Method of analyzing corrosion and corrosion prevention

An object to be analyzed for corrosion and corrosion prevention is divided into a plurality of adjacent regions of plural types by a dividing plane, with one of the adjacent regions being referred to as an attentional region with a boundary as the dividing plane and the other as a non-attentional region with a boundary as the dividing plane. An initial current density or an initial potential is imparted to each element of the boundary of the non-attentional region to effect a boundary element analysis for determining a relationship between a potential and a current density in each the element. A potential distribution and a current density distribution in the attentional region in its entirety are determined, using the relationship between the potential and the current density in each the element of the boundary of the non-attentional region as a boundary condition for the attentional region.

BACKGROUND OF THE INVENTION
 1. Field of the Invention
 The present invention relates to a method of effecting a computerized
 analysis for predicting corrosion and corrosion prevention of metals, and
 more particularly to a method of analyzing macro cell corrosion such as
 bimetallic corrosion (also referred to as "galvanic corrosion") and
 differential aeration corrosion and cathodic corrosion prevention, among
 various metal corrosion and corrosion prevention phenomena. The present
 invention is also concerned with an analytical method applicable to a
 system such a plating system, a battery, an electrolytic tank, etc. where
 a macro anode and a macro cathode exist across an electrolyte, developing
 a field of electric potential.
 2. Description of the Prior Art
 In solutions having high electric conductivity, such as seawater, metals
 are susceptible to macro cell corrosion such as bimetallic corrosion
 caused when different metals are used together or differential flow
 velocity corrosion, i.e., differential aeration corrosion, due to flow
 velocity distribution irregularities. It has been desired to predict those
 corrosions accurately in advance so that appropriate preventive measures
 can be taken. Cathodic corrosion prevention based on the positive use of a
 corrosion inhibiting phenomenon at a cathode in a macro cell is finding
 wide use as the most basic corrosion prevention process. There has been a
 demand for the prediction of a range of corrosion prevention and the rate
 of consumption of a sacrificial anode depending on the material of the
 anode, the position of installation of the anode, the shape and materials
 of devices to be protected against corrosion, and solution conditions
 including electric conductivity, flow velocity, etc.
 Experimental approaches to the precision analysis of macro cell corrosion
 suffer limitations because the configuration of the field has a large
 effect on the behavior of macro cells. Specifically, when an experiment is
 conducted on bimetallic corrosion to inspect in detail the effect of
 various factors including the ratio of areas, the combination of
 materials, and the electric conductivity of the solution, the experimental
 result applies only to the three-dimensional shape of a region occupied by
 the solution in the experiment. Since actual devices and structures are
 quite complex in shape, the liquid junction resistance in a macro cell
 cannot accurately be estimated, and the experimental result cannot apply
 directly to the actual situation. It is practically impossible to carry
 out an experiment on the particular shape of a device to be protected
 against corrosion each time the shape of the device is changed. For these
 reasons, it has heretofore customary to predict macro cell corrosion and
 cathodic corrosion prevention for actual structures mostly according to
 empirical rules.
 Many attempts have been made to achieve a more accurate and quantitative
 analysis of macro cell corrosion and cathodic corrosion prevention for
 actual structures. One effort has been to solve purely mathematically a
 Laplace's equation governing a potential distribution for determining a
 potential distribution and a current density distribution. Objects to be
 analyzed by this process are limited to relatively simple systems in the
 form of flat plates, cylinders, etc. Processes long known in the art for
 analyzing electric field problems including a conformal mapping process
 and a process using electrically conductive paper. These processes,
 however, handle two-dimensional fields only.
 With the development in recent years of the computer technology, various
 efforts have been made to apply numerical analyses using a difference
 method, a finite element method, and a boundary element method. The
 difference method and the finite element method are disadvantageous in
 that the time required for calculations is very long because an object to
 be handled needs to be divided into elements. According to the boundary
 element method, since only the surface of an object to be handled needs to
 be divided into elements, it is possible to greatly reduce the time
 required to divide the object into elements and the time required for
 calculations. Based on the belief that the boundary element method is most
 suitable for analyzing corrosion problems where physical quantities
 including a potential and a current density on a surface are important,
 the inventors have developed an analytical technique based on the boundary
 element method for the prediction of macro cell corrosion and cathodic
 corrosion prevention problems.
