HW #3: Conservation of Linear Momentum, Conservation of Energy, Conservation of Angular Momentum and Turbomachines, Bernoulli s Equation

Transcription

1 HW #: Cnevatin f Linea Mentu Cnevatin f Enegy Cnevatin f ngula Mentu and Tubachine Benulli Equatin Due: Mn pil 0 at the ISE bx. i Bunyajitadulya HW #: Cnevatin f Linea Mentu Cnevatin f Enegy Cnevatin f ngula Mentu and Tubachine Benulli Equatin Pble. RTT and Cnevatin f Linea Mentu f Cntl lue a beved f Statinay Inetial Fae f Refeence [dapted f Fx et al. 00 Pble 4.84 p. 49.] tea f wate flw teadily thugh a cuved nzzle aebly that dichage the tea t the atphee. The nzzle a i 4.5 kg and it intenal vlue i ue that all fictinal fce can be neglected.. Cnide the vlue inide the nzzle excluding the nzzle a the cntl vlue C and al cnide the cincident ateial vlue M... Find the tie ate f change f linea entu f the cincident ateial vlue M y. Nte that thi i a vect quantity... If it i nt ze tate clealy which fce i epnible f thi change. Daw a fee-bdy-diaga FBD t illutate yu anwe... Deteine the net fce vect due t wate peue n the nzzle inne wall.. Nw cnide the cuved nzzle itelf and the wate inide it a the cntl vlue C and al cnide the cincident ateial vlue M... Find the tie ate f change f linea entu f the cincident ateial vlue M y... I it equal t that f M in.?; and if why?.. Deteine the eactin fce vect exeted n the nzzle at it flange..4. I the fce in. equal t that in.? Explain. = / y p = 5 kpa gage D = 7.5 c g x D =.5 c = 0

2 HW #: Cnevatin f Linea Mentu Cnevatin f Enegy Cnevatin f ngula Mentu and Tubachine Benulli Equatin Due: Mn pil 0 at the ISE bx. i Bunyajitadulya Slutin = / = / g y x F y p = 5 kpa gage D = 7.5 c p = 5 kpa gage D = 7.5 c F x w g D =.5 c = 0 w g D =.5 c = 0 n g C C Cntl lue Obeve C ae tatinay and nn-defing. C include the fluid tea inide the nzzle nly. C include the fluid tea inide the nzzle and the nzzle. Obeve i in the inetial fae f efeence IFR. uptin. Incpeible flw denity field i bth teady and unif. Steady velcity field.. Unif velcity at each c ectin. 4. Fictinal/vicu fce can be neglected. 5. Wate denity = 000 kg/. Baic Equatin. RTT dn M t dnc t f d N t d t / dm M t dm C t. C-Ma: 0 f / d M t d. C-M F F b M C t t f t t / d P t d t B C Calculatin Q / / 8 / c in 0 9 x / Q 000 kg / / 8.84 kg / y /

3 HW #: Cnevatin f Linea Mentu Cnevatin f Enegy Cnevatin f ngula Mentu and Tubachine Benulli Equatin Due: Mn pil 0 at the ISE bx. i Bunyajitadulya C C-Ma F an incpeible flw f C-Ma we have : Q Q : Q. With the unif velcity ve each c ectin we find. RTT f Linea Mentu Unteady Te C F a tatinay and nn-defing C and teady denity and velcity field we have 0 C. Net Cnvectin Efflux Te d f / With unif velcity ve each c ectin and the eult f C-Ma : we al have f / d. RTT Thu the RTT Eq. bece M t. M x t x x c 0 c / c kg kg / N M y t y y in in / in kg / 6.9 N Thu M t ˆ ˆ / c i / in j ˆ ˆ kg / 7.7 i 6.9 j N NS If fictin/vicu fce i negligible the nly uface fce i peue and the nly bdy fce i gavitatin. Hence the fce that ae epnible f thi ae x -cpnent: peue ve the nzzle inne wall and inlet and exit c ectin. NS y -cpnent: peue a abve and the weight f the wate in the nzzle. NS Thi i illutated in the FBD f C abve. C-x-M Fx Fpn x p c p c pat c0 4. N whee F pn x i the x -cpnent f the net peue fce f the nzzle inne wall n the C. F bx 0 pply the C-M M x C x Fx Fbx x f / d t with the tie ate f change f linea entu f the cincident M given by Eq. we have

