Multi-load Optimal Design of Burner-inner-liner Under Performance Index Constraint by Second-Order Polynomial Taylor Series Method

Using maximum expansion pressure of n-decane, the aeroengine burner-inner-liner combustion pressure load is computed. Aerodynamic loads are obtained from internal gas pressure load and gas momentum. Multi-load second-order Taylor series equations are established using multi-variant polynomials and their sensitivities. Optimal designs are carried out using various performance index constraints. When 0.25 to 0.8 rectifications of different design variants are implemented, they converge under 4 5 10 d-norm difference ratio.


Introduction
According to the turbine combustion technique [1], maintenance costs on turbine and combustion chamber account for 60% of the whole aircraft maintenance.Therefore, high performance on the optimal design, operation reliability and structural safety are demanded on modern aeroengine thermal components.As burner inner liner (BIL) is the major component in combustion chamber, it becomes one of the most significant components of the aeroengine.BIL is a metallic thin-wall cylinder that controls the combustion, mixing and cooling processes.It guides the combustion chamber cylinder and rotors from thermal combustion products [2].From the maintenance survey [3], BIL accounts for the 63% of the combustion chamber faults.Combustion chamber cylinder is 5% and fuel nozzle is 4%.Since performance indices are determined by BIL mainly, it becomes the research focus on cannular combustion chamber design.
In the operation process of BIL, complicated pressure loads occurs which can be classified as follows: 1) Due to gas combustion, expansion pressure is exerted on its cylinder; 2) Pressure load is generated by the pressure different between the inner and outer cylinder surfaces.Most traditional optimization methods work on unique load where only one disciplinary analysis is involved.To improve the performance of BIL under these loads in different disciplinary, multi-load optimization technique is implemented in this design.
Multi-load optimization technique allows the multiple responses objective functions to be optimized simultaneously under the coupled design variants.This method caters for the complicated interactions among different disciplinary, and the requirements of various performance indices.Inequality constraints are imposed by these indices.

Expansion pressure load created in fuel combustion
The changes in reactant, product and composition mole fraction of n-decane and aviation fuel premixed combustion flame are basically consistent as indicated by Zeng [4].Although the aviation fuel contains complicated ingredients, n-decane can be used for numerical simulation as alternative of aviation fuel.Details of his proposed reaction mechanism can describe detailed dynamic characteristics of n-decane premixed combustion.Because percentages of C and H within hexane hydrogen classes are the same, one can infer its complete combustion product proportions are the same.As a result, n-decane, propane and cyclohexane possess same maximum expansion pressure of 0.86MPa.From this known combustion pressure, BIL inside pressure distribution is obtained using inverse-square law where D , E , J are the axial angles of ring 1,2,8 respectively.n F is the pressure load on ring n. 0 n F is the equivalent nodal load of ring n.

Aerodynamic load generated by gas pressure and gas momentum
According to the aerodynamic equation, when the gas passes the design model (Fig. 2), total thrust is the gas momentum difference between outlet and inlet.Set the gas pressure load inside as P .Taking the thrust direction as positive, aerodynamic force on BIL is where t is the gas hole area, n S is surface area of ring n.
Therefore the total thrust load T F is composed of these multi-load as , , , where (2) k S H is the residual error vector in its expansion.For the p-th order expansion of the k-th thrust response value, one gets H is the residual error value in the expansion.
Termination criterion is established to control the accuracy of iterative process.For the optimal design of BIL, the objective function is met when the sum of thrust response residual errors drop to the global minimum.As a result, the d-norm d a difference ratio is chosen as the Based on the 1 2 k k km u u ukm ordered sets, one can interpolate m-variate thust response vector polynomial function as where is the Lagrange factor function of the ith design variant at the kith interpolated stiffness value, given by Meanwhile, its thust response value polynomial function is First-order derivative terms can be obtained by direct derivative on Eqs. ( 7), (9) with respect to i Q .For multiple variants rectification, the first-order thust response vector and value derivatives [7] can be expressed as where On the other hand, the second-order derivative consists of two terms namely the repeated differential and the unrepeated differential.For the unrepeated term, it is given by the direct first-order derivative of Eqs.7,9 according to i Q involved.Special care is given to the repeated terms where direct second-order derivative with respect to i Q are encountered.For multiple variants, the second-order thust response vector and value derivatives [7] are expressd as ICMIT 2016 Substituting Eqs. ( 10), (12) into Eq.( 4), second-order Taylor series expansion of thust response vector is furnished.Moreover, second-order thust response value equation of Eq. ( 5) is completed by Eqs. ( 11), (13).

Normalization of taylor series equation using accuracy number
Here we carry out the normalization of the optimization equations.For the response vectors, we reduce the data precision as small as possible, while not affecting its variation characteristics.We multiply the equations in each discipline by accuracy numbers, so that they are normalized to the same precision level.In this case, the computation problems such as the singularity, accuracy and convergency can be avoided.For example, accuracies of the distributed stress and pressure are 19  10 and 6 10 respectively, then the normalization accuracy numbers are 19 6 [10 ,10 ] .Under this treatment, faster and more accurate convergences are attained.

Optimal design of BIL under performance index constraint
In this optimization approach, inequality constraints are imposed using various performance indices.Q , its rectification cycle is shorter leading to less significant peak and valley features.Clear cut pattern is observed as in Fig. 6.As its nominal value is relatively large among the other variants, its priority is high in the pattern.Thus its optimization is   % deviation.Using same design variants with 0.25 rectification, 87 estimations are required.Therefore, the convergence rate is lower for larger rectification level.When m=1, 97 estimations are needed in both 0.25 and 0.8 rectification cases.For m=4, there is 69 estimations in 0.25 rectification case.In the 0.8 rectification, 44 estimations are required.When rectification level is larger, convergence rate is decreased.Meanwhile, for the increase in m, the convergence rate is increased.For the illustrated cases, m=2,3, their convergence rates follow these trends.

Figure 1 .
Figure 1.Sectional dimensions in each range.
r r as in Fig.
stress due to expansion pressure and distributed air pressure on the cylinder surface.In general, one can use second-order Taylor series expansion[7]  to obtain the change of the k-th thrust response vector due to change in design variant i

Figure 3 . 2 Q and compressor outlet pressure 4 Q
Figure 3. Convergence of d a ' under 0.5 rectification of

Figure 4 . 1 Q 1 Q
Figure 4. Estimation pattern of 1 Q under 0.5 rectification From Fig. 4, rapid adjustment of 1Q occurs up to estimation 40.Then it becomes stable afterwards.As its rectification is relatively small, wave-like pattern having peak and valley is observed.
Q is illustrated in Fig.9.It increases from estimation 3 gradually near +0.8 rectification level of 89.4kg/s where it is constrained by the performance indices at 2.08%.As 3 Pa is observed in the initial stage.Then it drops gradually to the initial level with Q rectification is zero, wave-like pattern having large peak and valley of