Significance of the Balance-Based Averaging Theory for Application in Energy and Process Engineering

Transcription

1 Signifiane of the Balane-Based veraging Theory for ppliation in Energy and Proess Engineering Dissertation zur Erlangung des Grades Doktor-Ingenieur der Fakultät für Mashinenbau der Ruhr-Universität Bohum von Santi Wattananusorn aus Bangkok Bohum 7

2 Doktorarbeit eingereiht am : 8. pril 7 Tag der mündlihen Prüfung : 8. Juni 7 Referent : Prof. Dr. ès s. tehn. (EPFL) H. Stoff Korreferent : Prof. Dr.-Ing. V. Sherer

3 I do not know what I may appear to the world, but to myself I seem to have been only like a boy playing on the seashore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great oean of truth lay all undisovered before me. - Isaa Newton -

4 knowledgements I would like to express the most sinere gratitude to Prof. Dr. Horst Stoff who has been my thesis advisor, my respeted teaher, my mentor, and my hero, for his kind supervision, valuable guidane, and helpful omments of this researh. From him I learned to appreiate many topis in fluid mehanis, other sientifi issues, and also, beyond anything perhaps, the way to a good life. For this gain in knowledge and the ontinuous support and enouragement (with full inspiration) in all phases of my work, I am indebted to him indeed. I wish to thank all my olleagues in the Department of Fluid Flow Mahines and other people at the Institute of Thermo- and Fluid Dynamis for their warm hospitality and friendship throughout the wonderful years there, and for useful suggestions and unhesitated helps whatever tehnial or non-tehnial problems, espeially Dr. Werner Volgmann for his superb teahings some mathematis. I am so grateful to the Ministry of Siene and Tehnology for their finanial assistane via a sholarship from the Royal Thai Government on the purpose of developing an eduation offier to aomplish the dotoral degree at the Faulty of Mehanial Engineering, Ruhr-University Bohum, Germany. Finally, the speial and deepest appreiation goes to my beloved parents for providing the neessary atmosphere at a distane of understanding during the untold amount of hours required for writing this dissertation whih I dediate to them as a gift of love. To all of you, I truly say a most heartfelt thank you. Bohum, November 6 Santi Wattananusorn

5 bstrat This thesis presents the mathematial frameworks in averaging non-uniform flow distributions onerning some engineering appliations by using the tehniques of Dzung, Traupel, and Kreitmeier. The alulating results are then provided for user onveniene as sets of mathematial formulations of the orresponding average quantities of interest. lso the new possibility of averaging spae-dependent flow fields using a oupling fator that links the equations of momentum and energy is introdued effetively. The sheme is applied to the mean veloity, whih is derived straightforwardly through the ontinuity equation. It reates a small unbalane, whih an be eliminated later ompletely. Smaller disrepanies in the integration of systems of balane equations for inhomogeneous flow are the onsequene. The proedure is verified on various flow patterns, and omparisons are made with other onventional methods and with some available experimental data. Despite investigating only numerial examples of inompressible flows aessible to analytial solutions here, the tehnique, in priniple, is apable of dealing with ompressible flows as well. pproximate numerial solutions in ompressible flow would not allow to attribute disrepanies between the methods uniquely to distintions between the methods but suffer also from the influene of onvergene. Furthermore, the proposed method disards some variables required in other tehniques while still providing useful and aeptable results in pratial problems. Keywords : non-uniform, inhomogeneous flow, spae averaging

6 Contents Chapter Introdution. Motivation of Researh. Sope and Objetive of Study.3 Referenes Chapter Theoretial Considerations on veraging 3. veraging Proedure Based on Integral System Effets 4.. Corretion-Fator Requirements 5. veraging Proedure Based on Complete Equilibrium 5.. Consistent veraging Method 6.3 Reversibly and Irreversibly veraged Quantities 7.4 Referenes 9 Chapter 3 Couette Shear Flow 3. Relationship to Reynolds Perturbation 4 3. Traupel s and Kreitmeier s pproahes to verage Veloity Referenes Chapter 4 Flows in a Non-Rotating Pipe 4. Typial Laminar and Turbulent Flows 4. Exponential Shapes of Veloity Profiles Referenes 3

7 Chapter 5 Flows in a Rotating Pipe Solid-Body Swirl Potential-Vortex Swirl Referenes 4 Chapter 6 Flows through Diffuser and Nozzle 4 6. Fores Exerted on Curved Channels of Cirular Cross Setion 4 6. Performane of a Sudden-Expansion Pipe Flow Referenes 5 Chapter 7 Step-Veloity Profile Flows 5 7. Step-Veloity Forward Flow 5 7. Step-Veloity Inluding Reverse Flow 6 Chapter 8 Hydrauli Jump Referenes 8 Chapter 9 Periodi Flow in a Piston Pump 8 9. Effet of Stroke Frequeny 84 Chapter Wall-Shear-Stress Estimation 88. Referenes 96 Chapter Evidene of the λ-model 97. Momentum-Energy Linkage 98. Numerial Examples 4.3 Referenes 6 Chapter Conlusions 8. Contributions of the Thesis 9

8 Chapter Introdution. Motivation of Researh n averaging tehnique may be used with design data or with measured data. Design data, obtained by omputation, are always onsistent. But the experimental determination of the distributed flow and state quantities requires the appliation of extensive traversing instrumentation at all omponent interfaes []. To simplify steady-state omponent and system analysis and to verify the results experimentally, employing only limited fixed instrumentation at the omponent interfaes, the real, non-uniform, three-dimensional flows with time-periodi veloities and state quantities (rotating wakes and turbulene), it would be desirable to desribe the real flow by uniform steady flow models. The veloity and state quantities of the uniform flow models an be determined by applying suitable time- and spae-averaging proedures to the distributed veloity and state quantities of the real flow. Introdution of representative average veloities, momentum flows and state quantities allow one to evaluate omponent and system performane in a simple manner. The quantitative results obtained from suh simplified performane analysis methods an desribe the steadystate performane parameters with adequate auray. In addition, suitable averaging tehniques define loations within the flow field where loal veloity and state quantities oinide with the averaged veloity and state properties. Fixed instrumentation therefore should be positioned at these loations in order to obtain truly representative experimental performane data []. Many different averaging proedures of non-uniform internal flows are

