Page 202 - Modelling in Transport Phenomena A Conceptual Approach

182 CHAPTER 7. UNSTEADY-STATE MACROSCOPIC BALANCES

7.1 APPROXIMATIONS USED IN
MODELLING OF UNSTEADY-STATE
PROCESSES

7.1.1 Pseudo-Steady-State Approximation
As stated in Chapter 1, the general inventory rate equation can be expressed in
the form
Rate of Rate of Rate of Rate of ) (7.1-1)
( input ) - ( output ) + ( generation ) = ( accumulation
Remember that the molecular and convective fluxes constitute the input and output
terms. Among the terms appearing on the left side of R. (7.1-l), molecular
transport is the slowest process. Therefore, in a given unsteady-state process, the
term on the right side of Eq. (7.1-1) may be considered negligible if

Rate of
(7.1-2)
molecular transport ) ( accEX:ion
or,
Gradient of Difference in quantity
(Diffusivity) ( Quantity/Volume ) (Area) >> Characteristic time (7.1-3)

Note that the “Gradient of Quantity/Volume” is expressed in the form
Difference in Quantity/Volume
Gradient of Quantity/Volume = (7.1-4)
Characteristic length
On the other hand, volume and area are expressed in terms of characteristic length
8s

Volume = (Characteristic length)3 (7.1-5)
Area = (Characteristic length)2 (7.1-6)
Substitution of Eqs. (7.1-4)-(7.1-6) into Eq. (7.1-3) gives

(Diffusivity)( Characteristic time)
(Characteristic length)2 >> 1 (7.1-7)

In engineering analysis, the neglect of the unsteady-state term is often referred
to as the pseudo-steady-state (or, quasi-steady-state) approximation. However, it
should be noted that the pseudo-steady-state approximation is only valid if the
constraint given by Eq. (7.1-7) is satisfied.