The Energy Balance

The suffixes 1 and 2 represent the inlet and outlet points respectively. Q is the heat transferred across the system boundary positive for heat entering the system, negative 

Equation 3.6 is a general equation for steady-state systems with flow. 

In chemical processes, the kinetic and potential energy terms are usually small compared with the heat and work terms, and can normally be neglected. 

It is convenient, and useful, to take the terms U and Pv together defining the term enthalpy, usual symbol //, as  

Enthalpy is a function of temperature and pressure. Values for the more common substances have been determined experimentally and are given in the various handbooks (see Chapter 8). 

We do not introduce a term for the rate of generation of heat by reaction since this will arise naturally as an enthalpy change due to changing composition. However, if the reaction is exothermic there may well be provision for cooling it (or for heating it in case of an endothermic reaction), and this term must be included as 

The exothermic case is the one we are most concerned with if heating is required we make Q negative and if the operation is adiabatic Q = 0. Then the energy balance gives 

Now the differentiation of the first term must take into account that it is the sum of the products of which the second factors are functions of T and of. .., C.S-. In fact. 

In this expression the last term disappears, since 2 = 0 by 

Moreover, dhJdT = Cpj is the partial molar heat capacity, so that 2 is the total heat capacity per unit volume of the 

A reasonably general energy balance for a flow reactor can be written in English as 

Enthalpy of input streams — enthalpy of output streams -I- heat generated by reaction — heat transferred out = accumulation of energy 

This is an integral balance written for the whole system. The various terms deserve discussion. The enthalpies are relative to some reference temperature, Tref. Standard tabulations of thermodynamic data (see Chapter 7) make it convenient to choose rre/ = 298K, but choices of 7 re/ = 0K or Tref = Q°C are also common. The enthalpy terms will normally be replaced by temperature using 

For many purposes, the heat capacity will be approximately constant over the range of temperatures in the system. Then 

By thermodynamic convention, l Hp 0 for exothermic reactions, so that a negative sign is attached to the heat-generation term. When there are multiple reactions, the heat-generation term refers to the net effect of all reactions. Thus, the term is an implicit summation over all M reactions that 

According to the first law of thermodynamics, the internal energy U of a closed system changes due to the addition of heat Qu and work IV12 into the system 

We will apply these equations to a fluid element of given mass. As this will be considered to be a closed system we can immediately exclude mass transport over the system boundary and with that diffusion. 

With the help of the transport theorem (3.17), by putting in z = u, we obtain the following temporal change in the internal energy of a flowing fluid 

According to this, the change in internal energy over time in a flowing fluid is equal to internal energy stored inside the fluid volume V(t) at time t, and the internal energy flowing out over the surface A(t) of the fluid volume. 

Heat is transferred, by definition, between a system and its surroundings, and therefore the heat fed into the system via its surface is 

We see that we need another relationship relating X and T or T and V to solve this equation. The energy balance hUI provide us with this relationship. 

So we add another step to our algorithm, this step is the energy balance. Energy Balance  

In this step, wc wfll find the appropriate energy balance to relate temperature and conversion or reaction rate. For e tample, if the reaction is adiabatic, we will show the temperature-conversion relationship can be written in a form such a.s 

We now have all the equations we need to solve for the conversion and temperature profiles. 

Analysis The purpose of this example was to demonstrate that for non-iso-thermal chemical reactions we need another step in our CRE algorithm, the energy balance. The energy balance allows us to solve for the reaction temperature, which is necessary in evaluating the specific reaction rate constant k[T). 

We begin our balance by excluding from consideration magnetic, electro- tatic, surface, and nuclear energies. Thus, the only kinds of energy which 1 lb )f matter can contain are internal, kinetic, and potential w + ke + pe. Now, he general balance says that 

Accumulation is the differential of the energy contained within the system boundary. The only such energy is that associated with the matter within the boundary. If the matter is uniform (if all has the same u, ke, and pe), then accumulation is d[m(u + pe + ke)] where m is the mass in the system. We use this simplified form for now and consider nonuniform systems in Sec. 4.13. 

Energy can enter in three ways. One way is by matter coming in the inlet pipe. For every infinitesimal amount of matter which flows in, the amount of energy which flows in with it is (M + pe + ke)j Obviously, for matter 

This is a preliminary equation its final form will appear shortly. We now 

These stem from the early attempts of thermodynamicists to study the steam engine. For a steady-running or cyclical steam engine, Eq. 4.4a reduces to 

If the kinetic and potential energy terms are neglected, equation 3.6 simplifies to 

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Cracking reactions are endothermic the energy balance is obtained by the production of coke that deposits on the catalyst and that is burned in the regenerator. 

Anon. (1983), Assessment of the energy balances and economic consequences of the reduction and elimination of lead in gasoline . Working Group ERGA (Evolutions of Regulations, Global Approach). CONCAWE, La Haye. 

The selection of a process can be complex, requiring carehil evaluation of the many variables for each appHcation. The hemihydrate process is energy efficient, but this may not be an overriding consideration when energy is readily available from an on-site sulfuric acid plant. The energy balance in the total on-site complex may be the determining factor. 

The energy balance should analyse the energy flows by type and amount, ie, present summaries of electricity, fuel gas, steam level, heat rejected to cooling water, etc. It should include reaUstic loss values for turbine inefficiencies and heat losses through insulation. 

