Energy loss at a manhole which is at a junction of storm sewers or combined sewers is important for accurately estimating its discharge capacity. However, even in Japan, the energy loss at the manhole is usually ignored in designing sewers and flood inundation analysis. One of the reasons for the ignorance is difficulty to formulate the energy loss at the manhole because the number of variables which must be considered increases as the number of sewers connected to the manhole increases. The authors have formulated the energy loss of a four-way circular manhole with three inflow sewers and one outflow sewer at crossroad. The formula is applicable only to the equal flow rates in two opposite lateral inflow sewers. In this paper, a new formula for the energy losses is proposed based on experimental data on a four-way circular manhole with different flow rates on three inflow sewers. It shows that the energy losses calculated with it almost reproduce the measured ones.

  • Effects of the flow rate ratio and the diameter ratio of the inflow and outflow pipes were clarified by experiments.

  • The energy loss coefficients of a four-way circular manhole were formulated by seven dimensionless variables.

It is known that the energy loss in a manhole at a junction of storm sewers or combined sewers cannot be neglected in designing sewers and flood inundation analysis. In Japan, however, even now, the energy loss in the manhole is not taken into consideration in the design of the sewers (Sewerage facility planning/design guidelines, 2019). In almost all cases, it is neglected even in the inland flood analysis. In a two-way or three-way junction manhole, various studies were conducted to devise the energy loss characteristics in the manhole for the pipe flow, and some energy loss calculation formulae were developed (Sangster et al. 1961; Lindvall 1984; Marsalek 1985,; Kerenyi & Jones 2006; Arao et al. 2016; Li et al. 2021). In a three-way junction manhole, the pressure loss coefficient for the main straight-through pipe and the lateral pipes was cunducted in terms of the continuity equation and the momentum conservation equation at the entrance and the exit of the manhole (Sangster et al. 1961). Energy losses at a surcharged two-way junction manhole with a main inflow pipe and a 90° lateral inflow pipe were measured by Lindvall (1984). Detailed experiments were conducted by changing the flow rate of a main straight-through pipe and a lateral pipe for both open channel flow and pipe flow (Marsalek 1985). An experiment was conducted in which the horizontal connecting angle between the lateral inflow pipe and the outflow pipe was changed in a three-way junction circular manhole, and the validity of the calculated value of the energy loss coefficient by the formula devised by Arao et al. (2016) was verified (Li et al. 2021).

The authors conducted a consecutive experimental study on the energy loss in a manhole and formulated the energy loss in a manhole with one to three inflow sewers connected to it based on the experimental results (Arao & Kusuda 2006; Arao et al. 2016; Arao et al. 2019). However, the more the number of inflow sewers connected to a manhole increases, the more the number of variables that must be taken into account also increases in the energy loss estimation, and therefore, it results in making formulation difficult. In a four-way junction manhole laid under the crossroad intersection, which is the target of this study, there are many variables to be considered, and there are few studies on it. Some of them were formalized; however, the formulae have limitations on the range of application and inadequate ways of thinking in formulating (Hare & O'Loughlin 1991; Stein et al. 1999; UDDM 2009; Ryu et al. 2016).

The energy loss calculation formulae in Urban Drainage Design Manual (UDDM) seem to be the widest range of application on the energy loss at a four-way manhole. However, there is a problem that a negative value of the energy loss coefficient in the manhole cannot be calculated by the formula (UDDM 2009). The hydraulic experimental apparatus which can change the manhole shapes (square, circle) in surcharged four-way combining manholes and flow ratios (lateral inflow flow rate/outflow flow rate) were installed to estimate the head loss coefficients. In the experiments, the diameter of inflow pipes and the diameter of outflow pipe are the same (constant), and two inflows were varied. The variation of head losses were strongly influenced by the lateral inflow because the head loss coefficient increases as the flow ratio increases (Ryu et al. 2016).

Thus, at present there is no formula which gives an appropriate estimation. In order to overcome this problem on conventional design work, the authors have already formulated the energy loss on a four-way junction circular manhole at a crossroad (Arao et al. 2019). However, this calculation formula had still limitations on the calculation under the condition that the flow rates of the two opposite lateral inflow sewers are equal. By changing the planar arrangement of three inflow pipes with different pipe diameters and changing the inflow flow rates of the three inflow pipes, the energy loss characteristics of the four-way junction circular manhole were experimentally obtained (Okamoto et al. 2020). Furthermore, a formula for calculating the energy loss coefficient of the manhole with a drop height between one of the inflow pipes and an outflow pipe was devised. In this study, the measured and calculated values of the energy loss coefficient with the drop and without the drop are compared (Okamoto et al. 2021).

