ABSTRACT MATHEMATICAL MODELING OF SUPERCRITICAL ONCE THROUGH BOILERS By Itzhak Gotlieb The installation of supercritical once through boilers in modern power plants has presented problems of design and control, due to the high steam pressures and temperatures of operation. Sudden changes of the electrical load that occur during normal Operation may cause fluctuations of the steam conditions, which in turn may result in excessive wear of metal parts and in losses of thermal efficiency. The highly non linear and inter-related processes that take place in supercritical once through boilers require a non linear mathematical model in order that the system's dynamic response to various changes in the operating condi- tions may be adequately described. The objective of this study is to develop a mathemati- cal model of general applicability to all supercritical once through boilers. The model includes a mathematical formulation derived from the physical Laws of Conservation and includes thermo— dynamic relationships and transport properties of the flue Itzhak Gotlieb gas and of the working fluid (water or steam). The system of non linear partial differential equations, together with a non linear algebraic formulation of the Equa- tion of State for water and steam, is solved numerically by the method of Finite Differences with the aid of a digital computer. The model includes a computer program which solves for the variation with respect to both time and space of the fluid pressure, temperature, velocity, and specific volume, and of the gas temperature. The Equations of State, which are presented in this re- port as subroutines of the computer program, are based on the 1967 IFC Formulation of Thermodynamic Properties of Steam for Industrial Use. The open-loop, dynamic response of the system to varia- tion of the fluid flow rate, temperature and pressure, and to variation of the fuel firing rate and of the burner tilt are described. No limitations on the magnitude of the disturban— ces that may be solved for, or on their functional form, were found. A CDC-6500 digital computer required 80 seconds for the computation of 100 seconds of response time. Thus, the capability of the model to provide rapid solutions of the system's dynamic was demonstrated. MATHEMATICAL MODELING OF SUPERCRITICAL ONCE THROUGH BOILERS By Itzhak Gotlieb A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Chemical Engineering 1970 To my wife 1i ACKNOWLEDGMENTS The author wishes to express his sincere appreciation to his advisor, Dr. G. A. Coulman, whose guidance and assistance were invaluable throughout the course of this study. Thanks are also due the other members of the author's guidance committee: Dr. M. H. Chetrick, Dr. G. L. Park, and Dr. B. W. Wilkinson. The author is indebted to the Division of Engineering Research, Michigan State University, for providing financial support. iii LIST OF FIGURES LIST OF TABLES TABLE OF CONTENTS INTRODUCTION .............. ....... . .......... ........ CHAPTER 1: BACKGROUND ............................... §1.1 The Once Through Boiler .... ..... . ......... §1.2 The Combustion Process .................... §l.3 Heat Transfer ................... ..... ..... §l.4 The Equations of Change ................... §1.5 Mathematical Modeling ............. ........ 1.5-1 Charles P. Crane Unit No. l ........ 1.5-2 Simulation of Bull Run Supercritical 1.5-3 unit ......OOOCOOOOOO......OOOOOOOOO Canady's Subcritical Once Through unit NO. 3.....OOOOOOO0.0.0.0.0.... §l.6 The Equations of State for Water and Steam CHAPTER 2: §2.1 §2.2 DESCRIPTION OF THE SYSTEM ............... The System .................... ....... ..... The Main Assumptions ....... ......... ...... 202-1 The FIUid Side ooooooooo ...... 00.... 202-2 The 688 Side oooooooo oooooooo o oooooo §2.3 Heat Transfer and Generation in the Gas ... 2 2 3-1 3-2 Heat Generation ........ ..... .. ..... Absorption of Radiation within the Gas valume 0.00.00.00.00.00.00.00... iv Page viii xi 13 15 16 17 21 22 24 24 26 26 27 28 28 29 Page 2.3-3 Heat Transfer in the Lower Furnace 31 CHAPTER 3: EQUATIONS AND BOUNDARY CONDITIONS ........ 33 §3.1 Equations of the Fluid Side .. .......... ... 33 §3.2 Equations of the Gas Side ................. 36 §3.3 The Boundary Conditions ..... ..... ......... 39 CHAPTER 4: THE NUMERICAL SOLUTION ........ ....... .... 42 §4.1 The Numerical Method . ...... ....... ........ 42 §4.2 The Steady State Solution ......... . ....... 46 4.2-1 The Equations of Change .......... .. 46 4.2-2 The Computational Procedure ... ..... 47 4.2-3 The Gas Temperature ................ 49 4 o 2-4 The Boundary conditions 0 o o o o o o o o o o o 49 §4.3 The Unsteady State Solution ......... ...... 50 4.3-1 The Equations of Change ............ 50 4.3-2 The Computational Procedure ........ 51 4 o 3-3 The Gas Temperature 0 o o o o o o o o o o o o o o o 52 4.3-4 The Boundary Conditions ............ 53 §4.4 Gas-Side Energy Balances . ........ ......... 53 4.4-1 The Upper Furnace .................. 54 4.4-2 The Superheater Section ............ 56 4.4-3 The Lower Furnace .................. 58 4.4-4 The Steady State Gas Temperature PrOfile ......OOOOOOOOOOOO00.0.00... 60 4.4-5 The Unsteady State Gas Temperature PrOfile 0.000.000.0000.........OOOOO 61 CIiAPTER 5: STABILITY AND CONVERGENCE O O O O O O O O O O O O O O O O 63 §5.l Preliminary Tests ... ............... ....... 64 5.1-1 Tests of Convergence . ..... ......... 65 501-2 TCStS Of Stability 0.000000000000000 66 §5.2 Determination of the Mesh Size ............ 67 §S.3 Conclusions ............................... 68 V Page CHAPTER 6: RESULTS AND DISCUSSION ................... 80 §6.l Variation of Fluid Inlet Velocity ......... 81 §6.2 Variation of Fluid Inlet Pressure . ........ 82 §6.3 Variation of Fluid Inlet Temperature ...... 83 §6.4 Variation of Firing Rate ............ . ..... 84 §6.5 Variation of Burner Tilt ... ....... . ....... 84 §6.6 Combination of Inputs ..................... 85 CHAPTER 7: CONCLUSIONS AND RECOMMENDATIONS .......... 101 APPENDIX A ................... ............. .... ...... 106 § A.l The Equation of Continuity ......... . ..... . 106 § A.2 The Equation of Motion ... ........ ......... 106 §.A.3 The Equation of Energy ....... ........ ..... 108 § A.4 The Specific Heat of the Gas .............. 110 § A.5 The Heat Generation Function .............. 111 § A.6 Energy Equations in the Lower Furnace ..... 113 APPENDIX B ............... ...... ...... .............. . 115 The Main Program ..... ............. ........ 115 APPENDIX C ....................... ...... ....... ...... 136 § C.1 Subroutine ch - The Specific Heat of the Gas 000000000000000000000.00.000.000.000... 136 § C.2 Subroutines SVl and CPl State Equations for subregionl......OOOOOOOOOOOOOOOCC.... 136 § C.3 Subroutines 8V2 and CP2 - State Equations for SUbregionZ......OOOOOOOOOO00.0.0.0... 139 § C.4 Subroutines SP3 and CP3 State Equations for Subregion 3 ........................... 143 vi Page §C.5 Subroutines SP4 and CP4 - State Equations for Subregion 4 ........... ........ . ....... 148 APPENDIX D .......................................... 151 §D.1 The Equation of Motion .................... 151 §D.2 Heat Conduction in the Fluid ...... ........ 151 §D.3 Gas-to-fluid Convective Heat Transfer ..... 152 §D.4 The Gas Energy Equation ................... 154 NOMENCLATURE ........................................ 157 BIBLIOGRAPHY 0.00.00.00.00 00000 0 00000000000 00.00.0000 162 vii LIST OF FIGURES Furnace Heat Absorption Pattern, Tangential Firing 00.000000000000000... 0000000000 000 000000 Subregions of the Equation of State for Water and Steam ......................... ...... . ..... The Fluid Path ...... ............. . ............ The Gas Side ......... ...... ..... ....... ....... Flame Location and Heat Generation ....... ..... Combined Effect of Heat Generation and Absorption of Radiation in the Gas ..... ....... Gas Energy Balance in the Upper Furnace ....... Gas Energy Balance in the Lower Furnace ....... Time—distance Space in a Finite Differences Rectangllar Grid 00000000000000.00000000000000. Notation for the Time-distance Grid ........... Energy Balance in the Upper Furnace ........... Schematic View of the Superheater Section ..... Energy Balance in the Superheater Section ..... Energy Balance in the Lower Furnace ........... Fluid Temperature Profilesglkt = 5 seconds and Az=6ocm0.0.0.000000000000000000.00000000000 The Effects of Mesh Size Variation on the Fluid Temperature Profiles at t = 5 seconds ... ..... The Effects of Mesh Size Variation on the Outlet Temperature Response ........... ........ viii Page 23 25 25 32 32 37 38 43 44 55 56 57 59 7O 71 72 5.5: 5.6: 5.7: 5.8: 5.9: 6.8 6.9: The Effects of Mesh Size Variation on the Outlet Velocity Response ......... The Effects of Mesh Size Variation on the Outlet Specific Volume Response .... The Effects Temperature The Effects Temperature The Effects Temperature The Effects Temperature The Effects Outlet Temperature Response ............. of Variation Response, A z of Variation Response, A z of Variation Response, A t of Variation Response, A z of Mesh Size of At on the Outlet = 60 cm ............ of At on the Outlet = 60 cm ... of Az on the Outlet = 5 seconds of At on the Outlet = 15 cm ...... Variation on the Fluid Temperature Profiles, -11% Velocity Step Input Fluid Velocity Profiles, -22% Velocity Step Input Fluid Dynamics at Boiler Outlet, Step Input 00000000000000... Fluid Dynamics at Boiler Outlet, +22% Velocity Step Input ..... Fluid Pressure Profiles, Fluid Dynamics at Boiler Outlet, Pressure Step Input 0.000000000000000. Fluid Temperature Profiles, Input -1. -1. bar bar Pressure Step Input ............................ Temperature Ramp Fluid Dynamics at Boiler Outlet, Temperature Ramp Input Fluid Dynamics at Boiler Outlet, -20% Firing Rate Step Input ........ ix —22% Velocity Page 73 74 75 76 77 78 79 89 90 91 92 93 94 95 96 97 Page 6.10: Gas Temperature Profiles, Burner Tilt Inputs .. 98 6.11: Fluid Dynamics at Boiler Outlet, Burner Tilt Inputs 00000.00‘000000000000000000000.000.00.00 99 6.12: Fluid Dynamics at Boiler Outlet, Fluid Velocity and Firing Rate Step Inputs ................... 100 3.1: A Flowchart of the Main Program .... ...... ..... 134 8.2: A Flowchart of the Profiles Computation ....... 135 LIST OF TABLES Page Table 6.1: Steady State Profiles ................... 86 Table D.l: Calculated Values of Tube Wall-to-fluid Convective Heat Transfer Coefficient .... 