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"55.74;? 4'" my. w w.“- MICHIGAN I mllilillillilil\”“jiil 3 1291’: 01020 8% l This is to certify that the thesis entitled SIMPLE MEANS OF FLOW PREDICTION IN A VANELESS DIFFUSER presented by EZZAT SALAMA AYAD has been accepted towards fulfillment of the requirements for _MAS_IEES__degree in MECHANICAL ENGINEERING Major pagessor Date [0"96' 5% 0-7639 MS U is an Affirmative Action/Equal Opportunity Institution _ LIBRARY Mlchlgan State Unlverslty PLACE ll RETURN BOX to mnovo this checkout from you: ncord. TO AVOID FINES Mum on or More data duo. DATE DUE DATE DUE DATE DUE MSU It An Affirmative WM Opportunity Institution mm; ‘ SIMPLE MEANS OF FLOW PREDICTION IN A VANELESS DIFFUSER By Ezzat Salama Ayad A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Mechanical Engineering 1994 AN ABSTRACT OF A THESIS SIMPLE MEANS OF FLOW PREDICTION IN A VANELESS DIFFUSER By Ezzat Salama Ayad In designing vaneless diffusers, designers need a fast accurate way to predict the flow physics parameters without the need to go to complicated three dimensional computer codes. The work studies the simple means of predicting the flow properties in a vaneless diffuser, and assessing the validity of the assumptions in trying to solve simultaneously a series of equations relating the properties of the diffuser to the radius by using the computer softwares, Excel and MATLAB. The study focuses on the change in the properties through a vaneless diffuser with respect to the radius. Relations of all the properties are developed. The theoretical relations are determined under specified boundary conditions of aerodynamic and thermodynamic conditions. TABLE OF CONTENTS LIST OF FIGURES LIST OF TABLES NOMENCLATURE LITERATURE SURVEY CHAPTER ONE Introduction 1.1 The impact of diffusers on turbomachinery performance 1.2 Types of diffusers CHAPTER TWO Essential parameters on diffuser performance 2.1 Overall performance parameters 2.2 Aerodynamic parameters of machine performance CHAPTER THREE One-Dimensional inviscid method for Flow prediction 3.1 Classroom method 3.2 Point (*) method CHAPTER FOUR One dimensional viscous approach 4.1 Mathematical model and developing of equations 4.2 Solution of differential equations CHAPTER FIVE Order of magnitude analysis and validity of the model. CHAPTER SIX Summary & conclusion LIST OF REFERENCES iv vii 13 13 17 27 28 30 38 48 53 69 91 Figure 1 Figure 1.1 Figure 1.2.1 Figure 1.2.2 Figure 1.2.3 Figure 1.3.1 Figure 1.3.2 Figure 1.3.3 Figure 1.3.4 Figure 1.3.5 Figure 1.3.6 Figure 1.3.7 Figure 1.3.8 Figure 1.3.9 Figure 1.3.10 Figure 1.3.11 Figure 1.3.12 Figure 2.1.1 Figure 2.1.2 Figure 3.1 Figure 3.2 LIST OF FIGURES Vaneless diffuser Loss Map by J apikse Sketch of a channel diffuser Sketch of a conical diffuser Sketch of an annular diffuser Sketch of a volute diffuser Sketch of a vaneless diffuser Sketch of a return channel diffuser Sketch of a straight channel diffuser Sketch of a straight plate diffuser Sketch of a vaned island diffuser Sketch of a circular arc diffuser Sketch of a cambered diffuser Sketch of a twisted diffuser Sketch of a multiple cascade diffuser Sketch of a conical diffuser Sketch of a low solidity diffuser Diffusion on the h-s diagram Ideal pressure recovery vs. swirl angle Elementary view of velocities in the diffuser Radial velocity vs. the area iv \oxoxoooooqoxunha, NNv—tr—hr—Av—Ah—tt—Ir—tr—hr—I \DxltnUJNr—v—v—OOO Figure 3.3 Figure 3.4 Figure 3.5 Figure 3.6 Figure 3.7 Figure 4.1.a Figure 4.1.b Figure 4.1.c Figure 4.1.d Figure 4.2.a Figure 4.2.b Figure 4.2.e Figure 4.2.d Figure 4.2.e Radius ratio vs Mach number Change in angle alpha vs. Mach number Pressure ratio vs. the velocity in the diffuser Pressure vs Mach number for a radius ratio of 1.4 Pressure vs Mach number for a radius ratio of 1.6 Change in Mach number"2 vs. radius ratio for the three cases Change in pressure vs. Mach number vs. radius ratio for the three cases Change in swirl angle alpha vs. radius ratio for the three cases Change in the flow path vs. radius ratio for the three cases Change in diffuser height vs. radius for a design problem Change in Mach number"2 vs. radius ratio for a design problem Change in angle alpha vs. radius ratio for a design problem Change in pressure vs. radius ratio for a design problem Change in flow path vs. radius ratio for a design problem 31 33 35 36 37 53 54 55 56 59 6O 61 62 63 Table 1 LIST OF TABLES Influence coefficients 48 NOMENCLATURE Ar—abig A Flow area AR area ratio C Absolute velocity Cr Velocity component in the radial direction Cu Velocity component in the tangential direction Cz Velocity component in the z-direction Cf Skin-friction coefficient Cp Diffuser pressure recovery c Local speed of sound D Diameter dQ Heat transfer rate from the fluid H Non dimensional effective passage height, h/hT h Effective passage height h' Coefficient of heat transfer K Loss coefficient M Local Mach number N Rotational speed parameter n Polytropic exponent P Pressure ratio, p/po vii R* R r,9, z T Static pressure Radius radius ratio, r/rT perfect gas constant cylindrical coordinates in the r,6, 2 directions static temperature Tw Wall temperature firsek or 9 flow angle tangential direction, half-divergence angle of conical or channel diffusers 20 Full divergence angle * Boundary layer thickness density r E, non dimensional friction coefficient , c f [31-] T 11 efficiency u slip factor 1: shear stress due to skin friction to Rotational speed subssripts 1 inlet 2 outlet i ideal 0 total viii t local r radial direction u