4 HW #: Cnevatin f Linea Mentu Cnevatin f Enegy Cnevatin f ngula Mentu and Tubachine Benulli Equatin Due: Mn pil 0 at the ISE bx. i Bunyajitadulya 4 F pn. x M x p c M x Fpn. x p c F N pn. x 80.8 N C-y-M F F p p in y whee F g by pn y w F i the y -cpnent f the net peue fce f the nzzle inne wall n the C. pn y pply the C-M Fy Fby M y C y d 4 y t with the tie ate f change f linea entu f the cincident M given by Eq. we have M y Fpn. y p p in wg M y Fpn. y p p in wg N 9 N Thu the net fce due t wate peue n the nzzle inne wall i given by F F 8 iˆ 9 ˆj N. NS pn pn f / Phyical Intepetatin Phyically the peue fce at and the weight f the wate wk againt the net peue fce the peue fce at in de t caue the acceleatin f the M in the dwnwad diectin: M y p wg Fpn. y p in 6.9 N. F pn. y and C RTT If we wite the RTT f linea entu f C/M we find that M t M t ˆ ˆ / 7.7 ˆ 6.9 ˆ kg / c i / in j i j N That i the tie ate f change f the linea entu f M and M ae equal. C iew: The ean f the equality i that the lid nzzle i tatinay; theefe it cntibute t neithe t the tie ate f change C n the net cnvectin efflux f / d ve the t f C. M iew: In the wd thee i n cntibutin t the tie ate f change f linea entu f the nzzle ince the nzzle i tatinay and it linea entu de nt change. NS C-x-M F F x x F bx 0 pply the C-M 5

5 HW #: Cnevatin f Linea Mentu Cnevatin f Enegy Cnevatin f ngula Mentu and Tubachine Benulli Equatin Due: Mn pil 0 at the ISE bx. i Bunyajitadulya 5 C-y-M F F x F bx M x C x x t f / d with the tie ate f change f linea entu f the cincident M given by Eq. 5 we have M x Fx / c 7. 7 N y Fy p g [Recall the net peue fce due t unif peue.] F g by w n pply the C-M Fy Fby M y C y y t f / d with the tie ate f change f linea entu f the cincident M given by Eq. 5 we have M y Fy pg w n g M y Fy pg wg n g N 554 N Thu the eactin fce exeted n the nzzle at it flange i given by F 8 iˆ 554 ˆj N. NS The eactin fce exeted n the nzzle at i flange F 8 iˆ 554 ˆj 8 iˆ 9 ˆj N n 8 iˆ 9 ˆj N. F pn F pn N i equal t neithe Even thugh the tw M have the ae tie ate f change f linea entu each ha diffeent uce f fce bdy fce and uface fce at the bundaie n the a hwn in the FBD belw. Thu althugh they ae elated in geneal we d nt expect the t be equal. Thei elatin can be een e clealy f the FBD and the tatic equilibiu f the nzzle alne a hwn belw. NS F y F y F x F x F pn F pn w g w g C N eactin fce n the flange peent n g C N peue fce f nzzle inne wall peent n g FBD f the nzzle alne N peue fce at inlet and exit peent

6 HW #: Cnevatin f Linea Mentu Cnevatin f Enegy Cnevatin f ngula Mentu and Tubachine Benulli Equatin Due: Mn pil 0 at the ISE bx. i Bunyajitadulya 6 Pble. Cnevatin f Ma and Linea Mentu f Cntl lue a beved f Statinay Fae f Refeence [dapted f Munn et al. 00 Pble 5.4 p. 8.] The hydaulic dedge i ued t dedge and f a ive btt and the and/wate ixtue i dichaged a a fee-jet a hwn belw. The dichage ha a c ectinal aea. The jet dichage peed i which i iented at an angle with epect t the hizntal. ue that the bell-uth uctin ha elative lage c ectinal aea uch that the and/wate ixtue peed at the uctin can be neglected. The pecific gavity f the and/wate ixtue i SG and wate denity i... Etiate the thut needed f the ppelle t hld the bat tatinay... Unde thi tatinay peating cnditin i the buyancy fce n the bat equal t the weight W f the bat including all the lad n the bat? If nt hw de the buyancy fce change f that when the hydaulic dedge i nt peated? Slutin Cntl lue: The cntl vlue i tatinay and nn-defing. It i the egin ccupied by the bat nly. y x T B W uptin. Incpeible flw teady and unif field. Steady field.. Unif at each c ectin. 4. The bell-uth uctin ha elative lage c ectinal aea uch that the and/wate ixtue peed at the uctin can be neglected. a. We aue that the agnitude f the velcity field aund the bat i all uch that the peue ditibutin can be appxiated by the hydtatic peue vaiatin. M x C x C- x -M: F Sx FBx u f / d P x u d Unteady te C x d C u d : tatinay and nndefing C C x d u d C C Net Cnvectin Efflux Te u f / d : u d t teady and C 0 d 0