Therefore, in this paper, modified calculation formulae are constructed with a planar arrangement of three inflow sewers, changes in pipe diameter and changes in flow rate on the basis of experimental results on the energy loss in a four-way junction circular manhole.

The structural variables to be considered in this study are manhole diameter, pipe diameter, and horizontal joint angles of inflow pipes. The shapes of a manhole and pipes are circular, which is often used in the practical field. Water depth in the manhole is not of importance for the surcharged case, that is, pipe-flow. Therefore, the variables to be considered when formulating the energy loss coefficient Kei of a four-way junction manhole under surcharged conditions are as given in Equation (1). Where b is the manhole diameter, Di is the diameter of inflow pipe i, Do is the diameter of outflow pipe, Qi is the flow rate of inflow pipe i, Qo is the flow rate of the outflow pipe (Qo=Q1 + Q2 + Q3); θi is the horizontal connecting angle between the inflow pipe i and the outflow pipe. As a manhole at a crossroad intersection, the angle between the inflow pipe 1 and the outflow pipe is set to θ1 = 180°, and the inflow pipes 2 and 3 are set to θ2 = θ3 = 90° as shown in Figure 1. Moreover, the pipes are connected to the manhole in the way that is often used in the practical field without considering the drop between inflow and outflow pipes. In this study, as pressurized flow, that is, pipe-flow is targeted, the measurement results of the horizontal pipe gradient are the same for any slopes, and therefore, in the experiments, the pipes were equipped in horizontal. From the above, the manhole energy loss coefficient Kei is composed of a function of the seven dimensionless numbers as shown in Equation (1). When the Kei of Equation (1) is obtained by experiments, tap water is used as flowing water in the pipes. At actual sites, SS (suspended solids) concentration of flowing water in the pipes increases in the early stage of rainfall, but the SS concentration decreases with time elapsing, therefore the flowing water is considered to be almost the same as fresh water.
formula
(1)

The energy loss coefficient, Kei and the pressure loss coefficient, Kpi at a four-way manhole are defined by Equations (2) and (3), respectively (refer to Figure 2). If the energy loss coefficient Kei is obtained, the pressure loss coefficient Kpi can be obtained from Equation (3), and therefore, the manhole water level can be estimated. Considering application to an actual watershed, the energy head at the downstream end of the inflow pipe can be calculated by adding the energy loss at the manhole to the energy head at the upstream of the outflow pipe. As a result, the boundary conditions are set at the downstream end of the inflow pipe over the manhole when viewed from the outflow pipe. The definition of these loss coefficients is a common definition used all over the world, and researchers can compare and discuss their research results. Equation (4) shows the loss coefficient K that is the composite of the energy loss coefficients for three inflow pipes. The energy loss coefficient K is obtained by substituting the continuity equation to the energy conservation equation between the inflow pipe and the outflow pipe (Li et al. 2021).

On the subscript i of Kei and Kpi, the subscript 1 shows the flow direction from inflow pipe 1 to the outflow pipe, and subscript 2 shows the flow direction from inflow pipe 2 to the outflow pipe, where ΔE is the energy loss at the manhole; ΔP is the pressure loss at the manhole; Vi is the cross-sectional mean flow velocity in the inflow pipe; V0 is the cross-sectional mean flow velocity in the outflow pipe; and g is the gravitational acceleration.
formula
(2)
formula
(3)
formula
(4)

Experimental apparatus

The outline of the experimental apparatus used in this study is shown in Figure 3. The manhole and the pipes were made of transparent acrylic resin. The experimental model is a scale of 1/5 of the real scale (manhole diameter is 90 cm and pipe diameter is 25 cm), and the manhole shape of cylinder is simpler than the prototype. The invert corresponding to 1/2 of the pipe diameter generally used in Japan is set up in the manhole bottom.

Experimental procedure

  • (1)

    Flow in the three pipes was regulated pressurized flow by three flow rate control valves. The measurement accuracy of the flow rate has a range of error within ±1%. The flow rate was calculated by dividing the volume of water stored in the water storage tank by the measurement time. The water storage tank was installed under the overflow water tank connected to the end of the outflow pipe. The water stop valve was attached to this water storage tank for controlling the outflow of water.

