Material Balance Calculations Research Articles

Simulation Improvements in Mineral Processing

Since 1986, BRGM has been developing a powerful software package, USIM PAC. It is an easy to use steady-state simulator which makes it possible for the mineral processing specialist to profit from available experimental data to model plant operations, and then find an optimal plant configuration to meet required objectives. Designers of new plants can also work with this simulator to calculate the sizes and settings required to achieve a given circuit objective. This software package includes functions to manage experimental data, to calculate coherent material balances, sizes and settings of units of equipment, physical properties of the processed material, to simulate plant operations and display results on tables and graphics. Widely used to design and optimize industrial plants, with more than one hundred copies in twenty-six countries, this software has been improved, through successive versions, to make it more accurate and easier to use. Developed in the MS-DOS environment, the evolution was limited because of calculation time and memory management capacity. To overcome these difficulties, the Process Simulation Group has chosen a new environment, Windows 3, which allows powerful calculation possibilities, associated at a very friendly interface, making the software really intuitive for the mineral processing engineer. Since the first version, numerous mathematical models for unit operations have been included and today most of the physical treatments of mineral processing can be reproduced by the simulator: crushing, screening, grinding, flotation, gravity and magnetic separation, solid-liquid separation, etc. The new USIM PAC 2, contains all the functions of the previous version but also new algorithms such as a material balance calculation, the Objective Driven Simulation and the Supervisor of Simulation. Its structure is now open to new fields of activity such as hydrometallurgy, fine grinding, soil decontamination or mining waste treatment. New models have already been included for some of these fields. The improvements of the simulator will be presented and some applications of the new capabilities on complex ores in Portugal, India and France will be briefly described.

Examples of Petroleum Engineering Objects

Summary In recent years, object-oriented (OO) methods have gained prominence among software developers as a means of more closely modeling the real world.1,R2 Most OO applications to date have their roots in the field of computer science, and this may deter people without a strong computing background from further investigating OO. However, OO methods can be applied to concepts that may be more familiar to petroleum engineers. The intention of this paper is to demonstrate that the power of OO can be used to simplify the solution of practical problems by engineers having only a limited background in computing. We present three such petroleum engineering applications in this paper: probabilistic reserves evaluation, a gas recycling scheme, and material-balance calculations. In the first of these examples, we use OO methods to model a simple gas recycling scheme, taking into account explicitly vapor/liquid equilibrium, both in the surface separation train and in the reservoir. In the second example, we use OO methods instead of the well-known Monte Carlo method to derive a reserves expectation curve. In the third example, we analyze the Schilthuis material-balance method and represent it using simple objects. A brief introduction to objects will prepare the ground for the more detailed analysis of these applications. In each case, we briefly review the underlying principles and analyze them in terms of objects. We will demonstrate the structure and implementation of each of the resulting object classes using short C++ programs.3 Finally, we apply these object classes to the solution of some real-life problems that have appeared in the petroleum engineering literature.

Energetic investigation of Saccharomyces cerevisiae during transitions. Part 1. Mass balances

The calculation of material balances during the transitions implies the evaluation of cumulated quantities consumed and produced. This article presents the techniques to achieve this goal by evaluating the quantities involved in the reactions at each observation and by calculating the material balances (carbon, nitrogen, degrees of reduction) at each observation. Balance profiles are assessed by statistical tests and an experiment involving different metabolic pathways is analyzed.

On the causes of 18O-depletion and 18O/16O homogenization during regional metamorphism; the East Humboldt Range core complex, Nevada

