CN114462159B - Multi-working-condition blade dehumidification optimization design method of marine wet steam turbine - Google Patents
Multi-working-condition blade dehumidification optimization design method of marine wet steam turbine Download PDFInfo
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Abstract
The invention discloses a multi-working-condition blade dehumidification optimization design method of a marine wet turbine, which combines a blade parameterization reconstruction method, a Keli Jin Jinshi model and a multi-objective genetic optimization algorithm to form a wet steam unbalanced condensation flow characteristic result calculated by a three-dimensional flow field numerical simulation calculation method, takes the blade grid outlet humidity of two operation working conditions of a minimized turbine as an optimization target, takes the stage efficiency of two working conditions as a performance constraint condition, and takes parameterized blade geometry variables as design variables, thereby being capable of realizing multi-working-condition multi-constraint dehumidification optimization design of the wet turbine blade.
Description
Technical Field
The invention belongs to the technical field of marine steam turbine part design, and particularly relates to a multi-working-condition blade dehumidification optimization design method of a marine wet steam turbine.
Background
The turbine blade is a core component for heat-power conversion of the marine turbine, and the performance of turbine efficiency, power, safety, reliability, service life and the like are closely related to the blade profile. The development of the optimal design of the turbine blades has very important significance for guaranteeing the economy, reliability and safety of the turbine. The students have conducted extensive researches on the blade optimization design method, but most of the researches are directed to the design working condition. The marine wet steam turbine has complex operation conditions and high mobility requirements, certain high-power single-cylinder steam turbine blades bear the actions of high-humidity steam and complex stress, the output power and efficiency of the steam turbine can be reduced due to the fact that the steam humidity is too high, the blades are corroded for a long time, damage, fracture and failure and the like of the blades are extremely easy to cause, the operation of the direct crisis steam turbine is safe, the performance under the condition of multiple operation conditions must be considered in design, and the final-stage humidity is reduced as much as possible.
Disclosure of Invention
The invention aims to overcome the defects of the technology, and provides a multi-working-condition blade dehumidification optimization design method of a marine wet steam turbine, which can meet the performance requirements of a plurality of working points and can improve the multi-working-condition operation reliability of the steam turbine.
In order to achieve the purpose, the multi-working-condition blade dehumidification optimization design method of the marine wet steam turbine is as follows:
step one: based on the three-time Bezier curve, parameterizing the blade profile of the blade, and determining design variables, objective functions and constraint conditions of the optimized design of the dehumidification of the blade profile of the steam turbine;
step two: taking initial leaf type parameters as a central point, wherein the value range of each design variable meets the constraint condition of the second step, and establishing a sample matrix X of the leaf type design variable by using a Latin hypercube test design method P×n Sample matrix X P×n Is n design variables, sample matrix X P×n The number of lines of the matrix is P groups of samples, and each line of the matrix corresponds to one leaf pattern;
step three: any oneA group of design variables determine a blade profile, a blade grid and a flow field model are established according to the blade profile, and the flow characteristic of wet steam in the blade grid formed by the blades is calculated by a three-dimensional flow field numerical simulation calculation method to obtain a target value sample matrix Y P×2 ＝[Y 1 ,Y 2 ]The column number of the target value sample matrix Y is 2, and the line number is P;
step four: sample matrix X of design variables P×n And a target value sample matrix Y P×2 As basic data, establishing a proxy model ψ of a blade design variable and an objective function based on a kri Jin Jinshi model;
step five: optionally establishing a cascade model by M groups of design variables meeting the constraint condition of the step one, and calculating M groups of objective function value matrixes of the flow characteristics of the wet steam in the cascade by a three-dimensional flow field numerical simulation calculation method;
carrying out one-to-one correspondence comparison between the M groups of objective function value matrixes for calculating the flow characteristics of the wet steam in the blade grid through a three-dimensional flow field numerical simulation calculation method and the M groups of objective function value matrixes calculated by the agent model ψ, and if the agent model meets the calculation precision requirement, turning to the step six; otherwise, returning to the second step, and increasing the number of sample points until the agent model meets the calculation precision requirement;
step six: optimizing the constructed Kriging proxy model psi by a multi-target non-dominant genetic algorithm, and obtaining an optimal target function Y according to the constraint conditions in the step one 1best ,Y 2best And the optimal value of the corresponding leaf design variable, Y 1best For the optimum condition of the grid outlet humidity, Y 2best And the humidity of the blade grid outlet is the optimal second working condition.
