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  • Electronic Resource  (448)
  • 1970-1974  (448)
  • 1971  (448)
  • Chemical Engineering  (386)
  • Engineering General  (62)
  • 1
    Electronic Resource
    Electronic Resource
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 3 (1971), S. 63-87 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: The finite element method is applied to the stability analysis of structural systems subject to non-conservative forces. The development of the method is general, but the specific application considered here is the stability of thin-walled members subject to follower forces. The method predicts the type of instability, whether it be buckling or flutter. Example problems, for which exact solutions are known, illustrate the accuracy and convergence characteristic of the finite element formulation.
    Additional Material: 10 Ill.
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  • 2
    Electronic Resource
    Electronic Resource
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 3 (1971), S. 103-117 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: The application of the finite element rank force method for the dynamic analysis of redundant structures is presented. The applied loading consists of discrete and distributed loads. Both forced and free vibrations are considered, the latter yielding latent vectors corresponding to dynamic redundancies.It is shown that the dynamic properties of an element are given by a static flexibility matrix, an inverse mass matrix, a damping parameter and a displacement vector representing the effect of the applied distributed loading.
    Additional Material: 5 Ill.
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  • 3
    Electronic Resource
    Electronic Resource
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 3 (1971), S. 119-129 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: A stiffness matrix for a finite element having the planform of an annular segment is derived using the displacement approach. Numerical problems involved in the derivation are discussed and rapid convergence to exact solutions is demonstrated on three sample problems. It is concluded that the new element will be of great value to engineers concerned with the analysis of slabs of bridge decks curved in plan.
    Additional Material: 8 Ill.
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  • 4
    Electronic Resource
    Electronic Resource
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 3 (1971), S. 89-101 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: The point matching numerical method and its generalization, the method of boundary point least squares, have been successfully applied to numerous boundary value and eigenvalue problems. The present paper demonstrates the application of these techniques to problems in the micromechanics of fibrous composite materials, i.e. determination of elastic moduli and stress concentrations for parallel-fibre materials which are loaded transversely with respect to the fibres. The solution technique utilizes exact solutions of the governing equations of plane elasticity for each component fibre and its surrounding matrix material in a typical repeating section of the composite material. The continuity conditions for stresses and displacements between fibre and matrix and the repeatability conditions at the boundary of the repeating section are satisfied approximately in a pointwise manner. Some special numerical techniques which were found to be particularly useful in applying the point matching method to these problems are delineated. The method is demonstrated for composite materials having circular, elliptical and square fibres in regular, staggered arrays. Numerical results are given which show the accuracy of the method as well as stress concentration and composite elastic moduli data.
    Additional Material: 4 Ill.
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  • 5
    Electronic Resource
    Electronic Resource
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 3 (1971), S. 131-144 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: This paper discusses the mathematical foundations of a technique that has been used extensively in structural optimization.1-6 Two basic problems are considered. The first of these is the concave programming problem which consists of finding the global minimum of ‘piece-wise concave functions’ on ‘piece-wise concave sets’. Since any function can be approximated by a piece-wise concave function, this method could in principle be used to find the global minimum in non-convex optimization problems. The second one is the piece-wise linear programming problem in which the objective function is convex and piece-wise linear. The iterative method outlined for handling this problem is shown to be much more efficient than the standard simplex method of linear programming.
    Additional Material: 13 Ill.
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  • 6
    Electronic Resource
    Electronic Resource
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 3 (1971), S. 149-149 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Type of Medium: Electronic Resource
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  • 7
    Electronic Resource
    Electronic Resource
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 3 (1971), S. 154-154 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Type of Medium: Electronic Resource
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  • 8
    Electronic Resource
    Electronic Resource
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 3 (1971), S. 275-290 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: The solution of plate and shell problems by an independent specification of slopes and middle surface displacements is attractive due to its simplicity and ability of reproducing shear deformation. Unfortunately elements of this type are much too stiff when thickness is reduced.In an earlier paper a derivation of such an element was presented1 which proved very successful in ‘thick’ situations. Here a very simple extension is made which allows the element to be economically used in all situations.The improved flexibility is achieved simply by reducing the order of numerical integration applied to certain terms without sacrificing convergence properties. The process is of very wide applicability in improvement of element properties.
    Additional Material: 10 Ill.
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  • 9
    Electronic Resource
    Electronic Resource
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 3 (1971) 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Type of Medium: Electronic Resource
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  • 10
    Electronic Resource
    Electronic Resource
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 3 (1971), S. 295-296 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Type of Medium: Electronic Resource
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  • 11
    Electronic Resource
    Electronic Resource
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 3 (1971), S. 299-315 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: A system of finite element shell analysis codes, called SABOR/DRASTIC, is used to analyse a complex two-layered shell of revolution under static and dynamic asymmetric loads. The dynamic analysis is compared with experimentally measured response. In this linear elastic analysis, emphasis is placed on the inherent flexibility of the finite element method in modelling the complex structural geometry of a given test specimen. Static studies, which involve variations in important shell parameters, and dynamic studies, which provide a successful correlation with experiment, are used to illustrate both the detail and the generality with which shell analyses may now be performed with confidence.
    Additional Material: 11 Ill.
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  • 12
    Electronic Resource
    Electronic Resource
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 3 (1971), S. 317-325 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: The theory of conjugate approximations1 is used to obtain consistent approximations of stress fields in finite element approximations based on displacement assumptions. These consistent stresses are continuous across interelement boundaries and involve less mean error than those computed by the conventional approach.
    Additional Material: 1 Ill.
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  • 13
    Electronic Resource
    Electronic Resource
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 3 (1971), S. 327-347 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: Many of the known finite elements are based on a variational procedure in a functional basis of monomials. The present paper attempts to formulate completely a finite element procedure which is non-variational on a functional basis which is left to the choice of the user.
    Additional Material: 5 Ill.
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  • 14
    Electronic Resource
    Electronic Resource
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 3 (1971), S. 443-444 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Type of Medium: Electronic Resource
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  • 15
    Electronic Resource
    Electronic Resource
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 3 (1971), S. 425-441 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: A unified network-topological approach to the formulation and solution of the analysis and minimum weight design of rigid-plastic structural systems is presented. It is shown that both analysis and design can be formulated using network concepts, but that, in contrast to elastic networks, the formulation results in a mathematical programming problem. Formulations based both on the static and kinematic theorems of plasticity are presented, and it is shown that duality guarantees that the two types of formulation yield identical results. The problem is first formulated on the basis of a simple definition of yielding, and is then generalized to the case where yielding is defined in terms of a yield surface.
    Additional Material: 4 Ill.
