Shell energy balances

In this section we set up shell energy balances for flowing polymeric fluids. This material should be helpful in conceptualizing the nonisothermal equations of change. The basic principle that is used is the conservation of thermal energy as applied to a thin shell of fluid, which is stated below  

Applying this principle to a differential volume element and taking the limit as the volume element goes to zero leads to a differential equation for the temperature distribution. The procedure is described in more detail elsewhere (Bird et al., 1960, 2007, Chapter 9). We illustrate the use of Eq. 5.15 through the following example. 

We next try to determine whether Eq. 5.19 can be reduced or simplified. By writing Eq. 5.19 and the boundary and initial conditions in dimensionless form it is much easier to determine which terms in the differential equation are most important. The dependent and independent variables are written in dimensionless form by dividing them by an appropriate characteristic quantity. In particular, we introduce the following dimensionless quantities  

The term multiplying 39/3 is also dimensionless and is called the Peclet number, Pe, and represents the ratio of the heat transfer by forced convection to that by conduction. The boundary and initial conditions given in Eq. 5.20 become 

FIGURE 5.3 Model of region between the film die and cast- 39  

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In view of the relatively high conductivity of copper and the thin wall of the tubing, the average instantaneous heat flow from the wall to the fluid can be determined from a shell energy balance ... 

In general the heat of reaction, which is of the order of 100 kj/mol, caimot be ignored, and the mass balance must then be complemented by an appropriate shell energy balance. That balance must consider the heat conducted in and out of the shell, as well as the heat generated or consmned within the pellet. 

With the assumption of constant physical properties, Eq. 5.72 would be the same as that obtained by means of the shell energy balance (Eq. 5.48). ... 

An energy balance will be maintained over the sphere, and it will be assumed that there is no angular dependence on heat transport. The energy balance will be executed over a thin (Ar thickness) spherical shell and solved in essentially the same way as in Section III.A, except that we will work in spherical coordinates. The steady-state energy balance is given by... 

Figure 5-2 Energy balance in a PFTR. A shell balance is made on an element of volume dV between Z and + dz. Species flow Fj and enthalpy flow in and out of this dement of volume are balanced hy species and energy generated by reaction.

In Figure 4.20(b), the encounter of a closed-shell species M with the open-shell excited species M shows that the occupation of the very same molecular orbitals leads to a net stabilizing situation two electrons sink to the lower Pb orbital, with an energy gain — 2A E, while one electron is raised by +AE to I, but the excited electron now sinks into P so that the energy balance is approximately E = — 2AE + AE — AE = — 2AE. This simple picture of excimer formation applies to closed-shell organic molecules as well as to He atoms. The excimer is indeed an electronically excited dimer which is stable only so long as its electrons follow the distribution of M. ... 

The combined stream is preheated to 122°C in a FEHE. A heater (HX3) is installed after the FEHE so that inlet temperature of the coolant stream in REACT2 can be adjusted to satisfy the energy balance when the exit temperature of the coolant stream is specified in this countercurrent tubular reactor. This temperature is 150°C, and the heat load in HX3 is 9.34 x 106 kcal/h. The stream is further preheated to 265°C in the tube side of reactor REACT2 by the heat transfer from the reactions that are occurring in the hot shell side of this vessel. There is no catalyst on the cold tube side, so the feed stream does not react but its temperature is increased. The stream is then fed to reactor REACT 1, which contains 48,000 kg of catalyst. This reactor is cooled by generating steam. The coolant temperature is 265°C (51 bar steam). This vessel contains 3750 tubes, 0.0375 m in diameter, and 12.2 m in length. The overall heat transfer coefficient between the process gas and the steam is 244 kcal h-1 m-2 °C 1. The heat transfer rate is 42 x 106 kcal/h. 

Similarly, the mass balance equation for the shell side and the energy balance equations for both regions can be developed. 

Now consider a thin cylindrical shell element of thickness Ar in a long cylinder, as shown in Fig, 2-15. Assume the density of the cylinder is p, the specific heat is c, and the length is /,. The area of the cylinder normal to the direction of heat transfer at any location is A = iTrrL where r is the value of the radius at that location. Note that the heat transfer area A depends on r in this case, and thus it varies with location. An energy balance on this thin cylindrical shell element during a small time interval At can be expressed as... 

Starting with an energy balance on a spherical shell volume clement, derive the one-dimensional transient heat conduction equation for a sphere with constant thermal conductivity and no heal generation. 

Starting with an energy balance on a cylindrical shell volume element, derive the steady one-dimensional heal conduction equation fora long cylinder with constant tliemial conductivity in which heat is generated at a rate of... 

Reconsider steady laminar flow of a fluid m a circular tube of radius R. The fluid properties p, k, and Cp are constant, and the work done by viscous forces is negligible. The fluid flows along the.r-axis with velocity n. Tlie flow is fully developed so that i< is independent of, v and thus u = n(r). Noting that energy is transferred by mass in the A-direction, and by conduction in the r direction (heat conduction in the. v-direction.is assumed to be negligible), the steady-flow energy balance for a cylindrical shell element of thickness dr and length d. can be expressed as (Fig. 8-21)... 

Figure 8-17 illustrates the arrangement in which the fluid in the tank is heated by an external heat exchanger. The heating medium is isothermal, therefore any type of exchanger with steam in the shell and tubes can be used. The variable temperature out of the exchanger, t",will differ from the variable tank temperature, t. An energy balance around the tank and the heat exchanger gives ...

Consider the same irreversible first-order reaction A B used in Sec. 11 -6 to. obtain the isothermal rj. If the effect of temperature on D is neglected, the differential mass balance and boundary conditions, Eqs. (11-46) to (11-48), are still applicable. The energy balance over the spherical shell of thickness Ar (see Fig. 11-6) is... 

To derive an equation for 9, we write the energy balance equation over the heating fluid in the shell element. For co-current flow,... 

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