 Basic Equations and Boundary Conditions
 The corrosion of a metal in an aqueous solution develops due to
 electrochemical reactions which comprise a pair of anodic and cathodic
 reactions. For example, the reactions which corrode iron in an aqueous
 solution of neutral salt, such as seawater, proceed according to the
 following equations (1) and (2):
EQU Fe.fwdarw.Fe.sup.2+ +2e.sup.- (anodic reaction) (1)
EQU 1/2.multidot.O.sub.2 +H.sub.2 O+2e.sup.-.fwdarw.2OH.sup.- (cathodic
 reaction) (2)
 On a surface of metal, an area where an anodic reaction occurs is referred
 to as anode, and an area where a cathodic reaction occurs is referred to
 as cathode. With respect to the corrosion of iron in seawater, anodes and
 cathodes are usually very small and mixed together, and their positions
 are not fixed. Therefore, the corrosion progresses substantially uniformly
 over the entire surface while producing some surface irregularities. If
 the material, the surface state, and the environment are not uniform, then
 anodes and cathodes are localized, allowing corrosion to concentrate in
 certain regions (anodes). The former type of corrosion is referred to as
 micro cell corrosion, and the latter type of corrosion as macro cell
 corrosion. The type of corrosion which is often responsible for extensive
 damage to seawater pumps is the macro cell corrosion which includes
 bimetallic corrosion and differential aeration corrosion. The cathode in a
 macro cell is inhibited from corroding because only a cathode current
 flows in the cathode. Cathodic corrosion prevention is a corrosion
 prevention process which positively uses such a corrosion inhibiting
 phenomenon.
 Each of systems of macro cell corrosion and cathodic corrosion prevention
 may be considered as a cell comprising an anode and a cathode disposed
 across an electrolyte. A potential (.phi.) distribution in the electrolyte
 is governed by the following Laplace's equation (3):
 .gradient..sup.2.phi.=0 (3)
 It is assumed that, as shown in FIG. 1 of the accompanying drawings, an
 electrolyte is surrounded by boundaries .GAMMA..sub.1, .GAMMA..sub.2,
 .GAMMA..sub.3a, and .GAMMA..sub.3c. The boundary .GAMMA..sub.1 is a
 boundary where the value of a potential .phi. is set to .phi..sub.0, i.e.,
 a boundary where the potential is constant. The boundary .GAMMA..sub.2 is
 a boundary where the value of a current density q is set to q.sub.0, i.e.,
 a boundary where the current density is constant. The boundaries
 .GAMMA..sub.3a and .GAMMA..sub.3c are the surface of an anode and the
 surface of a cathode, respectively. Boundary conditions in the respective
 boundaries are given by the following equations (4)-(7):
EQU On .GAMMA..sub.1 :.phi.=.phi..sub.0 (4)
EQU On .GAMMA..sub.2
 :q{.ident..kappa..differential..phi./.differential.n}=q.sub.0 (5)
EQU On .GAMMA..sub.3a.phi.=-f.sub.a (q) (6)
EQU On .GAMMA..sub.3c :.phi.=-f.sub.c (q) (7)
 where .kappa. represents the electric conductivity of the electrolyte,
 .differential./.differential.n a differential in the direction of an
 outward normal line, and f.sub.a (q) and f.sub.c (q) nonlinear functions
 indicative of polarization characteristics of the anode and the cathode,
 respectively, the nonlinear functions being determined by way of
 experimentation. By solving the equation(3) under the boundary conditions
 (4)-(7), it is possible to determine a potential distribution and a
 current density distribution near the surface. The potential .phi. and an
 actually measured electrode potential E are related to each other by
 .phi.=-E.
 Analysis According to the Boundary Element Method
 According to the normal formulation of the boundary element method, the
 following boundary integration equation (8) is derived from the equation
 (3):
 ##EQU1##
 where .phi..sup.* represents the fundamental solution of a
 three-dimensional Laplace's equation,
EQU q.sup.* =.kappa..differential..phi./.differential.n,
 .GAMMA. represents a boundary (=.GAMMA..sub.1 +.GAMMA..sub.2
 +.GAMMA..sub.3a +.GAMMA..sub.3c) surrounding the electrolyte, and c is
 c=1/2 for a smooth boundary and c=.omega./2.pi. at an angle point of an
 angle .omega..