154 xi INTRODUCTION In the normal operation of power plants, variations of load and other types of perturbations may result in undesi- rable fluctuations of the power output and of the steam con— ditions at the boiler outlet. This requires effective means for controlling the power generation process, particularly where the systems are designed to Operate at high steam pres— sures and temperatures. The development of once through boilers for central power station, operating at supercritical pressures and high temperatures, has brought about improved thermal efficiencies and reduced costs. The trend to operate at still higher tem— peratures and pressures is subject to limitations of present materials of construction. Fluctuations of the steam tempera- ture during a transient state, resulting from some perturbation of the operating conditions, may cause excessive wear in the tube circuitry and in the turbine, as well as reduced efficien- cies. The purpose of this work is to develop a mathematical model of a supercritical once through boiler. The model should provide a description of the dynamics of the boiler which is the most important and the least well understood part of the power plant. The model is to be of general applicability to all super- critical once through boilers, rather than to a specific design. The model should include: a) A mathematical formu— lation of the process dynamics; b) A numerical method for solving dynamic problems, involving various types of distur— bances; c) A computer program which can perform the numerical solution. The mathematical formulation is to be derived from the physical laws of conservation, from the thermodynamic equation of state for water and steam, and from correlations of trans- port properties of the working fluid and of the gas. Considering the highly non—linear nature of the equation of state, particularly in the neighborhood of the critical point, a linearized approach is not expected to provide a use— ful tool for control. A linearized model is applicable to small perturbations only, whereas in practice large scale disturbances are normally encountered. Therefore it is requi— red that the equations should not be linearized, and that the numerical method for solving them would allow rapid computa— tion, so that the model may be used for control purposes. CHAPTER 1: BACKGROUND The subject of power generation has been covered exten- sively in the literature. The reader may find that general information, describing aspects of design, construction, Operation, and control of power plantslz, is a useful back— ground for this work. This Chapter contains a description of some basic fea— tures of once through boilers ( §1.1), and a discussion of factors affecting the rate of combustion in the boiler fur— nace ( §1.2). The various modes of heat transfer, and the laws govern- ing its rate, are discussed in §1.3. The laws of conservation of mass, momentum, and energy are stated in §l.4 in the form of the differential equations of change. Previous work on modeling of once through boilers is reviewed in §1.5. In each case, the major simplifying assumptions involved in the simulation are given, and the applicability of the model is discussed. A qualitative description of the equations of state for water and steam is given in §1.6. The explicit formulation is given in Appendix C, as computer routines. § 1.1 The Once Through Boiler In a once through boiler there is no recirculation of the working fluid (water or steam) within the unit. In ele- mental form, the boiler is merely a length of tubing through which the fluid is pumped. Heat is applied, and the water flowing through the tube is converted to steam, superheated to the desired temperature at the outlet. In practice, a single tube is replaced by a multiplicity of small tubes, arranged to provide effective heat transfer.17 In modern central station boilers, most, if not all, of the furnace enclosure consists of waterwalls, which are expo- sed to high flame and gas temperatures. The walls are made up Of panels with parallel tube circuits, all arranged in a single upward pass. They are fed from furnace wall inlet headers at the lower end, and terminate in the outlet headers at the upper end.17 The increased thermal efficiencies Obtained by Operation at higher steam pressures and temperatures, made the elimina— tion of the steam drum a necessity, and enhanced the develop— ment of the once through boiler. Rising fuel costs provide further incentive for operation at still higher temperatures and at supercritical pressures. This trend is subject to limitations of tube materials and costs. Thus, Operation at 3500 psi and lOOOOF (steam pressure and temperature at the Superheater outlet) is widely accepted today in the design Of central station boilers. When steam is generated above the critical pressure (3208 psi), there is no boiling, and the change of phase occurs in a continuous manner. The corresponding changes in the fluid properties (e.g., density, heat capacity) are more moderate than in subcritical operation. As a result, the distribution of flow in a bank of parallel tubes is not as strongly influenced by variation in heat absorption. This provides for further simplification of design by eliminating the requirement of flow distribution devices.17 Significant losses in plant efficiency result if the steam temperatures fall below turbine nominal admission design values.16 Steam temperature is controlled by desuperheating, gas recircula— tion, regulation of firing rate, and burner tilting.16 The first two methods involve mixing of fluid or of gas streams of different temperatures. Thermodynamically, this results in a net "degradation" of energy, with subsequent loss in thermal efficiency. The nature of the combustion process is discussed in § 1.2. The flame propagation is considered rapid enough, so that the dynamics of the fire is not important, and the heat release in the furnace is taken as proportional to the firing 16 rate. The change of heat absorption rate along the height of the enclosure wall, in a typical pulverized—coal—fired fur- nace, as affected by burner tilt, is shown in Figure 1.1. average absorption rate horizontal tilt no tilt up tilt —————— down tilt -—-~—-—q_ Lower -—————a 100 Higher per cent average absorption rate Figure 1.1: Furnace Heat Absorption Pattern, Tangential Firing § 1.2 The Combustion Process Combustion is an exothermic oxidation reaction. In boil— ers, the reaction vessel is the furnace, which is enclosed by heat absorbing surfaces, and is provided with means for con- tinuous discharge of the reaction products; namely, flue gases and ash. Fuel and air enter the furnace and are subjected to rapid heating until the mixture ignites. The heat released by combustion also serves to ignite the incoming fuel, and the process sustains itself. In the case of powdered-coal—fired furnaces, the reac— tion mixture is heterogeneous. Considering a coal particle suspended in the gaseous reaction mixture, the reaction may be described as a three stage process:11 1) Oxygen from the gas phase diffuses towards the sur— face Of the coal particle. 2) The oxygen adsorbs chemically on the particle's sur— face and reaction occurs. 3) The reaction products desorb from the solid surface and diffuse towards the bulk of the gas phase. Some of the factors affecting the kinetics of this pro— 1,13 cess are: particle size; concentration of oxygen and other gases (nitrogen, reaction products), in the bulk of the gas phase and at the surface of the burning particle; temperature variation from the bulk of the gas towards the particle's surface, and within the particle; total pressure; fuel quality and composition; local velocities Of the particle and of the gas; diffusivities; heat capacities; viscosities; etc. The overall rate of combustion in the furnace depends on 11’15’32 the geometrical arrangement some additional factors: of the combustion chamber and of the heat absorbing surfaces; total fuel to air ratio in the feed; fuel quality and parti— cle size distribution; angle of firing (burner tilt), etc. Some Of the factors enumerated above are strongly inter— dependent. For example, the temperature field in the furnace depends on the relative rates of heat generation by combus— tion and of heat transfer within and out of the furnace. This temperature distribution also affects the rates Of heat transfer phenomena, as well as reaction kinetics para- meters, and physical properties of the reaction components, thus determining the rate of heat generation. The flow characteristics in the furnace exert a profound effect on the overall rate Of reaction. While it is improba— ble that a rigorous detailed treatment of the complex flows of practical furnaces can be carried out, it is essential that the main features of real flows be taken into account. Attempts to describe a furnace using "stirred tank" models (thus neglecting transport resistances), have been made.26 It was suggested that real furnaces might be repre- sented by a combination of "stirred tank" and "plug flow" 1 . . reactors.2 This could also serve as a model for the radiative 19 heat transfer. Experimental methods, such as smoke table tests and 3— dimensional water and air models, have been found useful for . . . . 14 Visualizing flows in furnaces. In conclusion, a rigorous analytical treatment of the rate of reaction in a boiler furnace is not feasible at the present time. § 1.3 Heat Transfer Heat transfer in general, and in boilers in particular, 22,23,28,34 is the subject of many books and articles. The basic laws expressing the rate Of heat transfer are: 1) Newton's law of convection q = hAlkT ..... ............. (1-1) 2) Fourier's law of conduction in the x—direction q=-kA%I 0.0000000000000000 (1-2) 3) Stefan-Boltzmann law of radiation _ 4 q — eg - I) wv ....... (3 11) 23.125.) 92 (3—12) dT . — = + . T. Equation wC qtr 01(a1n T p v dz p dz To solve these equations, three boundary conditions are 40 required. One boundary condition would be the fluid tempera— ture at the Superheater outlet (which is also the inlet to the turbine). This value is usually included in the boiler's specifications. Similar consideration applies to the fluid pressure at that point. The fluid velocity determines the mass flow rate. Thus, these three variables specify the ener— gy input into the turbine. The pressure and temperature at steady state operation are those given in the boiler rating, and the velocity varies with the electrical load. The rela- tionship between the load and the velocity is outside the sc0pe of this work. Deviations from steady state, caused by load variations, are viewed as changes in fluid velocity at the inlet to the boiler. Similarly, changes of fluid pressure and temperature at the boiler inlet are considered "inputs" to the system, in the sense used in Systems Analysis. The values of the fluid pressure, temperature, and velocity at the boiler's outlet are the most interesting outputs, in the same sense, although the method of solution described in Chapter 4 provides those quantities at all points along the tube. It is, thus, possible to solve the steady state equations, using the boundary conditions at the boiler outlet, and Obtain steady state profiles. Then, the response of the system to some disturbance, or input, may be studied. The steady state 41 profiles constitute initial conditions for the dynamic problem, and fluid pressure, temperature, and velocity at the inlet of the boiler are its boundary conditions. The gas temperature profile is a result of the interaction between the rates of heat generation and heat transfer. At unsteady state, the gas temperature variation will be treated as a sequence of steady states. (This point is discussed in §4u4-5). The value of the gas temperature at the bottom of the boiler is taken to be a constant. CHAPTER 4: THE NUMERICAL SOLUTION § 4.1 The Numerical Method The fluid dynamics is described by a system of 3 diffe— rential equation (Eqns. 3-1,2,3), referred to as the Equations Of Change. The equations are first-order, nonlinear, partial with respect to time t and distance 2, and include 4 variables. These variables are the fluid temperature T, pressure p, velo- city v, and density p Using the equations of state, the specific volume 5 = 143, may be expressed in terms of p and T (Eqn. 3-53), or p in terms of V and T (Eqn. 3-5d). The number of independent functions is, thus, reduced (implicitly) to three, and the system is consistent. The highly complex form of the state equations does not permit elimination by substitution of any function from the equations of change. Neither is it deemed feasible to Obtain an analytical solution Of the system. Linearization of the equations will impose serious limitations on the model. Most of the variables and parameters show marked nonlinear behavior in the vicinity of the critical point. As a result, the accuracy of linearization will be limited to very small deviations from steady state. The technical limitations of analog computers make it necessary to use a numerical method which a digital computer 42 43 can handle in reasonable computing time. The method selected for this work is the method of Finite Differences, whereby first—order derivatives are approximated by ratios of finite increments. F : éfi F g 95 31;-“ , 3: t (4-1) The t and z coordinate axes are divided into equal in- crements of time and distance, respectively. The z-t Space may be mapped by a rectangular grid, as shown in Figure 4.1. 