Dulong formula

Representative data on the ultimate analysis of typical industrial and municipal-waste components are presented in Table 22-58. If energy values are not available, approximate values can be determined by using Eq. (22-29), known as the modified Dulong formula, and the data in Table 22-58. 

You can estimate the heating value of a coal within about 3% from the Dulong formula ... 

The values of C, H, S, and O can be taken from the fuel or flue-gas analysis. If the heating value of a fuel is known and the C and S are known, the approximate value of the net H can be determined from the Dulong formula. A general relation between the gross heating and net heating values is... 

Ultimate analysis-an analysis to determine the amounts of basic feed constituents. These constituents are moisture, oxygen, carbon, hydro- gen, sulfur, nitrogen, and ash. In addition, it is typical to determine chloride and other elements that may contribute to air emissions or ash- disposal problems. Once the ultimate analysis has been completed, Dulong s formula can be used to estimate the heating value of the sludge, Dulong s formula is ... 

At high F, when the spacing of vibrational energy levels is low with respect to thermal energy, crystalline solids begin to show the classical behavior predicted by kinetic theory, and the heat capacity of the substance at constant volume (Cy) approaches the theoretical limit imposed by free motion of all atoms along three directions, in a compound with n moles of atoms per formula unit limit of Dulong and Petit) ... 

For accurate heat values of solid and liquid fuels calorimeter determinations are required. However, approximate heat values may be determined for most cools if the ultimate chemical analysis is known. Dulong s formula gives reasonably accurate results (within 2 to 3%) for most coals and is often used as a routine check of values determined by calorimeter ... 

It was to explain these deviations from the law of Dulong and Petit that Einstein developed his theory of specific heats. He treated the vibrations of the separate atoms by quantum theory, just as we did in Sec. 5, Chap. IX, and derived the formula... 

A selection of values of the Debye function are tabulated in table 12.3. As T increases D Tj0) tends to unity, so that at temperatures above the characteristic temperature we have a theoretical justification of the empirical rule of Dulong and Petit, namely that 3R for atomic solids. On the other hand at low temperatures D Tj ) tends to zero, and for T/ <0T we have the simple approximate formula... 

Dulong s Formula and the Choice of Biomass Derived Fuel Form... 

Feedstock Characteristics. Feedstock characteristics are presented in Table I. Results of the metal analysis will be considered in Phase II of the project. Calorific values were approximated using Dulong s formula and not actually measured. The oat straw had a lower ash content than the stover and a correspondingly higher calorific value. Calorific values, calculated on a dry basis and including the latent heat of water vapour in the products of combustion (high heat value), were 17.9 MJ/kg for the straw and 16.0 MJ/kg for the stover. 

High heat value approximated using Dulong s formula... 

The value of coal expressed in calories evolved in its combustion can be ascertained either by calculation or by actual determination in a calorimeter. The chief constituents of coal which in combustion evolve heat are carbon and hydrogen. Those which absorb it during combustion are moisture and the inorganic matter which remains in the form of ash. Dulong s formula for calculating the number of calories (V) evolved by the combustion of 1 kg. of coal is—... 

It must be remembered that the foregoing vapour-pressure formula depends on the classical kinetic theory, and that none of the laws of the quantum theory have been used in its derivation. It holds therefore for a solid which strictly obeys the law of Dulong and Petit (C = 3R) and for which there is no difference between C, and C9 (cf. p. 65), i.e. which shows no thermal expansion. 

If we knew the numerical value of the constant a in formula (7), this expression could be employed to calculate the value of dO/dt for any given value of 0. To evaluate a, substitute the observed values of V and 6 in (7) and take the mean of the different results so obtained. Thus, a = 0 031. The third column shows the velocities of cooling calculated on the assumption that Newton s law is true. The agreement between the experimental and theoretical results is very poor. Hence it is necessary to seek a second approximation to the true law. With this object, Dulong and Petit have proposed... 

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