SECTION 5

NUMERICAL METHODS

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ASME 2012 Early Career Technical Journal - Vol. 11 196 ASME Early Career Technical Journal 2012 ASME Early Career Technical Conference, ASME ECTC November 2 – 3, Atlanta, Georgia USA

COMPARISON OF THE LENGTH FACTOR ARTIFICIAL NEURAL NETWORK AND FINITE ELEMENT METHODS FOR SOLVING BOUNDARY VALUE PROBLEMS

Kevin McFall Patrick McEnroe Southern Polytechnic State University The Pennsylvania State University Marietta, GA, USA State College, PA, USA

parameters [7] preferentially chosen for high performance. ABSTRACT While this does provide an understanding for the best possible The length factor artificial neural network method ANN approximation, it does not necessarily reflect typical (LFANNM) has been demonstrated as an alternative to the performance nor provide insight into how to properly choose finite element method for solving boundary values problems design parameters. When solving BVPs with the LFANNM, the (BVPs). Besides optimizing on a rectangular grid rather than practitioner must select the number and location of training the typical triangular mesh of the FEM, a primary advantage of points, the number of hidden neurons in the ANN, and the the LFANNM is that the resulting continuous approximate starting values for ANN parameters. This manuscript aims to solution eliminates the need for interpolation between grid compare the LFANNM against the FEM when solving three points. This manuscript compares the accuracy of the different BVPs of varying complexity, and explore the LFANNM and FEM for BVPs with known analytical solutions, dependence of performance against the FEM on all three design both for single partial differential equations (PDEs) and for parameters. coupled systems of PDEs. Unlike the FEM, results show that error for the LFANNM on and between grid points is roughly THE LENGTH FACTOR METHOD equivalent. This superior built-in interpolation allows the The LFANNM reduces the BVP to an unconstrained LFANNM to outperform the FEM in most cases even when optimization problem by posing the trial approximate solution trained with as much as an order of magnitude fewer nodal as points.

ψ t ()()()()xθ,,=A x + LN x xθ (1) INTRODUCTION Problems in science and engineering often involve where x is the position vector in the domain, θ is the ANN differential equations (DEs) sufficiently complicated to require parameter vector including weights and biases, A is a problem numerical techniques for approximating their solutions. specific ansatz designed to satisfy the BVP’s Dirichlet BCs, L is Traditionally, the finite difference [1], finite element [2], and the length factor, and N is the ANN output. The ansatz A can be boundary element [3] methods have been employed to a direct analytical expression known to satisfy the BCs or can numerically solve DEs. Although powerful and widespread, be defined using thin plate spline (TPS) interpolation [8] on these tools do have drawbacks including problematic domains of arbitrary shape [5]. The length factor L can also be discretization of the problem domain, the need for interpolation defined by inspection or with TPS interpolation. In either case, between nodes, and complications in solving nonlinear DEs. the length factor must have a zero value on the domain Artificial neural networks (ANNs) have emerged as an boundary and non-zero values inside the domain. As such, alternative method for numerical solution of DEs, with a survey Equation (1) reduces to the exact BCs for all x on the boundary [4] citing roughly 50 different publications employing ANNs to irrespective of N since the contribution due to the ANN solve DEs in any number of application areas. Solving vanishes when multiplied by the zero length factor, leaving boundary value problems (BVPs) with ANNs requires only the correct BC value encoded in A. optimizing an approximate solution to satisfy the DE in Two-dimensional TPS interpolation takes the form question under the constraint imposed by the boundary conditions (BCs). The length factor artificial neural network m method (LFANNM) [5], [6] removes the BC constraint by 22 f() x, y=∑ Friiln r i ++ F m++12 F m x + F m + 3 y (2) posing the approximate solution in a form where the BCs are i=1 automatically satisfied regardless of ANN output. Typically, researchers justify the value of ANN methods by The quantities showing they rival or even outperform traditional methods. The comparison is often done for a single combination of design

ASME 2012 Early Career Technical Journal - Vol. 11 197 222 EXPERIMENTAL DESIGN r xy, = x − x +− y y + d (3) i() () ii() Results are evaluated for values of H = 20, 30, 40, and 50 and training points chosen on 10×10, 20×20, 30×30, 40×40, represent a measure of distance for a given point (x,y) from the and 50×50 square grids. The twenty combinations of values for th i control point (,xyii ), where d is a small number chosen to H and n are then trained with 100 different random starting be 0.1 for all calculations here. The TPS is analogous to a point values for ANN parameters. The ANN results are then force F located at each of the m control points. The net force on compared with the Galerkin FEM [2] using a meshing the plate should be zero, as should the net moment about the x algorithm [10] designed to generate triangular elements of and y axes. These three conditions on the m+3 coefficients F roughly equal area. Performance of both the LFANNM and the are combined with the specification that f(x,y) be set to known FEM as a function of the number of training points is values at each of the at m control points. The resulting compared. Comparing performance with n for the LFANNM (m+3)×(m+3) linear system is then solved to determine the and the same number of mesh knots for the FEM is justified values for F. since both methods involve solving linear systems of the same Control points for generating A are chosen on the domain size. Figure 1 illustrates the FEM mesh with 99 knots for a boundary and the value of f(x,y) at each is specified according square domain as well as the points used to train the ANN with to the Dirichlet BCs. When using TPS interpolation to define n = 100. the length factor, control points are also chosen on the domain boundary with values for f(x,y) set to zero there. A final control point is selected inside the domain with f(x,y) set to unity to avoid the interpolation from producing the trivial solution of f(x,y) = 0. The ANN architecture is a feed-forward, fully-connected, multi-layer perceptron network with the position vector components as inputs, H neurons in the hidden layer employing logistic sigmoid transfer functions, and a single output N computed with a linear transfer function. The ANN parameters, including weights and biases, are initialized to small random values and then optimized using the Levenberg-Marquardt [9] Figure 1: Training mesh with 99 knots for the FEM on a gradient descent method to minimize the error function square domain, with comparable n = 100 ANN training points. n 2 2 EG=∑ (xxiti,,ψψ()()() ∇∇ ti x , ψ ti x) (4) i=1 The average ANN performance over the 100 runs is reported in order to provide an indication of the typical error where G is the differential operator defining the desired expected from the LFANNM. Box plots [11] are also generated, differential equation indicating the spread of performance over the 100 runs as well as the maximum and minimum possible errors with the 2 LFANNM. G xx,,ψψ()()()∇∇ x , ψ x = 0 (5) ( ) The FEM produces discrete approximations at the mesh knots, which are linearly interpolated to produce for the exact analytical solution ψ. The error in Equation (4) is approximations between knots. Error in the FEM is therefore evaluated at the collocation of n training points chosen on a expected to be higher for points between knots. Two types of square grid falling within the domain of the problem. error are computed: training error measured at the mesh knots, Systems of DEs are solved in the LFANNM [6] by and evaluation error measured at all points falling on a 75×75 defining an approximate solution of the form of Equation (1) square grid. Interpolation is unnecessary in the LFANNM since for each unknown function in the system with its own ANN. the approximate solution in Equation (1) is continuous over the The total error function in comprised of the sum of entire domain. As a result, training and evaluation errors are not contributions in Equation (4) for each DE to be satisfied. This significantly different, with evaluation error typically less than total error is a function of the ANN parameters for each of the 1.1 times training error. For this reason error is only reported approximate solutions, and is minimized similarly using the for the LFANNM on the 75×75 evaluation grid since error at same Levenberg-Marquardt training algorithm. the training points is only slightly better. Results are dependent on the number of training points n, the number of hidden neurons H, and the starting ANN PROBLEM 1: SQUARE DOMAIN LAPLACE EQUATION parameters. The first BVP explored is the Laplace equation

