There are two key fluid properties that influence the head loss in pipe: the fluid's density and viscosity, and these properties vary with temperature in such a way that they have competing influences on the head loss. To evaluate these competing influences, we need to begin with some fundamental equations and graphs and to analyze just fluid properties, we'll use water as an example to solidify the underlining principle and keep the flow rate, pipe roughness, pipe length, and pipe size constant.
There are several equations used to calculate the head loss in a pipe. The Reynolds Number must first be calculated, then the friction factor, and then the Darcy head loss equation can be used to determine the head loss. The Hazen-Williams head loss equation used in some industries is not considered in this article because it is used just for water close to 60° F, but the principles discussed will still apply.
Calculating the Reynolds Number:
The first equation to review is the Reynolds Number equation. This equation is show below.
Looking at this equation one can note the density ( 
) is in the numerator and the viscosity (μ) is in the denominator.
Determining the Friction Factor:
The equation to calculate the friction factor is complex, so the Moody diagram is used to show how the friction factor varies with the Reynolds Number. A Moody diagram is show below.
In the transition zone for a given relative roughness, the higher the Reynolds number results in a lower the friction factor. Conversely, the lower the Reynolds Number results in the higher the friction factor.
Calculating the Head Loss:
Following is the Darcy head loss equation. It is important to note the friction factor (f) is in the numerator.
Evaluating the Equations and Graphs:
How the calculated value changes for an equation depends on whether a variable is in the numerator or denominator. If a variable in the numerator increases, the overall calculated value increases and if the variable in the numerator decreases, the overall calculated value decreases.
If the variable in the denominator increases, the overall calculated value decreases and if the value in the denominator decreases, the overall calculated value increases. Using this relationship, the Reynolds Number equation would have the following correlation for all possible changes of the fluid properties.
For the Moody diagram, the following correlation applies.
For the Darcy head loss equation, the following correlation applies:
With these correlations in mind, the best way to understand the numerator-denominator increase-decrease relationship is by looking at how both properties change for a real fluid with an example. For this analysis, water will be used as the fluid and the overriding external condition to cause density and viscosity to change will be a temperature variation. Fluid temperature variation is very common in piping systems. From chilled water application in the HVAC industry to process water application in the chemical process industry, water temperature is always changing.
The temperature verses density relationship for water can be seen in the following graph.
This graph shows water density decreasing as the temperature goes from 33 °F to 210 °F.
Next, the temperature verses viscosity relationship is shown below.
This graph shows the water viscosity decreasing as the temperature goes from 33 °F to 210 °F.
Note the change in the magnitude of the density (62.42 to 59.75) is only a 4.28 % over the temperature range. Whereas the change in the magnitude of the viscosity (1.633 to .2736) is 83.25% over the same temperature range. Both density and viscosity decrease with an increase in temperature, which means they have competing influences on the head loss. To determine which has the dominant influence, the head loss must be calculated for a given change in temperature (and corresponding changes in density and viscosity).
Following is a table showing the water fluid properties at two different temperatures and the calculated results for the Reynolds Number, friction factor, and head loss.
Looking at the above table, as the temperature increases, the density and viscosity decrease and the overall head loss decreases. This demonstrates that viscosity has the dominant influence on the head loss for water.
One should be cautious about making this assumption for all fluids, however. This analysis should be done for the particular fluid that is being used because the fluid properties may change differently for different fluids (particularly gases).
It is necessary to have good understanding of the density-numerator and viscosity-denominator relationship in the Reynolds Number equation, including how a change in the Reynolds Number moves the friction factor value on the Moody Diagram either to the left or right. Once the friction factor is determined, then a direct relationship between friction factor and the head loss value can be observed. This is important for accurately balancing the fluid piping system.
This is just one topic covered in the Piping System Fundamentals book and class. For more information on the Piping System Fundamentals book go to: www.eng-software.com/products/books/psfbook.aspx .
For more information on the Piping System Fundamentals two day training course go to: www.eng-software.com/products/PSTraining/PSF/default.aspx .
