ECONOMIC DESIGN OF PIPE-NOZZLE DISCHARGE LINES DELIVERING FREE JETS Mohamed M. M. Amin
1. INTRODUCTION The pipe-nozzle discharge lines are known for a wide range of applications in practice, they are generally used to have a high-velocity water jet that can be used for firefighting, mining, and power developments (the impulse turbines) Featherstone and El-Jumaily (1983), Streeter and Wylie (1985), Sharp (1985). Most of the studies are based on hydraulic considerations, and in this study, an analytical solution has been reached using a derivative method associated with the economic considerations and comparison requirements Simon (1987), Somaida (1994), Somaida et al. (2011), Somaida et al. (2012). Little is found in the literature concerning the present problem. Most of the previous studies are directed toward having the maximum power of the jet delivered from the nozzle. However, the present study is based on economic considerations satisfying the choosing of the pipe-nozzle diameter ratio leading to the minimum cost of the discharge line. This will be presented within the scope here. However, an analytical solution is derived to solve the problem with comparison requirements John et. al. (2011), Mazzoleni (1994), Joseph et al. (2010), Somaida et al. (2013). In practice, pipe nozzles are widely used in many water and wastewater engineering applications, such as irrigation systems, water supply, and wastewater system arrangements Wright et al. (2003), Poeck (2008). Common applications of nozzles in water and wastewater treatment systems are as follows: Koirala et al. (2021), BETE for Nozzle Performance Engineering (2022): · Evaporative disposal, such as disposal of excess water/chemical solution through evaporation, usually over a large pond. · Foam control, Spray nozzles to break up foam that can cause tank overflow, poor drainage, or other problems. · Mixing and blending tank contents, homogenizing sediment off the tank bottom to aid in transportation and filtration, sweeping solids across the bottom of the tank, and preventing thermal stratification. · Filter nozzles, which can be installed in both open and closed filters, to ensure maximum efficiency with minimum head losses Wright (2003), BETE for Nozzle Performance Engineering (2022). · Also, water jet lines and nozzles are used for sewer cleaning Medan et al. (2017). There are analytical methods for the development of comprehensive costing dealing with the economic sizes of any pipeline Sharp (1985). This can be applied to the present problem, where the pipeline ends with a nozzle acting as a gravity main and should have the optimum diameter ratio Somaida et al. (1994), Somaida et al. (2011), Somaida et al. (2012), Somaida et al. (2013). Poeck et al. (2008), studied a performance evaluation of various nozzle designs for waterjet scaling in underground excavations. Design and optimization of discharge pipelines delivering free jets are of high concern in industry based on economic considerations Renjie (2020), Schwartzentruber et al. (2016). 2. MATERIALS AND METHODS 2.1. Minor nozzle loss It is usually considered that for pipe of length longer
than 1000 diameter (L/D>1000), the error incurred by neglecting minor losses
is less than that inherent in selecting the value of friction factor F Joseph et al. (2010). In the case of an
approach pipe ending with a nozzle, Figure 1, which has a known
assumed loss coefficient, the head loss as associated with the high issuing
velocity head and is therefore not as minor loss. But, in the present study, it
is suggested that the minor loss of the nozzle may be expressed in terms of the
equivalent length of approach pipe (L Equation 1 Where F = friction factor of approach pipe, L Note that from continuity equation, Equation 2 Therefore, the total length of the pipe will be: Equation 3 However, the solution will be built around the friction factor of the approach pipe, rather than minor loss of the nozzle. Figure 1
2.2. Pipe-nozzle discharge line cost The pipe cost C Equation 4 Where, a = pipe cost function and x = pipe cost exponent. However, considering the pipe cost C Where, K = loss coefficient of the nozzle, which is known
by ,
where C 2.3. Derivative optimum for D/d ratio In order to obtain the optimum D/d ratio, use the derivative method except that the cost gradient will be relative to the diameter of the approach pipe D, because the pipe now forms a major part of the scheme. However, for minimum pipe cost, differentiate Equation 5, with respect to D, and equate to zero, with the following assumptions: 1) Constant diameter of nozzle opening d. 2) Turbulent
flow conditions and Reynold’s number ranges from 10 3) Variable coefficient of friction. Differentiate Equation 5 with respect to D and equate to zero, then: Put Equation 6 in the following form: The partial in the L.H.S. of Equation 7, can be written in the form: or , However, Equation 7, can take the form: The form can be evaluated from Von Karman formula for F: . Introducing the diameter ratio, B , or , ؞
Or . The term in this equation is constant. Differentiate F in this equation with respect to d, ؞ , or Substitute by in Equation 8, Put the diameter ratio , in Equation 9 and rearrange for B, then: Where K is the minor loss coefficient of the nozzle = , since , Streeter and Wylie (1985) and C is the flow coefficient of the nozzle. However, the diameter ratio D/d depends on friction factor F of approach pipe, loss coefficient of nozzle K, relative distance L/D, and pipe cost exponent x. Equation 10 can be solved by trial and error. 2.4. Illustrative example In the present illustrative example, the following data are given L = 20, D = 0.1m, relative distance L/D = 200, relative roughness e/D = 0.02 (F = 0.05). In view of the pipe cost exponent x, it was taken 0.45 by Featherstone and El-Jumaily (1983) and 1.03 by Somaida et al. (2012), However, it is taken 2.5 in the present study. The values of C at different D/d are evaluated using the following equation, which is evaluated by linear regression analysis of the data concerning the flow ratio C and the diameter ratio being interpreted from Streeter and Wylie (1985)(“Fig. 8.16”, P. 467). Equation 11 The results obtained by solving Equation 10 by trial and error as shown in Table 1. Graphical solution of Equation 10 based on minimum pipe cost is shown in Figure 2. This figure and Table 2 show that, the solution of Equation 10 is satisfying at average value of D/d = 1.3. Table 1
