Dulong equation

Experiments on more than 100 bituminous coals revealed differences between calculations and bomb calorimetric determinations that were seldom larger than 2 %. A modified Dulong equation with the elemental composition of wood was used by Abe in his investigations of 40 fast-growing tree species [91], The Dulong equation together with some other models to calculate combustion heats from chemical compositions of wood is discussed in Chapter 7 "Bomb Calorimetry in Biology" in this Handbook. 

Calculate the net heating value (NHV) of methane, chloroform, benzene(g), chlorobenzene, and hydrogen sulfide. This assumes that the water product is in the vapor state. Compare these values with those calculated using Dulong s equation. Calculate the relative percent difference between the true NHVs as determined by thermodynamic calculations and the estimated values calculated using Dulong s equation. Dulong s equation can be written as follows ... 

The following results from Dulong s equation can be summarized as ... 

Based on these calculations, the difference between the thermodynamically based NHV values and those estimated using Dulong s equation can be summarized as ... 

In this table, a basis of 1000 Ib/h total waste flow of the four compounds was used. The NHV can be calculated for this four-compound mixture using Dulong s equation ... 

The relationship between the high and low heating values of a fuel having the empirical chemical formula is given by Dulong s equations. Note that the negative sign indi-... 

Just like gases, solids can lose their ability to absorb energy at low temperatures. According to equipartition Equation (11.54), each vibration contributes kT to the energy. If there are N atoms in a solid, and each atom has three vibrational modes (in the x-, y-, and z-directions), the heat capacity wall be C = 3Nk, independently of temperature. This is called the Law of Dulong and Petit, named after the experimentalists who first observ ed this behavior around 1819. But more recent experimental data, such as that shown in Figures 11.14 and 11.15, indicate that this law does not hold at low temperatures. As the temperature approaches zero, Cv 0. The Einstein model, developed in 1907, shows why. This work was among the first evidence for the quantum theory of matter. 

Because the solution of Eq. (63) is not analytically feasible when k is not an integer, the case discussed here is k = 514, the Dulong and Petit s heat transfer law. So one can take the reasonable approximations only for the exponents in Equation (63),... 

This relation, commonly known as Einstein s (1905) law of heat capacities, correctly corresponds to the limiting cases Cy = Oat T = 0 and to Cy = 3R at infinite temperature—the law of Dulong and Petit. It immediately rationalizes the deviations of the observed Cy values from this law at lower temperatures that had mystified scientists before the advent of quantum mechanics. However, the above equation generally deviates to some extent from experiment in particular, at low T, the above relation shows that Cy, while experiments conform to aCy T variation. 

Dulong block To calculate performance- and capacity-related parameters, the lower heating value (LHV) is assessed according to Dulong s equation using the ultimate analysis, including the fuel moisture (as-received basis). 

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