Prof. Dr. Rishi Raj Design of an Impulse Turbine Blades Hasan-1 The main purpose of this project, design of an impulse turbine is to understand the concept of turbine blades by defining and designing the parameters that requires to obtain inlet and outlet velocity for the known mass flow rate of the steam. Also another objective is to design the length, size and shape of the blades as well that allows achieving the maximum efficiency of the designed steam turbine. Therefore in the following report of the impulse stage of turbine blades for a power plant the concepts have been used from mathematics, thermodynamics and physics simultaneously to derive the relation between mass flow rate and the velocity triangles, to design the parameters such as velocity triangles, length, size and shape of the blades and also achieving the better efficiency of the steam turbine power plant cycle in its first two impulse stages.
Prof. Dr. Rishi Raj Design of an Impulse Turbine Blades Hasan-2 Abbreviation Property Unit h Enthalpy Btu lbm h Change in Enthalpy Btu lbm v Specific Volume 3 ft lbm V Absolute fluid flow velocity ft s U Velocity of blade ft s α 2 Angle at which absolute fluid flow hits the blade β Angle at which real fluid γ Angle at which real fluid leaves the blade gc Constant A Area of blade cm r i Inside diameter cm r o Outside diameter cm P Power KW m Mass flow rate of steam lbm η th Efficiency % Table-1 (shows the abbreviation, property and units)
Prof. Dr. Rishi Raj Design of an Impulse Turbine Blades Hasan-3 Basic Rankine cycle is practical but relatively inefficient in producing power, therefore in a modern steam plant a number of modifications are made to this basic cycle to improve the efficiency. A steam turbine operates when the thermal energy of steam is converted into kinetic energy and then into mechanical energy. The history of steam turbine development is a long one, it has first in 150 B.C. when Hero of Alexandria built his first crude device that contains a rotating-reaction, nozzled-equipped sphere. The following device was a pure reaction type and generated no useful work. Figure-1 (Alexandria Crude Device) In 1831 Foster and Avery obtained a United States patent for a reaction wheel similar to Hero s. In 1882 Gustaf De Laval applied the turbine principal to a prime mover for his cream separator and few years later produced a series of small impulse turbines. At the very same time Sir Charles S. Parsons has developed a reaction turbine and used it in a marine application. During the period of 1896, C. G. Curtis, an American has developed another kind of impulse turbine. Steam turbine has made it the principal prime mover of generating stations from the last several years. At the present the average maximum unit size is approximately 600,000 kw for a single shaft fossil unit. In 1920 s the progress was made from 5000 to 30,000 kw turbines frequently and the most units used 200 psi and 550 F with some significant changes in steam conditions also. But, at the present moment the most units are designed steam of 2400 psi and 1000 F. Generally steam turbines are classified into two main groups, an impulse turbine and reaction turbine. In Impulse Turbines steam expands in stationary nozzle to attain the high velocity and then flows over the moving blades, converting some of its kinetic
Prof. Dr. Rishi Raj Design of an Impulse Turbine Blades Hasan-4 energy into mechanical energy. Similarly in Reaction Turbine steam expands both in stationary nozzle and moving blades. The relative amount of expansion between impulse and reaction turbine varies from design to design. However, for the practice purposes in generating the power both impulse and reaction sections are required to be used.