ASME 2012 Early Career Technical Journal - Vol. 11 198 ∂∂22ψψ domain, including a sample of mesh knots and ANN training += 220 (6) points, appears in Figure 3. The analytical solution ∂∂xy

1 16rr2 ++ 1 8 cosθ on a square domain with boundary conditions ψ = ln 2 (11) 2ln 2 rr++16 8 cosθ ψ()()()()()0,y= ψψ 1, yx = ,0 = 0 and ψ x ,1= sin π x (7) defined in polar coordinates satisfies Equation (6) with BCs of which has the analytical solution unity value on the outer circle defined by

2 22 sin()()ππxy sinh ()xy−+=1 2.5 (12) ψ = (8) sinh ()π and zero value on the inner circle defined by

The ansatz 2 22 xy+=1 (13) Ay= sin ()π x (9) satisfies the BCs in Equation (7) and the length factor

L= xy()() x −−11 y (10) is zero on the boundary and positive inside the domain.

Figure 2 illustrates the mean absolute error over the domain for the FEM and LFANNM. Error with the FEM is between 2 and Figure 3: Training mesh with 262 knots for the FEM on 10 times worse on the evaluation grid than at mesh knots, the domain of Problem 2, with comparable n = 276 ANN something that could be improved with interpolation more training points. sophisticated [12] than the linear method used. The LFANNM still outperforms the FEM by an order of magnitude compared The ansatz A in Figure 4 used for this problem was developed with the best case error at the FEM knots. Note that LFANNM using TPS interpolation [8] with control points as shown. The performance is not strongly dependent on the number of hidden length factor neurons although slightly worse with H = 20. 2 =2 −− −2222 + − L2.5() x 1 yxy()1 (14)

was determined by inspection in order to be zero on both the inside and outside circles with positive values in between.

Figure 2: Error in approximating Problem 1 as a function of training points or mesh knots using the FEM as well as the average error over 100 runs with the LFANNM using Figure 4: Ansatz A developed using TPS interpolation at various numbers of hidden nodes. the marked control points, and used to satisfy the Problem 2 BCs of zero on the inside circle and unity on PROBLEM 2: IRREGULAR DOMAIN LAPLACE EQUATION the outside circle. The second BVP considered is also the Laplace equation but on a domain defined by two non-concentric circles. The

ASME 2012 Early Career Technical Journal - Vol. 11 199 Figure 5 presents the error in both methods for Problem 2. r =1.52 + 1.5cosθ (17) Performance with the LFANNM is not overwhelmingly superior to the FEM, although error on the evaluation grid is TPS interpolation was employed to produce A1 and A2 better than with the FEM for less than 1000 training points appearing in Figures 6a and 6b, respectively, designed to satisfy when using at least 40 hidden neurons. This domain shape is Equation (16) on the domain boundary defined by challenging for the ANN approach since choosing training Equation (17). Figure 7 illustrates the length factor used, also points on a square grid is not conducive to evenly spaced points developed by TPS interpolation. in the small space between the circles, as is apparent in Figure 3. Interestingly enough, increasing the number of training points beyond 200 apparently does not increase performance with the LFANNM. While not categorically better than the FEM, the LFANNM with sufficiently many hidden neurons and training points produces acceptably low errors on the order of 10-4; results are comparable to evaluation error when using the FEM with similar numbers of mesh knots.

a) b) Figure 6: TPS interpolated ansatz functions to satisfy the BCs for a) ψ1 and b) ψ2 in Problem 3. TPS control points are marked with dots.

Figure 5: Error in approximating Problem 2 as a function of training points or mesh knots using the FEM as well as the average error over 100 runs with the LFANNM using various numbers of hidden nodes. Figure 7: TPS interpolated length factor for Problem 3 with values at boundary control points set to zero and the PROBLEM 3: SYSTEM OF TWO COUPLED DES value at the internal control point set to unity. TPS control The final BVP investigated involves the coupled points are marked with dots. differential equations Figures 8 and 9 display error in Problem 3 when approximating ∂2ψψψψ ∂∂∂ the solutions of ψ1 and ψ2, respectively. With sufficiently many 112+++ 2 =0 hidden neurons, the average LFANNM run outperforms the ∂x2 ∂∂∂yxy (15) FEM for both ψ1 and ψ2, even for error evaluated at the mesh ∂∂ψψ2 knots. 12++BeAx12 + Be A y =0 ∂x ∂y2 12 RESULTS with analytical solutions Figures 2, 5, 8, and 9 indicate that increasing the number of training points beyond a minimum value does not significantly improve performance with the LFANNM. The indication is that BB ψ =−+12eeAx12 A y training points selected on a 30×30 rectangular grid, at roughly 1 A A2 1 2 (16) 250 to 300 points depending on the geometry, is sufficient to B solve two-dimensional BVPs of varying complexity. The ψ =BeAx12 − 2 e A y 21 superior built-in interpolation achieved by the continuous A2 approximate solution in Equation (1) eliminates the need to use excessively many training points. On the other hand, where A1 = 0.85, A2 = 0.2, B1 = B2 = 0.3 in the case reported here. BCs are determined by evaluating (16) on the limaçon performance of the FEM continues to improve until several thousand mesh knots are included. The LFANNM has been shown to achieve similar or better performance with an order of

ASME 2012 Early Career Technical Journal - Vol. 11 200 magnitude fewer training points. This translates to a sizable explored. Outlier runs are marked with a + and error at the savings in computational complexity since factorization of mesh knots for the FEM are included as a comparison. Open linear systems grows at O(n3). circles represent FEM error with a number of mesh knots comparable to the n used in the LFANNM, and filled circles indicate FEM error with the maximum number of mesh knots evaluated. These circles represent the lowest possible FEM error assuming an interpolation scheme could reduce error between mesh knots to the same level as at the knots.

Figure 8: Error in approximating ψ1 in Problem 3 as a function of training points or mesh knots using the FEM as well as the average error over 100 runs with the a) b) c) d) LFANNM using various numbers of hidden nodes. Figure 10: Box plots representing the mean absolute error

in the solution for the LFANNM over 100 runs using the minimum design parameters. Performance includes a) Problem 1, b) Problem 2, and Problem 3 solutions c) ψ1 and d) ψ2. FEM error at mesh knots for a similar number of training points is marked with an open circle and FEM error at mesh knots with the maximum number of training points marked with a filled circle.

