AN ANALYTICAL STUDY OF THE INTERIOR BALLISTICS PROBLEM, INCLUDING MOVEMENT OF SOLIDS AND WALL HEAT TRANSFER A THESIS. Presented to

Transcription

1 In presenting the dissertatin as a partial fulfillment f the requirements fr an advanced degree frm the Gergia Institute f Technlgy, I agree that the Library f the Institute shall make it available fr inspectin and circulatin in accrdance with its regulatins gverning materials f this type. I agree that permissin t cpy frm, r t publish frm, this dissertatin may be granted by the prfessr under whse directin it was written, r, in his absence, by the Dean f the Graduate Divisin when such cpying r publicatin is slely fr schlarly purpses and des nt invlve ptential financial gain. It is understd that any cpying frm, r publicatin f, this dissertatin which invlves ptential financial gain will nt be allwed withut written permissin. 7/25/68

2 AN ANALYTICAL STUDY OF THE INTERIOR BALLISTICS PROBLEM, INCLUDING MOVEMENT OF SOLIDS AND WALL HEAT TRANSFER A THESIS Presented t The Faculty f the Divisin f Graduate Studies and Research by Pradip Sana In Partial Fulfillment f the Requirements fr the Degree t «i,..' Master f Science!'in Mechanical Engineering Gergia Institute f Technlgy September, 1971

3 AN ANALYTICAL STUDY OF THE INTERIOR BALLISTICS PROBLEM, INCLUDING MOVEMENT OF SOLIDS AND WALL HEAT TRANSFER Apprved: Chairman / Date apprved by Chairman / **?/. z => 71

4 11 ACKNOWLEDGMENTS The authr is deeply indebted t his faculty advisr, Dr. S. V. Sheltn, fr suggesting the prblem and fr cntinuus guidance and encuragement withut which this wrk culd nt be finished. It gives the authr a great pleasure t thank Drs. C. W. Grtn and A. E. Bergles fr their sincere interest and cnstructive suggestins in curse f reviewing this wrk. The cmments made by Dr. P. V. Desai n the bundary layer analysis are als appreciated. The authr is grateful t the U. S. Air Frce fr financial aid and a few experimental data f great help. Many ther persns, t numerus t list, helped the authr bth directly and indirectly during the perid f executin f this wrk. Special thanks are t Mr. P. K.Raut fr his help in varius stages f cmputer prgramming. The authr wuld like t dedicate this wrk t his parents fr their immeasurable lve, training and. cnstant encuragement in cntinuatin f his educatin.

5 TABLE OF CONTENTS Page ACKNOWLEDGMENTS ii LIST OF TABLES v LIST OF FIGURES.. ' vi NOMENCLATURE ix SUMMARY.. xiv Chapter I. INTRODUCTION....,.,... 1 Definitin f the Prblem Related Wrk Present: Investigatin II. MATHEMATICAL ANALYSIS One Dimensinal Analysis Including Heat Transfer and Skin Frictin Bundary Layer Analysis Heat Transfer Analysis Nn-dimensinalisatin III. SOLUTION PROCEDURE Slutin f Inteirir Pints Slutin f Bundary Pints Determinatin f Wall Temperature Summary f the Prcedure IV. RESULTS AND DISCUSSION Standard Cnditins Parameter Variatin Cmparisn with Other Wrk V. CONCLUSIONS. <(.... '. 108 VI. RECOMMENDATIONS

6 IV TABLE OF CONTENTS (cntinued) Page APPENDIX A. Derivatin f Cnservatin Equatins Cmputatin f Burning Surface Expressins fr Enthalpies f Slids and Gases B. Derivatin f Bundary Layer Mmentum Equatin C. Flw Chart fr the Cmputer Prgram BIBLIOGRAPHY

7 LIST OF TABLES Table ^ Page 1. Pressure Vs. Burning Rate Data fr the Prpellant Cmparisn f Results fr Tw Limiting Cases f Slids Velcity » Mass and Energy Balance fr Case I and Case II Results fr Varius Input Parameters. 106 ii. i

8 VI LIST OF FIGURES Figure Page 1. Schematic f the Pistn-Cylinder Arrangement Numerical Scheme fr Interir Pints Scheme fr End Pints Numerical Scheme fr Determinatin f Tube Wall Temperature Heat Balance fr a Thin Circular Element at the Inside Surface f the Tube Assumed Initial Distributin f Slid Particles in Case II..,.... -" Pistn Path fr Case I Variatin f End Pressures, Pistn Velcity and Mass Fracti n f Slids with Time ( v Case I)., Variatin f Gas Densities and Temperatures at the Breech and Pistn Base End*with Time (Case I) ^ Spacewise Distributin f Velcity at Varius Times (Case I) Spacewise Distributin f Pressure at Varius Times (Case I) Spacewise Distributin f Gas Temperature at Varius Times (Case I) Spacewise Distributin f Gas Density at Varius Times (Case I) Spacewise Distributin f Vlume Fractin f Slids at Varius Times (Case I) Cmparisn f Pistn Paths in Case I and Case II... -j^ 16. Cmparisn f End Pressures, Pistn Velcities and Mass Fractins f Slids in Case I and Case II ,-,

9 Vll LIST OF FIGURES (cntinued) Figure Page 17. Spacewise Distributin f Gas Velcity at Varius Times (Case II) Spacewise Distributin f Pressure at Varius Times (Case II) Spacewise Distributin f Gas Temperature at Varius Times (Case II).... ; Spacewise Distributin f Gas Density at Varius Times (Case II) ' i ' 21. Spacewise Distributin f Vlume Fractin f Slids at Varius Times (Case II) Bundary Layer Grwth with Time (Case I) Spacewise Distributin f Heat Transfer Cefficient at Varius Times (Case I)....'" '» Spacewise Distributin f Wall Surface Temperature at Varius Times (Case I) Variatin f Wall Temperature at a Particular Psitin with Depth at Varius Times (Case I) Variatin f Heat Transfer Cefficients at Certain Fixed Lcatins with Time (Case I) Variatin f Wall Surface Temperatures at Certain Fixed Lcatins with Time (Case I) Variatin f Heat Fluxes at Certain Fixed Lcatins with Time (Case I) Cmparisn f Ttal Heat Lsses t the Tube Wall in Case I and Case II. B Variatin f Breech Pressure and Pistn Velcity with Time fr Varius Pistn Start Pressures (Case I) Cmparisn f Bundary Layer Thickness and Heat Transfer Cefficient fr Varius Prfile Shape Factrs at secnd (Case. I) Cmparisn f Ttal Heat Lss t the Tube Wall fr Varius Prfile Shape Factrs (Case I)

