Control of the freezing interface morphology in solidification processes in the presence of natural convection

This paper presents a methodology for the solution of an inverse solidification design problem in the presence of natural convection. In particular, the boundary heat flux q₀ in the fixed mold wall, δΩ₀, is calculated such that a desired freezing front velocity and shape are obtained. As the front velocity together with the flux history q_ms on the solid side of the freezing front play a determinant role in the obtained cast structure, the potential applications of the proposed methods to the control of casting processes are enormous. The proposed technique consists of first solving a direct natural convection problem of the liquid phase in an a priori known shrinking cavity, Ω_L(t), before solving an ill-posed inverse design conduction problem in the solid phase in an a priori known growing region, Ω_S(t). The direct convection problem is used to evaluate the flux q_ml in the liquid side of the freezing front. A front tracking deforming finite element technique is employed. The flux q_ml can be used together with the Stefan condition to provide the freezing interface flux q_ms in the solid side of the front. As such, two boundary conditions (flux q_ms and freezing temperature θ_m) are especified along the (known) freezing interface δΩ_I(t). The developed design technique uses the adjoint method to calculate in L₂ the derivative of the cost functional, ∥θ_m – θ( x , t; q₀)∥, that expresses the square error between the calculated temperature θ( x , t; q₀) in the solid phase along δΩ_I(t) and the given melting temperature. The minimization of this cost functional is performed by the conjugate gradient method via the solutions of the direct, sensitivity and adjoint problems. A front tracking finite element technique is employed in this inverse analysis. Finally, an example is presented for the solidification of a superheated incompressible liquid aluminium, where the effects of natural convection in the moving interface shape are controlled with a proper adjustment of the cooling boundary conditions.     

Fracture and fragmentation of simplicial finite element meshes using graphs

An approach for the topological representation of simplicial finite element meshes as graphs is presented. It is shown that by using a graph, the topological changes induced by fracture reduce to a few, local kernel operations. The performance of the graph representation is demonstrated and analyzed, using as reference the three‐dimensional fracture algorithm by Pandolfi and Ortiz (Eng. Comput. 1998; 14 (4):287–308). It is shown that the graph representation initializes in O(N) time and fractures in O(N) time, while the reference implementation requires O(N) time to initialize and O(N) time to fracture, where N_E is the number of elements in the mesh and N_I is the number of interfaces to fracture. Copyright © 2007 John Wiley & Sons, Ltd.     

Fast matrix exponent for deterministic or random excitations

The solution of ż=Az is z(t)=exp(At)z₀=E_tz₀, z₀=z(0). Since z(2t)=E_2tz₀=Ez₀, z(4t)=E_4tz₀=Ez₀, etc., one function evaluation can double the time step. For an n‐degree‐of‐freedoms system, A is a 2n matrix of the nth‐order mass, damping and stiffness matrices M, C and K. If the forcing term is given as piecewise combinations of the elementary functions, the force response can be obtained analytically. The mean‐square response P to a white noise random force with intensity W(t) is governed by the Lyapunov differential equation: Ṗ=AP+PA^T+W. The solution of the homogeneous Lyapunov equation is P(t)=exp(At) P₀ exp(A^Tt), P₀=P(0). One function evaluation can also double the time step. If W(t) is given as piecewise polynomials, the mean‐square response can also be obtained analytically. In fact, exp(At) consists of the impulsive‐ and step‐response functions and requires no special treatment. The method is extended further to coloured noise. In particular, for a linear system initially at rest under white noise excitation, the classical non‐stationary response is resulted immediately without integration. The method is further extended to modulated noise excitations. The method gives analytical mean‐square response matrices for lightly damped or heavily damped systems without using modal expansion. No integration over the frequency is required for the mean‐square response. Four examples are given. The first one shows that the method include the result of Caughy and Stumpf as a particular case. The second one deals with non‐white excitation. The third finds the transient stress intensity factor of a gun barrel and the fourth finds the means‐square response matrix of a simply supported beam by finite element method. Copyright © 2001 John Wiley & Sons, Ltd.     

