Solutions for Transient Heat Conduction With Solid Body Motion and Convective Boundary Conditions

2007 ◽  
Author(s):  
Robert L. McMasters ◽  
James V. Beck
Keyword(s):  
Boundary Conditions ◽  
Solid Body ◽  
Green's Functions ◽  
Green’S Functions ◽  
Separation Of Variables ◽  
Body Movement ◽  
Convective Boundary Condition ◽  
Body Motion ◽  
Convective Boundary Conditions ◽  
Convective Boundary

The analytical solution for the problem of transient thermal conduction with solid body movement is developed for a parallelepiped with convective boundary conditions. An effective transformation scheme is used to eliminate the flow terms. The solution uses Green’s functions containing convolution-type integrals, which involve integration over a dummy time, referred to as “cotime.” Two types of Green’s functions are used: one for short cotimes comes from the Laplace transform and the other for long cotimes from the method of separation of variables. A primary advantage of this method is that it incorporates internal verification of the numerical results by varying the partition time between the short and long components. In some cases, the long time solution requires a zeroth term in the summation, which does not occur when solid body motion is not present. The existence of this zeroth term depends upon the magnitude of the heat transfer coefficient associated with the convective boundary condition. An example is given for a two-dimensional case involving both prescribed temperature and convective boundary conditions. Comprehensive tables are also provided for the nine possible combinations of boundary conditions in each dimension.

Solutions for Transient Heat Conduction With Solid Body Motion and Convective Boundary Conditions

2008 ◽  
Vol 130 (11) ◽  
Author(s):  
Robert L. McMasters ◽  
James V. Beck
Keyword(s):  
Boundary Conditions ◽  
Solid Body ◽  
Green's Functions ◽  
Green’S Functions ◽  
Separation Of Variables ◽  
Body Movement ◽  
Convective Boundary Condition ◽  
Body Motion ◽  
Convective Boundary Conditions ◽  
Convective Boundary

The analytical solution for the problem of transient thermal conduction with solid body movement is developed for a parallelepiped with convective boundary conditions. An effective transformation scheme is used to eliminate the flow terms. The solution uses Green’s functions containing convolution-type integrals, which involve integration over a dummy time, referred to as “cotime.” Two types of Green’s functions are used: one for short cotimes comes from the Laplace transform and the other for long cotimes from the method of separation of variables. A primary advantage of this method is that it incorporates internal verification of the numerical results by varying the partition time between the short and long components. In some cases, the long-time solution requires a zeroth term in the summation, which does not occur when solid body motion is not present. The existence of this zeroth term depends on the magnitude of the heat transfer coefficient associated with the convective boundary condition. An example is given for a two-dimensional case involving both prescribed temperature and convective boundary conditions. Comprehensive tables are also provided for the nine possible combinations of boundary conditions in each dimension.

A Two-Dimensional Cylindrical Transient Conduction Solution Using Green's Functions

2014 ◽  
Vol 136 (10) ◽  
Author(s):  
Robert L. McMasters ◽  
James V. Beck
Keyword(s):  
Parameter Estimation ◽  
Boundary Conditions ◽  
Green's Functions ◽  
Bessel Functions ◽  
Numerical Solutions ◽  
Green’S Functions ◽  
Two Dimensional ◽  
Trigonometric Functions ◽  
Convective Boundary Conditions ◽  
Convective Boundary

There are many applications for problems involving thermal conduction in two-dimensional (2D) cylindrical objects. Experiments involving thermal parameter estimation are a prime example, including cylindrical objects suddenly placed in hot or cold environments. In a parameter estimation application, the direct solution must be run iteratively in order to obtain convergence with the measured temperature history by changing the thermal parameters. For this reason, commercial conduction codes are often inconvenient to use. It is often practical to generate numerical solutions for such a test, but verification of custom-made numerical solutions is important in order to assure accuracy. The present work involves the generation of an exact solution using Green's functions where the principle of superposition is employed in combining a one-dimensional (1D) cylindrical case with a 1D Cartesian case to provide a temperature solution for a 2D cylindrical. Green's functions are employed in this solution in order to simplify the process, taking advantage of the modular nature of these superimposed components. The exact solutions involve infinite series of Bessel functions and trigonometric functions but these series sometimes converge using only a few terms. Eigenvalues must be determined using Bessel functions and trigonometric functions. The accuracy of the solutions generated using these series is extremely high, being verifiable to eight or ten significant digits. Two examples of the solutions are shown as part of this work for a family of thermal parameters. The first case involves a uniform initial condition and homogeneous convective boundary conditions on all of the surfaces of the cylinder. The second case involves a nonhomogeneous convective boundary condition on a part of one of the planar faces of the cylinder and homogeneous convective boundary conditions elsewhere with zero initial conditions.

