F3. Mass and Energy Balances Formulas

Thermochemistry: The heat of a chemical reaction is the heat absorbed in the course of the reaction, or, in a more general sense it is equal to the change in enthalpy of the system for the reaction proceeding at constant pressure. The standard heat of reaction is defined as the change in enthalpy resulting from the procedure of the reaction under a pressure of 1 atm, starting and ending with all materials at a constant temperature of 25 °C. The value of the heat of reaction accompanying an equation is the heat of reaction resulting from the procedure of the reaction from the left to the right of the equation as written. Unless otherwise specified it is assumed that each reactant or product is in its normal state of aggregation at a temperature of 25 °C and a pressure of 1 atm. The aggregation-state of substance is indicated by a letter in parentheses, followed by its chemical formula. Where the initial and final temperatures of the system are the same, a subscript may be used to designate this temperature. When heat is evolved in a reaction, corresponding to a decrease in enthalpy, the reaction is termed exothermic; when heat is absorbed the reaction is said to be endothermic. The heat of formation of a chemical compound is a special case of the standard heat of a chemical reaction wherein the reactants are the necessary elements and the compounds in question is the only product formed. The molal heat of formation of a compound represents the heat of reaction, ΔH_f, when 1 mole of the compound is formed from the elements in a reaction beginning and ending at 25 °C and at a pressure of 1 atm, with the reacting elements originally in the states of aggregation which are stable at these conditions of temperature and pressure. The heat of combustion of a substance is the heat of reaction resulting from the oxidation of a substance with molecular oxygen. Since combustion proceeds with a reduction in the enthalpy of the system, the value of ΔH must be negative, and hence the heat of combustion is also negative. The usually accepted standard heat of combustion is that resulting from the combustion of a substance, in the state that is normal at 25 °C and atmospheric pressure, with the combustion beginning and ending at a temperature of 25 °C. The major final products are generally gaseous carbon dioxide and liquid water. The standard heat of reaction is equal to the algebraic sum of the standard heats of formation of the products less the algebraic sum of the standard heats of formation of the reactants. The standard heat of reaction is also equal to the standard-heat of combustion of the reactants minus the standard heat of combustion of the products.

If specific heat is given as

Where T is given in °K, and C_P is in cal/(gmol·°K). The values of a, b, and c for the feed and product can be calculated by using molar average.

Enthalpy of the feed at any temperature T_F is given as follows:

Where the integration constant, H_Fo can be calculated as

Note that enthalpy at base temperature T_B, is zero. Enthalpy of the products is given as:

Where the integration constant H_Po is given as follows:

An energy balance around a reacting system gives the following equation:

Where q is the heat added to the system, and ΔH₂₉₈ is standard heat of reaction. Plugging in the values of q, ΔH₂₉₈, and H_F, we can find the enthalpy of products.

Product temperature can be founding by solving the following cubic equation.

This function has three roots. We are looking for a positive real root of the equation. It is easier to use the Newton's method. The derivative of the function can be found to be

The temperature attained when a fuel is burned in air or oxygen without gain or loss of heat is termed the theoretical flame temperature. The maximum adiabatic flame temperature is attained when the fuel is burned with the theoretically required amount of pure oxygen.

Mole ratio of nitrogen to oxygen in air is 3.76.

Specific volume of a gas: It is very important to realize that measurement of the gas is reported at conditions, which are quite different from the standard conditions. Standard conditions are 29.92 in Hg and 32 °F. The measurement temperature is 60 °F and measurement pressure is 30" Hg. If the gas behaves ideally, then its specific volume can be calculated using ideal gas law and the results are presented for various selected conditions.

One pound-mole of a gas at 32 °F and 29.92 in Hg occupies 359.05 ft³.

One pound-mole of a gas at 60 °F and 30 in Hg occupies 378.48 ft³.

The vapor pressure of water at 60 °F = 0.52" Hg.

Mole fraction of dry gas = (30 - 0.52)/30 = 0.9827

If the gas is completely saturated at 60 °F and 30 in Hg, then mole fraction of dry gas is 0.9827:

The volume of 1 mole of dry gas is 378.48/0.9827 = 385.16 ft³.

Combustion: To perform combustion calculations it is necessary to apply the chemistry of the reaction. Good combustion requires that excess air be present. The percent of excess air may be as low as 5% in a good combustor. Excess air may also be used to control the temperature when burning a high heating value waste. The combustion reactions for some hydrocarbon gases are given as follows:

If one mole of a combustible material A is reacted with n_O2 moles of oxygen, then n_CO2 moles of carbon dioxide and n_H2O moles of liquid water will be produced. The following table provides heat of formation for various components. This table also lists moles of oxygen required for complete combustion and moles of combustion products.

Mean specific heat and combustion-air requirement: If the temperature of the combustion products is known, then a mean molal specific heat can be calculated as follows:

The mean molal specific heat data is available for a number of industrial gases over a wide temperature range.

Products of Combustion:

Hydrocarbon combustion produces CO₂, and water vapor and possibly some CO. The combustion of wastes containing sulfur produces SO₂ and possibly SO₃. Halogen containing wastes produce acid halogen gas of each in a combustion reaction HCl, HF, and HBr.

Plant Safety, Explosive Limits:

Waste gases usually consist of a hydrocarbon or a mixture of hydrocarbons in air. They must be in such high concentration that ignition is not possible until more air is added. Such mixtures are said to be above the upper explosive limit (UEL). Mixtures that have such low hydrocarbon concentrations that they cannot be ignited, are said to be below the lower explosive limit (LEL). Most hydrocarbon air mixtures are of this type.

Oxygen correction:

Oxygen analysis, when combined with stack gas volume flow, can be used as a means of determining total heat input to the system. The EPA has established a level of 100 ppm CO, rolling average corrected to 7% oxygen on a dry basis, as an acceptable level to ensure greater than 99.99% DRE.

Corrected concentration, P_c = P_m(14)/(21 - Y)

Destruction and Removal Efficiency (DRE):

Destruction and removal efficiency is defined as

DRE = (W_in - W_out)/W_in ´ 100

Where W_in = mass feed rate of a particular POHC

W_in = mass emission rate of a particular POHC