Acessibilidade / Reportar erro

STEADY-STATE modeling and simulation of pipeline networks for compressible fluids

Abstract

This paper presents a model and an algorithm for the simulation of pipeline networks with compressible fluids. The model can predict pressures, flow rates, temperatures and gas compositions at any point of the network. Any network configuration can be simulated; the existence of cycles is not an obstacle. Numerical results from simulated data on a proposed network are shown for illustration. The potential of the simulator is explored by the analysis of a pressure relief network, using a stochastic procedure for the evaluation of system performance.

pipeline networks; compressible flow; pressure relief networks


STEADY-STATE MODELING AND SIMULATION OF PIPELINE NETWORKS FOR COMPRESSIBLE FLUIDS

A.L.H. COSTA, J.L. de MEDEIROS and F.L.P. PESSOA

Department of Chemical Engineering - Federal University of Rio de Janeiro - UFRJ - Escola de Química - Bloco E Centro de Tecnologia - Cidade Universitária - CEP 21949-900 - Rio de Janeiro, RJ - Brazil - Phone: (021) 590-3192 E-mail: andre@h2o.eq.ufrj.br

(Received: May 14, 1998; Accepted: September 26, 1998)

Abstract - This paper presents a model and an algorithm for the simulation of pipeline networks with compressible fluids. The model can predict pressures, flow rates, temperatures and gas compositions at any point of the network. Any network configuration can be simulated; the existence of cycles is not an obstacle. Numerical results from simulated data on a proposed network are shown for illustration. The potential of the simulator is explored by the analysis of a pressure relief network, using a stochastic procedure for the evaluation of system performance.

Keywords: pipeline networks; compressible flow; pressure relief networks.

INTRODUCTION

Pipeline networks are structures built to transport fluids between several supply and demand points. Transmission and distribution of natural gas are examples of services provided by pipeline networks.

In order to guarantee the most efficient and reliable network design, computational codes that can predict network behavior are necessary engineering tools. There are many references in the literature on pipe network simulation with incompressible fluids; the main goal of these papers is to determine pressures and flow rates (Beck and Boucher, 1997; Houache et al., 1996; Gostoli and Spadoni, 1985). However, the literature on rigorous pipe network simulation with compressible fluids is scarce (Greyvenstein and Laurie, 1994).

The purpose of this paper is to describe the structure of a simulator developed for the prediction of the steady-state behavior of pipeline networks with compressible fluids. The mathematical model developed here and the algorithm used in the simulation are discussed. The model is general and is able to determine pressures, flow rates, temperatures and gas compositions at any point of the network. The simulator has auxiliary routines for the calculation of thermodynamic properties (enthalpy, entropy and volume) and transport properties (viscosity). There are no constraints on the network topology; cyclic networks can be simulated.

ELEMENT MODELS

In this work, pipelines and compressors are selected as the building elements of compressible flow networks. However, the general structure of the proposed network model allows the inclusion of other kinds of elements as well.

Pipelines

The model of a pipeline with one-dimensional compressible flow describes the relation between pressure and temperature along the length of the pipeline. It is based on the flow equation associated with the energy equation. Critical and multiphase flow is not considered.

Flow equation

The flow equation, in differential form, may be represented as

(1) The result of the substitution of velocity by mass flow rate and of density by molar volume, with the Darcy equation for head loss, is multiplied by

, which yields

(2)

Integration of Equation 2 between the ends of the pipeline gives

(3)

where the friction factor in Equation 3 is an average of the values of the Darcy friction factor at the ends of the pipeline, evaluated using the correlation proposed by Churchill (1977). The orientation of the integration defined by indices "1" and "2" is arbitrary; it is not necessary that it agree with the real orientation of the flow. The term that represents the integral of elevation is estimated by the following approximation:

(4)

With the adoption of a general relation between gas volume and temperature valid throughout the gas flow

(5)

it is possible to define a relation between the pressure and volume differentials by using the total differential of pressure with respect to temperature and volume, as follows:

(6)

At first, Equation 6 allows the evaluation of the pressure integral in Equation 3. However, it must be noted that the irreversibility generated within the gas flow, depicted by the energy dissipation term, makes it impossible to establish the necessary relation equivalent to Equation 5. So, the exact evaluation of the pressure integral in Equation 3 is virtually impossible. For this reason, three reversible idealizations of Equation 5 are proposed: isothermal, adiabatic and polytropic flow. These idealizations are an attempt to approximate the real process; the decision on which of them will be more adequate for the simulation will depend on additional information about the process.