7 HW #: Cnevatin f Linea Mentu Cnevatin f Enegy Cnevatin f ngula Mentu and Tubachine Benulli Equatin Due: Mn pil 0 at the ISE bx. i Bunyajitadulya 7 u f / RTT pat: M d u u Net Suface Fce F Sx F Sx T Net Bdy Fce F Bx F Bx 0 Thu C- x -M bece c f / u x C x d u f / u CMa d f / u d u i u 0 unif ve each c u f / d u T c SG c. NS M y C y C- y -M: F Sy FBy v f / d P y v d B C y d Unteady te v d : C Siila t the cae f x -entu we have C y 0. Net Cnvectin Efflux Te v f / d : Siila t the cae f x -entu we have v f / d v v in. f / d RTT pat: M y C y v f / in d Thu the RTT tate that thee i a tie ate f change f linea y -entu f the cincident ateial vlue M. ccding t the Newtn ecnd law the net fce in the y diectin n the M hence al the C ut nt vanih. Net Suface Fce F Sy F Sy B Buyancy i the eultant f peue ditibutin ve the uface f the ubeged bdy. Hee we al aue that the agnitude f the velcity field aund the bat i all uch that the peue ditibutin can be appxiated by the hydtatic peue vaiatin. Net Bdy Fce F By F By W Thu C- y -M B bece

8 HW #: Cnevatin f Linea Mentu Cnevatin f Enegy Cnevatin f ngula Mentu and Tubachine Benulli Equatin Due: Mn pil 0 at the ISE bx. i Bunyajitadulya 8 B W in B W in W SG. in When the dedge i peated we have B W SG in When it i nt peated 0 we have B W. Thu unde the peating cnditin the buyancy fce: B W SG in i nt equal t the weight W f the bat; the buyancy fce unde the peating cnditin i e than when the dedge i nt peated by the aunt f SG in. NS

9 HW #: Cnevatin f Linea Mentu Cnevatin f Enegy Cnevatin f ngula Mentu and Tubachine Benulli Equatin Due: Mn pil 0 at the ISE bx. i Bunyajitadulya 9 Pble. Benulli Equatin and Cnevatin f Linea Mentu [ HW #6.] nzzle i attached t a vetical pipe and dichage wate int the atphee a hwn in the figue belw. The dichage ate i 0. /. The nzzle ha a weight f 00 N and the vlue f wate in the nzzle i Find the gage peue at the nzzle inlet... Find the fce vect F F iˆ x F ˆ y j that i equied t uppt the nzzle at the flange uppt. ue that the wate in the nzzle i acceleated at a ate uch that the fictinal effect can be neglected. Nte: In the pat we have lved the cnevatin f linea entu with peue hence peue fce being given. With the knwledge f the Benulli equatin unde the auptin ang the f inviicid flw we can ue the Benulli equatin t lve f the unknwn peue fit then ue the peue infatin t find the peue fce f the C-M equatin. 0.5 Slutin ˆ c i in ˆ j H y F F iˆ F ˆj x y p g Wnz W wate ˆj x Cntl vlue The cntl vlue C i tatinay and nn-defing. It include the wate vlue and the nzzle a hwn abve. uptin. ll ppetie ae teady.. Incpeible flw. Invicid flw 4. Pint and ae n the ae tealine. 5. ll ppetie ae unif at each c ectin. The velcity i axial and unif at each c ectin. 6. Wate denity = 000 kg/. Baic Equatin:

10 HW #: Cnevatin f Linea Mentu Cnevatin f Enegy Cnevatin f ngula Mentu and Tubachine Benulli Equatin Due: Mn pil 0 at the ISE bx. i Bunyajitadulya 0 dm M dm C C-Ma: 0 f / d M d M x C x C- x -M: F Sx FBx u f / d P x u d B M y C y C- y -M: F Sy FBy v f / d P y v d C Benulli Equatin p gy p gy kg Q kg / Q 0. / Q 0. / 5 / 0 / kg W wate g N Pa.. Cnide the tealine a hwn in the figue. F uptin -4 we can apply the Benulli equatin and get p gy p gy p gy p gy p p pg / gh / gh C Ma : Thu p p pg / gh kg kg Pa 45.5 Pa 40.0 kpa.. M x C x C- x -M: F Sx FBx u f / d P x u d B C x d Unteady te u d : C tatinay and nndefing C C x d u d C C Net Cnvectin Efflux Te u f / d : u d t teady and C 0 d 0 NS