Previous stable isotope studies at Lizzies Basin revealed that metasedimentary rocks are 18O-depleted relative to protolith values, particularly in the lower parts of the section (Lower Zone) where the rocks are also isotopically homogeneous on a scale of hundreds of meters (quartz δ18O=+9.0 to +9.6 per mil). In contrast, metasedimentary rocks at higher levels at Lizzies Basin (Upper Zone) are less 18O-depleted and more heterogeneous in δ18O. In order to understand more fully the isotopic evolution of this terrane, a series of detailed, meter-scale traverses across various metamorphic and igneous lithologies were completed at Lizzies Basin, and at the structurally higher Angel Lake locality. Traverses in the Lizzies Basin Lower Zone and in the lower parts of Angel Lake (Angel Lake Lower Sequence) across various silicate lithologies, including abundant granitoids, reveal similar degrees of homogeneity, although the average δ18O values are higher at Angel Lake. In contrast, traverses which include substantial thicknesses of marble and calc-silicate gneiss and very little granitoid have more heterogeneous quartz δ18O values (+11.9 to +13.4 per mil), and also have a higher average δ18O (+12.9 per mil), than observed elsewhere. The scale of 18O/16O homogeneity in quartz observed at Lizzies Basin and Angel Lake (meters to hundreds of meters) requires fluid-mediated isotope exchange, which accompanied Tertiary metamorphism. There is a correlation between the degree of 18O-depletion in metasedimentary rocks, 18O/16O homogenization between lithologies, and the proportion of granitoids (leucogranites in particular) within any part of the section, and a corresponding anticor-relation with the proportion of marble. This points to a causal relationship, whereby the leucogranites (as well as the Tertiary hornblende diorite and biotite monzogranite) acted as both a relatively low-18O reservoir and a source of fluids to enhance exchange, while the marbles hindered isotope deplction and homogenization by acting as relatively high-18O reservoirs and impermeable layers. Material balance calculations help delineate the plausible mechanisms of exchange between granitoids and metasediments. Single-pass infiltration of magmatic fluids from the granitoids is not capable of reproducing all of the observations. Fluidmediated exchange by convective recirculation of magmatic fluids on a scale of meters is the mechanism which explains all of the observations. The generalized model for the isotopic evolution of the East Humboldt Range core complex provides and excellent opportunity to establish the main causes and controlling factors of 18O-depletion and 18O/16O homogenization during regional metamorphism.

A Generalized Approach to Reservoir Material Balance Calculations

Abstract This work completes the search for a generalized material balance equation. No restrictions are placed on reservoir fluid compositions. Unlike the conventional material balance equation, the new equation specifically accounts for volatilized-oil and, therefore, is uniquely applicable to the full range of fluids, including volatile-oils and gas-condensates. Equally important, the new material balance equation retains the same simplicity with which the conventional material balance equation has become known. The new, generalized material balance equation is featured here by showing how it is used to predict reservoir performance and estimate reserves in example volatile-oil and gas-condensate reservoirs. In comparison, the conventional material balance equation is shown to lead to erroneous results. The new, generalized material balance equation leads to an improved method of reservoir performance analysis. Introduction Material balance calculations are a useful method of reservoir performance analysis. They are routinely used to estimate oil and gas reserves and predict future reservoir performance. Schilthuis(l), in 1936, was among the first to formulate and apply material balances. Inherent in his and others' use of material balances were the following assumptions:at most, there are two hydrocarbon phases: oil and gas,at most, there are two hydrocarbon components: stock-tank oil and separator (surface) gas,the reservoir oil-phase consists of stock-tank oil and separator-gas,the reservoir gas-phase consists of only separator-gas and no stock-tank oil, andthere are no compositional gradients within the system. Assumptions 1 and 2 define the popular two-hydrocarbon- component formulation. Assumption 3 effectively accounts for the effects of dissolved- or solution-gas.Assumption 4 ignores the possibility of volatilized-oil and. therefore, restricts application to black-oils and dry-gases and precludes application to volatile-oils, gas-condensates, or wet-gases. Volatilized-oil is the stock-tank oil content of the free reservoir gas-phase, Assumption 5 means that the system is effectively treated as a tank with no gradients; accordingly, Coats(2) and others have sometimes referred to this type of model as a tank or zero-dimensional model. Despite their apparent oversimplicity, early tank models found widespread use in reservoir engineering applications(3–16). As time progressed, more sophisticated material balance models evolved, each striving for greater generality. Eventually, following the advent of digital computers, the material balance equations were discretized and the first generation of multi-dimensional, black-oil, finite-difference reservoir simulators were developed in the 1960's (17–19). These models, like their zero-dimensional counterparts, could effectively simulate black-oil and dry-gas reservoirs but could 1I0t model volatile-oil and gas-condensate reservoirs. To address this limitation, Cook et al 20 in 1974 introduced a volatile-oil, finite-difference simulator. Their formulation retained the simplicity of two hydrocarbon components, but allowed for volatilized-oil. Consequently, their formulation was applicable to the full range of reservoir fluids. The resulting formulation was applicable to the full range of reservoir fluids. Coats(21) later, in 1986, verified the applicability of the black- and volatile-oil approaches in reservoir simulation by showing agreement with a fully-compositional, equation-of-state (EOS) reservoir simulator.

