11.2.2 Evaluating the Work Term

It is customary to separate the work term, , into flow work and other work, . The term , often referred to as the shaft work, could be produced from such things as a stirrer in a CSTR or a turbine in a PFR. Flow work is work that is necessary to get the mass into and out of the system. For example, when shear stresses are absent, we write

11-4

where P is the pressure (Pa) [1 Pa = 1 Newton/m² = 1 kg·m/s²/m²] and is the specific molar volume of species i (m³/mol of i).

Let’s look at the units of the flow work term, which is

where F_i is in mol/s, P is Pa (1 Pa = 1 Newton/m²), and is m³/mol.

We see that the units for flow work are consistent with the other terms in Equation (11-3), i.e., J/s.

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 (11-4) into (11-3) and grouping terms, we have

11-5

The energy E_i is the sum of the internal energy (U_i), the kinetic energy , the potential energy (gz_i), and any other energies, such as electric or magnetic energy or light:

11-6

In almost all chemical reactor situations, the kinetic, potential, and “other” energy terms are negligible in comparison with the enthalpy, heat transfer, and work terms, and hence will be omitted; that is,

11-7

We recall that the enthalpy, H_i (J/mol), is defined in terms of the internal energy U_i (J/mol), and the product (1 Pa·m³/mol = 1 J/mol):

11-8

Typical units of H_i are

Enthalpy carried into (or out of) the system can be expressed as the sum of the internal energy carried into (or out of) the system by mass flow plus the flow work:

Combining Equations (11-5), (11-7), and (11-8), we can now write the energy balance in the form

The energy of the system at any instant in time, Ê_sys, is the sum of the products of the number of moles of each species in the system multiplied by their respective energies. This term will be discussed in more detail when unsteady-state reactor operation is considered in Chapter 13.

We shall let the subscript “0” represent the inlet conditions. Unsubscripted variables represent the conditions at the outlet of the chosen system volume.

11-9

In Section 11.1, we discussed that in order to solve reaction engineering problems with heat effects, we needed to relate temperature, conversion, and rate of reaction. The energy balance as given in Equation (11-9) is the most convenient starting point as we proceed to develop this relationship.