The use of the Moody diagram in engineering.
The most common application of the Moody diagram in engineering is in the selection of a diameter for a pipe for some purpose.
Suppose that a steel pipe is required to carry a given flow of say m3/s (660 Imperial gallons/min). We can get some understanding of the actual problem involved if we make a few calculations. In order to avoid unnecessary complication let us consider the friction loss in a single length of pipe and ignore all losses due to fittings etc. Then the loss could be calculated using the Darcy expression if we knew a value for . Now it is clear that to find a value of using the Moody chart we must know a value of the relative roughness and a value for Re. Then we could follow one of the curved lines for the correct value of relative roughness to the left until it intersects with the value of Re. The position of this point of intersection will give a value of . Unfortunately all we have is that the pipe is a steel pipe and we have no diameter from which to calculate either Re or the relative roughness. We have to find a way round these difficulties.
The first thing to do is to find out how the velocity of flow varies with the diameter of the pipe. This gives us a feel for the figures involved. It is easily found from and is plotted in graph 7-6 for diameters up to 300 mm. The velocity of flow in a pipe seldom exceeds 3 m/s in practical pipes so we can see immediately that the diameter will be greater than say 140 mm.
We need more information. If we are to use the Moody diagram it is best to start by manipulating the Darcy expression into a form with instead of . We have:
Combining these two gives :
The constant when expressed in S.I. units becomes and frequently the expression is used in the approximate form : Joule/Newton. (We must include units because the 3 has units associated with it.)
This is often a preferred expression because it combines the quantities that will be known in most applications especially where the friction loss in the pipe is large compared with the loss in the fittings or where allowing for the loss in the fittings complicates the problem too much.
If, as a starting point, we supposed to be 0.005 (the figure given by Darcy) we could find out how the loss of head would vary with diameter for a length of 100 m. The loss would be given by The resulting graph is shown in graph 7-7 where the head lost is in metres per 100 m of pipe run. We see immediately that the lower diameter of 140 mm corresponds to the turning point in this graph. We would expect to be looking to a larger diameter.
In order to narrow the possibilities more we can make an attempt at costing. There will be two costs, that of installation and that of running. The installation cost will have two essential elements, the cost of the pipe and the cost of its installation. If the cost of the pipe depends on its weight then its cost is roughly proportional to . The cost of installation might well depend on its volume, i.e. on . Perhaps the cost of the pipe will depend on say so that we can say that the cost of installation = where is some figure that can be found from the cost of similar installations. The value of is not a figure that can be quoted but we can just look at the shape of the graph 7-8 which is the graph of against where . Clearly the cost of installation will be rising rapidly with diameter for the size of pipe, especially above 130mm, that will be needed but it is all too tenuous to be directly useful but it does tell us that we are on the right track..
The cost of running will be a pumping cost which for this given flow will vary with the head loss, the cost of electricity and the efficiency of the pump-motor unit and sundry overheads. We cannot hope to quantify this in this sort of text but we can evaluate the annual electricity consumption
Then the annual consumption for each run of 100 m of pipe can be found if we suppose the pumping to be continuous. It will be equal to (energy lost in kilowatts ) x (hours per year) / (efficiency of pump). If we let the pump motor unit have an efficiency of 50% this reduces to: kw hr per year. The annual consumption for the appropriate preferred diameters is given in the table below. It can be assessed as currency by multiplying by the cost of electricity.
It is now evident the choice of diameter will not be made from engineering. Those who are to finance this pipe have to make a choice on the basis of all manner of conflicting factors such as the future cost of electricity, the cost of servicing any debt incurred to install the pipe, pollution laws and so on. The engineer might reasonably be expected to present the relevant figures for each size of pipe and then the engineering problem reduces to the use of the Moody diagram for pipes of known construction and equivalent sand roughness and of the preferred sizes.
We can see how this might be done If is m3/s and is m, the mean velocity is m/s. Taking, for water, = 1000 kg/m3 and = kg/ms, . The design value of the sand grained roughness for commercial steel is and so the relative roughness is
. The curve for and intersect at = . The original figures could then be reworked for this improved value of and the same calculations made for the other diameters.
However, once the selection of a diameter has been made the engineer will be expected to produce a pipe to match the predictions.
Pipes undergo changes in the nature of the internal surface as a result of several influences. The most troublesome is corrosion of the internal surface increases its roughness and reduces its effective diameter. (See Froude’s battle with this in his papers.) Various protective coatings are used to inhibit corrosion. Pipes may accumulate internal layers from the fluid being transported. (The lead of an old domestic lead pipe can be melted away to leave a cylinder of limestone that is physically strong enough to be handled.) Thick oil may leave a sticky coating of bitumen inside the pipe. Dirty liquids may deposit sediment if the velocity of flow is too low. The surface might even improve if the right algae grows on it. In iron based water pipes a reduction of flow of 25% in 20 years might be expected. So some allowance must be made for the change in as a result of accretion on the inside surface of the pipe. Anyone looking at the figures in the table would think in terms of using either the 200 mm or the 250 mm pipe and probably the latter.
In the light of the uncertainty in the various factors affecting the choice of diameter and the large increments between preferred sizes it becomes clear that a better value of derived from some further complication of the Moody diagram would not be justified The Moody diagram is easy to use and accurate enough in the circumstances in which it will be used by engineers.