General analysis of the problem
Manufacture and operation of multilayer items, namely chemical production apparatuses, pipelines, tanks, etc., was necessitated by the growing requirements to improvement of productivity, capacity, reliability and safety of pressure equipment.
A feature of the technology of non-destructive testing of such items is the need to test their internal welded joints for tightness. This is due to the possibility of development of through-thickness defects in inner welds in welding, thus leading to penetration of the working, stored or transported substance from the item cavity into the cavity of the interlayer gap. Its accumulation there can be the reason for disturbance of the normal operation of the item or its failure.
Detection of through-thickness defects in welded joints of multilayer items is possible on condition of substance transport through the interface gaps or its accumulation in the defects. In order to determine the flow through the interface gaps, it is necessary to know their shapes and geometrical dimensions. However, it is extremely difficult to determine the actual structure of an interface gap. Characteristics of contact of two unprocessed metal surfaces are calculated by statistical methods allowing description of deformation of averaged protrusions and taking into account the laws of distribution of protrusions, waves and macrodeviations over the surface. Thus, it is necessary to create a model of the gap which could with a specified accuracy describe and determine its parameters during the surfaces contact. On the other hand, knowing the gap parameters, it is necessary to know the mechanism of substance transport through the interface gaps.
The factors influencing the shape and dimensions of interface gaps are the force impact on the multilayer wall, namely expansion, physico-mechanical properties of the surface layer (material yield point, modulus of elasticity, hardness, microhardness, etc.), environmental parameters (pressure, temperature, toughness, specific gas evolution, etc.), relief of the surface layer which is characterised by macrodeviations from the regular geometrical shape, waviness and roughness. The law of the surfaces drawing together under the impact of the above factors is exactly the basis for the model of an interface gap, determining its geometrical parameters.
The mechanism of substance transport through the interface gap is determined by the mode of its flow, namely viscous, molecular or intermediate. If gas is sorbed on the gap walls, also gas diffusion proceeds over its surface. Other transport mechanisms are also possible. The existence of a particular transport mechanism depends on the geometrical parameters of the gaps and the parameters of the substance being transported.
Knowing the mechanism of substance transport through the gaps and their parameters, it is possible to determine the extent of the substance flow in the gap, rate of its filling, etc.
As shown by analysis of the schematic of interaction of the main factors influencing the substance transport through the interface gaps, three main problems should be singled out in determination of the extent of transport:
- determination of drawing together of the contacting surfaces;
- determination of the parameters of interface gaps;
- determination of the model of substance transport through the gaps.
Calculation of drawing together of two contacting surfaces
Parameters of interface gaps of multilayer items were determined by us through calculations based on the theory of contact interaction of surfaces. The issues of contact interaction of two surfaces are widely applicable. They are taken into account in evaluation of friction and contact rigidity, in determination of electrical and thermal conduction of contact, in calculation of the area of the bodies contact and study of mass transfer through the interface gaps.
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 Ar 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:
This leads to quite cumbersome theoretical expressions for contact drawing together of surfaces with microroughnesses under the impact of applied load, which can only be calculated in a computer.
- distribution of protrusions by height nr(y);
- cross-sectional area of protrusions on the level of ypl ;
- probability of meeting of a pair of protrusions of the two surfaces;
- probability of protrusion being in dn2 layer;
- distribution of different kinds of microroughnesses by height;
- gauge length and a number of other values.
A normal drawing together of two surfaces with the same mechanical properties of the surface layers can be represented in the form of an inequality:
where ypl are plastic contact deformations of the surface layers of contacting bodies, under the impact of force N; yel 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):
|ypl = [2pNRaHzWz /sTAa]1/3
|yel = pcsTSM(1-m2)ypl1/3 / ERa
where N is the load per the nominal contact area Aa; Ra, Hz, Wz are parameters of roughness, waviness and macrodeviations of shape; sT is the metal yield point; c is the constraint coefficient, c = 2.82; SM 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/Aa).
Characteristics of surface unevenness of the rolled steel strip used in fabrication of multilayer items, have the following values: roughness Ra = 1 - 2 mm; Sm 1 - 5 mm; waviness Hz = 5 - 20 mm; macrodeviations of shape Wz = 50 - 100 mm. The strength properties of the steel strip are as follows: sT = 600 MPa, E = 2.105 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/Aa) dependencies for different values of Ra, Sm, Hz and Wz.
Methods of calculation of the gas flow through the interface gaps
Analysis of the calculation methods of determination of the gas flow through a contact of two metallic surfaces allows three main directions to the singled out, namely:
- methods based on calculation of the gas flow by empirical dependencies;
- methods based on calculation of the gas flow by equations of smooth (flat) reduced channels of height d;
- methods based on the use of a porous model of the butt of two metallic surfaces having a certain roughness, waviness and macrodeviations of shape, applying the laws of filtration movement of gas.
The first direction makes use of empirical dependencies of the type of Q = f(q, p, l) characterising gas transport across a butt of two surfaces, knowing the values of specific loads q on the contacting surfaces, pressure p of gas flowing through the gap, as well as dimensions l and characteristics of the condition of the surface.