On the solution of generalized non-linear complex-symmetric eigenvalue problems

This paper brings an attempt toward the systematic solution of the generalized non-linear, complex-symmetric eigenproblem ( K ₀−iω C ₁−ω² M ₁−iω³ C ₂−ω⁴ M ₂−···) ϕ = 0 , with real, symmetric matrices K ₀, C _j, M _j ε R^n×n, which are associated with the dynamic governing equations of a structure submitted to viscous damping, as laid out in the frame of an advanced mode superposition technique. The problem can be restated as ( K _(ω)−ω M _(ω)) ϕ = 0 , where K _(ω)= K and M _(ω)= M are complex-symmetric matrices given as power series of the complex eigenfrequencies ω, such that, if (ω, ϕ ) is a solution eigenpair, ϕ ^T M _(ω) ϕ =1 and ϕ ^T K _(ω) ϕ =ω. The traditional Rayleigh quotient iteration and the more recent Jacobi–Davidson method are outlined for complex-symmetric linear problems and shown to be mathematically equivalent, both with asymptotically cubic convergence. The Jacobi–Davidson method is more robust and adequate for the solution of a set of eigenpairs. The non-linear eigenproblem subject of this paper can be dealt with in the exact frame of the linear analysis, thus also presenting cubic convergence. Two examples help us to visualize some of the basic concepts developed. Three more examples illustrate the applicability of the proposed algorithm to solve non-linear problems, in the general case of underdamping, but also for overdamping combined with multiple and close eigenvalues. Copyright © 2007 John Wiley & Sons, Ltd.     

Effect of casting/mould interfacial heat transfer during solidification of aluminium alloys cast in CO2-sand mould

The ability of heat to flow across the casting and through the interface from the casting to the mold directly affects the evolution of solidification and plays a notable role in determining the freezing conditions within the casting, mainly in foundry systems of high thermal diffusivity such as chill castings. An experimental procedure has been utilized to measure the formation process of an interfacial gap and metal-mould interfacial movement during solidification of hollow cylindrical castings of Al-4.5 % Cu alloy cast in CO₂-sand mould. Heat flow between the casting and the mould during solidification of Al-4.5 % Cu alloy in CO₂-sand mould was assessed using an inverse modeling technique. The analysis yielded the interfacial heat flux (q), heat transfer coefficient (h) and the surface temperatures of the casting and the mould during solidification of the casting. The peak heat flux was incorporated as a dimensionless number and modeled as a function of the thermal diffusivities of the casting and the mould materials. Heat flux transients were normalized with respect to the peak heat flux and modeled as a function of time. The heat flux model proposed was to estimate the heat flux transients during solidification of Al-4.5 % Cu alloy cast in CO₂-sand moulds.     

Simulation of thermophysical processes occurring in transition of the surface metal layer of a moving strong-current gas discharge electrode to an amorphous state

We carried out a numerical simulation of the melting and solidification of a surface layer for various metals and heating parameters with account for phase transitions. We determined the conditions enabling the formation of amorphous metallic structures. The conditions were shown to be realized when the heat source was a steady-state strong-current gas-discharge plasma.Notation W ₀ rate of cooling - q heat flux density - t _ef time of thermal effect - a thermal diffusivity of metal - t andx time and space coordinates - T _mel andL melting temperature and latent heat of the melting of metal - s function of the depth of melting - H thermal function (enthalpy) - T temperature - density - c specific heat - , ,B, ,h dimensionless quantities corresponding tot, x, L, T, H - dimensionless parameter having the meaning of the volumetric fraction of the melt - T _max maximum surface temperature - X _mel depth of melting - t ₀ time of surface cooling Insitute of Molecular and Atomic Physics, Academy of Sciences of Belarus, Minsk. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 68, No. 5, pp. 834–839, September–October, 1995.     

Thermal conductivity of a chemically reacting ternary gas mixture

On the basis of the thermodynamics of irreversible processes, an expression is derived for the thermal conductivity of a chemically reacting gas mixture.Notation x_i molar fraction of i-th component - J_u energy flux - c_i mass fraction of i-th component - J_q heat flux - J_i mass flux of i-th component - T absolute temperature - _i molar chemical potential of i-th component - L_ik, L_uu, L_ui phenomenological coefficients, associated respectively with the transfer of mass and energy and with the superimposed effects - X_u and X_k thermodynamic forces associated with the mass and energy fluxes - h_i enthalpy of i-th component - S entropy of i-th component - D_ij diffusion coefficient of a multicomponent mixture - D _i ^T thermodiffusion coefficient of i-th component in a multicomponent system - ₀(O, p) chemical potential in standard state - density - h number density of mixture particles - h_j number density of particles of i-th component - m_i molecular weigh of i-th component - p pressure of mixture - v_i specific volume of i-th component - H heat of reaction - r number of component of mixture - R gas constant - d_t thermodiffusion coefficient of a binary mixture     