1+1 ) j+2 j+l p g 1 fl At ‘r 3+2 B J j j—1 : 1-2 i-l Til—AZ“ i+1 i+2 Distance z Figure 4.1: Time-distance Space in a Finite-differences Rectangular Grid A point in the z-t space is specified by an index number i, for the distance z,and by an index number j, for the time t. Using this notation, Equations 4—1 become: 44 Q}; = Fi+1,j+1 ' FiJJ+1 if: = Fi,j+1 " Fi,j az Az ’ at At "°° (4_2) The differential equations can be written in the Finite Differences form, which lends itself to algebraic solution. For example, the C. Equation (Eqn. 3-1): 01.341 1.3 + v pi+1..-I+1 ‘ pi.g'+1 + At i+§~,j+—§ Az V. . 1+ILJ+1 1LJ+1 = O (4_3) pi_*_;’j+% AZ 0000000000 For brevity, a different notation, shown in Figure 4.2, will be adopted. . ==F F. . =F F1,J y ’ i,j+1 x i+1,j _Fz ’ Fi+1,j+1 —Fm F . =F 1%,J+% J t 1 3+1 x m 34% J :I y z 1 1% 1+1 N” Figure 4.2: Notation for the Time-distance Grid 45 Given the values Fy’ F2, and Fx’ the intermediate value Fj is approximated by linear interpolation: Pi = (Fx + Fz)/2 ..................... (4-4) Fm may be expressed in terms of Fx’ Fy, F2, and Fj by rearranging the differential equation. The physical proper- ties of the fluid appearing in the differential equation are evaluated at the point "j". It may be noted that the equations of change are "coupled"; i.e., each of them involves more than one function. Therefore, the whole system of equations, which includes: i) The Equations of Change (Eqns. 3-1,2,3), ii) the S. Equation (Eqns. 3-5), and iii) the gas energy balances (Eqns. 3-6 and 3-8), must be solved simultaneously. The thermodynamic and transport proper- ties are computed at each point along the fluid and gas paths by theoretical or empirical formulae, given as computer sub- routines. Initial conditions for the dynamic problem will be the steady state profiles. Their computation is described in §4.2. The solution of the dynamic problem is described in §4.3, and the computation of the gas tenperature profiles is discussed in §4.4. 46 § 4.2 The Steady State Solution 4.2-1 The Equations of Changg The steady state equations (Eqns. 3—10,11,12), are first- Order, nonlinear, ordinary differential equations. By the Finite-differences method, a derivative is approximated as follows: it;éi=Fi+1‘Fi=__Fm‘Fx dz Az A z 412 Using the above notation and rearranging, the Equations Of Change become: C. Equation p v = w = constant ................... (4-5) M. Equation 6 _ 10 _ - _ Vm " Vx ‘ 34"" (Pm PX) AZlig/vx + (2f/D)vxl (4 68) _ -6 pm - px - 10 {w(vm - vx) +Az[ pg + (2f/D)wvx]}.... (4—6b) T. Equation T = T + Az q +0 l(aln v) vx ( _ ) (4_7) m X WCpx t!‘ . m px Cpx pm Px .... The term qtr is the combined radiative and convective heat transfer rate across the tube wall per unit of fluid [‘9 47 volume. It may be recalled (§'3.1) that the superheater tube is different from the waterwall tube. As a result, the ratios AA ._2 AV AA and 23% , which appear in the expression of qtr (Eqn. 3-4), are different. It is desired, however, to maintain the expression of qtr unchanged throughout the boiler. This is done by defining Deq as the equivalent diameter (see Appendix A, §A.3), and we obtain: qt 1) 2) 3) D = ._ea 4 _ 4 .33. _ Acx {o[(Tgx/1000) (TX/1000) J + 2 U(Tgx TX) ..... (4-8) Remarks: By the notation used in Equations 4-5 through 4—8, Fx = Fi , Fm = Fi+l ; i being the index number of the z (distance) coordinate. Physical properties of the fluid and the gas temperature T , are evaluated at point i, the "entrance" to the volume element. The emissivity,e , taken to be equal to 1, is omitted from the expreSSion of qtr' 4.2-2 The Computational Procedure The simultaneous solution Of Equations 4-5 through 4-7, together with the Equations of State (Eqns. 3-5), requires an iterative computational procedure. This procedure is described 48 below for a single volume element of the tube. an V 1) Given vx, T , p , Tgx’ the values of vx, Cpx’ galn T)px , x x are computed from the S. Equation (Eqns. 3-Sa b C)- ) , 2) px = l/Vx 3) The iterative process: i) An initial guess is made on the value of pm. ii) vm is computed from Eqn. 4-6a. iii);pm is computed from Eqn. 4-5. v) Tm is computed from Eqn. 4-7. vi) V5 (sp. volume) is computed from the S. Equation, (Eqn. 3-53), using the values Of pm and Tm. vii) If Gm % vs, the difference Gm-GS is used to correct the initial guess of pm, by linear interpolation or extrapolation. This is repeated until Gm = GS The values p , T , 3 , and v are then recorded and used m m m m in the computation of the next point along the tube. When the end of the tube is reached, the steady state pro— files of p, T, 3, and v are given in tabulated form (i.e., as numerical values at discrete and equidistant points along the tube). An alternative stagewise computation may be used, whereby the initial guess is made on the value of vm. The value of pm is then computed from Eqn. 4-6b, F)m and Gm from Eqn. 4-5, and 49 Tm from Eqn. 4-7. Using Tm and.ym, the pressure pS is com- puted from the S. Equation (Eqn. 3-5d). ps is compared to pm’ and the initial guess of vm is corrected as in the former method until pm = ps. This procedure is used in subregions 3 and 4 of the state equations, where pressure is expressed as a function of v and T. 4.2-3 The Gas Temperature The solution of the gas-side energy balances provides the gas temperature profiles. This will be described in §4.4. 4.2-4 The Boundary Conditions Boundary conditions have to be assigned for the problem to be mathematically defined. The values Of fluid pressure and temperature are specified at the boiler outlet, and the value of fluid velocity is relative to the electrical load (see § 3.3). It is, therefore, convenient to start the computation at the~boiler outlet, and proceed "backwards" along the tube. This does not affect the form of the equations, as lfiz is given a negative value. 50 § 4.3 The Unsteady State Solution 4.3-1 The Equations of Changg Using the notation described in §4.l, the Equations of Change (Eqns. 3-1,2,3) may be written as follows: C. Equation - Az . OX -92 ii p m _ p x - v. At - v. (vm — vx) (4—9) 3 J M. Equation 1 vx " VI 106 = _ -— + + ._ _ vm vx Az vjl: At a] (2’f/D)vj pjvj (pm px) alternatively: p.v. v - v = _.Aldl .J. .;£___4K _ _ pm PX 106 AZEVj ( At + g) + (Zf/D)vj] (vm vx}... ..... (4—10b) T. Equation T - T q Az x y tr = +-—- +--—-——- + Tm Tx v. [ At p.C .3 PJ V- ‘ P P ' P ln v y m x + _—-— 000 -11 0.1 A2 C .( T)pj[px At vj Az J (4 ) PJ vJ where v -l/bj and D =‘_sa 4 _ 4 1L. _ - qt {a[(Tgy/1000) Tj/IOOO) J + 2 U (Tgy 15))” (4 12) 51 The gas temperature is evaluated at point "y" (point i,j of the grid). This, and the computation of Tg, will be dis- cussed in §4.4. 4.3-2 The Computational Procedure The simultaneous solution of Equations 4-9 through 4—12, together with the Equations of State (Eqns. 3-5), is carried out in a similar manner to the steady state problem. The pro— cedure for a single volume element of the tube, of length IAz, is described as follows: 9 px’ P 3 Pz: V y Y a V 2 the y x y z 1) Given the values T , T ,T x y 2 intermediate values Tj’ pj, vj, are computed from Equation 4-4. ~ In 3 . 2) v,, C ., €31“ T)pj , are computed from the S. Equation (Eqns. 3-5 ), using the values Tj’ and pj. a,b,c 3) pj = 1/5j by definition. 4) The iterative process: i) An initial guess is made on the value of pm. ii) vm is computed from Eqn. 4-10a. iii) pm is computed from Eqn. 4-9. iv) Tm = l/pm v) Tm is computed from Eqns. 4—11 and 4-12. vi) Is (sp. volume) is computed from the S. Equation (Eqn. 3-53), us1ng the values Tm and pm. ‘ n_f LL- — 52 vii) If Gm f GS, the difference Gm-GS is used (by linear interpolation or extrapolation), to correct the ini- tial guess of pm. This process is repeated until ~ _ ~ V — V . m S The values of pm, Tm’ vm, and Gm are then recorded and used in the computation of the next point along the fluid path. This is continued until the end of the tube is reached, and the unsteady state profiles of p, T, v, and P, at time t, are given in tabulated form. The profiles thus obtained, and the boundary conditions (see §4.3-4), will be used to compute the profiles at time t +Zkt in the same way. In subregions 3 and 4, where the pressure p is given as a function of 6 and T, an alternative computational procedure is used. The initial guess is made on the value of the velo— city vm. pm is then computed from Eqn. 4-10 and cm from b’ pm Eqn. 4—9, and Tm from Eqn. 4—11. Using Tm and cm, the pres— sure pS is computed from the State Equation (Eqn. 3-5d). pS is compared to pm, and the initial guess is corrected as in the former method, until pm = ps. The process is continued as described above. 4.3-3 The Gas Temperature The gas temperature profile is computed simultaneously with the equations of change, from the gas energy balances 53 (Eqns. 3-6,7,8,9), made over each volume element. The method is described in §4.4. 4.3-4 The Boundary Conditions The steady state profiles constitute the initial condi- tions of the unsteady state problem. The variation with res- pect to time of v, T, and p at the boiler inlet (2 = O) is chosen to be the boundary conditions of the unsteady state problem, and this completes the mathematical definition of the problem. The physical significance of the boundary conditions has been discussed in§13.3. Accordingly, the variations with time of v, T, and p at the boiler inlet are considered "in— puts" to the system. The dynamic, Open-loop, response Of the system to various types of inputs will be studied. In addition to the inputs described above, changes in fuel fir— ing-rate and in burner tilt, are considered to be inputs. This will be discussed in §4.4-5. §4.4 Gas-side Energnyalances The gas side is divided into 3 parts (see§ 2.1), the Low— er Furnace, the Upper Furnace, and the Superheater Section. The differences in gas flow and in tube geometry, result in different expressions for the energy balances in each Of these 54 parts. In all cases, it is assumed that heat accumulation within a volume element of gas, over a period IAt, is small relative to heat "generation" (see §2.3) and to heat trans- port. As with the fluid, where one tube is taken to represent the multiple tube system, so in the gas side, a part of the gas stream, corresponding to a single tube, is taken to re- present the entire stream. 