10 viii LIST OF FIGURES (cntinued) Figure Page 33. Cmparisn f Breech Pressure, Pistn Velcity and Ttal Heat Lss t the Tube Wall fr Varius Tube Diameters with Same Lading Density and Same Pistn Mass per Unit Area (Case I)., Cmparisn f Breech Pressure, Pistn Velcity and Ttal Heat Lss t the Tube Wall fr Varius Prpellant Charges (Case I) , Cmparisn f Breech Pressure, Pistn Velcity and Ttal Heat Lss t the Tube Wall fr Varius Pistn Masses (Case I) < Cmparisn f Breech Pressure, Pistn Velcity and Ttal Heat Lss t the Tube Wall fr Varius Web Thicknesses (Case I).., Schematic f Cntrl Vlume Chsen fr the Derivatin f Cnservatin Equatins..., A Typical Slid Particle Assumed in the Present Study Schematic f Bundary Layer Grwth in a Tube with a Sliding Pistn at One End..;' : :,f';,. 132

11 NOMENCLATURE English In. i ntatins A cefficient in wall stieat stress expressin (2.59) A p a crss-sectinal area f tube snic velcity in gas media a pistn acceleratin P B expnent f Reynlds number in (2.59) B functin defined by Equatin (2,,11) B functin defined by Equatin (2.,38) C 1 rati f bundary layer thickness and mmentum thickness, 6/6* C functin defined by Equatin (2,12) C functin defined by Equatin (2,39) C. lcal wall frictin cefficient c s specific heat f slids c specific heat f gas at cnstant pressure c specific' heat f gas at cnstant vlume D tube inside diameter D' functin defined by Equatin (2.78) D' functin defined by Equatin (2.79) Dj. functin defined by Equatin (3.21) E_ functin defined by Equatin (2.13) E^T functin defined by Equatin (2,40) E 1 functin defined by Equatin (2,77)

12 NOMENCLATURE (cntinued) English ntatins e internal energy per unit mass G f functin defined by Equatin (2.83) H prfile, shape factr, 6 /6 H? _ functin defined by Equatin (2.84) H f functin defined by Equatin (2.94) h h J j enthalpy per unit mass film heat transfer cefficient mean heat transfer cefficient at the inner surface, (h. n + h. n+j )/2 ij ij mechanical equivalent f heat axial psitin f a ndal pint in finite difference grid L initial pistn distance, fcm tube head end L. :... ' " = > " " ' ' ; p... f pistn distance frm tubel head end at any instant L M tube length mlecular weight f gas M pistn mass.. ; p m i initial charge f prpellant n P Pr q ff R time in finite difference- grid reciprcal f expnent in pwer-law velcity prfile pressure Prandtl number heat flux per unit area tube inside radius

13 NOMENCLATURE (cntinued) English ntatins R. tube utside radius R gas cnstant, R /M g u R u Re>. universal gas cnstant & Reynlds number based n mmentum thickness r radial distance frm tube axis r, linear speed f burning S, ttal burning surface b t St Stantn number, ( - -) * p.uc f. P s entrpy T ne-dimensinal gas temperature T T T^ explsin temperature tube wall temperature free stream gas temperature T.. film temperature, (T + T.)/2 r f '» w,i T. inner surface temperature f tube wall w,i t U U U u v p time ne-dimensinal velcity pistn velcity free stream velcity velcity within bundary layer specific vlume V initial chamber vlume v, vlume rate f decrease f slids a s

14 NOMENCLATURE (cntinued) English ntatins R T W ptential f prpellant per unit mass, / -. \ w x S i initial web thickness f a slid particle axial distance frm tube head end Greek ntatins a thermal diffusivity 8 cefficient f thermal expansin Y rati f specific heats, c /c P v AE additinal energy available per unit mass during cnversin f slids int gases Af change f functin f 6 bundary layer thickness <5 displacement thickness r) cvlume in Equatin f state (2.6) x] 9 K characteristic directins crrespnding t psitive and negative value f A respectively 0 functin defined by Equatin (2.61) 0 mmentum thickness K A y v p s p thermal cnductivity arbitrary multiplier t determine n, K characteristics viscsity vlume fractin f slids ne-dimensinal gas density density f prpellant material

15 NOMENCLATURE (cntinued) Greek ntatins p f gas density at film temperature T f p mixture density, [v p + (1-v )p ] m ss sg T w C wall shear stress characteristic directin alng a particle path Subscripts f g i value at film temperature T f value fr gas value at inner surface f tube wall 0 initial value p s w value at pistn base value fr slids value fr wall material. free stream value Superscripts 1 crrespnding nn-dimensinal frm Special ntatins ( ) first estimated value f a functin after time At 1 ( ) secnd estimated value f a functin after time At 1 ( ). value at nde n,j in time-space finite difference grid

16 xiv SUMMARY = The bjective f this thesis is t prvide a mathematical mdel that can be used t predict the perfrmance f devices, such as guns, which prduce high pressure in an enclsed, but expanding vlume by burning slid prpellant. The prpellaiit.is assumed t be in the frm f slid particles and is burned in a clsed cylindrical tube with a sliding pistn at ne end';' Due t the cmplexity in estimating the relative velcity between the gas phase and slid phase, tw limiting cases f slids velcity are examined in the present wrk. These are: (a) assume the slids have the same velcity as the gases arund the particle and (b) assume the slids have zer velcity, i.e. the slids remaining statinary at their initial psitins. Fr bth cases, the cnservatin f mass, mmentum and energy results in a set f fur cupled partial differential equatins expressing vlume fractin f slids, gas density, velcity and pressure as a functin f axial distance frm the tube head end and time. The equatin f state f Nble and Abel, with cnstant cvlume, is used fr the cmbustin gas. The heat transfer t the tube wall and pressure drp due t skin frictin have als been taken int cnsideratin. A bundary layer analysis is carried ut by deriving the bundary layer mmentum integral equatin fr a nn-steady, nn-unifrm, develping flw in a tube. The prfile shape factr (rati f displacement thickness and mmentum thickness) is intrduced and the Ludwieg-Tillmann frictin cefficient is used. As a first apprximatin, the shape factr is

17 XV assumed t be cnstant and, as the flw is in the high Reynlds number regin, the usual apprximatin f a thin bundary layer is made. The cnservatin equatins tgether with the bundary layer equatin are written In finite difference frm and the MacCrmack versin f the Lax-Wendrbff methd is used t calculate all the ballistic prperties, i.e. gas velcity, density, pressure, temperature, vlume fractin f slids and bundary layer thickness at each f the interir pints in the axial directin at every time step. Fr the tw end pints, namely the tube head end and the pistn end, the methd f characteristics is used. The film heat transfer cefficient is btained by using Clburn's analgy between heat and mmentum transfer. The wall temperature is als cmpletely determined by slving the unsteady heat cnductin equatin fr the tube wall with apprpriate bundary cnditins. The calculatin prcedure is repeated until the pistn reaches the end f the tube. Results are btained fr a set f ''standard cnditins," fr bth f the limiting cases f slids velcity. Althugh the final pistn velcity and time f travel are very clse in bth cases, the peak pressure in the case f statinary slid is apprximately 10 t 15 per cent higher than the crrespnding value in the case f mving slids. There is als a large pressure gradient alng the length f the tube and at the peak cnditin., the tube head end pressure can be 30 t 40 per cent higher than the pistn base pressure. The maximum bundary layer displacement thickness is less than three per cent f the tube radius in the typical case with the shape factr equal t ,, i.e. with the ne-seventh velcity prfile.