A Two-Dimensional Cylindrical Transient Conduction Solution Using Green’s Functions

2013 ◽  
Author(s):  
Robert L. McMasters ◽  
James V. Beck
Keyword(s):  
Parameter Estimation ◽  
Green's Functions ◽  
Bessel Functions ◽  
Numerical Solutions ◽  
Green’S Functions ◽  
Two Dimensional ◽  
Trigonometric Functions ◽  
Convective Boundary Conditions ◽  
Convective Boundary ◽  
One Dimensional

There are many applications for problems involving thermal conduction in two-dimensional cylindrical objects. Experiments involving thermal parameter estimation are a prime example, including cylindrical objects suddenly placed in hot or cold environments. In a parameter estimation application, the direct solution must be run iteratively in order to obtain convergence with the measured temperature history by changing the thermal parameters. For this reason, commercial conduction codes are often inconvenient to use. It is often practical to generate numerical solutions for such a test, but verification of custom-made numerical solutions is important in order to assure accuracy. The present work involves the generation of an exact solution using Green’s functions where the principle of superposition is employed in combining a one-dimensional cylindrical case with a one-dimensional Cartesian case to provide a temperature solution for a two-dimensional cylindrical. Green’s functions are employed in this solution in order to simplify the process, taking advantage of the modular nature of these superimposed components. The exact solutions involve infinite series of Bessel functions and trigonometric functions but these series sometimes converge using only a few terms. Eigenvalues must be determined using Bessel functions and trigonometric functions. The accuracy of the solutions generated using these series is extremely high, being verifiable to eight or ten significant digits. Two examples of the solutions are shown as part of this work for a family of thermal parameters. The first case involves a uniform initial condition and homogeneous convective boundary conditions on all of the surfaces of the cylinder. The second case involves a nonhomogeneous convective boundary condition on a part of one of the planar faces of the cylinder and homogeneous convective boundary conditions elsewhere with zero initial conditions.

Impact of surface temperature and convective boundary conditions on a Nanofluid flow over a radially stretched Riga plate

2021 ◽  
pp. 095440892110544
Author(s):  
K.V. Prasad ◽  
Hanumesh Vaidya ◽  
Fateh Mebarek-Oudina ◽  
Rajashekhar Choudhari ◽  
Kottakkaran Sooppy Nisar ◽  
...  
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Surface Temperature ◽  
Surface Concentration ◽  
Mass Transfer Process ◽  
Convective Boundary Condition ◽  
Convective Boundary Conditions ◽  
Convective Boundary ◽  
Analytical Methodology ◽  
Riga Plate

The current work provides the optimal homotopic analytical methodology for the steady circulation over a non-isothermal radially stretched Riga plate/disc unit. The attributes of the heat, along with the mass transfer process, are assessed in the existence of variable transport and magnetic features. Radial stretched Riga disc is considered along with additional realistic boundary heating conditions, namely, prescribed surface temperature as well as prescribed surface concentration, convective boundary conditions and also zero mass flux concentration on the surface area of the Riga disc. The model tracks Brownian motion as well as the thermal diffusion of nanoparticles in fluid circulation all at once. Regulating equations, which are highly coupled, are changed right into non-dimensional equations using appropriate transformations of similarity. Through assembling series solutions, the resulting framework is planned and examined. Graphic summaries are offered for the rheological qualities of various parameters in size for velocity, temperature, as well as nanoparticles. The modified Hartman number improves the velocity distribution and reduces the temperature distribution in both prescribed surface temperature and convective boundary condition cases. The effect of the chemical reaction parameter shows the reduced concentration distribution for the prescribed surface temperature case. In contrast, it is precisely the opposite in the convective boundary condition case.

Comparison of 1-D vs 3-D Combustion Boundary Conditions for SI Engine Thermal Load Prediction

2013 ◽  
Author(s):  
O. Iqbal ◽  
S. Jonnalagedda ◽  
K. Arora ◽  
L. Zhong ◽  
S. Gaikwad
Keyword(s):  
Heat Transfer ◽  
Boundary Condition ◽  
Temperature Distribution ◽  
Boundary Conditions ◽  
Cylinder Head ◽  
Convective Boundary Condition ◽  
Combustion Model ◽  
Convective Boundary Conditions ◽  
Convective Boundary ◽  
Simulation Code

