scholarly journals One-dimensional thermal equation modeled on a two-dimensional heat exchanger with variable solid-fluid interface

2021 ◽  
Vol 2090 (1) ◽  
pp. 012140
Author(s):  
Hideshi Ishida ◽  
Koichi Higuchi ◽  
Taiki Hirahata
Keyword(s):  
Heat Flux ◽  
Heat Exchanger ◽  
Model Equation ◽  
Computational Time ◽  
Two Dimensional ◽  
Fluid Interface ◽  
Thermal Equation ◽  
Fluid Equation ◽  
One Dimensional ◽  
Dimensional Equation

Abstract In this study, we are to present that a one-dimensional equation for vertically averaged temperature, modeled on a vertically thin, two-dimensional heat exchanger with variable top solid-fluid interface, recovers the two-dimensional thermal information, i.e. steady temperature and flux distribution on the top and temperature-fixed bottom faces. The relative error of these quantities is less than 5% with the maximum gradient of the height kept approximately below 0.5, while the computational time is reduced to 0.1–5%, when compared with direct two-dimensional computations, depending on the shape of the top face. The model equation, derived by the vertical averaging of the two-dimensional thermal conduction equation, is closed by an approximation that the heat exchanger is sufficiently thin in the sense that the second derivative of temperature with respect to the horizontal coordinate depends only on the coordinate. In this model equation, the fluid equation above the exchanger is decoupled by a conventional equation for the normal heat flux on the top surface. In principle, however, the coupling of the model and the fluid equation is possible through the temperature and heat flux on the top interface, recovered by the model equation. The type of mathematical modeling can be applicable to a wide variety of bodies with extremely small dimensions in some (coordinate-transformed) directions.

Modeling of Mixed Convection Between Vertical Parallel Plates in Electric Equipment Immersed in High-Pr Liquids

2019 ◽  
Vol 11 (5) ◽  
Author(s):  
Thomas B. Gradinger ◽  
T. Laneryd
Keyword(s):  
Heat Flux ◽  
Reynolds Number ◽  
Natural Convection ◽  
Boundary Conditions ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Two Dimensional ◽  
Parallel Plates ◽  
One Dimensional ◽  
Electric Equipment

Natural-convection cooling with oil or other fluids of high Prandtl number plays an important role in many technical applications such as transformers or other electric equipment. For design and optimization, one-dimensional (1D) flow models are of great value. A standard configuration in such models is flow between vertical parallel plates. Accurate modeling of heat transfer, buoyancy, and pressure drop for this configuration is therefore of high importance but gets challenging as the influence of buoyancy rises. For increasing ratio of Grashof to Reynolds number, the accuracy of one-dimensional models based on the locally forced-flow assumption drops. In the present work, buoyancy corrections for use in one-dimensional models are developed and verified. Based on two-dimensional (2D) simulations of buoyant flow using finite-element solver COMSOL Multiphysics, corrections are derived for the local Nusselt number, the local friction coefficient, and a parameter relating velocity-weighted and volumetric mean temperature. The corrections are expressed in terms of the ratio of local Grashof to Reynolds number and a normalized distance from the channel inlet, both readily available in a one-dimensional model. The corrections universally apply to constant wall temperature, constant wall heat flux, and mixed boundary conditions. The developed correlations are tested against two-dimensional simulations for a case of mixed boundary conditions and are found to yield high accuracy in temperature, wall heat flux, and wall shear stress. An application example of a natural-convection loop with two finned heat exchangers shows the influence on mass-flow rate and top-to-bottom temperature difference.

Transient response of an erodable heat flux gauge using finite element analysis

2002 ◽  
Vol 216 (8) ◽  
pp. 701-706 ◽  
Author(s):  
D R Buttsworth
Keyword(s):  
Heat Flux ◽  
Finite Element ◽  
Surface Temperature ◽  
Transient Response ◽  
Transient Heat Conduction ◽  
Two Dimensional ◽  
Element Analysis ◽  
Ic Engine ◽  
One Dimensional ◽  
Transient Heat Flux

The transient response of an erodable ribbon element heat flux gauge has been assessed using a two-dimensional finite element (FE) analysis. Such transient heat flux gauges have previously been used for measurements in internal combustion (IC) engines. To identify the heat flux from the measurements of surface temperature, it is commonly assumed that the heat transfer within these devices is one-dimensional. A corollary of the one-dimensional treatment is that only one value of the thermal product, , is needed for identification of the transient heat flux, even though erodable heat flux gauges are constructed from at least two different materials. The current results demonstrate that two-dimensional transient heat conduction effects have a significant influence on the surface temperature measurements made with these devices. For the ribbon element gauge and timescales of interest in IC engine studies, using a one-dimensional analysis (and hence a single value of ) will lead to substantial inaccuracy in the derived heat flux measurements.