18 XVI Average values f the heat transfer cefficient and heat flux per unit surface area are fund t be 50 kcal/m'"-sec- K and 50,000 kcal/ra -sec respectively. The tube inner surface temperature can reach a peak value f C during the first peratin in an initially cld tube. The ttal heat lss t the tube wall is fund t be five t six per cent f the input energy and has insignificant effect n the ballistic perfrmance f the device. A study f parameter variatin shws that the initial chamber pressure, i.e. the "pistn start pressure,'" has little effect n the ballistic slutin. An increase in prpellant charge r pistn mass, r a reductin in initial web thickness f the slids can imprve the ballistic efficiency f the device; but there is always an adverse effect f higher peak pressure and higher wall temperature which put a limit n such attempts., Therefre, a great deal f judgment and care is needed t determine the ptimum cnditin fr a particular applicatin.

19 1 CHAPTERI- INTRODUCTION Definitin f the Prblem Devices which prduce high pressure in an enclsed but expanding vlume by burning cmbustible mixture f gases r slid prpellant with the bjective f perfrming wrk are cmmn in practice. Internal ballistics f these devices, fr example the prblem f the gun, have been slved experimentally since furteenth century when gunpwder first ' ' r -i* came int use _1 J. But surprisingly enugh, an analytical slutin which may be used t accurately predict the perfrmance f such devices is yet t cme. This lack f a mathematical mdel cmpels a designer t chse the cmparatively expensive path f experimentatin, althugh nly limited infrmatin can be btained frm these experiments. Mrever, a large number f experiments have t be perfrmed befre a set f ptimum design parameters can be determined fr a particular purpse, and still the final result remains in questin as t whether a truly ptimum cnditin has been achieved. The prblem f internal ballistics requires a mdeling f the fluid flw phenmena and heat transfer t the wall inside the expanding vlume. Fr simplicity., thrughut this wrk we shall restrict urselves t the special gemetry f a clsed cylindrical tube with a sliding pistn Number in [ ] refers t the references in Bibligraphy.

20 2 at ne end as shwn in Figure 1. The cmbustible mixture is burnt inside the enclsed vlume whereby the pressure is increased and the pistn is set int mtin. The prducts f cmbustin which flw dwn the cylinder behind the pistn impart a cnsiderable amunt f its energy t the pistn and a fractin is lst, t the tube wall. This cls the cmbustin gases and mdifies the pressure and flw cnditins. While heat transfer has sme effect n the ballistic prperties, this is prbably mre imprtant with respect t the material prperties f the tube. Since the cmbustin gases are usually at a temperature f K, after repeated use f the device at high frequency the wall temperature f the tube may; reach a value high enugh t cause appreciable wear as the pistn slides dwn the tube. A mdel f heat transfer, which can be used t predict: the wall temperature, will help a designer t chse the ptimum design parameters which will minimize the ersin rate. The purpse f the present research is, therefre, t prvide a wrking analytical mdel which shall be able t predict all the ballistic prperties, namely velcity, pressure, temperature and density f the cmbustin gas mixture as a functin f space and time. The heat lss t the tube wall shall be cnsidered and the temperature distributin at the wall shall be determined. This mdel will then allw study and ptimizatin f varius parameters withut expensive trial and errr experimentatin.

21 Tube Head End (Breech) Pistn Base End A ^q b^ kx V NT N V V V V V k-a-v---v-v V V V V V V V V V vvvvvvvvvvvv K X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X XW W \ l x x x x x x x x x x x x x x x ^ x x xx x x x x x x x x x x x x x x x x i -L (t) Figure 1. Schematic f the Pistn-Cylinder Arrangement.

22 4 Related Wrk Theretical slutins t the prblem f interir ballistics have been attempted since the days f Lagrange wh in 1793 first tried t determine the spatial distributin f pressure, density and gas velcity in the tube at all times after the cmbustin. The wrk available until nw can be divided int tw brad categries: 1) Semiempirical slutins which may have practical utility in the study f familiar devices. 2) Exact theries which attempt t include the predminate phenmena up t a certain rder f magnitude by frmulating a simple mathematical mdel f the flw. Semiempirical Slutins The majr wrks in this area with special applicatin t the guns using slid prpellant are described in references [l] and [2j. The main purpse f these wrks is t btain a slutin which matches with the experimental values f peak chamber pressure and muzzle velcity f the prjectile. Only a few f the number f slutins shall be discussed here. Isthermal Slutin. The slutin as described by Crner \_2~\ is based n the fllwing assumptins; 1) The prpellant stays in the chamber burning under the tube head end (breech) pressure and the rate f burning is prprtinal t that pressure. 2) During the perid f burning f the prpellant, the prgressive cling f the cmbustin gases due t the wrk dne n the prjectile can be apprximated by taking a mean gas temperature ver this time

23 5 interval, crrespnding t an effective mean frce cnstant A. 3) Unifrm gas density and linear velcity distributin in the space between the tube head end and pistn base. h) Resistance t mtin f the prjectile can be taken int accunt by intrducing an increased effective prjectile mass instead f actual mass. 5) The cvlume r\ (vlume crrecting term in the equatin f state f the cmbustin gas) is equal t the specific vlume f the prpellant material. The expressins fr breech pressure P, prjectile velcity V, and prjectile distance frm breech face x,, are given as a functin f "cnvenient variable" f, the frm;;factr 6, the frce cnstant A, burning rate 3, and central ballistic parameter M. The central ballistic parameter M itself is a functin f A, 3, initial mass and web size f prpellant, effective prjectile mass, and tube diameter. The frm factr 6 depends n the gemetrical shaipe f the prpellant and the variable "f" ges frm ne t zer as the prpellant is burnt. Other parameters, namely A, M and 3 are chsen fllwing a trial and errr prcedure until gd agreement is btained with the experimental values f peak pressure and muzzle velcity. The slutin, hwever, des nt take int accunt the heat lss t the tube wall. Cppck's Slutin [2]. This is an extensin t the isthermal slutin described abve with the fllwing mdificatins: 1) Instead f taking a mean gas temperature during burning, the analysis takes int accunt the kinetic: energy f the prjectile and that f the gases, assuming that the cmbustin gases are unifrm in density