The thermal field generated in an engine block and cylinder head as a result of combustion loading is of paramount significance for structural durability. Computational fluid dynamics and heat transfer modeling provide strong tools; perhaps the best and most precise available for predicting thermal fields within cylinder head and engine block. However, an enduring challenge has been the temperature prediction on metal wall as a response to the time dependent fluctuations in the fluids. Fluid (coolant) flow in an engine is steady for a given engine speed and load, but combustion dynamics are inherently transient. In this study, an effective set of convective boundary condition data (as combustion load) is generated using two different approaches in a stand-alone simulation and mapped onto a decoupled Conjugate Heat Transfer (CHT) model to predict the temperature distribution in the engine. In the first approach, a predictive combustion model, tuned to dyno test data, is solved in a 1-D simulation code. This provides the cycle-averaged convective boundary condition that can be used for a CHT model as a uniform heat source. In the second, more detailed approach, in-cylinder combustion simulations involving transient piston and valve motion with flame propagation modeling are carried out using a 3-D simulation code. The 3-D methodology gives a detailed distribution of convective boundary conditions on the walls touching the combustion gases. In order to predict the gradients in heat transfer coefficient with high accuracy, the resulting temperature distribution from the CHT simulation is fed back into the combustion model to regenerate the set of convective boundary conditions. This process is repeated until a converged set of convective boundary conditions are obtained. In this paper engine temperature predictions obtained using combustion loads from both 1-D and 3-D approaches will be compared with the thermocouple data from engine dyno test.

Peristaltic flow of a Jeffery fluid over a porous conduit in the presence of variable liquid properties and convective boundary conditions

2019 ◽  
Vol 6 (2) ◽  
Author(s):  
G. Manjunatha ◽  
C. Rajashekhar ◽  
K. V. Prasad ◽  
Hanumesh Vaidya ◽  
Saraswati
Keyword(s):  
Boundary Conditions ◽  
Frictional Force ◽  
Variable Viscosity ◽  
Pressure Rise ◽  
Velocity Slip ◽  
Peristaltic Flow ◽  
Physical Parameters ◽  
Convective Boundary Conditions ◽  
Convective Boundary ◽  
Closed Form Solutions

The present article addresses the peristaltic flow of a Jeffery fluid over an inclined axisymmetric porous tube with varying viscosity and thermal conductivity. Velocity slip and convective boundary conditions are considered. Resulting governing equations are solved using long wavelength and small Reynolds number approximations. The closed-form solutions are obtained for velocity, streamline, pressure gradient, temperature, pressure rise, and frictional force. The MATLAB numerical simulations are utilized to compute pressure rise and frictional force. The impacts of various physical parameters in the interims for time-averaged flow rate with pressure rise and is examined. The consequences of sinusoidal, multi-sinusoidal, triangular, trapezoidal, and square waveforms on physiological parameters are analyzed and discussed through graphs. The analysis reveals that the presence of variable viscosity helps in controlling the pumping performance of the fluid.

Unsteady and porous media flow of reactive non-Newtonian fluids subjected to buoyancy and suction/injection

2015 ◽  
Vol 25 (7) ◽  
pp. 1682-1704 ◽  
Author(s):  
Tirivanhu Chinyoka ◽  
Daniel Oluwole Makinde
Keyword(s):  
Heat Transfer ◽  
Porous Media ◽  
Boundary Conditions ◽  
Flow Velocity ◽  
Flow System ◽  
Finite Difference Methods ◽  
Porous Media Flow ◽  
Convective Boundary Conditions ◽  
Convective Boundary ◽  
Content Type

Purpose – The purpose of this paper is to examine the unsteady pressure-driven flow of a reactive third-grade non-Newtonian fluid in a channel filled with a porous medium. The flow is subjected to buoyancy, suction/injection asymmetrical and convective boundary conditions. Design/methodology/approach – The authors assume that exothermic chemical reactions take place within the flow system and that the asymmetric convective heat exchange with the ambient at the surfaces follow Newton’s law of cooling. The authors also assume unidirectional suction injection flow of uniform strength across the channel. The flow system is modeled via coupled non-linear partial differential equations derived from conservation laws of physics. The flow velocity and temperature are obtained by solving the governing equations numerically using semi-implicit finite difference methods. Findings – The authors present the results graphically and draw qualitative and quantitative observations and conclusions with respect to various parameters embedded in the problem. In particular the authors make observations regarding the effects of bouyancy, convective boundary conditions, suction/injection, non-Newtonian character and reaction strength on the flow velocity, temperature, wall shear stress and wall heat transfer. Originality/value – The combined fluid dynamical, porous media and heat transfer effects investigated in this paper have to the authors’ knowledge not been studied. Such fluid dynamical problems find important application in petroleum recovery.

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