**Approximate flow through an orifice
****13-12**

There are several reasons you might want to install a restrictive device, or orifice in a piping system.

- To create a false head for a centrifugal pump, allowing you to run the pump close to its BEP.
- To increase the line pressure.
- To decrease the flow through a line.
- To increase the fluid velocity in a line.

The equation for flow through an orifice is a simple one to understand. Only the units are somewhat awkward.

**Q = AV**

Q = The flow in cubic feet per second (ft^{3}/sec).A = The area of the orifice in square feet (ft

^{2}).V = The velocity of the liquid in feet per second (ft/sec).

Experience shows that the actual flow is quite different than calculated because of the different shapes of the various orifices. Look at the following diagrams and you will see some of these popular shapes. Each has been assigned a "K" value.

We will enter that "K" value into our equation and the new equation becomes:

**Q = AVK**

To make the equation easier to handle we can express the velocity "V" as:

- g = 32.2 ft/sec
^{2} - h = Head across the orifice. If the downstream side of the orifice is pressurized use the differential head across the orifice.

If you do not know how to convert pressure to head, use this formula:

It would also make sense to convert some of the terms in our equation to terms that are more convenient to use. As an example:

- "Q" can be converted from cubic feet per second to gallons per
minute:
- 1 ft
^{3}/sec = 448.8 gpm.

- 1 ft
- "A" The area in square feet can be converted to square inches:
- 1 ft
^{2}= 144 square inches

- 1 ft

Putting all of this together gives us a new formula that looks like this:

Let's plug in some numbers and calculate a flow through a typical orifice.

Given:

- h = 20 feet
- A = 0.049 square inches
- K = 0.62

Q = 25 x 0.049 x 0.62 x 4.47 or

**Q = 3.40 gallons per minute**

If we want to solve for the orifice area:

If you are uncomfortable working with the orifice area in square inches you can use the diameter instead. Use the following equation:

Inserting the 0.049 square inches we calculated from the prior formula we get

or 1/4 inch

We made our formula more user friendly by substituting some conversions and now we can make our calculations in gallons per minute and square inches, but the formulas would be better if we could measure the orifice diameter rather than the orifice area

I took you through this exercise to show you how the formulas we use in these papers are derived. We will re-write the flow and orifice diameter formulas again and maybe this time they will be simple enough for anybody to use. We will start with the flow formula and then fix the orifice formula:

The formula for calculating the orifice diameter becomes:

Let's see if the formulas still work. Here are the numbers:

- d = .250 or 1/4 inch
- K = 0.620
- Q = 3.4 gallons per minute
- h = 20 feet

We will begin by solving for flow (Q)

Well that worked, now let's try for orifice size:

All of these above numbers were generated assuming that you were moving water through the orifice. If you are making calculations for a liquid other than water you will have to factor in the viscosity of that liquid compared to water.

We also made an assumption that the orifice diameter is not greater than 30% of the pipe diameter. There is another formula we use for a less restrictive orifice.

Any time the ratio of the orifice diameter to the pipe diameter is greater than 30%

(0.30) you should modify the formula. The modifier (M) looks like this:

- d
_{1}= orifice diameter - d
_{2 }= pipe diameter

When you are using the modifier, the formulas look like this:

Now we will see what happens when a 0.250 inch (1/4) orifice is put into a smaller cross section 0.500 inch (1/2) pipe, assuming the other numbers stay the same:

This means that you would have to multiply by 1.03 so the 3.46 gpm we got in the last calculation would become 3.56 gpm.

How accurate are these predicted numbers? Anytime you make a calculation using flow as a part of the equation, you will run into some variables that will affect your results:

- The roughness of the piping inside walls affects the friction loses.
- The piping material and allowable wall thickness tolerances.
- Solids buildup inside the piping. Calcium in water applications and coke in hot oil applications are typical. Higher temperature usually hastens the solids buildup.

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