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  • 16
    Electronic Resource
    Electronic Resource
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 3 (1971), S. 463-483 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: Dynamic relaxation, an iterative method for use with digital computers, is described and is shown to be suitable for the solution of a system of linear equations and in particular for such problems derived from structural frame analysis. It is further shown that the method may be modified to include non-linear equations relating to these problems.Some specific examples of linear and non-linear solutions are given and comparisons are made with another computer method which performs the same tasks.
    Additional Material: 12 Ill.
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  • 17
    Electronic Resource
    Electronic Resource
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 3 (1971), S. 529-552 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: The discrete element displacement method is used to analyse the finite-deflection behaviour of shallow arches. The arches are idealized as assemblies of shallow curved elements and the necessary properties of the displacement patterns in these elements are discussed. Numerical solutions of the geometrically non-linear problem are obtained by directly minimizing the total potential energy of the system.
    Additional Material: 9 Ill.
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  • 18
    Electronic Resource
    Electronic Resource
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 3 (1971), S. 565-574 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: Finite element procedures usually require more degrees of freedom for a specified accuracy than does a classical Ritz procedure if suitable coordinate functions are available. This paper develops a combined global and local dependent variable representation which couples the conventional and finite element Ritz methods. This hybrid method preserves much of the flexibility of the finite element method while increasing the solution accuracy for a specified system order. The method is illustrated by examination of a beam and a plate vibration problem.
    Additional Material: 5 Ill.
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  • 19
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    Electronic Resource
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 3 (1971), S. 553-563 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: For a hierarchy of polynomials on the triangle there is derived an algorithm for computing the stiffness matrix of the plate bending element. The algorithm is easy to program and means a considerable saving of the computing time. The same approach can be used for any elliptic equation with constant coefficients.
    Additional Material: 1 Ill.
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  • 20
    Electronic Resource
    Electronic Resource
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 3 (1971), S. 575-586 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: A quadratic thick shell element derived from a three-dimensional isoparametric element was first introduced by Ahmad and co-workers in 1968. This element was noted, however, to be relatively inefficient in representing bending deformations in thin shell or thin plate applications. The present paper outlines a selective integration scheme for evaluating the stiffness matrix of the element, in which each component of the strain energy is evaluated separately using a different Gaussian integration grid for each contribution. By this procedure, the excessive bending stiffness of the element, which results from the use of me quadratic interpolation functions, is avoided.The improved performance of this element, as compared with the original thick shell element, is demonstrated by analyses of a variety of thin and thick shell problems. Editors' note: A similar development was outlined by O. C. Zienkiewicz and co-workers in lnt. J. num. Meth. Engng, 3, 275-290 (1971). Some important details differ between the two papers which are thus complementary.
    Additional Material: 14 Ill.
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  • 21
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    Electronic Resource
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 3 (1971), S. 591-592 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Type of Medium: Electronic Resource
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  • 22
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    Electronic Resource
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 3 (1971) 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Type of Medium: Electronic Resource
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  • 23
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    Electronic Resource
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 3 (1971), S. 1-2 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
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  • 24
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    Electronic Resource
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 3 (1971), S. 3-3 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
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  • 25
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    Electronic Resource
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 3 (1971), S. 155-160 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: The conventional central finite difference equations for the plane stress extension of flat plates are derived as a localized Ritz process. A dual differential-variational discretization of this type enables common classification of the finite difference and finite element methods. Also, it provides alternative methods of establishing sufficiency conditions and relative rates of convergence for discrete systems derived from a localized Ritz process, and the existence of solution bounds for discrete systems derived using difference procedures.
    Additional Material: 7 Ill.
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  • 26
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    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 3 (1971), S. 161-179 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: This paper is concerned with the development of consistent discrete models, via the concept of finite elements, of linear and non-linear electrothermomechanical behaviour of continuous bodies. In the development, general energy balances are utilized to derive equations governing electromagnetic fields over an element and coupled equations of motion and heat conduction of a typical element of the continuum. These equations make it possible to study a general class of field problems involving arbitrary geometries and boundary conditions. Sample problems are included.
    Additional Material: 9 Ill.
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  • 27
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    Electronic Resource
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 3 (1971), S. 181-198 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: The dynamic finite element technique, which is referred to in the literature as ‘computer-analysis’, is applied to wave propagation problems occurring in finite and semi-infinite linearly elastic membranes of revolution. Both semi-infinite and finite versions of cylindrical and conical membranes are considered, and a finite membrane having a meridional curve which is parabolic is solved. The source of excitation is generally a constant velocity motion of one end of the membranes, but the results for a stress-pulse input at one end of a semi-infinite cylindrical membrane are also given. The results for the finite membranes are new, and the results for the semi-infinite problems are discussed with respect to previously published results. The two-dimensional state of stress in the membrane requires careful ordering of the calculations, and the boundary conditions for finite membranes are shown to follow logically from this ordering of the calculations. The difference in the solutions resulting from prescribing an axial or a tangential velocity excitation at the end of a conical membrane is presented, and the mesh size necessary for convergence to the solution is indicated. The graphs of the results clearly indicate regions in space and time where the membrane model should be replaced by the shell formulation to represent a realistic structure. The technique is shown to be self-contained and independent of any formal method such as the method of characteristics.
    Additional Material: 18 Ill.
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  • 28
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    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 3 (1971), S. 199-213 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: The numerical solution of the biharmonic equation in a rectangular domain is presented in the context of continuous dynamic programming techniques. The equations are specialized to the solution of elastic rectangular plates. A suitable approximate expression of a certain functional equation containing derivatives only in one direction is used to derive equations for the stiffness and flexibility matrices of the plate. It is shown that those matrices satisfy matrix Riccati equations subject to suitable initial conditions. It is also shown that the condition of optimality in the Hamilton-Jacobi-Bellman equation directly expresses a classical variational principle, i.e. the principle of complementary energy. Some numerical examples are finally presented.
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  • 29
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    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 3 (1971), S. 215-232 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: The paper presents an analysis of the response of plastic spherical and cylindrical shells subjected to loads applied through a rigid boss. Special emphasis is placed on following the load-deflection behaviour of these structures after the application of the yield-point load in order to predict the appearance of any snapping action.The method of analysis assumes that upper bound calculations are applicable and that the associated velocity fields can be represented by a series of piecewise continuous polynomials. The internal dissipation of energy for the structures are then written in terms of the coefficients of these polynomials and the values appropriate to the solution are found by using a non-linear programming routine.The results of this analysis are correlated with a non-dimensional boss size parameter p and compared with the results of experimental investigations. Mention is also made of the possible effects of initial imperfections.
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  • 30
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    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 3 (1971), S. 261-273 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: A study of bending of plates for small deformations by applying the finite element method is presented in this paper.A new class of displacement functions has been developed for rectangular bending elements. For the first time a method is proposed for the derivation of displacement functions in which trigonometric expressions that had apparently failed to yield satisfactory results in earlier attempts can be used along with polynomial terms.The derivation leads to conforming type of displacement functions which satisfy the convergence criterion. The approach also demonstrates how many more new displacement functions can be found.
