Exact solution for time exponentially varying pulsed laser heating: Convective boundary condition case

2001 ◽  
Vol 215 (5) ◽  
pp. 591-606 ◽  
Author(s):  
M Kalyon ◽  
B S Yilbas
Keyword(s):  
Boundary Condition ◽  
Exact Solution ◽  
Closed Form ◽  
Temperature Rise ◽  
Laser Heating ◽  
Closed Form Solution ◽  
Convective Boundary Condition ◽  
Form Solution ◽  
Heating Process ◽  
Convective Boundary

Laser heating offers considerable advantages over conventional methods. The closed-form solution for the temperature rise in the substrate during the laser heating process gives insight into the physical phenomena involving during the heating process and the material response to a laser heating pulse. In the present study, the exact solution for the temperature rise due to a time exponentially varying pulse and convective boundary condition at the surface is obtained. The closed-form solution to the solutions available in the literature for a step input intensity pulse with a convective boundary condition at the surface as well as a time exponentially varying pulse with a non-convective boundary condition at the surface is deduced. A Laplace transformation method is used in the analysis. In order to account for a pulse resembling a typical laser pulse, an intensity function resulting in exponentially increasing and decaying intensity distribution is employed in the source term in the governing transport equation. The effects of the pulse parameters β′, β′/γ′ and Biot number Bi on the resulting temperature profiles are presented and the material response to a pulse profile resembling a typical actual laser pulse is discussed. It is found that the closed-form solution obtained in the present study becomes identical with those presented in the previous studies for different pulse and boundary conditions. Moreover, the coupling effect of pulse parameter β and Bi is significant for the temperature rise at the surface.

Closed-form and numerical solutions to the laser heating process

1998 ◽  
Vol 212 (2) ◽  
pp. 141-151 ◽  
Author(s):  
B S Yilbas ◽  
M Sami ◽  
A Al-Farayedhi
Keyword(s):  
Closed Form ◽  
Laser Heating ◽  
Numerical Solutions ◽  
Closed Form Solution ◽  
Form Solution ◽  
Heat Of Fusion ◽  
Heating Process ◽  
Temperature Profiles ◽  
Depth Analysis ◽  
Analytical Approaches

The laser processing of engineering materials requires an in-depth analysis of the applicable heating mechanism. The modelling of the laser heating process offers improved understanding of the machining mechanism. In the present study, a closed-form solution for a step input laser heating pulse is obtained and a numerical scheme solving a three-dimensional heat transfer equation is introduced. The numerical solution provides a comparison of temperature profiles with those obtained from the analytical approach. To validate the analytical and numerical solutions, an experiment is conducted to measure the surface temperature and evaporating front velocity during the Nd—YAG laser heating process. It is found that the temperature profiles resulting from both theory and experiment are in a good agreement. However, a small discrepancy in temperatures at the upper end of the profiles occurs. This may be due to the assumptions made in both the numerical and the analytical approaches. In addition, the equilibrium time, based on the energy balance among the internal energy gain, conduction losses and latent heat of fusion, is introduced.

Thermal and Fluid Dynamic Model for Capillary-Driven Heat Pipe With Closed-Form Solution

2017 ◽  
Author(s):  
Jesse Maxwell
Keyword(s):  
Boundary Condition ◽  
Heat Flux ◽  
Closed Form ◽  
Heat Pipe ◽  
Closed Form Solution ◽  
Constant Heat Flux ◽  
Form Solution ◽  
Constant Heat ◽  
Micro Channels ◽  
Flow Through

A model is derived for the steady state performance of capillary-driven heat pipes on the basis treating fluid flow through miniature- and micro-channels and applied as bulk properties to a large aspect ratio quasi-one-dimensional two-phase system. Surface tension provides the driving force based on an equivalent bulk capillary radius while laminar flow through micro-channels and the vapor core are treated. Heat conduction is accounted for radially while isothermal advection is treated along the axis. A closed-form solution is derived for a steady state heat pipe with a constant heat flux boundary condition on the evaporator as well as a constant heat flux or a convective boundary condition along the condenser. Two solution methods are proposed, and the result is compared to empirical data for a copper-water heat pipe. The components of the closed-form solution are discussed as contributors to driving or frictional forces, and the existence of an optimal pore radius is demonstrated.