4.4-1 The Upper Furnace The gas in the Upper Furnace is assumed to flow axially upwards, in parallel to the fluid flowing in the surrounding waterwall tubes. On the basis of a volume of gas, correspond— ing to a tube Of lengthHAz, the energy balance equations may be written as follows: _ + _ :: _1 Hgi Hgo A2(qS qrc) 0 (4 3) where H . = W °C 'T . gi g g g z is the enthalpy of the gas evaluated at point 2 H = W -C -T is the gas enthalpy evaluated at point go g g g Z+A z z +-Az fmx q = -——-———- (L - z) is the rate of heat generation s Lt - zfu t within the gas, per unit length of the gas column. (See 55 Appendix A, §A.5 for definition of the parameters fmx’ zfu, and Lt’ and for the derivation of the expression). qrc = qtr-Acx is the rate of heat transport from the gas into the fluid, per unit length of the gas column. Equations 4-8 and 4—12 give qtr for steady and unsteady states, respec— tively. Acx is the cross-sectional area of the tube. (Note that in the Upper Furnace, the length of the gas column coin— cides with the length of the tube. This also is correct in the Lower Furnace but not in the Superheater Section.) H go z+Az 5 PC H . g1 Figure 4.3: Energy Balance in the Upper Furnace Substituting Tg into Equation 4-13 and rearranging, we have: = T z+ Az g T g Z Cg is the specific heat of the gas, given as a function of Tg’ (Appendix A, §A.4). The average lepe of the gas temperature profile may also be computed: ATg/Az = (qs - qu)/(wgcg) (4—15) 56 4.4-2 The Superheater Section The Superheater Section contains four horizontal passes of the Superheater tube. The fluid path in these passes is shown schematically in Figure 4.4. The gas temperature is assumed constant at each pass. Four energy balances are made: one for each pass. 0 T ge Pass D _____1 T I) gd P ass C T .__1. gc V A Pass B ____. I II Tgb Pass A ____ WP01ESF T ) outlet ga waterwall gas )4 Figure 4.4: Schematic View of the Superheater Section The gas energy balance is expressed as in Equation 4-13, with: H w c T , H = w c T , T . = T , gi g g ga go g g gb g1 ga Tgo - Tgb for superheat pass A, with similar expressions for superheat passes B, C, and D, respectively. 57 go go gi gi gas'flow Figure 4.5: Energy Balance in the Superheater Section The value of qrc is obtained by numerical integration of the terms qtr'A -Az over the entire length of the pass. cx The value of qs is Obtained by integration of the heat generation function over the length of the gas column, contain— ing the superheat pass. Denote this length by ls, and the length of the entire Superheater Section by LS, then 1S - Ls/4 , 4 being the number of superheat passes. The value Of qs is, thus, the heat generated within a gas column of length ls. The resulting expressions are: .7. fmx 2 958:: 2 ' L _ 2 1S ......... .. ..... (4-16 ) 58 f __ 5 mx 2 q _ . - 1 0000000000000000000 (4"1‘3 ) sb 2 Lt zfu s b ;§ fmx 2 q = . _ 1 co oooooo o ooooooooo (4—16 ) sc 2 Lt zfu s f _ 1 . mx 2 qu 2 L - z ls o ooooooooooooo on (4—16d) t fu The derivation of Equations 4-16 is given in Appendix A, §A.5. 4.4-3 The Lower Furnace The axial heat transfer in the Lower Furnace is described by the Dispersion Model (see §2.3 and §3.2). Consider a volume of gas, corresponding to a waterwall tube Of lengtthz. The energy balance may be written as follows: ..qa Z +Az(qs - q ) = O ............... (4-17) a PC z+Az where qa is the rate of axial heat flow. dT qa =- C. dz 0 oooooooooo 0.. (4-18) z dT qa =-Dc°:l-z-g (4-18b) z+Az z+Az and fmx q8 = z 'z is the rate of heat generation per unit length fu Of the gas C01umn 00000000000000. (4-19) q I) a z+Az I) Z qa Figure 4.6: Energy Balance in the Lower Furnace The variation of Tg with 2 over the length Az can be ex- pressed by an ordinary differential equation: dT fmx Dc'—g’2 =qm~z z (4-20) dz fu Assuming that qrc may be taken as a constant over Az, it is possible to carry out the integration of eqn. (5-20) and obtain: dT dT Az f Tf = if D— U“; 22‘" <22 +4.” ------ (HI) z+Az z c fu dT 2 f T = T + “—553 A2 + %L [q — 3m): (32 + Az)]..(4—22) g z+Az g z 2 C zfu The derivation of Equations 4-19 through 4—22 and the de- finition of the parameters appearing in the equations are given 60 in Appendix A, §A.6. 4.4-4 The Steady State Gas Temperature Profile The steady state gas temperature profile is a numerical solution of an ordinary differential equation. It is necessa- ry to provide a boundary condition. For example, the value of Tg at some fixed point along the gas path. This, however, is considered undesirable due to the definition of Tg' It has been assumed that Tg can be taken as an average over the cross- sectional area of the gas column. An experimental determina- tion of this average value would be difficult, both technically and conceptually. It is possible to circumvent this difficulty in a way compatible with the general concept which considers the effects of Tg on heat transfer rather than its "real" phy— sical significance. Noting that the fluid temperature at the boiler inlet is a fairly constant value (ca. 600°F), we Can use this value, denoted by Tin’ as a boundary condition for the gas equation. An initial guess is made of Tga’ the gas temperature at the fourth superheat pass, and the steady state profiles are computed. Denote the computed value of the fluid temperature at the boiler inlet by Tic and compare the values of Tin’ and Tic' If Tic # Tin’ then the difference Tic-Tin is used to correct the initial guess Of Tga by linear interpolation or 61 extrapolation. This is repeated until TiC = Tin' The steady state profiles, including the gas temperature profile, are then recorded. 4-4.5 The Unsteady State Gas Temperature Profiles The gas temperature at any instant is a result of both the heat generation rate and the cooling rate. At the same time, Tg determines the cooling rate (Eqn. 4-12). This pre— sents some difficulty which could be overcome by trial and error computation. A more serious difficulty arises from the discontinuity Of the gas path with respect to the fluid path (or vice versa) at the passage from the waterwalls to the Superheater. In real systems there are many such discontinuities, and it is desired to avoid compounded trial and error computations, such as would be required for the gas temperature profile at each discontinuity. Therefore, a simplifying assumption is made, whereby the variation of Tg with time is seen as a sequen- ce of steady states. (See Appendix D,§ D.4). Two boundary conditions are required for the Lower Furnace. dT The values T and ID? are considered to be z=0 z=0 boundary conditions, and, in general, "inputs" to the system. The computation of the unsteady state profile is, thus, essentially the same as of the steady state. We start at 62 z = O, and proceed along the positive direction of z (i.e., upwards), following the fluid path. The cooling rate qrc is based on qtr as defined in Equation 4-12. Additional "inputs", or disturbances, to the system are changes in fuel firing rate, and changes in burner tilt. A change in firing rate will be modeled as a change in the values fmx and Wg, and a change in burner tilt will be modeled as a change in the value of 2 (See Appendix A, fu' §A.5). CHAPTER 5: STABILITY AND CONVERGENCE A numerical solution to a differential equation is an approximation of its exact solution. It is obtained by neglec— ting high order terms in the Taylor Series expansion. An addi- tional source of inaccuracy is the truncation error which de- pends on the machine's precision and on the amount of computa- tion involved. Errors of the first type can be made smaller by reducing the numerical mesh size, i.e., the magnitudes Of the distance increment,IAz, and of the time increment,Z§t. It is necessary to establish that the numerical solutions converge to the exact solution as the mesh size is reduced. In many cases of practical importance, as in this case, an exact solution is not possible. It is proposed, therefore, to test the numerical solution for convergence by obtaining several solutions of the problem, using a different mesh size for each solution. If the solution curves tend to come closer together as the mesh size is reduced, this would indicate convergence to the exact solution. The repetitive computation process may tend to compound errors of both types. This results in an unstable solution which tends to oscillate and, eventually, blow up. Error pro- pagation can be treated analytically in some simple cases, but this is not considered feasible for the problem in this work. 63 64 From both the theory and the practice of numerical analy— sis we know that stability depends on the increments' sizes. It is prOposed, therefore, to establish by experiment the mesh size that would yield a stable solution. Preliminary tests of stability and convergence were con- ducted on a simplified system, comprised of an horizontal superheat pass, in order to study the major effects. The re- sults are described in §5.l. Further study of the complete model, as described in Chapters 1 and 2, and the conclusions are given in §5.2 and §5.3, respectively. §5.1 Preliminary_Tests The system under study is a horizontal tube, representing a superheat pass. At steady state, the fluid mass velocity is 60 g/(cm2)(sec), the outlet pressure is 240.0 bar, and the outlet temperature is 560°C. The inputs are step changes of the fluid inlet pressure and velocity, and a ramp change of the fluid inlet temperature. The fixed parameters are: Pressure step change —O.2 bar Velocity step change -40.0 cm/sec Temperature ramp change 0.2 OC/sec Tube length 1200. cm Tube inside diameter 4.0 cm 65 Gas temperature 1200. OC Overall convective heat transfer coefficient 6.0 Btu/(hr)(ft2)(°F) Friction factor 0.01 5.1-1 Tests of Convergence Four computer solutions to the problem were obtained, with different mesh sizes. The number of the distance increments is denoted by n. a) Az = 60.0 cm At = 5.00 sec n = 20 b) A2 = 30.0 cm At = 2.50 sec n = 40 c) A2 = 15.0 cm At 1.250 sec n = 80 d) Az 7.5 cm At 0.625 sec n = 160 The results are shown in the Figures 5.1 through 5.5. In Figure 5.1, fluid temperature profiles at various response times, with mesh size (a), are shown. The profiles are nearly linear for the system under study, and the variation with res— pect to time is observed to be largest during the initial 10 seconds of response time. The rate of temperature change le— vels off shortly afterwards, thus following the input ramp change. The effects of mesh size variation on the temperature profiles is shown in Figure 5.2. The profiles, at response time t = 5 seconds, are markedly convergent. The effects of mesh size on the outlet steam conditions 66 are shown in Figures 5.3, 5.4, and 5.5. The results indicate poor convergence during the initial 10 seconds of response time, and good convergence thereafter. 