Prof. Dr. Rishi Raj Design of an Impulse Turbine Blades Hasan-5 Most of the steam turbine plants use impulse steam turbines, whereas gas turbine plants do not use impulse very often. However, the working principles are the same whether the gas or steam is used as a working fluid. In impulse steam turbine the following facts are very important to remember; Blades are usually symmetrical Entrance and exit angles are around 20 o Enthalpy drop and pressure drop occur in the nozzle Used in the entrance high-pressure stages WORKING PRINCIPLE The steam is supplied to a single-wheel impulse turbine expands completely in the nozzle and leaves with absolute velocity (V) at an angle α and substracting the blade velocity vector (U) the relative velocity vector at entry to the rotor can be determined. The relative velocity makes an angle of β with respect to blade velocity vector. The increase in a value of an angle α decreases the value of the absolute velocity component, V cosα and increases the axial or flow component. In the following working principle the two important and particular points are the inlet and exit of the blades. Furthermore, vectorially substracting the blade speed results in absolute velocity. The steam leaves tangentially at an angle β with relative velocity. The important fact is that the impulse steam turbine can not have 180 o in an actual application, similarly blade entrance angle and blade exit angle cannot be zero. Figure-2 (blade entrance and exit angle)
Prof. Dr. Rishi Raj Design of an Impulse Turbine Blades Hasan-6 THEORY Motion (Newton s second law) The law relates the net force on an object to the acceleration of that object. It is given by; FF = mmmm Continuity The continuity equation in Cartesian coordinates is given as follows; δδ(ρρρρ) δδδδ + δδ(ρρρρ) δδδδ + δδ(ρρρρ) = δδδδ δδδδ δδδδ 1-D steady flow: δδ(ρρρρ) δδδδ = 0 Intergral form : ρρ ii AA ii VV ii = 0 ffffff 1 ii nn Momentum Momentum of the fluid as it expands in the turbine is integral to work output. Momentum equation. FF xx = mmvv xx dddd = mmvv 2 dddd mmvv 1 dddd Where V 2 and V 1 are given by: (mmvv xx ) 1 = (mmvv xx ) tt + VV ii dddd ii (mmvv xx ) 2 = (mmvv xx ) tt + VV ee dddd ee Substituting, (mmvv xx ) 1 (mmvv xx ) 2 = [(mmvv xx ) tt+dddd (mmvv xx ) tttt ] + [(VV xx dddd) ee + VV ii dddd ii ] Taking the time derivative for 1-D flow, FF xx = mm[(vv ee ) (VV ii )] xx Power : PP = δδδδ δδδδ = mmmmmm(vv 1cccccccc + VV 2 cccccccc)
Prof. Dr. Rishi Raj Design of an Impulse Turbine Blades Hasan-7 Analyzing and mechanically designing the first two stages of steam turbine will be done by obtaining the length of blades, shape of blades, efficiency and the other parameters and evaluate each by assuming the first two stages are impulse stages further more the velocity triangles need to be drawn. Analysis of this impulse stage is also based on the information that has been obtained from the optimizing and designing the steam turbine. Such information as a design specifications are given as, Power Plant Power output = 150000 kw Inlet Pressure = 1500 psi Inlet Temperature = 1000 o F Maximum Moisture Level = 13% Mass Flow Rate = 740045 lbm/hr = 12334 lbm/min = 206 lbm/sec FIRST STAGE ANALYSIS The very first stage of the impulse turbine is a velocity analysis, which is the very important analysis in the cycle. It is known that the before entering into the impulse stage steam has normally a velocity (v 1 ) of range between 100 ft/s to 200 ft/s (normally assumed to be at 100 ft/s). When steam enters the stator its velocity changes from v1 to v2 and when it goes out the velocity once again changes from v2 to v3. This change is required to be determined and it can be analyzed through the velocity triangle that is created in between stationary and moving blades. This following stage is based on few steps that can be shown in the analysis and calculations. VELOCITY ANALYSIS AT THE STATOR The entering velocity of the steam to the rotors is v2 that can be determined and is given as, vv oo = vv 2 = 222 h Where h is the amount of heat required to be converted, which can be selected from the range of 30 Btu/lbm to 70 Btu/lbm. Hence for the calculation purposes it is selected to be 30 Btu/lbm. Therefore the velocity to be found as, vv oo = vv 2 = 222 30 = 1216 ffff ss In determining the exit velocity v3 the relative velocity is other important parameters that is required to find before the exit. The stator velocity that has been found to be v2 = 1216 ft/s is the absolute steam velocity and through this velocity determining the exit velocity is not an appropriate analysis. Therefore using simple geometric transformations the relative velocity can be determined that can further required in analyzing the exit
