HMT-6-C8 8 th National & 7 th ISHMT-ASME Heat and Ma Tranfer Conference January 4-6, 6 IIT Guwahati, India Computation of Velocity, Preure and Temperature Profile in a Cryogenic Turboexpander Subrata K. Ghoh Mechanical Engineering Department National Intitute of Technology, Rourkela Rourkela 7698, Oria, India ubratarec@yahoo.co.in Ranjit K. Sahoo Mechanical Engineering Department National Intitute of Technology, Rourkela Rourkela 7698, Oria, India rkahoo@nitrkl.ac.in N. Sehaiah Mechanical Engineering Department National Intitute of Technology, Rourkela Rourkela 7698, Oria, India ehuet@yahoo.com Sunil K. Sarangi Cryogenic Engineering Centre Indian Intitute of Technology, Kharagpur Kharagpur 7, Wet Bengal, India arangi@hijli.iitkgp.ernet.in Abtract An indigenou programme on deign and development of a cryogenic turboexpander ha been taken up at NIT, Rourkela. Thi paper preent the detailed computational procedure for determining the velocity, preure and temperature profile in the turbine wheel, nozzle and diffuer of the turboexpander. The procedure allow any arbitrary combination of fluid pecie, inlet condition and expanion ratio, the fluid propertie being properly taken care of in the relevant equation. The computational proce i illutrated with an example. Keyword: turboexpander; preure; temperature; velocity. W relative velocity Nomenclature Z number of blade r, θ, z cylindrical coordinate fixed to rotor b channel width C abolute velocity Greek Symbol D tr turbine wheel diameter η efficiency D tip eye tip diameter ω rotational peed (rad/) D hub eye hub diameter ε ratio of tip diameter to turbine wheel diameter d pecific diameter λ ratio of hub diameter to tip diameter h enthalpy (J/kg) β relative velocity angle h in- adiabatic enthalpy drop acro turbine wheel τ time coordinate (J / kg) δ angle between meridional velocity component K e free parameter and axial coordinate K h free parameter ρ denity of fluid. m ma flow rate M mach number Subcript N number of revolution n pecific peed P power produced p preure Q volumetric flow rate (m / ) R m radiu of curvature of meridional treamline r radiu direction and arclength of a meridional treamline entropy (J/kg-K) t b blade thickne T temperature U circumferential velocity 5 in inlet to nozzle exit to nozzle inlet to turbine wheel inlet to diffuer (exit to turbine wheel) ex exit from diffuer hub hub of turbine wheel at exit tip tip of turbine wheel at exit m meridional u circumferential tr turbine ientropic condition tagnation condition
Fig.: Longitudinal ection of the expanion turbine diplaying the layout of the component.. Introduction The expanion turbine contitute the mot critical component of a large number of cryogenic proce plant like air eparation unit, helium liquefier and low temperature refrigerator. The primary function of thi unit i to produce cooling by expanding a precooled high preure ga tream where power i extracted from the fluid and enthalpy of the ga decreae. Compared with the other expanion device the ue of expanion turbine offer greater economy, afety and flexibility. We can eliminate problem like high maintenance cot, large ize, difficult valve operation and improper ealing by uing a turboexpander. In our laboratory a turboexpander ha been deigned a hown in Fig. having the following pecification. Working fluid :Nitrogen Turbine inlet temperature (T,in ) :K Turbine inlet preure (p,in ) :6 bar Exit preure (p ex ) :.5 bar Throughput ( m& ) :8 nm /hr Expected efficiency (η T-t ) : 75% Thi turbine i comparable in characteritic to that developed earlier at IIT Kharagpur [,] and draw heavily from that experience. A turboexpander aembly conit of the following baic unit: the turbine wheel, nozzle and diffuer, the haft, the brake compreor, a pair of journal bearing and a pair of thrut bearing, appropriate houing.. Computational Procedure The deign of turbine wheel ha been done following the method outlined by Balje [] and by Kun and Sentz [4], which are baed on the well known imilarity principle. The imilarity law tate that for given Reynold number, Mach number and Specific heat ratio of the working fluid, to achieve optimized geometry, two dimenionle parameter: pecific peed and pecific diameter uniquely determine the major dimenion of the wheel and it inlet and exit velocity triangle. Specific peed (n ) and pecific diameter (d ) are defined a: Specific peed Specific diameter n d ω Q = () ( h ) 4 in ( h ) 4 in Dtr = () Q The true value of Q and h, which define n and d are not known a priori. Kun and Sentz [4], however ugget two empirical factor k and k for finding out thee parameter. Q m = k Q k () ρ ex =. ex ρ = ρ ex / k (4) ( h h ) hin = k in ex (5) The factor k and k account for the difference between the tate and ex caued by preure recovery and conequent rie in temperature and denity in the diffuer. Following the uggetion of Kun and 6