Heat Transfer of Heat-Releasing Fluid in the Top Portion of a Closed Volume

The method of analytic estimates is used to determine the characteristics of steady-state free-convection heat transfer of a fluid with internal heat sources in the top part of a closed volume with different conditions of heat removal on the top horizontal boundary at the Prandtl number value on the order of unity. It is demonstrated that, in the case of adiabatic condition on the top boundary of the volume, the maximal heat flux q _max attained in the region of intersection of the top horizontal and vertical boundaries depends only on the maximal temperature in the volume T _max and on the thermal characteristics of the fluid. The correction to the bulk temperature (outside of the boundary layers) T _b z ^1/4, which is a function of the vertical coordinate z, significantly prevails over perturbations in the horizontal section. When the turbulent Rayleigh-Benard (RB) convection arises, the heat removal through the top boundary is defined only by the energy release in the RB-layer. Given a fixed power of heat release Q, the RB-layer thickness increases by the linear law h=q/Q with increasing heat flux q through the top horizontal boundary.     

An iterative defect‐correction type meshless method for acoustics

Accurate numerical simulation of acoustic wave propagation is still an open problem, particularly for medium frequencies. We have thus formulated a new numerical method better suited to the acoustical problem: the element‐free Galerkin method (EFGM) improved by appropriate basis functions computed by a defect correction approach. One of the EFGM advantages is that the shape functions are customizable. Indeed, we can construct the basis of the approximation with terms that are suited to the problem which has to be solved. Acoustical problems, in cavities Ω with boundary T, are governed by the Helmholtz equation completed with appropriate boundary conditions. As the pressure p(x,y) is a complex variable, it can always be expressed as a function of cosθ(x,y) and sinθ(x,y) where θ(x,y) is the phase of the wave in each point (x,y). If the exact distribution θ(x,y) of the phase is known and if a meshless basis {1, cosθ(x,y), sinθ (x,y) } is used, then the exact solution of the acoustic problem can be obtained. Obviously, in real‐life cases, the distribution of the phase is unknown. The aim of our work is to resolve, as a first step, the acoustic problem by using a polynomial basis to obtain a first approximation of the pressure field p(x,y). As a second step, from p(x,y) we compute the distribution of the phase θ(x,y) and we introduce it in the meshless basis in order to compute a second approximated pressure field p(x,y). From p(x,y), a new distribution of the phase is computed in order to obtain a third approximated pressure field and so on until a convergence criterion, concerning the pressure or the phase, is obtained. So, an iterative defect‐correction type meshless method has been developed to compute the pressure field in Ω. This work will show the efficiency of this meshless method in terms of accuracy and in terms of computational time. We will also compare the performance of this method with the classical finite element method. Copyright © 2003 John Wiley & Sons, Ltd.     

Simple time and space adaptation in one-dimensional evolutionary partial differential equations

A simple technique for time and space adaptation in one-dimensional evolutionary partial differential equations, suggested by Sanz-Serna and Christie,¹ is tested. It is found that the equidistribution of hu″(x_i) greatly improves on the equidistribution of arc-length used by those authors. The time-step control is found to perform poorly in the integration of rough solutions and the reasons for this behaviour are analysed.     

Bandwidth reduction by automatic renumbering

Consider a solid heat conductor with a non-linear constitutive equation for the heat flux. If the material is anisotropic and inhomogeneous, the heat conduction equation to be satisfied by the temperature field θ(x, t) is, Here L (θ, x ) [grad θ] is a vector-valued function of θ, x , grad θ which is linear in grad θ, In the present paper, the application of the finite element method to the solution of this class of problems is demonstrated. General discrete models are developed which enable approximate solutions to be obtained for arbitrary three-dimensional regions and the following boundary and initial conditions: (a) prescribed surface temperature, (b) prescribed heat flux at the surface and (c) linear heat transfer at the surface. Numerical examples involve a homogeneous solid with a dimensionless temperature-diffusivity curve of the form κ = κ₀(l + σT). The resulting system of non-linear differential equations is integrated numerically.     