Heavy Components Control Reservoir Fluid Behavior

The five reservoir fluids (black oils, volatile oils, retrograde gas-condensates, wet gases, and dry gases) are defined because production of each fluid requires different engineering techniques. The fluid type must be determined very early in the life of a reservoir (often before sampling or initial production) because fluid type is the critical factor in many of the decisions that must be made about producing the fluid from the reservoir. This paper summarizes the guidelines for determining fluid type from field data. If any of these three properties fails to meet the criteria described, the test fails and a representative sample of the reservoir fluid must be examined in a laboratory to establish fluid type. It also shows the expected results of laboratory analysis of the fluids. The paper also summarizes schematically the trends of producing GOR and stock-tank liquid gravity. These fluid types are defined for engineering purposes. They should not be confused with reservoir fluid descriptions defined by legal agencies that regulate the petroleum industry. Regulatory definitions of oil, crude oil, condensate, gas, natural gas, casing-head gas, etc., are not related to these engineering definitions and, in fact, often contradict them.

Building multivariable predictive control models by process simulation and data regression

This paper presents a multivariable model-based predictive control (MMPC) approach. It aims at processes such as distillation that have many difficulties such as multiple disturbances, long process lag and dead time and complex heat and material balance calculations. To implement the control strategy and control scheme recommended in the paper, a manipulated variable feedforward control model, a process performance prediction model and a process optimization model must be created. Well-planned rigorous process simulations can effectively and efficiently generate the data essential for model development. Statistical data regression is a powerful tool in selecting the model's independent variables and in establishing the quantitative relationships between the dependent variables and the independent variables. Process dynamics missing in the steady state simulation can be built into the models through theoretical analysis and practical process tests. When actually implementing the control system, the model accuracy can be improved simply by adjusting some pre-designed tuning factors. This approach is proven based on several successful installations at Arco gas processing facilities.

Analysis of Incinerator Performance and Metal Emissions from Recent Trial and Test Burns

Recent trial- and test-burn data from five rotary kiln incinerator facilities were analyzed for combustion performance and metal emissions. The incinerator facilities examined included: DuPont's Gulf Coast Regional Waste Incinerator in Orange, Texas; Chemical Waste Management's Incinerator in Port Arthur, Texas; Rollins Environmental Service's Incinerator in Deer Park, Texas; Martin Marietta's TSCA Incinerator in Oak Ridge, Tennessee; and EPA's Incineration Research Facility in Jefferson, Arkansas. The analysis involved the use of a PC-based computer program capable of performing material and energy balance calculations and predicting equilibrium compositions based on the minimization of system free energy. For each analysis, the feed data of waste and fuel and the corresponding operating parameters associated with incinerator and/or afterburner operation were input to the program and the program simulated the combustion performance under equilibrium conditions. In the analysis, the field-recorded performance data were compared with the simulated equilibrium results and the incinerator performance, including the quality of the field data, the combustion efficiency, the percent excess air, the heat loss, and the amount of air inleakage, was evaluated. In addition, the field-obtained metal data were analyzed for emission rate and metal balance.

Multidiscipline Studies of the Depletion Behavior of the F23 Gas Field, Offshore Sarawak, East Malaysia: ABSTRACT

The F23 gas field is located 178 km north-northwest of Bintulu in the central Luconia province, offshore Sarawak. The accumulation consists of a late Miocene layered platform-type carbonate buildup 22 km[sup 2] and a maximum gas thickness of over 1000 ft. Eleven development wells and one observation well were completed in 1983, with initial gas delivery to Malaysia LNG in October 1983. Annual TDT surveys are conducted in the observation well to monitor the movement of the GWC in the central area of the field. As of April 1992, a cumulative rise of 25 ft was observed. This contrasts with material balance calculations predicted rise of some 60 ft based on Sgr of 50% inferred from core measurements. Among the potential explanations were irregular bottom-water encroachment, preferential flank water advance, or larger GIIP. As all had potentially important consequences to depletion strategy and ultimate reserves, multidiscipline studies were initiated. Time-lapse seismic data were used to better determine the existing fluid levels across the field and reservoir simulation studies were used to match observation well data and predict future reservoir behavior. Concurrent reservoir and simulation studies suggested that the material balance and observation well data could be reconciled by (1) amore » reduced aquifer, and/or increased GIIP, and/or a lower residual gas saturation, or (2) preferential flank water influx due to internal low-permeability layers. As (1) would not lead to early water breakthrough, the simulation effort was aimed to investigate (2), the possibility and implications of early edge water breakthrough via the flank zone.« less

Gel Placement in Production Wells

Summary Straightforward applications of fractional-flow theory and material-balance calculations demonstrate that, if zones are not isolated during gel placement in production wells, gelant can penetrate significantly into all open zones, not just those with high water saturations. Unless oil saturations in the oil-productive zones are extremely high, oil productivity will be damaged even if the gel reduces water permeability without affecting oil permeability. Also, in field applications, capillary pressure will not prevent gelant penetration into oil-productive zones. An explanation is provided for the occurrence of successful applications of gels in fractured wells produced by bottomwater drive. With the right properties, gels could significantly increase the critical rate for water influx in fractured wells.

Modeling a Repressured Waterdrive Gas Reservoir

Summary Reservoir modeling confirmed reserve losses occurring in a waterdrive gas reservoir to justify blowdown project initiation. The material-balance models initially used could not match the water influx and repressuring observed in the reservoir. This paper shows that material-balance calculations need to be modified to account for trapped-gas compression that occurs with repressuring, and demonstrates the use of a "simple" numerical simulation model to predict performance. Numerical simulation enabled us to match reservoir behavior successfully, quantifying the effects of trapped-gas compression, and to identify the importance of critical blowdown rates and timing for optimizing reserve recovery from this repressuring waterdrive gas reservoir. Introduction The Tom O'Connor 5,100 Ft Sand is a gas reservoir located in Refugio County, TX. This fairly homogeneous Frio sandstone contains dry gas with a moderate directional waterdrive from the southwest. Fig. 1 gives a composite log and a structure map of the reservoir. At the start of this study, the reservoir was already in the later stages of depletion, having recovered 85% of the original gas in place. During a period of low gas market demand, the reservoir experienced significant water advancement and gas-zone repressuring from a lengthy production curtailment by the purchaser. Gas-zone pressures increased from 800 to 1,100 psig during this period (see Fig. 2). The operator successfully demonstrated that substantial reserve losses were occurring and obtained a special gas allowable to blow down the reservoir . Reservoir modeling was performed to confirm and quantify the gas reserve losses resulting from production curtailment and to predict the additional recovery that would result from blowdown initiation. Numerical simulation was used effectively to model the historical performance of the reservoir and to determine the optimum blowdown rate to maximize recovery. Model Development A simple tank-type model was developed with constant reservoir rock and fluid properties. The model incorporated radial flow from one central producer and contained a limited, directional aquifer as indicated by performance. Fig. 3 gives the reservoir input properties and model layout. Fig. 4 shows historical water encroachment and remaining free-gas volumes determined from known gas/water contact locations based on drill wells and gamma ray/neutron log surveys. Model accuracy was enhanced by the lengthy performance data available, which allowed excellent model calibration through history matching.

Upgrading of high-pollutant oil residues into environmentally acceptable hydrogen fuel by partial oxidation

Technological development in upgrading oil residues is becoming a major field of research in its own right. In this paper, upgrading approaches utilizing both the conventional chemical conversion processes and the proposed noncatalytic partial oxidation process are analyzed. The case of heavy oil residues-especially the one found in the Middle East-is presented. Material balance calculations indicate a hydrogen yield of about 13,000 SCF per bbl of feed of residue, when non-catalytically partially oxidized. The high-purity oxygen needed for the partial oxidation could be provided by coupling the process with thermochemical cycles. If air has to be used instead, the H 2/N 2 product makes an ideal feed for ammonia synthesis, hence urea production.

