Some Studies on the Thermodynamic Relations in Multi-Component Systems Citation Schaafsma, Jan Gerard (1934) Some Studies on the Thermodynamic Relations in Multi-Component Systems. Master's thesis, California Institute of Technology. doi:10.7907/8s06-q590. https://resolver.caltech.edu/CaltechTHESIS:09242024-152737146 Abstract No abstract. Item Type: Thesis (Master's thesis) Subject Keywords: (Chemical Engineering) Degree Grantor: California Institute of Technology Division: Chemistry and Chemical Engineering Major Option: Chemical Engineering Thesis Availability: Public (worldwide access) Research Advisor(s): Unknown, Unknown Thesis Committee: Unknown, Unknown Defense Date: 1 January 1934 Record Number: CaltechTHESIS:09242024-152737146 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:09242024-152737146 DOI: 10.7907/8s06-q590 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 16755 Collection: CaltechTHESIS Deposited By: Benjamin Perez Deposited On: 24 Sep 2024 16:47 Last Modified: 24 Sep 2024 16:57 Thesis Files PDF - Final Version See Usage Policy. 5MB Repository Staff Only: item control page

Up to the present time very little work has been done on the application of thermodynamics to liquid-gas multi-component systems. There are in existence mar,y tables and charts giving the thermal properties of pure substances, but the only notable example of a chart 1 for a mixture is the temperature-entropy chart for air. Due to the fact that the critical temperatures of the main constituents of air are so close together, the chart for air is quite similar to that for a _single component system. It does not show the critical phenomena peculiar to a mixture,; nor does it show very clearly the difference in behavior in the two-phase region. This thesis has been r1ri t ten with a view to pre sen.ting the thermal properties of a two-component system in v1hich the critic al temperatures of the components are sufficiently far apart to bring out clearly the properties peculiar to a mixture. A method is pre- sented nhich employs siru1;le thermodynamic relations to derive these quantities from complete pressure-volume-temperature data over the range studied, and from the change of one thermal quantity vii th temperature. The solutions have been obtained graphically, but could have been carried out analitically by fitting empirical equations to the pressure-volume-temperature data. This general method, which is valid for a1;y uul ti-compone ,:t system, no matter how complex, has been applied to the construction of a t eo perature-entropy chart for a fifty mol-percent mixture of methane and propane. -2- Scources of Data From the pre ssure-volume-temperature data alone one can calculate, by means of rigorous thermodynamic relations, the isothermal changes in all the thermal properties of a system, but not the changes with temperature. To be able to refer the values of the thermal properties under any condition to a single arbitrary basis, one must know the change in one thermal quantity with temperature holding one physical condition constant (e.g. at one constant pressure or one constant volume). obtained from two sources. For this study, the data has been The pressure-volume-temperature data were taken from the ~ata of Sage, Lacey, and Schaafsr!la on the methanepropane system. 2 Density data for the fifty mol-percent mixture were converted to volume data. When the experimental work was originally done, not much stre ss r,as placed on the low pressure region below about thirty atmospheres, ·ui th the result that the iata were not reported to as many places/'-or to as high an accuracy in this region as is required by thermodynamic calculations. Therefore the volumes a-c. these lovwr pressures were calculated from compressibility factors (.E.Y) by extrapolation of isotherms of these factors for the higher P,v; pressures to atmospheric pressure. The volumes were plotted as iso- bars on the temperature-volume plane, a portion of which is shown in Figure 1. C and c' are the critical and cricondentherm points. point of maximum pressure occurs at about 4o0 c. The The two-phase region is different from that of a pure substance in that the isobars are not horizontal. The thermal data consisted of an empirical equation for Cp for the lighter paraffins, taken frora the literature) This equation -3- has been verified recently for propane,4 as well as for othe r para ffins and pa raffin mixtures by various investi z;ators.3 The equation is: MCp = 4-. 