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  • 31
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    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 3 (1971), S. 249-259 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: This paper describes a search procedure for finding the unconstrained maximum or minimum of a function of many independent variables. If this function represents the volume, cost, stiffness, etc. of a structure and each of the independent variables is associated with a parameter governing the geometry of the structure then designs of optimum geometry may be made in certain cases with the assistance of this search procedure. The procedure consists of steepest gradient calculations used in conjunction with a reverse Fibonacci location process. A ridge-following technique is included to speed convergence in addition to localized exploration in the region of an optimum.The search procedure has been programmed for computer use and an outline of its structural design applications is presented together with an example of its efficiency in a specific case.
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  • 32
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    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 3 (1971), S. 293-294 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
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  • 33
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    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 3 (1971), S. 291-292 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
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  • 34
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    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 3 (1971), S. 446-446 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
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  • 35
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    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 3 (1971), S. 444-445 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
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  • 36
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    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 3 (1971), S. 447-448 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
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  • 37
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    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 3 (1971) 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
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  • 38
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    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 3 (1971), S. 451-462 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: Using the finite element method, a detailed study of the deformations and stresses produced in an elastic-perfectly plastic half-space indented by a rigid sphere was done. The analysis covers the transition region from the maximum elastic contact load to a state where this load has been increased one hundredfold. Experimental results available in the literature are in good agreement with the analysis. In solving repeatedly the large number of linear equations involved in the solution of the problem, it was found profitable, in order to save computer time, to modify the direct elimination method. This technique is described in some details in the paper.
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  • 39
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    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 3 (1971), S. 485-493 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: The convergence of nodal displacements obtained by the finite element method is investigated for special element patterns in plane elasticity with the aid of the differential equation derived from the equilibrium condition at each node. The convergence of strains is also discussed. It is shown that averaging is an effective tool for obtaining approximations to strain for some kinds of mesh arrangement. It is also shown that the approach becomes inaccurate along force boundaries.
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  • 40
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    International Journal for Numerical Methods in Engineering 3 (1971), S. 495-508 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: A refined axisymmetric curved finite element for the analysis of thin elastic-plastic shells of revolution is described in the paper. The improved element is obtained by employing cubic polynomials in terms of local Cartesian co-ordinates for the assumed in-plane and out-of-plane displacements. This introduces into the solution two internal degrees of freedom in the cord direction of each element. These internal degrees of freedom are removed by static condensation before assembling the individual element stiffness matrices, and are subsequently recovered after the nodal displacements are obtained. On comparison with the previous formulation, this procedure greatly improves the accuracy of the solution especially with regards to in-plane stress-resultants at discontinuities in the meridional curvature and interelement equilibrium of forces. The latter fact makes it possible to analyse shells with a discontinuous meridional slope. In using this element, improvement in the convergence of the elastic-plastic solutions has also been observed.Several examples illustrate the quality of solutions. The reported study is limited to axisymmetric loadings cum boundary conditions.
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  • 41
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    International Journal for Numerical Methods in Engineering 3 (1971), S. 509-517 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: This paper considers methods of solution of the relevant equations based on the Newton-Raphson process. Expressions are derived for calculation with both good and very poor initial guesses and for calculation of pressure, volume and temperature during an equilibrium process. A method of programming the calculations is outlined. Variations and modifications to suit specific situations are discussed.
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  • 42
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    International Journal for Numerical Methods in Engineering 3 (1971), S. 519-528 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: A computer orientated method is presented which generates meshes of triangular elements in plane and curved surfaces. Depending on geometrical and material variations, the region to be discretized is divided into a number of four sided zones. By using curvi-linear co-ordinate systems, nodes within and on the boundary of each zone are automatically positioned and referenced to a global Cartesian co-ordinate system. Elements are automatically assembled from these nodes. Input data is required to specify the positions and material properties of each zone and how they are connected. Examples are given which illustrate the range of meshes that can be generated. Extension of the method to generate three-dimensional tetrahedral elements is indicated.
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  • 43
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    International Journal for Numerical Methods in Engineering 3 (1971), S. 587-588 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
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  • 44
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    International Journal for Numerical Methods in Engineering 3 (1971), S. 591-591 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
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  • 45
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    International Journal for Numerical Methods in Engineering 3 (1971), S. 588-591 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
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  • 46
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    International Journal for Numerical Methods in Engineering 3 (1971), S. 233-248 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: A finite element technique is used to determine the natural frequencies and the mode shapes of a cantilever plate mounted on the periphery of a rotating disc. The plane of the plate is assumed to make any arbitrary angle with the plane of rotation of the disc. The distributed centrifugal force is resolved into two components - one acting in the plane of the plate and the other normal to the plate. The stresses produced in the middle surface of the plate due to the in-plane forces are first determined. The increase in the bending stiffness of the plate elements due to these in-plane stresses is obtained in a manner similar to that used in the stability analysis of plates. The component of the distributed centrifugal force normal to the plate surface is added to the inertia force.From the results of computations carried out for various values of the aspect ratio, the speed of rotation, the disc radius and the setting angle, empirical formulae are derived giving the effect of these parameters on the natural frequencies. These empirical formulae are observed to be in agreement with the corresponding known formulae for rotating cantilever beams, when the aspect ratio is high.
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  • 47
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    International Journal for Numerical Methods in Engineering 3 (1971), S. 445-446 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
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  • 48
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    International Journal for Numerical Methods in Engineering 3 (1971), S. 5-11 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: It has long been recognized that permeability, compressibility and viscosity vary during the consolidation of soil. However, the introduction of these variations into consolidation theory leads to formidable difficulties due to their rendering the equations highly non-linear. Numerical methods are given herein which enable the governing equation to be solved for a variety of different non-linear laws.
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  • 49
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    International Journal for Numerical Methods in Engineering 3 (1971), S. 13-24 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: The simultaneous iteration method of obtaining eigenvalues and eigenvectors is employed for the solution of undamped vibration problems. This method is of significance when a few of the dominant eigenvalues and eigenvectors are required from a large matrix, and hence is particularly suitable for vibration problems involving a large number of degrees of freedom. It is shown that advantage may be taken of both the symmetry and the band form of the mass and stiffness matrices, thus making it feasible to process on a computer larger order vibration problems than can be processed using transformation methods. A method of allowing for body freedom is given and some numerical tests are discussed.
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  • 50
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    International Journal for Numerical Methods in Engineering 3 (1971), S. 25-33 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: The general eigenvalue problem Ax = λBx as arising from vibration problems tackled by the finite element method is often solved by the economization method in which the two matrices are reduced to a more manageable size and approximate answers are obtained.This paper analyses the method in a more mathematical way than previous accounts and leads to a definition of the optimum set of variables to be retained during the reduction. An extension to the method is suggested which allows the approximate eigenvalue to be improved and bounds are obtained on the eigenvalues of the full problem before reduction.An estimate is made of the calculation involved in the method and it is concluded, by reference to examples, that the extended method leads to a more efficient algorithm.