9 urrently in use. Thus the question arises, if performane parameters derived by using different averaging methods remain omparable with eah other. Some alulations are used to illustrate this omparison.. Sope and Objetive of Study The main objetive for the work reported here is to make use of known averaging methods and to present their differenes, to quantify inauraies or unertainties inherently assoiated with different averaging proedures and to arrive at reommendations for pratial appliations. For the present study the sope of work was restrited to onsider only the fundamental problems that an be solved analytially. Furthermore, the averaging methods onsidered here are appliable only to steady-state omponent and system performane analysis. Not only the final average solutions will be reported, but also the routes to obtain some of those results will be demonstrated in the details of mathematial expressions along the steps of alulations. The dissertation is organized and subdivided in hapters depending on the similarities of the flow harateristis. This makes eah hapter being an individual aspet of onsideration and able to stand alone on its own ontent. Therefore, one an skip diretly to the hapter of interest without being interfered from other hapters at all..3 Referenes. Wazelt, F. (983): Suitable averaging tehniques in non-uniform internal flows. dvisory Group for erospae Researh and Development, GRD, report no. GRD-R-8 (Eng.), p... Ibid., p..

10 Chapter Theoretial Considerations on veraging In order to be able to treat the aerodynami and thermodynami proesses in a fluid flow, it is neessary to establish representative average values for the surfaes of a fixed ontrol volume. These average values desribe orretly the integral effet of the fluid flowing through this ontrol volume. This problem arises not only in the interpretation of test results, but also during the aerodynami design of omponents. lthough eah theoretial treatise for the formation of physially sensible average values starts from the idea that the nonhomogeneous flow onditions in the ontrol surfae are fully known, this ondition often presents onsiderable diffiulties in pratial test rigs. The arguments resulting therefrom, whih are more onerned with pratial matters in the formulation of rules for averaging, are not examined at this point. Thus, the following arguments always assume that the nonhomogeneous flow onditions are known aurately. The averaging methods given in the literature an be divided in priniple into two groups : The first group relies only on averaging definitions whih represent, qualitatively and aurately, the Integral System Effets of interest. Here an attempt is made to satisfy an additional ondition, namely, to establish basi equations by means of average values with as few orretion fators as possible. In the seond group, the non-homogeneous flow state in the measurement or alulation plane is onverted into a state of Complete Equilibrium with the aid of the onservation laws for mass, momentum and energy. This ondition should be attained asymptotially in an 3

11 infinitely long settling hannel, beause of the visosity of the fluid, in whih the fluid does not adhere to the wall but moves along without frition [,].. veraging Proedure Based on Integral System Effets In this method of averaging, the attempt is made primarily to represent the Integral System Effets of interest by representative average values. s a reasonable additional ondition, the requirement is to use as few orretion fators as possible in the formulation of the onservation laws by means of the defined average values. s suggested by Traupel [3,4], the onvetively transported flux of a mass-speifi flow quantity θ through area d in unit time is given by θ ρ d, where ρ and are respetively the fluid density and veloity. The mass flow rate through a ross-setion surfae is given by m = ρ d. (.) The average of θ per unit mass is therefore θ = d m θρ. (.) The proedure to average flow data with Traupel s method is to determine an average veloity from equation (.) by setting θ =, whih amounts to averaging the momentum flux. Next, the average speifi stati enthalpy h is obtained by setting average speifi total enthalpy results as θ = h = C T, and the P h t = h +. (.3) where the term is obtained by setting θ =. For averaging the stati pressure, Traupel reommends a simple (unweighted) area averaging in order to onserve pressure fores in momentum balanes as p = p d. (.4) With p and h known, the gas equations are used to alulate a mean density ρ as ( ) ( P )( ) ρ = ρ p, h = C R p h. (.5) nd the mean speifi kineti energy k is diretly represented by 4

12 k =. (.6).. Corretion-Fator Requirements The above momentum-based definition of does not satisfy the ontinuity equation in its simplest form, m ρ (.7) beause equation (.) requires a kind of averaging different from equation (.). The orretness of mass flow rate is restored by defining a ontinuity shape fator, ε K, for the veloity profile as ( K ) m ρ, (.8) and = ε ρ d ( ε K ) =. (.9) ρ Further, as in the ase of mass flux, the mean kineti energy flux also requires a orretion, beause is not idential to defined above. Traupel introdues an energy shape fator, ε E, to the average obtained from momentum averaging as ( ε E ) = yielding the relation ( ε E) (.) ht = h +. (.) In summary, Traupel defines all veloity omponents by momentum averaging, but needs various shape fators to onserve mass, and energy. In ase of uniform flow, these beome ε ε = =. K E. veraging Proedure Based on Complete Equilibrium Using this method of averaging, the non-homogeneous flow ondition in the measurement or alulation plane is onverted into a state of Complete Equilibrium by means of the 5

13 onservation laws for mass, momentum, and energy. The ondition of Complete Equilibrium is haraterized by the fat that the visous fluid is in a state of equilibrium, both mehanially and thermally. This means that neither momentum nor energy exhange takes plae between the individual fluid partiles. If, for instane, the fluid has a swirl omponent, then only pressure fores normal to the veloity vetor are allowable whih affet the required hange in diretion of the irumferential omponent of the veloity vetor [5]. In order to desribe the effets of a system in whose exit plane the averaging method is used quantitatively aurately by the state values of Complete Equilibrium, the integrals of the onservation quantities - mass, momentum, and energy - have to be equal at the inlet and exit of the ontrol volume in whih the irreversible mixing proess ours. Considering that the irreversible mixing proess ours in a real settling hannel of infinite length, then the ondition of Complete Equilibrium will not be reahed, beause of wall boundary layers. Thus, as a matter of definition, the no-slip ondition existing on a real wall should be negleted, without limiting the effet of visosity neessary for an irreversible mixing proess. In order to make this abstration learer, one should simply imagine that the mixing proess ours in a disontinuity plane similar to a normal ompression shok. Thus, the additional ondition that the inlet and exit plane areas of the ontrol volume are equal is satisfied automatially. The fat that the system of equations resulting from the onservation laws produes two solutions underlies the analogy to the ompression shok. For subsoni flow, one solution ontradits the seond law of thermodynamis and, for supersoni flow, a strong and a weak solution arise with respet to the irreversible entropy inrease... Consistent veraging Method Dzung s onsistently averaged values are defined to omply with mass, momentum, and energy onservation, and avoid shape fators [3,6]. For this purpose, the mean speifi kineti energy k is defined as k = (.) where is the ontinuum average obtained with equation (.) from m = ρ = ρ d. (.3) 6

