Fast Calculation of Thermodynamic Properties of Water and Steam in Process Modelling using Spline Interpolation

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1 P R E P R N T CPWS XV Berlin, September 8, 008 Fast Calculation of Termodynamic Properties of Water and Steam in Process Modelling using Spline nterpolation Mattias Kunick a, Hans-Joacim Kretzscmar a, and Uwe Gampe b a Department of Tecnical Termodynamics, Zittau/Goerlitz University of Applied Sciences, 076 Zittau, Germany b nstitute for Power Engineering, Cair of Termal Power Macinery and Plants, Tecnical University of Dresden, 006 Dresden, Germany mattiaskunick@otmail.com, j.kretzscmar@s-zigr.de, uwe.gampe@tu-dresden.de Te designing and optimising of advanced power cycles and processes requires fast and accurate metods for calculating termodynamic properties. Precise fundamental equations for most working fluids ave already been developed. n industrial software applications suc as process optimisations wit eat balance calculations, te calculation of tese equations requires enormous computing time. n Computational Fluid Dynamics (CFD), ideal-gas equations or oter simple algoritms are often used to acieve reasonable computing times, but tis procedure leads to inaccuracies in te process calculation. APWS terefore establised a task group in 007 to develop extremely fast and accurate property algoritms for water and steam. Tis paper gives an overview of te aims, initial results acieved at te Zittau/Goerlitz University of Applied Sciences and of future tasks. An advanced metod for te calculation of termodynamic properties using spline interpolation is currently in development. An initial study as been done wit te calculation of T(p, and (p, in te steam region of APWS- F97. ntroduction Optimising eat cycles and calculating nonstationary processes require extremely fast algoritms for termodynamic properties of working fluids because tey are frequently used in te inner iteration cycles of te process calculations. Te APWS-F97 [,] contains very fast and accurate equations. For Computational Fluid Dynamics (CFD), owever, even APWS- F97 is too slow. Terefore in CFD, fluid properties are often calculated wit simple equations, for example wit te ideal gas equation. Depending on te range of state tis procedure leads to inaccuracies in te process calculation. APWS terefore establised a task group in 007 for developing fast property algoritms for water and steam. n te past, table look-up metods ave been developed to calculate fluid properties faster and wit reasonable accuracy. One of tese metods, te Tabular Taylor Series Expansion Metod (TTSE) [], was adopted by APWS as a guideline in 00 []. One disadvantage of tis metod is tat it does not represent te property surfaces in a continuous form. n order to calculate fluid properties as fast as or faster as TTSE but wit continuously represented property surfaces, two-dimensional splines can be used in a table look up metod. n te first step a bi-quadratic spline surface for te function T ( p, for region of APWS- F97 was developed. Additionally, ( p, was obtained by solving T ( p, in terms of. Tis paper gives a brief introduction into splinebased fluid property calculations and discusses its caracteristics in comparison to APWS-F97 and TTSE. Te main focus of tis work is te development of an extremely fast spline-based algoritm for calculating termodynamic properties. An investigation to determine splineinterpolation algoritms for termodynamic properties not only for water and steam but also for oter pure fluids and mixtures is currently being conducted.

2 Aims of te Project n ligt of te requirements of modern calculation procedures suc as solver-based eat cycle calculation and CFD, a metod for fast and accurate calculation of fluid properties sould be developed. Te resulting algoritm sall substitute te calculation of properties from fundamental equations wen ig calculation speed and ig accuracy is required. Because of iterative procedures in te superior process calculation, steadiness of te supplied functions and of teir first derivatives is required. Anoter point of interest is te numerical consistency between forward and backward functions suc as (p, and T(p,. n complex calculations, suc as simulations of transient processes, numerical consistency is important. Te algoritm sould be usable for working fluid mixtures, as well. Tese requirements lead to spline-based table look-up metods. n order to minimize computation time, fast searc algoritms and data andling metods must be applied. For eac property function, te corresponding look-up table must be pre-processed. A software tool will be created wic will generate tese tables and necessary functions directly as usable source code or software libraries. Spline nterpolation of Termodynamic Properties Tis section gives a brief description of ow a spline-based calculation of fluid properties can be introduced into process calculations. f a property z is determined as a function of te variables x and x ten a grid of values of tese properties can be created as sown in Fig., were and J denote te number of grid lines along x and x, respectively. Consequently te grid consists of J nodes and ( ) ( J ) sub-rectangles. t is advantageous if te grid lines are equidistant, as tis enables us to determine te corresponding sub-rectangle in te grid from a given point (x, x ) witout a compreensive searc algoritm. Tis is important because it would slow down te computing time of z ( x, x). A subdivision of te grid into several equidistant grids can be done as sown in Fig.. Te density of te grid can ten be canged locally in order to optimise te accuracy of te resulting spline function. x j J j + z ( x, x ) j i i + i Figure : Grid of nodes Spline functions zij ( x, x) of te same structure must be defined for eac sub-rectangle (i, j) (see Fig. ). Depending on te requirements of te superior process calculation, a function must be cosen wic is able to represent te fluid surface properly. A certain accuracy of zij ( x, x) EOS comparison to z ( x, x) in, te property function derived from te equation of state, is required first. A second demand comes from te solver-based calculation algoritms, suc as Newton s metod for non-linear systems, wic are usually applied in eat cycle calculation programs. Normally te iterative procedures require continuous property functions and continuous first derivatives. Te simplest function wic is able to satisfy tese requirements is te bi-quadratic polynomial. t can be written as: k l ij (, ) = ijkl ( i ) ( j ) () k= l= z x x a x x x x or zij ( x, x ) = aij + aij Δ x i + aδx i + a Δ x + a Δx Δ x + a Δx Δx + a Δ x + a Δx Δ x + a Δx Δx ij j ij i j i j ij j ij i j i j wit Δ xi = ( x xi) and x j ( x x j ) () Δ =. Te coefficients for tis function must be determined so tat te resulting function z is continuously differentiable at least once. To determine te 9 ( )( J ) coefficients, 9 for eac sub-rectangle, te same number of conditions must be given. Tere are several possible ways of calculating te coefficients. Te coice of te conditions influences accuracy and steadiness of te resulting spline function. To create a spline function wile preserving its optimal sape, it is recommended in [6] to ave knots wic are different from nodes. Tis means tat a second grid of knots must be created. Tis sould be done as sown in Figure. ij x