The worst runs for Problems 1 and 3 (excepting a couple outliers) perform better than the best possible FEM, even with a nearly unlimited number of mesh knots. LFANNM performance in Problem 2 is typically somewhat worse than the best case FEM, primarily due to selecting training points on a square grid for the circular domain shape. However, 6% of the Figure 9: Error in approximating ψ2 in Problem 3 as a LFANNM runs do result in lower error than the FEM with a function of training points or mesh knots using the FEM similar number of mesh knots. as well as the average error over 100 runs with the LFANNM using various numbers of hidden nodes. CONCLUSION In general, the LFANNM is shown to significantly Like the number of training points, results show that a outperform the FEM, even when using as many as an order of minimum number of hidden neurons, H = 40, is necessary to magnitude fewer training points than mesh knots. The guarantee acceptable performance. Experience has indicated LFANNM is able to more accurately model the analytical that training points selected on a 30×30 rectangular grid and 40 solution to BVPs using relatively few training points due to the hidden neurons are the minimum required design parameters. built-in interpolation from a continuous rather than discrete Using smaller values risks inferior performance, and using approximate solution. As long as sufficiently many training more unnecessarily increases training time. points are included (two-dimensional problems require points Initial ANN weight and bias values are the final parameter chosen on at least a 30×30 square grid) the continuous nature of affecting performance. Figures 2, 5, 8, and 9 present error the approximate solution successfully models the solution in averaged over 100 runs with different random starting weights. the intervening spaces between training points. In fact, error As such, those errors represent performance for a typical run between training points is nearly identical to that at the training when any given run could produce higher or lower error points. Indeed, ANNs are known to be universal approximators depending on the location of local minima in the error space. [13], which can model any function to arbitrary levels of Figure 10 displays box plots for the 100 runs using minimum accuracy. Using more than a 30×30 grid of training points does design parameters for H and n in each of the three problems not improve error much, in contrast with the FEM which

ASME 2012 Early Career Technical Journal - Vol. 11 201 benefits significantly from increased numbers of mesh knots [2] G. R. Liu and S. S. Quek, The Finite Element Method: A and the accompanying increased computational complexity. Practical Course. Boston: Butterworth-Heinemann, 2003. Like the number of training points, results indicate a lower [3] C. Brebbia, J. Telles, and L. Wrobel, Boundary Element limit on the number of hidden neurons at about 40. Increasing Techniques: Theory and Applications in Engineering. Berlin: this number can improve performance somewhat, but all Springer-Verlag, 1984. problems were solved with acceptably low errors using no more [4] M. Kumar and N. Yadav, “Multilayer perceptrons and than 40 hidden neurons. radial basis function neural network methods for the solution of One of the strongest criticisms of ANN methods is the differential equations: A survey,” Computers & Mathematics dependence of performance on the choice of initial network with Applications, vol. 62, no. 10, pp. 3796–3811, Nov. 2011. weights. Even the worst runs of the LFANNM outperformed [5] K. McFall and J. Mahan, “Artificial Neural Network the FEM for two of the three problems explored here. FEM Method for Solution of Boundary Value Problems With Exact approximation was somewhat better in the other problem, Satisfaction of Arbitrary Boundary Conditions,” Neural presumably because the triangular mesh more accurately Networks, IEEE Transactions on, vol. 20, no. 8, pp. 1221–1233, represented the circular domain shape than points chosen on a 2009. square grid. [6] K. McFall, “Solving Coupled Systems of Differential ANN methods for solving BVPs have become more Equations Using the Length Factor Artificial Neural Network popular in recent years due to the simplicity of choosing Method,” American Society of Mechanical Engineers Early training points, their continuous approximate solutions, and Career Technical Journal, vol. 9, pp. 27–34, 2010. their ease of use with complex nonlinear DEs. This manuscript [7] G. P. Zhang, “Avoiding Pitfalls in Neural Network exhibits that this list of benefits should also include superior Research,” Systems, Man, and Cybernetics, Part C: approximation in general compared with the more traditional Applications and Reviews, IEEE Transactions on, vol. 37, no. FEM approach. Additionally, the increased performance is 1, pp. 3–16, 2007. achieved with a significantly lower computational complexity [8] L. Zagorchev and A. Goshtasby, “A comparative study of due to requiring far fewer training points than mesh knots. transformation functions for nonrigid image registration,” While the example problems explored in this manuscript Image Processing, IEEE Transactions on, vol. 15, no. 3, pp. represent relatively simple situations, the LFANNM has been 529–538, 2006. shown to accurately solve problems in three dimensions [5] and [9] K. L. Priddy and P. E. Keller, Artificial neural networks: more realistic thermal-fluids problems such as the Blasius an introduction. SPIE Press, 2005. boundary layer and the entrance length problem [6]. Even the [10] Per-Olof Persson and G. Strang, “A Simple Mesh apparently simplistic Laplace equation on a domain of non- Generator in MATLAB,” SIAM Review, vol. 46, no. 2, pp. 329– concentric circles in Problem 2 could represent heat dissipation 345, Jun. 2004. in a the insulation of a current carrying conductor where the [11] R. McGill, J. W. Tukey, and W. A. Larsen, “Variations of solution produces the temperature distribution rather than Box Plots,” The American Statistician, vol. 32, no. 1, pp. 12– simply a shape factor for computing overall heat transfer [14]. 16, Feb. 1978. Comparison for this study was limited to linear DEQs easily [12] H. Späth, Two Dimensional Spline Interpolation solved with the FEM. Future studies will look to confirm Algorithms. A K Peters/CRC Press, 1993. similar performance of the LFANNM with more complicated [13] K. Hornik, M. Stinchcombe, and H. White, “Multilayer DEQs modeling more realistic phenomena. Feedforward Networks are Universal Approximators,” NEURAL NETW, vol. 2, no. 5, pp. 359–366, 1989. REFERENCES [14] Incropera, et. al. Fundamentals of Heat and Mass Transfer, [1] G. Smith, Numerical Solution of Partial Differential Hoboken, NJ: Wiley, 2007 Equations: Finite Difference Methods. Oxford, UK: Clarendon, 1978.

ASME 2012 Early Career Technical Journal - Vol. 11 202 ASME Early Career Technical Journal 2012 ASME Early Career Technical Conference, ASME ECTC November 2 – 3, Atlanta, Georgia USA

MODELING OF GAS TURBINE-BASED COMBINED CYCLE POWER PLANT

Farshid Zabihian Alan S. Fung Department of Mechanical Engineering Department of Mechanical and Industrial West Virginia University Institute of Technology Engineering, Ryerson University Montgomery, WV, U.S.A. Toronto, Ontario, Canada

In a combined cycle power plant (CCPP), the high ABSTRACT temperature exhaust stream of the gas turbine is used to Gas turbine (GT)-based cycles generate a significant generate required steam for the steam cycle in a heat recovery portion of world’s electricity. They have been used in the steam generator (HRSG). This is possible because the inlet power generation industry for more than a century due to their temperature of the turbine in a gas turbine, higher than 1200°C low capital cost, short installation time, compact size, and for modern GTs, and as a result their exhaust temperature, is short start-up and shut-down time. Despite their widespread much higher than that for the steam turbines, between 500°C applications, the main disadvantages associated with using and 600°C. A unit of a CCPP consists of several GTs, usually simple gas turbines for power generation is that the exhaust two or three, a heat recovery steam generator, and a single temperature of GTs is very high, which means high-grade steam cycle (Figure 2). Very high efficiency, around 60% energy is wasted and results in the low efficiency of the (LHV), has been reported for recent modern CCPPs [2, 3]. system. Applications of gas turbines in cogeneration plants and combined cycle power plants can help to recover some of the wasted thermal energy from the GT exhaust stream to produce further electricity and/or useful thermal energy. This paper presents the modeling of the combined gas turbine and steam cycle. In the model, the thermal energy of the gas turbine exhaust is used to generate steam for the bottoming steam cycle. The efficiency of the model of the combined cycle power plant is about 47% (based on lower heating value, LHV), which is significantly higher than a simple gas turbine cycle operating at the same conditions with an efficiency of about 40% (LHV). Also, the important operational parameters of the cycle are presented to provide an insight to the cycle’s internal operation. Figure 1: Schematic of a simple steam power generation INTRODUCTION (Rankine) cycle In order to improve the efficiency of gas turbines (GTs), combining the gas turbine with a steam cycle is a suitable HRSGs can be designed with or without additional firing. option. A schematic diagram of a steam cycle, working based Additional firing, also called duct burning, is used to increase on the Rankine cycle, is illustrated in Figure 1. A the inlet temperature of the HRSG, leading to the higher comprehensive review of various gas turbine-based combined output power of the steam cycle, although the overall power plants was reported in Najjar [1]. efficiency of the cycle reduces. In the earlier CCPPs, this configuration was very common due to the low temperature of the GT exhaust stream. However, due to the simplicity of

