## Case Study Calcium Carbonate Scaling in Water Mains

A small layer of calcium carbonate scale on water mains protects them from corrosion, but heavy scale reduces the hydraulic capacity. Finding the middle ground (protection without damage to pipes) is a matter of controlling the pH of the water. Two measures of the tendency to scale or corrode are the Langlier saturation index (LSI) and the Ryznar stability index (RSI). These are:

where pH is the measured value and pHs the saturation value. pH is a calculated value that is a function of temperature (T), total dissolved solids concentration (TDS), alkalinity [Alk], and calcium concentration [Ca]. [Alk] and [Ca] are expressed as mg/L equivalent CaCO3. The saturation pH is pHs = A - log10[Ca] - log10[Alk], where A = 9.3 + log10(K/K2) + 2'+55 in which ^ is the ionic strength. Ks, a solubility product, and K2, an ionization constant, depend on temperature and TDS.

As a rule of thumb, it is desirable to have LSI = 0.25 ± 0.25 and RSI = 6.5 ± 0.3. If LSI > 0, CaCO3 scale tends to deposit on pipes, if LSI < 0, pipes may corrode (Spencer, 1983). RSI < 6 indicates a tendency to form scale; at RSI > 7.0, there is a possibility of corrosion.

This is a fairly narrow range of ideal conditions and one might like to know how errors in the measured pH, alkalinity, calcium, TDS, and temperature affect the calculated values of the LSI and RSI. The variances of the index numbers are:

Var(LSI) = Var(pHs) + Var(pH) Var(RSI) = 22Var(pHs) + Var(pH)

Given equal errors in pH and pHs, the RSI value is more uncertain than the LSI value. Also, errors in estimating pHs are four times more critical in estimating RSI than in estimating LSI.

Suppose that pH can be measured with a standard deviation a = 0.1 units and pHs can be estimated with a standard deviation of 0.15 unit. This gives:

Var(LSI) = (0.15)2 + (0.1)2 = 0.0325 oLSI = 0.18 pH units Var(RSI) = 4(0.15)2 + (0.1)2 = 0.1000 oRSI = 0.32 pH units

Suppose further that the true index values for the water are RSI = 6.5 and LSI = 0.25. Repeated measurements of pH, [Ca], [Alk], and repeated calculation of RSI and LSI will generate values that we can expect, with 95% confidence, to fall in the ranges of:

LSI = 0.25 ± 2(0.18) -0.11 < LSI < 0.61 RSI = 6.5 ± 2(0.32) 5.86 < RSI < 7.14

These ranges may seem surprisingly large given the reasonably accurate pH measurements and pHs estimates. Both indices will falsely indicate scaling or corrosive tendencies in roughly one out of ten calculations even when the water quality is exactly on target. A water utility that had this much variation in calculated values would find it difficult to tell whether water is scaling, stable, or corrosive until after many measurements have been made. Of course, in practice, real variations in water chemistry add to the "analytical uncertainty" we have just estimated.

In the example, we used a standard deviation of 0.15 pH units for pHs. Let us apply the same error propagation technique to see whether this was reasonable. To keep the calculations simple, assume that A, Ks, K2, and ^ are known exactly (in reality, they are not). Then:

Var(pHs) = (log10e)2([Ca]-2Var[Ca] + [Alk]-2 Var[Alk]}

The variance of pHs depends on the level of the calcium and alkalinity as well as on their variances. Assuming [Ca] = 36 mg/L, o[Ca] = 3 mg/L, [Alk] = 50 mg/L, and 0[Alk] = 3 mg/L gives:

which converts to a standard deviation of 0.045, much smaller than the value used in the earlier example. Using this estimate of Var(pHs) gives approximate 95% confidence intervals of:

This example shows how errors that seem large do not always propagate into large errors in calculated values. But the reverse is also true. Our intuition is not very reliable for nonlinear functions, and it is useless when several equations are used. Whether the error is magnified or suppressed in the calculation depends on the function and on the level of the variables. That is, the final error is not solely a function of the measurement error.

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