tangential direction LITERATURE SURVEY Due to the widespread use of the vaneless diffuser for centrifugal pumps and compressors, computational techniques for the vaneless diffuser emerged early. One of the earliest efforts to predict the performance of the vaneless diffuser was reported by Brown. In this work a modified Bernoulli equation Was written in the direction of the streamline for a constant area vaneless diffuser, and a skin friction coefficient was introduced. By comparing the measured wall static pressures with predicted results, a suitable value for skin friction was established. This work showed reasonable agreement between the theoretical model and measured results and could be considered a one-dimensional single parameter model . In subsequent work, Stanitz(1952), introduced a comprehensive treatment of the vaneless diffuser( Fig.1), and published a correct set of one- dimensional equations for the radial momentum, tangential momentum, conservation of mass, conservation of energy, and the equation of state. In this case a single empirical parameter was still employed but was introduced independently to both the radial and tangential momentum equations. Calculations of this type were reported by Johnston and Dean (1965). A further comparison of this computational approach was reported by Japikse. Fauldres (1954) reported results that showed that the skin friction coefficient in the inlet (developing) region of a vaneless diffuser is quite high compared to fully developed flow, but the fully developed flow level is x / Vaneless difquEr Fig. 1 Vaneless Diffuser approximately correct near the end of the vaneless diffuser. Thus the process is quite similar to pipe or channel flow where large values are to be expected in the inlet region, and the fully developed value only applies well downstream. Average values, as implemented as industrial standards, tend to fall higher than the fully developed flow level. W. Traupel (1977) has taken this analysis procedure further and has introduced a two-parameter modeling system which adds the mechanical energy equation to the system of equations established by Stanitz. Dean and Senoo developed a theory for the radial impeller discharge mixing process including wall friction, friction between wake and jet, and reversible work exchange. In their flow model, the flow out of each impeller passage was divided into two regions, the wake and the jet. The relative velocity was constant in each region with a high velocity for the jet and a low velocity for the wake. The mixing process of the jet and the wake was calculated by solving the momentum and the continuity equations for the jet and the wake simultaneously. They showed that a large part of the loss at the vaneless diffuser inlet was due to wall friction loss of an asymmetric flow. They also concluded that the asymmetric flow pattern became uniform rapidly by the reversible work exchange between jet and wake. Johnston and Dean presented a simple analysis based on an assumption of sudden expansion. In their flow model, the wake and the jet were mixed up at the inlet to the diffuser by by the sudden expansion. They compared their calculations for various sudden expansion and the flow was thereafter assumed axially symmetric in the diffuser. The large total pressure loss at the vaneless diffuser inlet was attributed to the mixing loss centrifugal blower outlet flows with those based on the Dean and Senoo model. Their simple model gave very similar predictions of total pressure xii loss to those predicted by the more precise method over a wide range of compressor parameters. Senoo and Ishida found that the flow at the exit of a centrifugal blower is artificially distorted so that decay of the asymmetric flow in the vaneless diffuser is experimentally examined. It is concluded that the shear force between the wake zone and the jet zone does not play the major role for the behavior of flow in the vaneless diffuser, and the behavior is mainly controlled by the reversible work exchange. Based on the above survey, we will present in this work a documentation of some simple means of flow predictions in a vaneless diffuser. Due to time limitations, we will only present a one-dimensional approach using computer software packages Excel and Matlab. CHAPTER ONE Introduction The design of turbomachinery is dominated by diffusion-the conversion of velocity or dynamic head into stream pressure. Every blade row in a typical axial compressor is a collection of parallel diffusers. In most centrifugal compressors both the rotor and the radial diffuser are limited by the diffusion capabilities of the flow channels. 1.1 The impact of diffusers on turbomachinery performance In the first decade of this century a strong debate raged in the academic society as to the practical utility of placing an exhaust diffuser downstream of a hydroelectric turbine. Experts argued back and forth whether the exhaust diffuser would, or would not, improve the performance of the turbine. The counter arguments essentially maintained that the fluid had already left the turbine and little good could be done; the proponents recognized the importance of increasing the expansion ratio across the turbine rotor by the reduction in rotor back pressure with the use of a well- designed diffuser. Today' 5 arguments and concerns over the role of the diffuser are significantly more advanced. Nonetheless, the details of the diffuser design and performance are in some instances as vague as the early debate on 1 2 diffuser application for