Terms in the Energy Balance

The design equations for a chemical reactor contain several parameters that are functions of temperature. Equation (7.17) applies to a nonisothermal batch reactor and is exemplary of the physical property variations that can be important even for ideal reactors. Note that the word ideal has three uses in this chapter. In connection with reactors, ideal refers to the quality of mixing in the vessel. Ideal batch reactors and CSTRs have perfect internal mixing. Ideal PFRs are perfectly mixed in the radial direction and have no mixing in the axial direction. These ideal reactors may be nonisothermal and may have physical properties that vary with temperature, pressure, and composition. 

Enthalpy. Enthalpy is calculated relative to a standard state that is normally chosen as Tq = 298.15 K = 25°C and Pq=1 bar pressure. The change in enthalpy with pressure can usually be ignored. For extreme changes in pressure, use 

The change in enthalpy with respect to temperature is not neghgible. It can be calculated for a pure component using the specific heat correlations like those in Table 7.1  

This table provides data for calculating molar heat capacities at low pressures according to the empirical formula 

Source Data selected from Smith, J. M., Van Ness, H. C., and Abbott, M. M., Introduction to Chemical Engineering Thermodynamics, 6th ed., McGraw-Hill, New York, 2001. 

The column marked Standard shows the calculated value of Cp/Rg at 298.15 K. 

Ideal gases obey the ideal gas law, Ppmoiar = RgT, and have internal energies that are a function of temperature alone. Ideal solutions have no enthalpy change upon 

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This section is concerned with the UA xtiT — Text) term in the energy balance for a stirred tank. The usual and simplest case is heat transfer from a jacket. Then A xt refers to the inside surface area of the tank that is jacketed on the outside and in contact with the fluid on the inside. The temperature difference, T - Text, is between the bulk fluid in the tank and the heat transfer medium in the jacket. The overall heat transfer coefficient includes the usual contributions from wall resistance and jacket-side coefficient, but the inside coefficient is normally limiting. A correlation applicable to turbine, paddle, and propeller agitators is... 

The accumulation term in the energy balance equation can be rewritten as... 

With only component B in the feed, the flow term in the energy balance becomes... 

The input and output terms of equation 1.5-1 may each have more than one contribution. The input of a species may be by convective (bulk) flow, by diffusion of some kind across the entry point(s), and by formation by chemical reaction(s) within the control volume. The output of a species may include consumption by reaction(s) within the control volume. There are also corresponding terms in the energy balance (e.g., generation or consumption of enthalpy by reaction), and in addition there is heat transfer (2), which does not involve material flow. The accumulation term on the right side of equation 1.5-1 is the net result of the inputs and outputs for steady-state operation, it is zero, and for unsteady-state operation, it is nonzero. 

Equation 12.3-16 is valid whether heat is transferred to or from the system, and whether the reaction is exothermic or endothermic. Note that each term on the left side of equation 12.3-16 may be an input or an output. Furthermore, CP is the molar heat capacity of the system, and is given by equation 12.3-13 as such, it may depend on both T and composition (through /A). The right side of equation 12.3-16 may also be expressed on a specific-mass basis (12.3-11). This does not affect the consistency of the units of the terms in the energy balance, which are usually J s-1. 

Any stirring within the reactor wiU generate heat, and we call this term Ws- We omit gravity and kinetic energy terms in the energy balance because these are usually very small compared to the other terms in the energy balance in a chemical reactor. 

Thus, even though each term in the energy-balance equation is now dimensionless, we associate each term with a term in the original energy-balance equation T — To)/J represents the heat flow out rninus the heat flow in, k T — Tc)I J represents the heat transfer through the walls, and X(T) represents the rate of heat generation by reactiou... 

Another potential model simplification involves assuming negligible energy accumulation in the gas phase as compared to that in the solid, which is equivalent to the earlier approximation [Eq. (66)] based on the relative magnitude of the energy accumulation in the gas and solid. For our system, the accumulation of energy in the solid is approximately 250 to 300 times that in the gas phase due to the relative thermal capacitance of the gas [Eq. (65)] and the similarity of the temporal behavior of the gas and catalyst temperatures (e.g., Fig. 19). Thus the accumulation term in the energy balance... 