  • (2)

    As shown in Figure 2, the distance from the inside wall top of the higher elevation inflow pipe to the water surface is the water depth h in the manhole. The water depth h in the manhole was adjusted by the weir in the overflow tank at the end of the outflow pipe.

  • (3)

    The water depth h in the manhole was measured by scales installed at four points along the wall outside the manhole. And the value of h was defined by averaged value of those. The pipe top connection was used in this study because it is often used in the field. So, the top of the inner wall of the pipe is set at 0 cm even if the pipe diameter connected to the manhole changes. Based on this, the water depth of the manhole was measured.

  • (4)

    As shown in Figure 2, the pressure heads of the pipe are measured twice with manometers installed in inflow pipe 1, 2, 3 and the outflow pipe, respectively, at three places. When the depth of water in the manhole is low, the water surface of the manometer also fluctuates up and down because the water surface in the manhole changes greatly by the generation of large-scale eddies. When the water depth in the manhole is high, the water surface of the manometer hardly changes. In either case, the average position of the water surface of each manometer was read. And the average of the two measurements was used. The energy loss ΔE at the manhole is obtained by calculated energy grade line; its line was calculated by adding velocity head (V2/2 g) to the measured pressure head (refer to Figure 2). In addition, the energy loss coefficient Kei is calculated by Equation (2) and the pressure loss coefficient Kpi is calculated by Equation (3). The average value of the energy loss coefficients when the water depth ratio h/Do is larger than 2 was used as representative value of loss coefficient at each flow rate ratio. As described later, the influence of water depth in the manhole was not considered when formulating the energy loss coefficient in this study. The reason is that changes in water depth have little effect on the loss coefficient (Li et al. 2021), and it is very difficult to formulate the effect of water depth for each condition. Also, it is considered that there is almost no difference even if the average value of the measured values for each water depth is adopted, especially at high water levels of manholes.

Experimental conditions

The combination for the pipe diameter (D1, D2, D3 and Do) including the previous research is indicated in Table 1 (see supplementary information). Three types of combinations for pipe inner diameter were used in this study; the three types were named Type E, Type F and Type G, respectively. In each type, the flow rate Q1 of the horizontal inflow pipe 1, the flow rate Q2 of the horizontal inflow pipe 2, and the flow rate Q3 of the horizontal inflow pipe 3 were set to various values. The 46 cases of flow rate set in Type F are shown in Table 2. In Type E and Type G, the flow rate condition is slightly reduced compared to the flow rate condition of Type F. In the previous study (Arao et al. 2019), as shown in Table 3, only six kinds of flow rate ratio Qlat/Qo (Qlat = Q2 + Q3, Q2 = Q3) were simply considered. The water depth h in the manhole was changed in the range from 5 to 30 cm in each flow rate ratio. When the inner diameter of the inflow pipe is 3 cm and the inflow flow rate exceeds 1.0 L/s (liter/s), the friction loss in the pipe becomes very large and the hydraulic gradient line exceeds the height of the upstream tank. Therefore, the maximum amount of inflow rate into the pipe was determined to be 1.0 L/s when the pipe inner diameter is 3 cm. The inflow rate into the other pipes was set again so that the flow rate ratio is the same as Table 2. This means that, for example, the target flow rate ratio is obtained by resetting Q3 = 1.75 L/s (more than 1.0 L/s) to half of flow rate Q3 = 0.875 L/s in the case No. 10 of the flow rate setting under the condition of Q1 = 0.25 L/s in Table 2. In addition, Tables 2 and 3 indicate the measured values and standard deviation of the energy loss coefficient. As shown in Tables 2 and 3, it can be seen that the standard deviations under most flow rate conditions are small and then the influence due to water depth change is small.