Recent advances in the numerical analysis of heat pipes

Numerical simulation of heat pipes has progressed significantly in recent years. The state-of-the-art has been advanced in steady state, continuum transient, and frozen startup simulation for high, moderate, and low temperature heat pipes of conventional cylindrical and nonconventional geometries such as wing leading edges and spacecraft nosecaps. This review summarizes these advancements and discusses the important results.List of symbols A cross-sectional area of the vapor channel, m² - c specific heat, J/(kg-K) - C _p specific heat at constant pressure, J/(kg-K) - C _v specific heat at constant volume, J/(kg-K) - D vapor space diameter, m - D _v coefficient of self-diffusion, m²/s - G vapor mass flux, kg/(m²-s) - h convective heat transfer coefficient, W/(m²-K) - h _fg latent heat of evaporation, J/kg - H latent heat due to melting or freezing, J/kg - k thermal conductivity, W/(m-K) - Kn Knudsen number, /D - L total length of the heat pipe, m - L _a length of the adiabatic section, m - L _c length of the condenser, m - M molecular weight, kg/kmol - Ma Mach number, - m _i mass flux at the liquid-vapor interface, kg/(m²-s) - P pressure, N/m² - q heat flux, W/m² - Q heat input at the active evaporator, W - Q _o heat output at the condenser, W - r radial coordinate, m - R gas constant, J/(kg-K) - R _o outer pipe wall radius, m - R _u universal gas constant, J/(kmol-K) - R _v vapor space radius, m - t time, s - T temperature, K - T _i,c interfacial temperature on the continuum vapor flow side, K - T _i,r interfacial temperature on the rarefied vapor flow side, K - T _rf reference (saturation) temperature, K - T _tr transition vapor temperature, K - v radial velocity, m/s Funding for this work was provided by a joint effort of the NASA Lewis Research Center and the Thermal Technology Center of Wright Laboratory under contract No. F33615-88-C-2820     

Surface wavelets: a multiresolution signal processing tool for 3D computational modelling

In this paper, we provide an introduction to wavelet representations for complex surfaces (surface wavelets), with the goal of demonstrating their potential for 3D scientific and engineering computing applications. Surface wavelets were originally developed for representing geometric objects in a multiresolution format in computer graphics. These wavelets share all of the major advantages of conventional wavelets, in that they provide an analysis tool for studying data, functions and operators at different scales. However, unlike conventional wavelets, which are restricted to uniform grids, surface wavelets have the power to perform signal processing operations on complex meshes, such as those encountered in finite element modelling. This motivates the study of surface wavelets as an efficient representation for the modelling and simulation of physical processes. We show how surface wavelets can be applied to partial differential equations, stated either in integral form or in differential form. We analyse and implement the wavelet approach for a model 3D potential problem using a surface wavelet basis with linear interpolating properties. We show both theoretically and experimentally that an O(h) convergence rate, h_n being the mesh size, can be obtained by retaining only O((logN) ^7/2N) entries in the discrete operator matrix, where N is the number of unknowns. The principles described here may also be extended to volumetric discretizations. Copyright © 2001 John Wiley & Sons, Ltd.     

An adjoint method for the inverse design of solidification processes with natural convection

This paper presents a finite element algorithm based on the adjoint method for the design of a certain class of solidification processes. In particular, the paper addresses the design of directional solidification processes for pure materials such that a desired freezing front heat flux and growth velocity are achieved. This is the first time that an infinite-dimensional continuum adjoint formulation is obtained and implemented for the solution of such inverse/design problems with moving boundaries and Boussinesq incompressible flow. The present design problem belongs to a category of inverse problems in which one is looking for the unknown conditions in part of the boundary, while overspecified boundary conditions are supplied in another part of the boundary (here the freezing interface). The solidification design problem is mathematically posed as a whole time-domain optimization problem. The gradient of the cost functional is calculated using the solution of an appropriately defined continuous adjoint problem. The minimization process is realized by the conjugate gradient method via the solutions of the direct, adjoint and sensitivity sub-problems. The proposed methodology is demonstrated with the solidification of an initially superheated liquid aluminum confined in a square mold. The non-uniformity in the casting product in the direction of gravity due to the existence of natural convection in the melt is emphasized. The inverse design problem is then posed as finding the appropriate spatial-temporal variations of the boundary heat flux on the vertical mold walls that can eliminate or reduce the effects of convection on the freezing interface heat fluxes and growth velocity. The numerical example demonstrates the accuracy and convergence of the adjoint formulation. Finally, open related research design problems are discussed. © 1998 John Wiley & Sons, Ltd.     