14 In order to onserve the speifi total enthalpy in adiabati flow aording to the first law of thermodynamis : h t = h + (.4) Dzung adapts the definition of speifi stati enthalpy as h = h ρ d m +. (.5) Compliane with momentum onservation is ahieved by adapting the definition of stati pressure as = p + ρ m p d d. (.6) With the stati pressure p and the speifi stati enthalpy h, the average density an be determined by the gas equations as ρ ( p, h ) = ρ. (.7) With the density ρ known, the average veloity an iteratively be alulated again by equation (.3)..3 Reversibly and Irreversibly veraged Quantities Kreitmeier introdues the averaging method that takes into aount all relevant balane equations and is haraterized by hypothetial reversible and irreversible equilibration proesses. The differenes between the reversibly [~] and irreversibly [^] averaged quantities are a measure of the flow inhomogeneity and are employed at all levels in its modelling [7,8]. The main advantage of Kreitmeier s tehnique is that orretion fators are not required. Kreitmeier formulates the following onservation equations : ontinuity equation m = ρ d ρˆˆ, (.8) linear momentum equation ( ρ ) d ( ρ ˆˆ pˆ ) I = + p +, (.9) energy equation 7

15 ˆ ĉ H d h ρˆˆ t = h + ρ +, (.) and non-onservation equations : H = hρ d h ρˆˆ, (.) K = k ρ d k ρˆˆ. (.) ording to Kreitmeier s philosophy, the mean speifi total enthalpy between both irreversible and reversible ones must be onserved, therefore, we have ĥ t With = h. (.3) ˆ t ˆ k, we an express terms in (.3) being a ombination of the stati and its kineti parts orresponding to eah average proess as hˆ + k ˆ = h + k (.4) k + h h = k. (.5) or, ˆ ( ˆ ) Next, the differene between two modes of speifi stati enthalpy in the above equation is onsidered to be a summation of dissipation ( j ) and speifi flow work ( y ) : ĥ h = y + j (.6) where pˆ p y = pˆ p d = ρ. (.7) ρ Substitute (.7) and (.6) into (.5), yields ξ + χ + ζ = (.8) where ξ is the kineti-energy oeffiient and equals to ˆk k, χ is the pressure-reovery oeffiient and equals to ( ˆp p) ( ρ k ) ζ is the pressure-loss oeffiient and equals to j k., The relations of average variables proposed in Kreitmeier s method an be illustrated as Figure.. In it, the entropy differene (the inreasing of entropy) due to averaging proesses will be straightforwardly determined by hˆ pˆ ŝ s = CP ln R ln h p. (.9) 8

16 h h t = hˆ t k ˆp ˆk ĥ h ŝ s p ŝ s s Figure. : Reversible and irreversible equilibration proesses..4 Referenes. Happel, H-W. (983): Suitable averaging tehniques in non-uniform internal flows. dvisory Group for erospae Researh and Development, GRD, report no. GRD- R-8 (Eng.), p. 5.. Prasad,. (4): Calulation of the mixed-out state in turbomahine flows. Proeedings of GT4 SME Turbo Expo 4: Power for Land, Sea and ir, Vienna, paper no. GT Köppel, P.; Roduner, C.; Kupfershmied, P.; Gyarmathy, G. (): On the development and appliation of the fast-response aerodynami probe system in turbomahines Part 3: Comparison of averaging methods applied to entrifugal ompressor measurements. Journal of Turbomahinery, vol., no. 3, pp Traupel, W. (977): Thermishe Turbomashinen I, Springer-Verlag, Berlin, 3rd ed., p Happel, H-W. (983): Suitable averaging tehniques in non-uniform internal flows. dvisory Group for erospae Researh and Development, GRD, report no. GRD- R-8 (Eng.), p

17 6. Dzung, L. S. (97): Consistent mean values for ompressible media in the theory of turbomahines. Brown Boveri Review, vol. 58, no., pp Kreitmeier, F. (99): Spae-averaging 3D flows using stritly formulated balane equations in turbomahinery. SME COGEN-TURBO, vol. 7, pp Kreitmeier, F.; Lüking, P. (3): Demonstration of balane-based spae averaging and modelling in view of blading-diffuser interation. Journal of Power and Energy, vol. 7, no. 4, pp

18 Chapter 3 Couette Shear Flow Consider the motion of inompressible fluid through a retangular ross-setional area, between two very long parallel plates, one of whih is at rest, the other moving with a onstant veloity as shown in Figure 3.. Let the distane between the plates be b, and there is no hange in pressure aross the fluid. The experiment teahes us that [] the fluid adheres to both walls, so that its veloity at the lower plate is zero, and that at the upper plate is equal to the speed of the plate, U. Therefore, = U. (3.) U x b a Figure 3. : Geometry of Couette shear flow.

19 Furthermore, the veloity distribution of the fluid between plates is linear, so that it is proportional to the distane x from the lower plate, then we have ( x) = x b. (3.) In general there are three typial approahes to determine the average veloities. Basis on mass : m = ρ Q = b ρa ( x) dx, (3.3a) m ρ. (3.3b) Basis on momentum : b I = ρ Q = ρa ( x) dx, (3.4a) [ ] I ρ. (3.4b) Basis on energy : b 3 E = ρ Q = ρa [ ( x) ] dx, (3.5a) 3 ρ. (3.5b) E Solving the sets of equations (3.3), (3.4), and (3.5), for, the values of.5,.57735, and.6996, are found respetively. In other words, all these average veloities, based on different approahes, are about 5, 58, and 63 % of the imum veloity (speed of moving the plate). Thus we an rewrite all speeds using the speified subsripts of their soures as K = m, I ρ = I ρ, and E E = ρ 3. To extrat the orret average values of stati pressure ( p ) and speifi stati enthalpy ( h ), theoretially one needs to put I, and E into the balane equations of momentum, and energy, respetively. However, aording to Kreitmeier s irreversible method, not only p and h but also the rest of desired variables an be alulated by the mass-based mean veloity K. With this sheme, the user is free to perform the omputations without knowing of I or E. The method generates a small error that an be pratially aepted and demonstrated here.