11 HW #: Cnevatin f Linea Mentu Cnevatin f Enegy Cnevatin f ngula Mentu and Tubachine Benulli Equatin Due: Mn pil 0 at the ISE bx. i Bunyajitadulya u f / d u u f / u d u CMa f / u d u i unif ve each u f / d u f / d RTT pat: M x C x u u u f / d c Net Suface Fce F Sx : F Sx F x [Except at peue i unifly p at thughut. Peue n de nt cntibute t the fce in the x diectin.] Bdy Fce F Bx : F Bx 0 Thu Eq. B bece Fx c kg 00 0 c0 866 N. M y C y C- y -M: F Sy FBy v f / d P y v d C C y d Unteady te v d : C tatinay and nndefing C C y d v d C C Net Cnvectin Efflux Te v f / d : v f / d v v f / v d v CMa v d t f / v d v teady and i C unif ve each 0 d 0 v f / d v f / d RTT pat: M y v C y Net Suface Fce F Sy : F Sy F y Bdy Fce F By : F By v v f / d p pat Fy pg Wnz Wwate in / in 4 Thu Eq. C bece Fy pg Wnz Wwate / in. 5 and

12 HW #: Cnevatin f Linea Mentu Cnevatin f Enegy Cnevatin f ngula Mentu and Tubachine Benulli Equatin Due: Mn pil 0 at the ISE bx. i Bunyajitadulya F y / in in W nz W wate N N N Thu we have the fce vect F that i equied t uppt the nzzle at the flange uppt F 866iˆ 48. ˆj N. NS p g

13 HW #: Cnevatin f Linea Mentu Cnevatin f Enegy Cnevatin f ngula Mentu and Tubachine Benulli Equatin Due: Mn pil 0 at the ISE bx. i Bunyajitadulya Pble 4. Cnevatin f Linea Mentu f Cntl lue a beved f Mving/Tanlating Fae f Refeence [dapted f Fx et al. 00 Pble 4.4 p. 55.] cket led weighing N and tavelling 960 k/h i t be baked by lweing a cp int a wate tugh. The cp i w = 50 wide.. Deteine the tie equied afte lweing the cp t a depth f h = 75 int the wate t bing the led t a peed f k/h.. Plt the led peed a a functin f tie.. a cnequence f yu cuent eult in te f deign ugget thee ethd t educe the tie equied. = Slutin = f U t iˆ y MFR x D W y IFR x Cntl lue Obeve C i tatinay and nn-defing in the ving fae f efeence MFR. The fae MFR i ving with the led. The C include the led and wate in the cp. n beve i in the MFR. uptin. Incpeible flw denity field i bth teady and unif. Neglect change in linea entu in the C.. Unif velcity at each c ectin. 4. N change in elative peed f wate n the cp f inlet t exit. 5. The cp cntain cntant aunt f wate ve tie. 6. Neglect a f wate in the cp M C M led 7. Neglect any the dag and/ fictinal fce. 8. Wate denity = 000 kg/. Baic Equatin dm M t dm C t. C-Ma: 0 f / d M t d t t *** Nte that hee we ue C-Ma f an beve in MFR and nt in IFR hence f / and nt f /. ***

14 HW #: Cnevatin f Linea Mentu Cnevatin f Enegy Cnevatin f ngula Mentu and Tubachine Benulli Equatin Due: Mn pil 0 at the ISE bx. i Bunyajitadulya 4. C-M: F F b M C a f M C t d P t d f / t B elcitie and Relative elcitie Let f U iˆ d f du. Then a f iˆ. f 0 Uiˆ Uiˆ U [N change in elative peed f wate n the cp f inlet t exit.] U c iˆ in ˆ j C-Ma dm t Unteady Te C Becaue C i tatinay and nn-defing in MFR the flw f wate i incpeible and the cp cntain cntant aunt f wate ve tie dm t C 0. The C-Ma Eq. then give : Q Q : Q. The a flwate i given by d U [incpeible flw unif velcity at each c ectin f / f U iˆ / ] C-x-M F 0. [Neglect any the dag and/ fictinal fce.] x F bx 0. du M C a f M C [ f U t iˆ d f du. Thu a f iˆ ] C x 0. [Neglect change in linea entu in the C.] d U c U U c x f / x x x x t [unif velcity at each c ectin C-Ma] C-M Eq. B bece du 0 0 M C U c du c U M U U U U du t 0 C c M c t U M C C U U c U t b hw bt M C U t U b kg k / c0 U c h.6 k / h b.4 M N C 9.8 /

15 HW #: Cnevatin f Linea Mentu Cnevatin f Enegy Cnevatin f ngula Mentu and Tubachine Benulli Equatin Due: Mn pil 0 at the ISE bx. i Bunyajitadulya 5 Hence t deceleate f U = 960 k/h t U t = k/h equie [Neglect a f wate in the cp.] 960 k / h t. 5 k / h.4. NS The plt f peed S tie i hwn belw. NS In de t educe the deceleatin tie Eq. ugget that we need t inceae b. Thi can be achieved by deceae the angle inceae by inceae the depth h inceae the wih w. NS Since c 0 i aleady equal t 0.87 while c 0 = the deceae in eult in little change in cpain t an inceae in. The plt belw hw the peed S tie f tw value f h ne i twice the the U k/h h = h = t

