothermal borehole heat exchangers (BHEs), accurate and reliable mathematical models can help advance their thermal design and operations. In this study, an analytical model with a time-dependent heat flux boundary condition on the borehole wall is developed, capable of predicting the thermal performance of single, double, and multiple closed-loop BHEs, with an emphasis on solar- and waste-heat-assisted geothermal borehole systems (S-GBS and W-GBS) for energy storage. This analytical framework begins with a one-dimensional transient heat conduction problem subjected to a time-dependent heat flux for a single borehole. The single borehole scenario is then extended to multiple boreholes by exploiting lines of symmetry (or thermal superposition). A final expression of the temperature distribution along the center line is attained for single, double, and multiple boreholes, which is verified with a two-dimensional finite-element numerical model (less than 0.7% mean absolute deviation) and uses much lesser computational power and time. The analytical solution is also validated against a field-scale experiment from the literature regarding the borehole and ground temperatures at different time frames, with an absolute error below 6.3%. Further, the thermal performance of S-GBS and W-GBS is compared for a 3-by-3 borehole configuration using the analytical model to ensure its versatility in thermal energy storage. It is concluded that our proposed analytical framework can rapidly evaluate closed-loop geothermal BHEs, regardless of the numbers of boreholes and the type of the heat flux on the borehole wall.

Transient Internal Forced Convection Under Arbitrary Time-Dependent Heat Flux

2013 ◽  
Author(s):  
M. Fakoor-Pakdaman ◽  
M. Andisheh-Tadbir ◽  
Majid Bahrami
Keyword(s):  
Heat Transfer ◽  
Heat Flux ◽  
Nusselt Number ◽  
Forced Convection ◽  
Analytical Model ◽  
Time Dependent ◽  
Bulk Temperature ◽  
Fluid Temperature ◽  
Flux Boundary Condition ◽  
Arbitrary Time

A new all-time model is developed to predict transient laminar forced convection heat transfer inside a circular tube under arbitrary time-dependent heat flux. Slug flow condition is assumed for the velocity profile inside the tube. The solution to the time-dependent energy equation for a step heat flux boundary condition is generalized for arbitrary time variations in surface heat flux using a Duhamel’s integral technique. A cyclic time-dependent heat flux is considered and new compact closed-form relationships are proposed to predict: i) fluid temperature distribution inside the tube ii) fluid bulk temperature and iii) the Nusselt number. A new definition, cyclic fully-developed Nusselt number, is introduced and it is shown that in the thermally fully-developed region the Nusselt number is not a function of axial location, but it varies with time and the angular frequency of the imposed heat flux. Optimum conditions are found which maximize the heat transfer rate of the unsteady laminar forced-convective tube flow. We also performed an independent numerical simulation using ANSYS to validate the present analytical model. The comparison between the numerical and the present analytical model shows great agreement; a maximum relative difference less than 5.3%.

Unsteady Laminar Forced-Convective Tube Flow Under Dynamic Time-Dependent Heat Flux

2014 ◽  
Vol 136 (4) ◽  
Author(s):  
M. Fakoor-Pakdaman ◽  
Mehdi Andisheh-Tadbir ◽  
Majid Bahrami
Keyword(s):  
Heat Transfer ◽  
Heat Flux ◽  
Nusselt Number ◽  
Analytical Model ◽  
Time Dependent ◽  
Bulk Temperature ◽  
Fluid Temperature ◽  
Tube Flow ◽  
Flux Boundary Condition ◽  
Arbitrary Time

A new all-time model is developed to predict transient laminar forced convection heat transfer inside a circular tube under arbitrary time-dependent heat flux. Slug flow (SF) condition is assumed for the velocity profile inside the tube. The solution to the time-dependent energy equation for a step heat flux boundary condition is generalized for arbitrary time variations in surface heat flux using a Duhamel's integral technique. A cyclic time-dependent heat flux is considered and new compact closed-form relationships are proposed to predict (i) fluid temperature distribution inside the tube, (ii) fluid bulk temperature and (iii) the Nusselt number. A new definition, cyclic fully developed Nusselt number, is introduced and it is shown that in the thermally fully developed region the Nusselt number is not a function of axial location, but it varies with time and the angular frequency of the imposed heat flux. Optimum conditions are found which maximize the heat transfer rate of the unsteady laminar forced-convective tube flow. We also performed an independent numerical simulation using ansys fluent to validate the present analytical model. The comparison between the numerical and the present analytical model shows great agreement; a maximum relative difference less than 5.3%.

scholarly journals Investigating thermal performance of an ice rink cooling system with an underground thermal storage tank

2017 ◽  
Vol 36 (2) ◽  
pp. 314-334 ◽  
Author(s):  
Hakan Tutumlu ◽  
Recep Yumrutaş ◽  
Murtaza Yildirim
Keyword(s):  
Energy Storage ◽  
Analytical Model ◽  
Thermal Performance ◽  
Thermal Energy ◽  
Thermal Energy Storage ◽  
Storage Tank ◽  
Cooling System ◽  
Operating Time ◽  
Ambient Air ◽  
Coefficient Of Performance

This study deals with mathematical modeling and energy analysis of an ice rink cooling system with an underground thermal energy storage tank. The cooling system consists of an ice rink, chiller unit, and spherical thermal energy storage tank. An analytical model is developed for finding thermal performance of the cooling system. The model is based on formulations for transient heat transfer problem outside the thermal energy storage tank, for the energy needs of chiller unit, and for the ice rink. The solution of the thermal energy storage tank problem is obtained using a similarity transformation and Duhamel superposition techniques. Analytical expressions for heat gain of the ice rink and energy consumption of the chiller unit are derived as a function of inside design air, ambient air, and thermal energy storage tank temperatures. An interactive computer program in Matlab based on the analytical model is prepared for finding hourly variation of water temperature in the thermal energy storage tank, coefficient of performance of the chiller, suitable storage tank volume depending on ice rink area, and timespan required to attain an annually periodic operating condition. Results indicate that operation time of span 6–7 years will be obtained periodically for the system during 10 years operating time.

scholarly journals Time-dependent solutions for daily-periodic slope flows driven by surface energy budget

2021 ◽  
Author(s):  
Mattia Marchio ◽  
Sofia Farina ◽  
Dino Zardi
Keyword(s):  
Heat Flux ◽  
Surface Energy ◽  
Surface Temperature ◽  
Wind Velocity ◽  
Analytical Model ◽  
Slope Angle ◽  
Time Dependent ◽  
Fourier Series Expansion ◽  
Ambient Atmosphere ◽  
Slope Wind

<p>Diurnal wind systems generated from daytime heating and nighttime cooling of valleys and slopes are a very common feature over mountainous terrains. Despite their frequent occurrence and relevance for a variety of applications, ranging from pollutant transport to convection initiation, slope winds are far from being fully understood and still provide an open research topic.</p><p>A well-known steady-state analytical model <span>is the one</span> developed by Prandtl (1942). Then, a first time-dependent analytical model was proposed by F. Defant (1949) and later extended by Zardi and Serafin (2015). These models provide slope-normal profiles of temperature and along-slope wind velocity as a response to a sinusoidal forcing representing the surface temperature. The resulting profiles exhibit sinusoidal oscillations at every distance from the surface, although with different phase lags under different regimes, determined by different combinations of slope angle and stability of the unperturbed ambient atmosphere. As a consequence, they can not explain the observed differences between daytime upslope and nighttime downslope winds in terms of magnitude and height of the peak of wind velocity, as well as the different timing characterising nighttime, daytime, and the two reversal phases.</p><p>In the present work, the solutions derived in Zardi and Serafin (2015) are extended to include a more realistic daily-periodic surface forcing taking into account the daily evolution of the surface temperature computed on the basis of a surface energy <span>budget</span>. Incoming solar radiation is represented by means of a Fourier series expansion derived from well-established relationships taking into account latitude, day of the year, slope angle, exposition and other astronomical and atmospheric factors. Based on <span>these</span> expansions, suitable harmonic solutions are derived for the heat flux into the ground and sensible heat flux in the atmosphere, and hence for the daily evolution of slope-normal profiles of along-slope wind velocity and potential temperature.</p><p>References:</p><ul><li><span>Prandtl L. 1942. Führer durch die Strömungslehre, Chapter 5. Vieweg und Sohn: Braunschweig, Germany. </span>[English translation: Prandtl L. 1952. Mountain and valley winds in stratified air, in Essentials of Fluid Dynamics: 422–425. Hafner Publishing Company: New York, NY]</li> <li><span>Defant F. 1949. Zur Theorie der Hangwinde, nebst Bemerkungen zur Theorie der Berg- und Talwinde. </span>Arch. Meteorol. Geophys. Bioklimatol. A 1: 421–450</li> <li>Zardi D., Serafin S. 2015. An analytic solution for time‐periodic thermally driven slope flows. Q. J. R. Meteorol. Soc., 141, 1968–1974, https://doi.org/10.1002/qj.2485</li> </ul>

Single Phase Liquid Cooling of High Heat Flux Devices with Local Hotspot in a Micro-Gap with Non-Uniform Fin Array

2020 ◽  
Author(s):  
Yuanchen Hu ◽  
Tom Sarvey ◽  
Muhannad Bakir ◽  
Yogendra Joshi
Keyword(s):  
Heat Flux ◽  
Thermal Performance ◽  
Single Phase ◽  
Heat Fluxes ◽  
Thermal Design ◽  
Liquid Cooling ◽  
Cooling Performance ◽  
Pin Fin ◽  
Phase Liquid ◽  
Pin Fin Array

