·Table of Contents ·Industrial Plants and Structures | Control of Tightness of Welded Joints on Multilayer StructuresYuriy PosypaykoE.O.Paton Electric Welding Institute, Kyiv, Ukraine Contact |
The theory of contact interaction proceeds from that the actual surface of the metal deviates from the nominal surface. The surface unevenness is divided into the macrodeviations of shape, waviness and roughness.
The surface of rolled sheets of which the walls of multilayer vessels are made, is characterised by irregular microrelief differing by a pronounced non-uniformity by the strip width and length. The non-uniformity of roughness and waviness is combined with a comparatively strong macrodeviation of shape (warpage and twisting, etc.).
The real area of contact A_{r} of two actual surfaces consists of individual discrete sites located at different height, at different angles, with different degree of elastic and plastic deformation of metal.
In order to describe the physical pattern of contact deformation of the microrelief of two uniform surfaces in their drawing together under the impact of load N, it is necessary to take into account:
y = 2(y_{pl} + y_{el}), | (1) |
where y_{pl} are plastic contact deformations of the surface layers of contacting bodies, under the impact of force N; y_{el} are elastic displacements of the adjacent lower lying layers of contacting bodies.
Omitting the description of theoretical calculations, let us give the expressions for engineering calculation of the summands of expression (1):
y_{pl} = [2pNR_{a}H_{z}W_{z }/s_{T}A_{a}]^{1/3} | (2) |
y_{el} = pcs_{T}S_{M}(1-m^{2})y_{pl}^{1/3 }/ ER_{a} | (3) |
where N is the load per the nominal contact area A_{a}; R_{a}, H_{z}, W_{z }are parameters of roughness, waviness and macrodeviations of shape; s_{T} is the metal yield point; c is the constraint coefficient, c = 2.82; S_{M} is the average pitch of the roughness protrusions; mis the Poisson's ratio, E is the modulus of elasticity.
The derived equations describe drawing together of two surfaces of a multilayer item as a function of pressure: y = f(N/A_{a}).
Characteristics of surface unevenness of the rolled steel strip used in fabrication of multilayer items, have the following values: roughness R_{a} = 1 - 2 mm; S_{m} 1 - 5 mm; waviness H_{z }= 5 - 20_{ }mm; macrodeviations of shape W_{z} = 50 - 100 mm. The strength properties of the steel strip are as follows: s_{T} = 600 MPa, E = 2.10^{5} MPa, m = 0.3.
Thus, knowing the values of characteristics of unevenness of the steel strip surface of which the multilayer item is made, and the acting force of pressing the surfaces together, we can calculate the drawing together of the contacting surfaces, namely y = f(N/A_{a}) dependencies for different values of R_{a}, S_{m}, H_{z} and W_{z}.
Q_{V} = hd^{3}(p_{2}^{2} - p_{1}^{2}) / 24h1 | (4) |
Solution of the equation derived for gas flow in the molecular flow mode (Knudsen equation) under the same conditions, has the following form:
Q_{M} = 2 h d^{2}v_{av}(p_{2} - p_{1})/3l | (5) |
where v_{av} = (8RT/pM)^{1/2} is the average velocity of molecules; R is the universal gas constant, R = 8,31 J/mole.K) = 6,34.10^{4} l.mm mm.Hg/(mole.K); T is the absolute temperature; M is the molecular mass of gas.
There is no clearly defined boundary between the molecular and viscous flow. In a wide range of pressures there exists the so-called intermediate flow for which the flow can be determined from the following equation (generalised Knudsen equation):
Q_{int} = Q_{V} +zQ_{M} | (6) |
where z is the empirical function z(d,p) of a complex form, introduced by Knudsen for bringing the calculated dependence in accordance with that found by him experimentally. In transition from the viscous flow to the molecular flow, coefficient z changes from 0.81 to 1.0 and in practical calculations it is often taken to be equal to 0.9. The boundaries between the flow modes are determined by the ratios of the molecule free path length l to characteristic dimension d of the cavity in which the gas flows (for the gap it is its height). The length of the free path is determined from the following equation:
l= kT /1,4pd^{2}p_{av} | (7) |
where k is the Boltsmann constant; d is the molecule diameter; p_{av} is the average pressure in the gap, p_{av} = (p_{1 }+ p_{2})/2.