Measurement of Temperature and Heat Flux Changes During the Fixing Process in Electrophotographic Machines

1996 ◽  
Vol 118 (1) ◽  
pp. 150-154 ◽  
Author(s):  
T. Mitsuya ◽  
K. Masuda ◽  
Y. Hori
Keyword(s):  
Heat Flux ◽  
Print Quality ◽  
Two Dimensional ◽  
Temperature Changes ◽  
One Dimensional ◽  
Temperature Heat ◽  
Rapid Temperature ◽  
Pressure Changes ◽  
The Relationship ◽  
Electrophotographic Printing

Increasingly higher speeds of modern electrophotographic printing force examination of the problem of retaining sufficient fixing strength without deterioration of print quality. In the nip region between the two rollers where fixing occurs, the significant parameters are temperature, heat flux, and pressure changes. Their optimization is necessary to maintain both speed and print quality. Difficulty in analyzing the relationship among these parameters occurs because of the complexity of two-dimensional phenomena in a rotating field and the rapidity of changes. Experimental equipment to measure relative heat flux in the nip region during rapid temperature changes was designed. Two sensors are installed in the heat roller. An adiabatic piece is buried under sensor 1. Sensor 2, without an adiabatic piece, detects temperature. Sensor 1 is electrically heated and always at the same temperature as sensor 2. Heat flux changes are obtained by noting the electric power supplied to sensor 1. The equipment was fabricated and measurements were made. They indicate an intermittent two-dimensional heat flux. Because of this, temperature decreases rapidly before the entrance to the nip region. Estimates of two-dimensional effects are made and modified for a one-dimensional case. From them, the temperature field in the nip region for actual fixing conditions is calculated.

Wave packets, resonant interactions and soliton formation in inlet pipe flow

1993 ◽  
Vol 252 ◽  
pp. 1-30 ◽  
Author(s):  
Igor V. Savenkov
Keyword(s):  
Wave Packets ◽  
Three Dimensional ◽  
Short Wave ◽  
Nonlinear Interactions ◽  
Two Dimensional ◽  
Inlet Pipe ◽  
One Dimensional ◽  
Resonant Interactions ◽  
Dimensional Equation ◽  
Axisymmetric Disturbances

The development of disturbances (two-dimensional non-linear and three-dimensional linear) in the entrance region of a circular pipe is studied in the limit of Reynolds number R → ∞ in the framework of triple-deck theory. It is found that lower-branch axisymmetric disturbances can interact in the resonant manner. Numerical calculations show that a two-dimensional nonlinear wave packet grows much more rapidly than that in the boundary layer on a flat plate, producing a spike-like solution which seems to become singular at a finite time. Large-sized, short-scaled disturbances are also studied. In this case the development of axisymmetric disturbances is governed by single one-dimensional equation in the form of the Korteweg-de Vries and Benjamin-Ono equations in the long- and short-wave limits respectively. The nonlinear interactions of these disturbances lead to the formation of solitons which can run both upstream and downstream. Linear three-dimensional wave packets are also calculated.

scholarly journals Analytical solutions for the neutron transport using the spectral methods

2006 ◽  
Vol 2006 ◽  
pp. 1-11 ◽  
Author(s):  
Abdelouahab Kadem
Keyword(s):  
Analytical Solutions ◽  
Spectral Methods ◽  
Chebyshev Polynomials ◽  
Neutron Transport ◽  
Two Dimensional ◽  
Dimensional Problem ◽  
One Dimensional ◽  
Multidimensional Problems ◽  
Dimensional Equation ◽  
Transport Problems

We present a method for solving the two-dimensional equation of transfer. The method can be extended easily to the general linear transport problem. The used technique allows us to reduce the two-dimensional equation to a system of one-dimensional equations. The idea of using the spectral method for searching for solutions to the multidimensional transport problems leads us to a solution for all values of the independant variables, the proposed method reduces the solution of the multidimensional problems into a set of one-dimensional ones that have well-established deterministic solutions. The procedure is based on the development of the angular flux in truncated series of Chebyshev polynomials which will permit us to transform the two-dimensional problem into a set of one-dimensional problems.