Isothermal flow: The equations equivalent to Equations 5 and 6 are

(7)

(8)

Thus, the pressure integral in Equation 3 can be evaluated as follows:

(9)

The expression of the Iv function depends on the equation of state used to determine the derivative of pressure with respect to volume present in Equation 9.

Adiabatic flow: The relationships equivalent to Equations 5 and 6 are

(10)

(11)

Along the isentropic path in the pressure integral, temperature is a complex function of volume. Thus, a numerical integration algorithm would be necessary. However, applied to the entire network, this procedure would result in excessive computing times. An alternative for the evaluation of this integral is to describe the gas flow with a reversible adiabatic process for an ideal gas with constant heat capacities. The expression of the integral becomes

(12)

This approximation can be partially corrected with the evaluation of the real gas volume in the integral and using parameter evaluated by an average of the parameter computed at the ends of the pipeline.

Polytropic flow: In this case, the relationship equivalent to Equation 5 is empirically defined, based on the particular flow process. The description of the process based on ideal gas behavior can also be used, yielding an expression similar to Equation 12, but now parameter becomes an empirical constant of the process. In the simulation results presented in this paper, this empirical constant is approximated by the ratio of the ideal gas heat capacities.

Moore et al. (1980) proposed the following expression for evaluating the pressure integral:

(13)

where T

f and V

i are the outlet temperature and the inlet volume of the gas flow, respectively, independent of the orientation defined. It must be noted that since the differential, dP/V, is not exact, Equation 13 is an approximation. This equation can be applied to isothermal (in this case, identical to Equation 9), adiabatic and polytropic flow regimes.

Energy equation

The energy equation is defined by a steady-state energy balance in a pipeline. This equation is used according to the three idealizations considered above.

Isothermal flow: The energy equation is substituted by the equality between the terminal temperatures of the pipeline:

(14)

Adiabatic flow: The energy equation is (Smith and Van Ness, 1987)

(15)

Substitution of velocity by mass flow rate in Equation 15, gives

(16)

Polytropic flow: Inclusion in Equation 16 of the term for heat transfer between the fluid and the environment gives

(17)

where the heat transfer rate is approximated by the following expression:

(18)

If the pipeline is very long, it should be divided into smaller parts to avoid errors caused by this approximation.

Compressors

The mathematical model for compressors is formed by the following relationships:

Equation between rise in pressure and mass flow rate: This equation is represented by a polynomial expression of the following form:

(19)

Constraints: To ensure that in the simulation the gas flows in the compressor in the direction allowed by the equipment (conventionally, ) and with a flow rate below the maximum limit (qlim), the following two equations are introduced:

(20)

(21)

Thermodynamic process: The change in gas temperature in the compressor can be determined by relating the real process to an adiabatic reversible one. This relation is defined using the real process efficiency (), which can be represented as a polynomial function of the mass flow rate. The process equations are

(22)

(23)

(24)

(25)

Physical Property Evaluation

The molar volume, enthalpy and entropy of gas mixtures are evaluated using autonomous routines based on the Redlich-Kwong equation of state and residual property calculation (Walas, 1985). The viscosity of gas mixtures is evaluated by the Chung et al. correlation (Reid et al., 1987).

