ABSTRACT
In Section 2.2, we derived the mathematical form of the macroscopic mass balance for a species in a mixture (Equation 2.39). In our analysis, we were not concerned with processes occurring at each point of the control volume but rather with an average view of mass transfer. That approach allowed us to formulate and solve problems in perfectly mixed systems, such as the CSTR. However, in many practical applications, it is necessary to quantify mass transfer at each point in the continuum. In this section we will derive the general point equations that represent species mass conservation. We start by writing an instantaneous species mole balance in a control volume along the lines of Equation 2.32. However, we will apply this mole balance to a control volume (Figure 4.1) that is a small parallelepiped of size Δx × Δy × Δz with the intention of eventually letting Δx, Δy, Δz → 0, to find an equation that is applicable at a point in the continuum. The species mole balance is
The time rate of change of the total moles of A in a control volume
Th{ } = e rate at which moles ofthe control volume
rat
net A enter
net
{ } +
The e at which moles of A are in the control volume by chemical rea
created ctions
(4.1)
Since the control volume is arbitrarily small, we will consider that the species molar concentration is uniform and, therefore, represented by the single value cA. In view of this, the accumulation terms in the balance can be written as follows:
The time rate of change of the total moles of A in a control volume{ } = ∂∂t c x y zA( )∆ ∆ ∆ (4.2)
Input and output of species A can occur only across the boundaries of the control volume, represented by the six faces. The moles per unit time that cross each of the
parallelepiped’s faces are equal to the corresponding component of the molar flux vector, NA, times the surface area. According to Figure 4.2, we can write
The rate at which moles of the control volume
N y zAx x net
A enter{ } = −∆ ∆ N y z N x z N x z N x y N x y
+
− + −
∆ ∆ ∆ ∆
∆ ∆ ∆ ∆ ∆ ∆
(4.3)
Note that components of the flux entering the control volume are positive whereas components that represent moles of A leaving the volume are negative.