ASME 2012 Early Career Technical Journal - Vol. 11 203 construction and the higher temperature of the exhaust stream CCPP MODEL IN ASPEN PLUS® of modern GTs, CCPPs with no additional firing in the HRSG, Figure 3 illustrates the model of the CCPP cycle where all fuel is combusted in the GT cycle, is also very developed for this work. The gas turbine side of the cycle is common these days. In the CCPPs without additional firing, exactly identical to the gas turbine side of the Whitby the efficiency and output power of the bottoming steam cycle cogeneration power plant. The Whitby cogeneration power are limited by the temperature and flow rate of the gas turbine plant, located in Whitby, Ontario, is a gas turbine-based plant, off-gas and by the temperature of the exhaust stream at the which is equipped with a gas turbine with rated capacity of 58 stack. The stack temperature is bounded by the exhaust stream MW (at the site’s standard conditions). This type of GT was dew point to prevent acid corrosion. originally designed for aviation applications, and when it was adapted for stationary power generation applications, two types of NOx control systems were added: a dry system and a wet system. In the dry system, the flame temperature is controlled by the flow rate of excess air. But in the wet system, demineralized water is injected to the combustion chamber to control the flame temperature. The gas turbine in this plant was originally equipped with a dry NOx control system. But due to the high maintenance cost of this system, it was replaced with a wet control system in February 2010. The main design and performance parameters of the gas turbines used in the plant were presented elsewhere [10-11]. They are summarized in Table 1. In the gas turbine, the pressure ratio of the compressor is distributed among the 3 stages. The low pressure compressor runs at 3,600 rpm with the generator synchronized with the grid. The intermediate and high pressure compressors are driven by the intermediate and high pressure turbines, around 6,500 rpm and 10,000 rpm, respectively. The individual pressure ratios are the result of the speeds, ambient temperature, power, fuel, and variable geometry control. Figure 2: Schematic of a simple combined cycle power Tables 2 and 3 list the model constants for the equipment plant of the gas turbine and steam cycle of the plant. The important

thermodynamic properties and operational parameters of the Based on the HRSG arrangement, CCPP can be a single-, gas turbine side of the cycle were reported in [10-11]. Table 4 two- or three-pressure cycle. In the simplest configuration, the shows the important thermodynamic properties of the major single-pressure cycle, the HRSG can consist of either an streams in the steam cycle at the ambient temperature of 10°C. economizer, an evaporator, and a superheater or a once- through boiler to generate steam with one pressure. The RESULTS AND DISCUSSION disadvantage of the single-pressure configuration is its poor In this cycle, since the temperature of the GT3 exhaust waste heat recovery efficiency. To improve this efficiency, a stream, COMBPRO4 (436.7°C), is not sufficient for steam multi-pressure HRSG can be used to produce steam with generation in the HRSG, supplementary firing equipment, also different pressures. These multiple-pressure steams can drive known as the duct burner, is used to increase the temperature high- and low-pressure (and possibly intermediate-pressure) of the HRSG inlet stream. For this purpose, the mass flow rate steam turbines with a higher efficiency. Figure 3 illustrates an of the inlet natural gas is increased to 12,293 kg/h of which Aspen Plus® model of a two-pressure CCPP. 10,793 kg/h is combusted in the gas turbine combustor, and Several studies reported the performance of CCPPs the rest (1,500 kg/h) is combusted in the duct burner. It should through energy and exergy analyses [4-9]. In this work, the be noted that due to the very high air-to-fuel ratio in the model of the gas turbine-based combined cycle power plants combustion chamber of the gas turbine, there is sufficient was developed in Aspen Plus®. It was essential to determine a oxygen in the duct burner for combustion (89,892 kg/h) such real world cycle with equipment specifications to model a that even after the duct burner, the mass flow rate of oxygen in specific power plant. Furthermore, performance results of the stream is 84,053 kg/h. The combustion products at a these real power plants were required to validate the models. temperature of 540°C and pressure of 1.048 bar enter the For this work, the data from the Whitby cogeneration power HRSG and pass through seven heat exchangers, each plant were used. representing different sections of the HRSG, and are discharged to the atmosphere at a temperature of 148.7°C.

ASME 2012 Early Career Technical Journal - Vol. 11 204 Table 1: Design and performance parameters of Rolls- Table 2: Constants for the gas turbine cogeneration plant Royce gas turbine at 15°C employed at Withby power model plant Air compressor 1 (AIRCOMP1) Parameter Unit Value Altitude m 91 Pressure ratio 1.7 Ambient pressure kPa 100.232 Isentropic efficiency (%) 85 Relative humidity % 60 Mechanical efficiency (%) 98.6 Installation inlet stack pressure mmH O 101.6 loss 2 Air compressor 2 (AIRCOMP2) Installation exhaust stack mmH O 254 Pressure ratio 4.5 pressure loss 2 Isentropic efficiency (%) 85 Generator efficiency % 98.35 Nominal gross output power MW 58.0 Mechanical efficiency (%) 98.6 Nominal gross output heat rate kJ/kWh 8,959 Air compressor 3 (AIRCOMP3) Nominal gross output thermal % 40.18 Pressure ratio 4.4 efficiency Auxiliary power off-take MW 0.2 Isentropic efficiency (%) 85 Nominal net output power - MW 57.8 Mechanical efficiency (%) 98.6 base load Nominal net output heat rate - Natural gas compressor (NGCOMP) kJ/kWh 8,990 base load Discharge pressure (bar) 56 Nominal net output thermal % 40.04 efficiency Isentropic efficiency (%) 85 Gas turbine inlet mass flow kg/s 159 Mechanical efficiency (%) 98 Fuel type Natural

Gas NOx water pump (NOXWPUMP) Fuel LHV MJ/kg 48.8 Discharge pressure (bar) 38 Fuel temperature °C 50 Isentropic efficiency (%) 80 Nominal fuel flow kg/h 10,866 Mechanical efficiency (%) 98 Fuel molecular weight g/mol 16.7 Water temperature °C 15 Gas turbine 1 (GT1) Water flow (for HRSG) kg/h 14,034 Pressure ratio 2.7 Water-to-fuel ratio - 1.292 Isentropic efficiency (%) 90 Gas turbine package exit kPa 103.3 pressure Mechanical efficiency (%) 98.6 Gas turbine package exit °C 431.8 Gas turbine 2 (GT2) temperature Gas turbine package exit flow kg/s 164.78 Pressure ratio 2.2 Exhaust stack exit total kPa 100.8 Isentropic efficiency (%) 90 pressure Nominal exhaust stack exit Mechanical efficiency (%) 98.6 °C 431.8 temperature Gas turbine 3 (GT3) Nominal exhaust stack exit kg/s 164.78 flow Pressure ratio 5.4 2 Exhaust stack exit area m 6.97 Isentropic efficiency (%) 90 Exhaust stack exit velocity m/s 49.15 Mechanical efficiency (%) 98.6

