Gas Turbines Dimensions Optimization by Scale Factor Design Principle Analysis

Transcription

1 Journal of Materials Science and Engineering A 7 (- (07-9 doi: 0.765/6-63/ D DAVID PULISING Gas Turbines Dimensions Optimiation by Scale Factor Design Principle Analysis endaima elkacem M sila, Algérie Abstract: In the present work and on the basis of a previous study, we expect to determine optimum methods of gas turbine design. We will look for a design of more efficient machines using the principle of scale factor. It is a question of modifying of machines dimensions that we are going to look for. Key words: Gas turbine, scale factor, rate flow, performances.. asic Equation, Equation esolution: [] From the meridien flow equation of the axial turbo machinery with single stage and if it is supposed that the forces of blading are negligible, the general equation in co-ordinates(, r, is written for multistage machines as Eq. ( []: 0 ( r r r is stream function; this is by holding account that the radial component of the force of blading is negligible, in particular in the following cases: ( The blades are radial, ( The flow is in free vortex, (3 We are outside the blade one. It is the Laplace equation in cylindrical co-ordinates which governs flows in free vortex, in which many works treat its resolution. We try on this study to proceed to a new resolution. y considering the boundary conditions of Dirichlet and Neumann, the expression of the steam function is written for nominal rate flow, which crosses the machine, in Eq. ( []:. ( ( r ( For two D: Corresponding author: endaima elkacem, Ph.D., research fields: turbo machinery performances and design, vibrations analysis and diagnostic. r ( (3 In Eq. (3, and are the hub and the casing radius of last stage of the machine. If we note Eq. (: ( Then the stream function does not depend on and depend on flow rate (of the flow velocity, then Eq. (5: ( (5 r And Eq. (6:. Stream Function (6 r Stream function: (r=f( is Eq. (7: r k r (7 0 k For gas turbine with two stages and if is the. casing radius of second stage and of the first. stage, we pose:. (8.

2 Gas Turbines Dimensions Optimiation by Scale Factor Design Principle Analysis 5 Let us suppose that the line which passes through the two points of the axis has the length. Flow is on free Vortex: r cte If we take the axis that passes through the center of gravity (Fig., the slope of this line is:.. a.( For all machines designed by scale factor, and by multiplying dimensions with the coefficient we have: a Cte Then:. r a. (. b For: 0 We have: r.. ( r a... If: Then:.. Then: r a. (. (9. If: We have Eq. (0: r a. (. (0 Casing radius do not depend on, it depends on.. 3. Dimensions Optimiation 3. ate Flow Optimiation ate flow expression is: V. S V. ( r r V. [( With r the inferior radius(hub radius. Maximal rate flow is for: Then Eq. (: r. r max V.[. ] V[.... ] (. V.. S. With. the radius of hub for the first stage of rotor and: S (.. Then Eq. (: max ( If we design a machine from an initial turbine and if we use the scale factor principle and with: This factor is constant for all machines designed. 3. Scale Factor The value of axial velocity V in the meridian stream plane is: [] V. (3 ( r Eq. (3 is negative because the large radius to the small direct sense of the velocity; the maximal value of the axial velocity is for r. For expression of the maximum of volume flow rate, we take the absolute value of axial velocity as Eq. (: max. With: V max : Maximal axial velocity. S : Section of exhaust turbine. : Total volume flow rate (nominal. ] ( When we multiply the hub and the casing radius by the coefficient such as, the maximal

3 6 Gas Turbines Dimensions Optimiation by Scale Factor Design Principle Analysis value of the flow rate is proportional to the square of, and the maximum variation of the flow rate according to the nominal flow rate is thus linear. The slope is the ratio of the square of casing to hub radius. The maximum of flow rate corresponds to a nominal of the new machine []. We have machines dimensions with: max n A geometric progression of expression of reason, the relative number will be the subect of a multitude of machines, among which we can choose which is appropriate for our need. We choose two factors: For n, we have For n, we have 0. 5 We have the same manner [0] by multiplying the dimensions with the factor, the rate flow will be: We can design machines with less dimensions, but better performances when we choose the conditions:.. And length : Constante The stream function is then preserved: r.. ; r. r When we use scale factor principle [], we can say that by varying all dimensions by time, we obtain stream function proportional to the square of the factor. The stream function is then preserved []: With: r.. ; r. r... The radius r is proportional to and the factor of proportionality is the same that of the initial machine. With identification, we obtain Eq. (5: And Eq. (6:.. (5. ait air. g. g (6. With air and g respectively the air flow rate and the flow rate of natural gas of the new machine, air and g are for the initial machine. We will proceed of the same manner to design the axial compressor of the new machine by multiplying its radius by the same factor. We can determine machine designs by combining the data in Table.. Design Cases ( We can do calculation for a new machine using our principle such as: max. and multiplying the dimensions of the machine designed by the coefficient 0. 5 ( We have designed another machine of the same rate as that of the initial machine but with smaller dimensions. ( With the same manner instead of using the scale factor ( [0], which satisfies:.., we use our concept, and some will have smaller dimensions. Table Coefficient and scale factors. Scale factor Coefficient Our machine rate flow ate flow [] max. 6...