The relationships between the energy loss coefficient Kei and the flow ratio Q2/Q0 for the combination of pipe diameters of Type E, Type F, and Type G as presented in Table 1 are shown in Figures 46. In each Figure, the flow rate of the inflow pipe 1 is constant, while the flow rates for inflow pipe 2 and the inflow pipe 3 are varied. As can be seen from Figure 4(a), when Q1/Qo = 0, the energy loss coefficient Ke1 is small because water does not flow in the inflow pipe 1, even if Q2 and Q3 are changed, and Ke1 takes a constant value near 0.4. Regarding the inflow pipe 2, the loss coefficient Ke2 decreases in a range of 0≦Q2/Qo≦0.5, and it increases again as Q2/Qo increases beyond 0.5. This is because the velocity head increases as Q2 increases. At Q2/Qo = 1, water flows only in the inflow pipe 2; therefore, the value of Ke2 is the same as the loss coefficient of the 90-degree bend. Further, at Q2/Qo = 0, when water flows only in the inflow pipe 3, the loss coefficient Ke3 of the inflow pipe 3 becomes large. The value of Ke3 decreases as Q2/Qo increases. Furthermore, when Q2/Qo = 0, the pressure in the inflow pipe 2 is increased because Q3 becomes the maximum, and thereby, the loss coefficient Ke2 is affected by this effect. As shown in Figure 4(a)–4(c), as the flow rate Q1 in the inflow pipe 1 increases, the energy head rises and the loss coefficient Ke1 increases. Since the flow rates of the inflow pipe 2 and the inflow pipe 3 decrease, the loss coefficient Ke2 and Ke3 decrease conversely.

For Type F, Figures 4(d) and 5(a)–5(d) show the change of energy loss coefficient when the flow rates through inflow pipe 2 and inflow pipe 3 are changed and the flow rate in the inflow pipe 1 is kept constant. As shown in Table 1, the diameters for inflow pipe 1 and inflow pipe 3 are mutually interchanged between Type F and Type E; however, the position of the inflow pipe 2 is not changed. In the case of Type F, at Q1/Qo = 0, Type F has a larger energy loss coefficient for all three types as compared with Type E. At this flow rate ratio, Q3 is the maximum, and the flow from the inflow pipe 3 into the manhole causes a pressure increase in the inflow pipe 1 and the inflow pipe 2. Therefore, when Q1/Qo and Q2/Qo = 0, Ke1 is about 0.8, which is about twice the value of Type E. Moreover, Ke2 is about 8 and Ke3 is about 16, these indicating that the energy loss coefficient is extremely large. Furthermore, the change in Ke2 and Ke3 with the increase of Q2/Qo for Type F is similar to Type E. As shown in Figures 4(d) and 5(a)–5(d), like Type E, as the flow rate Q1 in the inflow pipe 1 increases, the energy head of the inflow pipe 1 rises, and it results in the increase in the loss coefficient Ke1, while the loss coefficients Ke2 and Ke3 decrease, conversely.

The change in energy loss coefficient for Type G is shown in Figure 6(a)–6(f), when the flow rates through the inflow pipe 2 and inflow pipe 3 are changed and the flow rate in the inflow pipe 1 is kept constant. As shown in Table 1, Type G can be obtained by replacing the arrangement for the inflow pipe 1 and inflow pipe 2 with Type F. However, the position of the inflow pipe 3 is not changed. Therefore, the inflow pipe 3 has a smaller diameter of 3 cm, and this condition has a great influence on the pressure rise in the inflow pipe 2 facing the inflow pipe 3. As can be seen from Figures 4(d) and 6(a), Type G and Type F clearly differ in the magnitude of the energy loss coefficient Ke2 of the inflow pipe 2, and the energy loss coefficient Ke2 for Type G is larger at Q1/Qo = 0 and Q2/Qo = 0.

The reason for this is that Type G is more affected by the flow from inflow pipe 3 because the diameter of inflow pipe 2 is 4 cm, which is smaller than that of Type F. It can be seen that as Q2/Qo increases, the difference between both of Ke2 in Type F and Type G decreases. As shown in Figure 6(a)–6(f), when Q1/Qo increases, the energy loss coefficients Ke2 and Ke3 gradually decrease and reach their minimum. However, the change of Ke1 is very small. This is because the diameter of inflow pipe 1 of Type G is the largest.

The energy loss coefficient of a four-way junction circular manhole was formulated in this study. An overview of the authors’ previous research (Arao et al. 2019) related to the formulation is given as follows.