24 6 between the breech and the prjectile and that their velcity at any pint is prprtinal t the distance frm the breech face. The ttal heat lss t the tube wall up t a particular instance f time is assumed t be a certain fractin f the ttal kinetic energy f the prjectile and the gases at that instant. In practice, the effect f heat lss is incrprated in the energy equatin by a prper chice f Y (rati f specific heats at cnstant pressure and cnstant vlume). 2) The gases have a cnstant cvlume ri, nt necessarily equal t the specific vlume f the prpellant: material. Frm the bserved peak pressure it is pssible t back-calculate the central ballistic parameter M, and thence the burning rate (E. The slutin is superir t the isthermal slutin because there is nly ne arbitrary parameter, namely the burning rate 3, whse value is selected s that the peak pressure matches the experimental data. Mrever, the mdel takes int accunt the heat lss t the tube wall, thugh in a crude fashin. Gldie's Slutin [2]. The slutin fllws Cppck*s slutin described abve with the nly mdificatin that the prjectile is assumed t be mtinless until a "sht-start pressure" is prduced inside the chamber. If there is any resistance t mtin at later times, the effect is simulated by a change in effective sht weight. Apart frm these slutins, there are slutins which attempt t use a better relatinship between the burning rate and the crrespnding pressure. But the slutins still need trial and errr f ne r mre variables t match experimental data. Besides, there is n guarantee as t hw gd the slutins will be when predictin f perfrmance f

25 7 a new device is desired. Als t n infrmatin regarding the ballistic prperties in between the breech face and the prjectile is available frm any f these mdels. Even a recent publicatin [3] fails t prvide such infrmatins. Exact Theries As mentined earlier, Lagrange tk the initiative t slve the ne-dimensinal prblem f interir ballistics in He intrduced the "Lagrange apprximatin" which assumes that the gas velcity at any instant increases linearly with distance alng the tube, frm zer at the tube head end t the full prjectile velcity at the back f the pistn. It is further assumed that all the prpellant charge is in gaseus frm frm the start and at any time the gas density is the same at all pints. It can be shwn frm the equatin f cntinuity that if gas density is independent f psitin, the velcity distributin is linear; but the cnverse is nt necessarily true. In ther wrk, Hugnit in 1889 used the thery f waves f finite amplitude develped by Riemann in 1858, with the assumptin that all the prpellant was cmpletely burnt when''the pistn began t mve. He fllwed the resulting wave f rarefactin n its jurney t the tube head end. The methd was extended by Gsst and Liuville t fllw the wave as it travels back t the pistn after being reflected frm the tube head end. Finally, Lve [4 J carried the analysis as far as the third wave traveling tward the breech and Pidduck [4] applied Lve's slutin in the special case f internal ballistics. But all these slutins, thugh cmpletely analytical, hld gd under tw imprtant assumptins?

26 8 a) Instantaneus cmbustin, b) Adiabatic expansin f each element f gas. The assumptins may be applicable fr the devices which.use gaseus fuel as prpellant, say autmbile engines,, but fr the devices using slid prpellant the assumptins are far frm the real situatin. In this case, gradual burning f the prpellant must be cnsidered. Analytical wrk based n mst realistic assumptins has been dne by Carriere [5]. Fr simplicity he assumed the prpellant t be statinary in the cmbustin chamber at the time f burning which is a gd assumptin fr cast prpellant in a rcket-mtr. Frm the basic cncept f cnservatin f mass, mmentum and energy, he derived three partial differential equatins expressing gas density, gas velcity and entrpy as a functin f time and distance. He transfrmed thse equatins int three rdinary differential equatins alng three characteristic directins in the time-space c-rdinate. Then with prper chice f the equatin f state fr the cmbustin gas, he fllwed what is cmmnly knwn as the "methd f characteristics" t determine the gas prperties at any time and psitin. The effect f frictinal lsses and heat lss t the tube wall were disregarded in the analysis. The prblem f heat lss t the tube wall has been studied by Hicks and Thrnhill in England. A fairly elabrate descriptin f their methd has been given in bth references [l] and [2], This wrk is als based n the Lagrange apprximatin f linear velcity distributin and unifrm gas density in between the breech face and the pistn. It can be shwn that at high velcity, heat is mainly:transferred t the tube wall by cnvectin. It is als evident that a bundary layer

27 9 is frmed at the inner surface f the tube., The heat transfer rate per unit area thrugh the bundary layer can be given as h(t -T ), where h, 8 s T and T are the film heat transfer cefficient, temperature f the gas, g s and temperature f the inner surface f the tube respectively. three quantities depend n time as well as psitin alng the tube. All Hicks and Thrnhill cnsidered the flw in the bundary layer t be the same as the flw ver a flat plate. In internal ballistic applicatins the flw is in the turbulent regin mst f the time. Therefre, they used the analgy slutin, as extended by Vn Karman t cver Prandtl number ther than unity, t btain a relatin between the heat transfer cefficient h and wall shear stress T. T get the wall w shear stress they first fund a "best" pwer law fr the velcity prfile (nn-dimensinalized with respect t the shear velcity /T /p) inside the bundary layer which was capable f giving the lcal wall shear stress T within three per cent f the value that culd be btained by using mre rigrus lgarithmic frm f the velcity prfile when applied t steady and unifrm flw situatins. Then they used the bundary layer mmentum integral, including the terms due t nn-steady and nn-unifrm iiature f the flw, and used the "best" pwer law fund earlier t btain the lcal wall shear stress at all pints. The heat transfer cefficient h is then easily calculated frm the analgy slutin. They, hwever, mitted ne bundary cnditin that the bundary layer thickness at the base f the pistn be zer at all times,, The heat transfer in the tube wall has been calculated by using the differential equatin fr unsteady heat cnductin with prper bundary"cnditins. Fr the case studied by Hicks and Thrnhill, i.e.