Further, in the first step, the vane profile parameters include 12 parameters including 8 vane profile geometry parameters and 4 Bezier curve shape control parameters, and the 8 vane profile geometry parameters are the vane width B, the mounting angle gamma, and the leading edge arc radius r 1 Radius r of trailing edge arc 2 Geometric steam inlet angle beta 1 Geometric steam outlet angle beta 2 Wedge angle of leading edgeTrailing edge wedge +.>The 4 Bezier curve shape control parameters are h (1), h (2), h (3) and h (4), and each parameter corresponds to a design variable of a blade profile; n design variables are selected from 12 parameters, the other 12-n parameters are constants, and n is more than or equal to 2 and less than or equal to 12.
Further, in the first step, the objective function is:
minY 1 (X 1 ，X 2 ，……，X n )
minY 2 (X 1 ，X 2 ，……，X n )
constraint conditions:
l d ·X 1 ≤X 1 ≤X 1 ·l u
l d ·X 2 ≤X 2 ≤X 2 ·l u
……
l d ·X n ≤X n ≤X n ·l u
η 01 ≤η 1 ，η 02 ≤η 2
wherein X is 1 ，X 2 ，……，X n The design variables are 1 st, 2 nd, … … th and n th design variables respectively; y is Y 1 Represents the grid outlet humidity and eta of the working condition 1 The isentropic efficiency of the working condition is represented; y is Y 2 Represents the outlet humidity eta of the blade grid under the second working condition 2 The isentropic efficiency of the second working condition is shown; l (L) d Taking 0.8 to 0.9; l (L) u Taking 1.1 to 1.2; η (eta) 01 Representing isentropic efficiency, eta of prototype cascade during working conditions 02 And the isentropic efficiency of the prototype blade cascade under the second working condition is represented.
Further, the isentropic efficiency is expressed as η= (h in -h out )/(h in -h out,s )，h in For the actual enthalpy value of the cascade inlet, h out For the actual enthalpy value of the blade grid outlet, h out,s Is the isentropic enthalpy of the outlet.
Further, in the second step, P is not less than 3n.
Further, in the fourth step, the proxy model ψ is:
Ψ＝f(x k ) T β * +r(x k ) T γ *
β * ＝(F T R -1 F) -1 F T R -1 Y
γ * ＝R -1 (Y-Fβ * )
x k k=1, 2, … …, n, a parameter of the design variable;
wherein f (x) k ) As a row vector, f () is a first order polynomial function; r (x) k ) Parameter x being a design variable k And sample matrix X P×n A correlation function column vector between samples; parameter x with F matrix element as design variable k The function value f (x) i,k ) I=1, 2, … …, P, k=1, 2, … … n; r represents a correlation function matrix, and the expressions of elements in the matrix are as follows: r (X) i ,X j )＝Π k＝1 exp(-θ k d k 2 )，i＝1,2,……,P，j＝1,2,……,P，k＝1,2,……,n，θ k Is an anisotropic parameter, d k Is the distance between any two points in the sample.
Further, in the fifth step, M is more than or equal to 3 and less than or equal to P.
Further, in the fifth step, the calculation accuracy is 2% -3% of the relative error of the two calculation values.
Compared with the prior art, the invention has the following advantages: the design method fully considers the performance requirements of the steam turbine in a plurality of maneuvering conditions, takes the thermodynamic parameters of the condensing flow of the wet steam as optimization targets, and adopts an optimization method based on an approximate model to optimally design the prototype blade by changing the profile parameters of the blade; the design method can control the coagulation nucleation phenomenon, reduce the humidity of the through-flow part, and also can meet the performance requirements of a plurality of working condition points, and has important significance for improving the safety, reliability and economy of multi-working-condition operation of the blade.
Drawings
FIG. 1 is a graph of Dykas blade profile design optimization before and after comparison;
FIG. 2 is a cloud chart of an initial leaf type humidity distribution under a rated working condition;
FIG. 3 is a cloud chart of optimized leaf type humidity distribution under rated working conditions;
FIG. 4 is a cloud chart of the initial leaf profile humidity distribution under a high flow condition;
FIG. 5 is a cloud plot of optimized leaf profile humidity distribution under high flow conditions.
