Energy loss equation

^{Hassan (2017)} developed a mathematical model of linear programming in which he evaluated the costs and losses of energy in five nominal diameters of pipelines 3”, 3-1/2”, 4”, 5” and 6” with the equations of Hazen-Williams, Manning, Scobey and Darcy & Wesbach and determined that the best equations for energy losses were those of Darcy & Wesbach and Manning. For this reason, in this work the Manning and Darcy & Wesbach equations were used.

Frictional energy loss equation

Next, Manning’s frictional energy loss equation 1 is presented Hf=4103n2Lq2π2d163 1). Where: Hf= energy loss due to friction (m); q= pipe flow (m³ s^-1); L= pipe length (m); n= Manning’s roughness coefficient; and d= pipe diameter (m).

Localized energy loss equation

The Darcy & Wesbach localized energy loss equation. HL= KLv22g 2). Where: HL= localized energy loss (m); K_L= localized energy loss coefficient; v= velocity of the fluid (m); and g= acceleration of gravity (m s^-2). If the velocity is substituted in the previous equation as a function of flow and area, it results in: HL= KL8q2gπ2d4 3).

Analysis of an irrigating line

Figure 1 shows an irrigating line of emitters. When analyzing the irrigating line from right to left, a flow ‘q’ circulates in the section of pipe one, a flow ‘2q’ circulates in section 2, a flow ‘3q’ circulates in section 3, a flow ‘4q’ circulates in section 4 and so on. This means that the flow increases by a ‘q’ in each section of pipe until the maximum expenditure of the pipe ‘Nq’ in the ‘NL’ section is reached. Therefore, frictional and localized energy losses are accumulating in a ‘q’ in each section of pipe through which the water circulates until having the total energy losses in the ‘NL’ section and maximum flow ‘Nq’.

Figure 1 Diagram of an irrigating line of drippers. 

By adding the energy losses due to friction in each section from right to left, using equation 1, gives equation (4). HfT= 4103n2L1q2π2d163+4103n2L2q2π2d163+4103n2L3q2π2d163.......+4103n2LNq2π2d163= 4103n2Lq2π2d163∑i= 1Ni2 4). By adding the localized energy losses in each emitter from right to left, with equation (3), equation (5) results. HLT= KL81q2gπ2d4+KL82q2gπ2d4+KL83q2gπ2d4+KL84q2gπ2d4+............KL8Nq2gπ2d4= KL8q2gπ2d4∑i= 1Ni2 5). By adding the total energy losses due to friction ‘Hf_T’ (4), total localized energy losses HL_T (5) and if factored, the equation of total energy losses in an irrigating line (6) results. fTLR= 4103n2Lq2π2d163+KL8q2gπ2d4∑i= 1Ni2 6).

Solving the sum of equation (6) and substituting it into the same equation, equation (7) results. 4103n2Lq2π2d163+KL8q2gπ2d42N3+3N2+N6= HfTotal 7). Arranging terms “N” from the above equation, a polynomial of the total energy of the irrigating line results the equation (8). 2N3+3N2+N-6HfTotal4103n2Lq2π2d163+KL8q2gπ2d4= 0 8).

Solution of the polynomial of the irrigating line energy

The constant term is extracted from equation eight and if analyzed algebraically, equation 8.1 results. K= 34*HfTotal*π2gd1634116gn2sq2+KLq2d43 8.1). Compacting equation 8, it transforms into equation 9. FN=2N3+3N2+N-K= 0 9). The above equation consists of the following coefficients: a=2, b=3, c=1 and d= -K.

Next, equation (10) is presented, which solves equation (9) by generating the real solution of the equation. N=K4+3ac-b29a2+K4213+K4-3ac-b29a2+K4213-12 10). Substituting the coefficients a, b and c of equation (9) into equation (10), equation (11) is obtained, which is the solution of equation (9). N=K4+K42-112+13+K4-K42-11213-12 11).

Analysis of the real solution of the polynomial of total energy of the irrigating line

Equation 11 is composed of three terms (1°, 2° and 3°), the first two terms contain a variable and the third term is a constant, these terms are presented below (equations 11.1, 11.2 and 11.3, respectively). 1st=K4+K42-11213 11.1); 2nd=K4-K42-11213 11.2); 3rd= -12 11.3).

Statistical analysis of the number of outlets

The relative percentage error was determined, for this, the equation of ^{Olivares et al. (2019)} was adapted, obtaining the following equation. Relative error (%)=Ne-NN100 12). Where: relative error (%) = is the percent of relative error, N= number of emitters with equation (11); Ne= number of drippers estimated with equation 11.1.

To estimate this error, an electronic spreadsheet (Excel) was used for each of the equations that determine the number of outlets. Table 1 shows the numerical value of ‘N’ number of emitters (11), the error in the percentage of ‘N’ number of emitters in relation to the first term ‘1st’ (11.1), the partial solution of the terms (1°, 2° and 3°) to the total solution of the polynomial, as well as the contribution to the solution in percentage of each of the terms.