The intermediate flow of gases is characteristic for the range of 1/3³l/d³5.10^{-3}. At l/d>1/3 the flow is molecular, at l/d<5.10^{-3} it is viscous. For air at T = 293 K the limit of viscous flow is 0,02£p_{av}.d£1,33; at p_{av}.d>1,33 the flow is viscous, at p_{av}.d<0,02 it is molecular.
In gas transport through the interface gaps, the accepted belief is that the flows exceeding 10^{-3}m^{3} Pa/s are viscous flows, those below 10^{-5}m^{3} Pa/s being molecular.
Equations (4) and (5) are widely used in practical simulation of the mechanism of gas flowing through the interface cavity. They, however, do not take into account the features of gas flow through the gaps, related to a considerable pressure gradient along the gap length. In this case different modes of gas flow exist simultaneously in different sections of the gap. Equation (6) should be used for calculation of the gas flow through the interface cavity.
Gap height d(ord_{av}) should be determined based on the theory of contact interaction of surfaces or experimentally, having measured the volume of the cavity of interface gaps. Gap heightd is determined from the theory of contact interaction of two surfaces as follows:
d= W_{z} + H_{z} + 2R_{p} - y, | (8) |
where W_{z} is the average height of macrodeviations of shape; H_{z }is the average height of waviness; R_{p} is the average value of roughness; y is the value of surfaces drawing together under the impact of the load given by equation (1).
Thus, equations (1-8) permit calculation of the value of gas flow through a crevice-like gap, depending on the force of compression of the surfaces forming the gap and characteristics of surfaces at different gradients of gas pressure in the gap (p_{2 }- p_{1}).
We developed an interactive computer program for calculation of the flows, which envisages entering the parameters of the surfaces forming the crevice-like gap and displaying the plots of Q = f (N/A_{o}) dependencies. The proposed approach provides an indirect possibility to calculate the time of filling the blind gaps based on equation Q = V/t without allowing for the flow reduction as the gap is filled.
The third method of flow calculation has been widely studied and used lately, which is based on representation of a butt of two contacting surfaces as a certain porous structure and application of the main laws of filtration movement of gas to it. As the shape, height and distribution of unevenness on both the surfaces are random, therefore, the contact layer formed by the compressed surface relief, is characterised by the presence of a multitude of random microchannels and pores of various shape and extent. The overall set of microchannels and pores can be considered as a certain layer of a porous body whose parameters vary as the contacting surfaces are drawn together.
The equation of continuity of the filtration flow should be used, in order to determine the rate of the flow of gas through a butt of two surfaces. Assuming that the gas flow is a viscous steady-state one; flow velocity in the plane normal to its movement, is constant; porosity is uniformly distributed over the butt; gas pressure changes linearly along the butt length; coefficient of dynamic viscosity of gas is independent on pressure, the flow rate of the compressed gas will be described by Darsi equation:
Q = (A_{s}K_{p }/ RTm) pdp/dr | (9) |
where A_{s} is the cross-sectional area of the contact layer through which filtration proceeds; K_{p }is the permeability coefficient; R is the gas constant; T is the absolute temperature; m is the dynamic coefficient of viscosity; p is gas pressure.
Making certain transformations, after integration we will have:
Q = pV_{f}^{3}(p_{2}^{2} - p_{1}^{2}) / 4mkA_{a}^{3}RT(ln r_{2}/r_{1}) | (10) |
where V_{f} is the volume of the free space of pores; A_{a} is the nominal area of contact; k is the Carman constant dependent on the channel shape; usually k = 4.5 to 5.0.
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