Application of empirical dependencies for forecasting gas transport across a butt of contacting surfaces is only possible for an approximate assessment of the flow in well studied cases, for instance, in the case of assessment of pipeline fittings tightness. The attempts at deriving an empirical dependence for gas flow across a butt of contacting untreated metallic surfaces of a large diameter, did not yield positive results. Therefore, this method of calculation will not be further considered by us.
The second direction uses a calculation model based on the equation of movement of a viscous non-compressible liquid, the boundary condition for which is velocity speed of movement on the flow-limiting surface.
Solution of the equation for these conditions derived for gas flow in the viscous flow mode (Poiseil equation) through gap of width h, height d and length l with the pressure gradient between the gap edges (p2 - p1), has the form of:
|QV = hd3(p22 - p12) / 24h1
Solution of the equation derived for gas flow in the molecular flow mode (Knudsen equation) under the same conditions, has the following form:
|QM = 2 h d2vav(p2 - p1)/3l
where vav = (8RT/pM)1/2 is the average velocity of molecules; R is the universal gas constant, R = 8,31 J/mole.K) = 6,34.104 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):
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:
where k is the Boltsmann constant; d is the molecule diameter; pav is the average pressure in the gap, pav = (p1 + p2)/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£pav.d£1,33; at pav.d>1,33 the flow is viscous, at pav.d<0,02 it is molecular.
In gas transport through the interface gaps, the accepted belief is that the flows exceeding 10-3m3 Pa/s are viscous flows, those below 10-5m3 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(ordav) 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= Wz + Hz + 2Rp - y,
where Wz is the average height of macrodeviations of shape; Hz is the average height of waviness; Rp 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 (p2 - p1).
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/Ao) 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 = (AsKp / RTm) pdp/dr
where As is the cross-sectional area of the contact layer through which filtration proceeds; Kp 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 = pVf3(p22 - p12) / 4mkAa3RT(ln r2/r1)
where Vf is the volume of the free space of pores; Aa is the nominal area of contact; k is the Carman constant dependent on the channel shape; usually k = 4.5 to 5.0.
Selection of the methods of tightness control
The process of taking a decision on application of a particular method of control of tightness of a specific item or construction, understood to mean selection from several alternatives, is the main step in specialists' work.
Proceeding from his experience and intuition, taking into account the world experience and useful advice of his colleagues, a specialist acts in his own way based on his assessment of the situation. The selection conditions are influenced by a whole number of factors, namely:
The complexity of the selection problem increases with the number of alternatives and number of alternative evaluation criteria.
- novelty of the issue under consideration;
- difficulty of getting a full list of alternatives;
- multicriterial nature of alternatives assessment;
- difficulties in revealing all the criteria of alternatives comparison;
- difficulties of comparison of dissimilar criteria;
- subjective nature of many evaluations of alternatives.
The best known approach to solving complex selection problems was called the multicriterial analysis of alternatives. The methods of multicriterial analysis determine the precise sequence of actions, namely consideration of the goals and means, defining and successively considering the alternative variants of solving the problem, comparison of alternatives with each other by individual criteria, desire to make a rational selection from them by combining the estimates of individual criteria into a common estimate of usefulness of an alternative.
Thus, selection of the method of control of tightness of a specific item or construction can be reduced to calculation. This requires making a list of the alternative methods and a list of criteria for their comparison, making estimates by individual criteria for each alternative and combining these estimates into a common estimate of usefulness of an alternative. A list of alternatives and criteria for their comparison, are developed by experts. They also determine how each criterion can be evaluated, i.e. construct an estimation scale.
Transition to derivation of a common estimate of an alternative is usually based on equations combining the estimates by individual criteria into a common estimate of usefulness of an alternative. There exist several widely accepted methods of taking a decision in the case of many criteria, differing by the equation of transition to a common estimate of usefulness of alternatives.
Irrespective of the advantages and disadvantages of individual methods, it is possible to single out a certain common positive effect from application of a multicriterial approach to alternatives analysis.
First of all, the very decomposition of one quality into a set of components has unquestionable merits. Assessment by individual criteria is much easier to make. In the case, if these estimates raise doubts, they are easier to check. If several specialists evaluate one criterion, their opinions can differ. Practice shows that this discrepancy is much greater in assessment of an alternative as a whole. Now in assessment by individual criteria, coincidence of the experts' opinions is much higher. This is understandable, as evaluation by an individual criterion is not as complex and has a much more definite meaning.
We have selected ten of the most promising alternatives, i.e. methods of control of tightness of welded joints of multilayer structures, namely large diameter multilayer pipes; determined the main criteria of assessment of alternative methods and established weight coefficients for each criterion of assessment.
As shown by calculations, the highest values of the usefulness function are derived for several varieties of bubble control, namely the method of a vacuum chamber filled with liquid; method of pressure testing of the interface gaps with air and a combined method of pressure and vacuum chamber testing. Next comes a group of gas analysis methods. The indices for a group of chemical methods are lower.
We performed laboratory investigations and factory tests of all the above methods of tightness control. This permitted suggesting the most rational technologies of control of the tightness of welded joints of several kinds of large-sized multilayer structures.