Compressible Flow Modeling in a Constant Area Duct

Transcription

1 PDHonlne Course 98 ( PDH) Compressble Flow odelng n a Constant Area Duct Instructor: Nel Hcks, P.E. 0 PDH Onlne PDH Center 57 eadow Estates Drve Farfax, VA Phone & Fax: An Approved Contnung Educaton Provder

2 Compressble Flow odelng In a Constant Area Ppe Nel Hcks PE Course Content Fgure shows a system that s desgned to power a gearbox through the use of a turbne by expanson of a constant mass flow rate helum gas. In ths system the desgner wants to control the mass flow rate usng a regulator and a sonc orfce. It s mperatve that the orfce controls the mass flow to acheve the requred turbne output for the gearbox. Pout500 psg Po6000 Psg To540 R Helum Pressure Control Regulator Tout 545 R 60 n Smooth SS ppe Dameter? Insulated ppe Porfce out 950 psa To Atmosphere Requred Pturbne805 psa Tturbne n 533 R dot n 4.00 lbm/sec Gear Box Fgure : Example Compressble Flow System The requred turbne nput condtons are gven at the nlet of the turbne. The job of the engneer s to properly sze the ppe dameter to acheve the turbne nlet pressure and temperature of the helum gas. The regulator has been set to mantan an nlet orfce pressure of 500 psa and the orfce has been szed to choke and control the mass flow rate at 4.00 lbm/sec. The maxmum backpressure from the downstream system resstance s 050 psa. Any pressure above that wll unchoke the orfce and reduce the mass flow rate. To stay conservatve, the engneer s goal wll be to acheve a backpressure or orfce out recovery pressure of no greater than 950 psa. The followng nstructons wll revew compressble flow propertes of a gas assumng the flow wll be adabatc and wth frctonal losses (Fanno Flow). In addton, a Fanno Flow algorthm wll be derved to determne the requred ppe dameter for the system descrbed n Fgure. Page of 8

3 Adabatc Flow wth Frcton In a Constant Area Ppe Frcton cannot be gnored n a long constant area ppe. In addton, f the ppe s nsulated or the veloctes of the gases are hgh enough that gas temperature s not effected by heat transfer from or to the surroundngs, then modelng can be performed assumng adabatc flow. Adabatc flow cannot be assumed n long unsulated ppes n whch the velocty of the gas s slow and the gas temperature wll be affected by the heat transfer. The followng Fgure and Table summarzes the effects of Fanno flow propertes n a constant area ppe. The reader should revew Fgure and Table untl he/she has a complete understandng. It s mperatve at ths pont to understand statc pressure, total pressure, and temperature always decreases when velocty of a gas enterng a ppe s subsonc, and therefore the densty decreases. Because the densty of the gas s decreasng then the velocty of the gas partcles must ncrease to mantan a constant flow rate. If the velocty of the gas becomes equal to the speed of sound, then the flow rate wll choke at the end of the ppe, and the ppe can take control of the total flow rate by unchokng the flow-controllng orfce n the case of Fgure system. Thus the operator of the system wll lose the capablty of knowng the exact flow rate based on the orfce upstream pressure, temperature, and orfce Cd. T P Po ρ < To and ho constant Where TStatc Temperature PStatc Pressure PoStagnaton Pressure ρdensty > s Fgure : Schematc Ts Dagram Fanno Flow Lne for a Constant Area Duct Page of 8

4 Property Duct Entry Duct Entry Supersonc Obtaned from: Subsonc < > Stagnaton Temperature, To Constant Constant Energy Equaton Entropy, s Increases Increases Second Law Stagnaton Pressure, Po Decreases Increases Entropy ncreases Temperature, T Decreases Increases Shape of Fanno lne Statc Pressure, P Decreases Increases Equaton of state, and effects on ρ, T Densty, ρ Decreases Increases Contnuty equaton and effect on V Velocty, V Increases Decreases Energy Equaton and trend of T ach Number, Increases Decreases Trends of V, T, and defnton of Table : Summary of Effects of Frcton on a Compressble Gas Flowng In a Constant Area Ppe (Table coped from Introducton to Flud echancs by Fox & cdonald 4 th Edton) Fgure also shows what happens f the velocty of the gas enterng a ppe s supersonc. Frcton wll cause the pressure and temperature to ncrease n the drecton of flow. The densty wll ncrease and the velocty wll decrease to mantan the constant mass flow rate. Eventually the velocty n the ppe becomes sonc. In ths course, the dscusson and quz wll refer to modelng compressble flow wth the velocty of the gas enterng a constant area ppe at subsonc condtons. Fgure 3 s a dagram of gas flowng n a constant area ppe. The key to predctng gas propertes usng FANNO flow s knowng ether the nlet or the end condton pressure, temperature, and mass flow rate of the gas. In the case of ths course, the engneer knows the ppe end condtons. Lmax L L Lmax Inlet Outlet Fgure 3: Constant Area Ppe Revewng Fgure 3, one can see that L s some where n space downstream (hopefully) of the outlet of the ppe. L s a pont where the flow velocty wll be predcted to reach the speed of sound and the mach number wll be one (sonc). If Lmax or s calculated or predcted to be a length less than L (ppe length), then the sonc pont s nsde the ppe and the flow wll become sonc at the ext. Typcally ths s called Fanno chokng. As explaned n the precedng text, once the ppe becomes choked t could possbly take control of the mass flow rate. It s not uncommon to have more than one Page 3 of 8

