Formulas for calculating the approach to equilibrium in open channel flash evaporators for saline water

The evaporator used in multistage flash desalination plants, and considered for open cycle ocean-thermal energy conversion (OTEC) plants is the open-channel type, in which an essentially-horizontal stream of seawater is passed through an open channel in which it is exposed to a pressure lower than the saturation pressure which corresponds to its temperature. This paper reviews the equations used at present to determine the nonequilibrium allowance Δ′ which indicates how far from equilibrium the seawater stream is as it leaves the evaporator.

Flash evaporation in open channels is used in most of the water desalination plants, and the existing equations for Δ′ have been developed for that application. The equations are typically empirical correlations, each developed for one type of geometry and range of parameters. The equations are compared graphically, plotted side-by-side to express Δ′ as a function of the major parameters: stage saturation temperature, stage flash-down, flashing seawater flow rate, stage length, and flashing seawater depth.

It was found for water desalination applications that there exists a large spread between the non-equilibrium fraction values calculated by the different equations, of about one order of magnitude. The situation is even worse for OTEC conditions. Consequently, it was concluded that no general method exists for the adequately accurate prediction of Δ′, i.e. of the approach to equilibrium of flashing free stream channel flows.     

Fuel allocation in dual-purpose plants

Exergy provides the cost accounting equations for allocating energy in any plant. Sets of complex auxiliary equations can be derived to allocate energy cost for each plant item and system. Here a simplified alternate method has been developed utilizing the First Law of Thermodynamics with consideration of the cost factors to distribute fuel in dual-purpose boiler turbine generator (BTG)-MSF plants. The method is based on a formulation of an ideal point at which power and water products are equivalent.     

A New Approach to Investigation of vdW Type of Equations of State

A new approach to the investigation ofvdW type of equations of state (EOS) is developed by embedding a vapor pressure equation and a saturated liquid volume equation into vdW type EOS, which resuits in a new function A^s(T). The A^s(T) possesses the properties of an attractive parameter A(T), and if an EOS is accurate in the whole PVT space, then its numerical wahie equal8 A(T). As a useful tool for investigating EOS, the A^s(T) has been used to make comparisons among RKS, PRSVII, PT and ALS EOS, and to indicate where the shortcomings of the EOS are coming from. Based or. the A^s(T), a possible way to develop at real predictive equation of state is also suggested.     

Newton-type decomposition methods in large-scale dynamic process simulation

For the plantwide dynamic simulations in the chemical process industry, a parallel approach using a divide and conquer strategy is developed, and its realization on parallel computers with shared memory is discussed. The system of differential–algebraic equations (DAEs) is partitioned into blocks, and then it is extended appropriately such that block-structured Newton-type methods can be applied, which enable the application of relaxation techniques. Two types of block-structured methods are considered and compared with one another. Applied to industrial applications, the approach has gained considerable speedup factors for the dynamic process simulation of various large-scale distillation plants, covering systems with up to 60 000 equations.     

Cyclic-pressure freeze drying

Cyclic pressure freeze drying has previously been found to give faster drying than the constant pressure process. Equations have been developed to describe this process during the period when dehydration is occurring by sublimation. Two models are described, one which employs a continuously moving sublimation interface (C.M.I.) and a second which has a discretely moving interface (D.M.I). The equations were developed from the momentum, mass, and energy conservation equations. Because of the dependence of the heat and mass transfer parameters on the pressure, the resulting equations are non-linear, second order parabolic in type. They were solved by a number of analog and numerical methods, the most successful being a modified Crank-Nicholson approach. Correlation between theory and experiment was satisfactory. The resultant models have been used to describe drying conditions within various materials and are suitable for the determination of an optimal cycling policy.     

On the numerical solution of differential algebraic equations

A method is proposed to transform a system of differential algebraic equations (D.A.E.) to a system of ordinary differential equations (O.D.E.) which can be solved relatively easily by standard numerical techniques. Two examples, including a model of an absorption tower, are given to illustrate the utility of the method. The example problems reveal that this easily implemented technique offers significant savings in CPU time compared to the numerical solution of the untransformed D.A.E.s particularly when the algebraic equations are nonlinear. Furthermore, it appears to be faster and/or more reliable than other numerical schemes which have been recently developed for equations of this type.     

