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In “The Missing Hunter’s Curve” (Plumbing Engineer, October 2013), I demonstrated that the probability curve developed by Dr. Roy Hunter in the early 1920s is still the basis for sizing vertical drainage stacks in the Uniform Plumbing Code (UPC). I also mentioned that the method of probability he developed for vertical stacks was not applied toward horizontal branches, building drains, or building sewers. Rather, he utilized a mean flow rate based on an assumed frequency of use.
This article will discuss the methodology Dr. Hunter used for horizontal pipe sizing, demonstrating how he derived the capacities of horizontal drains in fixture units and how they have been adopted in the UPC drainage sizing table. When discussing the UPC, we will be revealing significant divergences from Hunter’s drain capacities that will need interpretation.
As with the probability model for drainage stacks, tables for horizontal drain capacities are published in “Recommended Minimum Requirements for Plumbing, Report BH13.” The 1928 edition is an updated revision of the original 1924 publication resulting from continued research at the National Bureau of Standards. A further progress revision was published in 1931, which distinguished the capacities of horizontal branches from the capacities of house drains and sewers. This article refers only to the 1928 edition since the UPC adopted the horizontal drainage table from this revision and not from the 1931 revision.
The development of horizontal capacities was not as convoluted and onerous as what we have previously seen for vertical stacks. Yet, we shall meet with some complexities and modifying factors that do not allow a straightforward formula such as Manning’s formula, which is used in other model codes. Instead of just presenting the tables from Report BH13, we will recreate the tables to reveal underlying assumptions. Although the very complex and cumbersome probability model for drainage stacks was not used, a probable modifying factor was introduced. Further modifications were needed based on empirical data, resulting in several steps of development before the final sizing table was realized. Discharge flow rates in horizontal piping are still converted in terms of fixture units based on the water closet value of six fixture units, as discussed previously in “The Missing Hunter’s Curve.”
Development of the horizontal pipe sizes
Dr. Hunter’s starting point was to calculate the capacities of pipe flowing full under its own head (i.e., without static head) at ¼-inch per foot slope using Darcy’s formula (see Equation 1).
where:
V = velocity in feet per second (fps)
g = acceleration due to gravity (32.2 ft/sec2)
D = diameter of pipe in feet
f = friction coefficient
h/l = head gradient per length of pipe in feet (slope)
Once determining the velocity, it is a simple calculation to find the flow rate using the formula shown in Equation 2.
Equation 2
q = 2.448d2v
where:
q = gallons per minute (gpm)
d = diameter of pipe in inches
v = velocity in fps
When evaluating the friction factor by solving for f in the Darcy formula, given Hunter’s computed velocities, we find that Hunter assumed turbulent (unsteady) flow for rough pipe conditions with a friction factor ranging from 0.06 for 3-inch-diameter pipe to 0.05 for 12-inch-diameter pipe. This yields an absolute roughness (ε) of approximately 0.01 for pipe sizes 3 inches to 6 inches in diameter, and approximately 0.02 for pipe sizes 8 inches to 12 inches in diameter when referencing the Moody chart. This indeed verifies what Hunter confessed in his report: his calculations assumed a rough pipe condition from old cast iron pipes lined with deposit, which needs a brief comment.
Dr. Hunter had argued for decreasing pipe capacity relative to changes of pipe roughness due to aging (Water-Distributing Systems for Buildings, Report BMS79, 1941). We see this allowance for aging influencing his calculations from the beginning of his published works, whether for drainage or water pipes, but is somewhat ignored today with pipe material other than cast iron or galvanized steel. Whether choosing a C-factor in the Hazen-Williams formula, or an n-factor in Manning’s formula, the tendency may be to choose the manufacturer’s recommended coefficient for smoothness under pristine conditions when calculating capacity without consideration of aging or conditions of actual service. The plumbing engineer should consider how to modify the friction factor relative to the actual conditions of service. This point is being emphasized only to raise awareness of the role the friction coefficient plays in calculating pipe capacity.
Table 1 capacities are derived using Equations 1 and 2 and reproduce the same results found in Report BH13. The column displaying the friction factor is added to exhibit the friction coefficient that Hunter used in the Darcy formula. When comparing to Manning’s formula, capacities are approximate when using n = 0.014 for smaller cast-iron pipe diameters and n = 0.016 for larger cast-iron pipe diameters.
Having established pipe capacities, Dr. Hunter’s next step was to transform the flow rate into fixture units. The previous article demonstrated that a fixture unit is a rate of flow—with 1 fixture unit being 1 cubic foot per minute, or 7.5 gpm. To transform the capacities in Table 1 into fixture units, we would expect to divide the capacity by 7.5.
However, this is not what Dr. Hunter did. This is where he introduced a factor of probability when transforming capacity into fixture units. In order to do this, Dr. Hunter translated the flow rate into the probable number of toilets that would generate the pipe’s capacity. Assuming that each toilet discharged a volume of 5 gallons and that this would occur once every minute during peak use, the estimated mean rate of flow is calculated by Q/T, or 5/1. Hence, pipe capacities were divided by five to yield the probable number of toilets equivalent to the pipe’s capacity (see Table 2). For example, 22 toilets discharging 5 gpm into a 4-inch pipe yields the pipe capacity of 110 gpm.
The results in Table 2 were problematic for Dr. Hunter considering that the estimates for the probable number of toilets were too high for 3-inch-diameter pipe and too low for 12-inch-diameter pipe. Testing had shown that only three toilets closely spaced on a 3-inch horizontal drain with a ¼-inch per foot slope discharge successfully. With an additional fixture discharging, the flushes became sluggish. Therefore, three toilets discharging into a 3-inch drain is the approximate limit of the drain. The probable number of 10 toilets would certainly overcharge the 3-inch drain. On the other end, Dr. Hunter considered that the storage capacity is greater in the 12-inch pipe and that runoff is the controlling factor and congestion is improbable since an average flush rate of once a minute per toilet cannot be maintained in a system large enough to require a 12-inch drain. Therefore, Dr. Hunter permitted the estimate for a 12-inch drain to be increased to twice its value assuming runoff based on a flush frequency of once every two minutes instead of one minute.
To resolve this problem, Dr. Hunter corrected Table 2 by grading the values between the tested value for 3-inch pipe and the assumed doubled capacity for a 12-inch pipe. Column three in Table 2 was adjusted by multiplying the number of estimated toilets by a graduating corrective factor that reduced the number of toilets to three for 3-inch-diameter pipe and that doubled the number of toilets for 12-inch-diameter pipe. These modifying factors are seen in Table 3 and reproduce the same results in column 4 published in Report BH13.
In recreating Table 3, we discover our first discrepancy with the published report, as noted in parentheses in column 3. In the third column, notice the values on the left side and the parenthetical values on the right side. The modifying factors on the left side yield the actual results in column 4 that appear in Report BH13, whereas Dr. Hunter cites the modifying factors on the right side. Obviously, the parenthetical values for 5-inch and 12-inch pipe are rounded values and will yield slightly different values than what the tables in the report show. We can only assume that 1.85 for 8-inch-diameter pipe may be a typographical error and that 1.9 for 10-inch pipe has been mistakenly rounded up since these would yield a significant difference from what is shown in the tables in the report.
Having adjusted the equivalent number of toilets for safe and practical discharges per pipe diameter, the number of toilets was then transformed into fixture units by multiplying by a factor of 6 (each toilet valued at 6 fixture units). This yielded the recommended pipe capacities in fixture units per pipe diameter (see Table 4). The pipe capacities in fixture units in Table 4 are for slopes at ¼-inch per foot. The fixture units will need to be adjusted for slopes at 1/8-inch and ½-inch per foot.
Before showing the adjustments made to the fixture units based on differing slopes, we need to point out another discrepancy found when recreating Table 4. The discrepancy is noted in parentheses in column 3. The BH13 Report shows the pipe capacity in fixture units for 8-inch pipe as 1,392. We cannot explain this other than as possible errata in the published report where the third digit should have been a 2. Unfortunately, this errata affected 8-inch pipe capacities in fixture units for varying slopes as will be shown below.
Considering that the head varies with respect to varying slopes of pipe and that runoff will vary as the square roots of the total heads, the use of Darcy’s formula suggests that slopes of 1/8, ¼, and ½ inch per foot will vary the pipe capacity by the ratio 36:51:80. Based on the estimates of 3-inch pipe discussed above, Hunter modified the ratios for smaller pipe diameters (Table 5).
To calculate the pipe capacity in fixture units for varying slopes using the ratios, multiply the pipe capacity in fixture units by the appropriate ratio. For example, to calculate the fixture unit capacity for three-inch pipe at 1/8-inch per foot slope, multiply 18 fixture units by the ratio 5/6. For ½-inch slope, multiply 18 fixture units by the ratio 7/6. As we continue this calculation for the rest of the pipe diameters we derive the final table for capacities of horizontal drains in fixture units (Table 6).
Once again, when recreating the table published in Report BH13 we see several discrepancies in comparison. The values posted on the left side of the columns are derived from using the ratios provided in Table 5. The values in parentheses are the actual values published in Report BH13. An explanation for each discrepancy follows.
Beginning with 4-inch-diameter pipe at 1/8-inch per foot slope, we notice that if we convert 77 fixture units to number of toilets (77/6 = 12.8 toilets) and round up the number of toilets to an even number (14), then we yield 84 fixture units as given in the table. Continuing across the table at ½-inch per foot slope, 115 fixture units convert to 19.2 toilets, and when rounded to 19 toilets it yields 114 fixture units.
The discrepancy noted for 5-inch pipe at ½-inch per foot slope is a difference of one toilet. If we accept the modifying factor in Table 3 as 0.9, the adjusted number of toilets would be 35, yielding 210 fixture units (although BH13 displays 36 and 216 respectively in the tables). Applying the ratios to 210 fixture units to calculate fixture units at ½-inch per foot slope would yield 262.5 fixture units, and rounding up to the nearest toilet yields what is given with 264 fixture units. Applying the same for the lesser slope still yields the same value as given.
Obviously, the error noted earlier for 8-inch pipe (1,392 fixture units) affected the values for 1/8-inch and ½-inch per foot slopes when the ratios were applied. Accepting the value of 1,392 fixture units for ¼-inch slope, the calculated values for 1/8-inch and ½-inch per foot slopes would have been 988 and 2,227 fixture units respectively. The calculated 988 fixture units are equivalent to 164.7 toilets, and rounding up to 165 toilets yields 990 fixture units. The calculated 2,227 fixture units, for 8-inch pipe, are equivalent to 371.2 toilets, which were rounded down to an even number (370), yielding 2,220 fixture units as given in the report.
Moving down the table for the remaining pipe sizes, notice the same pattern of rounding. For 10-inch pipe, converting 1,789 fixture units to number of toilets (298.2) and rounding up to 300 yields 1,800 fixture units; converting 3,956 fixture units to number of toilets (659.3) and rounding down to 650 yields 3,900 fixture units. For 12-inch pipe, converting 3,067 fixture units to number of toilets (511.2) and rounding up to 514 toilets, yields 3,084 fixture units.
Where calculated values translated to an even number of toilets, no rounding occurred. Hence, we conclude that the published values are rounded to the nearest toilet in fixture units or rounded to a chosen near even number. The rounding is somewhat inconsistent and arbitrary.
No rounding to an even number of toilets was applied for 3-inch pipe diameter. The possibility of overcharging the drain is greatest for 3-inch-diameter pipe, and no more than three toilets were permitted as mentioned earlier. For slopes less than ¼-inch per foot, only two toilets were permitted with an extra allowance of three fixture units. For ½-inch per foot, three toilets were permitted with the same extra allowance of fixture units as the lesser slope.
In “The Missing Hunter’s Curve,” we were able to calculate the estimated flow rate for drainage stacks by using Dr. Hunter’s estimating curve. For horizontal drains, calculating estimated flow rates is more problematic. Of course, we could work our way backward and calculate the pipe capacities shown in Tables 1 and 2, but that would be misreading Table 6. Table 6 is showing the practical pipe capacities in terms of toilets rather than flow rates. The modifying factors introduced in Table 3 implicitly alter the probable average mean flow rate of 5 gpm. For example, the modifying factor for 12-inch pipe altered the probable average mean flow rate to 5 gallons every two minutes, or 2.5 gpm for each toilet rather than 5 gpm. The modifying factor for 3-inch pipe altered the probable average mean flow rate to approximately 17 gpm for each toilet, which allows only three toilets to correspond to the pipe’s capacity of 51 gpm. Three toilets closely spaced seemed to indicate that the pipe’s capacity was reaching its limit, but was it reaching its limit of 51 gpm?
The problem the modifying factors were attempting to correct was the differing flow characteristics between surge flows and continuous flows yet to be realized by Dr. Hunter. Only later did Dr. Hunter begin to experiment and analyze surge flows. The surge flow is the temporary peak flow from the immediate discharge of a fixture into the drain. The surge flow will quickly flatten out and assume terminal velocity for the diameter and slope of the pipe, which then becomes a continuous flow and can be calculated as a mean flow rate. The 1931 revision of BH13 premiered this distinction, which later came to full development in “Plumbing Manual, BMS66,” published in 1940. What limited the 3-inch pipe capacity was the fact of the three toilets closely spaced together. The discharges were not average mean flows applicable for building drains and sewers, but were surge flows in horizontal branches. The modifying factors do not make this distinction, and calculating flow rates is a graduating mixture of surge flows in 3-inch pipe to mean flows in 12-inch pipe. For this very reason, Dr. Hunter later developed separate tables for sizing horizontal drains depending on whether they were primary or secondary: horizontal branches receiving surge flows or building drains and building sewers where the stream is tending toward continuous flow that can be expressed in a mean flow rate.
Look for the conclusion of “Hunter’s Horizontal Drain Capacities and the Uniform Plumbing Code” in the July issue of Plumbing Engineer.
Daniel Cole is the Technical Services Supervisor for the International Association of Plumbing & Mechanical Officials (IAPMO), which publishes the Uniform Plumbing Code. He can be reached at dan.cole@iapmo.org.