The table provides the relationship between the pressure of a waterjet system, the size of the nozzle that the water is fed through, and the resulting flow rate that is being used, the horsepower of the jet, and the thrust that the jet will exert back on the equipment/person holding the nozzle.
It is a very straightforward set of calculations, and I will build the table in two parts. The first will be a line-by-line explanation of how the calculations are made, and what the basis is, and then I will provide a tabular format (which is the one that I use) from which values can be read off. Because this is built in Excel the values on the edges of the table are changeable, to fit your own particular set of needs. Construction of the tables will be given through a series of 30 steps.
I am going to write about the parts that make up a system to deliver water under pressure in later articles, and so some things that will be explained then are going to be just stated at this point. The first of these comes when one considers nozzle size.
A nozzle, at its most basic, is a hole of a fixed size. Under just the force of gravity flow is quite low, and to get more water to flow through that hole some pressure must be applied to the water. The very simple relationship between the pressure at which the water is pushed, and the resulting speed of the water is given by the equation:
Speed (ft/sec) = 12.5 x square root of pressure (in psi)
Please note that water starts to compress significantly at about 15,000 psi. For the sake of this initial set of tabulations I am going to neglect that issue, though it will come up at some future date.
1. Since pressure is a value that is often chosen by the operator, the value for pressure is entered into cell c3. For this example, a value of 10,000 is used. (These come from the first system that I worked with, back in Leeds in 1965).
2. The equation to determine the velocity of the water is entered into cell c4 as
[ =12.5*sqrt(c3)].
Because the units need to be consistent going through the calculation, inches will be used initially. So the initial velocity value is multiplied by 12.
3. To convert into inches/second, the value in cell c4 is multiplied by 12 in cell c5 using the equation:
[ = 12*c4]
4. Nozzle diameter is the exit diameter of the nozzle, and this is sometimes referred to as the orifice diameter. This is a selected value and is entered into cell c6. I am using 0.04 inches in the initial example.
The cross-sectional area of the orifice is given by the equation:
Cross-sectional area = π x (radius) squared
5. Orifice cross-sectional area is calculated in cell c7, by entering the equation:
[ = 3.1412*((c6/2)^2)]
As water flows through a hole, the stream does not flow out of the hole at the same diameter as the hole. As the flow enters the hole it necks down to a slightly smaller diameter, which is a function of the nozzle shape, among other things. The reduction is known as the Coefficient of Discharge for the nozzle, and is a specific value for an individual orifice that can vary from a value as low as 0.61 to a high of around 0.95 or better. This is an input value, based usually on a manufacturer’s statement.
6. Enter a coefficient of discharge value, I have used a value of 0.81, in cell c8.
7. Calculate the effective area of flow by entering the equation into cell c9.
[ = c7*c8]
By multiplying the area of the flow by the velocity (the length of the water column that flows through the orifice in a second) then the volume of water that flows through the orifice in a second is calculated.
7. Calculate the volume flow each second, by entering the following into cell c10:
[ = c9*c5]
The volume flow rate is normally required in gallons/minute, and the conversion is to multiply by 60 (to convert from seconds to a minute) and then dividing by 231 (the number of cubic inches in a gallon).
8. The calculation is made in cell c11.
[ = c10*60/231]
Computers calculate to a high number of decimal values, and to keep this in normal perspective I usually trim this to show either one or two decimal points. The value shown should therefore be 3.97 gallons/minute, and the table to date should look like this:
Figure 1. The basic steps in calculating the volume flow of water through a nozzle.
There are two other values that are useful to calculate. The first is the horsepower that is being used in the jet. This calculation is a straightforward multiplication of the pressure of the jet (in psi) and the flow rate (in gpm) divided by 1714.
9. Enter into cell c14 the equation:
[=c11*c3/1714]
The other equation that is often useful to calculate (particularly where lances are being held-held in cleaning operations) is the reaction thrust that comes back from the nozzle. Some years ago we validated in the laboratory that this value can be calculated from the equation:
Thrust = 0.052 x flow (gpm) x square root of pressure (psi)
10. Enter into cell c 16 the equation:
[ =0.052*c11*sqrt(c3)]
This gives the basic form for the calculation of the basic values that are most useful.
Figure 2. The initial individual values calculated for the flow.
(You might want to SAVE at this point).
However most of the time I want to do some comparisons and so instead of carrying out a single calculation I would like to see the values in a table.
To make the table I use the same basic equations that are given in the steps above, but I lay out a table of values for pressure and nozzle diameter, which I will step through for those who are less familiar with some of the features of Excel.
The first step is to enter the values that are going to be most useful. In a general table this starts with the pressure that might be used to clean the siding of a house.