16 HW #: Cnevatin f Linea Mentu Cnevatin f Enegy Cnevatin f ngula Mentu and Tubachine Benulli Equatin Due: Mn pil 0 at the ISE bx. i Bunyajitadulya 6 Pble 5. Cnevatin f Enegy [Fx et al. 00 Pble 4.0 p. 60.] i ente a cpe at 96 kpa 7 C with negligible peed and i dichaged at 480 kpa 60 C with a peed f 5 /. If the pwe input i.8 MW and the flw ate i 9 kg/ deteine the ate f heat tanfe. Slutin W =.8 MW Q W = 9 kg/ p = 96 kpa T = 7 C 0 / p = 480 kpa T = 60 C = 5 / + Q =? Cntl lue C i tatinay and nn-defing. It cve the cpe f inlet t exit. uptin. Steady ppety field.. Unif ppetie at each c ectin.. Neglect all the wk except haft wk. 4. Neglect velcity/kinetic enegy at inlet. 5. Neglect change in ptential enegy. 6. Teat ai a a pefect ga with cntant c p = 005 J/kg-K R = 87 J/kg-K. Baic Equatin dm M t dm C t. C-Ma: 0 f / d M d. d. C-Enegy Q W e d e pv f / d B C : W W W hea W the e pv h gz C-Ma F teady ppety field f C-Ma we have :. C-Enegy Q? W W [Neglect all the wk except haft wk.] d 0 e d [Steady ppety field.] C e pv d f / e pv f / d h gz h h c p T T Thu the C-Enegy Eq. B bece h gz [pefect ga] t [unif ppetie at each c ectin C - a.] [neglect kinetic enegy at inlet neglect change in ptential enegy.]

17 HW #: Cnevatin f Linea Mentu Cnevatin f Enegy Cnevatin f ngula Mentu and Tubachine Benulli Equatin Due: Mn pil 0 at the ISE bx. i Bunyajitadulya 7 kw MW K K kg J kg W T T c Q T T c W Q p p / Thu thee i heat tanfe ut at the ate f 68.6 kw. NS

18 HW #: Cnevatin f Linea Mentu Cnevatin f Enegy Cnevatin f ngula Mentu and Tubachine Benulli Equatin Due: Mn pil 0 at the ISE bx. i Bunyajitadulya 8 Pble 6. Idealized Machine: I it a pup a tubine? Idealized xial-flw Tubachine F the given blade angle. qualitatively ketch the blade hape. qualitatively ketch the inlet and exit velcity diaga. ue apppiate gvening equatin t hw whethe the deign i uitable f a pup a tubine. 4. l tate neceay auptin yu ake in de t aive at thee eult. flw z U a b flw flw U U Idealized Radial-Flw Tubachine [dapted f Munn et al. 00 Pble. p. 86.] The t hwn belw ha taight thugh backwadly inclined blade and cntant paage wih f inlet t exit. It tate with an angula velcity f 000 p. ue that the fluid ente the ipelle with the ablute flw velcity puely in the adial diectin and the elative flw velcity i tangent t the blade ac the entie t. Ue apppiate velcity diaga and gvening equatin t hw whethe the device i a pup a tubine. Relevant velcity cpnent and angle huld be calculated in de t hw clealy that thi device i a pup a tubine.

19 HW #: Cnevatin f Linea Mentu Cnevatin f Enegy Cnevatin f ngula Mentu and Tubachine Benulli Equatin Due: Mn pil 0 at the ISE bx. i Bunyajitadulya 9 Slutin: Idealized xial-flw Tubachine a flw n U n U U : pup cncave ide leading fwad b b U ê b b b U b ê n flw U n U U : tubine cnvex ide leading fwad b U b ê NS

