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Time of Concentration

The time of concentration (t_c) is the time needed for water to flow from the farthest point in a watershed to the outlet of the watershed. Definition aside (which is taken straight from Wikipedia) there are a lot of formula's that solve for time of concentration. So many that I doubt it's validity in all cases and wonder if it will be on the breadth test :). Without further ado, the formula's:

In all of these examples the following variables are used:
L     Length of flow path
S     Average Slopw
C     Overall rational coefficient (C_{avg})
i      intensity (in/hr)
n     manning's roughness coefficient


Kirpich (general, mostly rural)

 t_c = 0.0078L^{0.77}S^{-0.385}

Kirpich's equation should be used for a lot of areas in general.

FAA Formula (Urban Areas)

 t_c = \dfrac{1.8(1.1-C)\sqrt{L}}{S^{1/3}}

Manning's Kinematic Wave Formula (Paved Areas)

 t_c = \dfrac{0.938}{i^{0.4}} \left( \dfrac{nL_o}{\sqrt{S}} \right)^{0.6}

NCRS Lag Equation (Small urban areas, most cities)

 t_c = \dfrac{1.67 L_o^{0.8} \left(\dfrac{1000}{CN}-9 \right)^{0.7}}{1900S^{0.5}}

Kerby's Equation

 t_c = 0.67 \left( \dfrac{nL_o}{ \sqrt{S} } \right)^{0.467}

I am not a hydrologist, so which of these to use is a shot in the dark. I think part of being a hydrologist is picking your favorite of these methods and modifying it to fit your own needs for your region.

For those of us just tackling this stuff on the breadth test, well, I am definitely taking these formulas into the test with me... will I use them? I hope not, but if I have to  I will pick one of them based on what variables I am given.

Some OSHA Regulations

The Occupational Safety and Health Administration has a few rules you should to be aware of:

Excavation

Banks over 4 feet must be shored

Trenches deeper than 4 feet and longer than 8 feet must be shored

Shoring slope cannot exceed 1:2 (H:V)

Bell bottom trenches (wider at the bottom) must have bracing on the thin part to prevent cave-in. Workers entering these should wear a life-line.

Ladders must be provided at 25 feet intervals for all trenches deeper than 3 feet.

Construction Site

Toilets must be provided at the site.

#People

< 20  1

>20   1 toilet and 1 urinal for every 40 workers

>200 1 toilet and 1 urinal for every 50 workers

The Prime contractor assumes all obligations. If the work has been subcontracted they may share obligations with the sub-contractor.

Employer must have an emergency action plan for evacuation

First aid supplies are required

Incident rate = number of injuries (annual) /  number of annual hours * 200,000

Employees must be protected form high sound levels

Employees must be protected from extreme light intensities

Proper signs must be placed.

The Critical Fall height is 6 feet. Anything over this height must have guardrails or other protective measures taken. Guardrails must be 42 inches.

Masonry Walls

While construction masonry walls a limited access zone must be set up adjacent to the wall. The size of it needs to be the wall height plus four feet. Only employees constructing the wall can enter it.

Site Layout

Construction sites need to have adequate access to roads around the site to ensure safe transportation of construction equipment and materials. Pedestrian and vehicle access in nearby roads must be maintained and controlled.

The site must be firm, properly drained and dry, and have adequate space to perform the work, operate equipment etc, and be large enough to store needed materials.

Rainfall Intensity Analysis

The intensity (I or i) of a rainfall event is a storm characteristic that measures the rainfall rate. In America this is in inches per hour ( \dfrac{in}{hr} ).

I think the primary reason to determine the intensity is for use in the rational method of determining peak storm flow.

The two main methods I have found to get the intensity are from Intensity-Duration-Frequency (IDF) Curves and from empirical formulas (Steel Equation).

IDF Curve

IDF Curves are curves that have been put together for a specific region using multiple years (decades) of recorded storm data. Here is an example.

To get intensity, you need to know the storm frequency as well as the storm duration. If the storm frequency is not one of the common ones (5, 10, 15, 20, 50 etc) you may have to interpolate on the graph and give your best estimate. The duration also may need to be interpolated.

With the above pictured IDF Curve (which is for some place in ReviewCivilPE land), a 10 year storm that lasts for d minutes would correspond to an intensity, i. Easy!

Steel's Formula

 I = \dfrac{K}{t_c+b}

The steel equation is an empirical equation for for estimating intensity and requires knowing the time of concentration (another process). You also need to determine values of the coefficients c and b (I can't seem to find names for these at the moment).

Both the CERM and AIO provide a map of the United States divided up into geographic regions for use with Steel's formula, and an accompanying table of c and b values.

  1. Find your region
  2. Find your return period (storm frequency)
  3. Get c and b values
  4. Success

With your time of concentration and c and b values, the Steel formula can be solved and your intensity obtained for use in the ration method.

When to Choose IDF or Steel Formula

Ah the golden question, what are the indicators of going with one method or the other? If you are given an IDF curve, and the frequency and duration I think this is a no-brainer, get it from the IDF curve!

If you are given a city or region, but no IDF curve, then you should probably shoot for the Steel Formula, they had better given you enough info to also calculate the time of concentration.

Rational Method for Peak Flow

The rational method determines the max flow (Q) at a point after a storm event.

 Q = CiA

Q, peak flow

C, runoff coefficient

i, storm intensity (\dfrac{in}{hr} )

A, area (acres)

To use the rational method you need to divide the watershed (area contributing flow to the point) up into separate areas depending on the runoff coefficients (C). The runoff coefficient changes based on different land types such as concrete, bare earch, turf meadow, residential etc. If the entire area is under one runoff coefficient then lucky you. If not, determine the area of each subregion and it's accompanying C value. Tables for C values are provided in both the AIO and CERM.

You should end up with (or be given) data similar to this:

Get the Average Runoff Coefficient

You can only enter one C value into the equation. It should be the weighted average of all of the C values.

 C_{avg} = \dfrac{ \sum{CA}}{ \sum{A} }

With the data above this would be:

 C_{avg} = \dfrac{25*0.2 + 15*0.3 + 2*0.4}{25+15+2} = 0.245

On a short question like on the breadth, I am thinking that the intensity MAY be provided to you, it is sort of a process to get it on your own. Either way, get the intensity. Once you have all three values, multiply for max Q!

Let's just say I solved for intensity (using information that has not been provided in this example) and it is 1.25 \dfrac{in}{hr} (on the less-intense end):

 Q = C_{avg} * i * A

 Q = 1.25 * 0.245 * 42

 Q = 12.875 \dfrac{ft^3}{s}

There you have it! This seems like a high flow to me, but it's a huge area. And it's an example problem, I made all these values up.

Speed Tip

You may have noticed that to solve for the C_{avg} value you have to divide by the total area. Why would you divide by the total area to get that and then turn around and multiply by the total area when you solve or Q?

To save some time, just multiply the sum of the region areas and coefficients by the intensity. It could be rewritten as Q=\left(\sum{ CA_{region}}\right)*i.