Balance equations elemental

Derivation of the working equations of upwinded schemes for heat transport in a polymeric flow is similar to the previously described weighted residual Petrov-Galerkm finite element method. In this section a basic outline of this derivation is given using a steady-state heat balance equation as an example. 

Formulate the constraining material-balance equations, based on conservation of the total number of atoms of each element in a system comprised of w elements. Let subscript k identify a particular atom, and define Ai as the total number of atomic masses of the /cth element in the feed. Further, let a be the number of atoms of the /cth element present in each molecule of chemical species i. The material balance for element k is then... 

If species i is an element, AG/ is zero. There are N equilibrium equations (Eqs. [4-355]), one for each chemical species, and there are w material-balance equations (Eqs. [4-353]), one for each element—a total of N + to equations. The unknowns in these equations are the (note that y, = of which there are N, and the Xi, of which... 

One of the key elements of the assessment phase of a pollution prevention program involves mass balance equations. These calculations are often referred to as material balances the calculations are performed via the consei vation law for mass. The details of this ofien-used law are described below. 

The air temperature of a room at any given time is given hy a heat-balance equation which includes the heat flux exchanged by convection at each wall element (A q ), the heat flow exchanged by ventilation (4>j,), the convective part of heat flow due to internal heat gains the convective part of heat flow due to the HVAC system (< >[,(.), and the variation of energy in the room air(r M ... 

The internal surface temperature of each wall element / is given hy a heat-balance equation ... 

The hunnidity ratio oJ a rotjm at any given time is given by a Jatent heat balance equation including the water vapor flows due to infiltration to ventilation to moisture transport through envelope elements... 

Nicmanis, N. and Hounslow, M.I., 1998. Finite-element methods for steady-state population balance equations. American Institution of Chemical Engineers Journal, 44(10), 2258-2272. 

One molecule (or mole) of propane reacts with five molecules (or moles) of oxygen to produce three molecules (or moles) or carbon dioxide and four molecules (or moles) of water. These numbers are called stoichiometric coefficients (v.) of the reaction and are shown below each reactant and product in the equation. In a stoichiometrically balanced equation, the total number of atoms of each constituent element in the reactants must be the same as that in the products. Thus, there are three atoms of C, eight atoms of H, and ten atoms of O on either side of the equation. This indicates that the compositions expressed in gram-atoms of elements remain unaltered during a chemical reaction. This is a consequence of the principle of conservation of mass applied to an isolated reactive system. It is also true that the combined mass of reactants is always equal to the combined mass of products in a chemical reaction, but the same is not generally valid for the total number of moles. To achieve equality on a molar basis, the sum of the stoichiometric coefficients for the reactants must equal the sum of v. for the products. Definitions of certain terms bearing relevance to reactive systems will follow next. 

Chemical reactions are represented by chemical equations, which identify reactants and products. Formulas of reactants appear on the left side of the equation those of products are written on the right In a balanced chemical equation, there are the same number of atoms of a given element on both sides. The same situation holds for a chemical reaction that you carry out in the laboratory atoms are conserved. For that reason, any calculation involving a reaction must be based on the balanced equation for that reaction. 

Cupric sulfide, copper(II) sulfide, reacts with hot nitric acid to produce nitric oxide gas, NO, and elemental sulfur. Only the oxidation numbers of S and N change. Write the balanced equation for the reaction. 

The coefficients a, b and c, which appear throughout these balance equations describe the extent to which these reactions occur relative to the growth reaction (ie 1 + o) and are written taken into account elemental balances for each reaction. 

Then we balance the equation. A good strategy is to balance one element at a time, starting with one that appears in the fewest formulas, such as carbon and hydrogen. Then specify the states. Because water is produced as a vapor, we write... 

H.9 In one stage in the commercial production of iron metal in a blast furnace, the iron(III) oxide, I c20 , reacts with carbon monoxide to form solid Fe 04 and carbon dioxide gas. In a second stage, the Fe304 reacts further with carbon monoxide to produce solid elemental iron and carbon dioxide. Write the balanced equation for each stage in the process. 

K.18 The industrial production of sodium metal and chlorine gas makes use of the Downs process, in which molten sodium chloride is electrolyzed (Chapter 12). Write a balanced equation for the production of the two elements from molten sodium chloride. Which element is produced by oxidation and which by reduction ... 

STRATEGY We expect a strongly negative value because all combustions are exothermic and this oxidation is like an incomplete combustion. First, add up the individual standard enthalpies of formation of the products, multiplying each value by the appropriate number of moles from the balanced equation. Remember that the standard enthalpy of formation of an element in its most stable form is zero. Then, calculate the total standard enthalpy of formation of the reactants in the same way and use Eq. 20 to calculate the standard reaction enthalpy. 

Describe and write balanced equations for the principal reactions used to produce hydrogen and the Period 1 through 3 elements in Groups 1, 2, 13/III, and 14/IV. 

The pragmatic consideration is that if a student were to undertake this reaction, then it would be important to react corresponding amounts of the two reactants. Amount here implies the number of moles, and the unbalanced version of the equation would imply that equal volumes of reactant solutions (if the same concentration) were needed, when actually twice as much alkali solution would be needed as acid solution because the acid is dibasic. The principled point is that the equation represents a chemical process, which is subject to the constraints of conservation rules matter (as energy) is conserved. In a chemical change, the elements present (whether as elements or in compounds), must be conserved. A balanced equation has the same elements in the quantities represented on both sides ... 

Students at senior high school/college level are expected to be able to undertake this type of manipulation of symbols to produce balanced equations from half equations (whilst also adding the electrode potentials to check if the reaction is feasible). The resultant equation balances in terms of both the quantity of each element represented (2Mn, 80, 16H, lOBr) and in terms of overall charge (—2 - -16 — 10 = - -4). 