The last term on the left-hand side of eq. (3.301) corresponds to the heat transfer to the external fixed-bed wall. The overall heat transfer resistance is the sum of the internal, external, and wall resistances. In an adiabatic operation, the overall heat transfer coefficient is zero so the corresponding term in the energy balance expression drops out, while in an isothermal operation this coefficient is infinite, so that 7 f 7 s 7W. 

The exiting temperature of the water for this operation can be computed from a total energy balance about the tower. Reviewing each term in the energy balance ... 

As an illustration, we will derive the viscous dissipation terms in the energy balance using a simple shear flow system such as the one shown in Fig. 5.6. 

Since the process is one of steady flow for which the shaft work and kinetic- and potential-energy terms in the energy balance [Eq. (2.10)] are zero or negligible, AH - Q. Thus, Q - -5,368.64 kJ, and this amount of heat is transferred to the boiler for every 100 mol dry flue gases formed. This represents... 

Let the system be the whole car. The accumulation term in the energy balance is not zero because the kinetic energy of the vehicle is initially not zero but after stopping is zero. Also, energy (heat) is transferred from the vehicle to the surroundings so that the energy transfer term in the energy balance is not zero. The rest of the terms presumably are zero. Consequently we get (in symbols)... 

C What are the two basic methods of solution of transient problems based on finite differencing How do heat transfer terms in the energy balance formulation differ in the two metliods ... 

The term Wj, often referred to as the sha/t work, could be produced from such things as a stirrer in a CSTR or a turbine in a PFR. In most instances, the flow work term is combined with those terms in the energy balance that represent the energy exchange by mass flow across the system boundaries. Substituting Equation (8-4) into (8-3) and grouping terms, we have... 

Figure 4.12 Terms in the energy balance for a closed system.

We note that the radiation from the glow has not been considered in this model. To include this effect requires a mass balance for each excited species, a contribution in the momentum balances for each excited species, and additional terms in the energy balances that represent the energy lost via the radiation field. These effects are smaller than the terms that are retained in the model and thus represent second order effects. As such they are expected to change the solution to the problem slightly and have not been retained in the description. 

The Carty et al. report specifies that reaction 1 cannot be thermally driven. Their experimental work indicates a minimum cell potential of 0.45V that might be achieved with an optimized cell design and electrocatalysts [1,6]. This value was used to calculate the electrochemical work. The corresponding efficiency with the electrochemical work is 46.0% (LHV) as shown in Table 2. The drop in efficiency illustrates the high cost of the work terms in the energy balance. 

When the gas velocity approaches the sound velocity, the kinetic energy and viscous work terms in the energy balance equation are not negligible (as assumed in Chapter 5). For these cases, we write the general energy balance equation for a differential plug-flow reactor with length dL (see Eq. 5.2.44),... 

As a last check to assess the validity of the method used, it is applied to Phillips (1985) saturation spectrum. In Phillips effort to derive a saturation spectrum (in his case B(k) oc k05), he defines forms for three source terms in the energy balance equation, viz. wind input, non-linear interaction and dissipation. When written in the form of Eq. 21, his energy balance has f2 = 0. In that case, Eq. 24 can actually be further reduced to p - 2 /. Save for a factor of 2, this yields the Plant and Wright relaxation... 

Steady-state temperatnres along the length of a PFR are governed by an ODE. Figure 5.3 illustrates the differential energy balance for a PFR. The terms in the energy balance are the same as in Equation 5.13, bnt the reactor is assumed to be at... 

ENTHALPY BALANCES IN HEAT EXCHANGERS. In heat exchangers there is no shaft work, and mechanical, potential, and kinetic energies are small in comparison with the other terms in the energy-balance equation. Thus, for one stream through the exchanger... 

This is an experimental fact reproducible in any well-equipped laboratory. Using the conversion factors in Eq. 4.22, we can easily convert all the terms in the energy balance to a common basis. In SI the use of the calorie is discouraged thermal energy quantities are to be expressed only in joules.. However, the use of the calorie (or kilocalorie) is quite common in countries using metric units. Today s student will have to be familiar with its use. 

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