28 10 the first rund f firing frm a cld gun, the curvature effect f the wall was neglected as the temperature rise was cnfined within ne millimeter f the inside surface. Cnsequently, there was n heat lss frm the uter surface f the tube which remained at ambient temperature. The heat cnductin alng the length f the barrel was als neglected. Knwing the tube material prperties, namely thermal cnductivity and diffusivity, it was pssible t btain the temperature distributin at the inner surface f the tube alng the length at all times. The free stream values f the gas velcity, density and temperature were taken frm the ne-dimensinal ballistic slutin., It has been indicated in reference [,2 ] that frictinal pressure drp is small cmpared t the inertia pressure drp needed t accelerate the gas. But n analysis until nw indicate quantitatively the effect f skin frictin n the ballistic prperties. Even the heat transfer slutin has nt been fed back t study its effect ri the ne-dimensinal slutin. Present Investigatin In the light f available theries, it is clear that a gd nedimensinal slutin is first required t replace the Lagrange apprximatin, r at least check its validity fr the particular prblem. The first and mst frmidable difficulty in writing dwn the ne-dimensinal cntinuity, mmentum and energy equatins during the burning f the slid prpellant is due t the uncertainty f the relative velcity between the gas phase and the slid phase. It is extremely difficult t estimate the drag exerted n the burning slid particles by the accelerating

29 k 11 cmbustin gases. Therefre, tw limiting cases f the slids velcity have been cnsidered in the present wrk: Case I. The slid particles mve at the same velcity as the gas phase. Case II. The slid particles remain at their initial psitins thrughut the perid f burning. Fr bth cases the cnservatin f mass, mmentum and energy results in fur cupled partial differential equatins expressing vlume fractin f slid v, gas density p, gas velcity U, and pressure P as s g a functin f axial distance x and time t. The heat release due t gradual burning f the prpellant is taken int accunt. A special prpellant gemetry, namely a hllw cylinder, is cnsidered whereby the ttal burning surface remains cnstant, althugh this assumptin is nt essential. The ballistic prperties at the internal pints are calculated frm these equatins after writing the same in finite difference frm. But t btain the prperties at the tw ends, namely the tube head end and the pistn base, the equatins are transfrmed int rdinary differential equatins alng the characteristic directins. The cvlume f the gas is assumed t be cnstant, and experimental data fr burning rate is used. As ne f the initial cnditins, it is assumed that the pistn des nt start until a certain specified pressure is reached inside the chamber and thereafter the pistn des nt experience any resistance t mtin. The bundary layer mmentum integral fr a nn-steady, nn-unifrm, develping flw inside a tube is derived. The prfile shape factr H

30 12 (rati between the displacement thickness <5 and mmentum thickness 6) is intrduced and the Ludwieg-Tillmann [6] frictin factr is used. As a first apprximatin, the shape factr is assumed t be cnstant in the present wrk. The flw is in the high Reynlds number regin fr which the bundary layer thickness is small cmpared t the tube radius. It is therefre legitimate t replace the free stream values f gas density and velcity by the values btained frm the ne-dimensinal slutin neglecting the bundary layer thickness. The lcal heat transfer cefficient h is calculated by using Clburn's analgy [7 ] between heat and mmentum transfer. It cvers Prandtl numbers ther than unity and is simple t use. The values f viscsity and gas density at the film temperature are used. The heat transfer In the tube wall is cmputed frm the unsteady ne-dimensinal (radial) heat cnductin equatin with apprpriate bundary cnditins. The wall temperature is als fund as a functin f axial distance and time. The heat lss term is entered int the ne-dimensinal energy equatin and a cmparisn f ballistic prperties is made with the slutin withut heat lss. Effect f wall shear stress is als included. The ballistic efficiency f the pistn-cylinder arrangement is cmpared by varying different design parameters,.

31 13 CHAPTER II MATHEMATICAL ANALYSIS The mathematical analysis cnsists f tw majr parts: 1) One-dimensinal analysis with gradual burning f the slid prpellant, including the effect f heat transfer and skin frictin. 2) Frmulatin f the bundary layer prblem and determinatin f heat transfer t the tube wall. As utlined in the previus chapter, the present analysis is carried ut fr tw extreme cases f slid velcity. In the first case, it is assumed that a burning slid particle mves with the same velcity as the cmbustin gases, In the secnd case, hwever, the slid particles are assumed t be statinary at their initial psitins thrughut the perid f burning. Hencefrth these tw cases are referred as Case I and Case II, respectively. One-Dimensinal Analysis Including Heat Transfer and Skin Frictin Case I The assumptins, ther than that regarding the slids velcity, which are made t simplify the mdel are as fllws: 1) At any instance f time, the linear speed f burning r ^s b same fr all the slid particles and it is a functin f the

32 14 average pressure in the chamber (space in between the tube head end and the pistn base). 2) The slid prpellants are single perfrated circular cylinders in shape whereby the ttal burning surface remains cnstant during the whle perid f burning. 3) The burning rate is fast enugh t cnsider that the temperature f the remaining slids at any instance f time remains cnstant at the initial temperature. 4) The prpellant: material is incmpressible and its cefficient f thermal expansin is negligible. 5) The pistn starts t mve nly when the chamber pressure reaches a certain value P, and thereafter the resistance t its mtin is negligible cmpared t the pressure frce exerted n it by the cmbustin gases in the chamber. The cnservatin equatins are as fllws (fr derivatin see Appendix A) : Slid cntinuity: 3v dx> T-S- + Ur-5- + V --'+ V, = 0 (2.1) dt 9x s 3x : d s Gas cntinuity: 8p g 3p g p * 3U (p s" p g ) 3t 3x ^ (1-v ) 9x (1-v ) V d s S S U ' Z) Mmentum:

33 15 M + u.is._^3 _iv (23) 9t 3x p 3x p R m m U " J; Energy: Dh Dh ntj v P s,,, v g DP s r s Dt + (1-v s ) p g --r* Dt Dt 2h. 2T U - p s (w h gfti - -r<^,i) + -r- s s (2-4) where v, is the vlume rate f decrease f slids per unit cylinder s vlume and is given by: v, (,,t)- ^ r 4 ' i "- (2.5) s p / P v (x,t)dx s ' ' The equatin f state f the gas isj P(v - n) = R T g r, P( - - n) = R T (2.6) P g where the gas cnstant R is btained frm the rati f the universal O gas cnstant R and the mlecular weight: f the gas M. u It has been shwn: in Appendix A that under assumptins three and fur as stated earlier, the differential f enthalpy f slids per unit

34 16 mass h and the differential f enthalpy f gases per unit mass h can be given by: dh = dp (2.7) S p s (Y-np ) P dh =, «dp - 7 f»; 7 dp (2.8) g p (Y-D (Y-1)P Z g Substituting equatins (2.7) and (2 8) int the energy equatin (2.4) (l-v s )(l-np g ) Dp (l-v s )VPDp Cy-1) Dt TY-1>P Dt = p (W + - h )v. s p s g d s 2h. 2T U --r^vi^tt- ( 2 -«) Using gas cntinuity, i.e. equatin (2.2) t replace := =- in equatin (2.9) the final frm f the energy equatin becmes: (l-v ) (l-np ) DP _ ^ m TP(P.-P ). (Y-D Dt (v-1) dx (Y-1)P d e g s 2h i P = p (W h )v, - ~^(T-T,) s p gd R w,i s s 2T U