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Figure 3: Schematic of an Aspen Plus® model of a two-pressure CCPP

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Table 3: Model constants for the steam cycle of the MW. This turbine discharges the saturated steam to the CCPP condenser at a temperature of 32.9°C and partial vacuum of High pressure steam turbine (HPSTTURB) 0.05 bar. The total output power of the CCPP is 79 MW of Discharge pressure (bar) 22 which 58.2 MW (73.7%) is generated in the gas turbine and Isentropic efficiency (%) 80 20.8 MW (26.3%) is generated in the steam cycle. The total Mechanical efficiency (%) 95 efficiencies of the cycle are 42.7% and 47.4%, respectively, Intermediate pressure steam turbine (IPSTTURB) based on the HHV and LHV. Discharge pressure (bar) 5 bar Isentropic efficiency (%) 80 CONCLUSION The macro level model of the gas turbine-based combined Mechanical efficiency (%) 98 cycle power plant was developed in ASPEN PLUS® to Low pressure steam turbine (LPSTTURB) evaluate the performance of the cycle. The results confirmed Discharge pressure (bar) 6.9 bar that the efficiency of the cycle can be improved from about Isentropic efficiency (%) 80 40% for a single gas turbine to about 47% for the combined Mechanical efficiency (%) 98 cycle. This work demonstrates that, although utilization of High pressure feed water pump (HPWPUMP) fossil fuels for power generation is inevitable, at least in the Discharge pressure 76.5 bar short- and mid-term future, it is possible and practical to carry Isentropic efficiency (%) 80 out such utilization more efficiently and in an environmentally Mechanical efficiency (%) 98 friendlier manner. Also, the modeling results of gas turbine- High pressure drum pump (HPDRUMP) based power plants signify that relatively simple model can Pressure ratio 73.5 bar predict plant performance with acceptable accuracy. Isentropic efficiency (%) 80 Mechanical efficiency (%) 98 ACKNOWLEDGMENTS Low pressure feed water pump (LPWPUMP) The authors gratefully acknowledge the financial support Pressure ratio 6.9 bar provided by the Natural Sciences and Engineering Research Isentropic efficiency (%) 80 Council of Canada (NSERC) through Discovery Grants (DG) and Ontario Graduate Scholarship Program (OGS). Also, the Mechanical efficiency (%) 98 support provided by Mr. Fabio Schuler (P.Eng.), the manager Low pressure drum pump (LPDRUMP) of the Whitby cogeneration power plant in the form of Discharge pressure (bar) 6 providing the data required from the Whitby cogeneration Isentropic efficiency (%) 80 power plant is appreciated. Mechanical efficiency (%) 98 REFERENCES The HRSG of the cycle is a two-pressure system. For the [1] Najjar, Y.S.H., 2001, “Efficient use of energy by utilizing high pressure section of the HRSG, the feed water with a mass gas turbine combined systems,” Applied Thermal Engineering, flow rate of 62,000 kg/h is pressurized at HPWPUMP to the 21, PP. 407-438. pressure of 76.5 bar. Then, it is heated through two [2] Cengel, Y. A., & Boles, M. A., 1998. “Thermodynamics: economizers (HRSG6 and HRSG3), one evaporator (HRSG2), An engineering approach,” (third ed.). McGraw-Hill. one drum (HPDRUMP), and one superheater (HRSG1). The [3] Farmer, R., 1992, “227 MW frame 9FA will power Epon’s superheated steam enters the high pressure steam turbine 1.7 GW combined cycle,” Gas Turbine World, June 1992, PP. (HPSTTURB) at a temperature of 500°C and pressure of 73.5 15-22 bar. While expanding to a temperature of 343°C and pressure [4] Hammond, G. P., Ondo Akwe, S. S., 2007, of 22 bar, the steam generates 4.6 MW of power. The steam “Thermodynamic and related analysis of natural gas combined further expands through the intermediate pressure steam cycle power plants with and without carbon sequestration,” turbine (IPSTTURB) to a temperature of 193.5°C and pressure International Journal of Energy Research, Volume 31, Issue of 5 bar and generates 4.5 MW of electricity. 12, pages 1180–1201. For the low pressure section of the HRSG, the water [5] Sanaye, S., Rezazadeh, M., 2007 “Transient thermal enters the system with a mass flow rate of 16,000 kg/h and is modelling of heat recovery steam generators in combined pressurized at LPWPUMP to the pressure of 6.9 bar. Then, it cycle power plants,” International Journal of Energy Research, is heated through one economizer (HRSG7), one evaporator Volume 31, Issue 11, September, PP. 1047–1063. (HRSG5), one low pressure drum (LPDRUM), and one [6] Pourbeik, P., 2003, “Modeling of combined-cycle power superheater (HRSG4). The low pressure superheated steam is plants for power system studies,” Power Engineering Society then mixed with the IPSTTURB exhaust stream before being General Meeting, 2003, IEEE, Volume: 3, PP. 308 – 1313 fed to the low pressure steam turbine (LPSTTURB). The [7] Hamid, E., Newby, M., Pericles, P., 2011, “Performance electricity generated in the low pressure steam turbine is 11.6 Modeling of Unfired Steam Cycle Using Single and Dual

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Table 4: Important thermodynamic properties of the major streams in the steam cycle at the ambient temperature of 10°C Technical

COMB COMB COMB COMB COMB COMB COMB COMBP HPSAT HPSH Streams NGDUCT NGIN PRO5 PRO6 PRO7 PRO8 PRO9 PR10 PR14 R15 STM STM Journal Temperature (°C) 540.2 495.2 362.6 293.2 290.3 240.0 140.6 148.7 151.1 52.0 289.1 500.0 Pressure (bar) 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 56.2 19.5 73.5 73.5 - Vol. Mass flow (kg/h) 593,990 593,990 593,990 593,990 593,990 593,990 430,643 593,990 1,500 12,293 62,000 62,000

11 Vapour fraction 1 1 1 1 1 1 1 1 1 1 1 1

H2O 39,033 39,033 39,033 39,033 39,033 39,033 28,299 39,033 0 0 62,000 62,000

N2 430,511 430,511 430,511 430,511 430,511 430,511 312,121 430,511 29 239 0 0 Mass flow rate (kg/h)

O2 84,053 84,053 84,053 84,053 84,053 84,053 60,939 84,053 0 0 0 0 CO 0 0 0 0 0 0 0 0 0 0 0 0

CO2 33,330 33,330 33,330 33,330 33,330 33,330 24,165 33,330 5 42 0 0 Argon 7,062 7,062 7,062 7,062 7,062 7,062 5,120 7,062 0 0 0 0 Methane 0 0 0 0 0 0 0 0 1,433 11,744 0 0 Ethane 0 0 0 0 0 0 0 0 31 254 0 0 Propane 0 0 0 0 0 0 0 0 2 14 0 0

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Technical Table 4: Important thermodynamic properties of the major streams in the steam cycle at the ambient temperature of 10°C (cont.)