An inverse problem of simultaneously estimating contact conductance and heat transfer coefficient of exhaust gases between engine's exhaust valve and seat

The conjugate gradient method using two search step sizes is used to solve the inverse problem of simultaneously estimating the periodic thermal contact conductance, h_c(t), and the heat transfer coefficient of the exhaust gases, h_g(t), between the exhaust valve and seat in an internal combustion engine. The importance of the determination of h_c(t) and h_g(t) lie in that they are the critical factors for designing the cooling system and the insulation of the exhaust valve. The inverse analysis is based on the temperature measurements taken from the sensors placed in both the valve and seat regions during the transient process of operation. In this study two unknown timewise-varying functions h_c(t) and h_g(t) are to be estimated at the same time, thus two search step sizes with each one corresponding to each unknown function are derived. The results show that the CPU time for the inverse solutions using two search step sizes are greatly reduced than using just one search step size¹ for the determination of two unknowns, besides, it also shows that the inverse solutions are reliable even when the measurement errors are considered. The advantage of the conjugate gradient method is that no a priori information is needed on the variation of the unknown quantities, since the solution automatically determines the functional form over the domain specified. The successful development of the present technique can be applied to any kind of two-dimensional periodic contact problems, such as the determination of a two-dimensional contact conductance problem² and the temperature or heat flux behaviour on the inside wall of internal combustion engines³.     

Stability of the interphase surface in the freezing of moist ground

It is suggested that the formation of ice layers should be regarded as a consequence of a loss of stability of the motion of the freezing front. The kinetics of the freezing process is investigated and a stability criterion is obtained.Notation s(t) coordinate of the moving front - L length of the specimen - k moisture conductivity - W moisture content - heat of phase transition - W_H amount of unfrozen water - q flow of moisture from the melted zone into the frozen zone - v velocity of motion of the front - T_i temperature - Q_i heat flux - _i thermal conductivity - a_i thermal diffusivity (i=1 is the frozen zone and i=2 is the melted zone) - mass transfer coefficient - T_H the initial temperature Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 39, No. 1, pp. 96–101, July, 1980.     

Analytical technique for the determination of solidification rates during the inward freezing of cylinders

An analytical/experimental approach which permits the determination of solidification rates during the inward solidification of cylinders is proposed. The technique is based on a previous analytical solution that treats the generalized problem of solidification of slabs. This solution is modified by a geometric correlation to compensate for the cylindrical geometry. A number of experiments have been carried out with a special experimental set-up, designed to simulate the inward solidification of cylinders in a water-cooled mould. A series of comparisons of experimental results, numerical predictions and calculations furnished by the proposed technique were made, showing good agreement for any case examined.Nomenclature a _s Thermal diffusivity of solid metal = k _s/c _s d _s (m² sec^–1) - A _i Internal surface area of the mould (m²) - b _s Heat diffusivity of solid metal = (k _s c _s d _s ^1/2(J m^–2 sec^–1/2 K^–1) - c _s Specific heat of solid metal (J kg^–1 K^–1) - d _s Density of solid metal (kg m^–3) - h Newtonian heat transfer coefficien (W m^–2 K^–1) - H Latent heat of fusion (J kg^–1) - k _s Thermal conductivity of solid metal (W m^–1 K^–1) - q Heat flux (W m^–2) - r Radial position (m) - r _o Radius of cylinder (m) - r _f Radius of solid/liquid interface (m) - S Thickness of solidified metal (m) - S _o Thickness of metal side adjunct (m) - t Solidification time (sec) - T Temperature (K) - T _i Surface temperature (K) - T _f Freezing temperature of metal (K) - T _o Temperature of the coolant (K) - T _s Temperature at any point in the solidified metal (K) - V ₁ Volume of remaining liquid metal during the solidification (m³) - V _s Volume of solidified metal (m³) - V _T Total volume of metal in the mould (m³) - x Distance from metal/mould interface (m) - Dimensionless solidification constant.     