The Axially Loaded Circular Cylinder

1962 ◽  
Vol 29 (2) ◽  
pp. 318-320
Author(s):  
H. D. Conway
Keyword(s):  
Exact Solution ◽  
Fourier Transform ◽  
Circular Cylinder ◽  
Closed Form ◽  
Cross Sections ◽  
Closed Form Solution ◽  
Form Solution ◽  
Concentrated Force ◽  
Transform Method ◽  
Fourier Transform Method

Commencing with Kelvin’s closed-form solution to the problem of a concentrated force acting at a given point in an indefinitely extended solid, a Fourier transform method is used to obtain an exact solution for the case when the force acts along the axis of a circular cylinder. Numerical values are obtained for the maximum direct stress on cross sections at various distances from the force. These are then compared with the corresponding stresses from the solution for an infinitely long strip, and in both cases it is observed that the stresses are practically uniform on cross sections greater than a diameter or width from the point of application of the load.

scholarly journals Buckling Behavior of Nanobeams Placed in Electromagnetic Field Using Shifted Chebyshev Polynomials-Based Rayleigh-Ritz Method

Nanomaterials ◽  
2019 ◽  
Vol 9 (9) ◽  
pp. 1326 ◽  
Author(s):  
Subrat Kumar Jena ◽  
Snehashish Chakraverty ◽  
Francesco Tornabene
Keyword(s):  
Boundary Condition ◽  
Closed Form ◽  
Chebyshev Polynomials ◽  
Ritz Method ◽  
Closed Form Solution ◽  
Form Solution ◽  
Buckling Load ◽  
Mode Shapes ◽  
Critical Buckling Load ◽  
Buckling Behavior

In the present investigation, the buckling behavior of Euler–Bernoulli nanobeam, which is placed in an electro-magnetic field, is investigated in the framework of Eringen’s nonlocal theory. Critical buckling load for all the classical boundary conditions such as “Pined–Pined (P-P), Clamped–Pined (C-P), Clamped–Clamped (C-C), and Clamped-Free (C-F)” are obtained using shifted Chebyshev polynomials-based Rayleigh-Ritz method. The main advantage of the shifted Chebyshev polynomials is that it does not make the system ill-conditioning with the higher number of terms in the approximation due to the orthogonality of the functions. Validation and convergence studies of the model have been carried out for different cases. Also, a closed-form solution has been obtained for the “Pined–Pined (P-P)” boundary condition using Navier’s technique, and the numerical results obtained for the “Pined–Pined (P-P)” boundary condition are validated with a closed-form solution. Further, the effects of various scaling parameters on the critical buckling load have been explored, and new results are presented as Figures and Tables. Finally, buckling mode shapes are also plotted to show the sensitiveness of the critical buckling load.

Degenerate Scale Problem for a Hypocycloid Hole in an Infinite Plate in Plane Elasticity

2016 ◽  
Vol 32 (4) ◽  
pp. N7-N10
Author(s):  
Y.-Z. Chen
Keyword(s):  
Boundary Condition ◽  
Conformal Mapping ◽  
Closed Form ◽  
Infinite Plate ◽  
Closed Form Solution ◽  
Plane Elasticity ◽  
Form Solution ◽  
Scale Problem ◽  
Exterior Region ◽  
Degenerate Scale

AbstractBased on the conformal mapping, this paper provides a closed form solution for the degenerate scale of the hypocycloid hole in plane elasticity. In the derivation, we assume the vanishing displacements along the boundary in the degenerate scale problem. Some functions in the boundary condition are decomposed into three parts with particular behavior. Even the displacements are vanishing along the boundary of an exterior region, the displacements and stresses are not equal to zero in the exterior region. This is a particular feature in the degenerate scale problem.

Calculation of Discharge Current Waveforms in High Voltage Spark Sources. II. Extensions and Limitations of Closed-Form Solution

1982 ◽  
Vol 36 (1) ◽  
pp. 25-29 ◽  
Author(s):  
Alexander Scheeline ◽  
T. V. Tran
Keyword(s):  
Exact Solution ◽  
Numerical Methods ◽  
Closed Form ◽  
High Voltage ◽  
Discharge Current ◽  
Closed Form Solution ◽  
Approximate Model ◽  
Form Solution ◽  
Dynamic Impedance ◽  
Numerical Integration Methods

Simulation of gap breakdown and dynamic impedance effects in high voltage spark sources is performed using an algebraically exact solution to an approximate model of source behavior. The importance of diode shunt capacitance in determining gap breakdown behavior is shown. Limitations in generality and implicit use of numerical methods in dynamic situations lead naturally to consideration of numerical integration methods. Comparisons to hardware sources are made.

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