Using new packages for modelling, equation oriented simulation and optimization of a cogeneration plant

The development and testing of an equation-oriented (EO) mathematical model for the optimization of heat and power systems using a new state-of-the-art sequential quadratic programming (SQP) method called filterSQP has been reported in an earlier paper. The model has been extended to include combined cycle cogeneration plants (CCCPs), the economic optimization of which involves adding equipment investment, cost functions and operating cost models to the stream and unit operation models of the earlier paper. These systems are of substantial industrial interest and are substantially larger than those in Rodríguez-Toral et al., Computers and Chemical Engineering, August 2000. Their size and complexity required the use of a new flexible modelling system (FMS) recently created within our research group. FMS can be used to set up large systems of equations, create networks to be modelled and give starting guesses easily for problems of any size. Also, FMS is able to interact with solvers for EO simulation and optimization, which have been developed here and elsewhere. A number of EO simulation and optimization examples, from simple unit operations to a whole real cogeneration plant involving a commercial gas turbine with 1275 variables and equations, are used to demonstrate the applicability of the CCCP model and the modelling package.     

Hydrodynamic Pressure Distribution Within a Complex Geometry Coating Unit

When a wire is pulled through a stepped parallel or conical gap, which is filled with a viscous fluid, a very high pressure is generated. The pressure is termed hydrodynamic pressure and can facilitate the coating of wire. In this paper analytical equations have been developed for predicting the pressure distribution within a combined unit, taking account of the changes in the viscosity and the shear rate of the polymer melt during the coating process. A finite difference type approach has been employed to obtain solutions for these equations. The change in the viscosity due to variations in pressure within the unit as well as the change in shear rate in the tapered part due to change in gap between the wire and the unit have been taken into account. Theoretical results are obtained for different wire speeds in terms of the pressure distributions within the unit. These results are compared with experimental results obtained from the literature.     

Application of the method of weighted residuals to mixed boundary value problems: Dual-series relations

Mixed boundary value problems arise in a number of chemical engineering models for heat conduction, simultaneous diffusion and reaction, and fluid flow whenever there is a change in the type of boundary condition along the same boundary. In this paper a general computer algorithm based on the method of weighted residuals (MWR) is developed for determining approximate solutions of all problems of the above type which lead to dual series equations. The solution procedure reduces the determination of the series coefficients to the solution of a large system of algebraic equations. This technique offers advantages over finite difference or artificial interface methods in the accuracy which can be obtained, the type of problem which may be treated, and the simplicity of calculations to be performed. Solutions obtained by the application of MWR are compared and analysed for three example problems which are of interest in diffusion, reaction and conduction.     

A solution technique for noncatalytic diffusion-reaction models

A technique for solving the model equations typical of noncatalytic fluid-solid reactions where both diffusion and kinetics are important is presented. The partial differential equations are transformed by orthogonal collocation into a system of ordinary differential equations of the initial-value type. These equations are then solved by a semi-implicit Runge-Kutta method developed by Michelsen. Both gas-solid and liquid-solid reactions as well as general reaction rate forms can be treated. From the numerical examples presented the validity of the pseudo-steady-state assumption is demonstrated for values of the accumulation term parameter ψ?0.1.     

Rapid phase determination in multiple-phase flash calculations

A formulation of the equations which describe the behavior of systems which can form three phases is presented. The method cna be extended to systems of more than three phases if desired. Key features of the method are : (1) rapid a priori determination of the number of phases actually present, based on a multi-phase generalization of the well-known bubble- and dew-point criteria; and (2) rapid solution of the equations once the number of phases has been determined. The method was developed for incorporation into a general purpose, modular flowsheet simulation system.     

DIVA – a flow-sheet oriented dynamic process simulator

The detailed dynamic simulation of coupled process units in chemical plants is gaining an increasing importance as a useful tool in plant engineering and operation. The outline of the program package DIVA ( D ynamische Si mulation v erfahrenstechnischer A nlagen) which is currently under development is presented in the following. The dynamic plant equations and the corresponding Jacobian matrix are generated automatically. The full exploitation of sparse matrix techniques in combination with stiff ODE ( o rdinary d ifferential e quation) solvers allows an efficient solution of all the equations simultaneously. The possibilities offered by the simulator are demonstrated by the simulation of two laboratory plants.     