 For numerically solving the above boundary integration equation, it is
 necessary to discretize the boundary integration equation. Specifically,
 the boundary is divided into a number of elements, and the potential .phi.
 and the current density q are approximated by a discrete value and an
 interpolating function at each node, providing the following simultaneous
 algebraic equations:
 ##EQU2##
 where b.sub.j (j=1, 2, . . . , p) represents the value of a known component
 of .phi. or q, x.sub.j (j=1, 2, . . . , p) an unknown quantity
 corresponding to b.sub.j, f.sub.j (q.sub.j) (j=1, 2, . . . , s) a
 nonlinear function indicative of polarization characteristics, and [A] and
 [B] matrixes determined by the geometrical shape of the boundary .GAMMA..
 Since the above equations are nonlinear, repetitive calculations are
 needed to solve these equations. The inventors of the present application
 employ the Newton-Raphson method.
 Analytic Method for Axially Symmetric Region
 Many actual devices to be analyzed, such as pipes and some pump components,
 include axially symmetric regions, and it is desirable to analyze those
 axially symmetric regions simply. Primarily, the following two processes
 are considered as effective to solve axially symmetric problems:
 (i) A process which uses a fundamental solution to an axially symmetric
 problem; and
 (ii) A process which uses an ordinary fundamental solution to a
 three-dimensional problem and reduces the number of elements in view of
 axially symmetry upon discretization.
 The former process of using a fundamental solution which satisfies the
 axially symmetric condition is problematic in that it involves more
 complex integrating calculations than the process of using an ordinary
 fundamental solution. According to the present invention, the latter
 process of reducing the number of elements in view of axially symmetry
 upon discretization is employed. This process will now be described below.
 For an ordinary three-dimensional analysis, it is necessary to divide all
 boundaries into elements in order to discretize the boundary integration
 equation (8). Since .phi. and q have the same value in the circumferential
 direction owing to the axial symmetry, the boundary integration equation
 (8) can be modified as follows:
 ##EQU3##
 where .GAMMA..sub.1D represents a range on a one-dimensional line. From the
 equation (10), it can obtain simultaneous algebraic equations by
 discretizing only .GAMMA..sub.1D. Therefore, using the axial symmetry, it
 is possible to greatly reduce the number of unknowns and expect an
 increase in the accuracy.
 Process of Dividing a Region
 For the sake of brevity, an area made up of two regions as shown in FIG. 2
 of the accompanying drawings is considered. If an inner boundary plane is
 indicated by .GAMMA..sub.B, then since the equations (9) are satisfied in
 each of the regions, the following equations are satisfied:
 ##EQU4##
 where I, II represent quantities relative to the respective regions I, II,
 B a quantity relative to the inner boundary surface .GAMMA..sub.B,
 {X.sup.M } (M=I, II) a vector having a component which is a quantity
 relative to a boundary other than .GAMMA..sub.B of x.sub.i and q.sub.i,
 and {b.sub.M } (M=I, II) a vector having a component which is a known
 quantity (or a function indicative of a polarization curve) corresponding
 to X.sup.M.
 Inasmuch as the potential and the current density are continuous in the
 inner boundary, the following equations are satisfied:
EQU .phi..sup.IB =.phi..sup.IIB (13)
EQU q.sup.IB =-q.sup.IIB (14)
 Transposing [H.sup.MB]{.phi..sup.MB } (M=I, II) from the right-hand side to
 the left-hand side in the equations (11), (12), and substituting the
 equations (13), (14) in the resulting equations, the following equations
 are produced:
 ##EQU5##
 These equations can be put together into the following equation (17):
 ##EQU6##
 As with the equations (9), the equation (17) constitutes a nonlinear
 equation. According to the present invention, a solution to the equation
 (17) is determined by the Newton-Raphson method.
 The inventors have developed six programs for analyzing an open region
 (such as an outer vessel surface surrounded by an electrolyte extending
 infinitely) and a closed region (such as an inner pump surface surrounding
 by an electrolyte extending in a limited space) with respect to each of
 two- and three-dimensional axially symmetric structures for the purpose of
 practically solving corrosion and corrosion prevention problems.
 In an actual system, some of six regions that can be modeled
 two-dimensionally (open and closed regions), three-dimensionally (open and
 closed regions), and axially symmetrically (open and closed regions) may
 exist continuously. FIG. 3 of the accompanying drawings shows a specific
 example. In FIG. 3, a seawater pump 10 made of stainless steel has three
 annular sacrificial anodes 11a, 11b, 11c of Zn disposed circumferentially
 on an inner pump surface and four prismatic sacrificial anodes 12 of Zn
 disposed on an outer pump surface at equally spaced locations. The inner
 and outer pump surfaces communicate with each other through seawater, so
 that the inner pump surface should electrochemically affect the outer pump
 surface, and the outer pump surface should electrochemically affect the
 inner pump surface. Since the seawater surrounding the outer pump surface
 occupies a wide region, and a boundary to be divided into elements is too
 large for the region to be handled as a closed region, it is practically
 impossible to model and analyze the outer pump surface as a
 three-dimensional closed region like the inner pump surface.
 For this reason, the inner pump surface is analyzed as a three-dimensional
 closed region, and the pump outer surface is analyzed as an open region.
 An inner surface of a guide casing is compartmented into seven flow
 passages by seven helical guide vanes. Since these flow passages are
 symmetrical in shape, one of them is removed, and divided into
 three-dimensional elements. Assuming that the prismatic anodes on the
 outer pump surface are regarded as web-shaped anodes having the same area,
 they are handled as axially symmetric anodes, and hence as axially
 symmetric models in the open region.
 FIG. 4 of the accompanying drawings shows by way of example a plurality of
 elements divided from the outer pump surface for an axially symmetric
 analysis of the open region. Inasmuch as the inner and outer pump surfaces
 electrochemically affect each other in reality as described above, the
 analysis needs to take such an electrochemically effect into
 consideration. However, because those regions are handled by different
 analyzing programs, i.e., the outer pump surface is handled by a program
 for the three-dimensional closed region and the inner pump surface by a
 program for the axially symmetric closed region, it has heretofore been
 impossible to analyze the inner and outer pump surfaces while taking the
 mutual electrochemically effect into consideration. The region dividing
 process developed by the inventors has been able to analyze regions
 modeled according to the same modeling principle.
 If the process of analyzing different regions in a related fashion is
 applied to a situation where three-dimensional and axially symmetric
 regions are present continuously, then it is necessary to determine a
 region where the axially symmetric analysis is applicable. However,
 experiences and skills have to be relied upon to determine such a region
 because there is no process available at present for quantitatively
 determining the region.
 SUMMARY OF THE INVENTION
 It is therefore an object of the present invention to provide a method of
 analyzing corrosion and corrosion prevention in two or more continuous
 regions of one or different types in a related fashion, among those
 regions which are modeled two-dimensionally (open and closed regions),
 three-dimensionally (open and closed regions), and axially symmetrically
 (open and closed regions).
 Another object of the present invention is to provide a method of analyzing
 corrosion and corrosion prevention in a situation where three-dimensional
 and axially symmetric regions are present continuously by determining
 quantitatively a region where continuous analysis of both regions is
 applicable.
 According to an aspect of the present invention, there is provided a method
 of analyzing corrosion and corrosion prevention of an object, comprising
 the steps of dividing an object to be analyzed into a plurality of
 adjacent regions of plural types by a dividing plane, with one of the
 adjacent regions being referred to as an attentional region with a
 boundary as the dividing plane and the other as a non-attentional region
 with a boundary as the dividing plane, imparting an initial current
 density or an initial potential to each element of the boundary of the
 non-attentional region to effect a boundary element analysis for
 determining a relationship between a potential and a current density in
 each element, determining a potential distribution and a current density
 distribution in the attentional region in its entirety, using the
 relationship between the potential and the current density in each the
 element of the boundary of the non-attentional region as a boundary
 condition for the attentional region, and effecting an element analysis on
 the non-attentional region to determine a potential distribution and a
 current density distribution in the non-attentional region in its
 entirety, using the relationship between the potential and the current
 density in each element of the boundary of the attentional region as a
 boundary condition for the non-attentional region, whereby a potential
 distribution and a current density distribution across the regions can
 continuously be analyzed.
