# Mathematical Modelling and Design of an Advanced Once-Through Heat Recovery Steam Generator

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1 Mathematical Modelling and Design of an Advanced Once-Through Heat Recovery Steam Generator Abstract Marie-Noëlle Dumont, Georges Heyen LASSC, University of Liège, Sart Tilman B6A, B-4000 Liège (Belgium) Tel: Fax: The once-through heat recovery steam generator design is ideally matched to very high temperature and pressure, well into the supercritical range. Moreover this type of boiler is structurally simpler than a conventional one, since no drum is required. A specific mathematical model has been developed. Thermodynamic model has been implemented to suit very high pressure (up to 40 bar), sub- and supercritical steam properties. We illustrate the model use with a 180 bar once-through boiler (OTB). 1. Introduction Nowadays combined cycle (CC) power plants become a good choice to produce energy, because of their high efficiency and the use of low carbon content fuels (e.g. natural gas) that limits the greenhouse gases production to the minimum. CC plants couple a Brayton cycle with a Rankine cycle. The hot exhaust gas, available at the output of the gas turbine (Brayton cycle) is used to produce high-pressure steam for the Rankine cycle. The element, where the steam heating takes place, is the heat recovery steam generator (HRSG). High efficiency in CC (up to 58%) has been reached mainly for two reasons : Improvements in the gas turbine technology (i.e. higher inlet temperature); Improvement in the HRSG design We are interested in the second point. The introduction of several pressure levels with reheat in the steam cycle in the HRSG allows recovering more energy from the exhaust gas. Exergy losses decrease, due to a better matching of the gas curve with the water/steam curve in the heat exchange diagram (Dechamps,1998). Going to supercritical pressure with the OTB technology is another way to better match those curves and thus improve the CC efficiency. New improvements are announced in near future to reach efficiency as high as 60%. In the present work we propose a mathematical model for the simulation and design of the once-through boiler. It is not possible to use empirical equations used for the simulation of each part of the traditional boiler. General equations have to be used for each tube of the boiler. Moreover there is a more significant evolution of the water/steam flow pattern type due to the complete water vaporization inside the tubes (in a conventional boiler, the circulation flow is adjusted to reach a vapor fraction between 0% and 40% in the tubes and the vapor is separated in the drum). Changes of flow pattern induce a modification in the evaluation of the internal heat transfer coefficient as well as in the pressure drop formulation. The right equation has to be selected dynamically according to the flow conditions prevailing in the tube.

2 The uniform distribution of water among parallel tubes of the same geometry subjected to equal heating is not ensured from the outset but depends on the pressure drop in the tubes. The disappearance of the drum introduces a different understanding of the boiler s behavior. Effect of the various two-phase flow patterns have to be mathematically controlled. The stability criteria has changed.. Mathematical Model.1. Heat transfer.1.1. Water Mathematical models for traditional boilers are usually based on empirical equations corresponding to each part of the boiler : the economizer, the boiler and the superheater. Those three parts of boiler are clearly separated thus it is not difficult to choose the right equation. In a once-through boiler this separation is not so clear. We have first to estimate the flow pattern in the tube then to choose the equation to be used. Liquid single phase and vapor single phase are easily located with temperature and pressure data. According to Gnielinski (1993) the equation 1 applies for turbulent and hydrodynamically developed flow. ( ξ )( ) / 8 Re 1000 Pr l α * d Nu = = ; ξ = 1+ 1,7 /8 Pr 1 λ ( ξ /3 )( ) 1 ( 1,8 log Re 1, 64) During vaporization different flow patterns can be observed, for which the rate of heat transfer also differs. In stratified-wavy flow pattern incomplete wetting has an effect on the heat transfer coefficient. A reduction could appear for this type of flow pattern. Computing conditions where a change in flow pattern occurs is useful. A method to establish a flow pattern map in horizontal tube for given pressure and flow conditions is clearly exposed by Steiner (1993). This method has been used in this study. The different flow pattern in the vaporisation zone of the OTB are given in figure 1. The heat transfer coefficient is estimated from numerous data. It is a combination of convective heat transfer coefficient and nucleate boiling heat transfer coefficient. 10 (1) 1.E+0 1.E+01 1.E+00 1.E-01 1.E-0 1.E-03 1.E-04 1.E-05 Flow Pattern Diagram for Horizontal Flow (VDI (1993)) Flow in tubes with 5.08t/h and 5.166t/h Annular Mist (5) Wavy () Stratified (1) *1e X-Martinelli parameter Figure 1: Flow pattern map in the boiling zone. Plug or Slug (4) Internal heat transfer coef (kcal/hr/m²/k) ECOV3 6A ECOV34A ECOV3A ECOV30A ECOV8A ECOV6A ECOV4A Once-through HRSG ECOVA ECOV0A ECOV18A ECOV16A ECOV14A ECOV1A ECOV10A ECOV08A ECOV06A ECOV04A ECOV0A SUPH06A SUPH04A SUPH0A Traditional HRSG Figure : Internal heat transfer coefficient.

