Small and Large Time Solutions for Surface Temperature, Surface Heat Flux, and Energy Input in Transient, One-Dimensional Conduction

2008 ◽  
Vol 130 (10) ◽  
Author(s):  
A. S. Lavine ◽  
T. L. Bergman
Keyword(s):  
Boundary Condition ◽  
Heat Flux ◽  
Surface Temperature ◽  
Surface Heat Flux ◽  
Energy Input ◽  
Approximate Solutions ◽  
Surface Heat ◽  
Dimensionless Time ◽  
Flux Boundary Condition ◽  
Dimensionless Energy

This paper addresses one-dimensional transient conduction in simple geometries. It is well known that the transient thermal responses of various objects, or of an infinite medium surrounding such objects, collapse to the same behavior as a semi-infinite solid at small dimensionless time. At large dimensionless time, the temperature reaches a steady state (for a constant surface temperature boundary condition) or increases linearly with time (for a constant heat flux boundary condition). The objectives of this paper are to bring together existing small and large time solutions for transient conduction in simple geometries, put them into forms that will promote their usage, and quantify the errors associated with the approximations. Approximate solutions in the form of simple algebraic expressions are derived (or compiled from existing solutions) for use at both small and large times. In particular, approximate solutions, which are accurate for Fo<0.2 and which bridge the gap between the large Fo (single-term) approximation and the semi-infinite solid solution (valid only at very small Fo), are presented. Solutions are provided for the surface temperature when there is a constant surface heat flux boundary condition, or for the surface heat flux when there is a constant surface temperature boundary condition. These results are provided in terms of a dimensionless heat transfer rate. In addition, the dimensionless energy input is given for the constant surface temperature cases. The approximate expressions may be used with good accuracy over the entire Fourier number range to rapidly estimate important features of the transient thermal response. With the use of the approximations, it is now a trivial matter to calculate the dimensionless heat transfer rate and dimensionless energy input, using simple closed-form expressions.

Mathematical Model of the Cooling of a Plastic, Thermoformed Part

2010 ◽  
Author(s):  
Thomas M. Aidich ◽  
Tien-Chien Jen ◽  
Yi-Hsin Yen
Keyword(s):  
Experimental Data ◽  
Mathematical Model ◽  
Boundary Condition ◽  
Heat Flux ◽  
Boundary Conditions ◽  
Surface Temperature ◽  
Surface Heat Flux ◽  
Surface Heat ◽  
Cooling Cycle ◽  
Transient Conduction

This paper presents research into the mechanism involved in the cooling of a plastic thermoformed part after it is formed onto a mold. The intent of this research is to develop a simple and practical mathematical model useful to small thermoforming companies without a large engineering staff that describes the transient heat conduction of the cooling process. The model should also be able to predict the temperature distribution within the thickness of the part during the cooling. This mathematical model, which began with simplified boundary conditions, was then compared to experimental cooling data and modified accordingly to properly fit that data and the actual boundary conditions of the cooling part. The research began by examining the cooling of a series of high molecular weight polyethylene thermoformed side panels for plastic, portable restrooms. These parts where chosen for this preliminary research because of their very simple, flat geometric shape that lends them to being modeled as simple plane walls in transient conduction. The shape of the parts also leads to near constant thickness over the vast majority of the part. Using the model of a plane wall in transient conduction, the governing partial differential equation was solved for two possible boundary conditions on the mold side of the part: constant imposed surface temperature and constant imposed surface heat flux. These two solutions were then compared to experimental data gathered on the temperature profile of the free surface of the part during a production environment. After the experimental data and simple mathematical models were compared the necessary changes to the assumed mold side boundary condition was made to adjust the mathematical model to the experimental data. The research found that the use of simple boundary conditions at the mold side of the part is incorrect. Neither the constant imposed surface temperature nor the imposed surface heat flux boundary conditions fit the data. Initial analysis of the experimental data showed that a time of 30 seconds into the cooling cycle an apparent change in that boundary condition occurs for the part and mold used to gather the data. Further analysis showed that the boundary condition begins as a constant surface heat flux and then changes to an imposed surface temperature that decays exponentially to the initial mold surface temperature. Using this boundary condition, a revised mathematical was developed that match the experimental data very well. The error of the new model compared to the experimental was less than 1.5% for all times during the cooling cycle.

Laminar Natural Convection About Downward Facing Heated Blunt Bodies to Liquid Metals

1976 ◽  
Vol 98 (2) ◽  
pp. 208-212 ◽  
Author(s):  
G. M. Harpole ◽  
I. Catton
Keyword(s):  
Heat Flux ◽  
Surface Temperature ◽  
Series Expansion ◽  
Surface Heat Flux ◽  
Liquid Metals ◽  
Surface Heat ◽  
Surface Shape ◽  
Boundary Layer Equations ◽  
Stagnation Points ◽  
Shape Dependence

The laminar boundary layer equations for free convection over bodies of arbitrary shape (i.e., a three-term series expansion) and with arbitrary surface heat flux or surface temperature are solved in local Cartesian coordinates. Both two-dimensional bodies (e.g., horizontal cylinders) and axisymmetric bodies (e.g., spheres) with finite radii of curvature at their stagnation points are considered. A Blasius series expansion is applied to convert from partial to ordinary differential equations. An additional transformation removes the surface shape dependence and the surface heat flux or surface temperature dependence of the equations. A second-order-correct, finite-difference method is used to solve the resulting equations. Tables of results for low Prandtl numbers are presented, from which local Nusselt numbers can be computed.

scholarly journals Seasonality of Decadal Sea Surface Temperature Anomalies in the Northwestern Pacific

2006 ◽  
Vol 19 (12) ◽  
pp. 2953-2968 ◽  
Author(s):  
Takashi Mochizuki ◽  
Hideji Kida
Keyword(s):  
Heat Flux ◽  
Surface Temperature ◽  
Sea Surface Temperature ◽  
Mixed Layer ◽  
Surface Heat Flux ◽  
Heat Budget ◽  
Surface Heat ◽  
Sea Surface ◽  
Northwestern Pacific ◽  
Sst Anomaly

Abstract The seasonality of the decadal sea surface temperature (SST) anomalies and the related physical processes in the northwestern Pacific were investigated using a three-dimensional bulk mixed layer model. In the Kuroshio–Oyashio Extension (KOE) region, the strongest decadal SST anomaly was observed during December–February, while that of the central North Pacific occurred during February–April. From an examination of the seasonal heat budget of the ocean mixed layer, it was revealed that the seasonal-scale enhancement of the decadal SST anomaly in the KOE region was controlled by horizontal Ekman temperature transport in early winter and by vertical entrainment in autumn. The temperature transport by the geostrophic current made only a slight contribution to the seasonal variation of the decadal SST anomaly, despite controlling the upper-ocean thermal conditions on decadal time scales through the slow Rossby wave adjustment to the wind stress curl. When averaging over the entire KOE region, the contribution from the net sea surface heat flux was also no longer significantly detected. By examining the horizontal distributions of the local thermal damping rate, however, it was concluded that the wintertime decadal SST anomaly in the eastern KOE region was rather damped by the net sea surface heat flux. It was due to the fact that the anomalous local thermal damping of the SST anomaly resulting from the vertical entrainment in autumn was considerably strong enough to suppress the anomalous local atmospheric thermal forcing that acted to enhance the decadal SST anomaly.

Instantaneous Local Heat Flux Measurements in a Small Utility Engine

2009 ◽  
Author(s):  
Terry Hendricks ◽  
Jaal Ghandhi ◽  
John Brossman
Keyword(s):  
Heat Transfer ◽  
Heat Flux ◽  
Numerical Model ◽  
Surface Temperature ◽  
Heat Transfer Rate ◽  
Surface Heat Flux ◽  
Transfer Rate ◽  
Surface Heat ◽  
High Load ◽  
Flux Measurements

Heat flux measurements were performed in an air-cooled utility engine using a fast-response coaxial-type surface thermocouple. The surface heat flux was calculated using both analytical and numerical models. The heat flux was found to be a strong function of engine load. The peak heat flux and initial heat flux rise rate increase with engine load. The measured heat flux data were used to estimate a global heat transfer rate, and this was compared with the heat transfer rate calculated by a single-zone heat release analysis. The measured values of heat transfer were higher than the calculated values largely because of the lack of spatial averaging. The high load data showed an unexplainable negative heat flux during the expansion stroke while the gas temperature was still high. A 1D and 2D finite difference numerical model utilizing an adaptive timestep Crank-Nicholson (CN) integration routine was developed to investigate the surface temperature measurement. Applying the measured surface temperature profile to the 1D model, the resultant surface heat flux showed excellent agreement with the analytical inversion solution and captured the reversal of the energy flow back into the cylinder during the expansion stroke. The 2D numerical model was developed to observe transient lateral conduction effects within the probe and incorporated the various materials used in the construction and assembly of the heat flux sensor. The resulting average heat flux profile for the test case is shown to be slightly higher in peak and longer in duration when compared with the results from the 1D analytical inversion, and this is attributed to contributions from the high thermal diffusivity constituents in the sensor. Furthermore, the negative heat flux at high load was not eliminated suggesting that factors other than lateral conduction may be affecting the measurement accuracy.

