Mollier Diagram Research Articles

A Theory of the Wilson Line for Steam at Low Pressures

An algebraic method for determining the onset of condensation—the position of the Wilson line on a Mollier diagram—in steam at low pressures is presented. The method is based on the assumptions that the exponent of the nucleation rate expression is large and that the nucleation pulse duration is small. Under these conditions the growth integrals can be evaluated for specific rate expressions for the formation of stable droplets and their subsequent growth. The onset of condensation for a specified expansion rate is then determined by the solution to a system of algebraic equations. The method is illustrated using the classical nucleation rate and the droplet growth rate expression of Gyarmathy. The analytical solution agrees well with a published exact numerical solution by Gyarmathy. An extensive comparison of the predictions of the present method with the experimental results of Yellot (1933), Gyarmathy and Meyer (1965), Barschdorff (1965), and Moses and Stein (1978) is presented. The favorable comparison suggests that the method provides an efficient means for predicting the onset of condensation for various initial states and various expansion rates. The predictions of the theory on the role of the expansion rate, the initial stagnation conditions, and the history of the expansion are discussed.

Homogeneous Nucleation in Steam Nozzle Condensation

AFTER the introduction of Laval nozzles into steam turbine design, it became apparent that steam may be expanded beyond saturation before it condenses. Stodola1 observed that condensation was delayed and appeared in the supersonic part of the nozzle and pointed out that the expansion time was too short for appreciable condensation to take place on any heterogeneous nuclei likely to be present. It was found2,3 that the steam attains a supersaturation ratio 3 < p/p ∞ < 8; the locus of the onset of condensation in rapid nozzle expansions when plotted in a Mollier diagram is called “the Wilson line”; this empirical line lies about 33 kcal kg−1 below the saturation curve. Oswatitsch4 suggested that such condensation is initiated by homogeneous nucleation5–9. By combining a solution of the equations of motion of compressible nozzle flow involving heat addition with the “classical” rate equation7 of homogeneous nucleation and his own expressions for the growth rate of droplets, Oswatitsch found agreement between theory and experiment2,3. Hill10 and other authors11 using improved computational techniques obtained similar agreement. Consequently it is generally considered that the “classical” theory of condensation predicts not only the condensation of water vapour when in the presence of air, as in a cloud chamber12–15, but the condensation of pure steam as well. We show here that in the case of pure steam in nozzles the use of the “classical” theory of nucleation does not lead to agreement with experiment.

Instability in Condensing Flow of Steam

The condensing flow of supercooled steam in a convergent–divergent nozzle is treated theoretically. It is shown that when condensation occurs in the supersonic section of the flow, if the heat released by condensation exceeds the quantity which reduces the flow Mach number to unity, the flow becomes unstable. As a result of the investigation, critical instability boundaries have been plotted as a locus on the Mollier diagram which divide the initial entry conditions to a nozzle into stable and unstable zones. The findings have been verified experimentally on a converging–diverging nozzle and good agreement obtained between theoretical and experimental results.

Thermodynamic properties and nozzle flow of hydrogen to 1000 atm and100,000 K

Thermodynamic properties and nozzle flow calculations for high temperature and pressure hydrogen, presenting results in Mollier diagram

The Mollier diagram of normal hydrogen at temperatures of 16 to 100° k and pressures up to 500 kg/cm2

Experimental data on thermal properties are used to construct the Mollier diagram of normal hydrogen at temperatures of 16–100° K and pressures of 5–500 kg/cm2.

Structure of Radiation-Resisted Shock Waves with Vibrational Nonequilibrium

The continuum equations of gasdynamics are used to study the effect of thermal radiation and vibrational nonequilibrium on the structure of a shock wave. Emphasis is on a description of the gasdynamic phenomena involved rather than on numerical results. For example, a Mollier diagram interpretation, useful for understanding radiation-resisted shock waves, is given. The principal results stem from an analysis of the singular-point structure far upstream and far downstream in the flow. This analysis indicates in a heuristic fashion that a unique solution exists. It also shows that two parameters, a Mach number and the ratio of a radiation length to a relaxation length, primarily govern the qualitative behavior of the flow. Thus, for a specific range of values of these parameters, the vibrational energy overshoots its final equilibrium value. For a different range, a net amount of radiant energy is transferred from the hot gas behind the adiabatic shock to the gas far downstream.