HP HP HP LP LP LP Journal Streams HPWIN IPSHSTM STMEXHA WATER1 WATER3 WATER6 SHSTM3 WATER1 WATER5 Temperature (°C) 10.8 291.9 289.1 10.0 343.0 198.2 10.0 158.9 32.9 - Vol. Pressure (bar) 76.5 76.5 73.5 6.2 22.0 5.0 6.9 6.0 0.1

11 Mass flow (kg/h) 62,000 62,000 122,437 62,000 62,000 78,000 16,000 31,583 78,000 Vapour fraction 0 0 1 0 1 1 0 1 1

H2O 62,000 62,000 122,437 62,000 62,000 78,000 16,000 31,583 78,000

N2 0 0 0 0 0 0 0 0 0 Mass flow rate (kg/h)

O2 0 0 0 0 0 0 0 0 0 CO 0 0 0 0 0 0 0 0 0

CO2 0 0 0 0 0 0 0 0 0 Argon 0 0 0 0 0 0 0 0 0 Methane 0 0 0 0 0 0 0 0 0 Ethane 0 0 0 0 0 0 0 0 0 Propane 0 0 0 0 0 0 0 0 0

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Pressure Once-Through Steam Generator,” ASME Conf. Proc. GT2011 (2011) [8] Karrabi, H., Rasoulipour, S., 2010, “Second Law Based Analysis of Supplementary Firing Effects on the Heat Recovery Steam Generator in a Combined Cycle Power Plant,” ASME Conf. Proc. ESDA2010. [9] Mahbub, F., Hawlader, M. N. A., Mujumdar, A. S., “Exergoeconomic Analyses of a Combined Water and Power Plant (CWPP),” ASME Conf. Proc. ES2009 (2009). [10] Zabihian F., Fung A., Schuler F., 2012, “Modeling of Gas Turbine-Based Cogeneration System,” ASME 6th International Conference on Energy Sustainability and ASME 2012 10th Fuel Cell Science, Engineering and Technology Conference, 2012, San Diego, California, USA. [11] Zabihian F., Fung A., Schuler F., 2012, “Performance Evaluation of Evaporative Compressor Inlet Air Cooling System in a Gas Turbine-Based Cogeneration Plant,” ASME 2012 Power Conference, 2012, Anaheim, California, USA.

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MATHEMATICAL ANALYSIS OF ROPE BRAIDING

Mitchell J. Isaac, Chad L. Rodekohr David J. Branscomb Department of Physics, Presbyterian College, Department of Polymer and Fiber Engineering, Clinton, SC, USA Auburn University, Auburn, AL, USA

ABSTRACT This paper explores the physics and mathematics of the processes involved in braiding rope. When braiding rope, the braid point comes to an equilibrium point that correlates with the ratio between the take-up speed and the angular velocity of the braid machine. We examined the transitions between equilibrium points in search of an analytical equation that will describe the motion of the braid point. To validate analytical equations, a 16-yarn, diamond braided rope was braided to transition between equilibrium points, digital images were taken, and image analysis was done to collect data. The large amounts of collected data was analyzed using Mathematica®, a powerful math software package. This research resulted in two analytical equations and several empirical equations. The empirical equations found from the data were used to validate the analytical equations.

INTRODUCTION Throughout history, rope making has been more of a skill than a science. The art of rope making has been passed down from generation to generation, and improved along the way [1]. Today we use large machines to produce braided rope; however, different parameters of the machine must be manually Figure 1. A front view of the braid machine shows 16 set to obtain the desired type of rope. Parameters that must be carriers loaded with bobbins and 16 horned gears manually set include 1) how many carriers are loaded with arranged in a circle to create a sinusoidal-circular path yarn, 2) the tension in the individual yarns, 3) the angular for the carriers to follow. velocity of the braiding machine, and 4) the take-up speed of When the braiding machine is on, and the capstan is taking the braided rope. up rope, the braid point comes to an equilibrium point at which Contemporary braiding machines consist of spur gears o it remains relatively stationary in space. However, if the braid attached to horned that have four slots, each at 90 from the is currently at equilibrium and either the take-up speed or the adjacent slot, to pass carriers between gears. Carriers are loaded angular velocity of the braid machine is altered, the braid point with bobbins, as seen in figure 1, of yarn and contain a tensioning system that consists of two springs and a ratcheting will transition to a new equilibrium to compensate for the mechanism [2]. As carriers are passed between gears, they altered parameter. When the equilibrium point shifts, the follow a sinusoidal path around a circular track. Half of the conical angle (φ) changes, which causes the internal braid angle carriers move in a clockwise direction, while the other half (θ) to change, and ultimately affects the overall properties of move in a counter clockwise direction [1]. This causes the the rope. The conical angle (φ) and the braid angle (θ) can be individual yarns to interlace, forming the braid. The individual seen in figure 3. The smaller the internal braid angle (θ) is, the yarns come together at the braid point to form the braid and more axial the yarns are in the rope, meaning that the axial braid is taken up by a spinning wheel called a capstan [1]. coefficient of elasticity will be smaller and the axial load bearing ability of the rope will be greater.

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Today, there are all kinds of rope being braided, ranging from very cheap rope, to very expensive. In industrial settings, the transition between equilibrium points can cost a lot of time and money. The purpose in conducting this research is to be able to predict where in space the braid point will come to equilibrium, and, in doing so, this would reduce the time and material wasted during the transition between two equilibrium points. This has been accomplished by deriving analytical equation based on first principles and the careful examination of the braiding process. These analytical equations have been validated via empirical observation.

Figure 2. A schematic diagram of the braid machine and capstan.

During braiding point formation there are a few operating parameters varying and depending on each other. Rawal and Polturi simulated various braided structures based on mandrel shapes [4]. They mentioned that the braided structure is determined by the rotating speed of carriers, take-up speed, and radius of mandrel cross-section. The relationship between motion parameters and braiding geometry parameters were derived by Du and Popper [5]. They use a model according to this relationship to predict a braid angle. Long presents a model for the braiding process based on a general mandrel cross- section, which is composed of a number of flat facets [6]. The calculation of braid angle is also based on the motion of the mandrel. These discussions, however, assume the braiding point is stable. Actually, individual carriers will affect the final Figure 3. A digital image to show the conical angle(φ) structure of a braided product and can make the braiding point formed by the yarns going from the bobbins to the braid unstable during the braiding process. Few publications can be point, the braid angle(θ) formed by the yarns within the found regarding braiding process analysis and motion of braided rope, and the braid point shown by the curved braiding point with respect to a key component of a braiding arrow machine, that is, the carriers. . ANALYTICAL EQUATIONS Previous models assume the braid formation point in Analytical equations were derived based on logic, physics, (2) steady state equilibrium and rely on the longitudinal velocity of and first principles. The transitioning of the braid point between the mandrel, the angular rate of the bobbin carrier, and the equilibrium points can be described based on the concept of radius of a cylindrical mandrel [7]. Furthermore, these models two opposing velocities: the take-up velocity, and the velocity do not consider the carrier tension. Additional work has been of braid formation. done to develop process models for the dynamic braid It logically follows that when the braid point is at formation [8-10]. However these models rely on a mandrel to equilibrium, the velocity of braid formation must equal the account for the braiding diameter. The contribution of this velocity of braid take-up. The follow general equation results: work is unique in that the models presented thus far consider the braiding formation process and individual carrier tensions and do not require a core or mandrel. V = V −V braidformation xo braidpoint (1) A mathematical model of braiding point motion during braiding (which affects the final structure and properties of **Note: During data collection a blue dot was marked on the braided products) will also benefit tension control schemes. If braid at the initial braid point (xo) before each trial (Figure 3)** tension changes during the braiding process and how tension affects motion of the braiding point are known, the braiding Vbraidformation is the velocity of braid formation, Vxo is the point motion could be controlled by carrier tension. Work €velocity of the blue dot as it is part of the rope being taken up, supporting this effort has been investigated previously [11,12]. and Vbraidpoint is the velocity of the braid point. Notice that at