Detailed Description
The invention will now be described in further detail with reference to the drawings and to specific examples.
The blade dehumidification optimization design method for the marine wet steam turbine under multiple working conditions comprises the following specific steps:
step one: the blade profile of the blade is parameterized based on a cubic Bezier curve, wherein the blade profile parameters comprise 8 blade profile geometric parameters and 12 parameters including 4 Bezier curve shape control parameters, and the 8 blade profile geometric parameters comprise a blade width B, a mounting angle gamma and a front edge arc radius r 1 Radius r of trailing edge arc 2 Geometric steam inlet angle beta 1 Geometric steam outlet angle beta 2 Wedge angle of leading edgeTrailing edge wedge +.>The shape control parameters of the 4 Bezier curves are h (1), h (2), h (3) and h (4), and the 12 parameters can uniquely determine a leaf profile, and each parameter corresponds to a design variable of the leaf profile; when the optimization design of the blade profile dehumidification of the steam turbine is carried out, n design variables are selected from 12 parameters, the other 12-n parameters are constants, n is more than or equal to 2 and less than or equal to 12, and two objective functions are determined as follows:
minY 1 (X 1 ，X 2 ，……，X n )
minY 2 (X 1 ，X 2 ，……，X n )
constraint conditions:
l d ·X 1 ≤X 1 ≤X 1 ·l u
l d ·X 2 ≤X 2 ≤X 2 ·l u
……
l d ·X n ≤X n ≤X n ·l u
η 01 ≤η 1 ，η 02 ≤η 2
wherein X is 1 ，X 2 ，……，X n The design variables are 1 st, 2 nd, … … th and n th design variables respectively; y is Y 1 Represents the grid outlet humidity and eta of the working condition 1 The isentropic efficiency of the working condition is represented; y is Y 2 Represents the outlet humidity eta of the blade grid under the second working condition 2 The isentropic efficiency of the second working condition is shown; l (L) d Generally taking 0.8 to 0.9; l (L) u Taking 1.1 to 1.2; η (eta) 01 Representing isentropic efficiency, eta of prototype cascade during working conditions 02 The isentropic efficiency of the prototype blade cascade under the second working condition is represented, and eta is calculated in the process of optimizing design 01 And eta 02 As a performance constraint. Isentropic efficiency is expressed as η= (h in -h out )/(h in -h out,s )，h in For the actual enthalpy value of the cascade inlet, h out For the actual enthalpy value of the blade grid outlet, h out,s Is the isentropic enthalpy of the outlet.
Step two: taking initial (before optimization) leaf type parameters as center points, wherein the value range of each design variable meets the constraint condition of the first step, and establishing a sample matrix X of the leaf type design variable by using a Latin hypercube test design method P×n Sample matrix X P×n Is n design variables, sample matrix X P×n The number of lines of the matrix is P groups of samples, P is not less than 3n, each line of the matrix corresponds to one leaf profile, namely P leaf profiles are all used;
step three: determining a leaf profile from any set of design variables, establishing a leaf based on the leaf profileGrid and flow field model, calculating flow characteristics of wet steam in the blade grid formed by the blades by a three-dimensional flow field numerical simulation calculation method to obtain a target value sample matrix Y P×2 ＝[Y 1 ,Y 2 ]The column number of the target value sample matrix Y is 2, and the line number is P;
step four: sample matrix X of design variables P×n And a target value sample matrix Y P×2 As basic data, a proxy model of the blade design variables and the objective function is established based on the kri Jin Jinshi model, and the proxy model ψ is:
Ψ＝f(x k ) T β * +r(x k ) T γ *
β * ＝(F T R -1 F) -1 F T R -1 Y
γ * ＝R -1 (Y-Fβ * )
x k k=1, 2, … …, n, a parameter of the design variable;
wherein f (x) k ) As a row vector, f () is a first order polynomial function; r (x) k ) Parameter x being a design variable k And sample matrix X P×n A correlation function column vector between samples; parameter x with F matrix element as design variable k The function value f (x) i,k ) (i=1, 2, … …, P, k=1, 2, … … n); r represents a correlation function matrix, and the expressions of elements in the matrix are as follows: r (X) i ,X j )＝Π k＝1 exp(-θ k d k 2 )，i＝1,2,……,P，j＝1,2,……,P，k＝1,2,……,n，θ k Is an anisotropic parameter, d k Distance between any two points in the sample;
step five: optionally, M groups of design variables meeting the constraint condition of the step one are used for establishing a blade grid model, wherein M is more than or equal to 3 and less than or equal to P; calculating M groups of objective function value matrixes of the flow characteristics of the wet steam in the blade grid by a three-dimensional flow field numerical simulation calculation method;