35 17 r i +u 3 +B M = c v - E S (T _ T > 3t ^x I 3x I d I R U w,i ; s - ' 2T. U + J T -~- (2.10) where B l - O^Td-nP -nc J. g (2,11) YP(P O -P CT ) + (Y-l)p <W + f- - h ) s g s g p g C «..._s (2.12) i P { l-v )(i-npj} g s g. E T = -il=ii (2.13) I (1-VJ(1-T1P 0 )!:» g The initial cnditins f the cnservatin equatins are: Psitin f the pistn, L (0) L P U(x,0) = 0 ; P(x,0) = P ; T(x,0) - T ; p (*,;0)! = p a S OQ (2.14) and v (x,0) = v at 0 < x < L s s where P is the pressure at which the pistn starts t mve, and T is -. the explsin temperature f the prpellant. By knwing P and T it is pssible t determine p frm the equatin f state (2.6): 8

36 18 g'-irr 1 < 2-15 > 0 -f-a + n Neglecting the initial mass f air in the chamber, a mass balance gives m = v V p + (1-v )V p s. ss s e :L O O & r, (m c /V ) - Q s. g n v - (2.16) s p -p s g (m /V ) is called the lading density. s. i The bundary cnditins are: at x = 0, U(0,t) =0 and (2.17) at x = L, U(L 9t) = U (t) P P P The pistn velcit}^ U (t) is btained frm the equatin f mtin f the pistn, which under the assumptin five takes the fllwing frm: du P A... E = a = EL.P- (2 18} dt a p M U - ltt; P The psitin f the pistn is btained frm::

37 19 A = a (2.19) dt 2? The unknwns, and the crrespnding equatins frm which they can be calculated are listed belw: Unknwn Equatin Vlume fractin f slids, v Slid cntinuity, (2.1) Gas density, p Gas cntinuity, (2.2) Velcity, U Mmentum equatin, (2.3) Pressure, P Energy equatin, (2.10) Gas temperature, T Equatin f state, (2.6) The cnservatin equatins, i.e (2.1), (2.2)., (2.3) and (2.10) are written in finite difference frm, and a numerical scheme which takes int accunt bth frward and backward space derivatives are used t calculate the crrespnding unknwns, i.e. v, p, U and P, at all the s 8 interir pints at an advanced time by knwing the present values at and arund thse pints. The gas temperature, T, is then calculated frm the equatin f state (2.6). The details f the slutin technique shall be discussed in Chapter III. The abve slutin technique, hwever, is nt applicable t the bundary pints, i.e. the pistn base end and the tube head end, as space derivatives n bth sides f these tw pints are nt available. This necessitates the transfrmatin f the cnservatin equatins t rdinary differential equatins alng characteristic directins, i.e. t fllw the "methd f characteristics" [8].

38 20 P-U Characteristic. The energy equatin (2.10) is: 8P. TT8P. 8U 8t Sx I 3x I d I R w.i 2hj 2T U + E ^» I R Multiplying the mmentum equatin (2.3) by an arbitrary cnstant X: J S*\St^- l T < 2 ' 20 > Adding equatin (2.20) t equatin (2.10): P 3u +(U+A) T + J. fi Xp + ( B T + X pu) S t 3x J I m 3t I ^m 8x 2h. = c i \ - E i TT (T - T w,i } 2T [v - x ] < 2-21 > T btain the characteristic directins, the value f X shall be such that: Ap m dt 1 dx U+A B T +Xp U I m A l,2 " * "V'V (2-22 >

39 Dividing equatin (2.21) by /l+(u+a) z and using 1 9,. (U+A) JL = /I+OJ^ at 3x Vfe^u+xF dn^ where n crrespnds t Xj i.e. +ve sign f X and crrespnds t A2 i.e. -ve sign f A the equatin (2.21) becmes % dp., ; du + X_ 0p dr,,5 1,2 m dn.c /] L+(U+A )2 Vd S 2h. 2T E I -TT^w.i'J + -T^8! 0 -"] Nw, An, = /(Ax) z +' ^At) z == At. /l+(u+x) z (2 J *. -1 Therefre, alng ^-characteristic, i.e. - - U + /B /p 1 m _ r 2h. AP + p /Op " AU = C T v. - E. r - (T-T.) p m I'm Lid - I R w,i + -g* (E r U - /y^)] At (2 and, alng -, ^-characteristic,,., '.. i.e.. dt - 1 dx U " ^lk

40 22 AP _ -V^mAU= r 2h - LVd " E i TT (T - T w,i } 2T ^ + -^ (EjU + ^V^)J At (2.26) P- g Characteristic. By rearranging equatin (2.9), 3p 3p ~. "T U"T 3t 3x yp 3t lit f. -. (h - - W) + (Y"DP s P g g P s (l-v s )yp (Y-l)p T2h. 2T U + _ - L i( T -T ) + (1-v ) Y P L R w.i R (2.27) Dividing this equatin by /l+u 2 and using, L 9 + _ U.3. d /i+u 7 dt /i+u Y 3x d and A? == /(Ax) z +(At) r == At /l+u 2 alng a particle path, i.e. dt dx I. U X P (l-np ) Ap = -a -_&- AP + P g YP / I N ' (h - S ^-^P-sPs (1-v )yp - W) ^ v. At a s + (Y-DP g (l-v s ) Y P f2h. 2T IT (T - T w,i> " TT 0 I At (2.2.8)

41 23 v s -P g Characteristic. Frm equatin (2 2): _3U ax (l-o fd P :3P C ;I (P C ~PJ g 3t dx J p g d s (2.29) - 3U. Substituting this expressin fr -r in slid cntinuity (2.1): 3v 3v v (1-v ) V-1 _ * s dt dx p g 3p 9p -l_ u & I at 9x v p +(l-v )p i s s s g g Vj = 0 (2.30) a s Prceeding in the same fashin as fr the P~ g characteristic, ne btains: Alng a particle path, i.e. ḏt 1 dx U * v (1-v ) p A S S. Ill Av = Ap - S P g 8 p g v. At a (2.31) The prcedure f slving the abve characteristic equatins are discussed in Chapter III. Case II In this case the slid prpellant particles are assumed t be statinary at their initial psitins thrughut the perid f burning. The linear speed f burning r,, is same fr all the slid particles and is a functin f the average pressure in the space between the tube head end and the initial psitin f the pistn L. The rest f the assumptins are the same as thse fr Case I. derivatin): The cnservatin equatins in this case are (See Appendix A fr