3 0 α( z) = 3α( z) 3 + α( z) 3 () conv B () z α conv ρ ( 1 x) 1.x ( 1 x) liq α go ρ x 1 8( 1 x) liq = α ρ α ρ lo vap lo vap (3) α ( z) B = { heatflow,pressure,roughness,geometry} (4) α lo α LO is the heat transfer coefficient with total mass velocity in the form of the liquid and α GO is the heat transfer coefficient with total mass velocity in the form of the vapor. The internal coefficients computed for all the tubes of the OTB are presented in figure..1.. Fumes There is no difference between the equations used for a conventional heat recovery boiler and a once trough heat recovery boiler. The main part of the heat transfer coefficient is the convective part (low fumes temperature). The effect of the turbulence has been introduced to reduce the heat transfer coefficient in the first few rows of the tube bundle. The main difficulty to evaluate the heat transfer coefficient for the fume side comes from the fins that enhance the heat transfer, but could also produce other sources of resistance in the heat transfer, such as fouling on the surface of fins or inadequate contact between the core tube and the fin base. There are two methods to evaluate the heat transfer coefficient: The first one is based on a general equation to evaluate the Nusselt number in cross flow over pipes and the efficiency of the fins. An apparent heat transfer coefficient is then computed with equation 6. The second one is based on empirical correlations derived from experimental data. For more than four banks in staggered arrangement, equation 7 can be used. It is not obvious to find the most appropriate correlation for a given fin geometry and tube bundle arrangement. The best is to ask finned tube manufacturers to provide their correlations for heat transfer coefficient and fin efficiency corresponding to the required finned tube LOAD (kw) ECOV36A ECOV34A ECOV3A ECOV30A ECOV8A ECOV6A ECOV4A ECOVA ECOV0A ECOV18A ECOV16A ECOV14A ECOV1A ECOV10A ECOV08A ECOV06A ECOV04A ECOV0A SUPH06A SUPH04A SUPH0A LOAD FUME IN WATER OUT Figure 3: Heat exchange diagram TEMPERATURE (C) Pressure drop (mmho) ECOV36A ECOV34A ECOV3A ECOV30A ECOV8A ECOV6A ECOV4A ECOVA ECOV0A ECOV18A ECOV16A Pressure drop ECOV14A ECOV1A ECOV10A ECOV08A ECOV06A ECOV04A vapour fraction ECOV0A SUPH06A Figure 4: Pressure drop and vapor fraction evolution SUPH04A SUPH0A

4 A A po fo α = α * + η app f A f A 0.15 A 1 Nu = 0.38 Re 0.6 Pr 3 d d A b (6) (7).1.3. Overall heat transfer coefficient Finally the overall heat transfer coefficient is obtained from equation 8. The global heat transferred for each tube is computed with equation 9. We call T sl semi logarithmic temperature difference. It is the best compromise between pure logarithmic temperature difference that has no sense here (only one tube) and pure arithmetic temperature difference that does not allow to follow the evolution of water properties along the tube. The heat exchange diagram of the OTB is presented in figure e 1 = + + (8) α α A A app λ* w α * i A i A Q = α * A* T ; sl.. Pressure drop..1. Water ( T T ) T = w w1 sl T T mf w1 ln T mf T w ; T + T f 1 f T = (9) mf 64 f = f ρ V l Re P = with g d i f = 4 Re (10) The coefficient f depends on the Reynolds number for flow within the tube. In laminar flow, the Hagen-Poiseuille law can be applied. In turbulent flow the Blasius equation is used. The main difficulty is the evaluation of water pressure drop during transition boiling. The pressure drop consists of three components : friction ( P f ), acceleration ( P m ) and static pressure ( P g ). In once-through horizontal tubes boiler P g =0. The Lockard-Martinelli formulation is used to estimate the friction term. P P = Φ L L ftt phases liquid (11)

