CN106503344A - The method for solving of open channel canal for water conveyance and its optimal hydraulic cross-section under a kind of ice sheet - Google Patents

Abstract

The method for solving of open channel canal for water conveyance and its optimal hydraulic cross-section under a kind of ice sheet, canal for water conveyance includes canal bottom, channel left side slope, slope, left levee crown and right levee crown on the right of channel, the left and right sides at canal bottom is connected with the lower end on slope on the right of channel left side slope and channel respectively, the upper end on channel left side slope is connected with left levee crown, and on the right of channel, the upper end on slope is connected with right levee crown；Under ice sheet, the section of open channel canal for water conveyance is trapezoidal cross-section, the method for solving when breadth depth ratio of optimal hydraulic cross-section is difference when the upper strata ice sheet zone of influence is identical with the current mean flow rate of lower floor's canal bed zone of influence for optimal hydraulic cross-section initially sets up solving model, then the differential equation of open channel canal for water conveyance optimal hydraulic cross-section under ice sheet is solved, the breadth depth ratio of open channel canal for water conveyance optimal hydraulic cross-section under ice sheet is finally calculated.The present invention is a kind of narrow deep formula channel, can have maximum conveyance capacity in ice age water delivery, it is possible to use geothermal energy, reduce ice face radiating, it is to avoid ice berg, the generation of ice dam accident.

Description

Water delivery channel of open channel under ice cover and solving method of hydraulic optimal section of water delivery channel

Technical Field

The invention relates to a water delivery channel, in particular to a water delivery channel of an open channel under an ice cover and a method for solving the hydraulic optimal section of the water delivery channel, and belongs to the technical field of irrigation area water delivery channel planning and design.

Background

The hydraulic optimum cross section is a cross section which maximizes the flow rate of water delivered from the open channel (or minimizes the cross-sectional area of water delivery when the flow rate is constant) when the area is constant. Hydraulically optimized sections are widely used in engineering practice. The hydraulic optimal section (also called optimal hydraulic section) is based on the hydraulic optimal section obtained under the condition of free water surface (without ice cover), and the optimal width-depth ratio is(wherein b is the width of the bottom of the trench, h is the depth of the water, and m is the coefficient of the slope). The optimum width-depth ratio of the trapezoid cross section commonly used in engineering is only related to the slope coefficient m, but not to the roughness and the longitudinal slope.

However, as the contradiction between water resources is increasingly prominent, in order to avoid the conflict with agricultural irrigation water in summer, many channels are designed to mainly deliver water under the ice cover in winter, such as yellow-leading and green-saving projects, and the water delivery period is designed to be 10 months to 3 months in the next year. At present, water delivery in winter becomes an important water delivery mode.

The known open channel hydraulic optimal section is designed according to the ice-free water surface, and the change of the wet cycle caused by an ice cover is not considered. However, in the actual winter, the water surface is frozen during water delivery in the ice period, the existence of the ice cover increases the wet cycle and resistance, the water delivery capacity is reduced, and the optimal section designed according to the non-ice period (free water surface) is not the optimal section when water is delivered under the ice cover. If the conventional hydraulic optimum section is applied to the under-ice-cover water delivery channel mainly for winter water delivery, it is obviously not suitable, and therefore, it is urgently required to design a hydraulic optimum section capable of increasing the under-ice-cover water delivery flow rate.

Disclosure of Invention

Aiming at the defects of the prior art, the invention provides the trapezoidal water conveying channel of the open channel under the ice cover and the solving method of the hydraulic optimal section of the trapezoidal water conveying channel, which can have the maximum flow capacity during water conveying in the ice season, reduce the heat dissipation of a water body, keep the water temperature and better utilize geothermal energy to reduce the icing, thereby avoiding the accidents of ice plugs and ice dams.

The technical scheme adopted for solving the technical problems is as follows: an open channel water delivery channel under an ice cover is characterized by comprising a channel bottom, a channel left side slope, a channel right side slope, a left bank top and a right bank top, wherein the left side and the right side of the channel bottom are respectively connected with the lower ends of the channel left side slope and the channel right side slope, the upper end of the channel left side slope is connected with the left bank top, and the upper end of the channel right side slope is connected with the right bank top; the section of the water delivery channel of the open channel under the ice cover is a trapezoidal section, and the average flow velocity V of water flow in the influence area of the ice cover on the upper layer_iAnd the average flow velocity V of the water flow in the lower layer channel bed influence area_bWhen water is delivered under the ice cover under the same condition, the width-depth ratio of the hydraulic optimal section of the water delivery channel of the open channel under the ice cover is as follows:average flow velocity V of water flow in upper ice cover influence area_iAnd the average flow velocity V of the water flow in the lower layer channel bed influence area_bWhen carrying out ice lid water delivery under inequality circumstances:wherein b is the width of the bottom of the channel, h is the depth of the water, m is the coefficient of the side slope, i is the gradient of the longitudinal slope of the bottom of the channel, n_bIs the roughness of the trench, n_iIs the ice cover roughness, Z₁＝n_i ^3/2(∈^5/2+∈^3/2)n_b ^3/2+∈⁴n_i ³+n_b ³，