Performance of Pipeline Models

Flow temperature and pressure at a point in a duct can be determined by the Multistep Method (Ouyang and Aziz, 1996). In this method, the duct is divided into small segments and the flow and energy equations, in finite difference form, are applied to all segments sequentially. This procedure can achieve very accurate results at the expense of a large computational effort. Thus, the Multistep Method is applied here to a pipeline flow example to check the accuracy of the flow models presented previously. The models proposed are solved by the application of the Newton-Raphson Method to the energy and flow equations. The implementation times are compared to show the relative performance of these models. The times are for a Pentium 200 Mhz with 32 Mb RAM.

The pipeline flow example has the following characteristics: a gas composition of 90 % methane and 10 % ethane (mass base), a pipe length of 150× 103 m, a pipe diameter of 0.3048 m, a pipe roughness of 46× 10-6 m, a gain in elevation of +200 m and inlet conditions of 50× 105 Pa and 298 K. In polytropic flow, additional parameters are defined: an overall heat transfer coefficient, 4 W/m2K apipe wall thickness of 0.005 m and an ambient temperature of 298 K.

The final pressure and temperature calculated by the Multistep Method for each flow regime are isothermal flow: 27.942× 105 Pa and 298 K; adiabatic flow: 28.701× 105 Pa and 285.4 K; and polytropic flow: 28.013× 105 Pa and 297.2 K.

Isothermal flow: The deviation of pressure loss predicted by the isothermal flow model from the Multistep Method result is only 0.03 % and the implementation time employed is 0.3 s.

Adiabatic/Polytropic flow: The two pipeline models proposed differ in the way that the pressure integral (Equation 3) is calculated, based on ideal gas behavior (Equation 12) or on the Moore et al. (1980) approach (Equation 13). Additionally, for comparison, the pressure integral along an isentropic path is considered using a numerical integration procedure. The results are presented in Table 1.

According to Table 1, the Moore et al. (1980) approach achieves a smaller pressure loss deviation than the ideal gas model. However, network simulations based on the Moore et al. (1980) approach suggest that this model can give rise to some convergence problems. The higher deviation observed for the ideal gas model in the polytropic flow regime occurs due to the approximation of the heat transfer term in the energy equation (Equation 18). The division of the pipeline into two or three intervals allows a large reduction of this deviation. The evaluation of the isentropic pressure integral with a numerical procedure gives results similar to those of the ideal gas model, but it requires larger computational times.

Table 1: Results of adiabatic/polytropic flow models*

NETWORK MODEL

Pipeline networks can be described with a digraph (Mah and Shacham, 1978). The network elements are the edges of the digraph and the connection points are the vertices. The goal of a simulation is to describe the behavior of networks formed by S edges (St pipelines and Sc compressors, ) and N vertices (Na open vertices which exchange fluid with a supply/demand site and Nf closed vertices which do not exchange fluid with a supply/demand site, such that ). There are C components in the gas mixtures flowing in the network.

The network structure is represented by the incidence matrix (size N x S) of the digraph (Mah, 1990). The arbitrary orientation of each edge of the digraph does not define the flow direction. The direction of the flow is defined by the simulation, and it may (q > 0) or may not (q < 0) agree with the edge orientation.

Parameters

The following parameters must be defined before the simulation:

Physical data: Molecular weight, critical coordinates, acentric factor, dipole moment and ideal gas heat capacity polynomial coefficients for each gas component present in the network.

Network structure: Network connections, represented by the incidence matrix; elevation of each vertex and indication of the subset of open vertices.

Physical characteristics of the network elements: Diameter, length, thickness and absolute roughness of each pipeline (thickness is only necessary for polytropic flow); suction and discharge diameters and maximum flow rates of the compressors; polynomial coefficients relating efficiency and the rise in pressure to the mass flow rate in each compressor.

Information about the environment: Overall heat transfer coefficient between the fluid and environment and the temperature of the environment (this information is only necessary for polytropic flow).

Variables

The variables defined in the network problem are:

Internal mass flow rates: (S x 1) - These variables represent the flow rates in the elements.

Pressures at vertices: (N x 1) - The pressures at the ends of the elements correspond to the pressures at the incidence vertices.

External mass flow rates at vertices: (N x 1) and (N x 1) - These variables represent the supply streams and the demand streams, respectively.