20 HW #: Cnevatin f Linea Mentu Cnevatin f Enegy Cnevatin f ngula Mentu and Tubachine Benulli Equatin Due: Mn pil 0 at the ISE bx. i Bunyajitadulya 0 flw c U c T T xial-flw achine Radial-flw achine Cntl lue: F bth axial- and adial-flw achine cntl vlue ae tatinay and nn-defing a hwn abve. It include the t/ipelle and cut thugh the lid haft. T i the haft tque. uptin. The flw i incpeible i bth teady and unif.. The velcity field i teady in ean. Ue and evaluate the ean ppetie.. Neglect all the ent/tque e.g. ent due t the uface fce fictin/hea and peue fictinal tque at beaing ent due t bdy fce g except haft tque. 4. i unif ve each c ectin. F axial flw achine: evaluate ppetie/velcitie at ean adiu. F adial flw achine: velcity and the cepnding velcity cpnent i unif with epect t the z cdinate. F futhe detail ee ftnte. 5. Shckle enty/exit cnditin. That i the elative flw i.e. the velcity f fluid elative t the tating blade b ente and leave the t at the geetic blade angle. Ue in the cntuctin f the velcity diaga. NS Baic Equatin. C-Ma: dm M t dm C t 0 f / d M t d. t t. C-ngula Mentu: dh M c t dh C c t M c f / d H c t d t t B : M T F g d c S C t uptin: t each inlet/exit c ectin i unif ve the c ectin. / d f exit ectin f inlet ectin. f eˆ zeˆ z eˆ eˆ z eˆ z z eˆ z z eˆ eˆ z eˆ n eˆ z axial f / d zdz f / d eˆ eˆ zeˆ z deˆ n eˆ n eˆ adial f / d d Given that we ae inteeted in the tque f the haft which i aligned in the z diectin the abve auptin iplie that xial-achine: de nt depend n d z ; i unif ve the c ectin. Radial-achine: de nt depend n d ; i unif ve the c ectin.

21 HW #: Cnevatin f Linea Mentu Cnevatin f Enegy Cnevatin f ngula Mentu and Tubachine Benulli Equatin Due: Mn pil 0 at the ISE bx. i Bunyajitadulya C-ngula Mentu: Unteady Te t H d c C ] 0 ae tedy and [ 0 [C i tatinay and nn - defing.] t d t d d t dh C C c C Net cnvectin efflux te f d / : : c ectin] ve each unif i [ / / / / / / / f f f f f f f d d d d d d d RTT pat: / d dh dh f c C c M Net extenal ent abut c c M except haft tque the ent Negelect all t C S c T d g F T M Then Eq. B bece the Eule Tubachine equatin Eule tubachine equatin: tque hydaulic : h T T C and we al have the aciated pwe equatin Eq. C The ciated Pwe Equatin: U U T W. D Nte that f C-Ma we al have :

22 HW #: Cnevatin f Linea Mentu Cnevatin f Enegy Cnevatin f ngula Mentu and Tubachine Benulli Equatin Due: Mn pil 0 at the ISE bx. i Bunyajitadulya Idealized axial-flw achine Becaue U U : U the aciated pwe equatin Eq. D bece W T U U U :. Thu if i in the ae diectin a U W U 0 and it i a pup enegy being input int the yte; if i in the ppite diectin t U W U and it i a tubine enegy being extacted f the yte. 0 The velcity diaga hwn abve theefe indicate that the cnditin f the angle and unde which the achine change type i a fllw: W U 0 pup. Hence a i a pup. W U 0 tubine. Hence b i a tubine. NS

23 HW #: Cnevatin f Linea Mentu Cnevatin f Enegy Cnevatin f ngula Mentu and Tubachine Benulli Equatin Due: Mn pil 0 at the ISE bx. i Bunyajitadulya Slutin: Idealized Radial-Flw Tubachine Becaue i puely adial while U i puely tangential U 0 and the aciated pwe equatin Eq. D bece W T U. Thu if U i in the ae diectin a W U 0 and it i a pup enegy being input int the yte; if i in the ppite diectin t U W U 0 and it i a tubine enegy being extacted f the yte. Inlet elcity Diaga b 0 ft/ n 0.8 U 04.7 ft/ U ˆ ev ad in U ft 04.7 ft / in ev 60 n 0 ft / 0 ft / tan tan 0.8 U 04.7 ft / Exit elcity Diaga e n b. ft/ n.7 = 0.8 U 88.5 ft/ U ˆ e ev ad in U ft 88.5 ft / in ev 60 C-Ma 0 n n b n n n n b cntant paage wih b n n ft ft 0 ft /. ft / n n. ft / tan tan.7 U 88.5 ft / Since f a taight backwadly-inclined blade ee nte belw. Thu The velcity diaga abve then hw that f i in the ae diectin a U and theefe W U 0. Hence the achine i a pup. NS Nte The elatin between the blade angle and f taight-bladed adial-flw achine. b

24 HW #: Cnevatin f Linea Mentu Cnevatin f Enegy Cnevatin f ngula Mentu and Tubachine Benulli Equatin Due: Mn pil 0 at the ISE bx. i Bunyajitadulya 4 F a iple taight blade geety we have the fllwing elatin. Backwadly-inclined blade / / Fwadly-inclined blade / / In bth cae we ee that when 0 / and we have a adial blade i.e. the blade i aligned in the adial diectin. Backwadly-inclined blade Fwadly-inclined blade

