## O

(a) Partial derivative (b) Total derivative

FIGURE 2 Partial derivative vs. total derivative. Partial versus Total Derivative

A world of difference exists between partial derivative and total derivative. Yet, the environmental engineering literature seems not able to distinguish the difference between the two. In a given instance of use of the term, either partial derivative or total derivative is employed when, actually, only one version should be used—not either. To illustrate the difference, first define the word property. Property is an observable quality of matter. For example, consider water. Water may contain sodium and chloride ions, and their concentrations are an observable quality. Thus, the concentration of the sodium and chloride ions are properties of water. Water, of course, has temperature, and temperature is an observable quality; therefore, teml perature is also a property of water. If the water is flowing, it will have velocity and velocity is an observable quality. Hence, velocity is also a property of water. In other words, to repeat, property is an observable quality of matter.

Now, to demonstrate the difference between partial derivative and total derival tive, first, consider the left-hand side of Figure 2. This figure is an oval container which has holes on its sides. As shown, because of these holes, a mass can enter and leave the container. Let O represent any value of the property of the mass and let the container move in any direction. As the mass enters and leaves the container, it will be carrying with it its property, so the value of O will vary with distance as the container moves. O is then said to be a function of distance or space. If the Cartesian coordinate space, xyz, is chosen, then O is a function of x, y, and z. In addition, it must also be a function of time, t. In mathematical symbols,

FIGURE 2 Partial derivative vs. total derivative. Partial versus Total Derivative

The variables x, y, z, and t are called independent variables, while O is called the dependent variable of these independent variables.

Now, if the derivative of O with respect to any of the independent variables is to be found, it can only be partial, because it is simultaneously a function of the four independent variables. In other words, it can only be varying partially, say, with respect to x, because it is varying, also partially, with respect to y, z, and t. Partial differentiation uses the symbol "d." Hence, in mathematical symbols, the partial derivatives are dO _ ¿O(x, y,z,t) dx dx

¿O _ ¿O(x, y, z, t) dz dz dO _ ¿O(x, y, z, t) dt _ dt y, z, t _ constant (70)

Now, consider the right-hand side of Figure 2. This is also an oval container similar to the one on the left but with no holes on the sides. Because of the absence of holes, no mass can enter the container. Let the container move in any direction. Because no mass is allowed to enter, the value of O will not be affected by any outside mass and, therefore, will not be a function of x, y, and z as the container moves in space. It is still, however, a function of time. It is no longer a function of space, so it has no partial derivative with respect to x, y, and z. It is now only a function of one variable t, so its derivative will no longer be partial but total. The symbol d is used for total derivative. In other words, dO = dO(x,y,z, t- = dO(t) („.)

As you can see, there is a world of difference between partial derivative and total derivative. The total derivative applies when the dependent variable is a function of only one independent variable. When it is a function of more than one independent variable, then its derivative with respect to any one of the independent variables can only be partial. From the previous derivation, total derivatives apply to a closed system, while partial derivatives apply to an open system.

### Reynolds Transport Theorem

The Reynolds transport theorem demonstrates the difference between total derivative (or full derivative) and the partial derivative in the derivation of the material balance equation for the property of a mass. It is important that this distinction be made here because, as previously mentioned, the environmental engineering literature does not appear to distinguish the difference between the two.

For example, the material balance for the microbial kinetics of the activated sludge process is often written as (Metcalf and Eddy, Inc., 1991):

 where