Hardy-Cross Pipe Network Tutorial

This program is intended as an introduction to the Hardy-Cross method of analyzing simple pipe networks. The Hardy-Cross method consists of the following procedure:

- Number each of the various loops

- Assume a flow direction (clockwise = positive ; counterclockwise = negative)

and assume an initial flow through each pipe.

- Calculate the head loss in each loop. Use the same sign convention as above.

- Check the closure of loop by summing head losses of all pipes in loop.

- Calculate flow corrections to improve headloss closure.

- Repeat process until head losses converge to desired accuracy.

Consider the following system:(See Figure A)

+-

Pipe AB: L = 2000 ft. Diameter = 8 in.

Pipe BC: L = 4000 ft. Diameter = 8 in.

Pipe CD: L = 2000 ft. Diameter = 8 in.

Pipe BD: L = 2000 ft. Diameter = 6 in.

Pipe AD: L = 4000 ft. Diameter = 10 in.

What is the first step? Should you:

Determine flow direction

Number each loop

Assume an initial flow

Balance flows at each junction?

Now...is the second step to:

Calculate head loss

Make an initial guess of roughness factors

Determine equivalent slope for the system Assume an initial flow and flow direction?

***Step 2 is the assumption of flows and their directions. Take a look at junction A. (See Figure B)

Hazen-Williams Equation

The Hazen-Williams formula is an empirically derivedequation for circular conduits flowing full in the turbulentflow regime.

The Hazen-Williams formula can be written as:

V= 1.318 Chw RH 0.63 S0.51

Where:

V = Average Velocity of Fluid (ft./sec.)

Chw = Hazen-Williams Friction Factor (no units)

RH = Hydraulic Radius (ft.)

Area Affected By Skin Friction = Area of PipeWetted PerimeterS = Slope of hydraulic grade line (no unit).

NOTES:• For metric units, 1.318 is to be replaced with a factorof 0.845• Hazen-Williams equation is valid for turbulent flowonly.• Equation and factors have been derived and are validfor water only. Properties for other fluids cannot betaken into account.

Using common sense, which of the following holds true?

(1) ab + ad + 1000 = 0

(2) ab - ad - 1000 = 0

(3) 1000 -ab + ad = 0

(4) 1000 -ab - ad = 0

As you can see, having assumed initial flows and directions (step 2), inflow must equal outflow for each junction, and therefore, (4) is correct.

Now...Step 3 gets a bit more difficult. Head losses are calculated with the Hazen-Williams formula:

h = L*Q1.85

17,076*C1.85*D4.87

h = headloss, feet

L = pipe length, feet

C = roughness coefficient

D = pipe diameter, feet

Q = flow, gallons per minute

Once the head losses are calculated for each pipe, do you remember just what comes next?

Check closure by summing h for each loop

Sum roughness coefficients for all pipes

Calculate correction factor

Re-initialize flows?

To do this, the calculated head losses are summed for each loop.

If each loop sums to less than one foot, that is:

If sum of h < 1.0 ftSTOP!!!!

You're close enough! But...

If the sum of the head losses are greater than one foot, then you must calculate and apply a correction factor, q, until head losses total less than one foot. This may mean LOTS of iterations!

For our example,

Does the sum of the heads for Loop I =: AB - BC - AC

AB + BC - AC

AC + BC + AB

AB + AC - BC

I

II

+

+-

**REMEMBER THE SIGN CONVENTION**

Now try this one:

Does the sum of the heads for Loop II =:

BD - CD - BC

BD + BC - CD

BD + CD + BC

CD + BD - BC

I

II

+

-

-

**REMEMBER THE SIGN CONVENTION**

If the heads do not close to within 1.0, the flows must be adjusted and the head losses recalculated until closure is attained.

The correction used is as follows:

Correction factor q = -H/ Σ A

where: H = sum of the head losses for loopA = 1.85*h/Q for a particular loop

Σ A = sum total of A in loop

The final flows and directions as calculated for this example with the Hardy-Cross method are found in figure D (Answer diagram).

Typical piping layout