20 From the expression of the rate of total linear momentum : m I = dm + p d. The result is to be found that I = ρ + p. ( ) pplying the definition of Kreitmeier : ˆ I ρ + pˆ, where ( ) ĉ =, this leads us to K p = p ˆ = p ρ. (3.6) The last term of this equation represents the error emerged from the averaging proess. For the average speifi stati enthalpy, the equation of the rate of total enthalpy is employed Ht ρ h = + ( d), and its solution is H =.5 ρ h +.5. t ( ) One again, applying the definition of Kreitmeier : ˆ ĉ H t ρ ĉ h +, still ĉ =, therefore, we have K h h ˆ = = h +.5. (3.7) Using information of the average veloity, one an diretly ompute the average speifi kineti energy by ˆ ĉ k = k = =.5. (3.8) On the other hand, if we follow Kreitmeier s reversible method, these parameters will be = K, p = p, h = h, and k = k, that require some extra equations to be solved. From the equation of rate of kineti energy : K k m m 3 3 = d = ρ d =.5 ρ. lso from Kreitmeier s theory : K ρ ĉk, 3

21 We found that k = k =.5. (3.9) Compared equations (3.8) and (3.9), between two kinds of speifi kineti energy predited by the theory, the irreversibly averaged value apparently is smaller than the reversibly averaged one by the exat fator of. This number, however, is valid only in the ase of Couette shear flow. It varies as the veloity profile hanges. 3. Relationship to Reynolds Perturbation One may speak of any suh veloity as onsisting of a spae-average omponent, and a flutuating funtion; so at any point ( x) = + ( x). (3.) Reproduing another set of averagings by replaing (3.) with (3.), we have b b d ( x) d m = ρa x + ρ x b. (3.) Due to the flutuation being both plus and minus, therefore the average value of ( x) is zero : b ( x) x = d. b Equation (3.) finally redues to m = ρ, the result by omparison with Kreitmeier s mass-based equation an be extrated : ĉ =. When the perturbation is applied to averaging of the total linear-momentum rate, we end up with ( ) I = ρ + ρ + p. (3.) pply the definition of Kreitmeier s irreversible mode to (3.), we obtain ˆp p = ρ. (3.3) Similar to averaging of the total enthalpy rate, it an be shown that ( ) H t = ρ h (3.4) 4

22 this leads us to 3 ĥ h = 3 +. (3.5) For irreversibly speifi kineti energy, exatly the same appears as equation (3.8) : ˆk ( ) =, (3.6) and its reversible mode is 3 k = ( ) (3.7) Table 3. summarizes all average parameters from Kreitmeier s approah, and the omparison with Reynolds perturbation : ( x) b = x = + ( x) K ( x) =.5 K = K ˆp p =.8333 ρ ˆp p = ρ ĥ h =.5 3 ĥ h = K ˆk =.5 ˆk =.5 ( ) K k =.5 ( ) 3 k =.5 K K Table 3. : veragings of the Kreitmeier (left) and the orresponding Reynolds perturbation approah (right). 5

23 3. Traupel s and Kreitmeier s pproahes to verage Veloity Similar to the appliation of Kreitmeier s method in Setion 3., here we disuss Traupel s method in omparison with the Reynolds perturbation approah for the Couette shear flow. The alulation based on equation (3.) for Traupel s average veloity results in ρ = d = m where m is the exat mass flow rate from the ontinuity equation (3.3). (3.8) Other average parameters required in the theory are shown in the following : ρ = ( d ) =.5 m, (3.9) h = h ( d) h m ρ =, (3.) h t = h + = h +.5 ( ) k = =., (3.), (3.) p = p d = p. (3.3) Likewise as in the previous setion, we have to rewrite equation (3.), when Reynolds perturbation is taken into aount, as ( x) = + ( x). (3.4) T Obviously, the inequality between the average veloities obtained by the methods of Traupel and Reynolds perturbation is found to be : T b ( x) x, and fores us to onlude that = d. (3.5) b This expression supports the reason for retaining the same profile, as used in Setion 3., in whih averaging of the flutuation requires the non-zero value. fterwards, with the help of equations (3.8), (3.), (3.) and (3.3), the other average parameters for Traupel s method, written in the ontext of Reynolds perturbation, an be alulated and are shown in Table 3.. 6

24 ( x) b = x = + ( x) T ( x) = T ( ) = = T K K K + p p = p p = h h = h h = ( K) k = ( K) ( K) K k = K ( K) K +.5 ( K ) Table 3. : veragings of the Traupel (left) and the orresponding Reynolds perturbation approah (right). One an generalize the relations of the averages of flutuations in Tables 3. and 3. as = γ = γ 3 3 = γ3, (3.6), (3.7), (3.8) 7

25 with an additional form of the normal average-veloity : = γ (3.9) where γ, γ, γ, and γ 3 are oeffiients of whih eah one has a ertain unique value as grouped in Table 3.3. Follow these speified values of both averaging methods, the equivalene of the average parameters approahed by the original standard proedure and the Reynolds perturbation proedure is maintained. Consequently, one an selet either the left or the right olumn in the Tables to alulate the desired variables independently. Coeffiient Kreitmeier veraging Method Traupel γ γ.6667 γ γ Table 3.3 : Coeffiients that are equivalent to Tables 3., and 3., respetively. Consider the inequality between the orret mass-flow-rate derived from the ontinuity equation and the one omputed by Traupel s average veloity : m ρ T. Using the orretion fator [], we an write T ( ε K ) m = ρ (3.3) where ε K is a ontinuity shape fator. 8