Variables of the compressor constraints: (Sc x 1) and (Sc x 1).

Outlet temperatures in the elements: (S x 1) - These variables indicate the gas temperature leaving each element.

Temperatures at vertices: (N x 1) - The temperature of a vertex is the initial temperature of all streams which leave that vertex.

Isentropic temperatures at discharge of the compressors: (Sc x 1).

Molar fractions at vertices: (N x C) - Each line of the matrix represents the gas composition at a vertex. The streams that leave a vertex have the composition of that vertex.

It can be seen that a subset of the model variables is related to elements and another subset is related to vertices. This arrangement is proposed in order to allow the whole set of variables to describe the initial and final conditions of all streams.

Definitions

It is possible to access the conditions of the gas flow at the ends of the elements, according to the orientation of the digraph or to the real orientation of the flow. These associations create new definitions that help to build the mathematical model of the network. The matrices used in these definitions (, , , and ) are described in the Appendix.

Gas conditions according to the orientation of the digraph: Initial state - "1" - and final state - "2"

Pressure:

and

Temperature: For k = 1, ... , S,

If , and

If , and

Gas conditions according to the orientation of the flow: Initial state - "i" - and final state - "f"

Pressure:

and

Temperature:

and

Equations

The system of equations of the entire network model is formed by the association of the element models, according to the network configuration, with equations that describe the gas state at the vertices. The structure of the system is divided into three blocks.

Block I

This block corresponds to the mass balance equations at the vertices for each component. In matrix form, it is

(26)

where (N x C) and (N x C) are the matrices of the mass fractions at the vertices and external supply streams, respectively. If the flow rates of the network are known, Block I will consist of C systems of N linear equations with respect to the mass fraction at the vertices. Since originally molar fractions were the compositional variables of the network, these equations must be complemented by the mass fraction/molar fraction conversion.

Block II

The equations of this block are:

Global mass balance equations at the vertices:

(27)

Flow equations: The structure of these equations is defined by the reversible idealization of the flow (isothermal, adiabatic and polytropic),

for k = 1, ... , St (28)

It is important to note that for each pipeline "k," Equation 28 is applied to a different gas mixture with composition .

Equations for rise in pressure and mass flow rate in the compressors:

for k = St +1, ... , S (29)

Equations related to external mass flow rates: These equations ensure that any open vertex will be exclusively a supply vertex or a demand vertex.

if t is an open vertex (30)

For closed vertices, both external flow rates are specified equal to zero.

Specifications:

(31)

where . Matrix is defined as if variable ym is specified with value en; otherwise .

Constraints:

for k = St+1, ... , S (32)

for k = St+1, ... , S (33)

Block III

Energy balance at vertices: The rate of energy transport by an internal stream "k" entering/leaving a vertex is

(34)

In Equation 34, the gravitational energy is not considered because there is no difference in elevation between streams related to a same vertex. Substitution of velocity by mass flow rate gives

(35)

The equivalent expression for external streams does not contain the kinetic energy term because it is assumed that these streams have their origin/destination in large reservoirs,

(36)

From Equations 35 and 36, the energy balance equations at the vertices are written

for t = 1, ... , N (37)

where the individual terms have the following expressions:

Rate of energy transport by external supply streams (Fws):

(38)

Rate of energy transport by external demand streams (Fwd):

(39)

Rate of energy transport by internal streams that flow out of the vertex (Fqout):

(40)

where is the set of streams actually leaving vertex "t" and is the flow area of stream ( for and for ).

Rate of energy transport by internal streams that flow into the vertex (Fqin):

(41)

where is the set of streams actually entering vertex "t" and is the flow area of stream ( for and for ).

Energy equations: The energy equations are defined according to one of the three reversible idealizations considered, isothermal flow (Equation 42), adiabatic flow (Equation 43) or polytropic flow (Equation 44).

For k = 1, ... , St:

(42)

(43)

(44)

where . In any pipeline, if .