This table shows that regarding the contribution of the first term ‘1°’ in the column of partial solutions, the decimal part of these values tends to be constant as N increases and in the column of contribution to the solution, it tends to decrease. The contribution of the second term ‘2°’ in the partial solutions column tends to zero when ‘N’ takes values equal to or greater than 87 and in the column of contribution to the solution, it tends to a small value when ‘N’ is equal to 87 outlets. Likewise, the contribution of the third term ‘3°’ serves as a correction factor to the solution and tends to decrease significantly in the same number of outlets in the column of contribution to the solution. This means that the compact solution of equation (9) is summarized in the first ‘1°’ and third ‘3°’ term of equation (11), resulting in equation 13. N=K4+K42-11213-12 13).

Equation 13 consists of two variable subterms, which are presented below. Subterm 1= K4 13.1); Subterm2=K42-112 13.2).

Table 2 shows the numerical value of ‘N’ number of emitters (11), the values of ‘subterm 1’ (13.1) and ‘subterm 2’ (13.2), the numerical value of ‘modified N’, calculated with the sum of equations 13.1 and 13.2 that represents equation (11.1); the ‘N’ error determined with the values of ‘modified N’ and the values of ‘N’, calculated with equation (12).

This table shows the contribution of subterm 1 and subterm 2, where it is observed that there is little difference between the two columns when ‘N’ takes values from one to seven and the values of the two subterms are equal when N is equal to or greater than eight. This is because the subtraction of one twelfth to the second term of the equation does not affect the value of subterm two, comparing the contribution of the two subterms when ‘N’ takes a value of 87 the ‘N’ error calculated with equation 12 is 0.6 percent. Therefore, the value of -1/12 of the ‘subterm 2’ is removed, giving rise to equation (13.21). Subterm 2= K4 13.21)

By merging subterm 1 (13.1) with subterm 2 (13.21), equation (14) results as follows: N=K213 14).

Deterministic equations

Joining equation (8.1) with equation (14) results in equation (15). N= 38HfTotalπ2gd1634113gn2sq2+KLq2d4313 15). Adding the constant term (11.3), which has a value of (-1/2), to equation (15) results in equation (16). N= 38HfTotalπ2gd1634113gn2sq2+KLq2d4313-12 16).

Table 3 shows the ‘N values’ calculated with equations (15) and (16) and their respective relative errors based on equation (11), calculated with equation (12). Equation (16), as N is greater than 87 emitters, errors are zero compared to equation (11), while in equation (15), errors tend to zero from 87 emitters. Equation (16) has a better fit than equation (15), with errors ranging from -37.28% for one emitter and -0.01% for 87 emitters. Equation (15) has a lower fit compared to equation (11), with errors ranging from 12.35% for one emitter and 0.56% for 87 emitters.

Validation of deterministic equations

For the validation of the deterministic equations (11, 15 and 16) it was necessary to design an irrigating line in an Excel spreadsheet. For this purpose, the following data and calculations are presented.

Design of an irrigating line

An irrigating line was designed with the methodology proposed by the National Center of Advanced Irrigation Methods (CANAMAR, 1979) belonging to the Secretariat of Agriculture and Hydraulic Resources (SARH).

Selection of the dripper

A non-self-compensated dripper with a consumption greater than one liter per hour was selected, the selected dripper was that of low spending Rivulis E1000 (Rivulis Irrigation Inc; San Diego, CA, USA), the technical data are presented in Table 4.

Hydraulic model of the dripper

The technical data of the dripper underwent a potential regression analysis in an Excel spreadsheet, the results are shown in Figure 2.

Figure 2 Regression analysis of the dripper (Rivulis E1000) to estimate the values of k and x. 

The results allowed determining the components of the hydraulic model of drippers, where the value of coefficient ‘k’ was equal to 0.6622 and the value of ‘x’ was 0.4875 exponent (Figure 2).

Calculation of the operating pressure of the irrigating line

The operating pressure of the dripper irrigation line was also determined with equation (17). H0=qmk1-%Vg1000.5  1x 17). Where: Ho= hydraulic load of operation of the dripper (m); qm= average flow of the dripper (2 L h^-1); %Vg= percent of flow variation between the first and last dripper (5%).

Calculation of permissible energy loss in the irrigating line

The calculation of the permissible energy loss was determined with equation (18). Dh=Ho1-100-%Vg1001x 18). Where: Dh= permissible energy loss (m).

Selection of diameters for the calculation of the number of drippers in the irrigating line

Eight diameters of PVC (polyvinyl chloride) pipeline were selected, for this purpose, the English series Amanco (Orbia Inc; San Francisco CA, USA) was selected because this type of pipeline has semi-equidistant internal diameters (Table 5).

Calculation of the number of drippers in different internal pipe diameters

An irrigating line was designed by determining the number of drippers that can be installed in different internal pipe diameters (Table 5) with equation 19. N=HfTotal10.29n2Lq2d163+KLq22gA13+12N+16N213 19). Where: N= number of drippers; A= cross-sectional area of the pipe (m²); K_L= dripper insertion coefficient (0.5); n= roughness coefficient of PVC pipe (0.0079); Hf_Total= total energy loss, for this case, a single irrigating line was designed Hf_Total= Dh. The number of drippers was calculated with equation 19 and equations 11, 15 and 16. The percent relative error was determined with equation 12 based on equation 19 (Table 6) (statistical analysis of the number of outlets).