Sentz [4], we have taken k =.. The factor k repreent the ratio Q / Qex, which i alo equal to ρ ex / ρ. The value of Q ex i known at thi tage, but that of Q i not known. A gue value of k i neceary to tart the calculation. By taking a gue value of k we find out the initial value of volume flow rate ( Q )and denity ( ρ ) at the exit of the turbine wheel by uing equation () and (4). Thi Q i ued for determining the initial value of Dtr and ω from equation () and (). After olving equation () and (), we get the abolute velocity C at the exit of turbine. Uing the tandard thermodynamic relation, h h = ex = ex and C h = h and uing the property table we calculate the value of ρ, and from that the value of the parameter k. The calculated ρ and initial ρ hould be ame. We tarted from a gue value of. for k. But that led to a final value of k much different from what we aumed. After examining the gueed and calculated value, we found that the aumed and calculated value agree cloely if we take.7 for k. From Balje [] the peak efficiency of a radial inflow turbine correpond to the value of: n =.54 and d =.4 (6) Rohlik [5] precribe that the ratio of inlet diameter to exit tip diameter hould be limited to a minimum value of.4 to avoid exceive hroud curvature. Correponding to the peak efficiency point []: For mall turbine, the hub circumference at exit and diameter of milling cutter available determine the number of blade. In ummary, the major dimenion for our prototype turbine have been computed a follow: Rotational peed: N = 8,8 r/min = 457.9 rad/ Wheel diameter: D tr = 6. mm Eye tip diameter: D tip = 8.9 mm Eye hub diameter: D hub = 6. mm Number of blade: Z tr = 7 Thickne of blade t tr = mm Blade height at entrance b tr =. mm. ( D + D ) tip hub U = ω 4 () C tan β = U () Q = A C = C ( D D ) ( D D ) π Zt tip tip hub 4 in β where, A i area at the exit of the turbine wheel. C U m () W = + () Equation () and () are now olved imultaneouly for exhaut velocity C and mean relative velocity angle β, giving: U = 88.75 m/ C = 75.84 m/ β = 4.5 hub State Point: In Inlet to nozzle (overall inlet) Exit of nozzle Inlet to wheel (exit from nozzle) Inlet to diffuer (exit from wheel) 4 Exit from diffuer (overall exit) ε = D tip D =.45, (7) tr According to Reference [5], the exit hub to tip diameter ratio hould maintained above a value of.4 to avoid exceive hub blade blockage and energy lo. Kun and Sentz [4] have taken a hub ratio of.5 citing mechanical conideration. λ = D / =.5 (8) hub D tip From continuity equation, the ratio of blade height at entrance to that of the wheel i computed a: b tr tr. = m ( πd Z t ) ρ C (9) tr tr tr mtr Wheel Nozzle Diffuer Fig. : The fluid flow path in a turbine and definition of the tate point. In 4 7
Table : Thermodynamic propertie at different tate of turboexpander Inlet State (tagnation condition) State State State State 4 Preure (bar) 6..7..6.5 Temperature (K)..6 7.8 9.47 96.7 Denity (kg/m ) 6.46.74 9.9 5.6 5.4 Abolute Velocity (m/) 88. 5.8 75.84. U,hub u,mean u,tip U =9.65 46 88.75.49.7 W =C m =C m =C 58.76 4.5 9.97 W,tip C =5.8 W,mean W,hub Fig.: Compoite velocity diagram for expanion turbine. (All velocitie are in unit of m/) The blade profile have been worked out uing the technique of Haelgruber [6] maintaining the aumption made there in uch a: i) contant acceleration of the relative velocity, ii) equal meridional velocity at the wheel inlet and at exit, iii) relative flow angle at the wheel inlet = 9. Haelgruber formulation lead to three characteritic function defined a follow. f = ( coec( )) + {( coec( )) ( coec( )) } A β β β (4) where, A = f ( k + ) coec( β ) + ( coec( β ) coec( β )) h = coec kh coec ( β ) + coec( β ) ( β ) + { coec( β ) coec( β )} ke kh + - (5) k h (6) f = f f (7) The function f depict the variation of the relative acceleration of the fluid from the wheel inlet to exit. The function f give the relative flow angle along the flow path while function f i a combination of f & f. The radiu of curvature of meridional treamline path i expreed in term of the three characteritic function a: f f r Rm = (8) r co() δ f r tan( β ) The angle between meridional velocity component and axial coordinate i derived to be: δ = d (9) Rm Other relation on the path of determining the velocity, denity, preure and temperature profile are ummarized below. r = (in δ )d () - f θ = d () r f z = (co δ )d () β = A in f () C m = C f f (4) ( πr in ztr tb ) / ztr b w = β (5) 8