5.1-2 Test of Stability Seven computer solutions to the problem were obtaired, with a fixed value A.z = 60 cm, and with different values of At. The results are presented in Figures 5.6 and 5.7, where the variation of outlet temperature with time is plotted for different values of At. In these graphs the time scale is different for each curve, and therefore the abscissa was chosen to be j, the time increment index number. The curves are observed individually for indications Of instability, such as oscillations. The curves appear to be stable for At = 5 seconds and for At = 1 second. Signs of instability are first noticeable when At = 0.2 seconds and become more pronounced as.At is decreased. The responsescfifvelocity and specific volume are similar to those of the temperature. The pressure response reveals no signs Of instability in the range of values of At under study. 67 §5.2 Determination of the Mesh Size The results of the preliminary tests indicate that for .Az = 60 cm and IAt = 5 seconds adequate stability and con— vergence are obtained, except for the initial 10 seconds of response time. Another conclusion is that stability may be improved by increasing the ratio zSt/Az. In order to establish the adequate mesh size for the complete model, which includes the interaction of the gas temperature with the steam conditions, further studies were made with the system as described in Chapters 2 and 3, using the numerical method described in Chapter 4. The system was disturbed from steady state by a -22% step input to the fluid inlet velocity. Different combina- tions of increment sizes were tried, and the variations of the fluid outlet temperature T0 with time were plotted in Figures 5.8 through 5.10. The effects of varying Az, with At being kept constant at 5 seconds, are shown in Figure 5.8. When le = 60 cm, some oscillation occurs during the initial 30 seconds of res- ponse time. The oscillation is reduced whenIAz is made small- er, and none can be observed when the values A2 = 15 cm and A2 = 7.5 cm are used. Convergence is also improved as Az is decreased. The effects of varying At, withmAz being kept constant at 68 15 cm, are shown in Figure 5.9. Slight oscillation during the initial 10 seconds is discernible for A‘t = 1.25 seconds. All three curves converge at the end of the transient period at response time 90 seconds and beyond. The effects of the magnitudes of both Az and At on con- vergence are shown in Figure 5.10. The ratio AtflAz was kept constant, and the increments' sizes used were At/Az = 5/15, 2.5/7.5, and 1.25/3.75 sec/cm, respectively. In general, higher values were obtained for finer mesh sizes with a maxi- mum deviation of 3.50C. § 5.3 Conclusions For a fixed ratio At/Az, improved convergence is Obtained for finer mesh sizes as seen from Figures 5.2 through 5.5, and from Figure 5.10. From the results shown in Figures 5.6 through 5.9 it ap- pears that stability is improved as the ratio AtflAz is incre— ased. Comparing the response curves corresponding to At/Az = 1.25/15 in Figure 5.9, and At/Az = 5/60 in Figure 5.8, we see that stability is improved in the finer mesh, (the ratio At/Az being the same in both cases). One may attribute the improvement to the reduction of either At, or Az, or both. However, the Observed stability of some response curves with 69 At = 5 seconds, and the Observed instability wheneverlsz = 60 cm, imply that stability is affected by the magnitude of Az, and not of At, in the range of values under study. When At is kept constant, the choice of A2 affects the final value of the variable (reached after approximately 90 seconds of response time), whereas when Az is kept constant, the final value is not affected by the magnitude of At. Both stability and convergence are considered satisfac- tory for Az = 15 cm and At = 5 seconds. With this mesh size, 80 seconds of computation (not including compilation time) were required for a CDC-6500 digital computer to solve the problem and provide the system's response during the 100 seconds after the step change was introduced. EU 00 H snutcm mpsooom m H pa.mmeawmocm oLSpMLeQEeH tends "H.m ousmflm 80 .N 0259 on» macaw oocmpmfln 00m OONH OwOH 000 a J) d J ovw Can 000 Omv Com ovN ONH . _ . u q u a a Iql . q q 4 q cam on 7O Aepmum hpmoumv o Omm 1 03 1 0mm 1 com 30 ‘I aanqeaadwel PInId mesooem m H 9 pm moaflmosm medumLOQEeH pedam was so sowpmfltm> eNflm awe: mo mooemwm 0:9 um.m ensues 50 “N 0939 0:9 mcoam museumfln 71 oee can owe ova o em. I _ 1 . . . . . . m3. . 00m 103 I cum 1 I 03. i 1 1 I e? mom 1 Heaven .I\ x. o. .IIIIIIIIt. m.e\m~e. 1 oem \“o. -.. I I I I BEN; . \\\\> IIIIIII em\m.~ I omm .\ .\ oe\m u ~<\p< A wll .I 1H“.V.‘.lu xxsx “ A com \\ . ‘I aanqeaadmal PTUIJ Do 72 ON omsoamem OLSpMLQQEoH uefipso map so coepmflhm> emflm :mez mo mpoemmm 0:9 "m.m etsmflm 00m .9 wasp omcoqmem NH 4 m a m.e\mme. ma\m~.~ em\m.m oe\m u ~<\u< 1 com mom cum I sangeaedwel detqno PInId O ‘ Do 73 emcoamom Sbaooao> peHDSO map so cospmflem> ONHm 3mm: mo muoemmm one "v.m Gunmen ON oem .0 mafia emcoamom NH v- I II I I l I. I )1 d — Ifi d u.) m.n\mmo. m~\ma.~ cm\m.~ ce\m u ~<\u< Own A T. n We. 0mm 0 n q T. 9 . 1. Con A W O O I. 4 00m .A A O 0 Cam MP S 9 0 Cam 74 ON omcoamom eESHo> oemaoeam peauzo 0:» co COfipmflLm> onem zwoz mo muoommm was "m.m Opsmflm com “a eeflfi omcoamom ma oa o m _ A a 4 q 4 e A a a . 4 . - III) m.e\m~e. -I I. I. m~\mm.~ ..... .2... \x \ .\ l oe\m u ~<\eq \ \ .\ .\ .\\ 1 0m.mH oo.mH Om.mH cm.mH oo.m~ oo.v~ o~.v~ 5/ mo ‘ 0A amnIoA Otgtoeds quqno ptntd 8 ~ . . . .lrIII-ns!l.l. a»: — ..‘IIA. nu- 575 u 0 8° 570 a) z. 3 +3 ‘6 $4 a) a. E cu 5-1 4.) a) ...I *3 o 565 p H :1 H a. 560 557 0 2 4 6 8 10 I l T T I l f I I 1 L At = 5 sec L L At = 1 sec t \ At = 0.1 sec At = 0.2 sec I I l l l 1 1 l l O 2 4 6 8 10 12 14 16 18 20 Time Increment Index j Figure 5.6: The Effects of Variation of At on the Outlet Temperature Response,le = 60 cm II) I. - II‘ 1‘ F} I.) Ir) II‘J ... . I . b HUFI-I, . .1 IIUNIO-hr'.~. .i.' i din! U 0 ... .Vulu-J'I a 1 O Fluid Outlet Temperature T 567 566 565 564 563 562 561 560 559 558 0 2 4 6 8 10 I I I I 7 I I I I I r At = .06 _ “//////// At = .04 sec L. I r—\ ’— jAt=.O8 )- “if 7 _ — ' I l l l l I l Figure 5.7: The Effects of Variation of At on the Outlet 8 10 12 14 16 18 2‘0 Time Increment Index Number j Temperature Response, A2 = 60 cm .°C 0 Fluid Outlet Temperature T 77 I) 730 7 AZ = 7.5 cm ’liw/“""'- AZ = 15. 720 . //’,,» """" AZ = 30. /,” AZ = 60 710 ’- .// .// / I // 6 -” 90 7 // / /, 680 . 7 h h 670 " [I p fl 660 I- "I -I h I 650 '_ 1", .II 640 P h I, II 630 *- [I p u 620 - V H H 610 e U N fl 600 — 4 4 590 ~ ’I 580 - 570 ‘ 560 550 ‘- 540 I 1 J 1 l I I I 1 l l 10 20 30 4O 50 6O 7O 80 90 100 Response Time t , sec Figure 5.8 The Effects of Variation ofAmzon the Outlet Temperature Response,.At = 5 seconds °C , Fluid Outlet Temperature T0 730 720 710 700 690 680 670 660 650 640 630 620 610 600 590 580 570 560 550 540 Figure 5.9 The Effects of Variation ofzn;on the Outlet Temperature Response, Az = 15 cm 78 Detail 600 ’1 590 r I' 580 b / / / / / 570 )— / /’ [I 560 I 1 I 0 5 10 15 20 I I 1 1 l I I L I O 10 20 3O 4O 50 60 70 8O 90 100 Response Time t , sec OC Fluid Outlet Temperature To , 730 700 650 600 550 540 79 Az/At = 15/5 Az/At = 7.5/2.5 —————— AZ/At = 3.75/1.25 ----—--.- I I I I l I 1 L, 1 1 i T 10 20 3O 4O 50 6O 70 8O 90 100 Response Time t , sec Figure 5.10 The Effects of Mesh Size Variation on the Outlet Temperature Response CHAPTER 6: RESULTS AND DISCUSSION The mathematical model is designed to provide the dynamic response of the system to various disturbances or inputs. The computer solution includes the steady state profiles (va- riation along the fluid or gas path) of the fluid temperatu- re T, the fluid pressure p, the fluid velocity v, the fluid specific volume V, and the gas temperature Tg in tabulated form. Similar profiles, at time intervals of 5 seconds, des- cribe the dynamics of the system in response to specified in- puts. Thus, the computer solution gives the variation of T, p, v, v, and Tg with respect to both time and distance. The fluid path is divided into 360 equal distance incre- ments, .Az = 15 cm. Thus, we have 361 values of each of the aforementioned variables describing the profile. Up to 26 time increments, Iat = 5 seconds, for a total of 130 seconds of response time, were found to be needed to describe the transient response. To include all that information in this report would require many volumes of tabulated data, and there— fore this was not done. Rather, selected portions of the data recorded by the computer were presented in graphical form. The most important information is the variation with res— pect to time of the steam conditions at the boiler outlet. In some cases profiles were also included in order to describe the variation along the fluid or gas path. 80 81 The steady state solution is shown in Table 6.1. The apparent discontinuity in the Tg profile at z = 3000 cm is due to the fact that the Tg data follows the fluid path which runs countercurrent to the gas path in the Superheater Section. §6.1 Variation of Fluid Inlet Velocity Changes in the fluid inlet velocity result in changes in the fluid flow rate, and consequently in the energy output of the boiler. Such changes are made by manipulating the throttle valve and the boiler feedpump. The effects of step changes of -11%, -22%, and +22%, respectively, are shown in Figures 6.1 through 6.4. The fluid temperature profiles at various response times are shown in Figure 6.1 for the -11% step input. The lepe of the curve depends on the local rate of heat transfer as well as on the heat capacity of the fluid at that point. Thus, in the neighborhood of the critical point (T = 370 to 390°C), the slope is almost zero because the specific heat is high. The variation of the slope at z = 3000 cm corresponds to the beginning of the Superheater Section where the fluid and the gas paths are countercurrent. The slope increases in this Section because the gas temperature increases along the fluid path. The fluid velocity profiles for the -22% step input are 82 shown in Figure 6.2. The velocity increases sharply in the zone of transition from a liquid to a vapor state. In the liquid region the profile is almost flat, and in the vapor region (above the critical temperature) the velocity is rough- . 1y proportional to the temperature. The dynamics of the fluid outlet temperature To’ pres- sure po, and velocity v0, in response to the -22% step input, are shown in Figure 6.3. The temperature rises to a new steady state at a higher level, T = 727°C. This is to be ex— pected, since fluid flow in the tube is decreased while the heat absorption remains essentially the same. The initial drOp of the outlet velocity V0 in response to the reduced inlet velocity is later offset by the decrease in the fluid density, associated with the rise in temperature. The outlet pressure po also rises, presumably as a result of the reduced flow rate. The responsestx>the +22% step input, shown in Figure 6.4 are inversely similar to the former case. §6.2 Variation of Fluid Inlet Pressure Fluid inlet pressure changes are associated with the boi- ler feedpump operating conditions. The effects Of a -1 bar (ca. 14.5 psi) step input to the fluid pressure at the boiler inlet are shown in Figures 6.5 and 6.6. 