5 0 1 Φ = 1+ + ftt X X with X = P L liquid P L vapor (1) The acceleration term is defined with equation 13 where α is the volume fraction of vapor (void fraction). It is recommended to discretize the tube in several short sections to obtain more accurate results (figure 4). x x ( 1 x) P = G * m α * ρ + ( 1 )* vap α ρ liq x 1 (13)... Fumes The pressure drop in a tube bundle is given by equation 14. In this case the number of rows (N R ) plays an important role in the pressure drop evaluation. The coefficient f is more difficult to compute from generalized correlations. The easiest way is once more to ask the finned tubes manufacturer to obtain accurate correlation. f ρ V P = N (14) R 3. Stability Stability calculation is necessary for the control of water distribution over parallel tubes of the same form and subjected to equal heating in forced circulation HRSG and particularly in OTB. The stability can be described with the stability coefficient S. HRSG manufacturers try to keep the stability coefficient in the range (0.7-). In OTB design inlet restrictions have been installed to increase single-phase friction in order to stabilize the boiler. Based on the π-criterion (Li Rizhu, Ju Huaiming, 00) defined as π P + P in liq =, the design has been realized with π about. This number has to Pboi + Pvap be reduced in near future when all various flow instabilities would be identified. relative change ( in pressure drop) relative change ( ) S = = with in flow rate S>0 stable S 0 unstable d( p) p d( M) M Pressure drop DP UNSTABLE STABLE Mass flow M Figure 5: Stability example.

6 4. An Example Results have been obtained for an OTB of pilot plant size (4 rows). WATER (10.5 t/h ;Tin=44 C ;Tout=500 C) FUMES (7.5 t/h ; Tin=59 C ;Tout= 197 C).In VALI- Belsim software in which the simulation model has been implemented, the simulation of the OTB needs 4 modules, one for each row of tubes. Since VALI implements an numerical procedures to solve large sets of non-linear equations, all model equations are solved simultaneously. The graphical user interface allows easy modification of the tube connections and the modelling of multiple pass bundles. 5. Conclusions and Future Work The mathematical model of the once-through boiler has been used to better understand the behaviour of the boiler. Future mathematical developments have still to be done to refine the stability criteria and improve the OTB design. Automatic generation of alternative bundle layouts in the graphical user interface is also foreseen. 6. Nomenclature A total area of outer surface (m²) A b bare tube outside surface area A fo fin outside surface area (m²) A i inside surface area (m²) A po free area of tube outer surface A w mean area of homogeneous tube wall c p specific heat capacity at constant pressure (J/kg/K) d i tube internal diameter (m) P pressure drop (bar) f pressure drop coefficient G mass flux (kg/m /s) H enthalpy flow (kw) N R number of rows in the bundle Nu Nusselt number Nu l l = α λ P pressure (bar) Pr Prandl number Pr = Q exchanged heat (kw) Re Reynolds number T temperature (K) V c p η fluid velocity (m/s) x vapor mass fraction x i component flow rate (kg/s) α heat transfer coefficient (kw/m²/k) α(z) local heat transfer coefficient λ thermal conductivity (W/m/K) ρ density (kg/m 3 ) η dynamic viscosity (Pa.s) or (kg/m/s) fin efficiency η f λ 7. References D. Steiner, 1993, VDI heat atlas, VDI-Verlag, HBB, Düsseldorf, Germany V.Gnielinski, 1993, VDI heat atlas, GA,GB, VDI-Verlag, Düsseldorf, Germany P.J. Dechamps, 1998, Advanced combined cycle alternatives with the latest gas turbines, ASME J. Engrg. Gas Turbines Power 10, Li Rizhu, Ju Huaiming, 00, Structural design and two-phase flow stability test for the steam generator, Nuclear Engineering and Design 18, Acknowledgements This work was financially supported by CMI Utility boilers (Belgium).

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