Preferably, the width-depth ratio of the hydraulic optimal section of the ice cover under-ice open channel water delivery channel is solved according to the following optimization model of the hydraulic optimal section of the ice cover under-ice open channel water delivery channel:

the objective function is to minimize the cross-sectional area:

A＝(b+hm)h (1)

the constraint condition is that the specified flow Q needs to be passed under the flow area:

wherein Q is the water delivery flow of the open channel under the ice cover, b is the width of the channel bottom, h is the water depth, m is the coefficient of the side slope, i is the gradient of the longitudinal slope of the channel bottom, n_bIs the roughness of the trench, n_iFor ice cover roughness, Φ is the constraint function.

Preferably, the water delivery channel of the open channel under the ice cover comprises a digging channel, a fill channel or a fill digging channel.

The invention also provides a method for solving the hydraulic optimal section of the water delivery channel of the open channel under the ice cover, which is characterized by firstly establishing a solution model of the hydraulic optimal section of the water delivery channel of the open channel under the ice cover, then obtaining a differential equation for solving the hydraulic optimal section of the water delivery channel of the open channel under the ice cover by using a Lagrange multiplier method, and finally calculating the width-depth ratio of the hydraulic optimal section of the water delivery channel of the open channel under the ice cover according to the differential equation.

Further, the method for solving the hydraulic optimal section of the water delivery channel of the open channel under the ice cover comprises the following steps:

the method comprises the following steps: establishing an optimization model of the hydraulic optimal section of the water delivery channel of the open channel under the ice cover:

A＝(b+hm)h (1)

in the formula, Q is the water delivery flow of the open channel under the ice cover, b is the width of the channel bottom, h is the water depth, m is the coefficient of the side slope, i is the gradient of the longitudinal slope of the channel bottom, n_bIs the roughness of the trench, n_iIs the ice cover roughness.

Step two: a differential equation for solving the hydraulic optimal section of the water delivery channel of the open channel under the ice cover is obtained by utilizing a Lagrange multiplier method, and the equation is shown as the formula (3):

step three: divide into the rivers under the ice cover in upper strata ice cover influence district and lower floor's bed influence district, have according to the conservation of mass theorem:

Q＝Q_i+Q_b

(4)

wherein Q is the water delivery flow of the open channel under the ice cover, Q_iWater flow, Q, for the upper ice cover affected zone_bWater flow of the lower-layer channel bed influence area;

setting dimensionless parametersV_iThe average flow velocity, V, of the water flow in the upper ice cover affected zone_bObtaining an expression of the comprehensive roughness rate for the average flow velocity of the water flow in the lower-layer trench bed influence area according to a theory of talent ability formula, wherein the expression is shown in formula (5):

in the formula, P_iWet cycle, P, for ice cover_bWet cycles caused by the bed;

step four: dimensionless parameter with width-depth ratioAnd solving the width-depth ratio of the hydraulic optimal section of the water delivery channel of the open channel under the ice cover by solving the equation after the derivative of A and phi in the formula (1) and the formula (2) is solved and the formula (3) is replaced.

Further, in step four, studies have shown (Chow Ven Te. (1959), Open channel Hydraulics, McGraw-Hill, New York.) that the average flow rate of water in the upper ice cover affected zone and the average flow rate of water in the lower bed affected zone are both approximately equal to the average flow rate of water across the entire cross-section, i.e., V_b＝V_iAt this time ∈ ≈ 1, the corresponding expression of the integrated roughness is:

the equation of the optimization model conversion of the hydraulic optimal section is as follows:

solving the equation shown in the formula (7) to obtain the width-depth ratio of the hydraulic optimal section of the water delivery channel of the open channel under the ice cover as follows:

and (3) substituting the formula (8) into a constraint condition formula (2) of the hydraulic optimal section optimization model of the water delivery channel of the open channel under the ice cover to obtain:

solving equation (9) to obtain water depth:

the width of the trench bottom is

In the formula, Q is the water delivery flow of the open channel under the ice cover, b is the width of the channel bottom, h is the water depth, m is the coefficient of the side slope, i is the gradient of the longitudinal slope of the channel bottom, n_bIs the roughness of the trench, n_iIs the ice cover roughness.