83 The pressure profiles at various response times are shown in Figure 6.5. The slope of the curve is steeper along the vertical section of the tube. A steady state is reached after 5 seconds throughout the length of the tube. The dynamics of To, p0, and v0, are shown in Figure 6.6. The transient response of the fluid pressure is terminated after 5 seconds. The outlet temperature undergoes slow fluc- tuations before reaching the final steady state value. §6.3 Variation of Fluid Inlet Temperature The fluid inlet temperature depends on the feedwater heat— ing system. This system's dynamics are reported to be slower than the boiler dynamics, and therefore the disturbances were described as ramp functions. The effects of a 0.20C/sec ramp input to the fluid tem— perature at the boiler inlet are shown in Figures 6.7 and 6.8. Fluid temperature profiles at various response times are shown in Figure 6.7. The general shape of the profile at t = 50 seconds is different from that of the steady state profile, indicating transient changes within the system. The resulting lag in the To response can be seen in Figure 6.8 in which the dynamics of To and v0 are shown. It appears that the outlet temperature response lags approximately 70 seconds behind the ramp input at the boiler inlet. At t = 100 seconds, the 84 rate of change of To becomes steady, and the shape of the pro- files becomes similar tO that of the steady state profile. §6.4 Variation Of Firing Rate Firing rate is a manipulated parameter in the Operation of power plants. This is simulated as a change in the value of fmx’ with a proportional change of the gas flow rate (See Appendix A, §A.5). The effects of a 20% decrease of the firing rate are shown in Figure 6.9. As expected, the outlet temperature drops to a lower level. The outlet velocity drops as a re- sult of the increase in the fluid density, associated with the temperature drop. §6.5 Variation of Burner Tilt Tilting the burner affects the flow pattern within the furnace and thus serves as a control parameter in the Opera- tion of power plants. This is simulated in this work as a change in the value of zfu ( See Appendix A,§ A.5). The steady state no—tilt gas temperature profile is com— pared with the steady state uptilt and downtilt profiles Ob— tained at t = 100 seconds, as shown in Figure 6.10. An up— ward tilt is represented by increasing 2 by 240 cm, and a fu downward tilt by decreasing z by 240 cm. The upward tilt fu 85 resulted in an upward "shift" of the gas temperature profile, and vice versa. The dynamics of T0 and v0 are shown in Figure 6.11. The response curves appear to be antisymmetric. The transient response lasts approximately 70 seconds after which a new steady state is reached. §6.6 Combination of quuts A combination of a -22% step input to the inlet velocity and a 20% decrease of the firing rate may be viewed as a simu- lated control action, following a reduction in the electrical load. The dynamics of To and v0 are shown in Figure 6.12. The initial temperature drop appears to be a result of the faster dynamics of the gas side. Thus, the effects of the decrease in firing rate result in the initial temperature drop. After 30 seconds, the reduced fluid flow rate causes the tem— perature to rise. 86 c..¢mm. ~c.cmm. co.ccm. mc.mnm_ cc..nm. on.n~m_ .o.n.m. ~m.mcm. nm.~oc_ Neopre— mwonoca __.bc¢. mm.omc. c¢.o.c. mr.oon~ nc.>on~ oc.ncn. oe.h.n_ .meooma omomoma b_ono~a .o.¢r~. mmecoma or..m~. me.>n~. .o.-~_ oo.>o~. .m.~o.. co.b>_. 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I J _ . _ . _ . L 00m 0mm. 00% Omv com .\A\. II.II.IIII mucooom OOH H u i \\.\ I 03 \. IIII I I.) mpcooom Om H p . >\ . \ A x .\ o u a . \.\ I 08 xx 1 \ I i ‘I eanqeaedwel pintd 3o oovm 00w? panes nopm sueooao> eNNI OONV 000m 000m OOQN so . a spam eesae map meos< oocmpmen A . q d U upcooem OOH mpcooom om O _ A 4 1 .meaemoei speooao> cease "N.e messes Om Om OOH 0mg OON 0mm cow 0mm oov owe ces/mo ‘ A AqtootaA PINIJ oC Fluid Outlet Temperature To , 720 700 680 660 640 620 600 580 560 540 91 Temperature L ’ o 8 ._ \E 55 o a - —460 “ ” 0 a? Velocity > 450 >, G) +3 9-1 ...; a ' r440 8 m H e Q’ A ._____.. -430 > p *3 .2 ~420 3 +3 =‘ 8 O «410 1, e .H ': 3 Q “400 E: LI... —240.20 Pressure 4 ~240.10 ~240.00 i 4 L I l I I g i O 10 20 3O 4O 50 6O 7O 8O 90 100 Response Time, seconds Figure 6.3: Fluid Dynamics at Boiler Outlet, —22% Velocity Step Input Fluid Outlet Temperature To , 0C 580 ~560 S40 520 500 480 460 440 92 8 h “460 m \ E 4450 ° 0 “440 > .3 —. T430 ...; O .3 —420 0 > ~410 g H p 1400 8 TS «390 5~ H« Li.- fl 1 1 1 4 1 1 1 1 1 0 10 20 3O 40 50 60 70 80 90 100 Response Time t , seconds bar Fluid Outlet Pressure pO , N p O O O 239.90 239.80 Figure 6.4: Fluid Dynamics at Boiler Outlet, +22% Velocity Step Input 93 OSQCH nopm madmmOLm Lmn .HI «wofifimoam mgzmmoum UHDHL "m.c oasmflm go a N spam vflsai wap mcoa< oocmumfla oovm 00m? OONV Doom DOOM OOVN OOmH OONH 000 O _ . q . a q _ 1‘ . d . . [A . _ . _ a Ow.wma 00.0mm O0.0VN u 1 o a h mtcooom oo~ -Illllll mucooom m o oo.~v~ 4i ov.Hv~ eanssaad PIUIJ d ‘ Jeq OC Fluid Outlet Temperature To , 565 560 555 550 Figure 6.6: Fluid Dynamic at Boiler Outlet, —1. 94 ‘* temperature rTFTj J I I I T K pressure //"\\\\\\\______‘: velocity 1 1 1 1 l 1 1 1 1 l bar 240.00 1 E]uid Outlet Pressure Dog? 239.00 Fluid Outlet Velocity v0 , cm/sec 4450 4440 ~430 420 O 10 20 3O 4O 50 60 7O 80 90 100 Response Time t , sec Pressure Step Input bar 95 DSQCH QEmM QLSOmgoQEOB amonflmOLm opspmpoQEOH UHSHm "5.0 ohsmflm ac . a spam .ufisam onp.mcoa< mocmumfla covm oowv ooNv 000m 000m oovm OOwH OONH ooo _ _ 4 _ 1 A _ 4 fi . _ . d \\ \\. \\ Iai?l.l.1. mccooom OmH H p x . . \ l.|:l:l:l. mucooom oo~ H p \ \\ \. \ \ \. llllll mucooom om H u «\ o H p \ 00m 0mm oov Omv 00m 0mm Goo ‘ I eanqeaedwel pintg 30 oC Fluid Outlet Temperature To , 640 630 620 610 600 590 580 570 560 550 540 96 4 Fluid Outlet Velocity v0 , cm/sec 1 p. c» O 470 460 450 1 1 1 l 1 1 l l 440 w 430 ~ 420 d 410 I 1 1 400 0 10 20 3O 40 50 6O 7O 80 90 Response Time t , SEC 100 110 120 130 Figure 6.8: Fluid Dynamics at Boiler Outlet, Temperature Ramp Input 0C ’ Fluid Outlet Temperature To 570 560 550 540 530 520 510 500 490 480 470 97 l _L 1 1 I l l l I O 10 20 3O 40 SO 60 7O 80 90 100 Response Time t, SCC 450 440 430 420 410 400 390 380 370 360 350 340 Fluid Outlet Velocity v0 , cm/sec Figure 6.9: Fluid Dynamics at Boiler Outlet, —20% Firing Rate Step Input 98 mpdmcH DHMH Locusm “moaflmoam oLSmeOQEOB moo "o~.© mpsmflm So a N Spam moo map wcoa< oocmpmfla OOOM OOAN OOVN OOHN OOwH OOm~ OON~ OOo OOo OOM O . W1 a _ A 4 J1 _ _ _ .\ 1 OONH \ \x \\ \\. \ .\\\ l OmNfi .\ x \\ ......... 1 paflpczoa \\ xxx 1 OOMH 1111111 adage: \\.\ \\.\ .\ . \\\ \ was» 02 xx \ 1 Ommfi \ .\ x \ x .\ \\ x .. oovfi \ .\ x \ \ .\ x /. \\\ .\. l OWVfi /. \ .\ x .\ // /. \\ \.\ /. \ . /. \ .\ J OOmH I I. \ .\ // /./ \ \.\ I, .I.’ \\ \. .I/ /.,.I \\\ .\\ [III]! H"!.\.D\“:WI .|\.\ l ommH 9 COOH x0 ‘ Bl eanqedadmal sea 0C Fluid Outlet Temperature To , 570 560 550 540 99 l Uptilt Downtilt temperature ~'~'—0~ ‘_o-o‘I-n---Id velocity ‘-‘-.—u~I---.-.—o-I-cd l l l L l l l l O 10 20 3O 4O 50 60 7O 8O 90 Response Time t , sec 100 450 440 430 420 Fluid Outlet Velocity v0 , cm/sec Figure 6.11: Fluid Dynamics at Boiler Outlet, Burner Tilt Inputs OC , Fluid Outlet Temperature To 630 620 610 600 590 580 570 560 550 540 530 100 l Fluid Outlet Velocity v0 , cm/sec ~ 460 d 440 — 420 =1 — 400 — 380 - 360 ‘ 340 — 320 1 1 1 l 1 1 1 1 I 1 1 300 O 10 20 3o 40 50 60 70 80 90 100110 120 130 Response Time t , sec Figure 6.12: Fluid Dynamics at Boiler Outlet, Fluid Velocity and Firing Rate Step Inputs CHAPTER 7: CONCLUSIONS AND RECOMMENDATIONS The results described in Chapter 6 demonstrate the im- portance of mathematical modeling for understanding and con- trolling the power generation process. It is unlikely that a system of such complexity, represented in this work by 3 non—linear partial differential equations, and by the highly non-linear State Equations, may be approximated by assuming overall average values for any of the variables. The response curves provide the necessary information for both design and control; namely: the final steady state values in response to step inputs, the final rates of change in response to ramp inputs, response time lags, fluctuations, etc. Of special interest is the observed response of the sys- tem to pressure inputs. The results (§6.2) indicate that a pressure step input is transmitted very rapidly and that steady state is attained within 5 seconds. Similar results have also 3’8 It may be concluded that been reported in the literature. pressure inputs at the boiler outlet, associated with varia- tion of the throttle valve position, may be simulated by pres- sure inputs at the boiler inlet, and thus a time consuming iterative solution of a split boundary condition problem will not be necessary. Also noteworthy is the considerable time lag (ca. 70 se— conds) associated with the response to the fluid temperature 101 102 input (§6.3). The apparent difference in the response time lags of the gas and of the fluid sides (§6.6) is significant for the design of a control system that will eliminate the resulting fluctu- ations. An important feature of the model is its versatility with respect to the nature and functional form of the input. Solu- tions for inputs of fluid flow rate, temperature, and pressure and of firing rate and burner tilt were described in Chapter 6. Both step and ramp inputs were tried, and other functional forms can be treated in a similar manner. A source of variation in a power plant operation which has not been accounted for in this work is the changing qua- lity and conditions of the fuel and air mixture. When the fuel is pulverized coal, the quality of the coal, the fuel to air ratio and the thermal conditions of the mixture may be eXpec- ted to vary under normal operating conditions. It is proposed to further develop the model by including an additional section in the gas side. This section, designated the Middle Furnace and located in the burners area between the Lower and the Upper Furnace sections, may be modeled as a stirred tank reac- tor. The fuel-air mixture may then be treated as a variable input to the Middle Furnace. The capability of the model to solve for large inputs, in 103 excess of 20%, is a marked improvement over linearized models. It should be noted that disturbances of such magnitude are to be expected under normal operating conditions. The availabi- lity of complete and thermodynamically consistent State Equa- tions for water and steam has been a major contribution to this work in facilitating the development of the non-linzari— zed model and in the derivation of additional thermodynamic relationships as required in the Equations of Change. In the design of central station boilers, some sections of the gas often contain more than one section of the fluid circuitry. For example, the Superheater Section may be en- closed in a waterwall type circuitry, in addition to the superheater tube