Balanced equation An equation using formulae where the number of atoms of each element involved in the reaction is the same on each side of the arrow. 

The basic scheme for the numerical solution is the same as that used for the 1 -D model, except that in this case the solid temperature field used to solve the DAE system for each monolith channel must be calculated from the three-dimensional solid-phase energy balance equation. The three-dimensional energy balance equation can be solved by a nonlinear finite element solver (such as ABAQUS) for the solid-phase temperature field while a nonlinear finite difference solver for the DAE system calculates the gas-phase temperature and... 

The description in the problem tells us what happens to the starting material NH4 NO3 breaks apart Into molecules of N2, O2, and H2 O. An unbalanced form of the equation can be written from this description. Then we must balance each element In turn by Inspection. 

C04-0117. The reaction of element U with element V is represented in the diagram shown below. Write the balanced equation for this reaction. 

Concentrated aqueous sulfuric acid, H2 SO4, is a strong oxidizing agent that can react with elemental carbon H2 SO4 + C CO2 + SO2 (unbalanced) What is the balanced equation for this process ... 

For the computation of compressible flow, the pressure-velocity coupling schemes previously described can be extended to pressure-velocity-density coupling schemes. Again, a solution of the linearized, compressible momentum equation obtained with the pressure and density values taken from a previous solver iteration in general does not satisfy the mass balance equation. In order to balance the mass fluxes into each volume element, a pressure, density and velocity correction on top of the old values is computed. Typically, the detailed algorithms for performing this task rely on the same approximations such as the SIMPLE or SIMPLEC schemes outlined in the previous paragraph. 

The system mass balance equations are often the most important elements of any modelling exercise, but are themselves rarely sufficient to completely formulate the model. Thus other relationships are needed to complete the model in terms of other important aspects of behaviour in order to satisfy the mathematical rigour of the modelling, such that the number of unknown variables must be equal to the number of defining equations. 

Thus considering a small differential element of liquid volume, dV, and depth, dZ, the balance equation becomes... 

A component balance equation can be derived for the element AV, based on the generalised component balance expression, where for any reactant, A... 

Consider a differential element of column volume, AV, height AZ and cross-sectional area, Ac, such that AV=Ac AZ. Component mass balance equations can be written for each of the liquid phases, where... 

The actual volume of each phase in element AV is that of the total volume of the element, multiplied by the respective fractional phase holdup. Hence considering the direction of solute transfer to occur from the aqueous or feed phase into the organic or solvent phase, the mass balance equations become ... 

Heat balance equations on the element of heat exchanger length AZ, according to enthalpy balance relationship. 

A balance equation can be written for each separately identifiable species present, elements, compounds or radicals and for the total material. 

Algebraic symbols are assigned to all the unknown flows and compositions. Balance equations are then written around each sub-system for the independent components (chemical species or elements). 

A more complete energy balance will be used that includes transport by conduction, convection, and radiation. The new energy balance equation over a small volume element takes the form... 

A chemical equation describes a chemical reaction in many ways as an empirical formula describes a chemical compound. The equation describes not only which substances react, but the relative number of moles of each undergoing reaction and the relative number of moles of each product formed. Note especially that it is the mole ratios in which the substances react, not how much is present, that the equation describes. In order to show the quantitative relationships, the equation must be balanced. That is, it must have the same number of atoms of each element used up and produced (except for special equations that describe nuclear reactions). The law of conservation of mass is thus obeyed, and also the "law of conservation of atoms. Coefficients are used before the formulas for elements and compounds to tell how many formula units of that substance are involved in the reaction. A coefficient does not imply any chemical bonding between units of the substance it is placed before. The number of atoms involved in each formula unit is multiplied by the coefficient to get the total number of atoms of each element involved. Later, when equations with individual ions are written (Chap. 9), the net charge on each side of the equation, as well as the numbers of atoms of each element, must be the same to have a balanced equation. The absence of a coefficient in a balanced equation implies a coefficient of 1. 

The equation states that elementary sodium reacts with elementary chlorine to produce sodium chloride, table salt. (The fact that chlorine is one of the seven elements that occur in diatomic molecules when not combined with other elements is indicated.) The numbers before the Na and NaCI are coefficients, stating how many formula units of these substances are involved. If there is no coefficient in a balanced equation, a coefficient of 1 is implied, and so the absence of a coefficient before the Cl2 implies one Cl2 molecule. The equation thus states that when the two reagents react, they do so in a ratio of two atoms of sodium to one molecule of chlorine, to form two formula units of sodium chloride. In addition, it states that when the two reagents react, they do so in a ratio of 2 mol of sodium to 1 mol of chlorine molecules, to form 2 mol of sodium chloride. The ratios of moles of each reactant and product to every other reactant or product are implied ... 

EXAMPLE 7.8. Write a complete, balanced equation for the reaction of each of the following pairs of elements ... 

In Sec. 13.2 we will learn to determine oxidation numbers from the formulas of compounds and ions. We will learn how to assign oxidation numbers from electron dot diagrams and more quickly from a short set of rules. We use these oxidation numbers for naming the compounds or ions (Chap. 6 and Sec. 13.4) and to balance equations for oxidation-reduction reactions (Sec. 13.5). In Sec. 13.3 we will learn to predict oxidation numbers for the elements from their positions in the periodic table in order to be able to predict formulas for their compounds and ions. 

We will apply the steady state momentum balance to a fluid in plug flow in a tube, as illustrated in Fig. 5-6. (The stream tube may be bounded by either solid or imaginary boundaries the only condition is that no fluid crosses the boundaries other that through the inlet and outlet planes.) The shape of the cross section does not have to be circular it can be any shape. The fluid element in the slice of thickness dx is our system, and the momentum balance equation on this system is... 

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