HUNGARIA JOURNAL OF INDUSTRIAL CHEMISTRY./ VESZPRÉM Vol. 14. pp. 22;'-236 (1986) COMPUTER DRAWING OF ENTHALPY-HUMIDITY CHARTS z. PAKOWSKI (Institute of Chemical Engineering, Lodz Technical University, Lodz, Poland) Received: September 3, 1985 EnthaJpy-humidity charta for a gas-vllpour mixture still provide a fast and easy tool for use in process culculations in drying, solvent recovery and related procesees. They have to be individualJy cons- tructed for each new system encountered. The computer graph- plotting method provides a fast and reliuble way of enthalpy-humidity chart production. Theoretical principles of computerized i-Y chart plotting are provided. Exemplary char ts for systema of nitrogen and benzene, n-heptane, and cyclohexane are given. Introduction Traditionally parametere of a humid gas were presented on a psychrometric chart made in rectangular linear coordinates: temperature V.8. absolute humid- ity Y. Such charta containing lines of constant relative humidity, and con- stant wet-bulb temperature are still in use in psychrometry, Le. humidity measuring by wet and dry-bulb ternperature read-outs, In 1923, R. MOLLIER [1] introduced an enthalpy-humidity chart (i-Y chart) allowing for simultaneous humidity, temperature and enthalpy read-outs. The chart was initially constructed in rectangular enthalpy-humidity coordi- nates. RAl\IZIN modified the chart by introducing oblique axes, which allow for better resolution of RH lines in low humidity area. Figure 1 presenta a comparison of both types of charta, i.e. psychrometric versus enthalpy-humidity chart. The relationships on which both charta are made can be directly used in a computer programme [2], however, the graphical form of a chart is handy in practical use and visually appealíng when used for presentation of the com- puted resulta. There are many industrial processes (e.g. drying) in which mixtures of vapour and inert gas other than water and air are dealt with. Enthalpy- humidity charta for such systems are only occaaionallyencountered in literature (e.g. [3]-tricWoroethylene-air and [4]-N2-methanol; N2-toluene). On the other hand, a fully computerized production of enthalpy-humidity charta 16 226 Z. PAKOWSKI y )4-",Ir / '-l -:V ./ -:;;... Vol. 14' .....y "Fig.l Two typ es of charta for humid gasea a - paychrornetric, b - enthalpy-humidity (Mollier chart) is feasible and has already been irnplemented by the leading dríer manufactur- ers [5]. This paper presents the principles of the computerized drawing of i- Y charta, together with ex mplary charta produced in this way. p rp=- p. Summary of Thermodynamic Relationships for a Gas-Vapour Mixture Absolute humidity Y is a number of kilograms of vapour per kilogram of dry gas. Relative humidily ep is a ratio of actnal partial pressure of vapour to that of saturated vapour at the same temperature: (1) On the basis of DALTON'S law, there is a following relationship between the two humidites: Y=D rpp. P-q,P. (2) where D=MAIMIJ. (3) Dew point temperaiure to is the temperature at which the vapour in gas becomes saturated. By cooling gas below this temperature, fog is formed. Enthalpy of humid qae i is calculated additively aceording to the simplified formula: i=(CR+CAY)t+.óh.oY (4) where specifíc heats CB and CA are averaged values. A veraging is perforrned between the reference ternperature and temperature t. For substances whose melting point is below O ° ,the reference temperature is taken as O oC. The latént heat of vaporization L1hvo is taken at the same temperature. Temperature of adiabatic saturation tas corresporrds to the equilibrium between gas and liquíd, and can be calculated aceording to the following equation: 1986 Computer Drawing of Enthalpy-Humidity Charta 227 . .. t -1.as y-y:' (5) Wel bulb temperalure tWb is reached by a small amount of liquid exposed to a large quantity of gas. Contrary to the la. it is controlled by