scholarly journals Thin rod heat flux transducer positioned in the earth having a uniform temperature gradient: a closed form solution

1977 ◽  
Author(s):  
H.F. Poppendiek ◽  
D.J. Connelly
Keyword(s):  
Heat Flux ◽  
Temperature Gradient ◽  
Closed Form ◽  
Closed Form Solution ◽  
Form Solution ◽  
Uniform Temperature ◽  
The Earth

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.

scholarly journals Closed-form solution of a thermocapillary free-film problem due to Pukhnachev

2015 ◽  
Vol 26 (5) ◽  
pp. 721-741 ◽  
Author(s):  
BRIAN R. DUFFY ◽  
MATTHIAS LANGER ◽  
STEPHEN K. WILSON
Keyword(s):  
Heat Flux ◽  
Closed Form ◽  
Closed Form Solution ◽  
Critical Value ◽  
Form Solution ◽  
Free Fluid ◽  
Parametric Solution ◽  
Free Film ◽  
Film Version ◽  
Flux Parameter

We consider the steady two-dimensional thin-film version of a problem concerning a weightless non-isothermal free fluid film subject to thermocapillarity, proposed and analysed by Pukhnachev and co-workers. Specifically, we extend and correct the paper by Karabut and Pukhnachev (J. Appl. Mech. Tech. Phys. 49, 568–579, 2008), in which the problem is solved numerically, and in which it is claimed that there exists a unique solution for any value of a prescribed heat-flux parameter in the model. We present a closed-form (parametric) solution of the problem, and from this show that, on the contrary, solutions exist only when the heat-flux parameter is less than a critical value found numerically by Karabut and Pukhnachev, and that when this condition is satisfied there are in fact two solutions, one of which recovers that obtained numerically by Karabut and Pukhnachev, the other being new.

Spontaneous potential log response expressed as convolution

Geophysics ◽  
1982 ◽  
Vol 47 (9) ◽  
pp. 1335-1337
Author(s):  
E. A. Nosal
Keyword(s):  
Closed Form ◽  
Signal Analysis ◽  
Closed Form Solution ◽  
Form Solution ◽  
The Other ◽  
Individual Contribution ◽  
Spontaneous Potential ◽  
The Earth ◽  
The Individual ◽  
Special Case

A special case of spontaneous potential (SP) logging, which has a closed‐form solution, will be expressed as a convolutional operation. Such a formal demonstration serves two purposes. First, it separates the individual contribution of the tool from that of the earth. Second, it places this logging device within the mathematical context of signal analysis. The special case for which a closed‐form solution is known is that where all resistivities are equal. Fourier analysis applied to this solution leads to a product of two functions, of which one is identified as the contribution of the earth and the other of the tool.

Laminar gravitational convection of heat in dead-end channels

1981 ◽  
Vol 110 ◽  
pp. 97-113 ◽  
Author(s):  
Terry W. Sturm
Keyword(s):  
Temperature Gradient ◽  
Closed Form ◽  
Closed Form Solution ◽  
Form Solution ◽  
Gravitational Convection ◽  
Longitudinal Temperature Gradient ◽  
Dead End ◽  
Longitudinal Temperature ◽  
Surface Heat Transfer Coefficient ◽  
Energy Equations

A closed-form solution of the coupled momentum and thermal energy equations is obtained for laminar gravitational circulation of water resulting from a longitudinal temperature gradient in a dead-end channel. The temperature gradient is determined by the rate of heat loss from the water surface. The solution is shown to be dependent on a modified Rayleigh number which involves the local surface heat-transfer coefficient. An experimental study was conducted, and the results are compared with the closed-form solution.

An Improved Closed-Form Solution for Shearing and Peeling Stresses of Trimaterial Assembly in Electronic Packaging

2008 ◽  
Vol 5 (1) ◽  
pp. 36-42 ◽  
Author(s):  
D. Sujan ◽  
M.V.V. Murthy ◽  
A.Y. Hassan ◽  
K.N. Seetharamu
Keyword(s):  
Numerical Simulation ◽  
Closed Form ◽  
Temperature Change ◽  
Closed Form Solution ◽  
Form Solution ◽  
Stress Model ◽  
Shearing Stress ◽  
Uniform Temperature ◽  
Applied Physics ◽  
Closed Form Solutions

Closed-form solutions for shearing and peeling stresses of trimaterial assembly were initially provided by Schmidt in 1999 [1] and Suhir in 2001 [2]. However, there exist some contradictions and inconsistencies the solutions of both Schmidt and Suhir. The contradiction arises in consideration of the exponent parameter k in the characteristic equation. Both Schmidt and Suhir showed that the exponent parameter k in the shearing stresses contains two roots, but for both cases they considered only one root for k; as a consequence, it leads to a mathematical inconsistency in the solution. In the present paper, a model for shearing stress for uniform temperature change is presented that considers both roots for k. Subsequently, a model for peeling stress is presented that considers moment equilibrium combined with the improved shearing stress model described above. The contradictions in Schmidt's and Suhir's solutions are highlighted in this paper. Analytical and FEM solutions are presented for the same trimaterial package as used by Suhir (cf. Journal of Applied Physics, ibid.) for comparison. The improved analytical results and the numerical simulation indicate better agreement compared with the solutions provided by Schmidt and Suhir.

On a Closed-Form Solution of Prandtl's System of Equations

2013 ◽  
Vol 40 (2) ◽  
pp. 106-114
Author(s):  
J. Venetis ◽  
Aimilios (Preferred name Emilios) Sideridis
Keyword(s):  
Closed Form ◽  
Closed Form Solution ◽  
Form Solution ◽  
System Of Equations
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