11. Insert pressure values starting with 1,500 psi in cell b23, and continuing along the row to that which is used for some of the more intricate cutting of metal, at 90,000 psi, which is in cell L23.
12. The discharge coefficient value is set just above the table in cell c21. I am using a value of 0.81. since this is a common value to all calculations in this table, it is put in a place where it is easy to find and change where needed.
13. Nozzle diameter values are also input as a column down from A24 to A35. I have used values from 0.005 inches to 0.1 inches to cover the range of likely interest, though these can be changed, after all the tables are in place. (Those following along might use the values I provide to create the table, after which use your own values for pressure and nozzle diameter, and don’t forget to change the coefficient of discharge.)
The result, at this point should look like this:
Table 3. Basic structure of the flow calculation table
14. Now, in cell b24 (or the relevant cell in your table) enter the following equation, which combines all the different stages outlined above into one single step.
(=$C$21*(3.1412*60*(A24/2)^2*12*12.5*SQRT($B$23))/231)
The $ sign means that the location after the sign is a constant. It can be selected by highlighting the location in the equation (c21) and then pressing the command and T keys at the same time.
15. Now select the column of values that run from b24 to l24, and then use the Fill Right command under the Edit command in the menu. This will give a series of numbers in the shaded squared that can be ignored for the minute, since we are now going to go through and change some of the values.
16. Go to c24 and change the B24 to A24. Change the $B$23 to $C$23. Tab to D24 and repeat (i.e. change the C24 to A24 and $B$23 to $D$23), and continue doing this along the row, ending in cell L24 changing the K24 to L24, and $B$23 to $L$23). (This is just correcting the calculation to using the right nozzle diameter, and the right pressure values). The table should now look like this:
Table 4. The flow table with the first row completed.
And now take advantage of the power of the table.
17. Select the cells from B24 to L35, and then go to Edit -> Fill Down. The table should be filled in. (You might want to SAVE at this point).
Table 5. Full flow table.
The next step is to create the table for the fluid horsepower contained in the jet.
Because all the calculations tie back to one another from now forward, I am going to use a copy function for the pressure and nozzle diameter values, so that if these values are changed in the above table, they will also change in the dependant tables which follow.
The first step then is to insert the pressure and nozzle diameter values.
18. Go to cell A40 and enter
[=A23)
19. Select row 40 cells A40 through L40. Use the EDIT -> Fill Right command to copy the pressure values into the new table.
20. Select column A cells A41 through A52. Use the EDIT -> Fill Down command to copy the nozzle diameter values into the new table.
21. In cell B41 enter the equation: [=B23*B24/1714] select the B23 and press command T which will change the equation to: [=$B$23*A24/1714)
22. Select the B row from B41 to B52 and use the EDIT -> Fill Down command to generate the first column.
23. Go back to the values in cell B41 and remove the $$ signs from $B$23, hit return and remove the $$ signs from $B$23 in cell B42. Continue down the column removing $$ signs from the cells. The numbers in the cells should not change as you go down.
24. Select the cells from B41 to L52. Enter EDIT -> Fill Right. The second table will generate. While the cells are still selected reduce the number of decimal places to 2. SAVE the file. You have just generated the fluid horsepower table, which should look like this.
Figure 6. Fluid horsepower table.
We will now use the same technique to calculate reaction force.
25. In cell A57 enter [=A23]
26. Select cells A57 to L57. Enter EDIT -> Fill Right, to enter the pressure values at the top of the table.
27. Select cells A57 to A52. Enter EDIT -> Fill Down, to enter the nozzle values along the left-hand side of the table.
28. In cell B58 enter the equation for reaction force in terms of pressure and flow.
[=0.052*B24*sqrt(B23)]
29. Select the cells B58 to L58 and enter EDIT - > Fill Right.
30. Enter cell B58 and select the term B23. Press the command key and T at the same time, which will change this from B23 to $B$23. Tab and repeat this for the C23 term in cell C58, for the D23 term in cell D58 and so across the row ending with changing L23 in cell L58 to $L$58.
31. Select cells B58 to L69. Enter EDIT -> Fill Down. The table should be complete. It should look like this:
Table 7. Reaction Force calculation table.
Congratulations, you now have your own table, and by changing the pressure, nozzle diameter and discharge coefficient values along the flow volume table the charts can be tailored for your own conditions.
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Hi,
ReplyDeleteIt would be nice if you would add the fact that your calculation of the velocity:
[ =12.5*sqrt(c3)]
depends on the hydrodynamic part from Bernoulli's equation. It took some minutes to verify this fact for me because normaly i'm calculating within the SI units ;)
I am in awe of your engineering prowess. It is a struggle for me to calculate in this VERY interesting problem. Since my needs are more mundane, you have solved all. I am watching your blog with great anticipation of good things to come.
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