kinetics of heat and mass transfer, and therefore depends on the Lewis number Le=Sc/Pr. Theoretical (thermodynamical) wet bulb temperature can be calculated from the following formula [4]: (6) where: (7) The coefficient {3is a correcting factor usually close to 1. Geometry of the Enthalpy-Humidity Chart An enthalpy-humidity chart is made in an oblique coordinate system. The vertical axis contains an enthalpy scale and the oblique axis an absolute humidity scale. Both scales are linear. In practice the projection of the oblique Y axis on a horizontalone is used, since the range of parameters between these two is of little intere t in indu trial applications. Finally on a rectangular chart we have a grid of vertical constant Y lines and oblique constant i lines (ísoenthalps). The angle between them i arbitrarily chosen as 1350 for an air-water system, although it can be selected otherwise. As each point in the chart simultaneou ly gives enthalpy, humidity and temperature, a grid of isotherms is also placed in the chart. On top of that, lines of constant relative humidity Y are superimposed with the most important one for ip= 100%- the saturation line. For fúrther information, lines of constant adiabatic saturation temperature and constant wet bulb temperature are added. AU these elements of i-Y chart geometry are shown in Fig. 2. Y Yci ,, y ~ x Yct 16· Fig.2 Elements of enthalpy-hwnidity chart geometry 228 Z. PAKOWSKI Vol. 14. When using a computerized graph-plotter, all these lines must be converted into x, y coordinates of the plotter. Therefore equations of all basic lines of i-Y chart must be first derived. For that purpose we assume that at the origin of graph-plotter coordinates (x = O and y = O) we also have i= O and Y = O. Additionally, scaling factors for enthalpy and humidity scales 81 and 8y are set in such a way that: Y=Byl (8) (9) where l is a certaln distance from the origin in unita of the plotter (mm or inches). Lsoenthalp Equation Isoenthalp as a straight line has the following equation: (10) The intercept Yci on the basis of (8) is given by: YC1=i/Bl (ll) The slope al can be calculated from both intercepts Xli and Ycl' Usually the slope is selected in such a way that: YCIBl = LlhvoXc!By (12) It is equivalent with an assumption that t = 0° isotherm coincides with the horizental axis x (as will be proven later). ome authars select other forms of condition (12), which leads to a different location of the isotherm 0°0. Oombining (IO), (ll) and (12) we have: al= - Llhvo(By/BI) (13) and therefore the isoenthalp equation has the following form: Y= - Llhvo(BY/Bt}X+i/BI (14) Constant Y Line Equation As vertical lines they have a trivial form of: 8yX=const (15) Isotherm Equation Local slope at of an isotherrn is given by the ratio Lly / Llx as can be seen in Fig. 3. The value of Lly can be found from isoenthaips as: Lly = alLlx + !li/BI (16) On the other hand, from (14) it follows that: (17) hence: Lly at=- Llx (18) 1986 Computer Drawing of Enthalpy-Humidity Charta 229 Fig.3 Auxiliary diagram for evaluation of slope of isotherm which on the basis of (13) is equal to: at = CAtBY/81 (19) As can easily be seen from (19) the slope is constant for given t (but changes with temperature). Therefore, the isotherma are straight lines. The full isotherm equation has the form: Y=at.X+Yct (20) where Yct can be easily found from the enthalpy scale (on the vertical axis Y =0) as: (21) Finally we have: 8y Y =CAt - x +CBf,/81 81 (22) It is worth noticing that for t= O Y= O so the isotherm O oC is identical with the x axis, In the fog area, isotherma follow a different equation (that of an adiabatic saturation line), which is explained in most textbooks. This area is of littIe industrial interest and will be omitted here. Other Lines The remaining lines are not straight (in principle it amo concerns tas and twb lines). Deriving a detaíled equation for these lines is not always simple or feasíble, taking into consideration changes of specific heats and other para- meters with temperature, Other plotting procedures are then employed. The scale factora 8y and 81 are selected on the basis of the required dimensions of the chart Áx and Áy (in graph-plotter unita) and maximum values of tem- perature and humidity to be covered by the chart. They are given by the formulas: Ymax 8y=-- Ax Cntmax 81=-- Áy (23, 24) Plotting of the individual elements of the enthalpy-humidity charta is per- formed by procedures described below. 