hydroturbines. Fluid machinery is conveniently divided into positive displacement and turbomachinery categories. The distinction follows directly along the lines of Newton' 5 Second Law of motion as applied either in Cartesian coordinate system or in cylindrical coordinate system. In the Cartesian coordinate system, Newton' 3 Second Law indicates that the force applied to an object will be equal to the change in linear momentum which is the basic principle behind positive displacement equipment. In the cylindrical coordinate system, Newton' 5 Second Law expresses the torque being proportional to the change in angular momentum. This principle leads directly to the Euler turbomachinery equation which expresses the energy transfer through turbomachinery as the change in UCu. Thus, the inherent function of turbomachinery involves the exchange of significant levels of kinetic energy in order to accomplish the intended purpose. As a consequence, very large levels of kinetic energy frequently accompany the work input and the work extraction processes, often on the order of 10-50% of the total energy transferred. Thus efficient diffusers are absolutely essential for good turbomachinery performance. With kinetic energy intensities of this level at the exit of the impeller; it is not hard to appreciate that the performance of a diffuser influences the overall efficiency level of a turbomachine. Thus the detailed processes that occur in diffusing elements must be carefully understood and thoroughly optimized if good turbomachinery performance is to be achieved. The range of diffuser performance levels can be appreciated by considering Fig 1.1. This diagram (by J apikse) presents a loss map which plots the loss coefficient K in a typical diffusing element versus the pressure recovery for the same diffuser. Maps of this type serve to focus many 3 important facts concerning diffuser performance in a compact fashion. One can immediately realize the impact of area ratio on diffuser performance, and high level of recovery will only be obtained if the area ratio is sufficiently high. There is also the implicit requirement that the diffuser must be designed so that the effectiveness is quite high. 1.2 Types of diffusers 1.2. 1. Overview In chapter 4, it will be shown that the basic equations of motion reveal the importance of both geometric and aerodynamic parameters on the ultimate performance of a diffuser. The specification of a wide variety of geometric and aerodynamic parameters is essential before the performance of a diffuser is uniquely given. In this section, the various geometric parameters are first reviewed for all classes of diffusers. A general definition of the different aerodynamic parameters is given in the next section. 1.2.2 Geometric Parameters: 1.2.2.1 Channel diffuser geometric specification The geometric specification of a channel diffuser is, at first appearance, comparatively straightforward. A simple schematic of a channel diffuser is shown in Fig 1.2.1, and the essential parameters which must be considered are defined in this figure. From these different geometric parameters, dimensionless parameters are formed as follows : W1=throat width, b =throat depth 5 L =centerline length, 20=divergence angle At=throat area=bW1, Ae=exit area=bW2 Sketch of Channel Diffuser Fig. 1.2.1. non-dimensional length W 1 aspect ratio AS = b 1/W 1 area ratio AR = A2/A1 Not all these geometric parameters are independent. There is a fixed relationship (correlation) between the area ratio and the other geometric parameters as follows : AR = l + 2 (W 1) tan 9 1.2.1.3 Conical diffusers The definition of the basic geometric parameters for conical diffusers is quite similar to that given previously for a channel diffuser, as illustrated below in Fig 1.2.2. Again, it is possible to define various dimensionless parameters as follows: D=throat diameter L=centerline length 29=divergence L angle At=throat area Ae=exit area 5 ._ +I throat exit Sketch of Conical Diffuser Fig. 1.2.2. dimensionless length [JD 1 area ratio AR = A2/A1 The area ratio for a conical diffuser is, of course, dependent on other geometric variables. The dependence is given as follows : 0AR=(1+2(L/D)tan® )2 1.2.1.4 Annular diffusers It is more difficult to define the essential geometric parameters for annular diffusers since the number of independent variables has increased. Here the essential variables are: 7 fl ' h “01— BM ___________ fl Straight Core Annular Fig. 1.2.3. non-dimensional length U A r or L/h1 area ratio AR = A2/A1 Consider Fig 1.2.3, where the types of straight wall annular diffusers are shown. For the equi-angular case, the geometry is specified by: AR = 1+2 (L/h) sin 9 For the straight core (constant hub) case the equation is more complex and becomes: r 11/4 V 1 _ 1 2L sinO L2 sin2 9 ( 0 j R. + 112 f R. N h 1+ )4 1+ % 0 K 0/, L AR=1+ and for more complex, but more common, case of independent changes in 6 land 6-) 2 one obtains: sinO +Iysin9 l—Iy (sinZO -sin2(~) ) Ml ._ h [1+%] hz (”1%) Unfortunately, many annular diffusers are even more complex and include curved walls. For such cases, the AR to L/h relationship must be derived for each specific case. 8 1.3 Examples of Common Diffusing Systems for centrifugal pumps or compressors Fig 1.3.1 Volute diffuser VOLUTE--used for low pressure ratio, mostly for single stage, and has radial thrust at off design flow. Even though widely used, still needs researching. Fig 1.3.2 Vaneless diffuser VAN ELESS DIFFUSER--tolerates large range of flow angles (60-80 degrees); simple annular