Thermodynamic process equations for compressors:

For k = St +1, ... , S:

(45)

(46)

(47)

(48)

Size of the system of equations

The proposed mathematical model for the pipeline networks with compressible fluids is formed by linear and nonlinear equations.

Specifications

Temperatures ( , N x 1) and compositions ( , N x C) of all external streams that can act as network supplies are considered to be known before the simulation. At the vertices, where there is no fluid supply, these values have no physical meaning.

The specification for the network simulation is completed with matrix and vector in Equation 31. The rule used to define and states that in each open vertex the value of a pressure or an external flow rate must be specified, whereas for closed vertices both external flow rates are defined as null.

NUMERICAL METHOD FOR SIMULATION

Due to the complexity of the system of equations, second-order methods such as Newton-Raphson and Quasi-Newton are natural candidates for solving the simulation, since these methods exhibit a faster performance when compared with ordinary linear iterative methods, starting from a reasonable estimate of solution (Westerberg et al., 1979). However, utilization of the Newton-Raphson Method gives rise to some obstacles. Solving the system of equations with a full numerical jacobian matrix would require an excessive computational effort; on the other hand, the large amount of variables and nonlinear equations involved makes it difficult to analytically evaluate the jacobian matrix.

The strategy adopted to avoid these obstacles is to use the Newton-Raphson Method within a successive substitution procedure. Thus, the evaluation of the analytic jacobian matrix for the whole system is substituted by the evaluation of the jacobian matrix for subsystems with smaller sizes. From among the several options studied (Costa, 1997), an algorithm formed by two nested loops is chosen. A graphical representation of this algorithm is shown in Figure 1.

Block II of equations forms the inner loop and is solved by the Newton-Raphson Method with respect to internal flow rates (), pressures , external flow rates ( ws and wd) and compressor constraints variables (a and b ). The other variables are kept constant. In the outer loop, Block I is a system of linear equations with respect to the vertex compositions () and is solved in each outer iteration. After the resolution of Block I, the algorithm does a Newton-Raphson iteration for Block III with respect to outlet temperatures (), temperatures at vertices () and isentropic temperatures at discharge of the compressors (). At the end of the process, convergence is checked with the norm of the vector of equation residuals.

The algorithm proposed does not employ sparse matrix procedures. These techniques can provide some improvement in algorithm performance, optimizing the resolution of the system of linearized equations.

NUMERICAL RESULTS

Pipeline Network Simulation

The numerical routines of the simulator are applied to the network presented in Figure 2. This network is composed of 28 pipelines connected with 26 vertices. The lengths and diameters of the pipelines are given in Table 2. The absolute roughness of the pipelines is 46 m m and the thickness is 0.005 m. The pressure at vertex 11 is 1× 105 Pa. The specified supply flow rates, compositions and temperatures are shown in Table 3. The other vertices are considered closed vertices. The network is simulated as a polytropic flow using the ideal gas model for the evaluation of the flow equations (Equation 12). The ambient temperature is 293 K and the overall heat transfer coefficient is 3.69 W/m2K.

Figure 1:
Simulation algorithm.
Figure 2:
Network structure.
Table 2: Lengths and diameters of pipelines

Table 3:
Flow rates, temperatures and compositions of supply streams

-4, the solution of the example above is obtained in 19.55 s of CPU time and takes 7 iterations in the outer loop and a sequence of 6, 4, 3, 2, 1, 1 and 0 iterations in the inner loop.

The simulation results, showing the pressures, temperatures and gas compositions at some selected vertices, are presented in Table 4. Table 5 shows the flow rates and the outlet temperatures in some pipelines.

Stochastic Analysis of a Pressure Relief Network

In order to demonstrate the potential of the simulator, an application based on a stochastic analysis of a hypothetical pressure relief network is formulated. Refineries and petrochemical plants have a pipe network that connects the industrial units with at least one flare. A flare is a tower where gas is burned permanently. If an accident occurs, the gas streams from the affected unit will be forced to deviate from their course to be burned at the flare. This procedure avoids the dispersion of dangerous gases in the industrial area.

The stochastic procedure is done using a large number of simulations. In each simulation, the set of supply flow rates related to the unit discharges are determined by random selection. Since it is not possible to predict the exact values of these flow rates in an accident, a description using a statistical procedure according to a probability density function (PDF) is adequate. This procedure is similar to the Monte-Carlo Method applied in molecular simulations.

The stochastic analysis consists in estimation of the PDF of the network outputs (pressures at the outlet of the relief devices - backpressures - and flow rate at the flare), given the PDF of their inputs (unit discharges). This estimation is represented by the mean and standard deviations of a random sample and histograms that describe the variable distributions.

The network analyzed is presented in Figure 2. It has a typical structure for a pressure relief network: the pipelines converge at the point where the flare is located (vertex 11). The unit discharges are controlled by c 2 distributions with four degrees of freedom and means given by the flow rates presented in Table 3. A sample with 298 successful simulations is chosen. An example of the input data used is represented in Figure 3 by a histogram of the discharge distribution at vertex 1.

Table 4:
Calculated pressures, temperatures and molar fractions at vertices
Table 5:
Calculated flow rates and outlet temperatures in pipelines

Figure

3: Histogram of discharge distribution at vertex 1.

The numerical results may be displayed by histograms of backpressure distribution along the network and flow rate distribution at the flare. Examples of these histograms are presented in Figures 4 and 5. The mean and standard deviations of the pressure at vertex 4 are 1.425

× 10

5 Pa and 0.374× 10

5 Pa, respectively; for the flow rate at the flare, these values are 35.80 kg/s and 7.77 kg/s, respectively.

This application of the simulator may be useful in the sizing of elements of a pressure relief network. With the unit discharge distributions, distributions of backpressures throughout the network can be numerically determined, as is shown. Thus, the diameters are defined to keep these distributions within acceptable ranges, avoiding (probabilistically) oversized (and expensive) projects or undersized (and dangerous) ones. Another important consequence of this analysis is the flow rate distribution at the flare, which can be used in the design of the burners and the height of the flare.


Figure

4: Histogram of pressure distribution at vertex 4.


Figure

5: Histogram of flow rate distribution at the flare - vertex 11.

CONCLUSIONS

A mathematical model for the rigorous steady-state simulation of pipeline networks for compressible fluids is formulated. This model is composed of a system of algebraic linear and nonlinear equations. The model is solved using the Newton-Raphson Method associated with a successive substitution procedure.

The simulator is used in a stochastic procedure for the performance analysis of a pressure relief network. This procedure can be useful as an auxiliary tool in the design of this kind of system.

ACKNOWLEDGEMENTS

The authors are grateful to the Research Support Foundation of the State of Rio de Janeiro (FAPERJ) and the Brazilian National Council for Scientific and Technological Development (CNPq) for their financial support.

NOMENCLATURE

A Flow area, m

2; D Pipeline diameter, m;

Ddis Discharge diameter of a compressor, m;

Dsuc Suction diameter of a compressor, m;

f Darcy friction factor;

F Rate of energy transport from a stream to a vertex - Fqin (internal affluent stream), Fqout (internal effluent stream), Fws (external supply stream), Fwd (external demand stream), W;

H Enthalpy, J/kgmol;

L Pipeline length, m;

h Head loss, m;

Average molecular weight, kg/kgmol;

Incidence matrix;

P Pressure, Pa;

q Internal mass flow rate, kg/s;

Q Heat transfer rate between fluid and environment, W;

S Entropy, J/Kkgmol;

T Temperature, K;

Text Ambient temperature, K;

Trev Isentropic temperature at discharge of a compressor, K;

v Fluid velocity, m/s;

V Molar volume, m3/kgmol;

wd External mass flow rate of a demand stream, kg/s;

ws External mass flow rate of a supply stream, kg/s;

We Compression work, J/kgmol;

Matrix of molar fractions at vertices (N x C);

Matrix of mass fractions at vertices (N x C);

z Elevation, m;

a Variable of compressor constraints, kg

0.5/s

0.5; b Variable of compressor constraints, kg

0.5/s

0.5; d Pipeline thickness, m; h Compressor efficiency.