5 chokng ponts n a system, but a problem can occur f the ppe chokng creates enough resstance to unchoke the upstream flow-controllng devce. If the upstream flowcontrollng devce s nstrumented to measure upstream pressure and temperature, then the pressure can be regulated to change the mass flow rate usng energy equatons. Unchokng such a devce due to excessve resstance negates the upstream flow rate predctng capablty. Usng the Fgure 3 as a reference and the equatons descrbed n the followng algorthm, one can easly solve for condtons at pont or pont by smply calculatng the mach number at ether pont usng the Lmax or Lmax respectfully. The remander of the text wll provde step-by-step drectons to demonstrate ths logc usng the descrbed problem n Fgure. Page 4 of 8

6 Solvng for the Dameter Step : Snce the condtons are known for pont of Fgure 3, then the analyss wll start by solvng for the ach number. However before dervng condtons at pont, an ntal guess of the ppe dameter s requred. Several guesses may need to be terated through the algorthm before achevng the proper pressure drop. For the frst guess, the author wll use a nomnal ppe dameter of.5 nch for a seamless stanless steel ppe wth a wall thckness of 0.0 nches and an nsde dameter of.0 nch. D A π D 4 p ρ ZRT () () where Z f the gas s assumed Ideal. However for ths exercse the author chose to treat the problem as Real Gas and Z was found usng NIST. velocty! m m ρa ρ! D () 3! V! m60 ρ std (4) ach# v krg T c v krt (5) c where k s the rato of specfc heats c p v Known Outlet Condtons INPUT P 805 psa T 533 R mdot lbm/s L 60 n Dh.0 n Gas Type 8 See Info on GASP Functon Calculated Inlet Condtons Gas Helum Gas Constant (ft-lbf)/(lbm-r) densty lbm/ft3 A 0.80 n vdot scfm velocty ft/s k.667 specfc heats rato ach Table : Input at Pont and Calculated Condtons (reference Fgure ) Page 5 of 8

7 Step : Once the mach number s known at pont, the followng condtons can be determned at the magnary sonc pont L of Fgure 3 usng the followng algorthm: Re ρvd µ where µ (6) lbm/ft -sec for helum ε.5 f.4 loglog + (7) 0.9 Re D h where ε for seamless smooth stanless steel duct T ( T)()[ + (0.5)( k ) + k ] (8) p ( P )( + k [ + (0.5)( k ) ) ] (9) fl max D h k + k ( + k) ln + k [ + (0.5)( k ) ] (0) D ( ) h + max ln + + k k L f k k [ (0.5)( ) ] + k () Table 3 gves the derved results of the condtons at Lmax. Note that Lmax s postve showng the sonc pont s downstream of the ppe. Havng ths data enables us to determne Lmax whch wll be explaned n step 3. T R P psa Darcy frcton factorƒ Lmax 99.0 n Lne Length Check Not Choked fl/dh.64 fl/dh.9 Table 3: Calculated Condtons at Imagnary Sonc Pont Lmax Page 6 of 8

8 Step 3: The followng step derves Lmax by smply addng the length of the ppe to Lmax derved n step. L max L + L max () Step 4: Once Lmax s calculated the followng equaton s used to derve the ach number at pont. Ths process can be smplfed greatly by usng some type of solver program or a spreadsheet. In table 4, the ach number s derved by teratng a mach number untl the dfference between the calculated Lmax and terated Lmax s equal to zero wthn sx sgnfcant dgts. D ( ) h + k + k L max ln + [ (0.5)( ) ] f k k + k (3) Calculated Inlet Condtons Calculated L + Lmax Lmax 59.0 n Guess ach Use Solver Iterated Lmax 59.0 n Dfference between Calculted and Iterated 4fl/Dh Iterates to 0000 Table 4: Calculated ach number at Pont Ppe Inlet Step 5: Now that the ach number s known at the nlet of the ppe, the condtons at the nlet can be derved. ost mportantly, the pressure and pressure drop can be determned usng a.5 n dameter tubng. p + k P ( 4 ) [ + (0.5)( k ) ] T ( + k) T ( 5 ) [ + (0.5)( k ) ] P psa T R Table 5: Ppe Inlet Condtons Condtons shown at Table 5 show that usng a.5 nch stanless steel tubng wth an nsde dameter of.0 nch provdes a downstream orfce recovery pressure of 948 psa. Our goal was a maxmum pressure of 950 psa. That s 0.0% dfference n pressure, and well wthn operatonal requrements for ths sample problem. Page 7 of 8

9 Course Summary Fanno flow modelng s used extensvely n one-dmensonal modelng of gases when the ach number s greater than 0.3, the process s assumed adabatc, and pressure loss due frcton s consdered. However, t not uncommon to use Fanno flow modelng for veloctes lowered than ach 0.3 as long as the system as long as the same assumptons are appled. Installng ths algorthm nto program or a spreadsheet reduces the workload greatly when teratng several dfferent ppe geometry confguratons. In addton, alternatng nlet condtons can lead to a quck understandng of the effect of pressure and temperature on the gas densty and gas velocty whch n turn effects chokng the flow nsde the ppe or chokng the flow outsde of the ppe at some magnary pont L. Once that understandng s understood, then desgnng or analyzng of compressble systems s smplfed greatly. Reference:. Robert W. Fox and Alan T. cdonald, Introducton to Flud echancs, 4 th Edton. Phlp Hll and Carl Peterson, echancs and Thermodynamcs of Propulson, nd Edton 3. chael R. Lndeburg PE, echancal Engneerng Reference anual for PE Exam, th Edton 4. Robert D. Blevns, Appled Flud Dynamcs Handbook, Reprnt Edton 99 Page 8 of 8