26 The orret mass-flow-rate also an be defined from m = ρ ρ T + ρ T (3.3) where T is the average flutuation orresponding to Traupel s method in equation (3.5). If (3.3) = (3.3), then ε K T =. (3.3) T However, in general, ε K will be alulated by ĉ ε = K K = T. (3.33) T One an simply dedue a relation from equations (3.3) and ( 3.33) : T = ĉ. (3.34) T This relation onfirms that the differene between the average veloities of Kreitmeier and Traupel is able to put into its ontribution for T, and being a non-zero value for any nonuniform veloity profile. Instead of using the average veloity alulated via Kreitmeier s method ( ĉ ), for any kind of profile, one an onfidently replae it by the average veloity obtained from the ontinuity equation ( K ), or even the average veloity defined in Reynolds perturbation ( ) without loss of any physial and mathematial information. tually we have demonstrated that ĉ = K =. nother onsideration of this setion is to take the root-mean-square (RMS) of, whih is analogous to the so-alled turbulene intensity. Now let φ be the equivalene intensity, written as ; we may also use the form : φ = γ for whih all parameters have been previously aquired. Therefore, the resulting ratio of Traupel-to-Kreitmeier equivalene intensity is determined by φtraupel =.5477, φ Kreitmeier φ or, Kreitmeier 87 % φ. Traupel 9

27 3.3 Referenes. Shlihting, H. (979): Boundary-Layer Theory, MGraw-Hill, New York, 7th ed., p. 6.. Köppel, P.; Roduner, C.; Kupfershmied, P.; Gyarmathy, G. (): On the development and appliation of the fast-response aerodynami probe system in turbomahines Part 3: Comparison of averaging methods applied to entrifugal ompressor measurements. Journal of Turbomahinery, vol., no. 3, pp

28 Chapter 4 Flows in a Non-Rotating Pipe The ase of fluid flows through a straight horizontal pipe of irular ross-setion having a onstant diameter ( D = R) is investigated in this hapter. Measurements show that [] the harateristis of the motions an be made learly visible by introduing a liquid dye into the low-flow-rate region in whih the pressure drop per unit length is proportional to the flow rate. The injeted dye forms a smooth, thin, straight streak down the pipe; there is no mixing perpendiular to the axis of the pipe. This type of flow, in whih all the motion is in axial diretion, is alled laminar flow (the fluid appears to move in thin shells or layers, or lamina). The fat that in the laminar motion under disussion fluid laminae slide over eah other, and that there are no radial veloity omponents, so that the pressure drop is proportional to the first power of the mean flow veloity ( ). By inreasing the flow veloity it is possible to reah a stage when the fluid partiles ease to move along straight lines and the regularity of the motion breaks down. In ontrast to the result in laminar flow, suh stage ours in the high-flow-rate region where the pressure drop per unit length is proportional to the flow rate of the power of.8 to. The injeted dye disperses rapidly throughout the entire pipe. The oloured thread beomes mixed with the fluid, its sharp outline beomes blurred and eventually the whole ross-setion beomes oloured []. On the axial motion there are now superimposed irregular radial flutuations whih affet a haoti motion in all diretions in the pipe and ause the rosswise mixing of the dye. This type of flow is alled turbulent flow. part from these two kinds of flow patterns, we also investigate some trial veloity profiles under the averagings of Kreitmeier and from others as well.

29 4. Typial Laminar and Turbulent Flows Firstly, we fous on a steady laminar flow of inompressible Newtonian fluid in whih its veloity distribution is paraboli : = ( r) r R where is a imum speed at the entre ( r = ) of a ross-setional area ( π R (4.) = ). The veloity at a no-slip wall ( r = R) is zero, beause of adhesion, and reahes a imum on the axis of symmetry. The veloity remains onstant on ylindrial surfaes whih are onentri with the axis, and the individual ylindrial laminae slide over eah other, the veloity being purely axial everywhere as shown in Figure 4.. s being a standard proedure in Kreitmeier s averaging, the average veloity is obtained from the ontinuity through the equation of mass flow rate : π R m = ρ r dr d ϕ ρ (4.) or, ĉ ( r) ˆ =.5. (4.3) The above result tells us that Kreitmeier s average veloity of a laminar-flow in a pipe has half the size or 5 % of its possible highest value and it is independent of the size of the pipe geometry. With the help of ĉ, we ontinue to extrat the average stati pressure using equation of total linear momentum flux : π R I = ρ r + p r r ρπ ( ) d dϕ R = 4 ( rr r R + r ) dr + π R p R ( p ) = ρ +. (4.4) ording to Kreitmeier, ( ρ ˆ pˆ ) I + (4.5) if (4.5) is substituted by (4.3), and ompared with (4.4), then we have ˆp p =.8333 ρ. (4.6) This shows the differene between two kinds of stati pressures of the theory : The irreversible ( ˆp ) and reversible ( p ) ones. We may onsider the disrepany as its error aused

30 by inserting the mass average veloity into the momentum equation sine all balane equations require their own orresponding average veloities, say, K for ontinuity, I for momentum, and E for energy as previously disussed in Chapter 3. Therefore, what we have from equation (4.6) is the average stati pressure based on the ontinuity-average veloity, not from the momentum. Furthermore, this onept of using only one single type of average veloity throughout the alulations still is extended to determine the speifi stati enthalpy via the equation of total enthalpy flux : π R ( r) H t = ( r) ρ h + r dr dϕ π ρ ρ R h = + + R 3 R ( rr 3r R 3r R r ) dr (.5 h.5 ) = ρ +. (4.7) π Comparing (4.7) with the definition of Kreitmeier, H t ρ ĉ hˆ + ĉ, yields. (4.8) ĥ h =.5 gain, here we have both average variables from the irreversible ( ĥ ) and reversible ( h ) proesses. The disrepany between these values depends on a imum speed of the profile. The faster fluid flows, the more error is produed. In the averaging of speifi kineti energy, there are also two speies of average variables in Kreitmeier s theory : ) The irreversible proess ( ˆk ) whih an be diretly alulated from equation (4.3) as ĉ ˆk = =.5. (4.9) It is remarked that the numerial value of ˆk oinidentally equals to the error generated by ĥ h from equation (4.8). ) The reversible proess ( k ) whih requires an extra formula to be performed : π R 3 K = ( r) ρ r r ϕ ρ d d ˆ k (4.) or,. (4.) k =.5 Next, we turn to a typial profile of veloity distribution in turbulent flow (Figure 4.). ll previous alulating proedures are repeated in a similar manner as in laminar flow, only replaing equation (4.) by 3

