An Integrated One-step Equation for Solving Duct/Pipe Friction Loss by Hand Calculator

ASHRAE Handbooks are the worldwide reference books for HVAC engineers. When we tried to develop a duct software, we also followed the steps shown in 2013 ASHRAE Handbook. Accidently we found that some friction loss data of a duct design example seemed contrary to the data obtained from duct friction chart. Then we go back to adopt Darcy’s and Colebrook’s equations that have been used to solve duct/pipe friction loss for decades. However, the calculation process needs to use complicated computer program. After doing huge trial and error processes by computerized program, we obtained one integrated equation that can be used to calculate duct/pipe friction loss by hand calculator. We own an HVAC&R consultancy firm and have the opportunity to contact many real duct/pipe projects. This empirical equation has been successfully applied to dozens of actual duct and pipe design projects. For Reynolds Number (Re) is greater than 10,000 (i.e. turbulent flow), our analysis shows the friction losses obtained from this integrated equation are within ±2.0% of those obtained from Darcy’s and Colebrook’s equations. The accuracy (±2.0%) is good enough for engineers doing realistic duct/pipe designs. Hence, this one-step equation can be the handy alternative for Darcy’s and Colebrook’s equations. For the practical duct/pipe designs, engineers can calculate friction loss easily, no need to use iterative method.


Introduction
Darcy's and Colebrook's equations always come into mind when discussing friction loss in ducts/pipes. For the fluid flow in duct or pipe, pressure drop due to friction loss can be calculated by Darcy equation [1]. Based on SI unit, Darcy equation can be rewritten as below. P f = * * * (1) where ∆P f =friction loss, Pa L=duct/pipe length, m f=friction factor, from equation (2) D=hydraulic diameter, mm ρ=fluid density, kg/m 3 V=fluid velocity, m/s For transitional and turbulent flows, friction factor (f) can be calculated by Colebrook equation [2]: √f = −2 log . + . √f (2) where ε=absolute roughness factor, mm Re=Reynolds number Reynolds number (Re) can be calculated by the following equation [2]: where ν=Kinematic viscosity, m 2 /s That means three equations are needed to calculate friction loss. In addition, because friction factor (f) appears on both sides of equation (2), solving equation (2) by hand calculator is almost impossible. Therefore, the Moody chart [3] (Figure 1) is adopted. However, using Moody chart is still tedious to obtain an exact f value. Then, in the real-design-world, engineers prefer to adopt friction chart; Figure 2 [4] for duct with ε=0.09mm, ρ=1.20kg/m 3 , and Figure 3 [5] for pipe with 20°C water and SCH40 commercial steel pipe.   [4].

Process of Developing an Integrated Equation for Solving Friction Loss
We tried to develop a duct design software in July 2017, and followed the steps as shown in the example 7 in chapter 21 in 2013 ASHRAE Handbook [6], as depicted in Figure 4. This example presented that the straight-duct friction loss (Pa/m) and friction factor (f) were calculated by Darcy equation (1) and Colebrook equation (2). The summarized values were shown in column 1 through column 6 in Table 1. Accidentally, we found that all the friction losses (column 6) were different from the friction losses (column 7) obtained from friction chart ( Figure 2) except SN 4 and SN 19. We cannot but help ask "Why".   In order to solve this problem, we reversed the calculation steps, i.e., using Darcy equation (1) and the friction loss ∆P 6 in column 6 to obtain friction factor (f) first. Then substitute f into Colebrook equation (2), and calculate both side values of Colebrook equation (2). The results were shown in columns 8 and 9 in Table 1. Obviously, there is something wrong with friction factor (f) because the left side values (column 8) do not equal the right side values (column 9).
We thought that maybe the friction factors (f) were just roughly read from Moody chart (Figure 1), not really calculated by Colebrook equation (2). Furthermore, in practice, when doing duct design, engineers still do not know the duct diameter yet. How can we use Colebrook equation to obtain friction factor (f)? Therefore, we tended to think from engineers' viewpoints, and hoped to find out if there is a better method to calculate friction factor (f) without diameter.
The process of solving problems is something like a reverse-thinking logic. We tried Steffensen method, iterative method and some mathematic techniques to solve Colebrook equation (2). After doing huge trial and error processes, we eventually obtained one integrated equation (4) by computerized programs. Equation (4) is a one-step method to calculate duct/pipe friction loss (Pa/m).