Prof. Dr. Rishi Raj Design of an Impulse Turbine Blades Hasan-8 velocity. Thus the relative velocity u can be calculated from the velocity ratio that is given as; μμ = uu vv 2 = 0.38 The above velocity ratio is always equal to 0.38 but, 0.37 or 0.36 can also be used for the designing purposes. However, it is known for the fact that the velocity ratio is equal to 0.5 only when the design is ideal. Thus due to the irreversibility that can be more likely a friction the velocity ratio is suggested to be 0.38 for the following designing the turbine blades, Thus, uu = 0.38vv 2 = 0.38 x 1216 = 462.08 ft s Since the blade velocity is determined to be u, therefore the relative velocity can be determined and is given as, ww 1 = vv 2 uu The relative velocity is an important parameter that is required to be analyzed through the velocity diagram that is shown below, u u V 2 w 2 Figure-3 (Velocity Triangle At The Entrance of A Stator) Let α 2 be the angle at which the absolute fluid flow hits the blade and β is the angle of the real fluid. For the designing purposes α 2 is the angle that can be chosen from 8 o to 180 o however the 16 o is the best to be chosen but, in this design the angle is chosen to be 20 o. Therefore the axial and tangential component can be determined with respect to the relative velocity and the angle of the real fluid, hence it is given as, Similarly, ww 2 ssssssss = vv 2 ssssssαα 2 (AAAAAAAAAA cccccccccccccccccc)
Prof. Dr. Rishi Raj Design of an Impulse Turbine Blades Hasan-9 ww 2 cccccccc = vv 2 ccccccαα 2 uu (TTTTTTTTTTTTTTTTTTTT cccccccccccccccccc) Thus by substituting the known parameters the above equation becomes, ww 2 ssiiiiii = vv 2 ssssssαα 2 = 1216ssssss(20 oo ) = 415.90 ffff ss Also, ww 2 cccccccc = vv 2 ccccccαα 2 uu = 1216cccccc(20 oo ) 462.08 = 680.59 ffff ss Thus knowing the trigonometric identities the angle of the real fluid can be found which either can be substituted in an axial or tangential component to determine the relative velocity. Thus the angle and the velocity is given as, ββ = 31.43 oo AAAAAA ww 2 = 797.57 ffff ss Therefore at the end of the first part of the analysis that is the velocity determination at the stator end, the steam velocity, the blade speed, the relative velocity and the real fluid angle was calculated and are found to be as, vv 2 = 1216 ffff ss ; uu = 462.08 ffff ss ; ww 2 = 797.57 ffff ss AAAAAA ββ = 31.43 oo VELOCITY ANALYSIS AT THE ROTOR Once the fluid leaves the stator it exits the blade with the velocity v3 through the rotor at an angle γ, an angle at which real fluid leaves the blade. At this point it is important to keep in mind that the parameters such as blade speed (u), relative velocity (w2) and the angle (β) remains constant through out the whole process. u v 2 w 2 Figure-4 (Velocity Triangle At The Exit of A Rotor) Since it is known that leaving velocity of the steam is absolute velocity and it can be determined in a similar way as the relative velocity and the real angle of the fluid has u
Prof. Dr. Rishi Raj Design of an Impulse Turbine Blades Hasan-10 determined through the axial and tangential component. Thus the tangential and axial components are given as, Similarly, vv 3 cccccccc = ww 3 cccccccc uu (TTTTTTTTTTTTTTTTTTTT cccccccccccccccccc) vv 3 ssssssss = ww 3 ssssssss (AAAAAAAAAA cccccccccccccccccc) Since the relative velocity is constant i.e. w2 = w3 = 797.57 ft/s, therefore by substituting the known parameters the equation becomes as, vv 3 cccccccc = ww 3 cccccccc uu = 797.57cccccc(31.43 oo ) 462.08 = 218.47 ffff ss Also, vv 3 ssssssss = ww 3 ssssssss = 797.57ssssss(31.43 oo ) = 415.90 ffff ss Thus knowing the trigonometric identities the angle of the real fluid can be found which either can be substituted in an axial or tangential component to determine the relative velocity. Thus the angle and the velocity is given as, γγ = 62.26 oo AAAAAA vv 2 = 469.91 ffff ss Therefore at the end of the second part of the analysis that is the velocity determination at the rotor end, the steam velocity, the blade speed, the relative velocity and the real fluid angle was calculated and are found to be as, vv 3 = 469.91 ffff ss ; uu = 462.08 ffff ss ; ww 2 = ww 3 = 797.57 ffff ss AAAAAA γγ = 62.26 oo Through the velocity analysis of the stator and rotor the results suggest that the absolute velocity of the steam has increased when it passes through the stator and decreases drastically when exits the blade through the rotor. This following change in the velocity is due to the change in pressure that creates the kinetic energy which further transform to mechanical work. This mechanical work is mainly responsible to generate the energy as an output by rotating the shaft through the high velocity moving through the blades. Once the blade speed, entrance and exit velocity with real fluid angle and leaving fluid angle with the relative velocity for the whole system is determined then the other important parameters are required to be calculated. The other parameters are length, shape, efficiency of the blade for the impulse stages.