4+4.4n +(O.O12+O. OO6n)t, whe re r. is the numbe r of carbon atoms i r- the molecule. In this cas e n equals 2. Methods of Calcul ation Froo Figure l value s of ( ~l)pwere obtained at a series of temperatures for each isobar by me a suring the slope g r aphically. plot wa s ma de of isothe r ms of these slopes against pressure. sho ws three such isothe r r:,s. A Fi gure 2 The va lue s of the slo pes decrease with i ncrease in pressure at constant temperature in the superheated re gion until close to the vapor dome. Ne a r the vapor dome the isobars on the temp e r ature-vo lume plane i ncre as e consi dera bly in slope before enteri ng the dome. This ef fect can be se en in Figure 2, where t he 50°and 50°0 isothe rms double ba ck on them selves i n the supe rhe ated re 0 ion. The slopes de cre as e almost a s rapi d ly i n the two pha se re gion a s in the sup e rheated region. Between 80 and 90 atmos phe res these two isotherms ente r t he conden sed re gi on with a sudden shift in slop e . At the point of maximum temperature at a temperature between _j0°and 50°c this shift doe s not occur. In the condensed region the slope decreases slowly. The slo pes in t he one pha se reg ions beha ve sin:il a rly to t hose for a sing le-component systeiu . I n the t wo phase r e;::;ion the slopes in the l atter c c..se ar e, of course, infinite. The 70°c curve is above the cricondenthe:rm t emperature, but still s how s t ne influence of the vapor dome. Figures 1 and 2 and the specific heat equation give all the value s ne eded for the complete dete rmination of all the t herma l -4- properties within the range covered. * From thermodynamics comes the relation5 ( dv 1\ = ( c) s 1\ at P op t Hence integration of the area under the isotherms in Figure 2 will give the isothe rmal changes in entropy from some arbitrary pressure to any other pressure and to the two-phase boundaries. In the case of a pure substance this relation becomes indeterminate in the twophase region. Hov,ever, the Clapeyron equation can be use d instead Another useful relation is given by the equation, v = T ( ~) + ( dh )t CJt P c3 p 5 • By reading off the appropriate values from Figures 1 and 2, one can calculate (~~)t for a series of pressures at a nwuber of temperatures and plot these values as isotherms against pressure, By integration of the area under these isotherms one can get the isothermal change in heat-content from some arbitrary pressure to any other pressure and to the two-phase boundaries. In this study both the change, in entro€! and the change in heat-content were calculated from one atmosphere, since the thermal data is given at this pressure. The values of Op were plotted against ·r, and a plot was also made of _£p against T. 'rhe change in entropy and the change in heat- T content at one atmosphere from 20°c to any other temperature could then be calculated by graphical integration of these two curves, since d s = ~ t and dh = Cpdt • 5 The above calculations su£fice to permit one to draw up a temperature-entropy diagram. The basis of such a chart is usually chosen as saturated liquid at some arbitrary temperature at the lower *Seepage 11 for key to symbols used in this thesis. -5- end of the temperature range studied, the entropy and heat-content being called zero at this point. as saturated liquid at 20°c. In this study the basis was chosen Since the change in entropy with pressure was known at 20°c froru the above calculations, the value of the entropy at the same temperature and at one atmosphere could be directly obtained. Knowing the change in entropy as a function of temp e rature at one atmosphere, a one atmosphere line could be drawn on the temperature-entropy plane, starting with the 20°c value. In like manner a one atmosphere line could be put on the heat-content temperature plane. Then the other constant pressure lines, as well as the two phase boundaries, could be put on both planes because the differences of entropy and heat-content from the one atmosphere values are known at a series of constant temperatures. The two phase boundary lines were rounded to meet at the cricondentherm temperature and the critical temperature was plotted. Constant heat-conent lines were drawn on the temperature- entropy plane by transferring temperG..ture-pressure intersectioi1s on the temperature-heat-centent plane to th.e temperature-entropy plane. 3 and 4 show the results of such plotting. Fi E;ures The basis of these charts has been changed somewhat, as will be explained later, but the principles of their construction are still just as valid. Lines of constant volume can also be put on the temperatureentropy plane by using the temperature inte rsections of the constant volume lines with the isobars in Figure 1. Unless there are a great many pressure lines, this method 6i ves only a fe w points for each line, since the volume lines and the pressure lines have somewhat the same -6- slope on the temperature-entropy plane. More points can be obtained by plotting constant entropy lines on Figure 1 by transferring temperature int ersections of the constant entropy lines with the isobars from Figure 5 to Figure 1. on Figure 1. Two such lines are shown as dashed lines Then temperature intersections of the constant volume lines with the constant entropy lines can be transferred to Figure 5 to give the necessary points. Another method of obtaining the constant volume lines is to draw isotherms on a pressure-entropy plane from points read from Figure 5. One can then find the entropy of each constant volume at a series of t empe ratures from the intersection of the equilibrium pre ssure for that volume and temperature with the corresponding isotherm. For a pure substance the volume lines in the two phase region are obtained from the relation vx=-- vg+ v (1-x) where xis 1 the weight percent in the vapor phase. Entropy and heat-content are not the only thermal quantities that can be obtained from such data as this. Having proceeded this far, one is in a very good position to calculate any of the other thermal properties of the system. For example, one can calculate the specific heat at constant pres sure for any pressure and tempe r ature in either one or tv,o phase regions by measu ring the slope of the pressure line on the t empe ratu re-entropy pland and applying the equation cP as T:;: (6-t )p . Similarly one can obto.in the specific heat at constant volume or the specific heat of the saturated gas or liquid. 'r ae Joule-Thompson coefficient may be obtained from the relation ft= (j~)h . It must be r emembered that the above method is applicable to iny system whatsoever that is in equilibrium with it's surroundings. -7- It could easily be applied to a corn~; lex mixture like a crude oil - natural gas system, or to a system in which solid phases occur. It is not limited to the case where the specific heat is a known function of temperature for one pressure in the gaseous r egion. One could just as easily apply a specific heat at either constant pressure or constant volume in the gaseous, two phase, or liquid region. One could use the Joule-Thompson coefficients if they were known as a function of temperature. Discussion of Accuracy In the construction of Figures 5 and 4, it was found that the basis originally chosen gave many negative values of the thermal quantities. Therefore the basis of the entropy values was shifted to 0.10 for saturated liquid at 20°0, and the basis for the heat-content values vias shifted to 100 for the same conditions. It v:as then found that the portion of the ci1~:,. rt including the liquid region and a part of the two phase region below 55°0 gave r c sults which were obviously incorrect. The heat-content lines recrossed the pressure lines, giving the peculiar effect of setting a lower limiting pressure below which a constant heat throttling could not occur. Similarly the pressure lines recrossed the entropy lines, thus setting a lower limit beyond which an isentropic expansion could not proceed. correct. 'l'he calculations ;·,ere rechecked and found The abnormality can be explained by the lack of precise data at low pressures. Since the themal vali..J.es at i1igher pressures are based upon those at lower pres sures, an error in these will throw off all the values at the same temperature. At the lower temperatures the vapor pressure was so low that none of the data in the gaseous region was as -8- accurate as was necessary for good thermal calculations. o The comp res- 0 sibility curves at 20 and 30 C were drawn in largely by extrapolation, and probably the values thus obtained were inaccurate. Fror:i other cal- culations on this system and from comparison with charts for pure hydrocarbons it is believed that the error is in the direction of too small a change in the heat content and the entropy in the gas region at the lower temperature. Probably the saturated gas line should be much more vertical, which would push the saturated liquid line and the liquid region sufficiently to the left to remove the abnormal features, It was not feasible to make further experiments to improve the data, so the lower left hand corner of the chart wa s omitted. This calculation points out the necessity of having good pressure-volumetemperature data, es pecially close to where the thermal data is taken. Errors in the experimental deterraination of thermal quantities will introduce a proportionate error in the final results. Srrors in the pressure-volume-temperature data will cause larger errors in the isothermal changes in therwal quantities because the errors are accentuated by the process of trucing derivatives. Discussion of Results Figure .3 is strikingly