Previous research on formulation (Arao et al. 2019)

The authors proposed Equations (5)–(15) based on the experimental results for Type A to Type D with all inflow pipes of same diameter so that the measured values can be reproduced, and these equations can be used to calculate the energy loss coefficients. Various coefficients and multipliers were introduced in these equations so that the functional form could represent the experimental results. For more details of this procedure, refer to Arao et al. (2019). The energy loss coefficients for the straight inflow pipe and the two transverse inflow pipes orthogonal to it are Ke1, Ke2 and Ke3, respectively, and these are calculated by Equations (5), (9) and (14). It should be mentioned that all of these equations are mainly the sum of the following two terms. The first one considers the influence of the manhole diameter ratio (b/Do), the flow rate ratio (Qlat/Qo) (Qlat = Q2 + Q3, Q2 = Q3), and the manhole bottom surface shape. The value CQi is calculated when the inflow pipe and outflow pipe have the same diameter. The second term CDi counts the pipe diameter ratio (Do/Di), and this term is added when the inner diameters of the inflow and outflow pipes are different. Similarly, Equations (8), (11) and (15) are obtained by the summation of the following two terms. The first term refers to the coefficient for the pipeline that is to be calculated, while the second term is for an other pipeline that may influence the flow rate ratio of the pipeline to be calculated. CB is a coefficient related to the shape of the bottom of the manhole. Here, CB is 0.95 because an invert corresponding to 1/2 of the pipe inner diameter is attached to the bottom of the manhole. In the above-mentioned studies, Equations (5)–(15) were constructed by simply setting only six types of flow rate ratio Qlat/Qo. Equations (12) and (13) reflect the pressure increase due to the difference in velocities between two inflow pipes with different diameter, which are positioned laterally opposite each other, and here, Qlat/Qo is greater than 0.5.

  • (A)
    Formulation of Ke1
    formula
    (5)
    formula
    (6)
    formula
    (7)
    formula
    (8)
  • (B)
    Formulation of Ke2
    formula
    (9)
    formula
    (10)
    formula
    (11)
    formula
    (12)
    formula
    (13)
  • (C)
    Formulation of Ke3
    formula
    (14)
    formula
    (15)

Development of formula considering flow rate changes in three inflow pipes

Here, we formulate the energy loss coefficient of a four-way junction circular manhole based on the experimental data obtained by varying the flow rates in three inflow pipes described above. In the previous study (Arao et al. 2019), the formulation was performed under the condition that the flow rates in two horizontal lateral inflow pipes were the same. Calculation formulae proposed below take into account the case in which the flow rates in two lateral inflow pipes facing each other are different, and the calculation formulae are shown in Equations (16)–(26). Here, if should be mentioned that Equations (17) and (18) for obtaining the energy loss coefficient Kei are the same as Equations (5) and (6) in the previous research (Arao et al. 2019). However, the only difference is that the flow rates in lateral inflow pipes 2 and 3 facing each other are not the same. In this study, as experiments were performed at different flow rates, Q1/Qo, Q2/Qo, and Q3/Qo were used as variables for the flow rate ratio.

Equations (19) and (22) are approximated by a quadratic curve that was obtained from Equation (16) based on the measured value Kei with D1 = D2 = D3 = Do = 5 cm as shown in Figure 7. Equation (25), obtaining CQ3 for the inflow pipe 3, has the same equation form as that of CQ2 for the opposing inflow pipe 2. To decide the coefficients and multipliers related to Equations (20), (23) and (26), first of all, a combination of Type A to Type D pipes in which three inflow pipes have the same diameter was used. Secondly, a target was set that the summation of squares of the difference between the calculated values and measured values for energy loss coefficients became the minimum. Therefore, the coefficients and multipliers were finely adjusted so that the calculated values for Type E to Type F in which the inflow pipes had different diameters were close to the experimental values of these types.
formula
(16)
  • (1)
    Formulation of Ke1
    formula
    (17)
    formula
    (18)
    formula
    (19)
    formula
    (20)
  • (2)
    Formulation of Ke2
    formula
    (21)
    formula
    (22)
    formula
    (23)
  • (3)
    Formulation of Ke3
    formula
    (24)
    formula
    (25)
    formula
    (26)

In the above formulae, when Q1 = 0 or Q2 = 0 or Q3 = 0, the formula can be applied to a three-way manhole. Further, when Q2 = Q3 = 0 or Q1 = Q2 = 0 or Q1 = Q3 = 0, it can be applied to a two-way manhole.