42 24 Slid cntinuity: 9v JT~ + *d z " s (2 * 32) Gas cntinuity: 3t ""^T P g 9x (1-v,) d (1-v ) 9x s s s (2.33) Mmentum: M + T I ^ - 1 9P I P s U. 3t ^x " p 3x (1-v )p~ V d " (1-v )p R U.W g : s /K g s s /K g 2T w Energy: P DP P U (1-v )p --& - (1-v )^r - p (W h )v\ s g Dt s Dt s p 2 g d s s 2h. 2T U -(T-T..) + R v ' w,i' R (2.35) As nne f the slid particles mves beynd L,' v s can be expressed as b t s r b \ -^Tl - < 2-36 > s P / v dx s 'O

43 25 The same equatin f state, i.e; equatin (.2.6), is used and by successive use f equatin (2.8) and (2.33) the final frm f the energy equatin (2.35) becmes: 8P, T_3P 9U 1 _ \, B II U 3v s 3t U 3x + B II 8x ~ C II v d + (1-v ) 3x, s s 2h. 2T U " *n-5t i, + '\i? E n -T- (2-37) where II yp (1-npJ (2.38) 2 P U YP(p -P ) + (Y-1)P P (W + P~ + 2~ " V C n P {(l-v )(l-np )> U,Jy; E = E = (Y-j-) -- (2.40) II I (l-v g )(l-^p ) The initial and bundary cnditins axe the same as thse fr Case I. The characteristic equatins are als required t calculate the ballistic prperties at the tw ends. P-U Characteristic. The prcedure is exactly same as Case I. Multiplying the mmentum equatin (2.34) by an arbitrary cnstant A, and adding t the energy equatin (2 37) ne btains:

44 26 ft + < u+x >f \ f + (B II+ Ap U)f 1 _ g dt II g 9xJ B TT U 3v p U p, Ii s, r s. c n v d The characteristic directins are such that: + "(i-v v "^r A a-v) v d s s) s' s 2h 2T " E xx -~(T~T.)+ -= II R v E_ U - w,i' R L II (1-v ) 2.41) < s Xp. dt 1 dx U+A B T _+Xp U 11 K g + /*E7 = + [_ YP X2 = + "W P g ""J^^V (2.42) Frm slid cntinuity, i.e. equatin (2 32) 3v s_ 3t ~ V d (2.43) By adding and substracting equatin (2.41): B U 3v II (1-v )(U+A) 3t n the right hand side f "f + < u+x >f 'i + xp g [f + <" +x) 3U' 3x' B H U (1-v )(U+A) r 3v 3v - r av L sr + < u+x)^rj - BU 3v II s (1-v )(U+X) 3t P U 2h. 2T..' _ + C^v., - XTI,v, - E TT -^i(t-t.) + S E u-7-i II d (1-v ) d II R w.i R IT (1-v )l S (2.44)

45 27 Dividing equatin (2.44) by /l+(u+a) z and using 1.3' (U+A) d /I+TU^AF 81 /i+tt^f Bx dt1 c yields the relatin: B dp,, du TT U dv II s dti,c + A l,2 K p. g dn,c (l-v ) (U+A) dri,c s /T+cu+xT 2 " p UX B_ U {c n" Ti=";r) + (i-v )(U+A) } ^d s s s 2h 2T (T-T.) + ~ v? {E TT U II R v w,i / (2.45) R II (1 ^y>] Therefre, alng n-character is tie, i.e : dx U+a B U AP + p g aau - =v~nu+at A V p U B U r. s, II i. + L c n - a ^rr * ^affj v d At 2h - E II R (T^i)At4.^[ EiiU._^y]At (2. 46),,.... dt 1 and alng ^-characteristic, i.e. -j - yrj- :

46 28 AP B T U " p g a AU " (1-v )(U-a) Av s p s U B II U + C am r + II (1-v ) (1-v )(U-a) s s v J At d s 2h E ii^t-\,i )At+ -/L E ii u+ TI^)j 2T At (2.47) P- g Characteristic. Using equatin (2.8) in equatin (2.35), an alter*- native frm f energy equatin is s 3p_ 3p p (1-np ) r 4. TT L B &- 1 + u2 " 3t 3x y? 3t 3x.2 (Y-l)p P (h - W - - ~~) s g.g P e 2 ^ ' s. (1-v s )yp " V d s (Y-1)P f 2h. 2T U + JL_ -(T-T ) - - W (1-v )yp R U w,i ; R J s (2.48) Prceeding in exactly the same manner as fr Case I, alng a particle dt 1 path, i.e. - - :.., (Y-l)P P (h - W - - ) p (1-np ) s M g g p 2 Ap = -S ^ 6- AP + -7= -r-5-" - v. At g YP (l-v s ) Y P d g (Y-DP e 2h. 2T U'l i (' T _ T ) J L At (2.49) (i-v )Y? I R w,r R

47 29 v s v s at the pistn base is always zer after the pistn starts mving and at the tube head end can be btained frm equatin (2.32) alne. Bundary Layer Analysis The bundary layer part f the entire analysis did nt receive much attentin in the past because f the nnsteady and nnunifrm nature f the free stream flw. The flw is generally in the turbulent regin with pressure gradient in the directin f the flw and a large temperature difference acrss the bundary layer. Als, in Case I a gas-slid mixture flws dwn the tube; hence the analysis is mre cmplicated. A number f attempts [9, 10, 11, 12 ] have been made in the past t mdel the mechanism f heat transfer in a gas-slid mixture with varius slid particle sizes and lading ratis (w /w ). It has s g been fund that the effect f the slids ri heat transfer is prminent fr micrn-size particles whereas fr millimeter size the effect is nt appreciable. The present prblem deals with the slid prpellant f millimeter size and mst f the time it burns ut cmpletely lng befre the pistn reaches the end f the tube. It has als been fund frm the study f Hicks and Thrnhill [2 J that the bundary layer thickness is small cmpared t the tube radius. Therefre, t simplify the mdel, it is assumed that the slids always stay in the cre f the flw and never enter int the thin bundary layer at the wall. In the present study, an integral apprach is preferred t a differential apprach t keep the mdel relatively simple and traceable. The bundary layer mmentum integral fr the nnsteady and nnunifrm cmpressible flw inside a tube as derived in Appendix B is:

48 30 3_ 3t ; : p, (U_-u)r dr R-6 + 3x R TpuCU^-u^ir dr R-6 + J p(u -u)r dr R-6 3.x 2 \ R6-f 3U 3U 3P -r + p_ " + p U -r 3x f 3t f 3x + T R w (2.50) Defining and 6 = Displacement thickness G = Mmentum thickness such that, f.r 2iTr dr = j p2irr(u -u)dr R-6 R-6 r, PfU^RS (1 R pr(u -u)dr R-6 (2.51) and p f U /" 2ur dr = { p 2ITI: u(u m -u)dr R-G R-6 r, r R PfufR9(l - ^) = J p P u r(u -u)dr R-6 (2.52) and using the definitin f the prfile shape factr H =, and fr r rs fl a thin bundary layer «1, -z? «1, ^:~ «1, the mmentum integral equatin (2.50) becmes,