banks contained in it. This would raise difficulties in the procedure of the numerical solution where the sequence of computations follows the fluid path and the gas temperature is calculated from the energy balance. In such situations, the following iterative procedure is prOposed: let the two overlapping fluid section be desig— nated Heat Exchanger A and B, respectively. Let the gas stream be conceptually divided into two streams, with flow rates Wga and W respectively. When computing along Heat Exchanger A, gb’ the gas temperature is calculated from the energy balance using the value of Wga for the gas flow rate. Denote the gas tempe— rature at the end point of A by Tga' The computation then 104 proceeds along the fluid path until Heat Exchanger B is reach- ed. In similar manner to A, the gas temperature is computed using the value of W for the gas flow rate. Let the gas gb temperature at the end point of B thus computed be denoted by T As both T and T refer to the same location in gb ' ga gb the gas path, they should be equal. If not, then the diffe— rence between them may be used in an iterative procedure to correct the initial guess of the relative magnitudes of W and ng (the sum of which is the actual gas flow rate). This may be repeated until Tga = Tgb' Whereas a rigorous test for the validity of the model can be made only by applying it to a real system, we may con— clude that the observed results are qualitatively compatible with known or predictable behavior. This applies to the ge— neral shape of the response curves, to the pressure dynamics, to the observed fast response of the gas side, relative to that of the fluid side, and to the effects of burner tilting on the gas temperature profiles. In summary, the major accomplishments of the prOposed model, compared to previous models, are: i) Absorption of ra- diation by the gas and by dust particles is accounted for; ii) The interaction between the fluid conditions and the gas temperature in the furnace is expressed by the modeling of the heat generation function. Thus, it is not necessary 105 to assume that a constant gas temperature exists in the fur— nace, and that variation of the fluid flow has no effect on the gas temperature; iii) The proposed model can solve for inputs that are up to an order of magnitude larger than in any previously reported work; iv) The computing time required is well within the practical limits for industrial use; v) the accuracy of the results, as seen from the convergence tests (Chapter 5), is highly satisfactory; vi) The introduc— tion of an accurate and thermodynamically consistent formula- tion of the equations of state for water and steam. APPENDIX A APPENDIX A §A.1 The Equation of Continuity The general form of the equation of continuity in rectan- gular coordinates is: <19+i at ax(pvx)+§;(ovy)+5§;(ovz)=o (A—1) Assuming variation in the z—direction only, we have §f+éa§(pvz)=0 (A-2) write v2 = v 35+93f+vgf=0 (A-3) Equation A-3 is identical with Equation 3—1. §A42 The Equation of Motion Assuming variation in the z-direction only, the equation of motion is: v av .12 an.z .. + — =- - 0000000000. _ 0% Vaz 62 az +Dg (1 9) In a horizontal tube, the graviational acceleration E = O; in a vertical tube with upwards flow E = -g = -981 cm/sec The normal stress T22 is related to the velocity gradient. 106 107 For Newtonian fluids with constant viscosity u , we have 2 8.52:“avz (A-4) 32 322 This term is small, compared to the other terms in Equation 1-9, and is neglected. Each term in Equation 1-9 eXpresses rate of momentum transfer per unit volume or, equivalently, force per unit vo— lume. Consider a fluid volume element AV, contained in a tube segment of lengthMAz and inside diameter D. We have: nDz “Dz AV = —-° Az , and cross sectional area A ='-- 4 ex 4 Area in contact with the tube wall Af == fiDAz. The friction force acting on the fluid in AV is given by: 2 2 F = f Af(pv /2) = f(flDAz)(pv /2) ........ ..... (A—5) Dividing F by AV and including it in the force balance, Equa- tion 1-9 becomes: 0855+Vg—Z)=-3§-pg-(2f/D)pv2 ............. (A-O) A factor of 106 multiplies the pressure term in Equation 3-2 6 2 in order that the units will be consistent (1 bar = 10 dyn/cm )- 108 §A.3 The Equation of Energy Assuming variation in the z-direction only, the equation of energy is: pep (33+ v37 -(v-a’) - (mm) + (593—172—1513? v35)... (1-11) Each term in Equation l-ll expresses rate of energy trans- fer or interchange, per unit volume of the fluid. The term (TaVv) is a tensor notation representation of the irreversible transformation of mechanical energy to internal energy by viscous dissipation. This effect is small, unless high velocity gradients are encountered. The term -CV°3) is the rate of heat transfer by conduc- tion across the boundaries of AV. In this work, this term is replaced by an expression of heat transfer from the gas into the fluid through convection and radiation. Consider a segment of the tube as in §A.2. Convective heat transfer is given by U(Tg - T)ZSAC, where AAC is the area of convective heat transfer. Radiative heat transfer is given by ¢;'c1'(Tg4 - T4)AAr, where AAr is the area normal to the radiative flux. The combined rate, per unit volume of the fluid is: = _ 4 _ 4 _ qtr U(:rg T)AAc/AV + eo'(Tg T )AAr/AV (A 7) 12 Equation 3-4'is obtained from Equation A-7 by using cz= o'°10 . 109 In the Superheater Section, we have: Dh = the inside diameter of the tube Acx = the cross sectional area of the tube 2 AV ”1tDl z/4 , AAc ‘nDlAz , AAr ZDhAz , hence AAc/AV = 4/Dh and AAr/AV = 8/(11Dh) With ez= 1, Equation A-7 becomes: __ 8 q —— tr fiDh JL 1 4 4 [2 U(Tg-T) +0 (Tg - T )J (A—8) The waterwall tube's inside diameter is Dv. There are n waterwall tubes per one superheater tube, but the total cross sectional area remains unchanged. Thus, Acx = flDi/4 = nnD3/4, and n = (Dh/Dv)2' Noting that only one side of the waterwall tubes is exposed to the gas, we have: _ 2 = = AV — nanAz/4 , AAC nanAz/Z , AAr anAz , AAC/AV = 2/Dv , AAr/AV = 4/111)v , and the rate expression becomes: qtr =;—g—;[-12‘-U(Tg - T) + c'(Tg4- T4)] (A-9) In order to have a uniform formulation throughout the boiler in the computer program,Deq is defined as follows: eq/Acx = 8/11Dh for the superheater tube, and Deq/Acx = 4/1rDv 2 for the waterwall tube. ‘With A =IID /4 , we get D = 2D cx h eq and Deq = Di/Dv for the superheater and waterwall tubes, h 110 respectively. The term (31-12%)1) was obtained explicitly from the other equations of state as follows: aln 3 __jL. G m)p_\7 (g?)p (A-10) When p is given as an explicit function of 4 and T, as in Equation 4-5d, then we can use the relation: (3? _ (EDP/8T); 519p _ .. (OP/(WM. (A-11) The resulting expressions appear in the computer subroutines (Appendix C), under the variable name DLNT. §A.4 The Specific Heat of the Gas Data on the variation of Cg vs. Tg for various fuels is given in the literature18 in the form of a chart. The curve corresponding to Bituminous Midwestern coal with 20% excess air was divided into segments, each of which was approximated by a straight line. The equations of these line segments are of the form: C =aT +b 0.0.0.0.........OOOOOOOOO (AI-12) g g The values a, b were obtained from the chart, and given as data in Subroutine SCG, Appendix C. lll §A.5 The Heat Generation Function Qualitatively, the form of the heat generation function f(z) is given in Figure 2.4. The area under the curve is equal to the total heat absorbed by the fluid in the boiler, under steady state conditions: Qf = (Houtlet — Hinlet)W0Acx 9.0000000000090000 (A‘13) Denote the height of the boiler by Lt' Then the area under the curve f(z) is L of /2 , where f is the maximum 1: mx mx value of f(z), occurring at z = zfu' Equating Qf to the area under the curve we obtain: 2WoA f =—-——-°" (H t outlet-Hialet) coco-0000000000000 (A-14) The function f(z) is given by two straight line equations: (fmx/zfu)z for O <. z < zfu ... (A-lSa) f(z) f mx L - 2 '(L - z) for 2 g z < L ... (A-15 ) t b t fu fu t In the Superheater Section of the boiler we have 4 super- heat passes, designated as pass A, pass B, pass C, and pass D, respectively. Each pass occupies a volume of gas, correspond- ing to a length 18 of the gas column. The amount of heat gene— rated within a gas volume of a superheat pass was calculated 112 by integration of Equation A-15b along an interval 15' Thus: 1 l f s 5 mx q =—-£f(z)____ +f(z)=_]=-—-(0+ 1) sd. 2 z Lt z Lt 1S 2 Lt zfu s _1. fmx 2 — i L - z 1s t fu 1s 1s fmx qsc =‘7T'[f(z) z=L -l + f(z) z=L -21 J _’7T' L - z (ls + 21s) t s s t fu = 4L”. fmx 12 2 Lt - zfu s The expressions of qSb and qsa are obtained in a similar manner. The results are given in Equations 4—16. The numerical values that were used are: 2 Lt = 3600 cm 18 = 150 cm hence l:/2 = 11,250 cm 153 The gas flow rate Wg is estimated from empirical data, correlating the heat of combustion, heat losses, and the ratio of air to fuel in the feed. When the firing rate is changed, then both fmx and Wg are changed, in the same proportion. The value of z u depends on the burner tilt. When the f tilt is downwards, zfu is reduced, and vice versa. 113 §A.6 Energy Equations in the Lower Furnace The heat balance in the Lower Furnace is given by the equations: qa - qa +Az(qs - qrc) = O ..... ............ (4-17) 2 z+Az dT T qa =-Dc-d-zg , qa =-D dg (4-18) +Az dz 2 z+Az qs = (fmx/zfu)z .................. (4-19) In order to obtain the variation of Tg within the gas volume element of lengthikz, replace .Az in Equations 4-17 and 4-18 by 82 and rearrange Equation 4—17: q ‘ q a z+Az a z - =- + 0.0000000000000000 -1 52 qs qrc (A 6) taking limits as 526:0 , we have - dqa/dz=-qs+qrc 000.000.000.000... (A-17) Substituting Equations 4-18 and 4-19, obtain: dzl‘ fmx Dc-—-2£=qu—T'z oooooooooooooooooo (A-18) dz fu Equation A-18 is identical with Equation 4-20. Integrating Equation A-18 twice between 2 and z+Az , taking qrc to be constant along the interval Az, we obtain 114 Equations 4-21 and 4-22. At 2 = zfu , the transition from the Lower Furnace to the Upper Furnace is accounted for by assuming eddy conductance at z and gas flow upwards at z+Az . The resulting heat ba- lance is: - T g ) = (q - qs) Az ...... (A-19) +w- - qa g Cg (Tg re 2 z+Az Substituting Equations 4-18 and 4-19, and dividing bytfiz, we have: T -T Dc dT gz+Az gz -—. - C 7 —q -f 0000.. 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[I @art (see Fig. B. {D— 135 Assign Boundary Conditions L = 1 subregi onstubregions l and 2 3 and 4 Compute 3., ij, Compute ij, DLNTj Continue DLNTi from S.Eqns. from S. Eqns. (See Fig. 3.1) ’ Yes Assume Assume Pm V No Compute vm, Vm, Compute pm,‘ym, T , from Eqns. Tm, from Eqns. of Change of Change L=L+1 Compute $8 Compute pS Compute Tg from S. Eqn. from S. Eqn. ‘ Correct Correct Distribution Pm Vm according to gas-side I No sections Figure 8.2: A Flowchart of the Profiles Computation APPENDIX C FOConwkofiQQUoNFOoN>OoP>OonaQ N .FZ40.UQU.a>oaaoak ..O~.U>..mu.Coflo.buo.mccuomuopcuoccu a ..o—cmuo.m.muoabcwu.ar~.OU.AU.mm.a~rcmCo‘m—c4m..mmc4u ZOIIOU Iflfluuufluluflfllflfl U 4mun 20—P4DCu U ~>w kZ—FDOQmDm 4 cofimocnsm com mcoflumsvm mumpm u Hmc cam 4>m mocflpsocnsm ~.o a Ozw ZQDFUC IGUtO¢m~o¢uFCQU .0fi.