Verification of formulation

  • (1)

    Ke1

Figures 8(a)–8(h) and 9(a)–9(c) show a comparison between the energy loss coefficient Ke1 obtained from Equations (17)–(20) and the measured values. As can be seen from the figures, Figure 8(a) and 8(b) confirm the validity of the formulae for four types of measured values in the previous study (Arao et al. 2019), in which inner diameters of the three inflow pipes were the same. It can be seen that the energy loss coefficients can be approximated through Figure 8(a) and 8(b) with a good accuracy. Figure 8(c)–8(h) show the comparison of calculated and measured values when inner diameters of the inflow pipes are all different. Figure 8(d), 8(f) and 8(h) were evaluated only by the measured values except the inflow rate Qi = 0 L/s under the experimental conditions of each type. This is because it is difficult to assume that no water flows in any one of the three pipelines when considering the actual rainfall. As shown in Figure 8(c) and 8(d), in Type E (3-5-4-6) when the measured energy loss coefficient Ke1 is less than 0.5, the calculated energy loss coefficient Ke1 is over the measured value. However, when the measured energy loss coefficient Ke1 exceeds 1, the formulae can generally produce a value close to the measured value. In Figure 8(c) and 8(d), the reason why the calculated value of the energy loss coefficient Ke1 does not increase with the increasing measured value when Ke1 is larger than 3 is as follows. In the calculation formula of Ke1, the energy loss coefficient changes according to Equations (19) and (20), the first term on the right side of Equation (20) is constant (Q1/Qo = 0.5), and when Q2/Qo increases, CQ1 of Equation (19) increases. On the other hand, Q3/Qo decreases and the third term decreases. As a result, Equation (17) did not show any change in the loss coefficient due to changes in the flow rates. Further analysis of the coefficients and multipliers in Equation (20) is required to bring the calculated values closer to the measured values. On the other hand, as shown in Figure 8(e)–8(h), the difference between the calculated value and the measured value is also slightly large in Type F (4-5-3-6) and Type G (5-4-3-6) under a different planar arrangement of the inflow pipe. Here, it should be noted that if only the measured value excluding the inflow Qi = 0 L/s is considered, the difference between the calculated value and measured value becomes small under these experimental conditions. The correlation between the calculated values and the measured values is not good in Figure 8(g). This is a further research subject.

Figure 9(a)–9(c) show the relationship between the flow rate ratio Q2/Qo and the energy loss coefficient Ke1 for Q1/Qo = 0, 0.25 and 0.375, respectively. The calculated value for Type E is slightly larger than the measured value when Q1/Qo = 0, as shown in Figure 9(a). For Type F and Type G, the calculated values are almost the same as the measured values when Q2/Qo is in a range of 0–0.4. As Q2/Qo exceeds 0.5, the calculated value is slightly larger than the measured value. When Q1/Qo = 0.25, it can be seen that though the calculated values for Type E are slightly different from the measured values, the calculated values are almost the same as the measured values for Type F and Type G. On the other hand, at Q1/Qo = 0.375, the calculated values for Type E and Type F are slightly smaller than the measured values; however, the calculated values can reproduce the measured values for Type G.

  • (2)

    Ke2

Figure 10(a)–10(h) show a comparison between the energy loss coefficient Ke2 obtained from Equations (21)–(23) and the measured values. As can be seen from these figures, both in cases where the inner diameters of three pipes are the same and where they are different, the calculated values are fairly matched with the measured values, and therefore, it can be said that the calculation can generally produce the measured values. On the other hand, in Figure 10(d), 10(f) and 10(h), the calculated values are compared with the measured values excluding the amount of inflow Qi = 0 L/s for each type when the inner diameters of the three inflow pipes are different. As can be seen from these figures, the calculated values and the measured values are close values except for the condition that no water flows in any of the pipelines.

Figure 11(a)–11(c) show the relationship between the flow rate ratio Q2/Qo and the energy loss coefficient Ke2 when Q1/Qo = 0, 0.25, and 0.375, respectively. When Q1/Qo = 0 and Q2/Qo≦0.3 in Figure 11(a), the calculated values of Type F are larger than the measured values, and the goodness of fit is a little poor; while when Q2/Qo > 0.3, the calculation well reproduces the measured values. Moreover, for Type E and Type G, the calculation reproduces the measured values. When Q1/Qo = 0.25 and 0.375, the measured values can be almost reproduced for all types except in the vicinity of Q2/Qo = 0.