25 HW #: Cnevatin f Linea Mentu Cnevatin f Enegy Cnevatin f ngula Mentu and Tubachine Benulli Equatin Due: Mn pil 0 at the ISE bx. i Bunyajitadulya 5 Pble 7. Radial-Flw Machine and The Effect f The Exit Blade ngle [ HW #6.] centifugal adial wate pup ha the dienin hwn in the figue belw. The vlue ate f flw i 0.5 ft / and the ablute inlet velcity i diected adially utwad. The angula velcity f the ipelle i 960 p. The exit velcity a een f a cdinate yte attached t the ipelle can be aued t be tangent t the vane at it tailing edge. Calculate the ideal pwe equied t dive the pup. Slutin Cntl lue: Cntl vlue i tatinay and nn-defing a hwn belw. It include the t/ipelle and cut thugh the lid haft. T i the haft tque. ê ê z uptin. Incpeible flw teady and unif denity field.. elcity field i teady in ean and evaluate the ean ppety.. Neglect fictinal tque and tque due t the uface fce except at haft.. 4. Neglect tque by bdy fce. 5. i unif ve each c ectin. F adial flw achine: velcity and the cepnding velcity cpnent i unif with epect t the z cdinate. 6. Shckle enty/exit cnditin. Ue in the cntuctin f the velcity diaga. Baic Equatin: C-Ma: C-ngula Mentu: dm 0 M t dm C t f / d M t t d.

26 HW #: Cnevatin f Linea Mentu Cnevatin f Enegy Cnevatin f ngula Mentu and Tubachine Benulli Equatin Due: Mn pil 0 at the ISE bx. i Bunyajitadulya F B S T C dh M C t dh C C t g d d H C t d 6 C-ngula Mentu: dh M C t dh C C t FS T g d d H C B dh C C t d Unteady te d : C tatinay and teadyin dh nndefing C and C C t d d d t C C Net Cnvectin Efflux Te f / d : f / d d f / i unif ve each evaluate at ean adiu f / d C ean t C d 0 d 0 f / d f / d CMa RTT pat: dh M C dh C C / d f Tque due t uface fce except at haft: FS F 0 S [Neglect fictinal tque and tque due t the uface fce except at haft.] Tque due t bdy fce: gd C g d 0 [Neglect tque by bdy fce.] C Shaft Tque: T Then Eq. B bece the Eule Tubachine equatin Eule tubachine equatin:. C and we al have the aciated pwe equatin T The ciated Pwe Equatin: W T U U. D

27 HW #: Cnevatin f Linea Mentu Cnevatin f Enegy Cnevatin f ngula Mentu and Tubachine Benulli Equatin Due: Mn pil 0 at the ISE bx. i Bunyajitadulya 7 Idealized adial-flw tubachine Becaue i puely adial while U i puely tangential U 0 and the aciated pwe equatin Eq. D bece W T U. b N 60 U Q Q 0.75 ft ft 0.80 ft ft ft lug ft ft.89 ft / ad ev in ad / ev in 60 ad 00.5 ft ft / lug / lug.94 ft elcity Diaga b U ê.89 ft / b : b in b b.696 ft / in in 55 : U b c U ct ft /.89 ft / ct ft / Thu f Eq. we have the ideal haft pwe W T U U lug ft ft lbf ft lbf ft hp. NS hp 550 lbf ft /

28 HW #: Cnevatin f Linea Mentu Cnevatin f Enegy Cnevatin f ngula Mentu and Tubachine Benulli Equatin Due: Mn pil 0 at the ISE bx. i Bunyajitadulya 8 Pble 8. Radial-Flw Machine [dapted f Fx et al. 00 Pble 0.4 pp ] Dienin f a centifugal pup ipelle ae Paaete Inlet Outlet Radiu Blade wih b 50 0 Blade angle deg The pup handle wate and i diven at 750 p. t the cuent peating cnditin the vlue flwate i 0.75 /. ue that the elative flw ente and leave the blade at the blade angle and epectively hckle enty/exit cnditin.. Daw the inlet and exit velcity diaga.. Find inlet and exit flw angle and.. Find the theetical haft tque. 4. Find the theetical haft pwe. 5. Find the theetical hydaulic pwe. 6. Find the theetical hydaulic head. 7. If intead the pup handle a fluid with denity 00 kg/ at the ae kineatical cnditin i.e. the ae velcity diaga find the theetical hydaulic pwe and head. Cpae thee t the f wate..