ASME 2012 Early Career Technical Journal - Vol. 11 212 equilibrium, the velocity of the braid point is zero. Therefore, at time as the braid point moves. Therefore, we must refer to equilibrium the velocity of braid formation must equal the Hooke’s law to redefine k in the equation describing the velocity of the initial braid point xo. The individual velocities in velocity of the initial braid point xo: the general equation (1) were then further examined and F Δl analytically defined. σ = Eε ⇒ = E

A l (4) o

Rearranging this equation, we get:

AE F = Δl € lo (5)

Finally lo is redefined and made a function of the conical angle(φ):

€ r l L bm o = − (6) tan(φ)

To finalize the analytical equation describing the velocity of the Figure 4. A digital image demonstrating the initial braid initial braid point xo we must find the limit as Δt⇒0, thus a point (xo) used to track the rope take-up as a function of time. derivative is taken to eliminate the discrete interval between φ1 and φ2€: VELOCITY OF THE XO POINT The analytical equation to describe the velocity of the initial braid point x was derived based on the concept that the ⎛ ⎞ o ⎜ ⎟ blue dot is part of the braided rope that is being taken up by the TN dφ capstan, which is rotating at an angular velocity of ω c. It V r ⎜ ⎟ sin( ) xo = cω c + φ follows that the velocity of the initial braid point xo should ⎜ AE(θ ) ⎟ dt equal the linear velocity of the edge of the capstan minus the ⎜ rbm ⎟ L − (7) stretch in the rope. The following initial equation results: ⎝ tan(φ ) ⎠

TN(cos(φ2 ) − cos(φ1)) V r xo = cω c − kΔt (2) This equation (7) describing the velocity of the initial braid € point xo is a differential equation that seems to be unsolvable. rc is the radius of the capstan, ωc is the angular velocity of the Therefore, we approximate the equation by assuming that the capstan, T is the tension in each individual yarn, N is the distance between equilibrium points is much smaller than the number of carriers loaded with a bobbin, φ is the conical angle, distance between the braid point and the capstan, and € k is the coefficient of elasticity, and t is time. The fallacy in the eliminating the term in the equation that accounts for the stretch equation (2) above, as it is written, lies in the coefficient of in the rope. Therefore the following equation results: elasticity. The coefficient of elasticity is defined by Hooke’s Law [3] as follows: V = V = r ω (8) xo capstan c c

F AE = k = VELOCITY OF BRAID FORMATION Δl lo (3) The analytical equation that describes the velocity of braid formation€ is derived based on the concept that the braid A is the cross-sectional area of the object, E is Young’s machine, loaded with 16 bobbins, lays 8 yarns right next to one another at the braid angle(θ), at a rate of N/2 yarns per second. modulus, and lo is the initial length. However, because the braid point moves during a transition between equilibrium points and

the capstan€ remains stationary, the initial length lo changes with

ASME 2012 Early Career Technical Journal - Vol. 11 213 was accomplished by setting the braid machine at a constant angular velocity while changing the capstan speed. Rope was first braided with the capstan set at one speed to allow the braid point to come to an initial equilibrium. The capstan speed was then either increased or decreased, and the braid point transitioned to its new equilibrium point. While the braid point transitioned between equilibrium points, digital images were taken at a rate of ten frames per second from a side view. Image analysis was performed on each individual image to track four points of interest: the blue dot, the braid point, and the upper and lower yarns; this is shown in Figure 5.

Figure 5. A diagram demonstrating two consecutive

yarns laid next to each other by counter-clockwise moving carriers.

The length of rope formed for every complete revolution made by the braid machine is defined as:

Nw

ltotal = 2sin(θ) (9) Figure 6. A sample digital image taken during the trials Taking into account that the period of one revolution of the and the four points of interest marked with black dots. braid machine is defined as 2π/ωbm, we get that the velocity of braid€ formation is defined as follows: The internal braid angle(θ) was measured from the actual braid produced in each trial. This was completed with the use of pencil, paper, a ruler, and a protractor. Each braid angle was Nω bm w Vbraidformation = (10) traced and then measured. 4π sin(θ)

N is the number of carriers loaded with bobbins on the braid machine, ωbm is the angular velocity of the braid machine, w is the €width of each individual yarn within the braid, and θ is the internal braid angle. If it is assumed that at equilibrium, the Figure 7. By coloring one yarn before braiding, the velocity of braid formation is equal to the velocity of take-up, internal braid angle (θ) of each yarn in the braided and the equation (10) is solved for θ, the result is an equation rope has been empirically observed. that produces the internal braid angle at equilibrium:

⎛ ⎞ Several plots were obtained directly from the data for each trial. Nω bm w These include the position of the xo point versus time (figures 8- θeq = Arc sin⎜ ⎟ ⎝ 4πrcω c ⎠ (11) 9), the position of the braid point versus time (figures 10-11), the conical angle versus time(figures 12-13), and the internal DATA braid angle versus length (figure 14). Each plot was fit with a To validate the analytical equations derived, they must be best-fit line and paired with the corresponding equation. compared€ to reality. To do this, data was collected, analyzed, and manipulated to produce meaningful graphs. Several trials were conducted with both nylon yarn and Kevlar yarn, in which rope was braided to transition between equilibrium points. This

ASME 2012 Early Career Technical Journal - Vol. 11 214 Nylon Nylon

Figure 8. Position of the initial braid point (xo) using

nylon yarns transitioning from a capstan angular Figure 10. Position of the braid point using nylon yarns

velocity of 0.038 rad/s to 0.063 rad/s has been transitioning from a capstan angular velocity of 0.038

observed to vary according to the function: rad/s to 0.063 rad/s has been observed to vary

−3.505t according to the function: xo = 20.94 −1.308e + 0.7691t (12) 2 −0.0294 t (15) R : 99.35% xbraidpoint = 28.34 −10.26e

Kevlar 2 R : 94.22%

€ Kevlar € €

€

Figure 9. Position of the initial braid point (xo) using

Kevlar yarns transitioning from a capstan angular

velocity of 0.038 rad/s to 0.063 rad/s has been

observed to vary according to the function: Figure 11. Position of the braid point using Kevlar x 15.06 0.6697 e−0.8277t 0.8073t (13) o = + + yarns transitioning from a capstan angular velocity of 2 R : 99.95% 0.038 rad/s to 0.063 rad/s has been observed to vary

according to the function: The plots of the position of the x point versus time o −0.0260t (figures 8-9) were best fit with an exponential function coupled xbraidpoint = 28.18 −12.68e € (16) with a linear function of the form: 2 R : 99.79% €

a + bt + ce−dt (14) The€ plots of the position of the braid point versus time (figures 10-11) were best fit with basic exponential functions The exponential terms in the best-fit equations (12-13) are of the form: € responsible for the slight curve in the plot from t=0s to t=2s, and is our empirical term that accounts for the stretch in the −ct rope. Notice€ that the linear term very quickly dominates, a + be (17) indicating that the stretch in the rope is negligible.

