U.P.B. Sci. Bull., Series D, Vol. 70, No. 4, 2008 ISSN 1454-2358 ASSESSMENT OF THE ERRORS DUE TO NEGLECTING AIR COMPRESSIBILITY WHEN DESIGNING FANS Adrei DRAGOMIRESCU 1, Valeriu PANAITESCU 2 Coform ormelor recete, vetilatoarele sut turbomaşii eumatice ale căror raoarte de comrimare u deăşesc valoarea τ = 1,3. Proiectarea vetilatoarelor se face, de regulă, î ioteza că aerul este icomresibil. Î rezeta lucrare este rezetată o aaliză a erorilor itroduse de eglijarea comresibilităţii aerului şi este stabilită o deedeţă a acestor erori de raortul de comrimare. Accordig to recet orms, fas are eumatic turbomachies, whose ressure ratios do ot exceed the value τ = 1.3. Whe desigig fas the assumtio is usually made that the air is icomressible. I the reset work a aalysis of the errors resultig from eglectig the air comressibility is made ad a relatioshi betwee these errors ad the ressure ratio is established. Keywords: comressibility, comutig errors, fas 1. Itroductio Accordig to ISO ad EUROVENT orms, fas are turbomachies that raise the secific eergy of a gas with u to 25 000 J/kg. For a fa hadlig air with a mea desity withi the fa of 1.2 kg/m 3, at a atmosheric ressure of about 100 000 Pa, the ressure rise ca ot exceed 30 000 Pa, or about 3 000 mm water. This limit of the ressure rise is higher tha that cosidered util recetly, of about 1200 mm water, above which turbomachies were called blowers. The ISO orms do ot itroduce aymore the deomiatio blower. Deedig o the rise of the secific eergy ad o the ressure rise, resectively, fas are classified as follows [1]: low ressure fas, u to 600 J/kg ad u to 720 Pa, resectively; medium ressure fas, betwee 600 J/kg ad 3 000 J/kg ad betwee 720 Pa ad 3 600 Pa, resectively; high ressure fas, betwee 3 000 J/kg ad 25 000 J/kg ad betwee 3 600 Pa ad 30 000 Pa, resectively. 1 Lecturer, Det. of Hydraulics, Hydraulic Machiery ad Evirometal Egieerig, Uiversity POLITEHNICA of Bucharest, Romaia 2 Professor, Det. of Hydraulics, Hydraulic Machiery ad Evirometal Egieerig, Uiversity POLITEHNICA of Bucharest, Romaia
168 Adrei Dragomirescu, Valeriu Paaitescu Above 25 000 J/kg ad 30 000 Pa, resectively, turbomachies are called turbocomressors. It ca be oticed that ressure ratios of fas do ot exceed the value τ = 1.3 (the ressure ratio beig the ratio betwee the absolute ressure at outlet ad the absolute ressure at ilet). Whe desigig eumatic turbomachies, u to this ressure ratio the air is usually cosidered a ideal ad icomressible gas, i.e. its desity is take as costat. Uder these coditios the questio arises to what extet eglectig air comressibility iflueces the erformaces of a fa ad its costructive characteristics. I the followig, a assessmet of the errors due to eglectig air comressibility whe desigig fa imellers is reseted ad a relatioshi betwee these errors ad the ressure ratio is established. 2. Assessmet of desigig errors due to cosiderig air as icomressible u to ressure ratios of 1.3 The aalysis will be accomlished uder the hyotheses that air is a ideal gas ad durig the flow through the imeller the heat exchage is egligible, i.e. the flow is adiabatic. To rove the validity of the first hyothesis it is required to kow the value of the comressibility factor of air. Air is actually a real gas, which fulfils the equatio of state = ρ Z RT, (1) where Z is the comressibility factor. This factor is a fuctio of two ideedet state roerties of the gas. However, the comressibility factor ca be take as costat ad equal to oe, Z = 1, if T/T cr > 2 ad / cr < 0.05 [2], where, for the gas take ito cosideratio, T cr is the critical temerature ad cr is the critical ressure. The critical values of air are T cr = 132.45 K ad cr = 37.9 bar [3]. I fas air temerature is usually higher tha 273.15 K ad the absolute ressure does ot exceed about 1.3 bar sice the ressure ratio is smaller tha 1.3. The followig limitig values are obtaied: (T/T cr ) mi = 273.15/132.45 = 2.06 ad (/ cr ) max = 1.3/37.9 = 0.034. It ca be oticed that these limitig values fulfil the aforemetioed coditios, which meas that the comressibility factor of air may be take as costat ad equal to 1 i the rage of fa ressure ratios. By this, the hyothesis that cosiders air as beig a ideal gas ca be acceted. The secod hyothesis may be also acceted because the time required by air articles to travel through the fa imeller is very short ad fas are ot foresee with coolers as turbocomressors are. I the followig the commo otatio will be used, with idex 0 deotig a oit located o the suctio edge of a imeller blade, just before the etrace,