R. Mollier und Das i- x-Diagramm

On the occasion of the hundredth birthday of R. Mollier and the fortieth anniversary of the publication of the i- x diagram, the essence of this diagram and the significance of the various families of curves are discussed. The article presents a survey of the diverse fields of application and a short synopses of problems which can be solved by use of the diagram. Recently suggested alterations, originated from the prolific influence of Mollier's diagram and intending both, further development of the diagram itself and extension of its applications are stated. Mollier's diagram is also useful in characterizing states of mass by states of air arising from a body surface. Resulting conclusions and evolutions are presently in the process of elaboration.

The Critical and Two-Phase Flow of Steam

The results of a digital computer and analytic study of the critical flow of a compressible fluid are presented in this paper. The expanding flow of a fluid in a single-phase region as well as the expansion of a fluid to a two-phase region is considered and described by analytic expressions relating choking velocity, critical pressure ratio, and flow per unit area characteristics. A comparison is made of the analytic results which assume a constant value of the isentropic expansion exponent, with the digital computer results using the actual properties of steam. All analyses assume the fluid to be in thermodynamic equilibrium. A skeleton Mollier diagram is presented for steam showing the exponent in the wet and superheated regions. The choking velocity is presented in plot form as a function of the inlet conditions as well as state point conditions; critical pressure ratio is presented as a function of inlet conditions. The critical flow per unit area is presented in the form of a factor K plotted versus inlet conditions; this factor K when multiplied by inlet pressure produces the desired value of critical flow.

A Note on the Impact Pressure Loading of a Rigid Plastic Spherical Shell

approximation hand computation utilizing normal-shock equations and graphical thermodynamic-state data. The accuracy of the method is dependent on the assumed curvature between data points in plotting the tabular data of Gilmore and the error in logarithmic interpolation between curves. I t was estimated that these errors are between 1 and 10 per cent, with the error of the major portion of the data less than 5 per cent. The charts, in general, are not as accurate as could be obtained with an electronic computer, but the method is justified on the basis of the relatively low cost in time and effort. The charts should, nonetheless, prove useful to those working in hypervelocity gasdynamics until more refined calculations are available. The series of gasdynamic charts of reference 1 include the following properties: flow velocity, density, pressure, temperature, internal energy and enthalpy associated with a traveling normal shock, stationary normal shock (bow wave), stagnation point, and reflected normal shock. Curves are plotted versus incident-shock velocity for initial air densities of 10 1 to 10 ~ times atmospheric density and an initial temperature of 273.2°K. The range of incident-shock speeds is from Mach 10 to Mach 50 or to the limit of the thermodynamic data, 24,000°K. Included is a plot of Gilmore's data in the form of a Mollier diagram. The thermodynamic-state data were plotted as enthalpy versus density with lines of constant entropy and temperature because the curves become more orthogonal and, therefore, greater accuracy in reading is obtainable. This general plot is presented in Fig. 1. I t is felt that the gasdynamic charts are of general usefulness to many scientists and engineers and may satisfy a current need. However, the volume of the charts precludes publishing in a technical journal. The report is available to those interested and copies may be obtained by request to the author.

A note on the compression of air through repeated shock waves

The results of calculations of the compression of air by repeated shock waves are compared with the perfect-gas values given by Evans &amp; Evans (1956). The comparison emphasizes the increasing divergence of real from perfect-gas results as shock strengths are raised. The equations relating conditions across a shock wave are obtained in a convenient form for solution using a Mollier diagram.

Hypersonic Flow Characteristics Including Real Gas Effects

N A RECENT NOTE1 a method has been proposed to extend the applicability of the perfect-gas compressible-flow equations for air to higher Mach Numbers and stagnation temperatures by utilizing suitable approximations for the variation of specific heat with temperature. It was also suggested1 that the foregoing method could be extended for normal and oblique shock proper­ ties. The purpose of the present note is to point out the existence of several references2-4 wherein the actual state of the gas, in­ cluding real gas effects, has been accounted for in the determi­ nation of hypersonic equilibrium flow properties and shock char­ acteristics. In reference 2, Feldman, using an IBM 650 computer, presents in chart form the equilibrium properties for normal and oblique shocks in argon-free air in hypersonic flow and in shock tubes. A Mollier diagram based upon National Bureau of Standards data and the statistical-mechanical calculations for the speed of sound by Logan and Treanor is also included, from which isentropic flow properties can be determined. In reference 3, a program following the methods of Feldman 2 and Moeckel,5 but using the more up-to-date 1956 ARDC Model Atmosphere, was run on an IBM 701 computer. The results,