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  • 51
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    International Journal for Numerical Methods in Engineering 3 (1971), S. 35-43 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: A technique is described for solving the compressible flow equations in subsonic flow. The general quasi-linear equation ∇.g∇v = 0 is considered with g a function of ∇v. ∇v, and iterations of the form ∇.gn∇vn+1 = 0 are analysed, where g0 is suitably chosen and gn defined from vn for n≥1. This approach is applied to the compressible flow equations in terms of a velocity potential ø: monotonic convergence is predicted and at each iteration the error is multiplied by a factor less than the square of the greatest Mach number in the solution.Reliable convergence has been obtained in practice solving the linear equation for øn+1 by a finite difference method. The alternative of working in terms of the stream function ψ is discussed, and also discretization by the finite element method.
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  • 52
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    International Journal for Numerical Methods in Engineering 3 (1971), S. 45-51 
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    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: A method of analysis is presented for two structures which have connections with clearance. It is assumed that the behaviour of the structures is described, by the movement of discrete nodes and that as the load on the structure changes, clearances open or close between corresponding nodes on the structures; each time this occurs the stiffness of the combined structure changes because its configuration has changed. Between these changes, the structure will behave linearly and the overall behaviour can be described as piecewise linear. Applications include diesel engine crankshafts with bearing clearances and the wrap-around problem in pin-and-eye connections.
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  • 53
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    International Journal for Numerical Methods in Engineering 3 (1971), S. 53-61 
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    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: A solution is presented for the computation of the transient thermoelastic stresses in a hollow cylinder with temperature boundary conditions given as a circumferential variation of surface heat transfer coefficient. The temperature distribution is solved explicitly. The problem is set up using the Airy stress function which leads to the biharmonic equation. This approach requires the satisfaction of three Michell integrals at the inner boundary in order to ensure single-valued displacements and rotation. An iterative method is described in which these integrals are all simultaneously satisfied and thus provide the necessary non-zero boundary conditions for the solution of the biharmonic equation which is rapidly solved by Gaussian elimination. Results are presented for the general case where the temperature is a function of r and θ. The computer program is checked by assuming a constant value of the surface heat transfer coefficients. In this case a closed form solution is obtained.
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    International Journal for Numerical Methods in Engineering 3 (1971), S. 145-147 
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    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: The Dynamic Relaxation (DR) method of solving a set of simultaneous linear equations requires an estimate of the spectral radius of the matrix. Dividing each equation by the corresponding row sum of moduli of the elements of the matrix gives a convenient upper bound of unity to this. This note shows that the DR method then gives a faster asymptotic rate of the convergence than the degenerate Chebyshev method which it closely resembles.
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  • 55
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    International Journal for Numerical Methods in Engineering 3 (1971) 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
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  • 56
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    International Journal for Numerical Methods in Engineering 3 (1971), S. 151-151 
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    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
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  • 57
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    International Journal for Numerical Methods in Engineering 3 (1971), S. 379-388 
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    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: A very simple equation solver of very large capacity is developed. The method described here does not require a very large in-core memory; however, a fast random access device like a magnetic disk drive is required.
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  • 58
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    International Journal for Numerical Methods in Engineering 3 (1971), S. 361-377 
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    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: The equations governing the axisymmetric dynamic deformation of an elastic solid are considered as a symmetric hyperbolic system of linear first-order partial differential equations. The characteristic properties of the system are determined and a numerical method for obtaining the solution of mixed initial and boundary value problems in elastodynamics is presented. Two examples are considered.
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  • 59
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    International Journal for Numerical Methods in Engineering 3 (1971), S. 349-360 
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    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: A computer program is described for the analysis of plates in extension using the equivalent plane-framework idealization. The conjugate gradient method is adopted to solve the resulting large number of simultaneous equations. Performance figures are quoted for the program when used on an ICL KDF9 machine with a restriction of 18K on the available core store. The method is generally shown to provide a satisfactory solution to a heavy computational task and thus permits the solution of very large structural systems on a small computer.
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  • 60
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    International Journal for Numerical Methods in Engineering 3 (1971), S. 389-403 
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    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: An important limitation of finite element analysis, namely, the need for a large number of small elements in regions of finite or infinite stress concentrations and the difficulties of convergence in such cases, is well known. Rao1 suggested a possibility of overcoming this by developing hybrid techniques combining continuum and finite element concepts. In such techniques, each region of stress concentration is covered by one large ‘primary element’ whose description includes term(s) identifying the type and order of concentration, while the remaining structure is split into a few ‘secondary elements’ which are conventional finite elements. In this paper a procedure incorporating this concept is developed and its effectiveness is clearly demonstrated by successful application to two important examples, one of them with stress singularities. The concept, in fact, can be applied equally well to other two- and three-dimensional problems of continua with discontinuities and concentrations.
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  • 61
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    International Journal for Numerical Methods in Engineering 3 (1971), S. 415-423 
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    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
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    Notes: The analysis is aimed at determination of displacements in elastic-perfectly plastic frames, in the case of monotonically increasing loads. Spread of partial plasticity from hinges and change of geometry on the conditions of equilibrium are not included. Extremum energy principles and linear programming methods are employed to solve the problem. Both the kinematic and the static formulations of the problem are presented. The mathematical models of the formulations constitute a dual pair of linear programming problems.
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  • 62
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    International Journal for Numerical Methods in Engineering 3 (1971), S. 405-413 
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    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
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    Notes: In convection heat transfer problems, it is convenient to solve the governing differential equations by using straightforward marching techniques for numerical integration. However, the boundary layer equations in these heat transfer problems are not of the initial value type even though they are parabolic equations. Consequently marching techniques are not successful until correct initial values are known. A means of determining these initial values is presented herein. Selected optimization methods are utilized in conjunction with marching integration techniques to solve third- and fourth-order ordinary differential equations which have three unknown initial boundary conditions. Two optimization methods, one deterministic and the other non-deterministic in nature, are used both independently and in combination as determined by the particular circumstances. Two applications are used to demonstrate the technique and comparison is made with existing solutions.
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  • 63
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    AIChE Journal 17 (1971), S. 3-4 
    ISSN: 0001-1541
    Keywords: Chemistry ; Chemical Engineering
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Chemistry and Pharmacology , Process Engineering, Biotechnology, Nutrition Technology
    Notes: Enthalpies of the coexisting equilibrium vapor and liquid mixtures of the ethane-n-butane-n-pentane system are computed at several pressures. Saturated liquid enthalpies are obtained by using a new multicomponent Clapeyron equation with the B-W-R equation of state and the experimental temperature-composition data for the ternary mixture. Saturated vapor enthalpies are calculated by applying the B-W-R equation to the vapor mixture. Ideal gas state enthalpies are used in both vapor and liquid enthalpy calculations.
