DMpdV dvPdV t

The left-hand side of the above equation is a derivative of the property in the control mass. Hence, in a control mass, the derivative is a total derivative. This is the derivative that would be observed on a given fluid property, irrespective of where the fluid is in space. Remember that the control mass system is closed; no mass can enter. Therefore, the property cannot vary with space, but only with time. This derivative describes what an observer would see if traveling with the mass inside the closed container—the control mass. This is called the Lagrangian method of describing the property. This derivative is also called the Lagrangian derivative.

The right-hand side of the equation shows the derivatives of the property inside the control volume, (diV.pdV)/ dt, and the property on the boundary of the control volume, f A.pv • ndA . Because f A .pv • ndA belongs in the same equation as (diV.pdV)/dt, it is also loosely called a derivative and, because this particular derivative is conveyed by a velocity, it is called a convective derivative. In other words, the convective derivative is conveyed from outside of the control volume into the inside of the control volume and vice versa. These derivatives describe what an observer from a distance will see. Standing at a distance, fluid properties as they vary from point to point may be observed (convective derivative), $A typV • n dA; in addition, standing at the same distance, the same fluid properties as they vary with time may also be observed (the derivative (diV^pdy)/dt). This mode of description is called the Eulerian method and the whole right-hand side of the equation may be called the Eulerian derivative.

Equation (85) portrays the equivalence of the Lagrangian and the Eulerian views. It is the Reynolds transport theorem. It states the values of the properties in one system, the control mass (the left-hand side), and states the same properties again in another system, the control volume (the right-hand side). These properties are one and the same things, so they must all be equal. They are said, therefore, to have a material balance between the properties in the control mass and the same properties in the control volume. The left-hand side of the equation is also called a material, substantive, or comoving derivative. The partial derivative on the right-hand side is called a local derivative. Because the theorem was derived by shrinking At to zero, it must be strongly stressed that the boundaries of both the control mass and the control volume coincide in the application of this theorem.

Kinetics of growth. In order to derive the correct version of Equation (75) using the Reynolds transport theorem, the kinetics of growth needs to be discussed. Let [X ] be the concentration of mixed population of microorganisms utilizing an organic waste. The rate of increase of [X ] fits the first order rate process as follows:

where f is the specific growth rate of the mixed population in units of per unit time and t is the time.

Now, the question is, what is that rate? Is it the total derivative d[X]/dt or the partial derivative d[ X ]/dt ? If one wants to determine experimentally the rate of growth of certain microorganisms, an obvious way is to put the culture in a large container reactor, take measurements, and calculate the rate of growth. Because culture will not be introduced and removed continuously to and from the container, the reactor is a closed system—it contains the control mass. Thus, the total derivative is used, instead of the partial derivative, and the rate equation becomes

When an organism is surrounded by an abundance of food, the growth rate represented by the previous equation is at a maximum. This growth rate is said to be logarithmic. When the food is in short supply or depleted, the organisms will cannibalize each other. The growth rate at these conditions is said to be endogenous. Monod (1949) discovered that in pure cultures f is a function of or is limited by the concentration [5] of a limiting substrate or nutrient and formulated the following equation:

where ¡m is the maximum j. In Equation (88), if ji is made equal to one-half of ¡m and Ks solved, Ks will be found equal to [5]. Therefore, Ks is the concentration of the substrate that will make the specific growth rate equal to one-half the maximum growth rate and, hence, is called the half-velocity constant. Although the equation applies only to pure cultures, if average population values of ¡im are used, it can be applied to the kinetics of mixed population such as the activated sludge process, as well.

In the dynamics of any population, some members are born, some members die, or some members simply grow en masse. In addition, in the absence of food, the population may cannibalize each other. The dynamics or kinetics of death and cannibalization may simply be mathematically represented as a mass decay of the population kd[X], where kd is the rate of decay. Incorporating Monod's concept and the kinetics of death into Equation (87), the model for the net rate of increase of [X] now becomes dJdX = l K^[ X] - kd[ X] (89)

This is the kinetics of growth, which, as can be seen, is expressed in total derivative. This represents the Lagrangian point of view (a closed-system point of view) of the process. Remember that we have used the total derivative, because cultures are not introduced into and withdrawn from the reactor, which means that the reactor is not an open system. The system is closed and, therefore, the total derivative is employed instead of the partial derivative.

Now, we are ready to derive the correct version of Equation (75) using the Reynolds transport theorem. First, let us determine the expression for (dJM<p dV)/dt in terms of [X]. Using the general dimensions of length L, mass M,

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