3 j J x i x Figure : Grid of nodes and grid of knots Fig. illustrates te relations between te grids; ere ( ) denotes te grid of knots. j J x j + j j j i i i i + Grid of nodes Grid of knots Grid of nodes Grid of knots i Figure : Sub-rectangle in grid of nodes and grid of knots Te interpolation requirement is to be fulfilled at ( i, j ). Te derivatives ( z/ x ) x at ( i, j) and ( i, j) as well as ( z/ x ) x at i j (, ) and ( i, j ), marked wit wite circles, must be equal to te corresponding derivatives of te neigbouring sub-rectangles. n addition, te crossed derivatives ( z/ ( x x) ) at te four corner points ( i, j ), ( i, j ), ( i, j ) and ( i, j ) must be equal to te corresponding derivatives of te neigbouring sub-rectangles. Te system of equations can be solved for given derivatives at te boundary knots to obtain te coefficients of te spline polynomials. Te continuous beaviour at te nodes as well as at te boundaries between te sub-rectangles and teir first derivatives can be matematically proven [6]. Since all necessary function values z i,j (x, x ) and derivatives ( z/ x ) x and ( z/ x ) x can be calculated directly from te fundamental equation, it is possible to create a spline wit tis approac. An inverse function for x (z, x ) or x (z, x ) can be obtained by solving zij ( x, x ), Eq. (), in terms of eiter x or x. x For example x (z, x ) in a sub-rectangle (i, j) can be calculated from (, ) ( B± B AC) x zx A x wit A = a +Δ x a + a Δ x,, ij = + i () ij j ij ij j B = a +Δ x a + a Δ x and ij j ij ij j C = a +Δ x a + a Δx z, ij j ij ij j were Δ x j is given in Eq. (). Te calculation of tis equation consumes more computing time tan te calculation of zij ( x, x) since te operations square root and division are slower tan multiplications. But te advantage is tat x, ij ( zx, ) is completely numerically consistent to te spline z x, x. ij ( ) Spline-Function T(p, and nverse Spline Function (p, for APWS-F97 Region Especially in eat cycle calculations termodynamic properties are frequently calculated from p and. n order to find out about te quality of a spline-based calculation of termodynamic properties wit te approac described in te section above, a spline function T ( p, for APWS-F97 region as been created. Furtermore a function ( p, was prepared by solving te spline function T ( p, in terms of. Consequently ( p, is completely numerically consistent to T ( p,. n te first step te p--grid was given and not optimised for accuracy. To reac te required accuracy te grid as been created as sown in F97 Fig. above. Te function T ( p, as been iterated from te fundamental equation for all nodes in te range MPa < p MPa and kj kg 6 kj kg. Extrapolation was necessary because te spline algoritm currently in use is usable for rectangular grids only. Te subdivision of te grid is also indicated in Fig.. Table sows te number of grid lines in te corresponding ranges. Te resulting grid consists of 60,000 nodes.