Concentration and temperature profiles for complex reactions in porous catalysts

A method of computation for stationary concentration and temperature profiles in porous catalysts was developed. These calculations can be used with highly complex reactions and with any type of rate equation or transport model. The differential equations are transformed by the collocation method to non-linear algebraic equations. These are solved by a method which uses the time-derivatives of the variables. The method finds stationary points for all problems which can be formulated as ? = f(y) and is demonstrated on two examples.     

Analytical study of type-1 facilitated transport through liquid surfactant membrane

In the present study, a mass transfer model for Type-1 facilitated transport in liquid surfactant membrane is developed by taking into consideration a size distribution of emulsion drops, and analytical solution of the model equations has been presented. The model takes into account the continuous phase and outer liquid membrane phase resistances along with diffusion through composite emulsion drop. Effort has been made to highlight the effect of the various system parameters on the extraction rate including computation of reaction front position. The results of this work are found to be in good agreement with the published experimental results on batch extraction of phenol using NaOH as internal reagent. The model would thus provide an insight of the separation mechanism involved in the mass transfer processes in this type of system.     

UNSTEADY MIXED CONVECTION WITH SORET AND DUFOUR EFFECTS PAST A POROUS PLATE MOVING THROUGH A BINARY MIXTURE OF CHEMICALLY REACTING FLUID

This study investigates the unsteady mixed convection flow past a vertical porous flat plate moving through a binary mixture in the presence of radiative heat transfer and nth-order Arrhenius type of irreversible chemical reaction by taking into account the diffusion-thermal (Dufour) and thermo-diffusion (Soret) effects. Assuming an optically thin radiating fluid and using a local similarity variable, the governing nonlinear partial differential equations have been transformed into a set of coupled nonlinear ordinary differential equations, which are solved numerically by applying shooting iteration technique together with fourth-order Runge-Kutta integration scheme. Graphical results for the dimensionless velocity, temperature, and concentration distributions are shown for various values of the thermophysical parameters controlling the flow regime. Finally, numerical values of physical quantities, such as the local skin-friction coefficient, the local Nusselt number, and the local Sherwood number are presented in tabular form.     

Designing reliable polymer coatings

A method to predict the ultimate adhesion performance of coatings subjected to biaxial, tensile stress is presented. A set of equations is developed to predict the onset of failure when the locus of failure is cohesive in the coating. The equations also have been extended to predict fatigue lifetime when a coating is cycled from a low temperature to room temperature. To test these equations, the Edge Liftoff Test (ELT) was developed. It consists of fabricating coatings of various heights on a rigid substrate and dicing the latter, so that the edges are 90° to the interface. The parts are cooled and the temperature of debonding is recorded. For thermal fatigue validation, cycles at which the first debond is observed are recorded. ELT results have been collected for a typical epoxy for both monotonic and cyclic cooling. The predictions are in agreement with results.     

THE DISSOLUTION OR GROWTH OF A SPHERE

The problem of the dissolution or growth of an isolated, stationary, sphere in a large fluid body is analyzed. The motion of the boundary as well as the resulting motion in the liquid are properly taken into account. The governing equations are solved using a recently developed technique (Subramanian and Weinberg, 1981) which employs an asymptotic expansion in time. Results for the radius of the sphere as a function of time are calculated. The range of utility of the present solution is established by comparison with a numerical solution of the governing equations obtained by the method of finite differences.     

THE DISSOLUTION OR GROWTH OF A SPHERE

The problem of the dissolution or growth of an isolated, stationary, sphere in a large fluid body is analyzed. The motion of the boundary as well as the resulting motion in the liquid are properly taken into account. The governing equations are solved using a recently developed technique (Subramanian and Weinberg, 1981) which employs an asymptotic expansion in time. Results for the radius of the sphere as a function of time are calculated. The range of utility of the present solution is established by comparison with a numerical solution of the governing equations obtained by the method of finite differences.