€

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Nylon The plots of the conical angle (φ) versus time (figures 12-13) were best fit with an exponential function coupled with sine function and resemble the form:

−ct a + be − dsin[ωt +δ] (20)

The exponential function in the equation (20) above accounts for the overall curve of the plot and the sine function in the equation (20) accounts for the fluctuations throughout the curve. €

Nylon

first page of the paper and is Arial font size 10, bold. Section titles are bold, Arial font type size 10 and in caps. Figure titles should be placed under the figure, centered Figure 12. Conical angle (φ) using nylon yarns transitioning from a capstan angular velocity of 0.038 and read as in this example below. Make sure each figure is referenced from within the text. Table titles use the same rad/s to 0.063 rad/s has been observed to vary according to the function: −0.021t MORE ON ASME PAPER FORMAT φ = 27.85 + 5.733e −1.684Sin(3.659t + 2.805) More information on ASME paper format may be found at 2 (18) R : 58.84%

the following link: Kevlar http://www.asme.org/kb/proceedings/proceedings/author-

€ templates

€ Nylon

Figure 14. Internal braid angle( θ) versus length using

nylon transitioning from a capstan angular velocity of

0.038 rad/s to 0.063 rad/s has been observed to vary

according to the function:

−0.02317l θ =12.33+ 6.744e (21) 2 R : 69.82%

A plot of length of rope formed versus time (figure 15) can € be obtained by subtracting the position of the braid point from

the position of€ the x point. o

Figure 13. Conical angle (φ) using Kevlar yarns

transitioning from a capstan angular velocity of 0.038

rad/s to 0.063 rad/s has been observed to vary

according to the function:

−0.0313t φ = 28.57 + 7.011e −1.629Sin(4.076t + 0.8566)

2 (19) R : 69.47%

€ **Note: The rate at which data was taken resonated, somewhat, with the rate at which the conical angle oscillated; therefore, the amplitudes seen in the data vary inconsistently, € causing the R2 values for the plots (figures 12-13) to be φ(t) lower than expected.**

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Nylon

Figure 17. Internal braid angle( θ) as a function of the Figure 15. Braid formed versus time for nylon yarns conical angle(Φ) has be found to vary according to the transitioning from a capstan angular velocity of .038 function: rad/sec to .063 rad/sec has been observed to vary −1.353E −26(. −26.75+φ )24.70 0.8240 according to the function: θ( ) =12.33+1.068e (−26.75 + φ) (22) φ (24) l = −1.168 +1.281e−0.4677t + 0.6735t R2 : 99.72% RESULTS AND CONCLUSIONS In the plots of the xo point versus time (figures 8-9), the By combining the best-fit equations from the θ(l) plot€ linear term becomes dominant very quickly. This indicates that (figure€ 14) and the length of rope(t) plot (figure 15), we produce it was a reasonable assumption to discard the term in the an equation that describes the internal braid angle(θ) as a analytical equation describing the velocity of the initial braid function of the€ length of braid formed, which is a function of point xo that accounts for the stretch in the rope. time; therefore, this results in an equation that describes the ⎛ ⎞ internal braid angle(θ) as a function of time(θ(t)) (23) (figure ⎜ ⎟ 16). Furthermore, we found the best-fit equation describing the ⎜ ⎟ conical angle(φ) as a function of time (18), for time, and ⎜ TN ⎟ dφ Vx = rcωc + sin(φ) inserted the resulting equation into the θ(t) equation (23). This o ⎜ AE ⎟ dt ⎜ ()θ ⎟ results in an empirical equation that describes the internal braid r ⎜ bm ⎟ angle( ) as a function of the conical angle( ) (24) (figure 17). ⎜ L − ⎟ θ φ ⎝ tan(φ) ⎠

Also, the plot of the internal braid angle versus time for

nylon transitioning from a capstan angular velocity of .038 € rad/sec to .063 rad/sec (figure 16) was extrapolated to find the

equilibrium internal braid angle. The equilibrium internal braid

angle found from extrapolating empirical data was found to be

12.33o. The parameters of that same trial were then inserted

into our derived analytical equation that describes the

equilibrium internal braid angle.

⎛ Nω w⎞ θ = Arc sin⎜ bm ⎟ eq 4πr ω ⎝ c c ⎠

Figure 16. Internal braid angle( θ) as a function of time For this trial (figures 8, 10, 12), N was 16, was 1.9 rad/sec, was found to varry according to the function: ωbm w was 0.0008 meters, rc was 0.14 meters, and ωc was 0.063 −0.02317(−1.168+1.281e −0.4677t +0.6735t ) θ =12.33+ 6.744e (23) €

ASME 2012 Early Career Technical Journal - Vol. 11 217 € rad/sec. The resulting equilibrium internal braid angle found [9] Nishimoto, H., Ohtani, A., Nakai, A., Hamada, H., using the analytical equation was 12.74o. This is a 3.32% Generation and prediction methods for circumferential difference between the analytical equilibrium internal braid distribution changes in the braiding angle on a cylindrical angle and the empirical equilibrium internal braid angle. braided fabric, Proceedings of the Institution of Mechanical The analytical equation (11) derived to describe the Engineers, Part L: Journal of Materials: Design and equilibrium internal braid angle is very significant when used in Applications, 2010 conjunction with the empirical equation (24) that describes the [10] Nishimoto, H., Ohtani, A., Nakai, A., Hamada, H., internal braid angle as a function of the conical angle. If the Prediction Method for Temporal Change in Fiber Orientation conical angle can be determined based on parameters that are on Cylindrical Braided Preforms, Textile Research Journal, manually set, then the distance from the braid machine at which 2010 the braid point will come to equilibrium can easily be [11] Pastore, Ko, F., (1988). CIM of Braided Preforms for determined using trigonometry. This would save time, Composties, Computer Aided Design in Composite Material materials, and money in the rope braiding industry. There is, Technology, Proceedings of the International Conference, 135- however, the equation (24) describing the internal braid angle 155 as a function of the conical angle has a limited range of [12] Ebler, N., Arnason, R., and Michaelis, G. (1993). Tension validity. By simply taking data over a longer transition between equilibrium points and repeating the process to obtain a more generalized equation relating the internal braid angle and the conical angle can remedy this issue.

FUTURE WORK Future work for this project includes estimating the trigonometric functions in the analytical equations using Taylor’s expansion. The trigonometric functions in the analytical equations can accurately be estimated using Taylor’s expansion because the change in θ is very small, with a range of about ten degrees. Using this technique we hope to bridge the empirical and analytical equation, and find an accurate best- fit curve of the experimental data using the derived analytical equations. This will further validate the analytically derived equations. Both the nylon data and the Kevlar data are in the process of being analyzed using this method.

REFERENCES [1] D. J. Branscomb, “New Directions in Braiding,” Fiber Society Meeting Presentation manuscript, 2010. [2] G. Ma, D. J. Branscomb, and D. G. Beale, “Modeling of the Tensioning System on a Braiding Machine Carrier,” Mechanism and Machine Theory, v.47, Jan, 2012. [3] Tipler, Paul Allen, and Gene Mosca. Physics for Scientists and Engineers. New York: W. H. Freeman, p 409-411, 2007. [4] Rawal, A., Potluri, P., Geometrical modeling of the yarn paths in three-dimensional braided structures, Journal of industrial textiles, v35, I 2, 115 (2005). [5] Du, G., Popper, P. (1994). Analysis of a circular braiding process for complex shapes Journal of the Textile Institute, v85, n3, 316-337. [6] Long, A., (2001). Process modelling for liquid moulding of braided performs, Composite Part A: Applied Science and Manufacturing, V32, I7 941-953 [7] Potluri, P., Rawal, A., Rivaldi, M. and Porat, I. (2003). Geometrical Modelling and Control of a Triaxial Braiding Machine for Producing 3D Preforms, Composites: Part A, 34(6): 481–492 [8] Mazzawi, A., The Steady State and Transient Behavior of 2D Braiding, University of Ottawa, PhD Dissertation, 2001

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