¢+l0h4aflfivdumwu “a «aunfi fin FNoKKoUN .m—tafic up c~+40mN\u0Fflafi oNfi+®o~4.W~ofiFNlX0k~flkOfi \QON. ~ o¢)No.¢DNoofl\No.\ONooDONoochoo®¢Noo¢¢Noo¢¢N¢oO¢Noo0¢No\m Ooh>Oonfln N ._ZJQoGQan>oGQoa~ ..GuCU).an—.0..OCFUoAECOUomUo~€UoO¢U — oaO~VManmvao.Fv—anfi~COanwcmmo.urchonfi—44moafichu ZCIZOU .m—cmoam—vd ZO—mZUZ—O fllflflllflfllflfluflfl U Fin ZO—Fo.b>o.naa N .PZ4ooaQU.a>.aa.ah ..©~.U>..«..0..0.bu..mcouowuowouoocu _ .ac—cruoamcmuoab._uo.m..Ou.aucmm..—mcmo.ar_c4m..mmc4u zozzcv "u"ufluuufluuuuflfluufluuuuuflfl umlfi 024 flfllfi mzcudecw «0U wZ_FDOGDDM 02w ZdDPma aa4au4aa4amk4m_hc\.fim.4U4o4+an4an4.akl.&n.4mc m 44NNC4U406+ 4nmu4anh+aoc4mc4C—F4.—N.4Ul w —,4u\.an4au4.04.404.n+aa4.o—.4U4on+am—.4ccl.o—4u\.h—.4U+ — oukmu+NF4am_c4U+ak4.¢~CuuN .aa4auc4m4om+ah4“4.4w4odlu>4u>4.n.4mchaomu44u Oh\.«c4mtmh4...4ml.~uu> 04404404umuh Mk4nbuOP 044mhunk ab4akumk CU 138 m.»4.~N. 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An estimate of the maxi- mum possible value of this term is presented to justify the assumption. From Table 6.1, the average variation of v along the tube is (av/adave. = (437 - 4s)/(90-60) = 0.0727 (cm/sec)/cm. Using the approximation azvz '_' (av/32) z+Az - (av/az) 2 A2 2 Q) and assuming Gavflaz) ave z+Az = (EV/32) 2 (av/a2) z = O ) we have azvzfiazz = .0727/60 = 1.21-10-3 (cm-sec)-l. 1 The maximum value of the viscosity encountered is:3 L1: 8.90'10-4 g/(cm)(sec). Thus, -erzzflaz) = 1.08-10—6 (dyn/cm3). In comparison, the friction force per unit volume is (from Table 6.1, at z = 600 cm) 12.8 dyn/cm3 , and the gravity force per unit volume is 612 dyn/cm3. §D.2 Heat Conduction in the Fluid 2 . Heat conduction in the z—direction. kfazTflaz ) is con— sidered negligible in the fluid energy equation. We shall 151 152 use the approximation azT .. (aT/az) z+Az - (aT/az) z az Az The average variation of T along 2 is (from Table 6.1), (am/azIave. = (560 - 320)/(60 90) = .0444 oK/cm. Taking (aT/az) z = 0 and (ST/32) z+Az = (air/anm, we have azTflazz = .0444/60 = 7.41-10-4 oK/cmz. The maximum 31 value of the thermal conductivity encountered is: k = 0.318 Btu/(hr.)(ft.)(°F) = 0.0055 Joule/(sec.)(cm)(°K) Hence k(azT/azz) = 4.07'10“6 Joule/(sec.)(cm3). In comparison, heat transfer by radiation is: 0.34-5.67-(1.24 — .834) D Egg-o=[(Tg/IOOO)4 - (T/1000)4] CX = 3.08 Joule/(sec.)(0m3) §D.3 Gas—to-fluid Convective Heat Transfer The overall resistance to heat transfer from the gas to the fluid is composed of 3 resistances in series: i) The gas- to-metal film resistance hgm; ii) The metal wall resistance; iii) The metal-to-fluid resistance hm 153 f. Experimental data for hgm ranges from 2 to 20 Btu/ (hr.)(ft.2)(°F). The minimum resistance is l/h = 0.05. gm 153 The thermal conductivity of the tube metal (steel) is:30a k = 20 Btu/(hr.)(ft.2)(°F). Assuming a wall thickness Ax = 1/4" we have the wall resistance (xx/k = 1.04-10_3. The fluid film resistance may be calculated from the Dittus-Boelter correlation (Equation 1-5), the Sieder—Tate correlation (Equation 1—6), and from the modified correlation for high temperature gradients (Equation 1—7). The results are given in Table D.l. The data31 corresponds to 3500 psi. Values of hm calculated by Equation 1-7 are invariably high- f er than those obtained from Equation 1-5, and therefore were not included in the table. The results show a maximum resistance of: l/(hmf) 1/278 = 3.6 10-3 . We see that the maximum resistance of the min. fluid film is 7% of the minimum resistance of the gas film, and that the tube wall resistance is smaller yet. It is, there- fore, permissible to ignore the variations of the fluid and of the wall resistances, and to lump them together with the gas film resistance as the overall convective coefficient U. 154 Table D.1: Calculated Value of Metal-tvaluid Convective Heat Transfer Coefficient T (OF) 600 700 800 900 1000 1100 1200 p.104 (poise) 8.90 4.84 3.35 3.21 3.35 3.49 3.69 kIBfgégllrnft) .318 .155 .088 .063 .060 .062 .064 CPEBtu/(lb)(°F)] 1.326 4.19 1.563 .941 .775 .701 .666 Pr Number 0.92 1.35 1.43 1.23 1.06 0.98 0.93 Re'IO-4 9.0 16.53 23.9 24.9 23.9 22.9 21.7 I h Bt ) mfkft3(f2;§] 775 718 560 390 338 327 318 Bt h flzftgflog] 618 1335 486 325 290 280 278 2) hm 1) Calculated by Equation 1-5 2) Calculated by Equation 1-6 §D.4 The Gas Energy Equation The kinetic and potential energy terms were neglected in the gas energy balances. Furthermore, an assumption was made that the energy dynamics may be modeled as a sequence of stea- dy states. To demonstrate the validity of these assumptions, consider a segment of the gas column, corresponding to a tube of length Az. The volume of the gas is AV. The energy balance over AV may be written in the following general form: 155 = - + +A +A ooooooooo '- AE/At QS Qrc A(wgcg'rg) 13k Ep (D 1) where, AE/At = (pgAN)Cg ATg/At = the rate of energy accumulation within AV Qs = qS'Az = the rate of heat generation within AV Q = q -Az = the rate of heat transfer out of AV rc rc A(W C T ) = W C °(T . - T ) = the net heat carried by g g g g g g,1n g,out the gas stream AB = %W '(vz . - v2 ) = the kinetic energy change k g g,1n g,out AB = Wg-g-Az = the potential energy change Using the values: Wg = 1800 g/sec .Az = 60 cm ‘Dg = .001 g/cm3 AV'= 2.16 105 cm3 vg = 500 cm/sec qS = 100 Joule/(cm)(sec) qrc = 50 Joule/(cm)(sec) Cg = 1.5 Joule/(g)(°K) We have: A(WgCng)= -5040 Joule/sec, assuming a variation of —20K over A z Qs = 6000 Joule/sec Q = 3000 Joule/sec rc AEk = 11.25 Joule/sec , assuming a 50% velocity change over A z Asp = 10.6 Joule/sec AE/At = 302 ATg/At 156 The results indicate that kinetic and potential energy effects are, indeed, negligible. The rate of change of Tg’ as obtained from the energy balance is .ATg/At = -6.7 oK/sec . Such a rapid change would result in a new steady state within a short period of time. This justifies the modeling of the gas dynamic as a sequence of steady states. NOMENCLATURE _1_‘e_xt-._ FORTRAN a — a constant parameter A — area ACx - ACX - cross sectional area of a superheater tube b - a constant parameter Cg - CGOT — specific heat of the gas Cp - specific heat at constant pressure Cpr - CPR - reduced (dimensionless) specific heat 0,8 - prefix, indicating differentiation D - diameter DC - DIC - dispersion coefficient (See § 3.2) Deq - DEQ - equivalent diameter (See § A.3) Dh - DH - inside diameter of the superheater tube Dv — DV - inside diameter of the waterwall tube E - energy f - function f - F - friction factor F — function; friction force fmx — FMX — maximum value of the heat generation function (See I§A.5) 157 Text g, g gm mf FORTRAN GAC PR PST 158 — gravitational acceleration convective heat transfer coefficient (film coefficient) enthalpy gas to tube wall film coefficient tube wall to fluid film coefficient distance index number in the Finite Differences grid time index number in the Finite Differences grid location in the Finite Differences grid (See Figure 4.2) thermal conductivity length of gas column containing one superheat pass length of the Superheater Section total length (height) of the boiler number of tubes pressure reduced (dimensionless) pressure Prandtl Number pressure as computed from the State Equation (See § 4.3) heat flux rate of heat absorption by the fluid in the whole boiler At ic in FORTRAN QRAC TIM TR VE VOL VR VST 159 rate of heat transfer per unit length of the tube rate of heat transfer by convection and radiation rate of heat generation per unit length of the tube rate of heat generation rate of heat transfer per unit volume of the tube Reynolds Number entropy time temperature time increment computed fluid temperature at boiler inlet (See § 4.4—4) fluid temperature at the boiler inlet (See § 4 4-4) reduced (dimensionless) temperature overall convective heat transfer coefficient velocity specific volume reduced (dimensionless) specific volume volume specific volume as computed from the State Equation (See1§ 4.3) Text fu FORTRAN W WAG EPS ZFU suffix A suffix B suffix C suffix D suffix J 160 — mass velocity of the fluid - mass flow rate of the gas distance coordinate distance coordinate; distance along the fluid or gas path - distance increment — location of the boundary of the Lower Furnace (See §A.5) Subscripts axial - for superheat pass A bulk — for superheat pass B convective; computed — for superheat pass C - for superheat pass D film; fluid; friction gas in; inlet; distance index number in the Finite Differences grid time index number in the Finite Differences grid - location in the Finite Differences grid (See Figure 4.2) Text 11 0,0' FORTRAN suffix suffix suffix suffix M X Y Z PAI RO SIG 161 kinetic location in the Finite Differences grid (See Figure 4.2) outlet; out potential radiative; reduced wall distance coordinate location in the Finite Differences grid (See Figure 4.2) distance coordinate location in the Finite Differences grid (See Figure 4.2) distance coordinate location in the Finite Differences grid (See Figure 4.2) Greek increment; interval; difference Del operator emissivity viscosity a number 3.1416... density the Stefan—Boltzmann constant stress BIBLIOGRAPHY 10. 11. 12. 13. 14. 15. 15 BIBLIOGRAPHY Abu-Romia, M.M., and Tien, C.L., J. of Heat Transfer, 89, 321 (1967). Adams, J., Clark, D.R., Louis, J.R., and Spanbauer, ].P., Trans. of IEEE, Power App., and Syst., 81, 146 (1965). Ahner, D.J., DeMello, F.P., Dyer, C.E., and Summer, V.C., 9th National Power Instrumentation Symposium Paper, May 1966. Bird, R.B., Stewart, W.E., and Lightfoot, E.N., Transport Phenomena, John Wiley, New York, 1960, p. 83. Ibid., p. 181. Ibid., p. 322. Chien, K.L., Ergin, E.I., Ling, C., and Lee, A., Trans. ASME, 82, 1809 (1958). DeMello, F.P., Trans. of IEEE, fingupplement, 664 (1963). Edwards, D.K., Glassen, L.K., Hauser, W.C., and Tuchscher, J.S., J. of Heat Transfer, 82, 219 (1967). Enns, M., J. of Heat Transfer, 84, No. 4, (1962). Essenhigh, R.H., Froberg, R., and Howard, J.B., Ind. Eng. Chem.,.iz, 32 (1965). Fryling, G.R., Ed., Combustion Engineering, Revised Edi— tion, Combustion Engineering Inc., New York, 1967. Ibid., Chapter 6. Ibid., Chapter 7. Ibid., Chapter 17. Ibid., Chapter 21. 162 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27 28. 29. 30. 30a 31. 32. 33. 34. 163 Ibid., Chapter 22. Ibid., Chapter 25. Ibid., Appendix C. Hottel, H.C., J. Institute of Fuel, 34, 220 (1961). Hottel, H.C., Sarofim, A.F., Evans, L.B., and Vasalos, I.A., J. of Heat Transfer, 29, 56 (1968). Hottel, H.C., Williams, G.C., and Bonnel, A.H., Comtus— tion and Flame, 2, 13 (1958). Jakob, M., and Hawkins, G.A., Elements of Heat Transfer, 3 ed., John Wiley, 1957. ' Kutateladze, 8.8., Fundamentals of Heat Transfer, Acade- mic Press, New York, 1963. Ibid., pp. 429-434. Littman, B., and Chen, T.S., Trans. of IEEE, Power App. and Syst., 85, 711 (1966). Longwell, J.P., and Weiss, M.A., Ind. Eng. Chem., 41, 1634 (1955)- Love, T.J., and Grosh, R.J., J. of Heat Transfer, 81, 161 (1965). MCAdams, W.H., Heat Transmission, 3 ed., McGraw Hill, New York, 1954. Ibid., p. 219. Ibid., p. 275. Ibid., p. 445. Meyer, C.A., MbClintock, R.B., Silvestri, G.J., and ngncer Jr., R.C., 1967 ASME Steam Tables, ASME, New York, Spalding, D.B., Combustion and Flame, 1, pp. 287—295 and 296-307, (1957). Viskanta, R., and Merriam, R.L., J. of Heat Transfer, 29 248 (1968). Wohlenberg, W.J., Trans. of ASME, 51, 531 (1935).