4 Gas Turbines Dimensions Optimiation by Scale Factor Design Principle Analysis 7 r -r = Stage Stage Fig. Gas turbine rotor longitudinal section (two stages machine. 5. Performances: [] The specific consumption of initial machine is Eq. (7: g.. PCI (7 P PCI is the lower calorific value of natural gas, is the density of the natural gas. The power delivered by an alternator driven by the machine according to the scale factor concept is the product of the power corresponding to the initial machine and square of the scale factor. [3] The variation in power is Eq. (8: P. (8 That in is Eq. (9: (9 6. Numerical Calculation 6. Numerical Calculation of Axial Velocity: [] In technique of the finished differences, we divided the flow field in a system of the point s grid; we take the differences between the values of the grid to adacent grid of points. The punctual coordinates of the flow are: r i. r and. ; ( r i i and. r And are the dimensions of the grid element, I and are the associated index to the points gird. The value of the variable r, can be represented by, the value of in the grid (. y Eq. ( and after the estimate of the first and second derivate of, we obtain Eq. (0 by taking r :, i (0 r r r r For m and, n We pose: c r The error on the stream function corresponds to the residues given by Eq. (: - res, i i i r c r c ( For: m And, n The approximate value of maximum axial velocity according to the stream function is Eq. (:,, V.max. ( r The errors on axial velocity can be obtained by Eq. (3: V ( (3 rr where is given by the expression Eq. (0. To estimate the errors on axial velocity, we use Eq. (: Max( res V ( rr 6. Limits Conditions: [] Dirichlet conditions: On ( ub: r, On ( Casing: r, m Neumann conditions: i r, r,,, i 0 On (3 In the upstream: 0, 0 On ( In the downstream:, n 0

5 8 Gas Turbines Dimensions Optimiation by Scale Factor Design Principle Analysis 6.3 Equation esolution: [] To resolve the equation of finished difference (0, the iterative method is used. Let us suppose the rectangular sector on the plan represented by Fig., on which the flow is divided into grid of 6 points. The values of are known on points to the borders of grid, and the values inside the grid indicated in black are unknown. For these internal points we use Eq. (5: c c., 3,,3, r r c c 3,,, 3,3 3, r3 r3 (5 c c,3,3 3,3,, r r c,3 r3 c r3 3, 3,3,3 3, On the previous equations, the known values are indicated by the indication. The Eq. (5 represents a system of equation to unknowns and it is soluble. They represent four equations with four unknowns. One of resolution techniques is the iteration method. 6. Scale Factor Calculation, Performances Gains Coefficient value cf. (3. can be calculated by Eq. (6: a max V max. S. (6 With the new section which verifies Eq. (7: (7 With: ( S And The first coefficient corresponds to flow rate of a new machine and a of the initial machine. V.max is given by Eq. (. Fig. Point s grid on rectangular region. We can easily calculate the value of (of using the value of rate flow, the specific consumption gain is like Eq. (8: (8 For Power Eq. (9: P (9 7. Conclusions The use of the scale factor notion and a result of the calculations carried out on our present study allow us to determine optimum design methods for gas turbines. From this study, we will have gain on dimensions and performances. We hope that our modest work will be the subect of scientific research in the future. eferences [] endaima,. 03. Meridian Flow Analysis in Gas Turbines: Design and Performances. Journal of Materials Science and Engineering A &. [] José, E. 00. Modélisation en régime nominaletpartiel de l écoulementméridiendans les turbo machines axiales et helicocentrifuges. Ecolenationalesupérieure d'art et métiers, Centre Paris. [3] randt, D. E., and Wesoric.. 99 Principe de la Conception des Turbines à Ga GE. GE Industrial &Power system Schenectady. New York: GE Company. [] endaima,. 0. Scale Factor Elaboration of Gas Turbines Design by Meridian Flow Analysis. International eview of Aerospace Engineering.

6 Gas Turbines Dimensions Optimiation by Scale Factor Design Principle Analysis 9 Table N0 Performances Gains Data: Scale factor Coefficient Gain in specific consumption 93.75% Gain in power P % Essays Data: Initial machine specific consumption: 336.9kcal / kwh Initial machine Electrical power: P 975. Kw Gains: New machine power: P Kw New machine specific consumption: ( Kcal / Kwh Gain in specific consumption: 0,8 Kcal / Kwh Difference in natural gas rate flow: g x( P P th Gain in power: P Kw Appendix