Verify the Validity of Equation (4)
Is Equation (4)  f(∆P L )=0.07136*∆P L *Q 0.5 /ρ/V 2.5 (5) where f (∆P L )=friction factor based on ∆P L ∆P L =friction loss (Pa/m) based on Equation (4) Moody chart can be used to verify Equation (4) by four steps. First of all, with given flow rate (Q) and velocity (V) to calculate friction loss (∆P L ) by Equation (4). Secondly, to calculate friction factor (f (∆P L )) by equation (5). Thirdly, to calculate Reynolds number (Re) by equation (3) and relative roughness (ε/D), and plot the junction point on Moody chart (Figure 1). Finally, to check if f (∆P L ) equals friction factor (f) on Moody chart (Figure 1).
For galvanized steel duct (ε=0.09) and standard air, the calculated f (∆P L ) values are shown in column 10 in Table 2.
Although you can only read an approximate f value from Moody chart, it is obvious to see the contrast between the f (∆P L ) values (column 10) and the friction factor (f) on Moody chart (Figure 1 points 1~5). You can clearly see that f (∆P L ) values (column 10) and f values (column11) in Table 2 are identical.
By the same token, for commercial steel pipe SCH40 (ε=0.065) and 20°C water, using the ε/D (column 8) and Re values (column 9) in Table 3, you can see both f(∆P L , column 10) and f (column 11) read from Moody chart are almost the same values. That means, Equation (4) is suitable for calculating pipe friction loss (Pa/m) also. ※Compare SN1~5 to points 1~5 in Figure 1 and    Figure 2 is the air friction chart suitable for galvanized steel round duct (ε=0.09mm) and standard air (20°C, ρ=1.204 kg/m 3 , ν=1.508*10 -5 m 2 /s). The friction losses (∆P L ) calculated by Equation (4) are shown in column 6 in Table 2, and the coincident points (1~5) are plotted on air friction chart ( Figure 2). You can see the ∆P L values (column 6) calculated by Equation (4) are almost the same as the friction loss (column 12) obtained by Q and V from Figure 2.
Similarly, Equation (4) can be used for the liquid Pipes. Figure 3 is the water friction chart suitable for 20°C water (ρ=998.2 kg/m 3 , ν=1.004*10 -6 m 2 /s). The friction loss (∆P L ) calculated by Equation (4) is shown in column 6 in Table 3, and the coincident points (1~5) are plotted on water friction chart (Figure 3). You can see the ∆P L values (column 6) calculated by Equation (4) are almost the same as the friction loss (column 12) obtained by Q and V from Figure 3 also. Therefore, the reliability of Equation (4) is verified by duct friction chart ( Figure 2) and water pipe friction chart ( Figure 3).

Darcy & Colebrook Equations vs. Equation (4)
Usually HVAC engineers determine flow rate (Q) based on cooling load calculation first. Duct/pipe diameter is still unknown at that time. Hence, it is not practical to use Moody chart ( Figure  1) to obtain exact friction factor (f), let alone Colebrook equation (2). The difference between Darcy's & Colebrook's equations and Equation (4) infers that the former is better for studying the relations between friction factor (f), Reynolds number (Re), relative roughness (ε/D) and friction loss (Pa/m), and Equation (4) is better for realistic duct/pipe designs. You can put different ρ, ν, ε, and D values into Colebrook equation (2) to obtain friction factor (f) iteratively, and use Darcy equation (1) to obtain friction loss (Pa/m). On the contrary, equation (4) is a one-step equation to obtain friction loss (Pa/m) by the known flow rate (Q) and velocity (V) that are determined by (HVAC) engineers.
The comparison between Darcy & Colebrook equations and equation (4) is shown in Table 4.

Equation (4) Apply to Equal Friction Loss Method
Engineers usually use given flow rate to decide duct/pipe diameter by friction chart. The most common design method is equal friction loss method. Actually, Equation (4) can be used for equal friction loss method by simple trial & error process. Here are the steps for applying Equation (4) Table 5 is the example showing the simple trail & error steps for duct and pipe designs; please refer to points 6, 7 in Figure 2 and Figure 3. Normally, the values ρ, νandεcan be found in common fluid mechanics books. Thus, equation (4) can be widely applied to most fluids and materials. Table 6 shows some applications for different fluids (ρ) and materials (ε). You can see that the friction loss (Pa/m, column 6) is variable depending on different fluids and materials (see Column 11).  Figure 2, SN7 can refer to point 8 in Figure 3.

Conclusions
The emphasis in this article is to verify if Equation (4) is coincident with Darcy and Colebrook equations. We own an HVAC&R consultancy firm and have the opportunity to contact many real duct/pipe projects. Equation (4) has been successfully applied to dozens of actual duct and pipe design projects since 2018. For the Reynolds number (Re) is greater than 10,000 (i.e. turbulent flow), our analysis indicates that the friction losses (Pa/m) obtained from Equation (4) is within ±2.0% of those obtained from Darcy's and Colebrook's equations. Therefore, the accuracy of equation (4) is good enough for engineers doing duct/pipe designs. Normally, engineers use given Q and V to obtain diameter D and friction loss ∆P f . We are professional engineers (P. E.) and satisfy with equation (4) applications. Besides, in real life duct/pipe applications, the Reynolds number is greater than 10,000 (see the Re in tables 2, 3, 6). We do not have the chance to try the situation with Re≦10,000. Maybe equation (4) is not reliable enough if it is used for the Reynolds number lesser than 10,000. For someone needs to differentiate laminar, transition and turbulent flow regions (see figure 1) when doing fluid dynamics research, you can use EXCEL worksheet to calculate ε/D, Re, f and ∆P L as we do in Table 6. Then, you can compare these values with the values obtained from equations (1), (2) and (3), or from Moody chart (Figure 1). There are many approximations of Colebrook's equation mentioned in public references, such as references from [7] to [15]. All these equations still need to calculate Reynolds Number (Re) first and just to solve Colebrook equation only. Not a similar equation like equation (4) using flow rate (Q) and velocity (V) to solve both of Colebrook equation and Darcy equation is found. Hence, we decide to release this article and let more engineers share our effort. Equation (4) can be the handy alternative for engineers to do realistic duct and pipe designs.