31 7 r ( r) = R. (4.) The power of the funtion that gives the best representation of the experiments varies from 6 for the Reynolds number ( Re ρ D µ ) at 3 4, to for 6 Re = 3.4. Prandtl, however, seleted 7 as the best average [3]. Therefore we use the 7-power-veloity distribution for our assumption. If the averaging method proposed by Kreitmeier is applied, we obtain ĉ = (4.3) The average veloity in the turbulent ase is about 8 % of the imum speed whih is higher than in the laminar ase due to the shape of the turbulent profile whih has less distortion than the laminar profile when ompared with the homogeneous profile. Other average variables are as follows : ˆp p =.36 ρ, (4.4), (4.5) ĥ h =.947 ˆk =.33348, (4.6). (4.7) k = Exponential Shapes of Veloity Profiles In this setion, we study the effet of varying flow patterns on its average parameters alulated from various averaging methods. Suppose a simple funtion of an exponential harateristi ( y ( r) n r = R n = x ) being a trial veloity distribution aross a flow area via the relation (4.8) where n varies from,, 3,, and 3. The profiles onsidered an be ategorized into three groups as n =, n <, and n > whih are graphially illustrated (not to sale) in Figures 4.3, 4.4, and 4.5, respetively. s n inreases, the flow pattern is more distorted from the uniformity of the veloity distribution. Hene the inhomogeneity beomes larger in the flow field. Tables 4. to 4.4 present suh effets influening the averaging proedures by Kreitmeier (irreversible), Kreitmeier 4

32 (reversible), Dzung, and Traupel, respetively. long with the results of the exponential veloity distributions, we summarize again for laminar and turbulent ases in Tables 4. and 4.. Furthermore, the averaging methods of Dzung and Traupel are also re-introdued to problems studied in Setion 4. where all alulated variables are listed in Tables 4.3 and 4.4. Our previous disussion has been involved with something alled the single-power profile, whereas, from this step on, we demonstrate how the averaging methods work under the situation of the multi-power profile. One example of the experimental data is ready to be analyzed : The veloity distribution in a vortex tube [4]. Here, the information of the rosssetional flow area at the distane of 98 % of the total length of the tube is hosen. Figure 4.6 reprodues the measurement results and also the additional polynomial funtion reated as its approximation of the reality, whih is = (4.9) α β β β β β β where α = ( r), β = r R. s a regular step in averages by the methods of Dzung and Kreitmeier, the mean veloity is omputed from the ontinuity equation : π m = ρ R α β β ϕ = ρ ρ d d.389. (4.) That means the average veloity is about 3 % of its imum throughflow speed (in this ase, the speed of the rotating wall). It seems to be an ordinary irumstane, as experiened previously however, a remarkable result emerges when Traupel s method is applied : π ρ R = α β dβ d ϕ = % m. (4.) It now sounds strange sine the value for the average veloity oupied the value exeeds its imum value! How ould it be? Not only this phenomenon ours when the reverse flow is present, but also other peuliar behaviours possibly take plae if Traupel s averaging method is performed for suh a type of the flow pattern whih will be later disussed in Setion 7.. The user should be aware and the appliation of other methods available are suggested in the range of <, for any appliation. Despite the multi-power formula in equation (4.9) apparently looking like the ombination of a separated single-power term onneted in series. Indeed there is not any relation among them at all as the following demonstration shows. If one tries to ollapse 5

33 the series of equation (4.9) to one equivalent power ( n eq maintains the original ontents, this an be expressed as ) through a simplified form and still α β n eq =. (4.) The averaging methods of Dzung and Kreitmeier (irreversible) give =.389 (4.3) [ ] and p = p (4.4) [ ] [ ] where [ ], p p ρ [ ], p p d ρ [ ]. If neq = 4 is assumed in equation (4.), the dimensionless average veloity obtained by Dzung s and Kreitmeier s methods equals to = (4.5) [ ] This onfirms that our seleted n eq is a good estimate sine its orresponding is about 33 % of the imum speed, whereas the alulated from the full power series (4.9) is about 3 %. We, therefore, onfidently assume both approahes to be in agreement with eah other in a sense of equivalene. Next, we extrapolate our known information to predit p [ in whih, aording to (4.4), the ] expeted dimensionless value should be more or less around.6, if the proposed n eq indeed represents the whole system of the entire flow field as desribed by equation (4.9). However, the result turns out to be p = p (4.6) [ ] [ ] s notied, there is an obvious differene between two rough numbers :.6 from (4.4) and.9 from (4.6), or about 35 % in error. This leads us to make a fair onlusion that, unfortunately, users must perform a new alulation for eah variable of interest, although both ases share the same [. No fundamental onnetion exists between them, they are just ] only oinidental in mathematial expressions beause the key ontent lies beneath their own funtions, in other words, the veloity profile itself. 6

34 r D R Figure 4. : Laminar profile. r D R Figure 4. : Turbulent profile. r D R Figure 4.3 : Exponential profile (n = ). r R D Figure 4.4 : Exponential profile (n < ). r D R Figure 4.5 : Exponential profile (n > ). 7

35 Profile Shape Veloity ĉ [ m s] Stati Pressure ˆp kg m s Speifi Stati Enthalpy ĥ m s Speifi Kineti Energy ˆk m s Laminar.5 p ρ h Turbulent.8667 ρ p +.36 h n = ρ p h n =.5 ρ p h n = 3.4 p +.9 ρ h n =.8 ρ p h n = ρ p +.53 h Table 4. : veragings of flows in stationary pipe by Kreitmeier s method (irreversible).

36 Profile Shape Veloity ĉ [ m s] Stati Pressure p kg m s Speifi Stati Enthalpy h m s Speifi Kineti Energy k m s Laminar.5 p d h d.5 Turbulent.8667 p d h d.3595 n = p d h d.3 n =.5 p d h d.5 n = 3.4 p d h d.78 n =.8 p d h d.3574 n = p d h d Table 4. : veragings of flows in stationary pipe by Kreitmeier s method (reversible).