Assessmet of the errors due to eglectig air comressibility whe desigig fas 169 ad idex 3 deotig a oit located o the ressure edge of the blade, immediately after the exit of the imeller. The dimesios of the imeller at exit are iflueced by the volume flow rate at imeller outlet. Uder the hyothesis of costat desity, the cotiuity equatio leads to the equality of the volume flow rates at ilet ad outlet. These two volume flow rates are actually differet, oly the mass flow rates beig equal. Because of this, the followig relative error aears whe comutig the volume flow rate: Q0 Q3 Δ Q =, (2) Q0 where Q is the volume flow rate. From the cotiuity equatio Q m = Q 0 m 3, (3) where Q m = ρq is the mass flow rate ad ρ is the air desity, the followig relatioshi betwee the volume flow rates at ilet ad outlet is obtaied: ρ0 Q 3 = Q0. (4) ρ3 At the outlet both the ressure ad the temerature rise, but these two rises have oosite effects o the desity, ad, cosequetly, o the volume flow rate. The ressure rise leads to a desity growth, while the temerature rise causes a decrease of the desity. Whether evetually the desity icreases or decreases deeds o which of the two arameters, ressure or temerature, has a stroger growth. Keeig a costat air desity corresods to a isochoric chage of state. But the chage of state that air actually udergoes i a fa is a olytroic oe, beig described by a equatio that ca be writte i the followig form: = cost. ρ, (5) where is the absolute ressure ad is the olytroic exoet. Cosiderig this equatio, the relatioshi betwee the volume flow rates becomes 1/ 1/ 0 3 1/ Q3 = Q0 Q0 = Q0 τ = 3, (6) 0 where τ = 3 / 0 is the ressure ratio. At this momet, assessig the error whe comutig the volume flow rate requires oly the olytroic exoet to be kow. O the other had, the
170 Adrei Dragomirescu, Valeriu Paaitescu olytroic exoet ca be exressed i terms of the olytroic efficiecy η ol ad the isetroic exoet k, sice, at desig momet, the olytroic efficiecy ca be estimated ad the isetroic exoet is kow. The olytroic efficiecy is give by the formula [4] l ol η ol =, (7) h 3 h 1 where l ol is the olytroic secific comressio work ad h is the secific ethaly of air. The olytroic secific comressio work is give by the relatioshi [4, 5] ( 1) / 0 3 l ol = 1 = R ( T3 T0 ) 1 0. (8) ρ 0 1 Sice air is cosidered a ideal gas, for which the secific heat at costat ressure c is costat, the differece of the ethalies at outlet ad ilet ca be writte i the form k h3 h0 = c ( T3 T0 ) = R ( T3 T0 ). (9) k 1 Itroducig this relatioshi i the exressio of the olytroic efficiecy, oe obtais k 1 η ol =, (10) 1 k from where the olytroic exoet results: η ol k =. (11) 1 k (1 η ol ) Relacig ow i equatio (6), the followig relatioshi is obtaied: 1 k (1 η ol ) Q3 = Q0 τ. (12) With this, the relative error of the volume flow as a fuctio of the ressure ratio ad the olytroic efficiecy ca be exressed as follows: ΔQ = 1 Q0 η ol ol k 1 η k 1 k (1 η τ ol ). (13)
Assessmet of the errors due to eglectig air comressibility whe desigig fas 171 18 16 14 80% 70% 12 Q /Q 0 [%] 10 8 6 4 2 0 1 1,05 1,1 1,15 1,2 1,25 1,3 Fig. 1. Relative error of the volume flow rate as a fuctio of the ressure ratio for olytroic efficiecies of 80 % ad 70 %. The relative error of the volume flow rate as a fuctio of the ressure ratio for two values of the olytroic efficiecy, η ol = 80 % ad η ol = 70 %, is lotted i figure 1. It ca be oticed that the error exceeds 2 % for ressure ratios higher tha 1.03 ad 5 %, resectively, for ressure ratios higher tha 1.09, icreasig u to about 15 % at a ressure ratio of 1.3. This meas that for low ressure fas, which have ressure ratios smaller tha τ = 1.0072, the errors ca be cosidered egligible, sice they have values below 1 %. I case of medium ressure fas, havig τ = 1.0072 1.036, the comutig error ca grow above 2 %, so that whe a icreased accuracy is required for such fas, the air comressibility should be take ito accout. Fially, eglectig the air comressibility whe desigig high ressure fas, havig ressure ratios τ betwee 1.036 ad 1.3, results i sigificat errors, the volume flow rate at the outlet beig overestimated (sice the desig flow rate, which is cosidered costat throughout the imeller ad equal to Q 0, is higher at the outlet tha the real flow rate Q 3 ). This meas that the imeller width b 2 at outlet, with which the volume flow rate is roortioal, will be also overestimated, accordig to the flow rate. The imellers of high ressure fas ad, geerally, the etire machie result also over dimesioed whe the air comressibility is eglected. Because of this, the roductio cost icreases ad the eergy cosumtio of the fa, whe oerated, will be higher. As a result, whe desigig high ressure fas it is advisable to cosider always the chage of the air desity i the imeller. τ
172 Adrei Dragomirescu, Valeriu Paaitescu 3. Coclusios I this work the errors that arise whe desigig fa imellers uder the assumtio of the air icomressibility were aalyzed. The results obtaied show that for low ressure fas the errors are egligible but they ca icrease above 2 % i case of medium ressure fas. For high ressure fas the errors become sigificat, growig u to about 15 %, deedig o the ressure ratio, ad thus causig a over dimesioig of the fa. As a result, whe the ecoomical asects related to roductio ad oeratig costs are of the first imortace, a roer desig should cosider the variatio of the air desity i the fa imeller. R E F E R E N C E S [1] *** EUROVENT 1/1 Fa Termiology, EUROVENT/CECOMAF. [2] M.J. Zucrow, J.D. Hoffma, Gas Dyamics, Joh Wiley & Sos, New York /, 1976. [3] *** Perry s Chemical Egieers Hadbook, editors R.H. Perry, D.W. Gree, J.O. Maloey, 7 th Editio, McGraw Hill, New York /, 1999. [4] *** ISO 5389:1992 Turbocomressors Performace Test Code. [5] C. Pfleiderer, H. Peterma, Strömugsmaschie, ed. a VI-a, Sriger Verlag, Berli /, 1991.