OILWAT - Microcomputer Program for Oil Material Balance With Gas Cap and Water Influx

Summary This paper describes an interactive computer program, OILWAT, for material-balance calculations in oil reservoirs with gas cap and water influx. A material-balance calculation uses production and pressure data to calculate original oil in place (OOIP) and aquifer characteristics. This formulation can be very useful in a history match to define reservoir and aquifer parameters before a simulation study. It also can be used to predict future pressure and primary production. The use of different material-balance methods usually results in different OOIP estimates. OILWAT helps solve this problem by combining several material-balance methods with various aquifer options (small pot, steady state, and unsteady state) and comparing them statistically. Panel input and graphical output are available in the program. Several important features are (1) variable bubblepoint, which enables very good history match during repressuring of the depleted reservoirs; (2) variable rock compressibility for abnormally pressured reservoirs; (3) an option to calculate original gas-cap or aquifer size provided that OOIP is known; and (4) applicability to saturated and/or undersaturated reservoirs. In contrast to most material-balance programs, OILWAT permits the use of a single data set for a reservoir initially at undersaturated condition that subsequently drops below bubblepoint.

Vulcanization Chemistry. Sulfur, Nt-Butyl-2-Benzothiazole Sulfenamide Formulations Studied by High Performance Liquid Chromatography

Abstract The paper reports on the sulfur vulcanization chemistry of cis-polyisoprene formulations accelerated with N-t-butyl-2-benzothiazole sulfenamide. High Performance Liquid Chromatography (HPLC) is utilized to examine the kinetics of accelerator—sulfur disappearance and the formation—disappearance profiles of extra-network cure intermediates across the vulcanization process. Cure profile measurements are reported on three formulations covering a wide range of sulfur—accelerator ratios. Also unique aspects of the analytical methodology are highlighted. The observed cure-intermediate concentration profiles reinforce earlier theories about the chemical details of sulfurization and subsequent crosslinking. On all formulations, benzothiazole polysulfides (S1−S4) reach a maximum concentration at the scorch midpoint coincident with slope changes in reactant and product profiles. Although pendant-accelerator crosslink precursors are not measured directly, their presence is implied from material balance calculations. Also, the formation of a small amount of benzothiazole early in the scorch period is a new observation in sulfenamide acceleration. It suggests that accelerator hydrolysis is a competitive side reaction early in the vulcanization process.

Long-Term Production Testing Improves Reservoir Characterization in the Oseberg Field

This paper reports that long-term oil or natural gas reservoir testing provides an improved understanding of static and dynamic reservoir behavior before full-field development. The long-term test also improves understanding of the geological model and the connected hydrocarbon volumes. The experience gained from long-term testing in the Oseberg Field has strengthened the basis for field development and reservoir management.

Two-Phase Compressibility Factors for Retrograde Gases

Summary The two-phase compressibility factor should be used in material-balance calculations for rich-gas-condensate systems. This paper presents a correlation for estimating this factor from field data. The correlation, which is based on 67 fluid-depletion studies with C7+ concentrations ≥4 mol% also applies to rich gases with large amounts of CO2, H2S, and N2. The correlation gives an average error of 3.66% but may result in larger errors for some rich gases.

Transient Behaviour Of Nonhomogeneous Linear Aquifers

Abstract This paper presents new analytical solutions for the dimensionless pressure drop or cumulative influx for nonhomogeneous aquifers whose thickness, permeability-viscosity ratio. or porosity-compressibility vary linearly with distance. These solutions apply to one-dimensional linear flow conditions with finite aquifer size. The inner boundary condition may be either constant rare or constant pressure. The outer boundary may be either dosed, or at constant pressure. The solutions presented; n this paper contain homogeneous aquifer solutions as limiting cases. Thus, this study extends the state-of-the-art of analytical aquifer modeling. Based on this study, a nonhomogeneous aquifer behaves as a homogeneous aquifer at early times. At late times, pseudosteady or steady-state behaviour is observed, depending on the outer boundary condition. But the dimensionless time to late-time state is dependent on the type and severity of the property variation. Limiting solutions at early and late times are presented for each case. Introduction Most hydrocarbon reservoirs are at least partly bounded by water bearing rocks called aquifers. The aquifer size may appear infinite or finite compared to the reservoirs they adjoin. In performing material balance calculations as proposed by van Everdingen and Hurst(1), and Hurst(2), the dimensionless aquifer response to a unit pressure drop or a unit fluid-withdrawal rate is needed. Also, aquifers are frequently represented in reservoir simulators using analytical response functions. Past studies have considered linear, radial, and spherical flow in the aquifers, Several studies of water influx have been described by Craft and Hawkins(3). Schilthius(4) first proposed a steady-state water influx model. A significant advance was made by the van Everdingen and Hurst(l) unsteady-state water influx model that used the response of a dimensionless aquifer v..ith time. They Considered the aquifer to be homogeneous and infinite in extent. Later, Hurst(2) presented a simplified solution of a material balance equation which used the response for linear and radial homogeneous, infinite aquifers. Miller(5), and Nabor and Barham(6) presented analytical solutions for linear, homogeneous aquifers. Nabor and Barham(6) considered both constant rate and constant pressure inner boundary conditions. They also considered the outer boundary to be infinite, closed or at a constant pressure. Mueller(7) used finite-difference techniques to compute dimensionless pressure drop or cumulative influx for few cases of a non-homogeneous aquifer. This paper presents analytical solutions for the transient behaviour of nonhomogeneous linear aquifers. Mathematical Model for Linear Aquifers A partial differential equation describing the flow of a slightly compressible fluid in porous media is: Equation 1 (available in full paper) In writing equation (1), we assume a horizontal formation, no gravity forces, isotropic properties, a single fluid of small but constant compressibility, etc. For a linear one-dimensional medium, equation (l) can be written as: Equation 2 (available in full paper) As indicated by Mueller(7), kh/ µ and Φc,h are two groups of variables which can be functions of the space variable. Thickness, appears in both groups. So one may study cases wherein either k/ µ (or k), h or Φc is a function of the space variable. Let us identify these as different cases:

Gravity recoverable gold from −13 mm crushed ore, Ester Dome, Alaska

-A pilot-scale study was performed to evaluate the potential for gravity concentration of gold from −13 mm (−1/2 in.) crushed ore from Ester Dome, AK. 9.04 t (9.94 st) of crushed ore were sampled and processed through a Pan-American jig-Wilfley table pilot plant. Plant products were sampled for assay and material balance calculations. Gold recovery through one stage of jigging and two stages of tabling was 17.4% at a concentration ratio of 615:1. Concentrate grade was 766.3 g/t (22.4 oz/st). The feed grade to the plant was calculated at 1.71 g/t(0.05 oz/st). Jig tailings, primary table tailings and secondary table tailings grades were 1.44 g/t (0.042 oz/st), 1.30 g/t (0.038 oz/st), and 5.47 g/t (0.16 oz/st, respectively. Significant coarse, free-milling gold values were shown to remain in the jig tailings. Concentrate mineralogy is discussed.

Oxygen isotopes in an oolitic ironstone and the determination of goethite δ18O values by selective dissolution of impurities: The 5M NaOH method

Treatment of iron (III) oxides with hot 5 M NaOH solution can selectively dissolve silicate impurities (Kampf and Schwertmann, 1982). In the current work 5M NaOH solutions enriched in 18O (+670%.) were employed to determine if this method could be used to purify goethites for δ18O analysis. These experiments suggest that the structural stoichiometric oxygen in well-crystallized goethites does not exchange with hot 5 M NaOH solution. Therefore, this selective dissolution method appears to be isotopically viable.18O-“normal” 5 M NaOH treatments were applied to goethite-dominated ooids of the Upper Ordovician Neda Fm. ironstone. While not completely removed by successive NaOH treatments, the impurities were incrementally dissolved in constant elemental proportions (within analytical error). Consequently, the δ18O value of the endmember goethite could be determined by material balance calculations. This goethite δ18O value is −1.0%. for all analyzed samples of Neda Fm. ooids, including those from occurrences about 200 km apart. The spatial uniformity of the oolitic goethite δ18O values suggests uniform conditions of goethite formation. The conditions might have been those of a low latitude continental weathering environment.

Design of packed towers for absorption and desorption of carbon dioxide using hot promoted K2CO3 solution

Abstract The mathematical model developed for the design of a packed column for the absorption of carbon dioxide in hot promoted potassium carbonate solution ( Gas Sep Purif (1989) Vol 3 p 152; (1990) Vol 4 p 58) is extended to take into account the condensation of water present in the gas feed to the absorption column. The revised model is further extended to obtain the design of the desorption column where the feed gas is pure steam and the exit gas consists of carbon dioxide and steam. The integration of the relevant differential equations for the desorption tower is done by starting the calculations from the top conditions of the column where the exit flow rate of steam and exit gas temperature are guessed and verified later after completing the tower calculations with the help of the overall material and energy balance calculations.