different from a temperature-entropy chart for a pure compound. The pressure lines in the two phase region are not horizontal as with a pure substance. Their extreme steepness is due to the great difference in critical temperatures of the two components. The highest temperature on the t,·10-phase boundary is the cricondentherm, 0 1 , while the critical tempero.tur~ C, lies considerably -9- lower and to the left of this point. The single phase region is very simil ar to that of a pure substance except near the critical region. The single phase gas region extends over on the left of the chart down to the critical temperature. This means that retrograde condensation will occur between the critical and cricondentherm temperatures. For instance, 0 if the mixture in the state of a gas at one atmosphere and 50 C were compressed isothermally, it would start to condense at an entropy of about ·o.22. The amount of condensate would at first increase and then decrease. At an entropy of 0.105 the mixture would be completely gaseous again, since it would enter the single phase region above the critical temperature. Pressure lines in the liquid region below 90 atmospheres enter the two phase region. 90 atmospheres is the maximum pressure at which two phases of this mixture can exist in equilibrium and a 90 atmosphere isobar would just be tangent to the two phase boundary. 88 atmosphere isobar would pass through the critical. An Between the critical pressure and the maximum pressure one would also have retrograde condensation. If the temperature were raised along an 89 atmosphere isobar the mixture would first start to vaporize and would then condense again and enter the liquid region a little below the critical point. Thermal charts for the usual crude oil-~atural gas systems under the same conditions of temperature and pressure would be considerably different because the critical region would fall far above this temperature range. Also, the two phase region would extend to pressures far below atmospheric because of the very low vapor pressures of some of the constituents of the crude oil. The lines in the liquid region would be much more pa rallel and evenly spaced, as would be the case for the mixture of -10- methane and propane at temperatures far below the critical. The slopes of the heat-content lines in the liquid region of the methane-propane mixture change in sign from negative to positive at sufficiently high pressures, so that a constant heat-content expansion would result in a rise in temperature. The inversion of this Joule-Thompson effect comes closer and closer to the two-phase boundary as the temperature is lowered. With a crude oil-natural gas system at temperatures far below the critical,one would probably not even notice this inversion,since it would occur so close to the two-phase boundary. From this study several conclusions can be drawn. In the first place the original data used in such calculations must be of a high degree of accuracy. The calculations involved are somewhat tedious, but are perfectly straightforward and involve no assumptions. Such calculations might be employed advantageously with other data to put them in a more useable fonn and to derive values of many useful properties which would otherwise not be available. Although the particular data here presented is not of ve ry much practical significance,a series of similar charts covering a range of compositions for a series of hydrocarbon systems would be of very great value in solving many of the problems occurring in production and refining. -11- List of Symbols p Pressure on system in standard atmospheres (760 mm.) v Volume in cubic centimeters per gram. t Temperature in degrees centigrade. T Temperature in degrees Kelvin. s Entropy in calories per gram per degree Kelvin. h Heat-content in calories per gram. C Heat cap a city in calories per gram per degree centigrade . . -12- Reference a 1 Hausen, Der Thomsen-Joule Effekt 2 Sage, Lacey, and Schaafsma, Ind. and 2ng. Chem. 26, 214 (1954) 3 Lewis and MacAda.ms, Chem. and Met. Eng., 456 (1929) 4 B. H. Sage, Ph.D. Thesis, California Institute of Technology (19.34) 5 Lewis and Randall, Thennodynamics and the Free Energy of Chemical Substances, pp. 152, 15.3, and 134, 0 0 n, 80 0 60 u w a:: :::> ~ a:: LLJ a. ~ LLJ 40 t- I 20 C.C. PER GRAM I 10 Figure 1 15 (/) w a:: 150 --iW-- - - - t - - - - - - - + - - - - - - - - t - - - - -----i w I Cl. (/) 0 ~ I- < w a: :::) (/) (/) w a:: Cl. .10 .30 Figure 2 u 0 160 w 80 Cl: ::::> ..... < a: w CL ~ w -1- ..... so % ENTROPY CALORIES PER GRAM Figure J PER °C 0 u w 80 - 0'.: - i- - - i ::::, ~ w ..-j----+----a.. ~ w Cl'.: I- r-- 20 - - + - - - HEAT CONTENT 100 I 120 CALORIES PER GRAM I 1"0 Figure 4 160