where m = cQ and Q = flow, a constant through the bed. (The minus sign of m results from the fact that at the inlet the unit normal vector n is in opposite direction to that of the velocity vector v.) Therefore, cv • nndA = ^dl = Q^dl = VsSodddl (7.31)
a dl dl dl
Vs is the superficial velocity of flow in the bed.
Also, over the same differential length dl and cross-sectional area So of bed, dj-qy - djqpr-- ddV — d-S0dl (7.32)
dt dt dt dt o dV in the second term has been taken out of the parentheses, because it is arbitrary and therefore independent of t.
Substituting Eqs. (7.31) and (7.32) in Equation (7.29),
(Since the solids are conservative substances, the total derivative is equal to zero.) Dividing out Sodl and rearranging,
The numerical counterpart of Equation (7.34), using n as the index for time and m as the index for distance, is qn+1Jm - qn,m — y ^-n,m+1 - Cn,m (7 35)
for the first time-step. Solving for qn+1,
AtVs ACn qn+1,m — qn,m a, (Cn,m-1 - Cn,m) — Qn,m - AtVs al (7.36)
The equation for the second time-step is
Ac Ac Ac qn+2,m = ?n+i,m - AtVs-Aj1 = qn,m - AtVs —" - AtVs(7.37)
Over a length Al, the gradients Ac/Al varies negligibly from time step to time step. Thus, Acn/Al = Acn+1/Al = Acn+2/Al and so on. Equation (7.37) may then be written as qn+2,m = qn,m 2AtVSAl
For the k th time step, the numerical equation becomes qn+k, m qn,m kA tV s a i
But the number of time steps k is t/At. Therefore, the numerical equation is simply qn+k,m is actually simply q at the end of filtration. Also, qn,m at the beginning of the filtration run is zero. Because the effect of the deposited materials has to start from a clean filter bed, qn,m may be removed from the equation. The final equation becomes
The nature and sizes of particles introduced to the filter will vary depending upon how they are produced. Thus, the influent to a filter coming from a softening plant is different than the influent coming from a coagulated surface water. Also, the influent coming from a secondary-treated effluent is different than that of any of the drinking water treatment influent. For these reasons, in order to use the previous equations for determining head losses, a pilot plant study should be conducted for a given type of influent.
Determination of constants a and b. As noted before, hd plots a straight line in logarithmic form with q. This means that only two data points are needed in order to determine the constants a and b. Let the data points be (hd1, q1) and (hd2, q2). The two equations for h that can be used to solve for the constants are then
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