Energy balance applied

It should also be noted that an overall energy balance applied to the control volume shown in Fig. 7.6 gives ... 

An energy equation like equation (58) also would be obtained from a phenomenological energy balance applied to the gas and condensed material simultaneously. A first integral of equation (58) is... 

The steady-state energy balance applied to the slnrry and sidestreams, excluding any pump in the line, is... 

If w e consider the constant-volume reactor with incompressible fluid (a = 0,Cv = Cp), Equation 6.16 reduces to Equation 6.15 as it should because Equation 6.15 is valid for any reactor operation with an incompressible fluid. We also notice that, in the constant-pressure case, the same energy balance applies for any fluid mixture (ideal gas, incompressible fluid, etc.), and that this balance is the same as the balance for an incompressible fluid in a constant-volume reactor. Although the same final balances are obtained for these two cases, the physical situations they describe are completely different. 

Three energy balances apply for each heat exchanger. For the first heat exchanger, E-lOO, they are... 

Energy balance. The energy balance applied to fluid flow must include... 

From an energy balance applied to the preheater the heat Qp required in the preheater is... 

Once the local concentration overpotential is known, the activation overpotential, ria, is obtained by subtracting Tjc from total Tj. The local activation overpotential is the actual driving force of the electrochemical reaction. It is related to the local current density at any point of the reaction zone by an electrochemical rate equation such as the Butler-Volmer equation (Eq. (10a)). Therefore, the rate equation, the Nernst equation (Eq. (37)), and the potential balance in combination couple the electric field with the species diffusion field. In addition, the energy balance applies also at the electrode level. Although this introduces another complication, a model including a temperature profile in the electrode is very useful because heat generation occurs mainly by electrochemical reaction and is localised at the reaction zone, while the... 

The same algorithms that solve NLP problems can be applied to solving sets of nonlinear equation problems (NEPs). Material and energy balances applied to chemical and biomolecular engineering problems often lead to NEPs. The following example is typical of such problems. 

Heat Transfer in Rotary Kilns. Heat transfer in rotary kilns occurs by conduction, convection, and radiation. In a highly simplified model, the treatment of radiation can be explained by applying a one-dimensional furnace approximation (19). The gas is assumed to be in plug flow the absorptivity, a, and emissivity, S, of the gas are assumed equal (a = e ) and the presence of water in the soHds is taken into account. Energy balances are performed on both the gas and soHd streams. Parallel or countercurrent kilns can be specified. 

Reaiianging equation 50, applying the energy balance, and assuming air at standard conditions enters the tower yield (38) ... 

A similar energy balance would apply to a vessel being cooled. 

For each stage J, the following 2C -1- 3 component material-balance (M), phase-equilibrium (E), mole-fraction-summation (S), and energy-balance (H) equations apply, where C is the number of chemical species ... 

A model can be defined as a set of relationships between the variables of interest in the system being investigated. A set of relationships may be in the form of equations the variables depend on the use to which the model is applied. Therefore, mathematical equations based on mass and energy balances, transport phenomena, essential metabolic pathway, and physiology of the culture are employed to describe the reaction processes taking place in a bioreactor. These equations form a model that enables reactor outputs to be related to geometrical aspects and operating conditions of the system. 

Fig. 6.2. The energy localization problem in shock-loaded porous powder mixtures involves a balance between the rate at which energy is applied locally and the rate at which it is removed from the site.

Many cliciiiical reactions evolve or absorb heat. When applying energy balances (consenatioit law for energy) in tccluiical calculations the heat (enthalpy) of reaction is often indicated in mole units so that tliey can be directly applied to demonstrate its chemical change. To simplify the presentation that follows, examine the equation ... 

Sometimes the failure occurs by propagation of a crack that starts at the top and travels downward until the interface is completely debonded. In this case, the fracture mechanics analysis using the energy balance approach has been applied [92] in which P, relates to specimen dimensions, elastic constants of fiber and matrix, initial crack length, and interfacial work of fracture (W,). 

With these assumptions, we take as a reference point the discharge level of the water, and apply Equation 2-110. Thus = 0, Zj = 600 ft and the energy balance becomes... 

If at time t the liquid level is D m above the bottom of the tank, then designating point 1 as the liquid level and point 2 as the pipe outlet, and applying the energy balance equation (2.67) for turbulent flow, then ... 

Then applying the energy balance equation between D and the liquid level in each of the tanks gives ... 

The energy balance equation can be applied between any two sections in a continuous fluid, If the fluid is not moving, the kinetic energy and the frictional loss are both zero, and therefore ... 

Considering a small filament of liquid which is brought to rest at section 2, and applying the energy balance equation between the two sections, since g Az, Wv, and F are all zero ... 

The time derivative is zero at steady state, but it is included so that the method of false transients can be used. The computational procedure in Section 4.3.2 applies directly when the energy balance is given by Equation (5.28). The same basic procedure can be used for Equation (5.25). The enthalpy rather than the temperature is marched ahead as the dependent variable, and then Tout is calculated from Hout after each time step. 

In the above discussions the only normal force on the sliding molecule results from the attractive interactions. When externally applied load or pressure is to be considered, the energy balance has to be modihed as... 

For both heating and cooling, the energy balance remains the same except that differing values of the overall heat transfer coefficient now apply. Thus for the coolant Us is replaced by Uf. 

Chapter 1 deals with the basic concepts of modelling, and the formulation of mass and energy balance relationships. In combination with other forms of relationship, these are shown to lead to a systematic development of models. Though the concepts are simple, they can be applied to very complex problems. 

In order to be able to explain the observed results plasma modeling was applied. A one-dimensional fluid model was used, which solves the particle balances for both the charged and neutral species, using the drift-diffusion approximation for the particle fluxes, the Poisson equation for the electric field, and the energy balance for the electrons [191] (see also Section 1.4.1). 

If the same procedure is applied to a tube only containing distilled water, the energy balance is shown in the next two equations ... 

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