The Finite Difference Method Applied for the Simulation of the Heat Exchangers Dynamics

Transcription

1 Prceedings f the 3th WEA Internatinal Cnference n YTEM The Finite Difference Methd Applied fr the imulatin f the Heat Exchangers Dynamics PAVEL NEVRIVA, TEPAN OZANA, LADILAV VILIMEC Department f Measurement and Cntrl, Department f Energy Engineering VŠB-Technical University f Ostrava 7. listpadu 5/7, Ostrava-Pruba, CZECH REPUBLIC stepan.zana@vsb.cz Abstract: - Heat exchangers that transfer energy frm flue gas t steam are imprtant units f thermal pwer statins. Their inertias are ften decisive fr the design f the steam temperature cntrl system. In this paper, the analysis and the simulatin f the dynamics f the steam superheater are discussed. uperheater is simulated as a unit f a cntrl lp that generates steam f desired state values. There are many types f steam superheaters. In this paper, the steam superheater f the cunterflw heating arrangement is presented as an example. The flue gas is the prduct f cmbustin f fuel. The heated medium is saturated steam, generated by the evaprating part f the biler. T simulate the steam superheater n the cmputer, the exchanger is described by the set f partial differential equatins. The equatins are then slved numerically by mdified finite difference methd. Discussin f methd and qualitative and quantitative results are presented. Key-Wrds: - imulatin, Heat exchangers, uperheaters, Partial differential equatins, Finite difference methd Intrductin Heat exchangers cnvert energy frm a heating medium t a heated medium. In this paper, the steam superheater is the heat exchanger that transfers energy frm flue gas t steam in the biler f a thermal pwer statin r a heating plant. The heating medium is usually the flue gas generated by cmbustin f sme kind f fuel. The heated medium is usually steam r the mixture f steam and air. Fig. presents the example f an arrangement f superheaters and reheaters in a big biler. In this paper, the analysis and the simulatin f the dynamics f thermal state variables f a superheater are discussed. uperheater is cnsidered t be the part f the cntrl lp that generates steam f desired state values. Nte that there are many types and cnfiguratins f superheaters. Mrever, ne energy blck f a pwer statin usually cntains several different superheaters. The intercnnectins f superheaters differ frm case t case. The mathematical mdel f the superheaters has t be therefre universal. It has t accmmdate different types f heat exchanging units and has t interface mdels f piping, valves, cntrllers, and ther parts f the cntrl system. The numerical methd applied t cnstructin f numerical mdels f superheaters shuld be als suitable fr numerical simulatin f thse very hetergeneus parts f a biler. Technical designs f the superheaters result in cnstructins that are cmplicated and cmplex. Accuracy f a three-dimensinal dynamical mdel f a superheater is limited by the accuracy f its parameters. T simulate the dynamics f the superheater, the nedimensinal mdel f the superheater was develped and tested. The fllwing paragraphs deal with superheaters that perate in a nrmal perating mde. Fig. The example f an arrangement f superheaters and reheaters in a big biler [8] IN: IBN:

2 Prceedings f the 3th WEA Internatinal Cnference n YTEM Mathematical mdel f a superheater In the basic frm, the thermal mdel f a superheater is defined by seven state variables. They are as fllws: ( x ( x ( x T, temperature f steam T, temperature f flue gas T, temperature f the wall f the heat exchanging surface f the superheater p pressure f steam p pressure f flue gas u velcity f steam u velcity f flue gas where x is the space variable alng the active length f the wall f the heat exchanging surface f the superheater and t is time. Fig. shws the principal schema f the physical state variables at a cunterflw steam superheater. Fig. Principal schema f the physical state variables at a cunterflw steam superheater. In the simplified mdel, presented in this paper, bth the pressure and the flw velcity f the flue gas are assumed t be the given functins independent f length. That is the pressure p = p ( acts usually as the input and velcity u = u ( is the functin f p () t. Three f the five remaining state variables are selected t be the input and utput variables, and the tw are the superheater s state variables. Input and utput variables are usually temperature f steam T, temperature f flue gas T, and pressure f steam p. uperheater s state variables are velcity f the steam u and temperature f the wall f the heat exchanging surface f the superheater, temperature f T x, t. the wall, ( ) Applying the energy equatins, Newtn s equatin, and heat transfer equatin, and principle f cntinuity the behavir f five state variables f superheater can be well described by five nnlinear partial differential equatins, PDE, as fllws: Reduced energy equatin fr flue gas: c ρf T T + ( ) 0 α u + T T = O Heat transfer equatin describes the transfer f heat frm burned gases t steam via the wall: T T T T T = 0 () cg cg αo αo Principle f cntinuity fr steam: ρ p ρ F p F Fu + u ρ F ρ p p F p F ρ p ρ u +ρ + F + = 0 p p Newtn s partial differential equatin fr steam: () (3) p u u + ρu + ρ + kρ + ρλ u u = 0 d (4) Energy partial equatin fr steam: u u ρ c T + + ρu ct + + { p. u} + { p. u. g. z} αo ( T T ) = 0 F (5) ( p T ) ( p T ) c = c, heat capacity f steam at cnstant pressure c = c, heat capacity f flue gas at cnstant pressure J.kg - K - J.kg - K - c heat capacity f J.kg - K - wall material d diameter f structural tube m F = F steam pass crssectin m F = F flue gas channel m crssectin g acceleratin f gravity m.s - G = G weight f wall per unit f length in x directin kg.m - IN: IBN:

3 Prceedings f the 3th WEA Internatinal Cnference n YTEM k cnstructinal parameter m.s - L active length f the wall m O O surface f wall per unit f m length in x directin fr steam O O surface f wall per unit f length in x directin m fr flue gas p = p pressure f steam Pa p = p pressure f flue gas Pa t time s T = T temperature f steam ºC T = T temperature f flue gas ºC T = T temperature f the wall ºC u = u velcity f the steam m.s - in x directin u = u velcity f the flue gas m.s - in x directin x space variable alng the active length f the wall m z = z grund elevatin f the superheater m α heat transfer cefficient between the wall and steam J.m- s - K - α heat transfer cefficient between the wall J.m - s - K - and flue gas λ = λ steam frictin cefficient ρ = ρ ( p,t ) density f steam kg.m -3 ρ = ρ ( p,t ) density f flue gas kg.m -3 3 Numerical methd The set f PDE ()-(5) can be slved by many methds. Equatins ()-(5) have tw independent variables. They are space variable length x [ 0, L] and time t [ 0, ). It fllws that PDE ()-(5) are nt suitable fr slutin by finite elements methd. Here, the methd f finite differences was used. The methd was mdified t facilitate the simulatin f cntrl prblems. The mdificatin can be demnstrated as fllws: Let PDE (6) is given. ( ) x, t + B [ 0, L], t [ 0, ) A = C (6) x where y is any physical variable A x, t, B x, t, C x, t are knwn ( ) ( ) ( ) functins PDE (6) is t be slved by the mdified methd f finite differences. The interval L f the space crdinate x is divided int n intervals f discretizatin f x. Intervals are f cnstant length L h =. n y x, t ( ) The derivatives f ( x y, are in the grid pints x i, i =,, L, n, apprximated by the differences f the secnd rder as fllws: yi 3y ( + 4y ( y ( y () t y () t y () t y () t y () t 4y () t + 3y () t where x x x3 M x n h = D = D3 n h () t = y, i =,, L, n 3 4 xi n h h 3 = D n ( x f ( x = Dn (7) y, The derivatives y, in grid pints x i, i =,, L, n can be assigned as dy ( dt x i i dt, i =,, L, n The PDE is apprximated by the set f n rdinary differential equatins, ODE. The independent variable f ODE is time. ( A( x D B( x i i, i + i, = C( xi,, i =,, L, n (9) Integratin f the set f ODE by sme standard numerical methd, Euler methd, Runge-Kutta methds, linear multistep methds, r any ther numerical frmula can be applied. The rder f the mdel may be rather large. Let us neglect the cmplex mdel f biler and turbine and cnsider nly its part cnsisting f fur heat exchangers and five intercnnecting steam lines. Representing the length crdinate f every exchanger and every intercnnecting steam line at twenty nds, we btain (4+5)*5*0=900 ODE f the first rder. (8) IN: IBN:

4 Prceedings f the 3th WEA Internatinal Cnference n YTEM Adaptatin f the methd cvers the advantages f the standard finite element methd and is applicable t simulatin f the cmplex superheater cntrl prblems, where the integratin in time is frequent. Dynamics f superheater s state variables is unequal. The change f the input pressure f steam p ( 0, prpagates thrugh the superheater s tract with velcity f the sund. Velcity f bth, the steam and the flue gas is up t ten meters per secnd. On the ther hand, the time inertia f the steam utput temperature with respect t the change f the flue gas input temperature may be measured in tens f minutes. It results in small integratin step f numerical methd used fr integratin f the system f ODE. As fr the cmputatin time, numerical methds fr the stiff equatins have certain advantage ver the standard methds. In example presented belw, the MATLAB tiff/ndf frmula was used. metimes, in perating pint, the derivatives f parameters f in PDE (3) can be neglected. Then, als the flw velcity and the pressure f steam can be assumed t be the knwn functins f time. Under these presumptins, the mathematical mdel f superheater describes nly the relatively slw heat transfer between media. Fr cnstant steam pass crssectin F = F steam velcity and steam pressure act as knwn inputs independent f length x, u ( x, = u(, p ( x, = p (. Fr hrizntal wall the equatins ()-(5) are replaced by equatins (0)-(): Reduced energy equatin fr flue gas: cρf T T u + + αo Heat transfer equatin f the wall: T T T cg α O T T cg α O ( T T ) = 0 = 0 Reduced energy equatin fr steam: c ρf u + + T α = O ( T ) 0 (0) () () Numerical calculatin f equatins (6)-(8) runs abut 000 times faster than runs the calculatin f the riginal set f equatins ()-(5). 4 Example The dynamics f the first superheater f the medium-size experimental steam generatr was simulated. The superheater is f a cunter-flw arrangement. At timet = 0 the superheater is at its steady-state. Temperatures f the steam, flue gas, and the wall are stabilized. It fllws that pressure f the steam and flue gas is stabilized t. The steady state f the superheater is defined by PDE ()-(5) fr actual parameters f the superheater and actual parameters f bth heat transferring media at the inputs t the superheater. Exact descriptin f all parameters f the superheater is beynd the extent f this paper. Typical parameters f the selected superheater are as fllws: active length f the wall L 60 m, weight f wall per unit f length in x directin G 350 kg.m team velcity u m.s.flue gas velcity u 7m.s. uch superheater represents an unit f an extremely great thermal inertia. team input temperature T ( 0,0) = 95 C. Flue gas input temperature T ( L,0) = 460 C. The Finite Difference Methd apprximates the PDE ()-(5) by 5 0 = 00 nnlinear ODE f the first rder. Calculating the steady state f the superheater, 00 steady-states f its numerical apprximatin were fund. Nte that in presented example, the steady-state steam utput temperature T ( L,0) = 44 C, steady state flue gas utput temperature T ( 0,0) = 75 C. lutin f PDE ()-(5) is usually made by methd f simulatin, parameters f superheater are ptimized with respect t technlgy. T study the dynamics f the superheater, tw simulatin experimens are presented belw. 4. tep change f steam At time t = 0, there is the step change f parameters f the steam at the input t the superheater. The temperature f the input steam is increased by the step value ΔT ( 0, = + 0 Ct 0. Change f ne state variable results in changes f all remaining state variables. Fig.3 shws the temperature increase ΔT ( L, f the steam at the utput and the temperature increase ΔT ( 0, f the flue gas at the utput. IN: IBN:

5 Prceedings f the 3th WEA Internatinal Cnference n YTEM value ΔT (L, t ) = +0 C t 0. Fig.5 crrespnds t Fig.3. It shws the temperature increase ΔT(L, t ) f the steam at the utput and the temperature increase ΔT (0, t ) f the flue gas at the utput. Fig.3 Increase f temperatures f media at the utputs f the superheater as the respnse t the step increase ΔT(0, t ) = +0 C t 0 f the temperature f the steam at the input. ΔT (L, t ) is increase f the temperature f the steam at the utput, ΔT (0, t ) is increase f the temperature f the flue gas at the utput Fig.5 Increase f temperatures f media at the utputs f the superheater as the respnse t the step increase ΔT (L, t ) = +0 C t 0 f the temperature f the flue gas at the input. ΔT (L, t ) is increase f the temperature f the steam at the utput, ΔT (0, t ) is increase f the temperature f the flue gas at the utput. Fig.4 shws, in lgarithmic scale, the temperature wave T t ) in the beginning f the prcess. Fig.6 shws, in lgarithmic scale, the temperature wave T (x, t ) in the beginning f the prcess. Fig.4 Increase f temperature ΔT t ) f steam in the superheater as the respnse t the step increase ΔT(0, t ) = +0 C t 0 f the temperature f the steam at the input. Fig.6 Increase f temperature ΔT t ) f flue gas in the superheater as the respnse t the step increase ΔT (L, t ) = +0 C t 0 f the temperature f the flue gas at the input. 4. tep change f flue gas At time t = 0, there is the step change f parameters f the flue gas at the input t the superheater. The temperature f the input flue gas is increased by the step IN: IBN:

6 Prceedings f the 3th WEA Internatinal Cnference n YTEM 5 Cnclusin The mathematical mdel f superheater presented in this paper describes the heat exchanger as a dynamic unit f a large distributed and diversified cntrl lp. The gal f the cntrl lp is t generate steam f desired state values. Generally speaking, the end f cntrl is t generate ptimally energy. The system cvers many aggregates and technlgical sets. Mathematical mdel f the prcess is very vast. The biler with its heat exchangers is in the middle f the mdel. The cmplexity f the system and demand fr the realtime simulatin led t the cnstructin f mdels in MATLAB / imulink / C++ envirnment. There are many cefficients in the equatins ()-(5). The accuracy f simulatin results depends n bth accuracy and crrectness f these cefficients. The prblem is there are n sufficient tls t quantify the accuracy f simulatin quickly. Ideal technique wuld be the cmparisn f the simulated and actual measured values. There is the big time lag between the biler design and pwer plant actuatin. In the meantime, the requirement n simulatin crrectin may expire. Cmparisn between simulated and measured data cnverges with technlgical research tasks. It is nt easy t cnvert the ptimized parameter values frm ne superheater structure t ther superheater designs. Nte that the required measurement at pwer plant is extremely expensive. Adequate methd f accuracy estimatin results frm experience. It cmpares selected steady-state values f physical variables btained by simulatin with values specified by the thermal and hydraulic biler calculatin, T&HBC. uch quantificatin f accuracy is partial und incmplete. The T&HBC defines perating parameters f the superheater primarily. It als defines varius perating steady-state values f state variables at bth the input and the utput f the superheater. It wuld be pssible slightly rectify cefficients f PDE t btain the slutin f system f ODE that generates the slutin that has the steady-state inputs and utputs equal t thse defined by the T&HBC. In practice, nly heat transfer cefficient between steam and steal structure f the wall f the heat exchanging surface f the superheater is adapted. Thus, there remains a little difference between bth simulated and T&HBC data. Acknwledgement: The wrk was supprted by the grant imulatin f heat exchangers with the high temperature wrking media and applicatin f mdels fr ptimal cntrl f heat exchangers, N.0/09/003, f the Czech cience Fundatin. References: [] Chen Chi-Tsng, ystem and ignal Analysis, ecnd Editin, undere Cllege Publishing, 989 [] Ck R. D. at al., Applicatins f Finite Element Analysis, Jhn Wiley and ns, 00, IBN [3] Dukelw. G., The Cntrl f Bilers, nd Editin, IA 99, IBN X [4] Dmytruk I, Integrating Nnlinear Heat Cnductin Equatin with urce Term, WEA Transactins n Mathematics, Issue, Vl.3, January 004, IN [5] Filipvá B., Mdelvání a regulace rázu pružnéh média v pružném ptrubí, VŠB TU Ostrava FEI, VŠB Ostrava 00 [6] Haberman R., Applied Partial Differential Equatins with Furier eries and Bundary Value Prblems, 4th Editin, Pearsn Bks, 003, IBN3: IBN0: [7] Hanuš B., Regulační charakteristiky příhřívačů páry u ktlů českslvenské výrby. trjírenství, 96, č.3., str [8] IHI, team Generatrs, Technical Nte f Ishikawajima-Harima Heavy Industries C., Ltd. Japan, * R(Y) [9] Jaluria Y., Trreance K.E., Cmputatinal Heat Transfer. Taylr and Francis, New Yrk, 003 IBN [0]Kattan. P.I., MATLAB Guide t Finite Elements: An Interactive Apprach. ecnnd Editin. pringer New Yrk 007. IBN []Nevriva P., Plesivcak P., Grbelny D., Experimental Validatin f Mathematical Mdels f Heat Transfer Dynamics f ensrs. WEA Transactins n ystems, Issue 8, Vl.5, August 006. IN []Nevriva P., Plesivcak P., Grbelny D., imulatin and experimental verificatin f heat transfer dynamics f sensrs. 0th WEA Int.Cnf. n YTEM 006, Athens, Greece, 006, IBN IN: IBN: