Instantaneous Quantities

It is obvious from (10.13) that the rate of polymerization is an instantaneous quantity it depends on the particular values of [M], [/], and T (through the temperature dependence of the rate constants) that exist at a particular instant (and location, for that matter) in a reactor. In a uniform, isothermal batch reaction (Example 2), the rate of polymerization decreases monotonically because of the decreases in both [M] and [/] with time. In a similar fashion, according to (10.35) is a function of [M], [/], T, and [R H], all of which may vary with time (and/or location) in a reactor. But is the concept of an instantaneous jc valid How much do these quantities change during the lifetime of individual chains This important point is clarified in the following example. 

Example 11. Consider the uniform, isothermal batch polymerization of Example Z For conditions at the start of the reaction, calculate 

The lifetime of free-radical chains is, in fact, so short that changes in concentrations are entirely negligible during a chain lifetime. Hence, it is perfectly proper to characterize chains formed at any instant when [M], [7], T, and [R H] have a particular set of values. AU the quantities defined to this point (x , q), and therefore the distributions, are just such instantaneous quant- 

Solution. Using the rate constants from Example 9.1 along with [M = M and [/] = [7]o from above in Equation 9.39 gives q = 0.9954. This illustrates the point made above that q 1 right from the beginning of a typical free-radical addition polymerization. 

Because termination is by combination, we use Equation 9.55 to get x =434 (M =45,200), so we are getting high molecular weight polymer right off the bat, too. 

---

Following the procedure of Reynolds averaging, the instantaneous quantities are written in terms of time-averaged and fluctuating quantities as... 

Equations (5) and (6) were derived for solid-liquid dispersions. These can be easily adapted to any other multiphase system such as gas-solid and gas-liquid systems. The one-dimensional equation of motion for solid phase in terms of instantaneous quantities is given by... 

In most of the early PUT attempts, the streaming velocities did not evolve continuously over time instead, they were instantaneous quantities that were freshly reassigned at each time step of the simulation. Such PUT procedures, however, might introduce other problems. At one extreme, PBTs interpret all secondary flows as thermal fluctuations and therefore weaken them. In the other extreme, if the u j s in a PUT scheme are heavily influenced by the... 

Reynolds [127] provided the fundamental ideas about averaging and was the first to accomplish the formulation of the governing equations for turbulent flows in terms of mean and fluctuating flow quantities rather than instantaneous quantities. Reynolds stated the mathematical rules for forming mean values. That is, he suggested splitting a turbulent velocity field into its mean and fluctuating components, and wrote down the equations of motion for these two velocity quantities. 

To reformulate the governing equations in terms of mean flow quantities rather than instantaneous quantities, Reynolds postulated the fundamental ideas of averaging. In the averaging procedure devised by Reynolds [127] the instantaneous quantities are decomposed into the sum of mean and fluctuating quantities. 

The mean quantities were defined by time-averaging the instantaneous quantities over a sufficient time period At), i.e., being long enough to smooth out the turbulent fluctuations, separating the turbulent parts from the non-turbulent parts. 

Introducing a generalized instantaneous quantity, -tp, the corresponding time average quantity is defined by ... 

The generalized instantaneous quantity V fc is decomposed into a weighted mean component and a fluctuation component in analogy to the Favre averaging procedure for compressible flows [75, 131] ... 

In order to calculate the mean frequency v of system cycles and the effective diffusion coefficient defined by eqs. (2.2) we first consider the corresponding instantaneous quantities... 

Equation (3) takes a similar form for both time-averaged and instantaneous quantities. In the time-averaged case, F is a turbulent eddy flux pu c, where overbars and primes denote time means and fluctuations, respectively. In the instantaneous case, F is a molecular diffusive flux which in practice can be neglected in the open air (on Sq) relative to fluxes arising from fluid motion (the high Peclet number approximation). In this case the fluid-motion fluxes appear in the term pc(u — v), which is unaveraged and includes transport by turbulent fluctuations as well as by the mean flow. In contrast with the situation in the open air (on Sq), molecular fluxes can never be neglected at solid boundaries (S,), where they are responsible for all the scalar transport. 

It is easily shown that any instantaneous quantity Q(x) is related to its cumulative value q(x) via ... 

For the turbulent flow, according to the basic concept of Reynolds-Averaged Navier-Stokes Equation (RANS), any instantaneous quantity can be resolved into two parts the time-averaged quantity and the fluctuating quantity, the latter is oscillating positively and negatively around the former. Thus, m,- and p can be expressed as follows ... 

Unfortunately (for the sake of simplicity), as conditions vary within a polymerization reactor, so do the instantaneous quantities, making it very difficult to accurately determine v and Xy, as functions of time. The polymer in the reactor is a mixture of material formed under varying conditions of temperature and concentrations, and therefore must be characterized by cumulative quantities, which are integrated averages of the instantaneous quantities of the material formed up until the reactor is sampled. The cumulative number-... 

Calculate and compare the average chain lifetime at the start of the reaction with the monomer half-life for the system in Example 9.10. Is the concept of an instantaneous quantity valid for this system kf = 14.5 — lO L/mol/s. 

Reynolds [128] postulated that the Navier-Stokes equations are still valid for turbulent flows, but recognized that these equations could not be applied directly due to the complexity and irregularity of the fluid dynamic variables. A true description of these flows at all points in time and space was not feasible, and probably not very useful at the time. Instead, Reynolds proposed to develop equations governing the mean quantities that were actually measurable. To reformulate the governing equations in terms of mean flow quantities rather than instantaneous quantities, Reynolds postulated the fundamental ideas of averaging. 

Small correction of a variable in FVM discretization Double prime, fluctuation component of an instantaneous quantity around it s mean weighted value. Used in the time-and ensemble averaging approaches Fractional step index, or intermediate value... 

---