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  • 64
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    AIChE Journal 17 (1971), S. 30-37 
    ISSN: 0001-1541
    Keywords: Chemistry ; Chemical Engineering
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Chemistry and Pharmacology , Process Engineering, Biotechnology, Nutrition Technology
    Notes: For certain classes of chemical reactions, it is possible to increase the yield of the desired products by choosing appropriate variable-volume operating policies. An analysis of the steady state and semibatch operation of a stirred tank reactor demonstrates operating policies for both the isothermal and adiabatic case to maximize the steady state yield and to produce an improved semibatch yield. Computations carried out by using an analogue computer with a digital logic expansion system demonstrate further relationships between yield increase and operating policy.
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    AIChE Journal 17 (1971), S. 51-56 
    ISSN: 0001-1541
    Keywords: Chemistry ; Chemical Engineering
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Chemistry and Pharmacology , Process Engineering, Biotechnology, Nutrition Technology
    Notes: Models for the laminar sublayer of turbulent pipe flow are reviewed. The observations of the sublayer by Popovich and Hummel and the experimental fluctuation frequencies obtained by Shaw and Hanratty are used with the models to evaluate experimental heat and mass transfer data. The eddy diffusivity model and a parallel conductance model appear to be consistent with the experimental data.
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  • 66
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    AIChE Journal 17 (1971), S. 81-85 
    ISSN: 0001-1541
    Keywords: Chemistry ; Chemical Engineering
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Chemistry and Pharmacology , Process Engineering, Biotechnology, Nutrition Technology
    Notes: The effect of viscosity and diffusivity variations on concentration polarization during the ultrafiltration of protein solutions in two-dimensional stagnation flows is investigated by numerical integration of the equations of continuity and motion. It is found that these effects are surprisingly small and insensitive to the forms of concentration dependence postulated. The high Schmidt numbers and low permeation rates characteristic of these systems result in considerable simplification of the analysis and permit use of a modified Chilton-Colburn relation for predicting the onset of sludge formation at the membrane-solution interface. The predicted dependence on Schmidt number agrees with published experimental data for other boundary-layer flows and suggests the possibility of improved prediction of equipment performance.
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    AIChE Journal 17 (1971), S. 246-248 
    ISSN: 0001-1541
    Keywords: Chemistry ; Chemical Engineering
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Chemistry and Pharmacology , Process Engineering, Biotechnology, Nutrition Technology
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  • 68
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    AIChE Journal 17 (1971), S. 255-256 
    ISSN: 0001-1541
    Keywords: Chemistry ; Chemical Engineering
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Chemistry and Pharmacology , Process Engineering, Biotechnology, Nutrition Technology
    Additional Material: 1 Ill.
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  • 69
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    Keywords: Chemistry ; Chemical Engineering
    Source: Wiley InterScience Backfile Collection 1832-2000
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  • 70
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    AIChE Journal 17 (1971), S. 272-280 
    ISSN: 0001-1541
    Keywords: Chemistry ; Chemical Engineering
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Chemistry and Pharmacology , Process Engineering, Biotechnology, Nutrition Technology
    Notes: The kinetics of simultaneous, noncatalytic solid-gas reaction systems has been analyzed in terms of the selectivity and effectiveness factor, based on the unreacted core shrinking model. The reaction systems discussed include the first-order independent, parallel, and consecutive reactions with the simultaneous influences of heat and mass transfer. The effects of various chemical and transport properties on the stability, selectivity, and effectiveness factor of the system are discussed. The pseudosteady state approximation is employed to reduce the mathematical difficulties.
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  • 71
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    AIChE Journal 17 (1971), S. 291-294 
    ISSN: 0001-1541
    Keywords: Chemistry ; Chemical Engineering
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Chemistry and Pharmacology , Process Engineering, Biotechnology, Nutrition Technology
    Notes: An analysis is made of the Van der Waals forces of two different species when separated by a third continuous component. Results show that under certain circumstances, attractions as well as repulsions can exist between the two different species. The theory is applied to a fibrous bed coalescer to determine what type of material would best serve as a coalescence medium.
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    AIChE Journal 17 (1971), S. 335-342 
    ISSN: 0001-1541
    Keywords: Chemistry ; Chemical Engineering
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Chemistry and Pharmacology , Process Engineering, Biotechnology, Nutrition Technology
    Notes: The periodic process utilizes a rapid pressure swing cycle in an adsorbent bed to effect the separation of gas mixtures. During the first portion of a cycle the compressed gas mixture flows into the adsorbent-filled column. Next, while the feed gas is restrained, an exhaust orifice is opened at the feed end of the column providing depressurization. The product stream is enriched in the component exhibiting the lowest coefficient of adsorption.A mathematical model based upon the assumption of instantaneous equilibrium between the gas phase and the adsorbed gas was formulated and solved to simulate the periodic, adsorption process. The measured nitrogen content of the product gas stream was found to correlate with the ratio of the product gas rate to the feed gas rate.At 24°C. the calculated pressure response, feed gas flow rate, and product gas composition correspond favorably with related experimental measurements for all values of the feed gas pressure, cycling frequency, and product gas flow rate within the ranges investigated.
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    AIChE Journal 17 (1971), S. 365-371 
    ISSN: 0001-1541
    Keywords: Chemistry ; Chemical Engineering
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Chemistry and Pharmacology , Process Engineering, Biotechnology, Nutrition Technology
    Notes: The optimal control problem for multiple beds is formulated and a distributed maximum principle derived. Gradient methods are then used to synthesize the optimal control policies for some examples in three-bed isothermal and adiabatic reactors. Finally, examples are worked to illustrate the computational algorithm in adiabatic beds when optimal catalyst loading is to be determined, and when it is desired to distribute the feed in an optimal fashion between the beds. Reasonably good convergence was obtained in all the examples, so that the method appears quite feasible for determining optimal control policies in multiple reactors subject to catalyst decay.
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    AIChE Journal 17 (1971), S. 383-390 
    ISSN: 0001-1541
    Keywords: Chemistry ; Chemical Engineering
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Chemistry and Pharmacology , Process Engineering, Biotechnology, Nutrition Technology
    Notes: The nature of flow choking has been studied as steam-water mixtures are passed through a tube with ever increasing pressure gradient. Choked flow for two-phase, one-component mixtures has been characterized with a separated flow model as resulting from maximization of the ratio of gas to liquid velocities, or slip ratio. Slip ratios at choking were measured and found to increase from 1.2 to 5.0 at 30 lb./sq.in. exit plane pressure with decrease in quality from 0.95 to 0.02. Separated models predict either 9.3 or 28.4 with no quality dependence. The difference is attributed to liquid entrainment. Flow regimes were observed to vary from purely entrained to annular entrained to slug entrained as quality decreased. Normal shock waves were observed in the free jet at qualities above 0.25. Choking flow rates were predicted with an average error of less than 2% with a stagnation energy balance model using an empirically developed slip ratio relation corrected for entrainment.