channel, but bulky. Fig 1.3.3 Return channel RETURN CHANNEL--employed in multistage applications, yet much is not known about its performance. E Fig 1.3.4 Straight channel STRAIGHT CHAN NEL/WEDGE—-has simple geometry, easy to manufacture ; very popular, but large in size. E? 3 Fig 1.3.5 Straight plate STRAIGHT PLATE-has large number of vanes, Z>30, and not so good pressure recovery. Fig 1.3.6 Vaned island VAN ED ISLAND-is a refined straight channel for high pressure ratio and Mc3>1; has good pressure recovery, but again large in size. 11 Fig 1.3.7 Circular arc CIRCULAR ARC--has simple geometry, but no outstanding aerodynamic characteristic. Fig 1.3.8 Cambered diffuser CAMBERED/AEROFOIL--used for transonic and subsonic applications, small size and good pressure recovery. Its design is based on axial cascade data. 12 l _ _I Fig 1.3.9 Twisted diffuser I WISTED--is a refined cambered vane to produce good efficiency, wide range and high pressure ratio. l _I Fig 1.3. 10 Multiple cascade MULTIPLE CASCADEnis a cambered van in cascade for higher efficiency with more manufacturing process. Fig 1.3.11 Conical diffuser 13 PIPE/CONICAL--used for higher pressure ratio and transonic applications, large in size, but with good pressure recovery. \ Fig 1.3.12 Low solidity diffuser LOW SOLIDITY—-used for low flow angles, but with good pressure recovery and range. Currently of great interest. CHAPTER TWO ESSENTIAL PARAMETERS OF DIFFUSER PERFORMANCE In this section we will refer to general types of diffusers, Radial diffusers will be discussed in section 2.3. 2.1 Overall Performance Parameters 2.1.1 Ideal Pressure Recovery The pressure recovery of a diffuser (actual or ideal) is most frequently defined as the static pressure rise through the diffuser divided by the inlet dynamic head; in other words: Cp=(p2—%>2‘P1) which is a very simple way of thinking about the fundamental purpose of a diffuser. An ideal pressure recovery can be set if the flow is assumed to be isentropic and the Bernoulli equation is used both in the numerator and the denominator to reduce the expression to a velocity in and a velocity out. Then, by employing the conservation of mass, the relationship can be converted to an area ratio for incompressible flow. We obtain the following: 0 = - 2 2 = _ 2 Cp 1 vexit/vinlet ”MR 14 15 This expression is very well known to most engineers, and it does show the ideal pressure recovery as a useful reference level. However, it can also be used to deduce some very important functional relationships. For example, in an annular diffuser, a number of different variables can influence the variation of pressure recovery under the conditions of swirling flow. Thus if we write a general expression for the ideal pressure recovery in an annular diffuser, with inlet swirl, one obtains: 2 2 2 2 tan a1+ b1 /b2 tan2a1+l This equation shows that diffuser inlet to diffuser outlet radius ratio is very r1 r2 Cpi=1~ important if high recovery is to be achieved. It also shows that the inlet to exit passage depth ratio plays a role. The swirl term, in practice, can only be suppressed by designing a diffuser with large radius ratio; another way of saying the same thing is to realize that the swirl component must be recovered in accordance with the law of conservation of angular momentum: 9=constant. The above expression shows maximum recovery with respect to swirl angle (an/aal = 0) when b1/b2=1; this result is independent of a. In fact when b1/b2 tanOt dR* Mz—secza 2 1; 1 dH M2 1+ —1 M ) -—-———— .( (Y ) Hcosa H dR* R* J Equations (14a), (14b) and(15c) are three differential equations that can be solved simultaneously for T , Mzand or. 0 Pressure After the variations in T 0’ M2and or with radius R* are known, the pressure P can be obtained from the continuity equation [31C1 cosorerbT = pC cosarb where the subscript 1 refers to known conditions at the diffuser inlet. From the equation of state and from the definition of Mach number P _1_ -2 T M1 T1 cosorerbT — TMx/Tcosoub 1 finally from equations (6) and (13e) 7-1 2 C080. M T0(1+ M1) 33—: 1 1 1 2 (17b) :1: _. P1 RHcosa M T (1+7 1M2) 01 2 Equation (17b) determines P from the known conditions at the diffuser inlet and from the known values of T 0, M2and or determined by the simultaneous solution of equations (14a), (14b), (15c) and (16). 49 Flow Path The flow path on the mean surface of revolution in the vaneless diffuser can be obtained from the known variation in tana with R* given by the solution of equation (16). R*do tan or = dR* or (K: = tan or (18) dR R* because the angle alpha is a known function of R, equation (18) determines the flow path. Influengs ggeffisient In some analysis problems it may be convenient or desirable to solve directly for one or more of the other dependent quantities rather than To, M"2, and alpha. Also, in the design problem, it may be desired to specify one of these quantities as a function of R and solve for the required value of 1/I-l*dH/dR. For these cases the change in the dependent variables P, Rho, T, C, Cr and Cu with radius R along the mean surface of revolution, as well as the change in To, M"2, and alpha, must be expressed in terms of the known quantities 1/T0*dT0/dR, g —1— 9g —1- which quantities are H cosa ’ H dR ’ R multiplied by influence coefficients. Thus, if X is any one of the dependent variables, (MZ—secza)l§=l (1+1'—1M2)—1—d—T—9+1 g XdR 1 2 To dR 2Hcosoc ldH 1 +1 ——+1 — 3HdR 4R where 11 through 14 are influence coefficients that are determined in the same way that equations (15c) and (16) were developed. The influence