37 Profile Shape Veloity [ m s] Stati Pressure p kg m s Speifi Stati Enthalpy h m s Speifi Kineti Energy k m s Laminar.5 p ρ h Turbulent.8667 ρ p +.36 h n = ρ p h n =.5 ρ p h n = 3.4 p +.9 ρ h n =.8 ρ p h n = ρ p +.53 h Table 4.3 : veragings of flows in stationary pipe by Dzung s method.

38 Profile Shape Veloity [ m s] Stati Pressure p kg m s Speifi Stati Enthalpy h m s Speifi Kineti Energy k m s Speifi Kineti Enthalpy m s Laminar p h..5 Turbulent p h n =.75 p h.85.3 n = p h..5 n = 3.65 p h n = p h n = p h Table 4.4 : veragings of flows in stationary pipe by Traupel s method.

39 . [4] approx..8.6 α β Figure 4.6 : Veloity distribution in a vortex tube. 4.3 Referenes. De Nevers, N. (99): Fluid Mehanis for Chemial Engineers, MGraw-Hill, Singapore, nd ed., p. 8.. Shlihting, H. (979): Boundary-Layer Theory, MGraw-Hill, New York, 7th ed., p De Nevers, N. (99): Fluid Mehanis for Chemial Engineers, MGraw-Hill, Singapore, nd ed., p Bruun, H. H. (969): Experimental investigation of the energy separation in vortex tubes. Journal of Mehanial Engineering Siene, vol., no. 6, pp

40 Chapter 5 Flows in a Rotating Pipe n inompressible Newtonian fluid flow through a straight horizontal pipe whih is rotating in irumferential diretion around its entreline (lokwise or ounterlokwise) is studied. Therefore, an extra omponent of veloity must be added to the main streamwise diretion. nd also another balane equation for angular-momentum flux is now required in our alulations. s the pipe keeps rotating, stati pressure aross flow area ( ) is affeted via [] p( r).5 ρ ϕ ( r) pr = o = + (5.) where ϕ is the veloity in the tangential diretion whih is perpendiular to the axial diretion. However, we assume that the veloity distribution remains unhanged during the rotation. Here, Prandtl s 7 turbulent power profile is still hosen. In general, there are two main types of rotating movements. If the tangential speed at the inner wall of the rotating ross-setion has a imum value and its distribution deays linearly toward the pipe entre, suh movement is said to be solid-body swirl and expressed by ϕ r or, ϕ = k r where a irumferential speed (ω ) of a symmetrial pipe rotating around its streamwise axis is suggested to take plae a onstant k in the above relation, therefore, we have [] ϕ = ω r ; r R. (5.) 33

41 Under this flow pattern (Figure 5.), the fluid oupies the highest possible veloity at r equals to the pipe radius (R) that is = ω R. (5.3) ϕ r = R ϕ, nd the fluid has zero veloity at the entre of pipe as ϕ r = o =. (5.4) ϕ,min nother type of rotation is alled potential-vortex swirl of whih the imum speed deays exponentially outward the pipe entre to the pipe wall unlike the solid-body swirl. Hene we an write ϕ r k or, ϕ = r and propose a onstant k as the funtion of a irumferential speed (like with the solid-body swirl), and also a referene radius ( R ) by R ω ϕ = ; r R r R. (5.5) This expression has two major advantages. Firstly, our domain of alulation is free from the infinite solution where r approahes zero at the pipe entre (Figure 5.). Choosing the R as small as possible, but not zero, are efforts to side-step the singularity in equation (5.5) and still maintain reliable solutions from suh approximations. Seondly, the imum value of the tangential veloity has the same form and dimension as in the solid-body swirl : = ω R. (5.6) ϕ r = R ϕ, n origin of the potential-vortex swirl may be onsidered as a rotation of the fluid itself starting at the ore of a stationary pipe and then propagate radially outward to the outer adjaent fluid-layers under the influene of visous fore. Suh an effet indues a motion of the whole ross setion, inluding the pipe, to move irularly or being rotated together. This kind of movement produes non-zero veloity in the flow field with its lowest possible value loated at the inner pipe wall, and equals R R ϕ ω r = R ϕ,min where R R R = (5.7) =. 34

42 = π R = π R r R ω r R ω R π ( ) = R R Figure 5. : Solid-body swirl. Figure 5. : Potential-vortex swirl. 5. Solid-Body Swirl pplying Kreitmeier s method, the average variables for turbulent solid-body swirling flow are listed as follows : ĉ =.8667, (5.8) ˆp.36 ρ.5 ρω R pr = o = + +, (5.9) ĥ = h, (5.) ˆk = ω R, (5.) k = ω R. (5.) In alulations of a rotating flow, the angular-momentum onsideration must be taken into aount : m π R r 7 3 ϕ d ρ ω d d R L = r m = r r = ρ R. (5.3) ϕ, nd, aording to Kreitmeier, we have L ρ r ˆˆ ˆ (5.4) ϕ where ˆr is alled the Dibelius radius or Euler radius. The definition of angular-momentum flux in equation (5.4) ontains two unknowns : ˆr, and ĉ ϕ. Equation (5.3) alone is not suffiient to solve for those unknowns simultaneously. t this step, the somewhat ad ho assumption may be proposed to overome the problem by ϕ 35

43 ˆ ϕ = ω rˆ ; r ˆ R. (5.5) Substitute (5.5) and (5.8) into (5.4), yields.8667 ρω rˆ (5.6) L = whih now ontains only one unknown, thus the onventional omparison of equations (5.6) and (5.3) leads us to ˆr = R 68 % R. (5.7) With known ˆr, the average tangential veloity of the turbulent flow with solid-body swirling is determined : ĉ = ω R (5.8) ϕ or, ĉ ϕ is about 68 % of ϕ, (the imum speed loates at the rotating pipe-wall). Likewise in Kreitmeier s theory, the average tangential veloity ϕ obtained from theories of Dzung and Traupel requires the known orresponding average radius r E whih is also alled the Euler radius. ording to Dzung : r (5.9) m where m, in ase of the turbulent flow with the 7-power law for the veloity distribution, is E = r dm.8667 ρ. Solving equation (5.9), the result of.468 R is to be found and Dzung s average radius equals R, leading us to = ω r = (5.) ϕ E ϕ, However, in Traupel s theory, we have to ompute the quantity r ϕ rather than r E or ˆr via r ϕ = ρ ( r) ϕ d =.468 m r ωr. (5.) nd the Traupel s average tangential veloity ϕ an be determined later by = r r E (5.) ϕ ϕ where r E stritly follows equation (5.9). Table 5. summarizes the mean variables alulated from various averaging methods when the flow is fully developed turbulent. dditionally, the results orresponding to laminar flow are also presented in Table