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  • 75
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    AIChE Journal 17 (1971), S. 257-257 
    ISSN: 0001-1541
    Keywords: Chemistry ; Chemical Engineering
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Chemistry and Pharmacology , Process Engineering, Biotechnology, Nutrition Technology
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  • 76
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    AIChE Journal 17 (1971), S. 259-265 
    ISSN: 0001-1541
    Keywords: Chemistry ; Chemical Engineering
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Chemistry and Pharmacology , Process Engineering, Biotechnology, Nutrition Technology
    Notes: Relationships have been developed for the calculation of second virial coefficients of polar fluids from the critical constants, acentric factor w, and fourth parameter x of the substance. Coefficients for the expansion of the reduced second virial coefficient in w and x are presented for 0.64 ≤TR ≤ 0.94. Comparisons with available experimental data indicate that good results are obtained by the method of this study for most polar fluids. The second virial coefficient for water is found to exhibit anomalous behavior. Procedures have also been established for extending the relationships for pure fluids to the second virial coefficient of mixtures containing nonpolar or polar components.
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  • 77
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    AIChE Journal 17 (1971), S. 281-285 
    ISSN: 0001-1541
    Keywords: Chemistry ; Chemical Engineering
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Chemistry and Pharmacology , Process Engineering, Biotechnology, Nutrition Technology
    Notes: The modification of the Blake-Kozeny equation for porous media flow using the power law has been shown to hold for molten polymers as well as for the previous cases for polymer solutions. The present work extended the correlation of friction factor with Reynolds number an additional three decades.It was also shown, however, that the modified Blake-Kozeny equation broke down when bed shear rates were in the range of excessive curvature on the polymer flow curves (shear stress-shear rate). In such cases it is suggested that the Newtonian Blake-Kozeny equation, together with apparent viscosity data, could be used to correlate such porous media flow of non-Newtonians.Appearance of viscoelastic effects at the critical Deborah number value of 0.05 predicted earlier were not found to hold. The data of the present investigation show that such a critical value must be at least 0.19 or even higher. This result helps to resolve partially the anomalous results obtained by previous investigators.
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  • 78
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    AIChE Journal 17 (1971), S. 322-325 
    ISSN: 0001-1541
    Keywords: Chemistry ; Chemical Engineering
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Chemistry and Pharmacology , Process Engineering, Biotechnology, Nutrition Technology
    Notes: Velocity profiles were calculated for transient fluid flow in the hydrodynamic entrance regions of a long narrow slit and of a long tube. The solutions were obtained by use of a linearized momentum equation. An estimate was also made of the manner in which the resistance to flow of the entrance region develops with time.
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  • 79
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    AIChE Journal 17 (1971), S. 349-352 
    ISSN: 0001-1541
    Keywords: Chemistry ; Chemical Engineering
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Chemistry and Pharmacology , Process Engineering, Biotechnology, Nutrition Technology
    Notes: Torque measurements were made in experiments carried out using a cylindrical Couette apparatus with the inner cylinder rotating and the outer one stationary. The fluid systems employed consisted of distilled water, 50% distilled-water glycerine solutions, and these with several concentrations of suspended polystyrene spheres (through 40 on 70 mesh). The parameters employed by Bjorklund and Kays were found to correlate the data successfully when suspension viscosities were used for the two-phase systems.
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  • 80
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    AIChE Journal 17 (1971), S. 419-424 
    ISSN: 0001-1541
    Keywords: Chemistry ; Chemical Engineering
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Chemistry and Pharmacology , Process Engineering, Biotechnology, Nutrition Technology
    Notes: A three-stage procexsss of reduction of acid chromate (HCrO4-), with NaHSO3 followed by precipitation of Cr(III) with NaOH followed by flotation with an anionic surfactant, provides 97% removal of chromium from aqueous suspensions containing 48.3 mg./liter Cr (0.929 mM). The required dosage of surfactant (sodium laurylsulfate) is 0.093 mole SLS/mole Cr, compared to a dosage of 1.1 mole cationic surfactant/mole Cr, used to ion float soluble acid chromate. The precipitate flotation process is pH dependent: below pH 6.3, soluble chromium species become appreciable and the flotation results agree with calculated soluble chromium concentrations; above pH 9.7, the charge of the precipitate is reversed, as indicated by surface potential measurements. The optimum pH range for flotation is 7.0 to 8.8. The effect of initial surfactant concentration on the flotation efficiency is established. Small quantities of calcium enhance the flotation of Cr, while higher concentrations, 0.5 mM and greater, decrease flotation efficiency. For suspensions with a doubled chromium concentration (96.6 mg./liter), the optimum pH range for flotation is lowered and narrowed to 6.3 to 6.5, indicating modifications in particle surface characteristics validated by measurements of surface potential and surfactant adsorption. Eighty-seven percent flotation is achieved at a 0.093 mole/mole ratio.
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  • 81
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    AIChE Journal 17 (1971), S. 2-2 
    ISSN: 0001-1541
    Keywords: Chemistry ; Chemical Engineering
    Source: Wiley InterScience Backfile Collection 1832-2000
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  • 82
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    AIChE Journal 17 (1971), S. 25-30 
    ISSN: 0001-1541
    Keywords: Chemistry ; Chemical Engineering
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Chemistry and Pharmacology , Process Engineering, Biotechnology, Nutrition Technology
    Notes: A numerical study was performed to investigate improvement in mass transfer rates obtainable by introduction of secondary convection into a proposed membrane blood oxygenator. The three-dimensional Navier-Stokes equations were solved numerically for the velocity field in channel and the results used in the numerical solution for the local Po2 in the oxygenator. The convective diffusion equation solved is nonlinear, in addition to being three-dimensional, because of the presence of red cell oxygen sinks.The convection introduced was generated by moving one wall of the blood flow channel laterally to the main flow. Decreases in blood side film resistance up to 5 × 104 were predicted. In the presence of moderate to strong convection, the limiting factor on oxygenation rate is imposed by the membrane situated between blood and gas phases. It appears that a considerable decrease in required blood priming volume would be possible if the method discussed here were implemented.
