I first learned about dimensional analysis from my grandfather. He taught me a simple way to verify that mathematical calculations were done correctly: make sure the dimensions work out correctly. If the dimensions are wrong, then the numerical value is wrong too.
I've never memorized equations for calculating welding costs — rather, I use dimensional analysis to figure them out. For this two-part feature, I'll use welding cost analysis to illustrate the principle of dimensional analysis; the same idea can be used for many computations.
Let's start a list of welding-related factors, along with some numerical values and importantly, some dimensions for measurement of these variables. Assume we want to know the cost per pound of deposited weld metal. The Table contains an assortment of welding-related data.
First, we identify all the factors in the table that have a cost ($) associated with them. These would include the labor and overhead rate, electrode cost, shielding gas cost and electrical cost. We can construct the basic equations shown in Step 1A to calculate the total cost.
After a review of the dimensions for this equation as seen in Step 1B, it is immediately clear that even though the numerator for these four factors is the same (all dollars), they can't be added together since the denominators are different. Further, only the second factor, the electrode cost, is in the units appropriate for the final calculation ($/pound). However, even this is not quite right; it is in units of pounds of filler metal, not pounds of weld metal.
Let's use some dimensional analysis, starting with labor and overhead costs as shown in Step 2. To get from $/hour to $/pound, we need to multiply the labor and overhead rate by something with units of hours per pound. A quick review of the variables in the Table shows that nothing has these units, although deposition rate has the inverse units (pounds per hour). Dividing the labor and overhead rate by the deposition rate yields the proper dimensions, as shown in Step 2A and 2B.
At this point, we need to apply a bit of welding engineering. Manufacturing personnel know that the welding arc can't be maintained constantly — so in the Table, an operating factor is included. This is a ratio of the arc time to the total time. Whether measured in hours or minutes, the dimensions cancel. We can therefore evaluate the labor and overhead portion of the cost by using the equations shown in Step 2C and 2D.
With the labor and overhead worked out, we move to the filler metal portion of cost. As we look at the dimensions expressed in Step 3B, it might appear as if the job is complete. However, notice that the dimensions shown are dollars per pound of filler, not dollars per pound of weld. Again, some welding engineering is required. The ratio of the weight of the deposited weld metal to the weight of filler metal consumed is called the electrode efficiency. The weld weight and filler metal weight are both expressed in pounds, and thus it is convenient to express this ratio as a unit-less percentage. The relationship is easily modified as shown in Step 3C, and Step 3D affirms that our dimensions are correct.
Next, we turn to the shielding gas costs shown in Step 4A and 4B. Somehow, we need to get from units of dollars per cubic foot to dollars per pound. In the list of variables, the only item with cubic feet is the gas flow rate, measured in cubic feet per hour. If we multiply these two items together, we obtain the relationship shown in Step 4C. As shown in Step 4D, we now have dimensions of dollars per hour.
The dimensions still are incorrect, but the term “dollars per hour” has been seen before, namely when the labor rate was calculated in Step 2. If the deposition rate is incorporated, we obtain the equation shown in Step 4E. Step 4F confirms that the dimensions are correct.
Finally, we come to Step 5 and the calculation of electrical costs. The dimensions of dollars per kilowatt-hour are far from the desired units of dollars per pound of weld metal. Recalling that a watt is a volt-ampere, we can take the welding machine input voltage and multiply it by the input current to get watts. That product, divided by 1000, yields kilowatts, and multiplying this by the welding time yields kilowatt-hours. Incorporating those steps into our generalized equation, one obtains the equation in Step 5A.
Most of the time, however, welding personnel don't have values for welding machine input power: we work with welding voltages and welding currents. Further, as shown in Step 5B, we still don't have the right dimensions; we also need to find a way to address the welding time, and also to introduce the weight of the weld metal.
The deposition rate, in units of pounds per hour, will address the second part of this problem. Dividing the equations in Step 5A and 5B by the deposition rate will introduce the weight of weld metal, and link the welding time to the amount of weld deposit.
It is more convenient to use welding voltages and currents to determine welding costs. However, we must consider the efficiency of the welding power source as well, since we pay for input power, not output power. The power source electrical efficiency can be used to account for power losses. Step 5C incorporates these concepts, and Step 5D affirms that our dimensions are correct.
To find the total cost per pound of weld metal using this procedure, these four components are added, and we obtain the total cost per pound shown in Step 6.
Dimensional analysis is a quick way to make sure your calculations are right, and an excellent tool to help refine calculations. I'm very grateful that my grandfather taught me the technique. In Part 2, we will explore some of the broader implications of the equation we have derived.
Omer W. Blodgett, Sc.D., P.E., senior design consultant with The Lincoln Electric Co., struck his first arc on his grandfather's welder at the age of ten. He is the author of Design of Welded Structures and Design of Weldments, and an internationally recognized expert in the field of weld design. In 1999, Blodgett was named one of the “Top 125 People of the Past 125 Years” by Engineering News Record. Blodgett may be reached at (216) 383-2225.