Subscripts

1,2 Initial and final points of an element according to the orientation of the digraph;

i,f Initial and final points of an element according to the orientation of the flow;

k Element index;

s Variable related to the external supply streams;

t Vertex index;

v Variable related to the vertices.

APPENDIX

This Appendix presents the definition of some matrices derived from the incidence matrix that were used in this paper.

Matrices and : These matrices indicate the edges that leave and enter the vertices, respectively,

If

and

If

and

If

Matrices , and : Matrix is analogous to the incidence matrix, referring to the actual orientation of the streams.

if stream "m" flows into vertex "n."

if stream "m" flows out to vertex "n."

if edge "m" is not linked to vertex "n."

Matrices and are derived from according to the same concept used for matrices and .

  • Beck, S. B. and Boucher, R. F., Steady State Analysis of Fluid Circuits Containing Three Port Devices, Transactions of the Institution of Chemical Engineers, 75, part A, p. 73 (1997).
  • Churchill, S. W., Friction-Factor Equation Spans All Fluid-Flow Regimes, Chemical Engineering, 84, No. 24, p. 91 (1977).
  • Costa, A. L. H., Modeling and Simulation of Steady-State Pipeline Networks, M.Sc. Thesis (in Portuguese), Escola de Química, Federal University of Rio de Janeiro/UFRJ (1997).
  • Gostoli, C. and Spadoni, G., Linearization of the Head-capacity Curve in the Analysis of Pipe Networks Including Pumps, Computers and Chemical Engineering, 9, No. 2, p. 89 (1985).
  • Greyvenstein, G. P. and Laurie, D. P., A Segregated CFD Approach to Pipe Networks Analysis, International Journal for Numerical Methods in Engineering, 37, p. 3685 (1994).
  • Houache, O., Khezzar, L. and Benhamza, M. H., Matrix Methods for Pipe Network Analysis: Observation on Partitioning and Cut Methods, Transactions of the Institution of Chemical Engineers, 74, part A, p. 321 (1996).
  • Mah, R. S. H., Chemical Processes Structures and Information Flows, Butterworth Publishers (1990).
  • Mah, R. S. H. and Shacham, M., Pipeline Network Design and Synthesis. In: Drew, T. B. et al. (editors), Advances in Chemical Engineering, 10, Academic Press (1978).
  • Moore, R. G., Bishnoi, P. R. and Donelly, J. K., Rigorous Design of High Pressure Natural Gas Pipelines Using BWR Equation of State, The Canadian Journal of Chemical Engineering, 58, p. 103 (1980).
  • Ouyang, L. and Aziz, K., Steady-state Gas Flow in Pipes, Journal of Petroleum Science and Engineering, 14, p. 137 (1996).
  • Reid, R. C., Prausnitz, J. M. and Poling, B. E., The Properties of Gases and Liquids, McGraw-Hill Book Company (1987).
  • Smith, J. M. and Van Ness, H. C., Introduction to Chemical Engineering Thermodynamics, McGraw-Hill Book Company (1987).
  • Walas, S. M., Phase Equilibrium in Chemical Engineering, Butterworths Publishers (1985).
  • Westerberg, A. W., Hutchinson, H. P., Motard, R. L. and Winter, P., Process Flowsheeting, Cambridge University Press (1979).

Publication Dates

  • Publication in this collection
    07 Dec 1998
  • Date of issue
    Dec 1998

History

  • Received
    14 May 1998
  • Accepted
    26 Sept 1998
Brazilian Society of Chemical Engineering Rua Líbero Badaró, 152 , 11. and., 01008-903 São Paulo SP Brazil, Tel.: +55 11 3107-8747, Fax.: +55 11 3104-4649, Fax: +55 11 3104-4649 - São Paulo - SP - Brazil
E-mail: rgiudici@usp.br