  • (3)

    Ke3

Figure 12(a)–12(h) show a comparison between the measured energy loss coefficient Ke3 and the values obtained by Equations (24)–(26). From these figures, it can be seen that the calculated values generally can reproduce the measured values whether the diameters of the three inflow pipes are the same or different. Figure 13(a)–13(c) show the relationship between the flow rate ratio Q2/Qo and the energy loss coefficient Ke3 at Q1/Qo = 0, 0.25, and 0.375, respectively. The calculated values can well be reproduced by these equations that are formulated based on the measured values under any inflow pipe diameters and any flow rate conditions in this experiment.

Comparison of calculated results with this research

Here, the values for energy loss coefficients calculated by Equations (16)–(26) in this study are compared to measured values, and they are shown in Figure 14(a)–14(c). The horizontal axis shows the total flow rate Qlat (=Q2 + Q3, Q2 = Q3) of the two horizontal inlet pipes normalized by the flow rate Qo of the outlet pipes. The calculated values in this study are almost the same as those in the previous study (Arao et al. 2019), and therefore, the usefulness of the formulae (16) to (26) was confirmed.

Verification in a three-way junction manhole using experimental data from Li et al. (2021) 

Comparison of the measured values of the energy loss coefficient by Li et al. (2021) in the three-way junction manhole where one inflow pipe joins the straight-through pipe and the calculated values by the formula obtained in this study.

There are no experiments other than this study under the condition that the diameter of three inflow pipes are different. In order to verify this formula, the measured value of the energy loss coefficient by Li et al. (2021) in the three-way junction manhole where the lateral inflow pipe joins the straight-through pipe is compared with the calculated value by the formula obtained in this study. The dimensions of the manhole and conduit used by Li et al. (2021) are manhole diameter b = 0.15 m and pipe diameters D1 = D2 = Do = 0.05 m. D1 is the diameter of the straight-through pipe in the upstream of the manhole, D2 is the diameter of the lateral inflow pipe connected to the straight-through pipe at an angle of 90 degrees, and Do is the diameter of the outflow pipe.This condition for pipe diameters is the same as Type A in this study.

Figure 15 shows the comparison results between the measured values and the calculated values. In the straight-through pipe, the calculated values become smaller as the flow rate ratio Q2/Qo increases beyond 0.33. It is presumed that this decrease in the energy loss coefficient is due to the calculation in Equation (19); therefore, further analysis is required in the future including other formulae. On the other hand, the calculated values of the energy loss coefficient for the lateral inflow pipe almost reproduce the measured values at any flow rate ratio. The Reynolds number of the inflow pipe is 25,300 in the experiment of Li et al. (2021), and that value is 6,400–51,000 in.this study, and the Reynolds number is sufficiently large and has no effect. Since other research institutes have not conducted experiments with three inflow pipes having different diameters as in this research, it is not possible to verify the calculation formula obtained in this research.

New formulations for energy loss coefficents in a four-way junction circular manhole with three inflow pipes and one outflow pipe were developed based on experimental results. These new formulae are available to calculate the energy loss coefficients under the consideration of flow rate changes in three inflow pipes, and the calculated values reproduced the measured values under almost all experimental conditions. A brief summary of the findings obtained in this study are as follows.

  • (1)

    The energy loss caused in the manhole was investigated experimentally by changing the diameters of three inflow pipes and changing the planar arrangement of the inflow pipes.

  • (2)

    The calculation formulae for energy loss coefficients Ke1, Ke2, Ke3 in a manhole connected with three inflow pipes and an outflow pipe were newly contrived based on the experimental results obtained in the previous study and in the present study.

  • (3)

    In these formulae, the inner diameter of the manhole, the inner diameters of three inflow and one outflow pipes, and the flow rates of three inflow pipes were considered as dimensionless parameters.

  • (4)

    As a result obtained from these formulae, under variously changing flow rate ratios of three inflow pipes, it was shown that the calculated values can adequately reproduce the measured values.

In the future, we would like to consider the drop height between an inflow pipe and an outflow pipe. Finally, if the energy loss formulae developed in this study are used for flood analysis, a more accurate prediction could be performed. Moreover, the flow capacity of storm sewers including manholes can be accurately grasped with these equations.

All relevant data are included in the paper or its Supplementary Information.

The authors declare there is no conflict.

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