scholarly journals Surface Heterogeneity Effects on Regional-Scale Fluxes in Stable Boundary Layers: Surface Temperature Transitions

2009 ◽  
Vol 66 (2) ◽  
pp. 412-431 ◽  
Author(s):  
Rob Stoll ◽  
Fernando Porté-Agel
Keyword(s):  
Boundary Layer ◽  
Heat Flux ◽  
Surface Temperature ◽  
Surface Heat Flux ◽  
Regional Scale ◽  
Potential Temperature ◽  
Surface Heterogeneity ◽  
Surface Heat ◽  
Average Surface ◽  
The Mean

Abstract Large-eddy simulation, with recently developed dynamic subgrid-scale models, is used to study the effect of heterogeneous surface temperature distributions on regional-scale turbulent fluxes in the stable boundary layer (SBL). Simulations are performed of a continuously turbulent SBL with surface heterogeneity added in the form of streamwise transitions in surface temperature. Temperature differences between patches of 6 and 3 K are explored with patch length scales ranging from one-half to twice the equivalent homogeneous boundary layer height. The surface temperature heterogeneity has important effects on the mean wind speed and potential temperature profiles as well as on the surface heat flux distribution. Increasing the difference between the patch temperatures results in decreased magnitude of the average surface heat flux, with a corresponding increase in the mean potential temperature in the boundary layer. The simulation results are also used to test existing models for average surface fluxes over heterogeneous terrain. The tested models fail to fully represent the average turbulent heat flux, with models that break the domain into homogeneous subareas grossly underestimating the heat flux magnitude over patches with relatively colder surface temperatures. Motivated by these results, a new parameterization based on local similarity theory is proposed. The new formulation is found to correct the bias over the cold patches, resulting in improved average surface heat flux calculations.

Effect of Subcooling and Nozzle Diameter on Heat Transfer Characteristics of Downward Facing Hot Surfaces Using Mist Jet

2018 ◽  
Author(s):  
Avadhesh Kumar Sharma ◽  
Monika Meena ◽  
Anirudh Soni ◽  
Santosh K. Sahu
Keyword(s):  
Heat Flux ◽  
Surface Temperature ◽  
Stagnation Point ◽  
Surface Heat Flux ◽  
Jet Impingement ◽  
Surface Heat ◽  
Nozzle Diameter ◽  
Initial Surface ◽  
Ir Camera ◽  
Impingement Cooling

The jet impingement cooling is always preferred over the other cooling methods due to its high heat removal capability. However, rapid quenching may lead to the formation of cracks and poor ductility to the quenched surface. Mist jet impingement cooling offers an alternative method to uncontrolled rapid cooling, particularly in steel and electronic industries. In mist cooling, the droplets are atomized by compressed air. Experiments are performed under transient conditions using two full-cone spray nozzles (Lechler Pneumatic atomizing nozzle 136.115.xx.A2 and 136.134.xx.A2) to study the effect of subcooling and nozzle diameter on surface heat flux. The hot surface used for the experiment is a stainless steel foil (AISI-304) of thickness 0.15mm. The initial surface temperature of the plate is maintained at 500±10°C with the help of an AC transformer. Infrared thermal image camera (A655sc, FLIR System) is used for data estimation. The IR camera and the nozzle are positioned on either side of the plate. The variation in surface temperature has been acquired at 8 different spatial points. It has been observed that that as we move away from the stagnation point then irrespective of air and water flow rates surface heat flux decreases. The maximum surface heat flux obtained at the stagnation point. With the increase in diameter surface heat flux increases irrespective of pressure values. The correlation between qm/qstag experimental and predicted values has been shown.

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