Generally, q is small because the outside area is not large in comparison to the amount of heat being transferred, and the energy balance can be simplified. In these conditions it is also convenient to write balances over a differential section of the column. These balances yield the following ... 

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

In the manufacture of explosives, sodium nitrate is used mainly in blasting agents. In slurries and emulsions, sodium nitrate improves stabiUty and sensitivity. It also improves the energy balance because sodium nitrate replaces water, so that more fuel can be added to the formulation. Sodium nitrate reduces crystal size of slurries, which in turn increases detonating speed. In dynamites sodium nitrate is used as an energy modifier. Typical content of sodium nitrate is 20—50 wt % in dynamites, 5—30 wt % in slurries, and 5—15 wt % in emulsions. Sodium nitrate is used also in permissible dynamites, a special type of dynamite for coal (qv) mining. 

A further enhancement to the HRS process whereby the exhaust from a gas fired turbine is used to superheat steam from the HRS process is also possible (129). The superheated steam is then fed through a turbogenerator to produce additional electricity. This increases the efficiency of heat recovery of the turbine exhaust gas. With this arrangement, electric power generation of over 13.6 kW for 1 t/d (15 kW/STPD) is possible. Good general discussions on the sources of heat and the energy balance within a sulfuric acid plant are available (130,131). 

Thermodynamics. The first law of thermodynamics, which states that energy can neither be created nor destroyed, dictates that the total energy entering an industrial plant equals the total of all of the energy that exits. Eeedstock, fuel, and electricity count equally, and a plant should always be able to close its energy balance to within 10%. If the energy balance does not close, there probably is a big opportunity for saving. 

The energy balance for a steady-state steady-flow process resulting from the first law of thermodynamics is... 

The continuity equation gives V2 = V AJa, and Vj = Q/A. The pressure drop measured by the manometer is pi —p2= (p — p)gA . Substituting these relations into the energy balance and rearranging, the desired expression for the flow rate is found. 

The viscous or frictional loss term in the mechanical energy balance for most cases is obtained experimentally. For many common fittings found in piping systems, such as expansions, contrac tions, elbows and valves, data are available to estimate the losses. Substitution into the energy balance then allows calculation of pressure drop. A common error is to assume that pressure drop and frictional losses are equivalent. Equation (6-16) shows that in addition to fric tional losses, other factors such as shaft work and velocity or elevation change influence pressure drop. 

Operating Lines The McCabe-Thiele method is based upon representation of the material-balance equations as operating lines on the y-x diagram. The lines are made straight (and the need for the energy balance obviated) by the assumption of constant molar overflow. The liqmd-phase flow rate is assumed to be constant from tray to tray in each sec tiou of the column between addition (feed) and withdrawal (produc t) points. If the liquid rate is constant, the vapor rate must also be constant. 

Many different forms of the energy balance have been used in fixed-bed adsorption studies. The form chosen for a particular study depends on the process considered (e.g., temperature swing adsorption or pressure swing adsorption) and on the degree of approximation that is appropriate. 

The isoteric heat of adsorption qf is composition-dependent, and the sum of integrals Eq. (16-60) is difficult to evaluate for multicomponent adsorption if the isosteric heats indeed depend on loading. Because each isosteric heat depends on the loadings of all components, the sum must be evaluated for a path beginning with clean adsorbent and ending with the proper loadings of all components, if the isosteric heat of adsorption is constant, as is commonly assumed, then the energy balance (Eq. 16-55) becomes... 

Estimation of operating data (usually consisting of a mass and energy in which the energy balance decides whether the absorption balance can be considered isothermal or adiabatic)... 

The energy balance across a pipe from the stagnation point 0 to a point 2 downstream is ... 

Choked Flow by Two-Phase Energy Balance From the energy balance, Eq. (26-88), taking= 0 (stagnation) and ( = 0 ... 

Other methods for implementing the energy balance have been proposed and lead to moderate differences in predicted fragment size (Grady, 1982 Glenn and Chudnovsky, 1986). We contrast this earlier kinetic energy based theory with a more recent development in the following section. 

In Fig. 8.18, we illustrate this just sufficient distribution in comparison to a hypothetical flaw distribution for an actual material. In this example, we envision a solid of finite extent, which will have a single critical flaw that activates at a minimum stress tensile stress, the population of flaws which activate should increase rapidly, perhaps as illustrated in Fig. 8.18. In contrast, a flaw distribution just sufficient to satisfy the energy balance criterion increases smoothly as JV [Pg.294]

Bond dissociation energies such as those in Table 12.6 are also useful for estimation of the energy balance in individual steps in a free-radical reaction sequence. This is an... 

A similar statement, or equation, can be used to express the energy balance ... 

These relationships enable the combination of the mass and the energy balances. For constant volume and density, with m = pV, ... 

The energy balance from Equation 6-39 yields a heating system... 

For a known reactor and kinetics, and at a given feed temperature Tq, the intersection of the energy balance line with the S-shaped mass... 

The numerical solution of the energy balance and momentum balance equations can be combined with flow equations to describe heat transfer and chemical reactions in flow situations. The simulation results can be in various forms numerical, graphical, or pictorial. CFD codes are structured around the numerical algorithms and, to provide easy assess to their solving power, CFD commercial packages incorporate user interfaces to input parameters and observe the results. CFD... 

The energy equation of a continuing system can be presented by means of the first law of thermodynamics and the energy balance of a flow system as... 

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