AN ACCURATE AND COMPUTATIONALLY EFFICIENT APPROXIMATION TO PSYCHROMETRIC CALCULATIONS

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1 Inernaional Journal of Laes Research in Science and Technology Volume 3, Issue 3: Page No May-June 014 h:// ISSN (Online): AN ACCURATE AND COMPUTATIONALLY EFFICIENT APPROXIMATION TO PSYCHROMETRIC CALCULATIONS 1 Emmanuel D. Rogkis, Dimirios Nik. Teriis Naional Technical Universiy of Ahens, Faculy of Mechanical Engineering, Laboraory of Alied Thermodynamics 9, Iroon Polyechniou Sr Ahens Absrac The resen aer deals wih a comuaionally efficien rocedure o calculae he sychromeric roeries of mois air. Taking ino accoun he simlified aroximaions by ASHRAE and he accurae correlaions for he dry air (Lemmon e al., 1999) and uraed seam (Wagner & Krezschmar, 008) roeries, a simle model is buil o redic he hermodynamic behaviour of any mois air working sysem. All roeries are calculaed using olynomial correlaions of higher grade or of lower grade, according o heir accuracy: hus, he comuaional efficiency is achieved, while he s of each roery is isfacorily accurae (average errors are lower han 05%). Aar from imroved correlaions for exensive roeries (volume, enhaly and enroy) and inensive roeries (arial ressure, dew oin emeraure) of mois air, a new aroximaion is resened o calculae he we-bulb emeraure wihou solving any ieraion based comlex equaion. The resuls of he resened mehod are comared o he ASHRAE aroximaion and he accurae correlaions by Lemmon and Krezschmar, in erms of error, average error and sanrd deviaion of error in a range of dry bulb emeraure beween 0 ï C and 80 o C. Keywords Psychromerics, Dry air roeries, Sauraed air roeries, Secific enhaly, Secific enroy, We-bulb emeraure I. INTRODUCTION The sychomeric calculaions are necessary for each air based energy sysems: HVAC lans, evaoraive coolers, cooling owers ec. The sychromeric chars are widely used o calculae quickly he hermodynamic roeries of mois air, while he sychromeric correlaions are used o design comlex cooling devices, as evaoraive coolers and cooling owers. Given ha he accuracy of he calculaions is direcly conneced o he qualiy of design of hese comlex devices, he accuracy of he comuaional rocedures is raher criical, esecially when hese devices are modelled using large comuaional grids, which require high comuaional ower and significan accuracy. The mos known, common, and widely used sychromeric calculaion has been develoed and resened by ASHRAE [1]. This mehod is based on an exonenial aroximaion of uraion ressure of waer and on a firs grade olynomial correlaion of secific enhaly. The comuaionally efficiency of his mehod is quie isfacory, alhough an ieraive calculaion of we-bulb is needed. Aar from correlaions, ASHARE also rovides ables for dry air, uraed air and uraed seam roeries; he ables a do no generally agree wih he corresonded s calculaed using he correlaion, however he errors are no significan (see Table I). TABLE I: SATURATION PRESSURE OVER LIQUID WATER, ACCORDING TO ASHRAE Temeraure [ o C] Temeraure [K] Sauraion ressure over liquid waer (eq. 6 [1] ) [Pa] Sauraion ressure over liquid waer (Table 3 [1] ) [Pa] error % % % % % % % Arnold Wexler and Richard Hyland resened [] a comrehensive sudy abou he roeries of dry air, mois air and waer. In his sudy, exremely comlex correlaions (based on Virial formulas) have develoed, in order o sudy he roeries of mois air wihin a wide range of ressure and emeraure condiions. The accuracy of his aroximaion is undoubedly high, however he comuaional efficiency is very low (as an significan number of derivaions has o be carried ou) and he whole ISSN:

2 mehodology is no based on he newes emeraure sanrds, as i is older han ITS-1990 [3] {4} 3 The mos recen and well-known works abou he roeries of dry air and waer seam have been resened by Eric Lemmon e al. [4] and Wolfgang Wagner & Hans Joachim Krezschmar [5], resecively. The firs sudy deals wih he roeries of dry air under differen condiions, roviding exensive correlaions for each roery; he second sudy focuses on he waer seam roeries and is based on he indusrial formulaion IAPWS IF1997. As he sychromeric calculaions are alied o devices working under amosheric ressure (so he dry air behaves as an almos ideal gas), he abled a for dry air roeries under 1am can be used wihou any significan error, while he waer seam roeries should always be calculaed as a funcion of emeraure and ressure. II. WATER VAPOUR CALCULATIONS According ASHRAE [1], he uraion ressure over liquid waer, as he emeraure varies beween 0 o C and 00 o C, is calculaed as a funcion of emeraure: T T 5 ex T T T lnt {1} A similar exression is roosed by Perry s Chemical Engineer s Handbook [6]: {} T ex T ln T T 758. The comarison resuls are resened on Table III. Given ha he IAPWS correlaion is considered o be he mos accurae among he aroximaions, he mehods roosed be Perry s and ASHRAE, as well as he resen mehod are comared o IAPWS correlaion. As shown, he roosed correlaion leads o an average error equal o 001% (while Perry s and ASHRAE s mehods lead o average errors 044% and 01%, resecively). Addiionally, he roosed mehod is he mos efficien in erms of comuaional ime: i requires abou 68% less ime in relaion o IAPWS mehod, while Perry s and ASHRAE s mehods need abou 63% and 56% less ime, resecively). III. HUMIDITY RATIO, RELATIVE HUMIDITY & DEGREE OF SATURATION The humidiy raio [kg w /kg ] of a given condiion of mois air is defined as he raion of waer vaour mass o he dry air mass. This raio can alernaively be exressed as a raio of he mole fracions, x and x, and he raio of molecular masses, ë Ì. The mole fracion of waer vaour in he uraed air is equal o he raio of waer vaour arial ressure o he air ressure (which is assumed o be equal o 1am): x air x x 1 x 1 {5} Thus, he humidiy raio of he uraed air is exressed as a funcion of emeraure: air The IAPWS based correlaion [5] rooses he following comlex correlaion: T a 10 6 PT T b {3} where ã è, P è c d, âè â è 4 áèãè e f g and h i j. The olynomial coefficiens a-j are given on Table II. TABLE II: POLYNOMIAL COEFFICIENTS OF EQ. {3} a f b g c h d i e j Using he IAPWS based correlaion, he following of 5 h grade is roosed: 4 W x ëm x 0. {6} air The humidiy raio of he uraed air is calculaed using he four correlaions for uraion ressure. The resuls are resened on Table IV and show ha Perry s mehod is of lower accuracy and ha ASHRAE s mehod is as accurae as he roosed mehod. However, in erms of comuaional ime, he roosed mehod is abou 73% faser han he analyical mehod based on IAPWS correlaion, while he average error of he calculaions does no exceed 001%. The relaive humidiy is defined as he raio of he arial ressure of waer vaour o he uraion ressure of waer vaour under he same emeraure: ö, {7} ö ISSN:

3 According o he above formula, he humidiy raio of he W air öw, {9} mois and non-uraed air is exress as a funcion of W ëm emeraure and relaive humidiy: The relaive humidiy mus no be confused wih he degree ö, W ö, ëm {8} of uraion, which is exressed by he following raio: air ö, W ö {10} The correlaion above can be inversed o calculae he relaive ìw, ìö, W 1 ö W humidiy: 1 ë M W&K [Pa] TABLE III: WATER VAPOUR SATURATION PRESSURE Perry ASHRAE [Pa] [Pa] [Pa] Average Comuaional ime [%] ISSN:

4 W&K [kg w/kg ] TABLE IV: WATER VAPOUR SATURATION PRESSURE [kg w/kg ] Perry ASHRAE [kg w/kg ] [kg w/kg ] Average Comuaional ime [%] IV. SPECIFIC VOLUME The secific volume [m 3 /kg ] of dry air under amosheric ressure is given as a funcion of emeraure is abled in Lemmon s sudy [4]; ASHRAE [1] also rovides a general equaion of he secific volume of mois air. Using he abled a by Lemmon s sudy, a 1 s grade olynomial can be wrien: v {11} The comarison beween Lemmon s a, he ASHRAE formula and he olynomial exression above is resened on Table V. As he boh exressions are of 1 s grade, here is no difference on comuaional ime, however he roosed olynomial is much more accurae. The secific volume of uraed air equal o [] v v {1} x ISSN:

5 and he secific air of mois non-uraed air is given by he 8704 vw, W {14} relaionshi: , v ìw, v v {13} v 1 ìw, ìw, v v W As ASHRAE exresses he secific volume as a 1 s grade funcion of wo variables: The equaion {13} is obviously much slower han he equaion {14}. As shown on Table VI, boh mehods lead o similar resuls (he average deviaion is abou 04%), hus he ASHRAE aroximaion is referable. TABLE V: SPECIFIC VOLUME OF DRY AIR Lemmon [m 3 /kg ] [m 3 /kg ] ASHRAE [m 3 /kg ] Average Comuaional ime [%] TABLE VI: SPECIFIC VOLUME OF SATURATED AIR ASHRAE [m 3 /kg ] [m 3 /kg ] Average 04 V. SPECIFIC ENTHALPY & SPECIFIC ENTROPY According o ASHRAE mehodology, he secific enhaly [kj/kg ] of mois air is he sum if individual arial enhalies of he comonens:, h W h W {15} h W On he oher hand, using he Lemmon s a for dry air and IAPWS s a for uraed seam, a very accurae relaion can be formed:, h ìw, h {16} h W As a ar of he resen sudy, an imroved olynomial is roosed (of nd grade for dry air and of 3 rd grade for waer vaour): ISSN:

6 error of ASHRAE aroximaion. For he calculaion of he hw, h W h {17} secific enhaly as a funcion of emeraure and relaive humidiy, he combinaion of he eqs. {8} and {17} is needed A similar formula can be wrien o calculae he secific enroy of mois air [kj/kg K]: W The comarison of hree aroximaions (eqs. {16}, {15} and {17}) is resened on Table VII, for a consan humidiy raio 010kg w /kg and a emeraure range 15 o C 80 o C. In erms of comuaional efficiency, he ASHRAE aroximaion and he roosed olynomial lead o 97% less ime comared o he analyical correlaion, based on he Lemmon s a for dry air and IAPWS s a for uraed seam. However, in erms of accuracy, he roosed olynomial lead o an average error abou 00%, which is much lower han he s W s s W, W {18} The resuls of he correlaion above are shown on Table VIII; he average error is abou 058%. TABLE VII: SPECIFIC ENTHALPY (FOR CONSTANT HUMIDITY RATIO W=010kg w/kg ) Lemmon and W&K [kj/kg ] [kg w/kg ] ASHRAE [kg w/kg ] ISSN: Average Comuaional ime [%]

7 TABLE VIII: SPECIFIC ENTROPY (FOR CONSTANT HUMIDITY RATIO W=010kg w/kg ) Lemmon and W&K [kj/kg K] [kj/kg K] Average 058 Comuaional ime [%] VI. WET-BULB TEMPERATURE The we-bulb emeraure is defined as he emeraure a which liquid waer evaoraes ino he air o bring i o uraion a exacly his same emeraure and oal ressure. By solving he following comlex equaion, he we-bulb emeraure is accuraely solved, however he comuaional ime will be obviously high, as a number of ieraions is needed o converge. h W W c, h {19} wb w wb where c,w =4.186kJ/kgK is he secific hea of waer. Insead of he above comuaionally inefficien relaion, a olynomial exression is roosed, o calculae he we-bulb emeraure as a funcion of emeraure and relaive humidiy; he combinaion of eqs. {9} and {0} lead o he wb calculaion of we-bulb emeraure as a funcion of humidiy raio and emeraure. 36 l mii wbö, kii ö ii {0} ii 1 The above correlaion allows he direc calculaion of webulb emeraure, wihou a loo calculaion; he coefficiens k, l and m are given on Table IX for various s of amoshere ressure (as a funcion of aliude), while on Table X and XI he resuls of eqs. {19} and {0} for a consan relaive humidiy 50% and 0%, resecively (a he sea level). The resuls rove ha, keeing he average error lower han 5% (as he emeraure varies beween 0 o C and 80 o C, he roosed olynomial aroximaion needs abou 87% less comuaional ime; he accuracy of he olynomial exression {0} is isfacorily (error <1%) for emeraures higher han 80 o C and lower han 110 o C. ISSN:

8 TABLE IX: WET-BULB TEMPERATURE CALCULATION COEFFICIENTS ISSN:78-599

9 TABLE X: WET-BULB TEMPERATURE (ö=50%) ASHRAE [ o C] [ o C] error [%] Average 554 Comuaional ime [%] TABLE XI: WET-BULB TEMPERATURE (ö=0%) ASHRAE [ o C] [ o C] error [%] Average 094 Comuaional ime [%] ISSN:

10 VII. CONCLUSIONS Exac analyical exressions for all sychromeric roeries of mois air have been derived using he comlex equaions of sae for dry air and waer vaour. However, he efficiency of hese exressions in erms of comuaional ime is low, making difficul he fas modelling of energy sysems, working wih mois air. Uilizing he secialiies of each hermodynamic arameer, he analyical exression of each of hem can be simlified, roducing a comrehensive and comuaionally efficien sychromeric model, he accuracy of i is abou equal o he model based on analyical mehods. Thus, he modelling of aarauses, such as evaoraive coolers and cooling owers is much faser, wihou any significan deviaion in comarison wih models using he analyical sychromeric equaions. The greaes deviaion beween he roosed aroximaion and he widely used ASHRAE s aroximaion in erms of comuaional efficiency is observed during he we-bulb calculaion; his hermodynamic arameer is criical for he oeraion of evaoraive devices. In his aer, he analyical exressions were subsiued by simler olynomial equaions, based on he analyical ones, which were characerized by an R greaer han 999. Using he olynomial equaion, a significan reducion of comuaional ime was achieved, wihou significanly affecing o he roduced resuls. Esecially and in order o evaluae he generic comuaional efficiency of he roosed aroximaion, an evaoraive cooler uilizing he novel Maisosenko cycle was modelled [7]. Under he same condiions, he new aroximaion leads o abou 65% less comuaional ime han using he analyical sychromeric correlaions (based on Lemmon and IAPWS formulaions), while he modelling resuls were almos equal for boh sychromeric aroximaion. Conclusively, he roosed aroximaion can hel engineers and sysem designers model more effecively novel devices and focus in deh on how he mois air roeries characerize he erformance of hem. Inernaional Journal of Laes Research in Science and Technology. REFERENCES 1. ASHRAE (009): Psychromerics. Funmenals, Wexler A., Hyland R., Sewar R. (1983): Thermodynamic Proeries of Dry Air, Mois Air and Waer and Psychromeric Chars. ASHRAE 3. Preson-Thomas H. (1989): The inernaional emeraure scale of 1990 (ITS-90). Merologia, 7, Lemmon E., Jacobsen R., Penoncello S., Friend D. (1999): Thermodynamic roeries of Air and mixures of Nirogen, Argon and Oxygen from 60 o 000K a ressures o 000MPa. Journal of Physical and Chemical Reference Daa, 9(3), Wagner W., Kreschmar H.J. (008): Inernaional Seam Tables. Sringer, ISBN Poling B., Thomson G., Friend D., Rowley R., Wilding W. (008): Physical and Chemical Daa. Perry s Chemical Engineers Handbook. 7. Rogkis E., Koronaki I., Teriis D. (014): Exerimenal and comuaional evaluaion of a Maisosenko evaoraive cooler a Greek climae. Energy and Building, 70, NOMENCLATURE c secific hea [kj/kgk] h secific enhaly [kj/kg ] ressure [Pa] s secific enroy [kj/kg K] emeraure [ o C] T emeraure [K] v secific volume [m 3 /kg ] W humidiy raio [kg w /kg ] x mole fracion [-] ë Ì waer molecular mass o dry air molecular mass raio [-] ì degree of uraion [-] ö relaive humidiy [%] Subscris air db w wb wb mois air dry air dry-bulb (refers o emeraure) waer we-bulb (refers o emeraure) waer vaour Suerscri uraion ISSN: