Hans Agarwal

Chemical equation balancing: An integer programming approach

Presented here is an integer linear program (ILP) formulation for automatic balancing of a chemical equation. Also described is a integer nonlinear programming (INP) algorithm for balancing. This special algorithm is polynomial time O(n^3), unlike the ILP approach, and uses the widely available conventional floating-point arithmetic, obviating the need for both rational arithmetic and multiple modulus residue arithmetic. The rational arithmetic is unsuitable due to intermediate number growth, while the residue arithmetic suffers from the lack of a priori knowledge of the set of prime bases that avoids a possible failure due to division by zero. Further, unlike the floating point arithmetic, both arithmetics are not built-in/standard and hence additional programming effort is needed. The INP algorithm has been tested on several typical chemical equations and found to be very successful for most problems in our extensive balancing experiments. This algorithm also has the capability to determine the feasibility of a new chemical reaction and, if it is feasible, then it will balance the equation and also provide the information if two or more linearly independent balancings exist through the rank information. Any general method to solve the ILP is fail-proof, but it is not polynomial time. Since we have not encountered truly large chemical equations having, say, 1000 products and reactants in a real-world situation, a non-polynomial ILP solver is also useful. A justification for the objective functions for ILP and INP algorithms, each of which produces a unique solution, is provided.

COMPARISON AND VALIDATION OF OHM AND SCEM MEASUREMENTS FOR A FULL-SCALE COAL-FIRED POWER PLANT

Mercury emission measurements were performed at a 250 MW coal-fired power plant using the Ontario Hydro method (OHM) and semi-continuous emission monitors (SCEM). Flue gas sampling was performed at the inlet of the air preheater and at the outlet of the electrostatic precipitator. The results indicated that there is some agreement between the OHM and SCEM measurements on the total mercury species. However, the SCEM results were not always in good agreement with the OHM measurements on the elemental mercury species. These discrepancies in elemental mercury concentrations are probably the result of the differences in the location of the SCEM and OHM probes, the temperature difference between the SCEM sampling probe and the flue gas, and the nonuniformities in mercury concentration over the flue gas duct cross section. The other factor that contributed to the deviation between the SCEM and OHM measurement results is the sampling method: the SCEM measurements were performed at a single point while the OHM probe was traversed over multiple points over the duct cross section and the results were averaged. The effect of the SCEM sampling probe temperature was investigated by designing a sampling probe that could be heated to the sampled flue gas temperatures. This resulted in improvements in the accuracy of the elemental mercury measurements by the SCEM system.

Birth, growth and computation of pi to ten trillion digits

The universal real constant pi, the ratio of the circumference of any circle and its diameter, has no exact numerical representation in a finite number of digits in any number/radix system. It has conjured up tremendous interest in mathematicians and non-mathematicians alike, who spent countless hours over millennia to explore its beauty and varied applications in science and engineering. The article attempts to record the pi exploration over centuries including its successive computation to ever increasing number of digits and its remarkable usages, the list of which is not yet closed.

Interpolation for nonlinear BVP in circular membrane with known upper and lower solutions

A successive nonextrapolatory linear interpolation is described to solve a singular two-point boundary value problem arising in circular membrane theory. The problem is associated with a second-order nonlinear ordinary differential equation for which the upper and lower bounds of the solution is analytically established/known. The importance and the scope of these bounds in solving the problem is stressed. Also depicted graphically are the lower and upper solutions as well as the true and iterated solutions. In addition, discussed are the reasons why linear interpolation, and not nonlinear interpolation or bisection which are possible procedures, has been employed.

2n in scientific computation and beyond

Certain decimal numbers have special characteristics unlike those of most others. These sometimes have enormous physical significance, or very interesting mathematical/scientific properties, or both at the same time. The present article attempts to explore the significance of the decimal number 2^n, the logarithm to base 2, and the binary equivalence (representation) corresponding to 2^n both in the physical/natural world and in the pure mathematical environment, specifically in the area of computer/computational science. Also, among the number systems in different bases, the status of the base 2^n, specifically for n=1, in the realm of computational/computer science is also stressed.

Gas flow problem: solving non-linear BVP with given lower-upper solutions interactively

Modelling the unsteady isothermal flow of a gas through a semi-infinite porous medium results in a sensitive non-linear two-point boundary value problem (BVP) over an infinite interval for which the lower and upper solutions have been established analytically. We present an interactive solution procedure based on the lower and upper solutions (LUSOL) for this BVP whose conversion to an initial value problem (IVP) without making use of the LUSOL is usually not numerically possible. In other words, the gas flow problem considered here is an excellent example of a case where knowledge of easily computable LUSOL is extremely important if one does not want to land in a situation in which all non-LUSOL-based methods fail.