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  • 83
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    AIChE Journal 17 (1971), S. 57-65 
    ISSN: 0001-1541
    Keywords: Chemistry ; Chemical Engineering
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Chemistry and Pharmacology , Process Engineering, Biotechnology, Nutrition Technology
    Notes: The effects of aliphatic alcohols on mass transfer and drag coefficients of a carbon dioxide bubble were investigated in very dilute aqueous solutions. A high speed photography technique provided an almost continuous record of the bubble volume, shape, and oscillation, as related to the height of the bubble in the liquid and the time elapsed from the moment of its release. The results showed that the concentration, molecular size, and structure of the investigated alcohols have a pronounced effect on mass transfer as well as drag coefficients of the rising bubble. The significance of these effects with respect to idealized models of a highly circulating bubble and a solid sphere was considered, and a functional relationship was found to exist. This relationship may be applied to systems more complex than a single bubble.
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  • 84
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    AIChE Journal 17 (1971), S. 75-81 
    ISSN: 0001-1541
    Keywords: Chemistry ; Chemical Engineering
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Chemistry and Pharmacology , Process Engineering, Biotechnology, Nutrition Technology
    Notes: A steady state experimental technique for measuring fluxes of the components and diffusivities in binary liquid systems has been developed. Steady state diffusion is established through a sintered glass diaphragm of ultrafine porosity with a known concentration difference across the diaphragm. The concentrations are maintained by a flow of solution on one side of the diaphragm and an infinite sink on the other side. The fluxes and diffusivities are calculated from the measured flows and compositions. A syringe pump capable of delivering extremely small flow rates with good reproducibility is used to pump the solution past the sintered glass diaphragm. The diffusion cell design permits an accurate control of pressure drop across the diaphragm so that fluxes can be measured both with and without pressure gradients. The diffusion cell was calibrated by using the hydrochloric acid-water system.Three binary systems, ethanol-benzene, aniline-benzene, and aniline-carbon tetrachloride were chosen for diffusion studies. Diffusivity-composition data for these systems at 25°C. were used to test the existing correlations for the concentration dependence of binary diffusivity. Of these, the empirical correlation of Vignes (11) represents the data satisfactorily. In addition, experimental molar diffusion flux ratio data for the above systems at 25°C. with no pressure gradients are reported and found to be essentially equal to the ratio of molar volumes of the components.
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  • 85
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    AIChE Journal 17 (1971), S. 101-103 
    ISSN: 0001-1541
    Keywords: Chemistry ; Chemical Engineering
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Chemistry and Pharmacology , Process Engineering, Biotechnology, Nutrition Technology
    Notes: Analyses of stretching of a viscoelastic liquid have been interpreted as showing that there is a maximum stretch rate to which the material can be subjected, and several physical phenomena have been explained on this basis. It is shown here theoretically and experimentally that the concept of a limiting stretch rate does not have general validity.
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  • 86
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    AIChE Journal 17 (1971), S. 115-121 
    ISSN: 0001-1541
    Keywords: Chemistry ; Chemical Engineering
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Chemistry and Pharmacology , Process Engineering, Biotechnology, Nutrition Technology
    Notes: The stability of a capillary liquid bridge of given volume between two small, solid, equal, separated spheres is investigated by formulating and treating a minimum energy problem in the calculus of variations and by experiment. A conjecture is made that in the case of two solutions one and only one is minimizing, and that the case of one solution represents the limiting stable bridge. This theory agrees accurately with our stability experiments. Furthermore, it is possible to predict the cohesive force. For the case of spheres in contact, the theory presented here is in agreement with some experimental work and also with the theory of Fisher and calculations of cohesive force based on Melrose and Wallick's solution to the bridge problem. For the case of separated spheres, the agreement with the only available experimental data is excellent except for close separations.
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  • 87
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    AIChE Journal 17 (1971), S. 122-130 
    ISSN: 0001-1541
    Keywords: Chemistry ; Chemical Engineering
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Chemistry and Pharmacology , Process Engineering, Biotechnology, Nutrition Technology
    Notes: Bubble size and distribution of bubble sizes were studied in an air fluidized bed, 11.5 in. in diam. and 3 ft. high, by taking motion pictures of the bubbles breaking the top surface of the bed and by determining the size of the bubbles from the film. The distribution of bubble volumes at a given bed height was expressed as a gamma distribution with parameters obtained from the observations. A statistical mechanical method proposed by Hulburt and Katz for studies in agglomeration of particles was used in the mathematical formulation of the problem. A proposed mechanism of bubble growth by coalescence in gas fluidized beds resulted in a one parameter model. Comparisons of the distributions predicted by the model to the experimentally obtained profiles yielded the parameter of the model.
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  • 88
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    AIChE Journal 17 (1971), S. 148-153 
    ISSN: 0001-1541
    Keywords: Chemistry ; Chemical Engineering
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Chemistry and Pharmacology , Process Engineering, Biotechnology, Nutrition Technology
    Notes: A new approach to separation calculations has been developed which has many practical advantages. The equations of conservation of mass and energy and of equilibrium are grouped by stage and then linearized. The resulting set of equations has a block-tridiagonal structure which permits solution by a simple technique. Thus a new way to apply the Newton-Raphson technique to separation calculations is devised. The method has been tried on a number of problems chosen to exhibit characteristics which cause other solution technique to fail. Most problems are solved with ten iterations or less.
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  • 89
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    AIChE Journal 17 (1971), S. 171-176 
    ISSN: 0001-1541
    Keywords: Chemistry ; Chemical Engineering
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Chemistry and Pharmacology , Process Engineering, Biotechnology, Nutrition Technology
    Notes: Direct thermal parametric pumping separations of a NaCl-H2O-ion retardation resin system have been investigated. Experimental breakthrough curves and batch parapump runs were used to determine the lumped-parameter mass transfer coefficient λ and its dependence on velocity and temperature. Computer simulation of additional batch runs verified both the model and the STOP-GO algorithm used to solve it. Computer estimates of open-system behavior are presented.
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  • 90
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    AIChE Journal 17 (1971), S. 191-198 
    ISSN: 0001-1541
    Keywords: Chemistry ; Chemical Engineering
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Chemistry and Pharmacology , Process Engineering, Biotechnology, Nutrition Technology
    Notes: A correlation based on conformal solution theory has been developed for the excess properties of the nitrogen-oxygen-argon systems. A better method for calculating the conformal parameters makes possible the simulataneous correlation of the three excess properties FE, HE, and VE. Each binary is characterized by a k12, c12 set which represents corrections to the geometric and arithmetic mean combination rules. These binary constants predict the ternary data. Also, the correlation based on data at one atm. (77° and 84°K.) predicts the FE data accurately up to the critical points (110° to 120°K.).