C r U = (6) r tanβ W = C f (7) C u = U W coβ (8) C = ( Cm + Cu ) (9) C M = () V Where r = radial coordinate of mean tream line θ = tangential coordinate of mean treamline z = axial coordinate of mean treamline β = relative velocity angle C m = meridian component of abolute velocity b w = width of the flow channel between two blade U = circumferential velocity W = relative velocity C u = circumferential component of abolute velocity C = abolute velocity V = ound velocity M = Mach number of the abolute velocity The thermodynamic quantitie are computed uing the following relation: ρ = ρ + m ρ P = P ρ m ρ = T ρ ( m ) γ C + U W m m C M () () T () Figure 4 preent a flow chart of the computational proce. tart T, in ; p, in Input ; p ex. ; m;η and working fluid T t Compute thermodynamic propertie ρin ; h in ; in at the inlet and ρ ex ; h ex ; ex ; Tex at the exit tate of the turboexpander by uing input data and property chart A Aume the initial value of K and K Compute Q and ρ from equation () and (4) Compute from equation (5) hin Compute n and d from Balje [] 9 B
B Compute ω and D tr from equation () & () Compute Dtip and D hub from equation (7) & (8) Compute β and C m from equation () & () Compute h by uing C m from tandard thermodynamic relation Compute ρ by uing property chart I thi ρ and initial ρ i ame No A Ye Compute U by uing equation () Compute W by uing equation () Compute C m, U, W, β, C u and C at turbine wheel by uing equation (4), (6), (7), (), (8) and (9) repectively. Compute denity, preure and temperature at turbine wheel by uing equation (), () and () repectively. Stop Fig. 4: Flow chart of the computer program for calculation of velocity, preure, temperature profile in cryogenic turboexpander
. Reult and Dicuion Temperature (K) Abolute velocity (m/) 5 5 5 Preure Temperature Nozzle Ab. Velocity V a n e l e p a c e Wheel Diffuer 4 6 8 7 6 5 4 Preure (bar) Fig. 5: Variation of preure, temperature, and abolute velocity in turboexpander (length not to cale) Temperature (K) Rel. Velocity (m/) 7 7 97 9 87 8 77 7 Preure Temperature.5 Relative velocity.5.5.5 5 5 Meridional Streamlength (mm) Preure (bar) Fig. 6: Preure, temperature, relative velocity ditribution along the meridional treamline of the turbine wheel. Figure 5 preent qualitatively the velocity, temperature and preure profile from inlet to exit, while Figure 6 how the ame profile along the middle treamline in the wheel. The reult may be ued to compute the net axial thrut by integration of the preure over the projected area of the turbine wheel. r Fa = πrpdr + πr p r (4) ( r r ) p Fa = π (5) where, F a r Net axial force = 5.8 N 4. Concluion = axial thrut force (Newton) = radiu of the haft The turboexpander i an important mechanical device in the cryogenic plant, which ha very wide applicability. The turboexpander i in the proce of fabrication. Thi paper give detailed computational procedure with an example. Suitability of the computational proce would be confirmed by conducting experiment.
Reference. Chakravarty, A. Analytical and Experimental Studie on Ga Bearing for Cryogenic Turboexpander Ph. D. diertation, IIT Kharagpur (). Ghoh, P. Analytical and Experimental Studie on Cryogenic Turboexpander Ph. D. diertation, IIT Kharagpur (). Balje, O. E. Turbomachine John Wiley and Son (98) 4. Kun, L.C. and Sentz, R. N. High efficiency expanion turbine in air eparation and liquefaction plant International Conference on Production and Purification of Coal Ga & Separation of Air, Beijing, China (985) 5. RohliK, Harold E. Analytical determination of radial inflow turbine geometry for maximum efficiency NASA TN D-484 (968) 6. Haelgruber, H. Stromunggerechte getaltung der laufrader von radialkompreoren mit axialem laufradeintrict Kontruction (958) () (in German).