230 Z. PAKOWSKI Vol. 14. Procedure: ISOENTHALP Draws subsequent isoenthalp for agiven increment Lli. Line drawing sub- routine of graph plotter is used. Each time end points of an isoenthalp are calculated prior to plotting. The end points correspond to an intersection of the isoenthalp and chart frame. Procedure: ISOHU MID-Y Using the line plotting subroutine of graph-plotter plots vertical lines for a given increment LlY. Procedure: ISOTHERM Using the line plotting subroutine plots isotherms for agiven increment Llt. The left end of eaoh isotherm corresponds to the left side of the frame, the right end oorresponds to the saturation line or the frame side (right or upper). The intersection point of an isotherm with saturation line is computed using Eq. (2). Additionally a boiling point isotherm (dashed line in Figs. 4-6) is drawn. Procedure: ISOHU MID-RH Draws ep = const lines for given values of ep tarting with ep = 1. They are indicat- ed on the right-hand side of the chart frame. It employs a three-point parabole fitting procedure of the plotter. For a given ep the following prooedure is performed: three values of temperature are taken (starting with t= O) corresponding values of Yare computed from Eq. (2) coordinates of these three points are converted into the x, y coordinates of the plotter a ourve is drawn through these points and the procedure repeated with the next three values of t unti! t = tb or unti! Y:> Ymax' Procedure: AST LINE A function plotting subroutine of the plotter is used in general to aooommodate ohanges of speoific heats with temperature. If high aoouraoy can be saorificed, then a straight line procedure may be used. Procedure: WBT LINE Beoause of the implicit form of the relationship (6), a similar procedure as for relative humidity lines was employed. For agiven twb three subsequent values of t are taken, starting with a point on the saturation line. Then by a numerical method Eq. (6) is solved to find corresponding values of Y. After conversion of ooordinates, a curve is drawn through these points and new values of tem- perature taken. This procedure was not employed in Figs. 4-6.-: 1986 Computer Drawing of Enthalpy-Humidity Charta 231 M~LLIER CHART F~R BENZENE-NITR~GEN PRE S S U R_~ 1 O 1325. O .p A 5 C ALS RHY. S. 10. 20. 30. 40. 50. 60. 70. 60. 90. 100. \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \ \ "\ \ < \ \ \ \ \ \ \ a. \ \ \ \ \ \ .. \ \ \~ \ \ \ \ \ N\ \ \ \ \ \ \ \ \ \\ \ \ \ \ \ \ \ \ \ \\ \ \ \ \ \ \ o \ \ cb. o 200.0 400.0 600.0 600.0 HUMIOIT'r. G/KG Fig.4 Enthalpy-humidity chart for nitrogen-benzene 232 Z. PAKOWSKI Vol. 14. M~LLIER PRESSURE CHRRT F~R 101.~25. O N-HEPTRNE-NITR~GEN PRSCRLS RH~ 5. 10. 20. 30: 40. 50. 60. 70. 80. 90. 100.': N CI> '"•• '"'":II: D X a: eL ••• .L \ \ \ 200.0 \ \ lOD.O ~CO.O 8 HUM 10 1TT. G/ KG Fig.5 ,/ Enthalpy-humidity chart for nitrogen-n-heptane 1~00_0 \200.0 1986 Computer Drawing of Enthalpy-Humidity Charta 233 MOLLIER CHRRT FOR PRESSUR~101325. O CYCLOHEXRNE-NITROGEN PRSCRLS t!>~ <,-, ~ )- c, -'CI: :r: l- z: lLl RMI 5. 10. 20. 30. 40. 50. 60. 70. 80. 90. 