44 Variable Method Kreitmeier (irreversible) Kreitmeier (reversible) Dzung Traupel p p r = o ρ ρω R.5 + ρω R p r = o p r = o ρ ρω R.5 + ρω R p r = o r R R R R ϕ ω R ω R ω R ω R h h h h h k ω R ω R ω R ω R Table 5. : verage physial quantities aording to different methods for solid-body swirling (turbulent veloity distribution).

45 Variable Method Kreitmeier (irreversible) Kreitmeier (reversible) Dzung Traupel p p r = o ρ ρω R.5 + ρω R p r = o p r = o ρ ρω R.5 + ρω R p r = o r R R R R ϕ ω R ω R ω R ω R h.5 + h h.5 + h h k ω R ω R ω R ω R Table 5. : verage physial quantities aording to different methods for solid-body swirling (laminar veloity distribution).

46 5. Potential-Vortex Swirl In order to deal with suh kind of rotation, we modify the original pressure distribution (5.) to be properly posed with our boundary ondition desribed by equation (5.5) as p( r) =.5 ρ, ( r) ϕ ϕ + pr = R. (5.3) The average axial veloity, aording to Kreitmeier, is alulated from R ( r) d π m = ρ r r = ρ β R or, ĉ [ ] β K K = (5.4) where β K =.75 R R, for R refer to equation (5.7). Subsript K in the above oeffiient implies the origin where it is derived from, in this ase, the ontinuity. With this average veloity, we an determine the average stati pressure through the flux equation for the total linear momentum : π R R r r R R r I = ( ) ( ) ρ + p r dr d ϕ = ρ r dr dϕ R π R R 4 ω R ϕ, r = R r R + ρ + p r dr d ϕ ρ dr dϕ R I ρ βi, ln = ϕ ρ + ϕ, + pr = R R or, { [ ] [ ] } I K ˆp p = β β ρ (5.5) where β I = R.875 R and, [ ] p = ρ ϕ ρ ϕ,min ln p + r = R. R Next, we onsider the angular momentum of this kind of rotation : m π R r L = r d m = ρ ω R r dr dϕ R ϕ R π = β ρ R (5.6) L ϕ,min 7 π, 7 39

47 where β L has the same mathematial expression as β K, therefore, β L β K =. Now omes our trouble if we attempt to follow the strategy used in equation (5.5) for solvings ˆr and ĉ ϕ. The reason is whenever applying a similar assumption in term of the potential-vortex swirl by ĉ ϕ ω R = ; R ˆr R (5.7) ˆr our desired unknowns eventually are disappeared by its own anellation in equation (5.4). Consequently there is not any unknown left to be solved, however, we may find a way out of this trouble with the onept of an average swirling mass-flow rate. Here, we propose that ϕ l R ω R R ϕ = ρ ϕ d ϕ = ρ d d = ρ ω ln r R R m r l l R (5.8) where ϕ is the flow area that ingests fluid, and l is the length of rotating pipe. nd also, with an approah of the Kreitmeier s definition, we have ( ) m ρ ˆ = ρ l R R ˆ. (5.9) ϕ ϕ ϕ ϕ Comparison of (5.8) and (5.9) results in ĉ ϕ = ln ( R ) R ϕ,min. (5.3) With this ĉ ϕ known, we are ready to turn bak to equations (5.4) and (5.6) for determining the Kreitmeier s average radius, whih is ˆr R =. (5.3) ( R ) ln R From the balane equation of total enthalpy flux, using the averaging of Kreitmeier s method, the speifi stati enthalpy an be dedued as follows. π R ( r) ĉ ĉ H ϕ ( ) d d ˆ hˆ ϕ t = r ρ h + + r r ϕ ρ + + R or, ĥ [ ] R β β β h = + β E K E, ˆ ϕ K βk r (5.3) where β E =.7 R.477 R, 4

48 β E ln R R =.36 R R. The speifi kineti energy (reversible) an be alulated from π R ( r) K ϕ = ρ ( r) + r dr d ϕ ρ ˆ k R or, β β E E k = + ϕ,. (5.33) β K β K nd its irreversible part will be diretly obtained from equations (5.4) and (5.3), through the expression of ˆk [ ] β R ĉ = + = + rˆ ĉ ϕ K ϕ,. (5.34) 5.3 Referenes. Trukenbrodt, E. (98): Fluidmehanik I, Springer-Verlag, Berlin, nd ed., p. 7.. Ibid., p. 8. 4

49 Chapter 6 Flows through Diffuser and Nozzle This hapter fouses mainly on the flow of an inompressible fluid through a non-onstant irular area along a streamwise diretion. The enlargements and ontrations of ylindrial hannels are onerned, disrepanies in results are observed and ompared for both types of the average parameters, irreversible and reversible, whih are part of the method of Kreitmeier. 6. Fores Exerted on Curved Channels of Cirular Cross Setion urved pipe as skethed in Figure 6. is of interest. When the fluid flows through it, the solid inner wall of the pipe onfines the flow to move along the ar, hene the flow path is bent gradually or heavily as ontrolled by its radius of urvature. However, only effet of the radius of the pipe on the flow is studied rather than the radius of urvature of whih the origin lies outside the pipe. Firstly, let us all the ratio of the radii of the pipe inlet and outlet as Rin Ω =. (6.) R out When Ω =, we have a bend. However as Ω inreases, or Ω >, this means that the flow area at the inlet is larger than at the outlet. ording to the onservation of mass, the flow will be aelerated. t the outlet of the pipe, the fluid possesses higher veloity than at the 4