Heat exchange with different schemes for introducing the heat-exchange agent into a furnace

The effect of the scheme for introducing the heat-exchange agent into a furnace for heat exchange in a selective nonisothermal gaseous medium is investigated taking into account the heat loss due to convection depending on the emissivity of the metal and the furnace lining.Notation T temperature - Er specific resulting radiative flux - q_c convective heat flux density - q_Me overall density of the resulting heat flux to the metal - emissivity - a_g0 absorbing capability of the gas relative to the radiation of the lining - a_g4 same for the metal - a _go /a+b+c absorbing ability of a layer of gas a+b+c... relative to the radiation of the lining - a_gj/ikmnp absorbing ability of the gas layer i+k + m + n + p with radiation from layer j - A emissivity of the walls - l _ef effective length of a ray path - coefficient of convective heat transfer - k heat-exchange coefficient from the internal surface of the lining with temperature T_o in the medium with temperature T_m - B fuel expenditure - Q_H ^P heat of combustion - v volume of the combustion fuel products per 1 m³ of gas - c_g specific heat capacity of the gas - h overall height of the channel - x instantaneous height coordinate in the channel The indices for the temperatures and heat fluxes are as follows: 0, lining; 4, metal; 1, 2, and 3, gas zones; m, media.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 39, No. 4, pp. 692–698, October, 1980.     

Heat and mass transfer of a multiatomic gas in a cylindrical capillary with arbitrary Knudsen numbers

Processes of heat and mass transfer of a multiatomic gas in a cylindrical channel of circular cross section with arbitrary Knudsen numbers are considered on the basis of a model kinetic equation, taking account of the excitation of rotational and vibrational degrees of freedom of the molecules.Notation Kn Knudsen number - f, f^tr total and translational Eucken factors - R_o capillary radius - m molecular mass - k Boltzmann's constant - n, T numerical density and temperature of gas - v_i i-th component of the molecular velocity - h_ij perturbation function - E_i ^(r), e_j ^(v) energy of the i-th rotational and j-th vibrational levels - E_o ^(r), E_o ^(v) equilibrium values of the rotational and vibrational energy - P_i ^(r), P_i ^(v) probability of rotational and vibrational states of energy E _i ^r and E _j ^v - , logarithmic pressure and temperature gradients - T_o mean gas temperature - R rarefaction parameter of gas - C _V ^r , C _V ^v contributions of rotational and vibrational degrees of freedom of the molecule to the specific heat at constant volume - U macroscopic gas velocity - q^(t), q^(r), q^(v) components of the heat flux density due to translational, rotational, and vibrational degrees of freedom of the molecules - P, pressure and dynamic viscosity of the gas - l free path length of molecules - up velocity of Poiseuille flow - u_T rate of thermal creep - cross-sectional area of capillary - I_n, I_q numerical and heat fluxes averaged over the channel cross section - universal index characterizing the thermomolecular pressure difference - ^t, ^r, ^v thermal conductivities due to translational, rotational, and vibrational degrees of freedom of the molecules - mass density of the gas - D_rr, D_vv diffusion coefficients of rotationally and vibrationally excited molecules among the unexcited molecules - Z_r rotational collisional number Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 47, No. 1, pp. 71–82, July, 1984.     

Capability assessment for processes with multiple characteristics: A generalization of the popular index Cpk

Process capability index C_pk is the most popular capability index widely used in the manufacturing industry. Existing research on the yield‐based measure index C_pk to date is restricted to processes with single characteristics. However, many manufacturing processes are commonly described with multiple characteristics, for example, the gold bumping process in the TFT‐LCD (thin film transistor‐liquid crystal display) manufacturing industry. In the gold bumping process, gold bumps have multiple characteristics all having effects on the process yield. Obtaining accurate gold bumping manufacturing yield is very important for quality assurance and in providing guidance toward process improvement. To obtain accurate yield assessment for processes with multiple characteristics, we propose a new overall yield‐measure index C, which is a generalization of the index C_pk, and a natural estimator of C. For the purpose of making inferences on the process capability, we derive a quite accurate approximation of the distribution of since the distribution is analytically intractable. With this distribution, we tabulate the lower confidence bounds of the new index under various sample sizes for in‐plant applications. In addition, we construct a statistical test on the new yield‐measure index in order to examine whether the yield meets the customers' requirements. For illustration purpose, a real case in a gold bumping factory located in the Science‐based Industrial Park at Hsinchu, Taiwan is presented. Copyright © 2011 John Wiley & Sons, Ltd.