Flow of two‐phase carbon dioxide through orifices

AbstractExperimental flow rate data are presented for saturated liquid, saturated vapor, and two‐phase liquid‐vapor carbon dioxide through a convergent nozzle and a square‐edged orifice. The data cover the range from the triple‐point pressure to the critical pressure. Charts have been prepared for this complete range at critical flow. Results are also presented for subcritical flow.The tests at various back pressures indicate that the saturated liquid behaved as a cold liquid without evaporation ahead of the throat.Saturated vapor became supersaturated in the nozzle, and the vapor behaved as if no condensation occurred.Equations are presented for the flow rates of saturated vapor, and two‐phase mixtures in the critical flow region.A Mollier (pressure‐enthalpy) diagram is used to determine the flow rates of saturated vapor and two‐phase mixtures where supersaturtation takes place. In these cases, the lines of constant specific volume or density are extrapolated from the superheated region into the normal two‐phase region to obtain values corrected for supersaturation.

Mollier Diagrams for Water Near Bubble Point

Abstract In many problems of engineering interest a knowledge of the thermodynamic properties of water in the heterogeneous region near bubble point is of value. A series of Mollier diagrams for water at states from bubble point to a quality of 0.05 was prepared and similar diagrams relating to the behavior of the condensed liquid were included. The Mollier diagram was employed because of its utility in connection with graphical operations of a thermodynamic nature pertaining to the flow of homogeneous and heterogeneous fluids near bubble point. An example of the application of the diagrams to the solution of the simple problem of flow in short tubes was included.

Calculation of a Mollier Diagram for the Decomposition Products of Aqueous Hydrogen Peroxide Solutions of 90 Weight Per Cent H2O2 Content

Calculation of a Mollier Diagram for the Decomposition Products of Aqueous Hydrogen Peroxide Solutions of 90 Weight Per Cent H2O2 Content

PHYSICAL FACTORS AFFECTING THE KILNING OF BARLEY AND MALT

The main physical factors affecting the kilning of barley and malt are the quantity of moisture to be removed, the temperature and humidity of the air entering and leaving the kiln, the temperature to which the air is heated, the hygroscopic state of the grain and the volume of air passing. The effect of weather on natural draught kilns can be expressed in an arbitrary Drying-Power Table. The hygroscopic state of barley or malt has been measured by determining the vapour pressures at various stages of kilning. Restriction of vapour pressure due to shrinkage of internal pores begins in barley at about 17% and in malt at about 12% moisture content. Knowing the degree of such restriction, it is possible using the Mollier Diagram to calculate the theoretical volume of heated air required at any stage of kilning and hence the theoretical fuel consumption. With suitable recording instruments, the actual volumes of heated air passing can be measured and the loss of efficiency due to imperfect saturation of the out-going air, and to heat losses, etc., can be determined. Draught control, combined if possible with recirculation of heated air during curing, is the principal factor affecting fuel economy.

New Equation of State for Air valid to 1, 000at.

A new equation of state for air was obtained on the basis of the experimental results of Amagat and Witkowski as follows : ν=29·27 T/P+1·09059·10-3-8·50053·10-3r1·64+4·25852·10-9Pr24·9-6·27588·10-16p2r2·92+2·78867·10-23P3r3·17 where P is pressure kg/m2, T=273·16+t is absolute temperature °K, t is temperature °C, ν is specific volume m3/kg, r=100/T. This equation is valid to 1, 000 at above 0°C, to 120 at above -35°C, to 80 at above -80°C and to 40 at above -100°C. Calculation was made of the numerical values of ν, i and s from this equation and its derived equations, and it was constructed Mollier's diagram and temperature-entropy diagram. The calculated values of specific heat at constant pressure is in accord satisfactorily with various experimenal results.

A mollier diagram for the internal-combustion engine

Summary Quantitative calculations of the effects of flame propagation and of piston movement on the temperatures attained in the various parts of on Otto-cycle engine charge during combustion are made relatively simple by the use of a Mollier diagram of the properties of the combustion products of octane and air. The method is applied to a consideration of the effect of spark advance on the temperature attained ahead of the flame front, and it is concluded that the results are significant in studies of engine knock.