meanwhile, the M groups of design variables are carried into a proxy model psic in the fourth step to calculate an objective function value matrix, the M groups of objective function value matrixes of the flow characteristics of the wet steam in the blade cascade calculated by a three-dimensional flow field numerical simulation calculation method are compared with the M groups of objective function value matrixes calculated by the proxy model psic in a one-to-one correspondence manner, and if the proxy model meets the calculation accuracy requirement (the relative error of the calculated value is taken to be 2% -3%), the step is transferred to the sixth step; otherwise, returning to the second step, and increasing the number of sample points until the agent model meets the calculation precision requirement;
step six: optimizing the constructed kriging proxy model ψ by multi-objective non-dominant genetic algorithm (NSGA-II) under the constraint condition as described in the first step until obtaining the optimal objective function Y 1best ,Y 2best And the optimal value of the corresponding leaf design variable, Y 1best For the optimum condition of the grid outlet humidity, Y 2best And the humidity of the blade grid outlet is the optimal second working condition.
The invention combines a blade parameterization reconstruction method, a Keli Jin Jinshi model and a multi-objective genetic optimization algorithm to form a wet steam unbalanced condensation flow characteristic result calculated by a three-dimensional flow field numerical simulation calculation method, takes the blade grid outlet humidity of two operation working conditions of a minimum steam turbine as an optimization target, takes the stage efficiency of two working condition points as a performance constraint condition, and takes parameterized blade geometry variables as design variables, so that a multi-working condition and multi-constraint dehumidification optimization design of the blade of the wet steam turbine can be realized, and powerful guidance is provided for intelligent optimization design and operation of the blade grid of the wet steam turbine.
According to the invention, the in-line blades and the torsion blades of the marine turbine can be optimally designed, and after the blade profiles at different positions along the blade height are respectively optimized, a three-dimensional solid blade with excellent performance can be constructed, and the multi-working-condition dehumidification optimal design of the dynamic and static blades of the turbine stage can be realized; the optimization design of the comprehensive performance and structural characteristics of the steam turbine such as power, efficiency, humidity and the like can be easily realized by changing optimization targets, constraint conditions and design variables, and the method has wide application prospect, can give consideration to performance requirements of a plurality of working condition points, improves the reliability of multi-working condition operation of the steam turbine, and can provide technical support for intelligent optimization design of variable working condition through-flow blades of the wet steam turbine.
Taking Dykas turbine blade as an example, the unbalanced condensation flow of wet steam affecting the blade profile moisture loss of the blade mainly occurs at the suction surface of the blade, so 7 parameters (the installation angle gamma, the geometric inlet angle beta are selected 1 Front edge wedge angleRadius r of arc of front and rear edges 1 And r 2 Bezier curve control variable h 1 And h 2 ) As a design variable, i.e., n=7. On the basis of the initial design variable, adding a floating value of +/-10% as the value range of the design variable (the floating variable is obtained through a large amount of analysis, the variation of the blade performance cannot be reflected due to the fact that the floating variable is too small, the blade is excessively deformed, and the blade profile difference of the actual turbine is obvious), namely l d ＝0.9，l u =1.1. At the same time, to reduce the influence of design variable units on the result, the variable h is controlled by the Bezier-removal curve 1 And h 2 And carrying out normalization processing on all the external variables by taking the initial leaf type parameters as references.
After a sample space generated by a Latin hypercube experimental design method is adopted to obtain a design variable matrix of a blade, dykas blade profile curve and blade grid flow field channels are established, unbalanced condensation calculation is carried out by adopting a three-dimensional flow field numerical simulation calculation method, and response values (Y) related to the blade profile under rated working condition (working condition I) and high-flow working condition (working condition II) are obtained 1 、η 1 、Y 2 、η 2 ) The method comprises the steps of carrying out a first treatment on the surface of the And according to the calculation result, establishing a proxy model of the blade design variable and the objective function based on the kriging Jin Jinshi model. Optionally, 5 groups of geometric parameters in the sample are used as input parameters to be put into a constructed Kriging proxy model, and the maximum value of the model predicted value and the three-dimensional flow field numerical simulation calculation error is 1.7%, so that the precision requirement is met; the degree of influence of each structural parameter on the humidity and efficiency of the objective function is obtained at the same time, as shown in table 1.
