Resistance constriction

Fig. 13. Schematic of a microscopic view of contact interface where constriction resistance originates in the constriction of current flow through the touching metallic junctions (a spots) of the mating surfaces. The arrows and lines indicate the flow of current.

Workers have shown theoretically that this effect can be caused both at the microstructural level (due to tunneling of the current near the TPB) as well as on a macroscopic level when the electrode is not perfectly electronically conductive and the current collector makes only intermittent contact. ° Fleig and Maier further showed that current constriction can have a distortional effect on the frequency response (impedance), which is sensitive to the relative importance of the surface vs bulk path. In particular, they showed that unlike the bulk electrolyte resistance, the constriction resistance can appear at frequencies overlapping the interfacial impedance. Thus, the effect can be hard to separate experimentally from interfacial electrochemical-kinetic resistances, particularly when one considers that many of the same microstructural parameters influencing the electrochemical kinetics (TPB area, contact area) also influence the current constriction. 

Fig. 24. Ratio of the additional current constriction resistance to the conventional spreading resistance for different ring widths (7>nng) normalized to the diameter of the electrochemically inactive inner part (dme), indicating that for small active rings the additional resistance becomes rather important.

Landauer R (1989) Conductance determined by transmission probes and quantised constriction resistance. J Phys Condens Matter 1 8099... 

The dimensionless shape factor for the isothermal rectangular annulus is derived from the correlation equation of Schneider [89], who obtained accurate numerical values of the thermal constriction resistance of doubly connected rectangular contact areas by means of the boundary integral equation method ... 

Spreading (constriction) resistance is an important thermal parameter that depends on several factors such as (1) geometry (singly or doubly connected areas, shape, aspect ratio), (2) domain (half-space, flux tube), (3) boundary condition (Dirchlet, Neumann, Robin), and (4) time (steady-state, transient). The results are presented in the form of infinite series and integrals that can be computed quickly and accurately by means of computer algebra systems. Accurate correlation equations are also provided. 

The general expression for the dimensionless spreading (constriction) resistance 4kaRs for a circular contact subjected to an arbitrary axisymmetric flux distribution f(u) [131-133] is obtained from the series... 

The dimensionless spreading (constriction) resistance coefficient C0 is the half-space value, and the correlation coefficients C through C7 are given in Table 3.15. 

Spreading Resistance Within Isotropic Finite Disks With Conductance. The dimensionless constriction resistance for isotropic (k= 1) finite disks (ii < 0.72) with negligible thermal resistance at the heat sink interface (Bi = °°) is given by the following solutions. 

The solution for the isoflux boundary condition and with external thermal resistance was recently reexamined by Song et al. [156] and Lee et al. [157]. These researchers nondimen-sionalized the constriction resistance based on the centroid and area-average temperatures using the square root of the contact area as recommended by Chow and Yovanovich [15] and Yovanovich [132,137,144-146,150], and compared the analytical results against the numerical results reported by Nelson and Sayers [158] over the full range of the independent parameters Bi, e, and x. Nelson and Sayers [158] also chose the square root of the contact area to report their numerical results. The analytical and numerical results were reported to be in excellent agreement. 

Lee et al. [157] recommended a simple closed-form expression for the dimensionless constriction resistance based on the area-average and centroid temperatures. They defined the dimensionless spreading resistance parameter as p = ViikaRc and recommended the following approximations. 

The effect of single and multiple isotropic layers or coatings on the end of a circular flux tube has been determined by Antonetti [2] and Sridhar et al. [107]. The heat enters the end of the circular flux tube of radius b and thermal conductivity k3 through a coaxial, circular contact area that is in perfect thermal contact with an isotropic layer of thermal conductivity k, and thickness This layer is in perfect contact with a second layer of thermal conductivity k2 and thickness t2 that is in perfect contact with the flux tube having thermal conductivity k3 (Fig. 3.22). The lateral boundary of the flux tube is adiabatic and the contact plane outside the contact area is also adiabatic. The boundary condition over the contact area may be (1) isoflux or (2) isothermal. The dimensionless constriction resistance p2 layers = 4k3aRc is defined with respect to the thermal conductivity of the flux... 