Abstract Single-phase liquid cooling in micro-channels and micro-gaps has been successfully demonstrated for heat fluxes of ~1 kW/cm2 for silicon chips with maximum temperature below 100 °C. However, effectively managing localized hotspots in heterogeneous integration, which refers to the integration of various components that achieve multiple functionalities, entails further thermal challenges. To address these, we use a non-uniform pin-fin array. Single-phase liquid-cooling performance of four silicon test chips, thermal design vehicles (TDVs), each with a non-uniform pin-fin array, are experimentally examined. We evaluate multiple combinations of hotspot and background heat fluxes using four background heaters aligned upstream to downstream, and one additional hotspot heater located in the center. We examine the thermal performance of cylindrical fin-enhanced TDVs and hydrofoil fin-enhanced TDVs, both with two designs: one with increased fin density around the hotspot only, and another with increased fin density spanning the entire width of the channel. The resulting heat flux ratio of the localized hotspot to background heaters varies from 1 to 5. TDVs with spanwise increased hydrofoil fin density (spanwise hydrofoil) exhibit the best thermal performance with 6%-14% lower hotspot temperature than others. TDVs with spanwise increased cylindrical fin (cylindrical spanwise) maintain a balance between hotspot cooling performance and pressure drops. In general, as the temperature of the hotspot remains around 70? with a heat flux of 625 W/cm2, the non-uniform fin-enhanced micro-gaps appears to be a promising hotspot thermal management approach.

Development of an Analytical Model to a Temperature Distribution of First Level Package With a Non-Uniformly Powered Die

2007 ◽  
Author(s):  
Abhijit Kaisare ◽  
Dereje Agonafer ◽  
A. Haji-Sheikh ◽  
Greg Chrysler ◽  
Ravi Mahajan
Keyword(s):  
Heat Flux ◽  
Temperature Distribution ◽  
Analytical Model ◽  
Functional Integration ◽  
Thermal Design ◽  
Junction Temperature ◽  
Direct Result ◽  
Uniform Heat Flux ◽  
Chip Surface ◽  
Functional Components

Microprocessors continue to grow in capabilities, complexity and performance. Microprocessors typically integrate functional components such as logic and level two (L2) cache memory in their architecture. This functional integration of logic and memory results in improved performance of the microprocessor. However, the integration also introduces a layer of complexity in the thermal design and management of microprocessors. As a direct result of functional integration, the power map on a microprocessor is typically highly non-uniform and the assumption of a uniform heat flux across the chip surface has been shown to be invalid post Pentium II architecture. The active side of the die is divided into several functional blocks with distinct power assigned to each functional block. Previous work has been done which includes numerical analysis and thermal Based optimization of a typical package consisting of a non-uniformly powered die, heat spreader, TIM I &II and the base of the heat sink. In this paper, an analytical approach to temperature distribution of a first level package with a non-uniformly powered die is carried out for the first time. The analytical model for two layer bodies developed by Haji-Sheikh et al. is extended to this typical package which is a multilayer body. The solution is to begin by designating each surface heat flux as a volumetric heat source. An inverse methodology will be applied to solve the equations for various surfaces to calculate maximum junction temperature for given multilayer body. Finally validation of the analytical solution will be carried out using developed numerical model.

Development of Analytical Model to a Temperature Distribution of a First Level Package With a Nonuniformly Powered Die

2009 ◽  
Vol 131 (1) ◽  
Author(s):  
Abhijit Kaisare ◽  
Dereje Agonafer ◽  
A. Haji-Sheikh ◽  
Greg Chrysler ◽  
Ravi Mahajan
Keyword(s):  
Heat Flux ◽  
Temperature Distribution ◽  
San Francisco ◽  
Analytical Model ◽  
Functional Integration ◽  
Thermal Design ◽  
Junction Temperature ◽  
Direct Result ◽  
Functional Block ◽  
Functional Components

Microprocessors continue to grow in capabilities, complexity, and performance. Microprocessors typically integrate functional components such as logic and level two cache memory in their architecture. This functional integration of logic and memory results in improved performance of the microprocessor. However, the integration also introduces a layer of complexity in the thermal design and management of microprocessors. As a direct result of functional integration, the power map on a microprocessor is typically highly nonuniform, and the assumption of a uniform heat flux across the die surface has been shown to be invalid post Pentium II architecture. The active side of the die is divided into several functional blocks with distinct power assigned to each functional block. Previous work (Kaisare et al., 2005, “Thermal Based Optimization of Functional Block Distributions in a Non-Uniformly Powered Die,” InterPACK 2005, San Francisco, CA, Jul. 17–22) has been done, which includes numerical analysis and thermal based optimization of a typical package consisting of a nonuniformly powered die, heat spreader, thermal interface materials I and II, and the base of the heat sink. In this paper, an analytical approach to temperature distribution of a first level package with a nonuniformly powered die is carried out for the first time. The analytical model for two-layer bodies developed by Haji-Sheikh et al. (2003, “Steady-State Heat Conduction in Multi-Layer Bodies,” Int. J. Heat Mass Transfer, 46(13), pp. 2363–2379) is extended to this typical package, which is a multilayer body. The solution is to begin by designating each surface heat flux as a volumetric heat source. An inverse methodology is applied to solve the equations for various surfaces to calculate the maximum junction temperature for a given multilayer body. Finally validation of the analytical solution is carried out using previously developed numerical model.

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