TABLE 1 ratio of influence of blade profile design variables on blade profile efficiency and outlet humidity
The optimized results and initial results of the blade profile design parameters calculated by the method of the present invention are shown in table 2. Compared with the initial blade cascade, the optimization of the blade profile reduces the geometric steam inlet angle and reduces the steam outlet side control parameter h of the suction surface Bezier curve 2 Therefore, the curvature from the throat to the tail edge of the suction edge of the blade profile is reduced, the molded line of the suction surface of the blade near the throat is more gentle, the expansion of the steam flow is more uniform, the local expansion rate and the turning of the steam flow are smaller, and fig. 1 is a comparison diagram before and after optimization of the blade profile.
TABLE 2Dykas cascade design variable Multi-task dehumidification design optimization results
Table 3 shows the comparison of the Dykas cascade coagulation property optimization results and the initial results calculated by the three-dimensional flow field numerical simulation calculation method. Compared with the initial blade cascade, the humidity of the outlet is reduced by 6.1 percent, the entropy is increased by 5.1 percent, the maximum droplet diameter is reduced by 11.4 percent, and the isentropic efficiency is improved by 0.6 percent after the optimization under the rated working condition; the humidity of the outlet is reduced by 8.9 percent, the entropy is increased by 15.2 percent, the maximum droplet diameter is reduced by 15.8 percent, and the isentropic efficiency is improved by 0.9 percent after the optimization under the large-flow working condition. As can be seen from comparison of humidity distribution cloud charts before and after the optimization of the working conditions in fig. 2 and 3, the optimized cascade can inhibit the unbalanced condensation flow process at two working condition points, and the humidity of steam in the cascade channels is reduced.
TABLE 3Dykas cascade design variable multiple task design optimization results for dehumidification
Claims (8)
1. A blade dehumidification optimal design method for a marine wet turbine under multiple working conditions is characterized by comprising the following steps of: the optimization design method comprises the following steps:
step one: based on the three-time Bezier curve, parameterizing the blade profile of the blade, and determining design variables, objective functions and constraint conditions of the optimized design of the dehumidification of the blade profile of the steam turbine;
step two: taking initial leaf type parameters as a central point, wherein the value range of each design variable meets the constraint condition of the second step, and establishing a sample matrix X of the leaf type design variable by using a Latin hypercube test design method P×n Sample matrix X P×n Is n design variables, sample matrix X P×n The number of lines of the matrix is P groups of samples, and each line of the matrix corresponds to one leaf pattern;
step three: determining a blade profile by any set of design variables, establishing a blade profile and a flow field model according to the blade profile, and calculating the flow characteristic of wet steam in the blade profile formed by the blades by a three-dimensional flow field numerical simulation calculation method to obtain a target value sample matrix Y P×2 ＝[Y 1 ,Y 2 ]The column number of the target value sample matrix Y is 2, and the line number is P;
step four: sample matrix X of design variables P×n And a target value sample matrix Y P×2 As basic data, establishing a proxy model ψ of a blade design variable and an objective function based on a kri Jin Jinshi model;
step five: optionally establishing a cascade model by M groups of design variables meeting the constraint condition of the step one, and calculating M groups of objective function value matrixes of the flow characteristics of the wet steam in the cascade by a three-dimensional flow field numerical simulation calculation method;
carrying out one-to-one correspondence comparison between the M groups of objective function value matrixes for calculating the flow characteristics of the wet steam in the blade grid through a three-dimensional flow field numerical simulation calculation method and the M groups of objective function value matrixes calculated by the agent model ψ, and if the agent model meets the calculation precision requirement, turning to the step six; otherwise, returning to the second step, and increasing the number of sample points until the agent model meets the calculation precision requirement;
step six: optimizing the constructed Kriging proxy model psi by a multi-target non-dominant genetic algorithm, and obtaining an optimal target function Y according to the constraint conditions in the step one 1best ,Y 2best And the optimal value of the corresponding leaf design variable, Y 1best For the optimum condition of the grid outlet humidity, Y 2best And the humidity of the blade grid outlet is the optimal second working condition.