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  • 91
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    AIChE Journal 17 (1971), S. 215-219 
    ISSN: 0001-1541
    Keywords: Chemistry ; Chemical Engineering
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Chemistry and Pharmacology , Process Engineering, Biotechnology, Nutrition Technology
    Notes: The problem treating an irreversible, first-order chemical reaction in an isothermal, tubular reactor operating under steady, non-Newtonian, laminar flow is studied. Numerical results for the eigenvalues, eigenfunctions, expansion coefficients and bulk concentration are reported for Ostwald-de Waele and Prandtl-Eyring fluids. Results show that chemical conversion in a tubular reactor may be significantly influenced by the non-Newtonian characteristics of a fluid.
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    AIChE Journal 17 (1971), S. 252-254 
    ISSN: 0001-1541
    Keywords: Chemistry ; Chemical Engineering
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Chemistry and Pharmacology , Process Engineering, Biotechnology, Nutrition Technology
    Additional Material: 2 Ill.
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  • 93
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    AIChE Journal 17 (1971) 
    ISSN: 0001-1541
    Keywords: Chemistry ; Chemical Engineering
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Chemistry and Pharmacology , Process Engineering, Biotechnology, Nutrition Technology
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  • 94
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    AIChE Journal 17 (1971), S. 258-510 
    ISSN: 0001-1541
    Keywords: Chemistry ; Chemical Engineering
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Chemistry and Pharmacology , Process Engineering, Biotechnology, Nutrition Technology
    Additional Material: 2 Ill.
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  • 95
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    AIChE Journal 17 (1971), S. 285-288 
    ISSN: 0001-1541
    Keywords: Chemistry ; Chemical Engineering
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Chemistry and Pharmacology , Process Engineering, Biotechnology, Nutrition Technology
    Notes: The following nucleation rate and growth rate relationships for the aqueous, continuous, cooling crystallization of potassium dichromate were determined from experimental crystal size distributions: J = 3.9 × 1010 (C - Ceq)0.9; r = 3.0 × 103 (C - Ceq)1.7. Since nucleation rate is of lower kinetic order than growth rate, higher levels of supersaturation will yield crystal distributions with a larger mean size. Therefore for this system, batch or plug-flow reactors are preferred to continuous-stirred tanks.
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  • 96
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    AIChE Journal 17 (1971), S. 308-312 
    ISSN: 0001-1541
    Keywords: Chemistry ; Chemical Engineering
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Chemistry and Pharmacology , Process Engineering, Biotechnology, Nutrition Technology
    Notes: An experimental facility for vapor-liquid equilibrium measurements at high pressures (100 to 15,000 lb./sq. in. abs.) and ordinary temperatures (100° to 300°F.) on multicomponent mixtures is described. The vapor-liquid phase behavior of a six-component hydrocarbon mixture, which simulated a natural gas and a condensate, was studied with this apparatus. Measurements were made at 150° and 250°F. and pressures ranging from 100 lb./sq. in. abs. to the single-phase pressures for these systems and temperatures.
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  • 97
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    AIChE Journal 17 (1971), S. 326-335 
    ISSN: 0001-1541
    Keywords: Chemistry ; Chemical Engineering
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Chemistry and Pharmacology , Process Engineering, Biotechnology, Nutrition Technology
    Notes: A method has been developed for applying classical minimization techniques to forms of algebraic performance indices for use in optimal adaptive control systems. Essentially, the derivative of a general objective function is constrained to be equal to the integrand of a desired integral performance index. This generates a set of linear algebraic equations, the solution of which results in an algebraic objective function which is explicit in the system parameters. Derivatives of this function can be taken with respect to the controllable parameters, set equal to zero, and solved for the settings which minimize the performance index over time. Alternatively, the function itself may be searched for its minimum on the controllable parameters.The method has been applied to a stirred tank chemical reactor with an exothermic first-order reaction in which heat removal was accomplished by cooling coils. The cooling water flow rate was controlled by a proportional-plus-integral controller and by a three-mode controller. The adaptive control system adjusted the controller settings periodically. The plant, which was third-order, was controlled to a second-order dynamic reference model. The responses to both initial offsets (start-up problem) and disturbances in system parameters were investigated. In all cases, the adaptive control system performance was markedly superior to that for the unadapted, ordinary feedback control system.
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    AIChE Journal 17 (1971), S. 353-356 
    ISSN: 0001-1541
    Keywords: Chemistry ; Chemical Engineering
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Chemistry and Pharmacology , Process Engineering, Biotechnology, Nutrition Technology
    Notes: Available generalized correlations of fugacity coefficients of nonpolar liquids are reexamined in their lower temperature ranges, and these correlations are extended to a reduced temperature of 0.35. Generalized functions are constructed from literature data on vapor pressures and liquid densities. Two sets of functions are presented with w and zc, respectively, as third parameter. The results fulfill a need for basic information in the engineering of cryogenic systems.The same vapor pressure data are correlated in a generalized form down to Tr = 0.35 to give a one-parameter family of reduced vapor pressures curves. The correlation is compared with data on simple fluids, normal fluids, polar substances, and quantum fluids.
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    AIChE Journal 17 (1971), S. 379-383 
    ISSN: 0001-1541
    Keywords: Chemistry ; Chemical Engineering
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Chemistry and Pharmacology , Process Engineering, Biotechnology, Nutrition Technology
    Notes: An analysis is made of diffusion and chemical reaction in an isothermal multicomponent fluid. Transient gas-phase reaction in a one-dimensional nonflow system and steady state gas-phase or surface reaction in a tubular reactor are considered. The equations are linearized around some state away from equilibrium. Multicomponent and binary diffusivity expressions are used and a series of reactions of varying complexity is considered. The possibility of oscillations increases as the complexity of the reactions is increased. These oscillations occur with time in the nonflow system and with space-time in the tubular reactor, and it appears that any reacting mixture which oscillates in one of these two systems will also oscillate in the other.
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    AIChE Journal 17 (1971), S. 402-409 
    ISSN: 0001-1541
    Keywords: Chemistry ; Chemical Engineering
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Chemistry and Pharmacology , Process Engineering, Biotechnology, Nutrition Technology
    Notes: Pressure drag coefficients of spheres in a cubic packing of equal spheres have been determined at Reynolds numbers of 27,000, 10,000, 5,000 and 2,500, where NRe is based on superficial air velocity and a sphere diameter of 7 cm. Two cubic arrangements were used: in the regular arrangement, the mean flow was parallel to one of the principal axes, while in the skewed arrangement, the mean flow made equal angles with the three principal axes of the packing.The pressure drag coefficient of the central sphere has been measured for each of the 12 banks in the regular arrangement, and the effect of bed length has been determined for lengths varying between 1 and 12 banks. The pressure drag coefficient for the central sphere and the overall pressure drop in the skewed arrangement were found to be lower than for the regular arrangement at the same Reynolds number. From the distribution of local pressure measurements, the separation and reattachment points on the central sphere in each bank were determined, indicating that the boundary-layer behavior on a sphere in a packing is similar to that over a single sphere, when allowing for the effects of turbulence and of pressure gradient.
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