E = 629,977.93

= ^-jOVVlyf * = 2(0.76)(1-0.42)(0.001352)(629,977.93) L n 2g^Jpidl 0.423 (2)(9.81)

7.4.2 Head Losses Due to Deposited Materials

The head loss expressions derived above pertain to head losses of clean filter beds. In actual operations, however, head loss is also a function of the amount of materials deposited in the pores of the filter. Letting q represent the deposited materials per unit volume of bed, the corresponding head loss hd as a result of this deposited materials may be modeled as hd = a(q)b (7.23)

where a and b are constants. Taking logarithms, lnhd = lna + bln q (7.24)

This equation shows that plotting lnhd against lnq will produce a straight line. By performing experiments, Tchobanoglous and Eliassen (1970) showed this statement to be correct.

Letting hIo represent the clean-bed head loss, the total head loss of the filter bed hL is then hi = hio + hd = hio + Xa(q/ (7.25)

where hd is the head loss over the several layers of grains in the bed due to the deposited materials.

Now, let us determine the expression for q. This expression can be readily derived from a material balance using the Reynolds transport theorem. This theorem is derived in any good book on fluid mechanics and will not be derived here. The derivation is, however, discussed in the chapter titled "Background Chemistry and Fluid Mechanics." It is important that the reader acquire a good grasp of this theorem as it is very fundamental in understanding the differential form of the material balance equation. This theorem states that the total derivative of a dependent variable is equal to the partial derivative of the variable plus its convective derivative. In terms of the deposition of the material q onto the filter bed, the total derivative is total derivative = d qdV^j (7.26)

V is the volume of the control volume. The partial derivative (also called local derivative) is partial derivative = d qdV^j (7.27)

and the convective derivative is convective derivative = <£ cv ■ nndA (7.28)

The symbol j>A means that the integration is to be done around the surface area of the volume. The vector n is a unit vector on the surface and normal to it and the vector v is the velocity vector through the surface for the flow into the filter. Now, substituting these equations into the statement of the Reynolds theorem, we obtain d [iqdV) = i {lqdV)+°?v^n ndA (7.29)

In the previous equation, the total derivative is also called Lagrangian derivative, material derivative, substantive derivative, or comoving derivative. The combination of the partial derivative and the convective derivative is also called the Eulerian derivative. Again, it is very important that this equation be thoroughly understood. It is to be noted that in the environmental engineering literature, many authors confuse the difference between the total derivative and the partial derivative. Some authors use the partial derivative instead of the total derivative and vise versa. As shown by the previous equation, there is a big difference between the total derivative and the partial derivative. If this difference is not carefully observed, any equation written that uses one derivative instead of the other is conceptually wrong; this confusion can be seen very often in the environmental engineering literature. Thus, caution in reading the literarture should be exercised.

Over a differential length dl and cross-sectional area A of bed, cv

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