2.5. Rigidity of the derived equation For this purpose, the total pipe cost C Figure 2
Table 2
3. RESULTS AND DISCUSSIONS 3.1. Evaluation of minimum cost 3.1.1.
Variation of C The data given within the illustrative example are x =
2.5, F = 0.05, and L/D = 200, while the pipe cost factor a is taken 7000, 7400,
and 8000, Table 3. The corresponding
plots of C 1) The plots exhibit similar trends. 2) The
increase of diameter ratio D/d leads to increase of C 3) At
a fixed value of D/d, C 4) All
the plots indicate that the minimum pipe cost C Figure 3
Figure 4
3.1.2.
Variation of C The data given within the illustrative example are: x = 2.5, F = 0.05, a = 7400, and L/D = (100, 200, 300, and
400), Table 4. The corresponding
plots are shown in Figure 5. Investigation of
these plots leads to the following: (A) The plots exhibit similar trends. (B)
The pipe cost C Table 3
3.1.3.
Variation of C The data given within the illustrative example are x = 2.5, a = 7400, and L/D = 200, The variable is e/D (0.01, 0.02, 0.04) which corresponds to F (0.038, 0.05, 0.065), Table 5. The corresponding plots are shown in Figure 6. Investigation of these plots lead to the following conclusions: (A) All the plots exhibit similar trends. (B) In each
plot, C Table 4
Figure 5
Figure 6
3.1.4.
Variation of C The data given within the illustrative example are x = 2.5, a = 7400, and L/D = 200, The variable is e/D (0.01, 0.02, 0.04) which corresponds to F (0.038, 0.05, 0.065), Table 5. The corresponding plots are shown in Figure 6. Investigation of these plots lead to the following conclusions: (A) All the plots exhibit similar trends. (B) In each
plot, C Table 5
Table 6
3.2. Comparison between minimum cost and conventional formulae computing D/d For this purpose, Table 6 is constructed showing
the important parameters to be compared using both formulae, where C Investigation of Table 6 indicates that: (A)
The derived Equation 10
shows a unique value for D/d = 1.3 which satisfies the minimum C Table 7
Figure 7
On the other hand, Table 7 is constructed for
D/d, P Figure 8
Figure 9
Figure 10
Finally, it may be stated that, the results obtained for
computing the minimum cost of pipe-nozzle diameter ratio in discharge lines
delivering free jets, by the derived Equation 10,
indicate the validity of the equation in estimating the economic pipe-nozzle
diameter ratio, D/d, the corresponding power of the jet P 4. CONCLUSIONS and RECOMMENDATIONS The following conclusions can be reached as follows: 1) Equation 10, derived for computing the economic nozzle-pipe diameter ratio D/d in discharge lines delivering free jets, is applicable over practical ranges of relative distance L/D and relative roughness e/D under rough, turbulent flow conditions (in approach pipe). For the time being, L/D ranges from 50 to 500, e/D ranges from 0.01 to 0.05, and Rn from 105 to 106. 2) In
the given illustrated example, Equation 10derived
for D/d ratio, is solved by trial and error, and shows that the economic
diameter ratio is close to 1.3, where the total pipe cost has a minimum value.
Also, the results of C 3) Investigation of Equation 10, shows that the economic diameter ratio D/d depends on; the relative distance L/D, the coefficient of friction F in the approach pipe, the minor loss coefficient of the nozzle K, and the pipe cost exponent x. 4) On
comparison basis between Equation 10
and the conventional formula, at (F = 0.05, L/D = 200), the first shows that at
the economic ratio D/d = 1.3, the pipe cost C 5) On
the other hand, at F = 0.05 and by increase of L/D to 400, still economic D/d =
1.4, P 6) Considering
the friction in the approach pipe, it has the effect of decreasing P 7) It
may be stated that, the evaluation study of Equation 10
reflects the rigidity and reliability of this equation in computing the
economic pipe-nozzle diameter ratio D/d in discharge lines delivering free jets
and for the time being, good results are obtained at F = 0.05 and L/D = 400,
higher PJ and lower C 5. NOMENCLATURE F = Coefficient of friction in the approach pipe, D = Mean diameter of approach pipe, L g = Gravitational acceleration, V D/d = Pipe-nozzle diameter ratio V d = diameter of the nozzle opening. K = Minor loss coefficient of the nozzle, L = Length of approach pipe, C C a = pipe cost function, x = Pipe cost exponent, C = Flow coefficient of the nozzle, L/D = Relative distance, P e = Roughness height in approach pipe, e/D = Relative roughness of pipe, P
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