The two-layer solution can be used to obtain the solution for a single layer of thermal conductivity fcj and thickness t, in perfect contact with a flux tube of thermal conductivity k2. In this case the dimensionless constriction resistance yi )ayer depends on the relative contact size e, the conductivity ratio k21, and the relative layer thickness x, ... 

For regular polygons having sides N > 3 as depicted in Fig. 3.14, the area is A = Nr] tan nIN, where r, is the radius of the inscribed circle. The dimensionless constriction resistance based on the centroid temperature kVAR0 is given by the following double integral [152] ... 

The contact resistance of the sphere-flat contact shown in Fig. 3.23 is discussed in this section. The thermal conductivities of the sphere and flux tube are /c, and k2, respectively. The total contact resistance is the sum of the constriction resistance in the sphere and the spreading resistance within the flux tube. The contact radius a is much smaller than the sphere diameter D and the tube diameter. Assuming isothermal contact area, the general elastoconstriction resistance model [143] becomes ... 

The dimensionless constriction resistance is R = L and the dimensionless joint resistance in a vacuum is... 

The radiative resistance was approximately 10 times the constriction resistance at the lightest load and 30 times at the highest load. The largest difference between the theory and experiments is approximately -4.7 percent, within the probable experimental error. These and other vacuum tests [47] verified the accuracy of the elastoconstriction and the radiation models. 

Thermal contact, gap, and joint conductance models developed by many researchers over the past five decades are reviewed and summarized in several articles [20,23,50,58,143,147,148] and in the recent text of Madhusudana [59]. The models are, in general, based on the assumption that the contacting surfaces are conforming (or flat) and that the surface asperities have particular height and asperity slope distributions [26, 116, 125]. The models assume either plastic or elastic deformation of the contacting asperities, and require the thermal spreading (constriction) resistance results presented above. 

J. R. Dryden, The Effect of a Surface Coating on the Constriction Resistance of a Spot on an Infinite Half-Plane, ASME Journal of Heat Transfer, Vol. 105, pp. 408-410,1983. 

N. J. Fisher and M. M. Yovanovich, Thermal Constriction Resistance of Sphere/Layered Flat Contacts Theory and Experiments, ASME Journal of Heat Transfer, Vol. Ill, pp. 249-256,1989. 

K. A. Martin, M. M. Yovanovich, and Y. L. Chow, Method of Moments Formulation of Thermal Constriction Resistance of Arbitrary Contacts, AIAA-84-1745, AIAA 19th Thermophysics Conference, Snowmass, CO, June 25-28,1984. 

K. J. Negus and M. M. Yovanovich, Constriction Resistance of Circular Flux Tubes With Mixed Boundary Conditions by Linear Superposition of Neumann Solutions, ASME 84-HT-84,1984. 

K. J. Negus, M. M. Yovanovich, and J. C. Thompson, Thermal Constriction Resistance of Circular Contacts on Coated Surfaces Effect of Contact Boundary Condition, AIAA-85-I014, AIAA 20th Thermophysics Conference, Williamsburg, VA, June 19-21,1985. 

M. R. Sridhar and M. M. Yovanovich, Elastoplastic Constriction Resistance Model for Sphere-Flat Contacts, ASME J. of Heat Transfer (118/1) 202-205,1996. 

P J. Turyk and M. M. Yovanovich, Transient Constriction Resistance for Elemental Flux Channels Heated by Uniform Heat Sources, 84-HT-52,1984. 

F. C. Yip, Thermal Contact Constriction Resistance, PhD thesis, Department of Mechanical Engineering, University of Calgary, Calgary, Alberta, Canada, 1969. 

M. M. Yovanovich, General Thermal Constriction Resistance Parameter for Annular Contacts on Circular Flux Tubes, AlAA Journal (14/6) 822-824,1976. 

M. M. Yovanovich, General Expressions for Constriction Resistances of Arbitrary Flux Distributions, in Radiative Transfer and Thermal Control, AlAA Progress in Astronautics and Aeronautics, Vol. 49, pp. 381-396, New York, 1976. 

M. M. Yovanovich and G. E. Schneider, Thermal Constriction Resistance Due to a Circular Annular Contact, AIAA Progress in Astronautics and Aeronautics, Vol. 56, pp. 141-154,1977. 

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