Prof. Dr. Rishi Raj Design of an Impulse Turbine Blades Hasan-11 SHAPE OF THE BLADE Figure-5 (One Rotor Blade) The shape of the blade is another important feature in analyzing the stage of the impulse turbine. It is important to have a shape of turbine blade in such a way that the flow of the fluid remains to be continuous. Therefore the symmetric blades with the curvature allows the steam to transfer most of the energy and saved enough amount of the energy to reach the next rotor; thus the cycle is kept moving in a similar fashion from one stator to another and so on. The suggested shape of the blade is shown in the figure-5. Once the shape of the blade is decided the next parameter is to analyze its dimensions that includes the axial velocity, mass flow rate (based on project-1, cycle-5), and area. The area of the blade is a very critical dimension that is required to be evaluated very carefully, thus the mass flow rate will help in obtaining the required area of the blade. Therefore the mass flow rate can be calculated and is given as, Thus the area becomes, mm = ρρvv xx AA AA = mm ρρvv xx Let mm be the mass flow rate and can be found in connection to the project-1 from the cycle-5, and it has found to be as, mm = 740045 llllll hrr = 12334 llllll mmmmmm = 206 llllll ssssss Let ρ be the density of the steam and can be found from the given inlet pressure and inlet temperature that has given as 1500 psi and 1000 o F respectively. Therefore the 1 lbm of steam and the specific volume at the given inlet pressure and temperature can be obtained from the superheated tables require in determining the density of the steam and is given as,
Prof. Dr. Rishi Raj Design of an Impulse Turbine Blades Hasan-12 mm 1 ρρ = = vv @1500pppppp,1000 0.54031 = 1.851 llllll ffff 3 The axial velocity that is perpendicular to the inner rotor blades can be determined and is given as, VV xx = VV 2 ccccccαα 2 = 1216cccccc(20) oo = 1143 ffff ss Therefore the area of the blade can be determined from the above equation that has derived from the mass flow rate equation and is given as, AA = 206 1.851 x 1143 = 0.09737ffff2 In above determination of the area the mass flow rate has been obtained from the project- 1 of cycle-5. However, it is important to know that the mass flow rate in general can be determined from the power equation and is given as, mm = PP x 3412 h nnnnnn From the above calculated area of the blade the other parameters such as length, mean radius, inner and outer radius of the blade will be required to determine and these parameters can be calculated as; As the area of the blades known to be 0.09737 ft 2 ; however, the area can be mathematically is given as, AA = ππ(rr oo 2 rr ii 2 ) = 0.09737ffff 2 Algebraically the equation can be further reduced and solved to determine the inner and outer radius of the blade and is given as, (rr oo rr ii )(rr oo + rr ii ) = 0.09737 ππ As we know that the overall length (L) and the mean radius (r m ) of the blade is determine as, LL = rr oo rr ii And, rr mm = rr oo + rr ii The mean radius can be determined from the calculated blade speed running at the certain designed rpm, n in rpm is required to assume either 1800 rpm or 3600 rpm. Thus for the following calculation purposes rpm has to be assumed as 1800 rpm. Therefore mean radius is to be determined as, uu = 2ππππ 60 rr mm
Prof. Dr. Rishi Raj Design of an Impulse Turbine Blades Hasan-13 Solving above equation for r m the equation becomes, rr mm = 60uu 60 x 462.08 = = 2.45ffff = 29.4iiii 2ππππ 2 x π x 1800 Substituting the length and the mean radius of the blade in the area equation that has been obtained in terms of inner and outer radius of the equation becomes, LLrr mm = 0.09737 ππ Solving the equation in terms of L to obtain the desired length of the blade and is given as, LL = 0.09737 = 0.09737 = 0.01265ffff = 0.1518iiii ππrr mm ππ x 2.45 Substituting the length and mean radius in equation the outer radius and inner radius of the blade can be determined and is given as, Also, rr oo rr ii = 0.1518iiii rr oo + rr ii = 29.4iiii Solving the equation simultaneously allow to obtain the inner and outer radius of the blade as, rr oo = 14.8iiii AAAAAA rr ii = 14.6iiii Now once the length, inner, outer and mean radius for the blade is calculated, the number of blades is the other required parameter and it can be determined using the following formula that is given as, NN = 2ππrr mm ss The Zweifel factor (Ψ) is required to be assumed in determining the s-axial chord, in general for the designing requirements it is suggested to assume and is given to be 0.85. But, mathematically the Zweifel factor is given as, Ψ = 2ss bb (ttttttαα 2 + tttttttt)cccccc 2 γγ Solving the above equation in terms of s the equation becomes, ss = ψψψψ 2(ttttttαα 2 + ttttttγγ)cccccc 2 γγ
Prof. Dr. Rishi Raj Design of an Impulse Turbine Blades Hasan-14 Substituting Ψ = 0.85 and b c (chord Length) = 2.0 in = 0.166 ft the equation determines the axial chord and is given as, ψψψψ ss = 2(ttttttαα 2 + tttttttt)cccccc 2 γγ = 0.85 x 0.166 2 x (tttttt(20) oo + tttttt(62.26) oo ) x cccccc 2 (62.26) oo = 0.14374ffff Therefore the axial chord value can be substituted in number of blades determining equation to calculate the number of blades are required for the following design and is given as, 2ππ x 2.45 NN = 0.14374 = 107 The above analysis shows that there 107 blades are required to design the impulse turbine blade for the power plant. Finally the efficiency is required to determine for the single impulse stage and is given as, ηη = 4 uu vv 2 ccccccαα 2 uu vv 2 ηη = 4 x 462.08 1216 x cos(20)o 462.08 = 0.85 1216 Therefore the efficiency for the following designed impulse stage turbine is found to be 85%
Prof. Dr. Rishi Raj Design of an Impulse Turbine Blades Hasan-15 Over a last decade the importance of the efficiency of the stem turbine has been much improved and increased. In this modern era the manufacturers are very much interested in using the combination of the impulse and a reaction turbine due to the fact that there is hardly pure impulse turbine is seen in the market these days. In determining the equation in analysis part the law of conservation of energy has been used to derive those equations. Since the energy is always conserved therefore the main purpose of the steam turbine is to transform the heat to the mechanical work by means of the rotational motion. Another important aspect of designing the stages of turbine is to understand the relationship between the velocity and the mass flow rate that are directly proportional to each other. Therefore both the mass flow rate and the velocity will affect the momentum and the force at the blades, which finally provides the work out put as a result. Finally the designing the first two stages are the important in such a way that allows the designer to choose the parameters such as the velocity of the steam that is coming in and leaving the blades, length of the blade, shape and size of the blade as well. All these parameters are designed during these two first stages of an impulse turbine helps in understanding the whole steam turbine cycle regardless the process consists of many cycles.
Prof. Dr. Rishi Raj Design of an Impulse Turbine Blades Hasan-16 Acknowledgement 1. Cengel A. Yunus, Boles A. Michael, Thermodynamics An Engineering Approach, Vapor and combined Power Cycle, pg#551-576, 5 th edition. 2. http://en.wikipedia.org/wiki/hero_of_alexandria