In its simplest terms, a numerical model is provided with input data that is used to fix specific operating parameters and return results. Without further un- derstanding, this simplistic view of modeling can lead to unsatisfactory results. More appropriately, a multi- step process is used:
1. Obtain a complete situational description includ- ing physical geometry, process flow, physical prop- erty data and the level of detail needed. It is im- portant to obtain detailed information because seemingly small differences can have a significant effect on numerical solutions.
2. Define the modeling assumptions appropriate for the specific flow system and computer model se- lected while making appropriate tradeoffs; cost and time are balanced against level of detail and information required.
3. Prepare the input data by converting the general technical information obtained in step 1 into the detailed inputs required by the computational model selected. Much of this is accomplished with the use of various computer programs such as com- puter assisted drafting (CAD) software and mesh generation software. Verification of the input data is an important part of this process.
4. Run the numerical computational model until an acceptable solution is obtained.
5. Analyze the results to verify the initial model as- sumptions, to check the results against known trends, to benchmark the output with known field data, and to present the results in a usable form. In application, the computer programs or software used to perform the modeling function are broken down into three general groups that work together to complete the analysis:
1. Pre-processing: generation of the calculational mesh or grid representing all boundary conditions (part of step 3 above and discussed later under
Mesh generation),
2. Solution: execution of the numerical model to de- rive an acceptable solution (step 4 above), and 3. Post-processing: generation of typically graphical
or tabular key results from the numerical model to permit interpretation and evaluation of the re- sults (part of step 5 above).
Limitations
Despite recent advances in technology, increased understanding of physics, and improvements in de- scribing input conditions, limitations remain in apply- ing numerical modeling to engineering problems. Nu- merical modeling can only be applied where there is an adequate understanding of the physics involved. In situations where there is not an appropriate math- ematical description of the physics, numerical model- ing is not possible. Even when a description exists, it may be too complex to be readily used in a model and a simplified approach is required. In this case, results will reflect the simplifying assumptions of the model. Computer technology continues to limit the level of detail that can be modeled with numerical methods. Our understanding of the physics of systems that are routinely modeled with CFD far exceeds the compu- tational resources (size and speed) that are available to model them. A considerable amount of effort is ex- pended on developing simplified descriptions of the physics to make the problem manageable with current computer technology.
The precision and accuracy of the input data also represents a significant limitation to numerical mod- eling. Sources where this may be significant include the level at which the geometry is described and rep- resented, the accuracy of imposing an inlet condition, and the assumptions made in specifying other bound- ary conditions and modeling parameters.
Despite these limitations, numerical modeling can be used in conjunction with other engineering analy- ses. When applied appropriately, numerical modeling can provide invaluable information.
Uses
Many applications for CFD and combustion mod- eling exist within the design and evaluation of steam generators (or boilers) and related equipment. Nu- merical models of the flue gas and steam-water flows
are used to predict boiler behavior, evaluate design modifications, or investigate localized phenomena. Ex- amples of flue gas applications include predicting tem- perature distributions within a furnace, evaluating fluid mixing due to the retrofit of systems to control nitrogen oxides (NOx) emissions, and improving air
heater flow distributions to increase heat absorption. Water-side applications include determining flow rates for boiler furnace circulation systems and evaluating system stability, among others. Many of the uses are summarized in Table 1.
Theory
The foundation of numerical modeling is the devel- opment of a mathematical description of the physical system to be modeled. Whether this is as simple as heat transfer through a wall or as complex as a pul-
Table 1
Sample Numerical Model Applications
Application Purpose
Windboxes Evaluate flow field within windbox, determine expected air distribution to combustion equipment, and determine pressure losses throughout system
Burners Accurately determine boundary conditions for furnace models, evaluate flame and burner flow characteristics
Overfire air ports Accurately determine boundary conditions for furnace models, determine flow characteristics and pressure losses through port Pulverized Examine combustion characteristics coal-fired boilers throughout the entire furnace;
evaluate fuel/air mixing, furnace performance, heat transfer, emissions and flow characteristics Recovery boilers Examine combustion characteristics
throughout the entire furnace; evaluate fuel/air mixing, furnace performance, heat transfer, emissions, carryover and flow characteristics within the furnace Waste-to-energy Examine combustion characteristics boilers of the entire furnace; evaluate
fuel/air mixing, furnace performance, heat transfer, emissions and flow characteristics Selective catalytic Determine inlet flow and
reduction systems temperature distributions; evaluate flow correction devices to meet specified velocity and temperature criteria
Wet scrubbers Determine flow and pressure drop conditions, evaluate scrubber emission removal performance
verized coal flame, the first step is to adequately de- fine the mathematical description.
The description is derived from first principles and physical laws and is primarily based on a set of con- servation relationships that result in a series of ordi- nary and partial differential equations (ODE and PDE). The PDEs describe such things as the conser- vation of mass, momentum, energy, and others. In addition, fundamental relationships are used to complete the description of the system. The complete description is made up of these PDEs and algebraic relationships.
Combustion modeling results in a particularly com- plex mathematical description of the overall process. Each physical process involved in a combustion sys- tem is described individually; however, they interact with other physical processes. This interaction creates a coupling between all the descriptions of the indi- vidual processes.
To demonstrate this coupling, consider a simple dif- fusion flame. Fluid dynamics describe the process of mixing two streams of reactants. The resulting reac- tion alters the constituents of the fluid, and heat re- lease from the reaction increases the local tempera- ture. The change in temperature and chemical compo- sition has a strong effect on local density. This change in density, in turn, has a strong effect on the fluid flow.
The system of processes, equations and interrela- tionships in a coal-fired boiler is far more complex, as shown in Fig. 1. Five fundamental processes must be addressed while providing for all key interactions: 1. Fluid transport: fluid motion, component mass
and energy transport in a turbulent mixing envi- ronment.
2. Particle transport: particle (in this case coal) or dis-
crete phase motion in a fluid.
3. Homogeneous chemical reactions: gaseous species combustion.
4. Heterogeneous chemical reactions: particle combustion. 5. Radiative heat transfer: radiative heat transfer in
a particle-laden participating media.
The second step to modeling the system is to use an appropriate technique to solve the set of equations that has been chosen to describe the physical system. It is not possible to analytically solve the partial differential equa- tions typically encountered in modeling combustion sys- tems. Thus, the differential equations are discretized to obtain a set of non-linear algebraic equations that can be solved with known numerical techniques. The last step in the process is to obtain the final solution.
Following is a more detailed description of each of these processes.
Number and Velocity of Particles Particle Velocities Gas Velocities Particle Size and Density Gas Velocities Number and Location of Particles Gas Properties Gas Velocities and Pressure
Complex Coupling Phenomena Between Subprocess Modules
Change in Gas Composition and Enthalpy Gas Composition and Temperature Gas Composition and Temperature Radiation Heat Transfer Particle Size, Temperature, and Composition Radiative Heat Transfer Module Fluid Transport Module Heterogeneous Chemical Reaction Module Homogeneous Chemical Reaction Module O2 NOX CO2 CHX Particle Transport Module