coefficients for various dependent variables X are given in the following 50 table: Tabble l-Influence coefficients x Influence Coefficients Ir I2 13 14 2 P 7” 11+ <7-1>M217M '7”: ‘Wzmz" p secza A42 (7secza- tanza) —M2 -Mzsecza M2 _ _ 2 T 1&2“ (y_ 1)M2(7M2_m2G) (7 1)M2 —(7- l)M286c a 2 2 _ 911 (7-1) 2 (7— 1) M tana 1 2(1: 2 M2) 2(l+—2—M") 2(l+—2—M2) 4M2 (1.. mm) 5.3. c:2 —2 2 [maze-7M2) 2 2 secza q -secza Mzban‘a—7seczaj secza secza-i-Mltanza Cu 0 sec a-—M2 0 seczot—M2 tana seczot sec‘a[1+(7—1)M2] -secza —Mzsecza Small stage efficiency mean surface of revolution in a vaneless diffuser is defined as the ratio of the ideal (ignoring friction and heat transfer) to the actual differential change in static enthalpy with radius required to accomplish the actual differential change in static pressure with radius. This definition leads to the following The small stage or polytropic efficiency at a given radius R on the expression for the small-stage efficiency 11. 51 1‘13. .1: PdR 2 (20a) PdR 7—1 2 To dR Hcosor Equation (20a) indicates that in the absence of heat transfer (dTo/dR=0) and friction (§=0), the small-stage efficiency is 100 percent. Also, for heat transfer from the fluid to the diffuser walls, (1/T0*dT0/dR) is negative and therefore results in an apparent increase in the small-stage efficiency. Thus, in the presence of heat transfer, the small-stage efficiency, as just defined, is not a good measure of the performance of vaneless diffusers in that it is not a measure of the magnitude of the losses involved. In the absence of heat transfer (dTO/dR=0) and equation (20a) reduces to 1";(M2 - sec2 or) n = 1 — (20b) 2 2 2 ) dH Hsec or 7M —tan or —cos0t + g( [dR R J Numerical Procedure In the analysis problem , the variation in fluid properties with R are determined for a specified geometry of the vaneless diffuser. In the design problem, the variation with R in one of the fluid properties is prescribed and the remaining fluid properties together with the variation in diffuser height H with radius R are determined. In both approaches we shall use MATLAB to solve for the different parameters, given the three unknown equations (14a) or (14b), (15c) and (16). We shall first solve for the three unknowns To, M"2 and the angle alpha. For this numerical example R varies from a value of 1 to a value of 2. 52 After the distribution of To, M"2 and the angle alpha with R Have been determined the distribution of P, p , T, C, Cu and Cr can be determined from equation (17b) and (4), (5), (6) and the equation of state (10a). M The flow path on the mean surface of revolution in the vaneless diffuser is given by e as a function of R along the surface. Because tanoc is a known function of R, the flow path (O=(~)(R)) can be determined by the integration of equation (18) assuming 6:0 at R=1.0. Design Problem In the design method, the variation in effective diffuser wall spacing with radius is determined for a prescribed variation in one fluid property. For efficient diffuser designs the selection of the one fluid property and its optimum prescribed variation will depend on viscous flow effects that are considered in boundary-layer studies. In the design problem the variation in H with R is unknown and must be determined to satisfy a specified variation in one characteristic of the flow (Cr, for example) with R. For this specified variation in one characteristic of the flow fig can be determined. Again using Matlab, we solve for H as a function of R, with H=1 at R=1. Numerical Examples The numerical examples are divided into two groups: 1- effects of some operating conditions 2- vaneless diffuser design problem The first group of numerical examples shows the effects of heat transfer and friction on the flow in vaneless diffusers. Three numerical examples are given: 53 - Isentropic compressible flow - Compressible flow with friction - Compressible flow with friction and heat transfer. t n ' ' n : For the first group of numerical examples the flow conditions at the diffuser inlet (R=1.0) are: P1 = 3.022 M12 = 1.37 (To)1= 941 °R (tan «)1: 3.829 These conditions were estimated for the following design and operating conditions of the impeller : Compressor flow coefficient, (1) 0.75 Impeller tip Mach number, MT 1.5 Impeller slip factor, 11 0.9 Impeller polytropic efficiency, 11 0.9 Compressor stagnation inlet temperature, To, ° R 520 Diffuser gesign: The design characteristic of the diffuser are : Passage height 1/R Wall temperature, Tw 750 Friction parameter, E 0.03 Results The results of the first group of three numerical examples are given in figure (4.1). In Figure (a) the change in Mzwith R is shown for the three numerical examples. The effect of friction is to reduce Mzat each R, and 54 the effect of heat transfer from the fluid is to increase M2 slightly (primarily because of the reduced speed of sound at the lower temperature) for the magnitudes of To and Tw involved in these examples. In Figure (b) the change in P with R is shown. As expected the effect of friction is to reduce P at each radius (primarily because of the decreased values of Cu, which require a smaller pressure gradient for equilibrium). The effect of heat transfer from the fluid is to raise P slightly or the magnitudes of To and Tw involved in these examples. In Figure (c) the change in flow direction awith R is shown. The effect of friction is to reduce or because Cu is reduced and Cr is increased to satisfy continuity with lower density due to lower P. The effect of heat transfer from the fluid is to increase or slightly because