100. gl~~t~'~~\11~~í~~t~1~~1~1~~ís)t~~=jf\~\{t~~~~i\~\i\t~it\~~t~J~~f~~~\~\t~Jt~f~~~~f;j\ \ \ \ \ \ \ \ \ \ \ \ o \ \ \ \ ~. o 200_ ji 400. o 600. o- 800. a 1000. aHUMIOITY.G/KG N., a> 7 '"\ •.. \ "\ o \ '"a:... N. Fig.6 Enthalpy- humidity chart for nitrogen-cyclohexane 234 Z. PAKOWSKI Vol. 14. A uxiliary procedurea These procedures draw picture frame, temperature, enthalpy and humidity scales and print the chart's title. Closure The above described procedure were used in the programme GRAFITE written in FORTRA for a ZETA 3653 C plotter. A series of charta for most common systems were dawn [6J. Eiqures 4-6 contain three charta for selected systems. Data used for plotting th Ee charts are given in the Appendix. The method is highly efficient and can be used for chart production in the design of elosed cycle dryers for pharmaceutical and other industries. Appendix Data Used for Plotting the Enclosed Charts 1. Specific heat of gas and vapour was calculated from the following equation: el=a+ bX lO-2T+ c x lO-aT2+ a»: lO-GTS with coefficients: b o d ben zene -0.4716 6.2177 -4.079 1.013 n-hept.mo Q. 24 5 6.0392 - 2.739 0.4144 cyclohexane -0.71171 8.0170 -4.4361 0.91344 nitrogen 1.05679 - 0.1973 0.4948 - 0.18834 a b o d 2. Specific heat of liquid was ealcula ted from the following equation: cl=a+ sx lO-2T+ex 1O-4T2+dx lO-7T3 with coefficients: ben zene -6.2007 6.472 -1.8296 1. 459 n-heptane - 0.12796 1.651 -0.4262 0.46061 cyclohexane - 5.3759 5.6061 -1.4695 1.3510 3. Saturated vapour pressure was oalculated from the following equation: /' IOgIO (P./133.322}=a-b/(e+t) 1986 Computer Drawing of Enthalpy-Humidity Charta with coefficients: 235 a b c ben zene n-heptane cyclobexane 6. 9272 6.90253 6.84941 1203.531 1267.82 1206.001 219.888 216.823 223.148 4. Molar masses and latent heat of vaporization at OoC were as follows: AxAy ll:! at CA CB cAl ca D L1hv L1hvo i Le l MA. MB P p p. 81 8y T t tms" x YYmax ~ ffJ M ben zene n-heptane cyclohexane nitrogen 78.11 100.20 86.17 28.01 446 381 406 YMBOLS dimension of chart in x-direction, mrn dim nsion of chart in y-dlrection, mrn slope of the isoenthalp, - slope of the isotherm, - specific heat of vapour (moisture), kJ/kgK specific heab of gas (inert), kJ /kgK specifío heat of liquid (moisture), kJ/kgK humid heat - Eq. (7), kJ /kgK rnolar mass ratio, - latent heat of vaporízation, kJ /kg lo.tent heat of vaporization at standard temperature (O OC),kJ /kg enthalpy of humid gas, kJ/kg dry gas LEVlS number, - distance from the origin of chart, mm molar mass of moisture, kmol/kg molar mass of gas, kmol/kg ambient pressure, Pa vapour pressure, Pa saturated VapOlll' pressure, Pa scaling factor for enthalpy, kJ/kg mm scaling factor for humidlty, kg/kg mm temperature, K temperature, oC maximum temperacure on the chart, oC horizental coordinte of graph-plotter, mm absolute humidity, kg/kg dry gas maximum absolute humidity on the chart, kg/kg dry gas vertical coordinate of graph-plotter, mm correction factor in Eq. (6), relative humidity (RH), - 236 Z. PAKOWSKI Vol. 14. Subscripts and superscripts as wb * adiabatic saturation point wet bulb point saturated averaged between standard and given temparature REFERENCES 1. MOLLIER,R.: Zeits. VDI, 1923, 67, 869. 2. BROOKER,D. B.: Trans. of the ASME, 1967, 558. 3. MROWIEO,M., CIESIELOZYK,W.: Inz. iAp. Chem., 1976, 1, 22. 4. KEEY, R. B.: Iriteoductíon to Industrial Drying Operutions, Pergamon Press, 1978. 5. MOLLER,J. T., HANSEN, O.: Proc. Eng., 1972, 53, 84. 6. PAKOWSKI, Z.: Enthalpy-Humidity Charts. In: Handbook of Industriul Drying, ed. A. S. Mujumdar, Marcel Dekker, 1986. PE3IOME .llHarpaMMbI nma sararmax-arraroconepscaaae raaoao-napossrx cMeceA OKa3bIBaeTCIIsecswa rro- ne3HblM HHCTpyMeHToM B pacxérax npoueccos cynrxa H H3Bne'feHHll paCTBOpHTeneJ:!.3TH nnarpass- Mbl .nOnlKHbl 6bITb cocraanensr B COOTBeTCTBIDfmy'leHHoJ:! CHCTeMblB Ka)l(.nOMKOHKpeTHOMcnyxae. BblpHCoBbIB31U1e.nHarpaMM aaransnaa-anaroconepseaaae c nOMOIUblKH:bIMMeTo.nOM. B nacroamea CTaTI! o6cy)I(.ZJ.elfbIreoperasecsae npennocsrrnor COCTaBneHHll .D.HarpaMM i- Y. B xasecrae rrpasrepoa nposeneusi .D.lIarpaMMhI CBCTeM33oT-6eH30n, -rerrraa -uaxnorexcaa nony- '1eHHl>IeHa 3BM.