Total energy balance, closed system

This relation is recognized from introductory subjects on thermodynamics. Recall that in equilibrium thermodynamics a local formulation is generally not needed, since the intensive state variables are independent of the space coordinates. This fundamental formulation of the total energy balance is known as the first law of thermodynamics for a closed system, which expresses the fundamental physical principle that the total energy of the system, Etotab is conserved (a postulate). 

We begin with the application of the first law of thermodynamics first to a dosed system and then to an open system. A system is any bounded portion of the universe, moving or stationary, which is chosen for the application of the various thermodynamic equations. For a closed system, in which no mass crosses the system boundaries, the change in total energy of the system, dE, is equal to the heat flow to the system. 8Q. minus the work done by the system on the surroundings. W. For a closed sy.sreni. the energy balance is... 

The general criterion of chemical reaction equiUbria is the same as that for phase equiUbria, namely that the total Gibbs energy of a closed system be a minimum at constant, uniform T and P (eq. 212). If the T and P of a siagle-phase, chemically reactive system are constant, then the quantities capable of change are the mole numbers, n. The iadependentiy variable quantities are just the r reaction coordinates, and thus the equiUbrium state is characterized by the rnecessary derivative conditions (and subject to the material balance constraints of equation 235) where j = 1,11,.. ., r ... 

Suppose sys(f) is the total energy (internal + kinetic + potential) of a system, and ihm and /hout are the mass flow rates of the system input and output streams. (If the system is closed, these quantities each equal zero.) Proceeding as in the development of the transient mass balance equation, we apply the general energy balance equation (11.3-1) to the system in a small time interval from t to t + 1st, during which time the properties of the input and output streams remain approximately constant. The terms of the equation are as follows (see Section 7,4) ... 

A closed system consists of a fixed mass. The total energy E for most systems encountered in practice consists of the internal energy V. This is especially the case for stationary systems since they don t involve any changes in Iheir velocity or elevation during a process. The energy balance relation in that case reduces to... 

Finally, to get the total work, we do an overall energy balance (system = two tanks adiabatic, closed, constant volume). 

The initial temperature profile used in the simulations is taken from the experiments. As mentioned earlier, the initial bed temperature is not totally uniform, but is increasing slightly from the inlet towards the outlet. This is caused by the heat leak into the system. To account for this heat leak in more detail, the tube was first cooled down, then cooling was stopped and the temperature rise was measured as a function of time in the radial centre and close to the tube wall. It was found that the temperature difference between these two locations was minimal and the temperature rise could be well described by an additional radiative energy infiux. Therefore, the following contribution was added to the energy balance ... 

Chemical equilibrium in a closed system at constant temperature and pressure is achieved at the minimum of the total Gibbs energy, min(G) constrained by material-balance and electro-neutrality conditions. For aqueous electrolyte solutions, we require activity coefficients for all species in the mixture. Well-established models, e.g. Debye-Htickel, extended Debye-Hiickel, Pitzer, and the Harvie-Weare modification of Pitzer s activity coefficient model, are used to take into account ionic interactions in natural systems [15-20]. 

The first law of thermodynamics states that the total energy in the universe is a constant. Energy balances have been developed for closed systems and for open systems. For example, the integral equation of the first law for a closed system, written in extensive form, is ... 

Equation (1.11) is now examined closely. If the s (products) total a number / , one needs (// + 1) equations to solve for the // n s and A. The energy equation is available as one equation. Furthermore, one has a mass balance equation for each atom in the system. If there are a atoms, then (/t - a) additional equations are required to solve the problem. These (// a) equations come from the equilibrium equations, which are basically nonlinear. For the C—H—O—N system one must simultaneously solve live linear equations and (/t - 4) nonlinear equations in which one of the unknowns, T2, is not even present explicitly. Rather, it is present in terms of the enthalpies of the products. This set of equations is a difficult one to solve and can be done only with modem computational codes. 

To understand why and how chemical reactions happen it is necessary to consider also intermolecular interactions. It is only in the hypothetical case of an ideal gas that intermolecular interactions are totally absent. In all other systems they represent an important factor that affects molecular conformation, reactivity and stability. Whenever molecules co-exist in equilibrium it means that intermolecular forces are not sufficient to pull the molecules apart or together into larger aggregates. Equilibrium implies a balance of thermodynamic factors, and when these factors change, intermolecular interactions may overcome the integrity of a partially holistic molecule, and lead on to chemical reaction. Onset of the reaction is said to be controlled by an activation energy barrier. This barrier must clearly be closely allied to the quanmm potential of the molecule. 

The route to the solution of problems in chemical and phase equilibria is indirect it derives from a formalism developed over a century ago by the American pliysicist J. W. Gibbs. Let G be the total Gibbs energy of a closed, multiphase system of constant and uniform T and P. Equilibrium slates are those for which G is a minimum, subject to material-balance constraints appropriate to the problem ... 

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