| OCR Text |
Show three additional input sets: (1) refractory wall temperatures, (2) water wall steam heat losses, or (3) bulk gas temperature/refractory wall temperature differences. In essence, the choice determines the boundary condition for each zone that governs the radiative transfer solution. The first two input options, though giving greatest accuracy, are the hardest to specify in practice. The third option is far simpler to use; at a minimum, one need only specify a single average difference between the expected bulk temperature and refractory wall temperature, averaged over the entire furnace length. The program fixes the zone length at one-quarter the hydraulic diameter of the furnace. The following parameters are calculated for each zone: mass fractions of CO2, H2O, O2, and N2; surface areas of refractory and waterwall; volume fraction occupied by luminous flame; and chemical heat release within the zone. The solutions to the zone energy balances (Equation 1) are then computed. The algorithm is a modification of the ZEROIN FORTRAN code developed in 1973 (Ref. 9). It solves a nonlinear equation by combining three different standard techniques: the bisection method, the secant method, and inverse quadratic interpolation. The approach used here chooses the next iterate from two candidates - one given by bisection and one by the secant method (interpolation). Inverse quadratic interpolation is used when the three previous evaluations are distinct and linear interpolation is used when they are not. The interpolation answer is chosen only if it is within the current interval and is "reasonably" far from the endpoints; otherwise the bisection guess is carried over. In this way, the interval length is guaranteed to decrease at each step. The number of function evaluations (energy balances) per zone is no greater than log2 [(TB -T^)/(l/2 r + 2 EM|Tg|)3, where E is the convergence tolerance and e^ is the relative machine precision. The value of e was set to 1.0 x 10*4; i.e., the net energy flux into a zone is zero to four decimal places. For the Apple 11+ computer, eM equals 1.0 x 10~H. With the initial guesses Tg and T^ set to 4,000° and 1,000°R, the actual number of function evaluations required is usually between six and nine. The real time requirements is only on the order of several seconds, even using a slow BASIC interpreter. Once the calculation is completed for each zone, the bulk gas temperatures and refractory wall temperatures are passed to a disk file which is used as input to the boundary layer program (module 2). 5.5.9 |