4 p / MPa C MPa 50 C δ kj kg.kj kg c - 97 T B ( p ) 80 kj kg δ.65kj kg 0 MPa 6 kj kg - 50 MPa 0.9MPa C 0. MPa = 0.MPa MPa 0.9kPa kPa /kjkg Figure : lg p--diagram wit sub-divided p- grid according to Table, were δp and δ indicate te distance along p and, respectively Te calculation of te entire spline function T ( p, gives a maximum relative deviation of 0 F97 % in comparison wit T ( p, iterated from te fundamental equation of APWS-F97 region. Tis means tat te values of F97 T ( p, of APWS-F97 are represented by 5 significant figures. Table : Number of grid lines in p--grid for region Range p-grid lines MPa p 0.0 MPa 0.0 MPa p 0. MPa 0. MPa p 0 MPa 0 MPa p MPa Range -grid lines kj kg 80 kj kg kj kg 6kJ kg Computing time comparisons Te computing time comparisons were carried out wit a Pentium Xeon. GHz PC and Microsoft Windows XP operating system. Using te APWS software NFBENCH [], te computing speed of te developed splineinterpolation algoritms was compared to APWS- F97 and to te TTSE metod. First, te function T ( p, was compared to te corresponding APWS-F97 backward 7BW equations T 9 ( p, and to te TTSE function TTSE T ( p, [5]. Table sows te average computing times. Tey were determined for,000 state points arbitrarily distributed in APWS-F97 region. As can be seen, te spline function is twice as fast as APWS-F97 backward equations. Te TTSE is even slower tan APWS- F97. Te reason for tis is te searc algoritm of TTSE and its internal cell-finding logic using previous results. For given state points ( p, ) close to eac oter, owever, te TTSE function would be. times faster tan APWS-F97. Table : Computing times for T(p, in μs T ( p, T p, 97BW TTSE T p, An interesting result of te investigations is tat te computing speed of te spline interpolation is nearly independent of te grid size and terefore of te number of nodes. Furtermore te computing speed of te inverse spline function ( p, was investigated. Te computing times of p T N V p, T are F97 (, ) and

5 listed in Table. t can be seen t at ( p, is. times faster tan F97 ( p, T ), but completely numerically consistent to T ( p,. Te reason for tis relatively low factor is tat for evaluating PL ( p, from T S ( p, ) an auxiliary spline function, ( p,, is used for generating an initial estimate for. Ten, in te p--grid, te corresponding sub-rectangle is determined by solving T ( p, in terms of an d comparing to te neigbouring nodes until ij ij + is fulfilled. Te square root operation in Eq. () also slows down computation speed. f greater computing speed for ( p, T ) is required and numerical consistency wit T ( p, does not need to be %, a sep arate p T could be generated. spline function (, ) Table : Computing times for (p, in μs ( p, F97 TTSE p, T ( p, T ) Summary and Outlook Te first results of tis project sow tat spline functions can be used to represent termodynamic properties. t as been proven tat a reduction in computing time is possible wile at te same time acieving ig accuracy and complete numerical consistency. Te computing speed is nearly independent of te size of te grid. Tis enables bot ig accuracy and low computing times. A continuous spline function T ( p, for region as been developed. By solving T ( p, in terms of, te function ( p,, wic is completely numerically consistent to T ( p,, could be obtained. Te computational speed of tese two functions is considerably faster tan tat of te APWS-F97 fundamental or backward equations. Te data grid of te splinepolynomials was created so tat bot functions represent APWS-F97 wit ig accuracy. Due to special data andling and simple searc algoritms, tis metod is even faster tan te TTSE metod. Now te algoritm must be modified to enable te creation of spline functions from nonrectangular grids. An algoritm for grid optimisation is also necessary in order to reac te required accuracy and to reduce te amount of memory needed. n order to make use of te described metod in calculations of non-stationary processes, spline functions from v-u and v- grids will be created. Te use of oter spline functions, suc as bicubic polynomials, is also intended for twodimensional functions. Spline algoritms are a very promising metod for increasing te computing speed in calculating termodynamic properties of mixtures. To make spline-based table look-up metods available for industrial software applications, te necessary algoritms need to be provided in a convenient form. Te software tool FluidGrid, currently in development, sould meet tis need. Tis tool will provide an interface wic allows te user to prepare spline-based property libraries. Tis will be possible for available equations of state provided by te user in a dynamic link library. For given range of state and required accuracy FluidGrid will create and optimise a data grid and te corresponding spline function. Finally te software will provide automatically generated source code wic can ten be linked to and compiled in te user s application. Literature [] APWS: Release on te APWS ndustrial Formulation 997 for te Termodynamic Properties of Water and Steam, available at te APWS website ttp:// (007). [] Wagner, W.: Te APWS ndustrial Formulation 997 for te Termodynamic Properties of Water and Steam, Journal of Engineering for Gas Turbines and Power Volume, 50-8 (000). [] Miyagawa, K. and Hill, P. G.: Rapid and Accurate Calculation of Water and Steam Properties Using te Tabular Taylor Series Expansion Metod, Journal of Engineering for Gas Turbines and Power Volume, ssue, pp (00). [] APWS: Guideline on te Tabular Taylor Series Expansion (TTSE) Metod for Calculation of Termodynamic Properties of Water and Steam Applied to APWS-95 as an Example, available at te APWS website ttp:// (00). [5] Miyagawa, K.: Software for te Tabular Taylor Series Expansion Metod (TTSE), Personal communication (008). [6] H. Spät: Two-Dimensional Spline nterpolation Algoritms, Oldenburg (995). 5