of the reduced value of Cr resulting from the increased value of p. In Figure (d) the flow path in the vaneless diffuser is shown. The effect of friction is to shorten the flow path because Otis decreased (figure (c)) . The effect of heat transfer is to lengthen the path slightly. A Vaneless Diffuser Design Problem The second part of the section on numerical examples is a simple vaneless diffuser design problem. The design variable in a vaneless diffuser is H: H(R), and the design problem will be to determine H(R) for a prescribed variation in Cr. For purposes of demonstrating the design method it is assumed that the deceleration of Cr, is the criterion for boundary-layer separation in a vaneless diffuser , so that a safe rate of deceleration is Mach number"2 1.4 I F T I I I I I f ...Isentropic —-Friction 1.2 - . --Friction & Heat transfer 1 .. 0.8 ~ 0.6 ~ 0.4 >- 0-2 I 1 l I 1 l I I I 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 Radius ratio Figure 4.1.a Change in Mach number"2 vs. the radius ratio 55 6 I I I I I I I ..lsentropic --Fn‘ction —Friction 8r Heat transfer 5.5 Pressure A 01 3.5 3 l l I I l l l l l 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 Radius ratio Figure 4.1.b Change in pressure vs the radius ratio 56 Angle alpha 3‘ 76 r I i I i i r i u I ........... 75 \I A \J O) r \1 to r ...lsentropic -—Friction 69 - -Friction 8 Heat transfer 68 - 67 1 l 1 1 I 1 1 l 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 Radius ratio Figure 4.1.c Change in angle of swirl alpha vs. the radius ratio 57 902 60 1.6 120 . 1 .2 ...lsentropic 150 \ 30 --Friction 0,3 ——Friction 8! Heat tra sfer . \ O €( \‘9 Figure 4.1.d Change in flow path vs. radius ratio 58 59 5 dC — —L = —0. 05 C dr r where 8 is proportional to the boundary-layer thickness. For purposes of the design example we assume 5 is equal to b/2, which is the effective thickness of a fully developed boundary layer in the vaneless diffuser. Thus, mm“, Cr dR rT and 1 dCr_ 1 "CTR“? r if b—T is equal to 10. Because of the assumption involved, this specified T dC variation in j— with H may have no practical significance with regard to vaneless diffuser performance and has been selected only to demonstrate an application of the design method. It should be pointed out that design variations in H affect primarily the velocity component Cr and through this component the flow direction or. Inlet cgng'tigns The impeller design and operating conditions are the same as for the first group of numerical examples and so the diffuser inlet conditions are the same P1 = 3.022 M12 = 1.37 (TO)1= 941 o R (tan (1)1: 3.829 60 mm The variation in H with R is to be determined. Heat transfer effects are neglected, and the value of the friction parameter is the same as for the first group of numerical examples(0.03). Mtg The results of the design problem are given in Figure (4.2). In the figures is shown the variations in H, -Cl— an, M2 , P, on and n with radius R. as r dC specified , E— at is equal to -1/H. In order to accomplish this variation, r H at first decreases with increasing R and then increases to approximately its initial value at R eqUal 2. The variation in on with R was slightly more than 3 degrees so that the flow path is approximately a logarithmic spiral. Height 1.02 0.98 0.96 0.94 0.92 0.9 0.88 0.86 0.84 A L 1 J_ l l l L l 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 Radius Figure 4.2.a Change in diffuser height vs. radius ratio for a design problem 61 1.4 1.2 Mach number“?! 0 a: 0.6 0.4 0.2 1 l l I l l l 2 1 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 Radius Figure 4.2.b Change in Mach number"2 vs. radius ratio for a design problem 62 Alpha angle 79.5 79 78.5 \I m V .‘1 0| \1 \1 76.5 76 75.5 L l l l 75 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 Radius Figure 4.2.c Change in angle alpha vs. radius ratio for a design problem 63 Pressure 1 .5 Radius Figure 4.2.d Change in pressure vs. radius ratio for a design problem 64 902 120 60 180 ‘ ; 210 330 240 300 270 Figure 4.2.e Change in flow path vs. radius ratio for a design problem 65 CHAPTER FIVE Order of Magnitude analysis & Validity of the one-dimensional model In this chapter, we shall present the full set of equations and conduct an order of magnitude analysis, to determine whether the terms canceled based on our previous assumptions could affect the solution. We shall first start by presenting the continuity, r-momentum and G-momentum including the unsteady term due to the start-up: Mass equation a a 1 l ;$(rcr)+;%(cu)=o.o r-momentum a la 192Cr 2 _ __ rC _ 3%....(3 £+Cu acr_Cu =_§£+ ar(rar( r))+r2 392 at I' at 1' 89 I' at 2 ac __ u _ r2 89 66 67 B-momentum we non-dimensionalize the equations with the following scaling Cu*=Cu/2 nan) t*=t/ts R=r/b 0 *= 92 1t Cr*=Cr/Cs P*=P/Ps we need to use the equation to tell us Cs,Ps Substituting into mass we obtain C81 6 121mm1 au ___( v) _ _ = b RaR b 21: 1139* 01' C * Ali RV*)+iau =0 awRBR R80* to keep both terms we must have C:s — as 0(1) are which gives C =aco s Now substituting into r-momentum 68 a0) 8v + aza)2 3v + 21ta2m2 11 av 41t2a2m2 u2 1 PS BP* tsar* b vaR 21th Rae b R pb 3R vfli(li( V} am 32v_2a(o21t13u b2 3R RBR 41:21)2 392 2an R2 39* we now divide by 4n2a2m2 /b bavlavluavu2 P331” + v +— — — 41t2acotsat* 41t2 3R 41:2Rae R 41:22:2me 3R 6 [ii(Rv))+;§3:_;i 8R R311 4,,2 392 R2 39* + v 41t203ab Now let us look at the B-momentum 21mm Bu 21ta2m2 8v +411:2a2(o2 u Bu ——+ ts at’“ b vaR 21th R39 F21mm) 3(1 3 ) 1 + -—(RU) +2na2w2vu PS BP“ b 3R RBR _— +V 2 b R 21tpr 39 21mm 3 u 2am 8v 2 2 2 + 2'5“ -41: b as 2R1tb 9- 21ta2rl)2 Now divide this equation by b an 3.. u an vu PS aw —+ —+——+—=— + acotsat“ Van R39 R mzpmzR ae (nab 8R RBR 4,,2 392 2an ae Now to decide on which equation to use to scale the pressure and the time. First note that each momentum equation would give us a different expression for ts and Ps if we simply set the coefficients equal to one. 69 In the O-direction we would expect that the time rate of change of momentum would be balanced with inertia and viscous forces, but not necessarily the pressure. Since the inertia terms are of order one we make is of order one and t = b arms 3 A0) 1 We note that the last two terms of the viscous term are of order— -—2 , and 41; can be neglected. For Ps, we consider the r-momentum equation, and note that inertia can be balanced with the pressure term. Then Ps 2 2 2 $2 isoforderoneand PS=41t a pa) 41: a pro l . note that the transient term becomes of order— 2 , and all the VISCOUS terms 41; of order %n , %{e so they can all be neglected. Our scaled equations become : continuity: 18u_ Rv R 1315" )+ R 86 r-momentum: i _ a_P R ER G-momentum : at vaR Rae R=ReBR RaR From these scaled equations, we can see that our previous assumption that there is no variation of any of the properties with time(steady state), proved to be partially wrong and that there is a time dependency of the 70 tangential velocity which will affect our results to a certain degree. Also our assumption of asymmetric flow, there is a variation of the tangential velocity with the 6 direction which also adds some inaccuracy in our model. Further study in that area is recommended to investigate to what extent our one dimensional model' s results vary from that of a two dimensional that includes all the time and tangential variations that were not previously included in our model. CHAPTER SIX Summary of Results and Conclusions This chapter concludes the work with a summary of the findings and a statement of the conclusions. Summary The work shows some simple means for flow predictions, some of the methods presented earlier in this work are very simple because most of the affecting terms were eliminated by assumptions and are only good for classroom purposes. In later chapters,analysis methods have been developed for one dimensional model that takes into account the compressibility, friction, heat transfer, and area changes in vaneless diffusers. In the analysis method, the variation in fluid properties, including the velocity and flow direction can be determined as a function of radius for a prescribed variation in diffuser height with radius. In the design method, the variation in diffuser height and all fluid properties except one can be determined as a function of radius for a prescribed variation in the one fluid property. For efficient diffuser designs the selection of the one fluid property and its optimum prescribed variation will depend on viscous flow effects that are considered in boundary-layer 7 1 72 studies. Three groups of numerical examples are presented in which the effects of friction, heat transfer, and diffuser height are investigated; and a simple design problem is presented. As a result of these examples it is concluded that: 1- Heat transfer from the fluid has the opposite effect of friction on pressure rise in vaneless diffusers and is therefore to be desired. On the other hand, heat transfer to the fluid has the same effect as friction and is to be avoided. 2- If the friction coefficient is unaffected by the diffuser height, and if flow separation does not occur, the diffuser efficiency is slightly improved by increasing the diffuser height. 3— With relatively low friction coefficients and neglecting mixing losses at the impeller tip, the friction losses in most vaneless diffuser designs are considerable, as indicated by computed diffuser efficiencies in the low 80's , and these losses result from the usually large ratio of wetted surface to flow area in vaneless diffusers. APPENDICES APPENDIX A Momentum in the r-direction Cracr _ C3 _ 1 aP 8r r par Momentum in the (*)-direction 8C C Cr ‘51; + CI —I'l1- = _C_u _ dCu r _ ar reduces to 911 _ _ acu r - 3r Substitute (1) into (2) BC BC 1 3p __1' J = __ Cr 81- + Cu at at which becomes 8P F+cracr + cuacu = 0 the flow is adiabatic and frictionless then ds=0, and 74 or 1 p2 =_RT2 P2 Substitute into (3) to obtain but we have cracr + CuBCu = %d(C? + (2121) = %d(C2) = CdC Integrate from P/P2=1 to P/P2, and from C2 to C P r— _l C I2(1>)yd(1>)=_ICdC _=1 2 P2 evaluate at the end points _ 'Y _ 1 _ _L _P_ Y _ ___1_ 2_ 2 y-l (P2) 1 ‘ 2RT2(C C2) Further reduced with C M:— w/yRT finally becomes 7 1 L - Y__-_ 1 2 C2 Y— (P2)_[l._ 2 M,[. 3]] .1. Recall from velocity triangle ~C2=C2+C2 . C2 = C2 + c2 2 r2 U2 also from continuity equation with the diffuser thickness (b) as a constant -C = ——(3 =rC r 9 r2 r2 D where l. = _r_ D r 2 from the velocity triangle at the impeller exit (diffuser inlet) 0 Cr2 = C2 smut2 0 Cu2 = C2 cosor2 substituting into equation (8) we get 7 _ i. =1+——Y1M2(1-‘I:2 sin2 or - 7:2 cos2 0t ) 1:2 2 2 2 2 Consider the vaneless space between r2 and r3, with a constant b 76 from Continuity prCr = const Angular momentum conservation Cur = const Usually the flow leaving the impeller is supersonic M2>1, and the flow leaving the vaneless diffuser is subsonic M3<1. Denote the radial position at which M=1 by r* and all properties at this position by (*) Cr =Ccosoc continuity equation prC cosor = p’r‘C' cosa' Angular momentum equation rC sina = r*C* sinr Dividing (3) by (2) tan or = tan or’“ * o 9 Assuming frictionless adiabatic flow ds=0 Talc p* Y-1 7%?) From the energy equation for the case of M=1 2T T = ——0 y + 1 back to Tar: par: 7‘1 _T— = [—9—] rearranging _L p- - 1 2 7—1 ] 1+— —1M p 7+1 TO l... .. Recall substituting (3) into (7) we get _1_ tanor = tana*[i[1+[Y—_—1)M2 H151 (8) 7+1 2 (1*can be evaluated by substituting or = or 2 and M = M 2 which are set at design also from (3) 78 substituting for T/T* 1 3k . 