Calibration of a 1D/1D urban flood model using 1D/2D model results in the absence of field data

2011 ◽  
Vol 64 (5) ◽  
pp. 1016-1024 ◽  
Author(s):  
J. Leandro ◽  
S. Djordjević ◽  
A. S. Chen ◽  
D. A. Savić ◽  
M. Stanić
Keyword(s):  
Field Data ◽  
Computational Time ◽  
Necessary Condition ◽  
Two Dimensional ◽  
Time Step ◽  
Urban Flood ◽  
One Dimensional ◽  
1D Model ◽  
Surface Network ◽  
2D Model

Recently increased flood events have been prompting researchers to improve existing coupled flood-models such as one-dimensional (1D)/1D and 1D/two-dimensional (2D) models. While 1D/1D models simulate sewer and surface networks using a one-dimensional approach, 1D/2D models represent the surface network by a two-dimensional surface grid. However their application raises two issues to urban flood modellers: (1) stormwater systems planning/emergency or risk analysis demands for fast models, and the 1D/2D computational time is prohibitive, (2) and the recognized lack of field data (e.g. Hunter et al. (2008)) causes difficulties for the calibration/validation of 1D/1D models. In this paper we propose to overcome these issues by calibrating a 1D/1D model with the results of a 1D/2D model. The flood-inundation results show that: (1) 1D/2D results can be used to calibrate faster 1D/1D models, (2) the 1D/1D model is able to map the 1D/2D flood maximum extent well, and the flooding limits satisfactorily in each time-step, (3) the 1D/1D model major differences are the instantaneous flow propagation and overestimation of the flood-depths within surface-ponds, (4) the agreement in the volume surcharged by both models is a necessary condition for the 1D surface-network validation and (5) the agreement of the manholes discharge shapes measures the fitness of the calibrated 1D surface-network.

scholarly journals Simplified models of the symmetric single-pass parallel-plate counterflow heat exchanger: a tutorial

2018 ◽  
Vol 5 (3) ◽  
pp. 171617 ◽  
Author(s):  
William F. Pickard ◽  
Barbara Abraham-Shrauner
Keyword(s):  
Steady State ◽  
Heat Exchanger ◽  
Plug Flow ◽  
Thermal Processes ◽  
Two Dimensional ◽  
One Dimensional ◽  
Trade Offs ◽  
The One ◽  
Difference Time ◽  
Approach Unity

The heat exchanger is important in practical thermal processes, especially those of (i) the molten-salt storage schemes, (ii) compressed air energy storage schemes and (iii) other load-shifting thermal storage presumed to undergird a Smart Grid. Such devices, although central to the utilization of energy from sustainable (but intermittent) renewable sources, will be unfamiliar to many scientists, who nevertheless need a working knowledge of them. This tutorial paper provides a largely self-contained conceptual introduction for such persons. It begins by modelling a novel quantized exchanger, 1 impractical as a device, but useful for comprehending the underlying thermophysics. It then reviews the one-dimensional steady-state idealization which demonstrates that effectiveness of heat transfer increases monotonically with (device length)/(device throughput). Next, it presents a two-dimensional steady-state idealization for plug flow and from it derives a novel formula for effectiveness of transfer; this formula is then shown to agree well with a finite-difference time-domain solution of the two-dimensional idealization under Hagen–Poiseuille flow. These results are consistent with a conclusion that effectiveness of heat exchange can approach unity, but may involve unwelcome trade-offs among device cost, size and throughput.

A one-dimensional self-gravitating stellar gas

1966 ◽  
Vol 25 ◽  
pp. 46-48 ◽  
Author(s):  
M. Lecar
Keyword(s):  
Gravitational Field ◽  
Phase Space ◽  
Motion Picture ◽  
Space Distribution ◽  
Two Dimensional ◽  
One Dimensional ◽  
Phase Space Distribution ◽  
Dimensional Phase Space ◽  
The Mean

“Dynamical mixing”, i.e. relaxation of a stellar phase space distribution through interaction with the mean gravitational field, is numerically investigated for a one-dimensional self-gravitating stellar gas. Qualitative results are presented in the form of a motion picture of the flow of phase points (representing homogeneous slabs of stars) in two-dimensional phase space.

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