THE CALCULATING METHOD OF HEAD LOSS COEFFICIENT FOR LOCK CULVERT MANIFOLD

Xu Qinghua

Nanjing Hydraulic Research Institute, Nanjing 210029, China

Abstract: This paper discusses the component of head loss coefficient for lock culvert manifold, and some simple calculating formulas are derived. Compared with the prototype measurements of several existing locks, the calculating results show rather good agreement.

1    INTRODUCTION

Navigation locks have two main types of filling and emptying system, namely, the end filling and emptying system and the longitudinal culvert filling and emptying system. The main part of longitudinal culvert system is the lock culvert manifold with a series of discharge ports. To determine total head loss coefficient or lock discharge coefficient in hydraulic calculation, the head loss coefficient of culvert manifold must be defined. But few data have been published in literatures before. Usually designers used the data of similar projects or of special hydraulic experiments. So it is necessary to have a simple and practical calculating method to determine this head loss coefficient.

2    COMPONENTS OF LOCK CULVERT MANIFOLD

Fig. 1    Sketch of culvert manifold

The total head loss of flow in culvert manifold (Fig.1) consists of friction head loss, diversion head loss and head loss through ports.

The first two losses can be calculated by the conventional method [1], assuming that the discharge of each port is equal .

The friction head loss  between port m and port i can be expressed as:

                        (1)

where,                                            (2)

                         (3)

Set eqns. (2) and (3) into (1), and consider that the number of ports is rather large, we

Have                                        (4)

                                      (5)

Hence,                                                          (6)

where  is the friction head loss coefficient of culvert manifold, L is the total length of manifold, c is Chezy Coefficient, R is the hydraulic radius of culvert, and g is the acceleration of gravity.

 The diversion head loss  between port m and port i can be expressed as:

                               (7)

                             (8)

                                  (9)

where  is the diversion head loss coefficient of culvert manifold.

As to the head loss of flow through port, it is briefly related to culvert-to-port velocity ratio  (Fig.2), and related to types of port as well. For the rectangular culvert with top slit port and sharp inlet edge, this head loss coefficient  can be expressed as follows [2]:

Fig. 2    Sketch of single port

                              (10)

                        (11)

where  is the head loss through port; u and v are the average velocity in port and culvert, respectively.

By the analysis of various experimental and theoretical data in other literatures, the following formula was suggested by author [3]:

                               (12)

Obviously, in formulas (11) and (12), when , . This is the head loss coefficient of the last port of manifold (See Fig.1), which has the same value of a nozzle. Therefore, in equation (12), the first term is a constant that is related to port shape, while the second term reflects the influence of approach velocity in culvert before that port.

3    DETERMINATION OF TOTAL HEAD LOSS COEFFICIENT FOR CULVERT MANIFOLD

There are two methods to calculate total head loss coefficient of culvert manifold

(1) Assume that each port of manifold has equal area  and equal discharge q, and as flow passes through each port of manifold, its head loss is all the same, which means that the friction and diversion head losses between port m and port i must be considered besides the head loss through port itself for port i. But for the first port m, only the head loss through port needs to be taken into account. So from equation (12), we obtain

        (13)

where  is the total port area of manifold,  is the culvert area, and  is the culvert discharge.

From equation (10), we have

Therefore,                  (14)

where  is the head loss coefficient of port m related to , and is also the total head loss coefficient of culvert manifold.

(2) The other calculating method is based on neglect of friction and diversion head losses in manifold. Equal port area for manifold is normally adopted, and this leads to unequal port discharges or velocities. Suppose that all ports have the same size, and a continuous discharge slot is used for calculation (Fig.3) instead of a series of ports,

where            p and v are the pressure head and mean velocity of culvert at section S,

respectively;

and are the pressure head and velocity of culvert at the initial

section separately;

and are the pressure head and velocity of culvert at the end section

respectively;

b is the width of discharge slot;

is the culvert section area;

u is the velocity of port at section S.

As the friction and diversion head losses can be omitted, we have:

Fig. 3    Sketch of culvert manifold for calculating

                               (15)

                            (16)

As                                                        (17)

where  is the port discharge coefficient related to the culvert pressure head before the port, and if the inlet head loss is neglected, we have

                               (18)

Thus,                                       (19)

From formulas (16) and (18), we get

                           (20)

where                                                                    (21)

Thus, we get

                           (22)

Consider boundary conditions

, ,

we have                                                   (23)

                    (24)

As                                       (25)

Form (17) and (18), we have

                      (26)

When           ,

We get        

where E is the energy head of flow at the initial section of culvert.

So                                                       (27)

Setting equation (27) into equation (23), we have

                               (28)

As s=L,

                             (29)

From definition

we get                                                  (30)

 As they are mentioned before,  is the port-to-culvert area ratio, while  is a constant. However, depends also on  in fact. According to the experimental data of culvert manifold with top slit ports, can be expressed as:

                    (31)

Thus                                       (32)

In formula (30), the coefficient  mainly depends on port shape. Therefore, on base of a series of experimental and prototype data of locks, the appropriate values of  for various port shapes can be given as follows:

 For the culvert manifold with side rectangular ports, =0.63 (with sharp inlet edge), =1.15 (with rounded inlet edge), and =1.90 (with bell mouth). For the culvert manifold with top slit ports, =0.80 (with sharp inlet edge and without energy dissipation cover plate), =0.65 (with sharp inlet edge and energy dissipation cover plate), =1.15 (with rounded inlet edge and without energy dissipation cover plate), and =0.75 (with rounded inlet edge and energy dissipation cover plate).

It is interesting to compare the calculating results from formula (14) and formula (32) based on different assumption. Fig.4 shows that for the culvert with top slit ports of sharp inlet edge, a rather good agreement is presented except that small difference exists when  becomes larger.

Fig. 4    Relation between  and

4    DISCUSSION AND CONCLUSIONS

The calculating formulas for head loss coefficient of culvert manifold given by this paper are quite simple. Only two major culvert manifolds are discussed, i.e. the rectangular culverts with side ports and with top slit ports.

As the main factors that influence the head loss coefficient of manifold are port-to-culvert area ratio and port type, the above-mentioned formulas can only give an approximate value.

For the port with sharp inlet edge, the value of  is more or less precise; but for the port with rounded edge, the given value of  is appropriate only for certain rational range of , i.e. 0.5~1.0 or so, because ( =inlet radius, b =port width) may be changed from 0.2 to 2.0. Otherwise, the calculating error might be larger.

Table 1    Comparison between calculated and measured values of head loss coefficient for culvert manifold

As to the value of , designers usually take 0.8~1.2. In this case, the error should not be significant.

In Table 1, some comparisons are made between calculated and measured head loss coefficient of culvert manifold for several locks. In Tab.1, all lock culvert manifolds have top slit ports with energy dissipation cover plates. It shows enough precision of the result from the calculating formula of total head loss coefficient for lock culvert manifold (equation 30). So it can be used in practice.

References

[1]    Качановский, Б.Д,, ГИДРАВНИКА　 СУДОХОДНЫХ ШЛЮЗОВ, 1951, Речиздат, 1951.

[2]    Zhang RuiKai, Xu Qinghua, Computer model and test verification of unsteady flow for a lock manifold system, Journal of Nanjing Hydraulic Research Institute, 1987, (3) (in Chinese).

[3]    Xu Qinghua, Lian Hengduo, Hydraulic research of lock culvert manifold[R], Nanjing Hydraulic Research Institute, 1965 (in Chinese).