49 31!TFP* U RHe] + ~r p.u 2 Re 3t _ f J 3x J f a> 8U CX' + (PfU^RHG) ad 3U 3U 1,,3P cci R6 h~- + p - + pjj - '3x w f 9t H f «8x + T R w r, 3fi 3H ^U d Pf 9 Sfl p.u^rh-^ + p U R&~ + p,rh6- + U RHfcr-^ + P-lTl 2. I dt f dt f 3t dt f 3x 31^ ' 3p 3U + 2p _U RG- + U RGr + p. -U RH9r-^ f dx dx ' f 3 X = R6 'ar> du _ J 8x p f 3t 3U Pf u -ar T R w (2.53) Dividing equatin (2.53) by p U^RHB : du U i a T au~ i i d P* du U d P* ±. JL. 4. J: Hi J. :L_ ' 1 f : c>_6 2 f 9 3t H 3t U 3t 'p 3t H0-'3x~ H 8x p^h 3x 3x «f f du 'a- 3U 1 p u He 3P, +.PrU w (2.54) f f 3x p U H9 f du Frm the study f steady cmpressible turbulent bundary layers by Reshtk and Tucker [13],, it is likely that fr mderate Mach number flw encuntered in this prblem (M < 1.5), the percentage change in 1 8H the shape factr, i.e. rr > is small cmpared t the percentage change n. d t 1 9 fi in mmentum thickness, 777. As a first apprximatin, therefre, the a d t shape factr, H, is assumed t be a cnstant:,. A mre rigrus apprach wuld be t derive anther auxiliary equatin, say mment f mmentum

50 32 integral 14 t btain an expressin fr ~. Hwever, derivatin f L J clt such an equatin fr the nnsteady case is extremely cmplicated and therefre neglected in the present wrk. temperature, Fr thin bundary layers, U - U ; T - T and the film T 4 T. T f = f*± (2,55) The gas density at the film temperature, p f, can be evaluated frm the equatin f state (2.6),, and the final frm is: p f l +n Pg ( - -ft] (2.56) Equatin (2.54) finally becmes: 86 _ U 80 T w at H ax p.uh i_ ^1 +. ] L f + i l + (H+2) _8U p 3t ' p" f H 3x"~ U 3t H 3x + dh[i + p f f+p f ] f The initial cnditin is: 6(x,0) = 0. The bundary cnditin at the pistn end is 9(L,t) = 0, which is P bvius frm the fact that all the particles at the pistn base are at the full pistn velcity all the time. The cnditin at the tube head end shall be established later. It is assumed that the entire flw is in the turbulent regin and the wall shear stress can be btained frm the Ludwieg-Tillmann

51 33 frictin factr [6], which was develped frm a series f experiments with all types f pressure gradients. The riginal expressin which hlds gd fr incmpressible flw with small temperature differences acrss the bundary layer is: r T w n 0/> Vn" - 678H /^~V.268,. f = H U z ' = * 246 x 10 '"""T ) (2.58) ^00 OC) 00 In the present wrk, the expressin is slightly mdified by using the fluid prperties (p, y) at the film temperature, T f, instead f the free stream temperature, T^, t take int accunt the effect f prperty variatin acrss the bundary layer. The expressins fr lcal frictin cefficient. C,., and lcal shear stress at the tube wall, T, used in f. w the present wrk are: C c = 2A f 1 <*-/? 1 T = A p U» -' <«.,)» (2.59) where A = ^ M B > P f U8 B = Re 9= Using the abve expressin fr wall shear stress in equatin (2.57), and p multiplying equatin (2.57) by (1+B)9 ;

52 34 30 at -ii+^fv (1+B)0 3t U 3p f 1 3U (H+2) _3U p H 3x U 3t H 3x r + (1+B) C 1 Q i <w_ i w, m,p f U 3x U 3t 3x (2.60) where 0=9 (1+B) r f -'-e---:ewand, (2.61) Cl = 5* At the tube head end, U! = 0 and the equatin (2.60) becmes 30 il = - (1+B) 0 1_ ^1 + JL. ^1 + I i + (H+ 2) _au p f 3t p f H 3x" U 3t H 3x + (1+B) 0 C. 1 _3J? 1 3U 3U LP U 3x U 3t ',3x" (2.62) and at t =0, 0 = 0. This implies that at the tube head end, 9, i.e. mmentum thickness r bundary layer thickness is zer at all times. The equatin (2.60) is applicable t bth Case I and II fr cmputing mmentum thickness 9, and thence the frictin cefficient,

53 35 C f, at each statin in the axial directin at each time step. Heat Transfer Analysis As the flw is in the turbulent regin, the analgy between the mmentum transfer and the heat transfer prvides the easiest way t determine the heat transfer cefficient, h., at the tube wall. Because f its simplicity, Clburn's analgy [?] has been used fr Prandtl numbers ther than unity as fllws: C f St Pr 2/3 = / r, C h, = p f Uc (~)/Pr 2/3 (2.63) y -c where Pr = ( B?).. K : - g The heat transfer in the tube wall is cnsidered as a nedimensinal (radial) unsteady heat cnductin prblem in a hllw cylinder. Lngitudinal heat cnductin is neglected because the temperature gradient in the radial directin is expected t be steeper by several rder f magnitude than in the axial directin. The differential equatin can be written as [l5j: 9T w _ 3t: w 8 2 T w, 9T 1 w.2 r 3r dr (2.64) The bundary cnditins are

54 36 at r = R, q. = h.(t - T.) == - K -5-=- * n i i v 00 w,i w 3r r=r ST at r = R, q «h (T - T,) = - K -r-^ (2.65)' w, amb w 3r r=r at t = 0, T(r) «T, amb It is pssible t slve equatin (2.64) numerically and btain the temperature at the inner surface f the tube T. at each statin alng r w,i the length f the tube at each time step. The lcal heat transfer rate t the wall per unit surface area is given by h.(t-t.), and integrating ver the entire surface, and the time, the ttal heat transfer t the tube wall can be determined» The values f lcal wall shear stress T, w heat transfer cefficient h. and inner surface temperature T. as 1 w,i calculated frm (2.59), (2.63) and (2.64) are used in the ne-dimensinal analysis fr the subsequent time step. Nn-dimensinalizatin Befre prceeding t the slutin technique that can be applied t slve the equatins derived s far, it is advantageus t nndimensinalize the equatins t btain a general slutin fr the gemetrically similar devices with the same initial cnditins. The nndimensinalized parameters arex Axial distance, x' = ~ L t Pressure, P» = p/p 0 Temperature, T 1 = T/T 0