2. The multi-working-condition blade dehumidification optimal design method of the marine wet steam turbine according to claim 1, which is characterized by comprising the following steps of: in the first step, the blade profile parameters include 12 parameters including 8 blade profile geometrical parameters and 4 Bezier curve shape control parameters, wherein the 8 blade profile geometrical parameters are the blade width B, the mounting angle gamma and the leading edge arc radius r 1 Radius r of trailing edge arc 2 Geometric steam inlet angle beta 1 Geometric steam outlet angle beta 2 Wedge angle of leading edgeTrailing edge wedge +.>The 4 Bezier curve shape control parameters are h (1), h (2), h (3) and h (4), and each parameter corresponds to a design variable of a blade profile; n design variables are selected from 12 parameters, the other 12-n parameters are constants, and n is more than or equal to 2 and less than or equal to 12.
3. The multi-condition blade dehumidification optimization design method of the marine wet steam turbine according to claim 2, wherein the method comprises the following steps of: in the first step, the objective function is:
minY 1 (X 1 ，X 2 ，……，X n )
minY 2 (X 1 ，X 2 ，……，X n )
constraint conditions:
l d ·X 1 ≤X 1 ≤X 1 ·l u
l d ·X 2 ≤X 2 ≤X 2 ·l u
……
l d ·X n ≤X n ≤X n ·l u
η 01 ≤η 1 ，η 02 ≤η 2
wherein X is 1 ，X 2 ，……，X n The design variables are 1 st, 2 nd, … … th and n th design variables respectively; y is Y 1 Represents the grid outlet humidity and eta of the working condition 1 The isentropic efficiency of the working condition is represented; y is Y 2 Represents the outlet humidity eta of the blade grid under the second working condition 2 The isentropic efficiency of the second working condition is shown; l (L) d Taking 0.8 to 0.9; l (L) u Taking 1.1 to 1.2; η (eta) 01 Representing isentropic efficiency, eta of prototype cascade during working conditions 02 And the isentropic efficiency of the prototype blade cascade under the second working condition is represented.
4. The multi-working-condition blade dehumidification optimal design method of the marine wet steam turbine according to claim 3, wherein the method comprises the following steps of: the isentropic efficiency is expressed as η= (h in -h out )/(h in -h out,s )，h in For the actual enthalpy value of the cascade inlet, h out For the actual enthalpy value of the blade grid outlet, h out,s Is the isentropic enthalpy of the outlet.
5. The multi-working-condition blade dehumidification optimal design method of the marine wet steam turbine according to claim 1, which is characterized by comprising the following steps of: in the second step, P is not less than 3n.
6. The multi-working-condition blade dehumidification optimal design method of the marine wet steam turbine according to claim 1, which is characterized by comprising the following steps of: in the fourth step, the proxy model ψ is:
Ψ＝f(x k ) T β * +r(x k ) T γ *
β * ＝(F T R -1 F) -1 F T R -1 Y
γ * ＝R -1 (Y-Fβ * )
x k k=1, 2, … …, n, a parameter of the design variable;
wherein f (x) k ) As a row vector, f () is a first order polynomial function; y represents a target value matrix; r (x) k ) Parameter x being a design variable k And sample matrix X P×n A correlation function column vector between samples; parameter x with F matrix element as design variable k The function value f (x) i,k ) I=1, 2, … …, P, k=1, 2, … … n; r represents a correlation function matrix, and the expressions of elements in the matrix are as follows: r (X) i ,X j )＝Π k＝1 exp(-θ k d k 2 )，i＝1,2,……,P，j＝1,2,……,P，k＝1,2,……,n，θ k Is an anisotropic parameter, d k Is the distance between any two points in the sample.
7. The multi-working-condition blade dehumidification optimal design method of the marine wet steam turbine according to claim 1, which is characterized by comprising the following steps of: in the fifth step, M is more than or equal to 3 and less than or equal to P.
8. The multi-working-condition blade dehumidification optimal design method of the marine wet steam turbine according to claim 1, which is characterized by comprising the following steps of: in the fifth step, the calculation accuracy is 2% -3% of the relative error of the two calculation values.
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