3k _— — 2 ——r Sin“ =M —2 (H(L 1)M2] (9) rsmor 7+1 2 to determine r* substitute r=r2 and M=M2 to determine or 3 from a known M3 use equation (8) to determine r3 from a known M3 and 0t 3 use equation (9) APPENDIX B By definition of the Mach number 2 M2 = C... (5) 7R T Differentiating both sides of the equation with respect to r 2 :1: (1C :1: 2dT T———— c — dr 2 (WT) Dividing both sides by C"2 2 * (YR T) M2 _ 7R*T dc2 _ 7R*Td_T (:2 dr (:2 dr T dr weget 1 sz 1 dc2 ldT = _ _ _ 5a M2 a. C2 dr T dr ( ) By definition of the total temperature To=T(1+Y—2——1M2) (6) Differentiating both sides of the equation with respect to r _ _ 2 fl=(1+—7 1MZJd—T+1‘(Y 1) 1 W dr 2 dr 2 M2 dr 79 80 Dividing both sides by T(l + 'Y_;1_ M2 ) (7-1) 2 —— M _1__dT° = ld_T + 2 (6a) _ 2 Todr Tdr (1+71M2)M dr from 5 from6a 7-1 2 — M _1_d_T_1dTo (2) 1c1M2 ‘77—" _ 2 Tdr Todr (1+71M2JM dr 2 from5,6a 7-1 2 — M 1dC2_1dM2+idT_o_(2) lsz c2 dr M2 dr Todr (1+y_;_l_M2)M2 dr rearranging ( (7-1)2\ 1c1C2 1dM21_ 2 M +11110 2 _ 2 - C dr M dr \ (1+7 1M2) To dr ) simplifying _1__d£2_= 1 1 “War—Lg (6b) C2 dr ”YT—IMZ M2 dr To dr From continuity equation we have pCrrb = constant Differentiating both sides of the equation with respect to r 81 d(pC,.rb) = 0 dc %rE(rCrb) + —dr_r(prb) + $(pCrr) + pCrb = 0 Dividing both sides by pCrrbwe obtain dC ld_p+_l__£.+.l_+-l__d_b.=0 (7) pdr c dr r bdr The differential pressure forces (opposed to the direction of Cr) are equal to the differential change on the end forces on the particle minus the component of the differential side forces on the particle in the direction of Cr. dP Differential pressure forces = (P + Edrjrdeb - Pr d8!) = 21: rdrdGb dr where the component of the differential side forces in the direction of C1- (last term in the equation) is equal to the pressure P multiplied by the projected area (in the direction of C r) of the side surface of the particle. The differential shear stress (1') on a diffuser wall is opposed to the direction of C and is given by 2 T = C EC— f 2 where c f is the skin friction coefficient. The differential shear forces in the radial direction on the fluid particle in Fig. are opposed to the direction of Cr and act on both walls of the diffuser. Differential shear forces = 2Trd®drcosa 2 = 6 PC f 2 = przrdercosa 2rd®dr cos a 82 The acceleration of the fluid particle in the direction opposed to Cr is made 2 C up of: 0 the component of the centripetal acceleration _u_ I' dC 0 the negative of the acceleration 7r- differentiating with respect to r instead of t . dCr =dCr gzc dCr dt dr dt T dr The differential force required for acceleration of the fluid particle becomes -=mxa - =pV x a (23 dcr . . . - =pbrd®dr — — Cr F (drfferentlal force requrred for r acceleration in a direction opposed to Cr). the sum of the differential pressure force and the shear forces must equal the force required for acceleration 2 C dC ' l-d—Pbrd(-9dr + c pC2rd®dr cosa = pbrd®dr[—u — Cr _r] p dr f r dr dividing both sides by (brder) we get the equation of radial equilibrium p dr b r r dr To get the tangential equilibrium equation the differential shear forces = 2trd®dr sin or substituting for the shear coefficient 2 9: 2rd®dr sin or Cf which gives 83 cprZrdersinor the tangential acceleration of the fluid particle opposed to the direction of Cu consists of 0 the negative of the tangential acceleration C dt r 0 the negative of the Coriolis equation 2C C u r r But E El}. -ldcu_gu_$ dt r r dt r2 (it changing the dependency of the right hand side of the equation to r instead of t 1[C0]=lfc_u$_529£ dt r rearranging d C C dC C _ __u_ = r u _ u dt r r dr r substituting into the equation d [Cu] 2Cucr {Cr dcu _ CrCu ]_ 2CuCr dt r r r dr r2 r rearranging d C 2C C dC C C ...1-_ __L1. ___1l_£_=_c __i- u f dt r r r dr r Acceleration is defined as mxa 84 or pV x a substituting for each term C —pbrd®dr[ r r ] equating the acceleration toC the differential shear force —pbrd@dr[Crd: —+— urcr]=cpr2rd®dr sinoc we finally get 2 . c C sm or dC C C _ f = C u + u r (9) b r dr r Equation of state P = pRT (10a) differentiating with respect to r dP =de—T+TRd—p dr dr dr Dividing both sides by pRT we get the differential form of the equation of state 1 dP _ 1 (IT +_1 d_p 35-5-7; p dr “0") Heat transfer equation: The hate Transfer rate to the diffuser casing is given by: dQ = 2h’(Tw — TO)27rrdr (11a) where (h') is the coefficient of heat transfer, Tw is the wall or diffuser casing temperature, and dQ is the heat transfer rate. The heat transfer rate from the fluid is given by: dT dQ= pC 21trbc ——0dr (11b) p dr equating both equations together 85 dT , _ _ 0 2h (TW T0)211rdr—pCr21trbcp _dr dr rearranging terms 1%: 2h’ Tw -1 TO dr pCrbcp TO substituting (h') from Reynolds's analogy I C h = _f_ cppC 2 from which the equation becomes 1 dT0 =cf seca[Tw _1] 0 TOdr b T developing an auxiliary equation from radial equilibrium equation rearrange C r = C cos 0: we have C -—11—=sinor C rearrange Cu =Csinoc substitute into the equation (11c) (12) 86 f . ldP c C2cosa_C2 ”pdr b r - 2 sm d_Ccosad(Ccos0t) from tangential equilibrium 2 . —ch smut=C dCu+CuCr b r dr r substituting for the velocities c C2 sinoc . 2 . _ f =Ccosord