 According to an aspect of the present invention, there is also provided a
 method of analyzing corrosion and corrosion prevention of an object
 including continuous regions which can be modeled three-dimensionally
 (open and closed regions) and axially symmetrically (open and closed
 regions), comprising the steps of extracting a candidate region which is
 axially symmetric, modeling the extracted region into a pipe having a
 radius R at a dividing plane A, imparting a current density distribution
 expressed by a delta function having an intensity a to the modeled pipe at
 r=R on a plane z=0, analytically determining a potential .phi. in the
 pipe, and determining a position z where the magnitude of a change of the
 potential .phi. is smaller than an allowable value to determine a region
 which can be modeled axially symmetrically. The method may further
 comprise the steps of dividing the object into a plurality of adjacent
 regions of plural types by a dividing plane, with one of the adjacent
 regions being referred to as an attentional region with a boundary as the
 dividing plane and the other as a non-attentional region with a boundary
 as the dividing plane, imparting an initial current density or an initial
 potential to each element of the boundary of the non-attentional region to
 effect a boundary element analysis for determining a relationship between
 a potential and a current density in each the element, determining a
 potential distribution and a current density distribution in the
 attentional region in its entirety, using the relationship between the
 potential and the current density in each the element of the boundary of
 the non-attentional region as a boundary condition for the attentional
 region, and effecting an element analysis on the non-attentional region to
 determine a potential distribution and a current density distribution in
 the non-attentional region in its entirety, using the relationship between
 the potential and the current density in each element of the boundary of
 the attentional region as a boundary condition for the non-attentional
 region, whereby a potential distribution and a current density
 distribution across the regions can continuously be analyzed.
 The regions of plural types may include regions which can be modeled in
 two-dimensional, three-dimensional, and axially symmetric open and closed
 spaces.
 The initial current density or the initial potential imparted to each
 element of the boundary of the non-attentional region may be uniform.
 The regions may include at least two non-attentional regions present
 contiguously to one attentional region.
 The above and other objects, features, and advantages of the present
 invention will become apparent from the following description when taken
 in conjunction with the accompanying drawings which illustrate preferred
 embodiments of the present invention by way of example.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS
 FIG. 5 schematically illustrates a situation where two regions are
 continuously present. One of the regions is referred to as an attentional
 region and the other as a non-attentional region. The two regions are
 divided by a dividing plane which is referred to as a boundary
 .GAMMA..sub.T viewed from the attentional region and as a boundary
 .GAMMA..sub.N viewed from the non-attentional region.
 A uniform current density q is imparted to each element .GAMMA..sub.k of
 the boundary .GAMMA..sub.N of the non-attentional region, and the
 non-attentional region is analyzed according to a boundary element process
 to determine a potential .phi..sub.k in each element .GAMMA..sub.k of the
 boundary .GAMMA..sub.N. A uniform current density q.sub.a is imparted to
 each element .GAMMA..sub.k of the boundary .GAMMA..sub.N, and a potential
 response .phi..sub.ak is determined at this time. Since the equation
 .phi..sub.ak =f.sub.k (q.sub.a) between the current density and the
 potential in each element .GAMMA..sub.k of the boundary .GAMMA..sub.N is
 also applicable to the boundary .GAMMA..sub.T viewed from the attentional
 region, it is used as a boundary condition on the boundary .GAMMA..sub.T.
 Therefore, using this boundary condition, the attentional region can be
 analyzed taking the non-attentional region into account. The above
 relationship between the current density and the potential is referred to
 as an equivalent boundary condition. If the non-attentional region is
 analyzed again using a current density or a potential on the boundary
 .GAMMA..sub.T which is obtained from the analysis of the attentional
 region, then the two continuous regions can be analyzed in their entirety.
 For analyzing the non-attentional region, a uniform potential, rather than
 the uniform current, may be imparted as an initial condition to the
 non-attentional region. The initial current density or potential to be
 imparted to the non-attentional region may not necessarily be uniform, but
 may differ slightly from element to element.
 FIG. 6 specifically shows an example in which an axially symmetrical region
 and a region that can be modeled three-dimensionally are present
 continuously. In this example, it is preferable that a non-attentional
 region which needs to be analyzed a plurality of times with different
 potentials or current densities should be the axially symmetrical region
 that requires a shorter analyzing time. The relationship between a
 potential and a current density on elements .GAMMA..sub.N1,
 .GAMMA..sub.N2, .GAMMA..sub.N3 obtained by an analysis of the axially
 symmetrical region is used as a boundary condition for corresponding
 elements .GAMMA..sub.A11 -.GAMMA..sub.A18, .GAMMA..sub.A21
 -.GAMMA..sub.A28, .GAMMA..sub.A31 -.GAMMA..sub.A38 of the
 three-dimensionally modeled region.
 According to the present invention, a region that can be molded axially
 symmetrically, i.e., a region to which axially symmetric elements are
 quantitatively applied, is determined using an analytic solution to the
 Laplace's equation governing corrosion problems. As shown in FIG. 7, a
 symmetric member of a complex-shape device is extracted, and the extracted
 member is modeled into a pipe having a radius R at a dividing plane A. A
 current density distribution expressed by a delta function having an
 intensity a is imparted to the modeled pipe at r=R on a plane z=0, and a
 potential .phi. in the pipe is analytically determined at this time. If it
 is assumed that polarization characteristics in the pipe are expressed by
 .phi.=-(ki+.phi..sub.0) (i represents the current density), then the
 analytic solution of the potential in the pipe is given as follows:
 ##EQU7##
 An example of the present invention will be described below.