A Series of Enthalpy-entropy Charts for Natural Gases

Abstract Enthalpy-entropy diagrams are presented for natural gasses of 0.6, 0.7, 0.8,0.9, and 1.0 gravity over the pressure range of 5 to 10,000 lb. per sq. in. andtemperature range of 32 degrees to 700 degrees Fahrenheit. The charts indicatedirectly the work requirement and temperature rise for adiabatic compression ortemperature change for free expansion of natural gases. Computation and Uses of Charts The Mollier diagram in which the enthalpy (heat content) is plotted against theentropy with lines of constant temperature, pressure, and in some cases volume, has been found most convenient when dealing with the compression, expansion, and flow of fluids. In dealing with the flow of fluids, the sum of the increasein the enthalpy plus the increase in kinetic energy plus the increase inpotential energy of the fluid in flow, is equal to the sum of the heat and workadded to the fluid while flowing between the entrance and exit of the flowsystem. In cases where there is no significant change in potential energy or in kineticenergy (velocity), it follows that the increase in enthalpy is equal to thetotal energy supplied to the fluid. Under such conditions the changes in theproperty of the fluid as it flows through a throttling valve, choke, or anyother similar arrangement, may be read directly from the enthalpy-entropydiagram by following a horizontal line between the known pressures. When compression or expanding a gas by which means of a compressor or engine inwhich no heat is added to or subtracted from the gas, but only work done, thechanges in the properties of the gas may be determined along a vertical line ofconstant entropy between the entering and exit pressures. The power requiredfor the compression of the gas may be readily determined by converting theincrease in enthalpy into the desired units. The enthalpy-entropy diagram for natural gases is to the gas engineer what thesteam diagram is to the steam-power engineer. For this reason it would beextremely convenient of a reasonably satisfactory enthalpy-entropy diagramcould be prepared as a function of the gravity of the gas. A careful study ofthe known properties of natural gas indicates that this is possible. The effect of temperature upon the enthalpy at consistent pressure is expressedas the ?heat capacity? or ‘specific heat’ of the gas. The best available datafor natural and petroleum refinery gases indicates the relationship shown. Fromthis it is clear that the specific heat of natural gases is a function only ofthe gas gravity and the temperature at atmospheric pressure. The effect ofpressure on the enthalpy of natural gases is dependent upon the pressure-volume-temperature relationships. T.P. 1747

An electrical detector of condensation in high-velocity steam

A new method has been developed for finding where condensation in a convergent-divergent steam nozzle commences. When a short length of fine wire, mounted on a rod, was traversed axially through the nozzle, it was found th at its resistance altered sharply at a certain point. The position of this point agreed closely with that of a small sudden pressure rise in the nozzle, which an earlier investigation had shown to indicate the beginning of condensation of the supersaturated steam. The mean temperature of the wire upstream from this point was found (as predicted by Griffith’s theory) to be nearly the same as that of the steam at entrance to the nozzle; it was greatly in excess of the temperature of the high-velocity steam passing the wire. The wire was heated electrically, and the Nusselt number of heat transfer to the steam was measured under the prevailing supersonic conditions. The Wilson line, showing the position of condensation on the Mollier diagram, was determined, and is compared with the line previously obtained for another nozzle of different shape.

RECOMPRESSION PHENOMENA IN STEAM NOZZLES: PART I

The purpose of the investigation was to find when and how recompression occurs in the flow of steam through a nozzle, its causes and effects, how the results can be predicted, and what losses are caused by these phenomena. Four types of recompression were observed: equilibrium, latent, vena contracta, and shock recompression.It is shown that steam in flowing through a nozzle of varying cross-section responds to the changes of area within certain limits and recompression phenomena may be expected. The Venturi effect has been observed in single nozzles, and in both parts of a composite nozzle.The behaviour of steam in equilibrium recompression in convergent-divergent nozzles can be predicted by means of an equation from which a valuers obtained for the pressure pr at which overexpansion ends and an increase in pressure begins; the pressure pr depends on the inlet pressure to the nozzle and on the difference in pressure at inlet and outlet.The recompression due to the contraction of the section can be controlled and avoided by suitable provision in the design. The compression following a change of state of a flowing fluid can be controlled by adjusting the rate of expansion of the fluid in specified pressure ranges. Novel evidence of latent recompression is found in a break or notch in the pressure expansion curve plotted from search tube observations.The experiments were carried out at inlet pressures within the range at which turbine-condition curves may cross the saturation line between the superheat and wet regions on the Mollier diagram, and the effects of recompression to be expected under varying load conditions can be interpreted for particular nozzle designs.Comparative studies of the effect of the nozzle shape on recompression phenomena and the losses resulting therefrom have indicated that the conditions for favourable efficiency require a larger inlet radius at the throat and a length shorter than the lengths commonly found in practice.