29 HW #: Cnevatin f Linea Mentu Cnevatin f Enegy Cnevatin f ngula Mentu and Tubachine Benulli Equatin Due: Mn pil 0 at the ISE bx. i Bunyajitadulya 9 Slutin c T Radial-flw achine Cntl lue: Cntl vlue i tatinay and nn-defing a hwn abve. It include the t/ipelle and cut thugh the lid haft. T i the haft tque. uptin. The flw i incpeible i bth teady and unif.. The velcity field i teady in ean. Ue and evaluate the ean ppetie.. Neglect all the ent/tque e.g. ent due t the uface fce fictin/hea and peue fictinal tque at beaing ent due t bdy fce g except haft tque. 4. i unif ve each c ectin. F adial flw achine: velcity and the cepnding velcity cpnent i unif with epect t the z cdinate. 5. Shckle enty/exit cnditin. That i the elative flw i.e. the velcity f fluid elative t the tating blade b ente and leave the t at the geetic blade angle. Ue in the cntuctin f the velcity diaga. 6. Wate denity = 000 kg/. Baic Equatin. C-Ma: :.. Eule tubachine equatin: T : hydaulic tque Th B. The ciated Pwe Equatin: W T U U. C 4. Relative elcity Relatin: U b D elcity Diaga: U b. Inlet velcity diaga Rtatinal peed N = 750 RPM = 78.5 ad/ Inlet adiu = 75. Inlet wih b = 50. Inlet blade tangential peed U 78.5 ad / / Inlet blade angle = 65 lue flwate Q = 0.75 /. Ma flwate kg Q / 750 kg / F the vlue flwate equatin we have Q d whee b Q 0.75 / Thu.64 / The velcity diaga can then be dawn a fllw..

30 HW #: Cnevatin f Linea Mentu Cnevatin f Enegy Cnevatin f ngula Mentu and Tubachine Benulli Equatin Due: Mn pil 0 at the ISE bx. i Bunyajitadulya 0 ˆ n e.64 / b U.74 / ê NS F the elative velcity elatin U velcity diaga we have b : : in c U b b in c / b in tan in.64 / 5. / in 65 U.74 /.64 / ct 65 U b c U in b.74 / ct / c b c U.64 / 5.5 / c 8.4 c U in 7.8 / ct U ct 0.54 ct NS. Exit velcity diaga Rtatinal peed N = 750 RPM = 78.5 ad/ Outlet adiu = 500. Outlet wih b = 0. Outlet blade tangential peed U 78.5 ad / / Outlet blade angle = 70 F C-Ma we have 0 b b b b b / 7.96 / b The velcity diaga can then be dawn a fllw. 7

31 HW #: Cnevatin f Linea Mentu Cnevatin f Enegy Cnevatin f ngula Mentu and Tubachine Benulli Equatin Due: Mn pil 0 at the ISE bx. i Bunyajitadulya ˆ n e 7.96 / U 9. / b ê NS F the elative velcity elatin U velcity diaga we have b : : in c U b b in c /9 0 : 9 tan b in 7.96 / 8.47 / in 70 U c U 9. / 8.47 / ct 70 U b c in b 9. / ct / c b U 7.96 / 7. / c ct ct 6.4 / 4.57 U ct NS. Theetical haft tque F the Eule tubachine equatin: T : hydaulic tque Th B We have T kg Theetical hydaulic and haft pwe gh p In thi ideal cae the hydaulic pwe and haft pwe ae equal p : W gh p W p.7 kn 78.5 ad / 5. Theetical hydaulic head W p U U T H p g g g 995. kw kg / 9.8 / / /.7 kn NS U U T 995. kw p 4 5 NS kg / NS 6

32 HW #: Cnevatin f Linea Mentu Cnevatin f Enegy Cnevatin f ngula Mentu and Tubachine Benulli Equatin Due: Mn pil 0 at the ISE bx. i Bunyajitadulya 6. If intead the pup handle a fluid with denity 00 kg/ at the ae kineatical cnditin i.e. the ae velcity diaga find the theetical hydaulic pwe and head. Cpae thee t the f wate. F the equatin f hydaulic head W p U U Enegy H p Length 6 g g Weight we ee that the hydaulic head which ha the dienin f Enegy/Weight depend nly n the kineatical cnditin and nt n denity f the fluid. Thu even thugh we change fluid denity t 00 kg/ unde the ae kineatical cnditin hence ae vlue flwate - we till have the ae hydaulic head: Hydaulic head H = 5. but nw at new Q 00 kg / 0.75 / 900kg /. p On the the hand the hydaulic pwe de change t Hydaulic pwe kg gh p kw. NS 7 That i cpaing t the f wate we ee that the hydaulic head - which ha the dienin f Enegy/Weight eain unchanged while the hydaulic pwe inceae linealy with denity.