 An analyzed object was a vertical-shaft pump having a diameter of 200 mm
 and a length of 6000 mm, as shown in FIG. 8A. As shown in FIG. 8A, the
 pump was divided into inner portions 15, 16 and an outer portion 17. The
 inner portions included a guide casing 15 of complex three-dimensional
 shape composed of a complex assembly of parts and having a helical flow
 passages, and a column pipe 16 which can be modeled axially symmetrically.
 In order to determine a region to which axially symmetrical elements are
 applicable among the inner portions, calculations were made with the
 allowable error .epsilon. in the equation (21) being set to 0.02. As a
 result, the region to which axially symmetrical elements are applicable
 was determined as being spaced 192 mm from the upper end of the guide
 casing 15. Therefore, an inner column pipe surface handled as an axially
 symmetric region was spaced 200 mm or more from the upper end of the guide
 casing 15. Three regions which were divided were an outer pump surface
 handled as an axially symmetric open region, an inner guide casing surface
 as a three-dimensional closed region, and an inner column pipe surface as
 an axially symmetric closed region. The inner guide casing surface was
 compartmented into seven flow passages by seven helical guide vanes. Since
 these flow passages were symmetrical in shape, one of them was removed,
 and divided into three-dimensional elements.
 In order to determine an equivalent boundary condition for boundary planes
 .GAMMA..sub.a, .GAMMA..sub.b between the outer pump surface, the inner
 column pipe surface, and the inner guide casing surface, a boundary
 element analysis was conducted on the outer pump surface and the inner
 column pipe surface. Specifically, an axially symmetric open region
 analysis was effected on the outer pump surface, and an axially symmetric
 closed region analysis was effected on the inner column pipe surface. A
 current density ranging from -2.0 to 2.0 A/m.sup.2 was applied in
 increments of 0.2 A/m.sup.2 to each element of the boundary planes
 .GAMMA..sub.a, .GAMMA..sub.b. Using the determined equivalent boundary
 condition, i.e., the relationship between the current density and the
 potential, as a boundary condition, a three-dimensional closed region
 analysis was carried out on the guide casing 15. Using an obtained current
 density at the boundary planes .GAMMA..sub.a, .GAMMA..sub.b as a boundary
 condition, the outer pump surface and the inner column pipe surface were
 analyzed again. In this manner, all the analysis was completed. The
 potential distribution of the inner guide casing surface, etc. which was
 obtained as a result of the analysis was considered as highly close to an
 actual potential distribution.
 Heretofore, for analyzing a situation where two or more of six regions that
 are modeled two-dimensionally (open and closed regions),
 three-dimensionally (open and closed regions), and axially symmetrically
 (open and closed regions) exist continuously, the regions have to be
 analyzed separately. According to the present invention, however, all the
 regions can be analyzed in a related fashion, so that a potential
 distribution or a current density distribution in boundaries can be
 determined accurately.
 For example, it has been customary to separately analyze inner and outer
 pump surfaces of a vertical-shaft pump though they are electrochemically
 affected by each other, and hence the pump cannot accurately be analyzed
 for corrosion and corrosion prevention. According to the present
 invention, it is possible to analyze such inner and outer pump surfaces in
 a related manner. Furthermore, while the entire inner pump surface has
 heretofore been analyzed with a three-dimensional closed region model, it
 is possible according to the present invention to analyze an inner column
 pipe surface of simple configuration with an axially symmetric model,
 allowing it to be divided easily into elements.
 In the case where a three-dimensional region and an axially symmetric
 region are existing continuously, it has conventionally been unable to
 determine accurately a region which can be modeled axially symmetrically.
 According to the present invention, however, such a region can be
 determined quantitatively.
 The present invention has been described as being applied to a method of
 analyzing corrosion and corrosion prevention of metals. However, the
 principles of the present invention are also applicable to the plating of
 metals, the designing of batteries and electrolytic tanks, etc.
 Although certain preferred embodiments of the present invention have been
 shown and described in detail, it should be understood that various
 changes and modifications may be made therein without departing from the
 scope of the appended claims.