Further, when the cross section of the water delivery channel of the open channel under the ice cover is a rectangular cross section, namely m is 0 (the trapezoidal cross section becomes the rectangular cross section), the width-depth ratio of the hydraulic optimal cross section of the rectangular cross section is obtained by substituting m into 0 in formula (8):

at this time, the water depth isThe width of the trench bottom isComparing the formula (11a) with the formula (8), the hydraulic optimal section of the trapezoidal section under the ice cover provided by the invention has the optimal width-depth ratio which is not only related to the slope coefficient m, but also related to the ice cover roughness n_iRoughness n of trench_bRelated (to the relative relationship between ice cover roughness and trench roughnessRelated) whose base width b is the depth of water hDoubling; whereas the conventional hydraulic optimum profile is independent of roughness (Zhao Zheng xing, He Jian Jing, hydraulics, Qinghua university Press);

substituting m-0 into formula (10) to obtain the water depth of the hydraulic optimal section of the rectangular section:

in the formula, Q is the water delivery flow of the open channel under the ice cover, b is the width of the channel bottom, h is the water depth, m is the coefficient of the side slope, i is the gradient of the longitudinal slope of the channel bottom, n_bIs the roughness of the trench, n_iIs the ice cover roughness.

Furthermore, in the fourth step, when the average flow velocity of the water flow in the upper ice cover influence area is not equal to the average flow velocity of the water flow in the lower canal bed influence area in the process of delivering water under the ice cover, namely V_b≠V_iAt this time, ∈ ≠ 1, the equation for the optimization model transformation of the hydraulic optimal section is as follows:

solving the equation shown in the formula (12) to obtain the width-depth ratio of the hydraulic optimal section of the water delivery channel of the open channel under the ice cover as follows:

wherein,

Z₁＝n_i ^3/2(∈^5/2+∈^3/2)n_b ^3/2+∈⁴n_i ³+n_b ³

wherein b is the width of the bottom of the channel, h is the depth of the water, m is the coefficient of the side slope, i is the gradient of the longitudinal slope of the bottom of the channel, n_bIs the roughness of the trench, n_iIs the ice cover roughness.

The invention has the following beneficial effects: the optimal hydraulic cross section of the trapezoidal cross section under the ice cover of the water delivery channel of the open channel under the ice cover has the optimal width-depth ratio which is not only related to the slope coefficient, but also related to the roughness of the ice cover and the roughness of the channel bed, and the conventional optimal hydraulic cross section is unrelated to the roughness; the cross section of the water delivery channel of the open channel under the ice cover can ensure that the channel has the maximum flow capacity under the condition of a certain flow area when the channel delivers water under the ice cover, or the flow area is the minimum under the condition of a certain flow.

The invention has the following characteristics:

(1) the traditional optimal hydraulic fracture surface is a solving algorithm aiming at the optimal hydraulic fracture surface without an icing channel and lacking water delivery under an ice cover. The method establishes a solving model of the hydraulic optimal section of the water delivery under the ice cover, obtains an algorithm for solving the hydraulic optimal section under the ice cover by using a Lagrange multiplier method, and can easily obtain the width-depth ratio of the hydraulic optimal section under the ice cover.

(2) The hydraulic optimal section provided by the invention has the maximum flow capacity when water is delivered under the ice cover. Whereas the conventional hydraulically optimal section is not an "optimal" section when delivering water under an ice cover.

(3) The invention provides an optimal section algorithm of water power under an ice cover, which comprises the following steps:when n is_iWhen the water surface is not frozen n is 0_i0) becomes:therefore, the algorithm of the hydraulic optimal section under the ice cover provided by the invention is more universal, and the conventional hydraulic optimal section (without icing) is only n_iA special case when 0.

(4) The optimal hydraulic fracture surface under the ice cover provided by the invention not only enables the open channel to have the maximum flow capacity when water is delivered under the ice cover, but also is compared with the conventional optimal hydraulic fracture surface; the hydraulic optimal section provided by the invention is a narrow-depth section, the width-depth ratio and the water surface width of the narrow-depth section are smaller, so that the narrow-depth section is favorable for geothermal energy to keep water temperature, the heat dissipation from the water surface or the ice surface in winter is reduced, the water temperature is favorably kept, the icing time is delayed, the thickness of an ice cover is reduced, the geothermal energy is better utilized to reduce icing, and thus the accidents of ice plugs and ice dams are avoided.

Drawings

FIG. 1 is a schematic diagram of a hydraulic optimum section of a water delivery channel of the present invention;

FIG. 2 is a schematic view showing the distribution characteristics of water delivery flow rate under an ice cover;

FIG. 3 is a schematic diagram of the shape of a conventional (when not frozen) hydraulically optimum cross-section;

FIG. 4 shows that when n_i＝n_bThe shape schematic diagram of the optimal hydraulic section of the water delivery under the ice cover;

in the figure, the position of the upper end of the main shaft,

1 is a channel bottom, 2 is a channel left side slope, 3 is a channel right side slope, 4 is a left bank top, 5 is a right bank top, and 6 is an ice cover;

b is the width of the bottom of the channel, h is the depth of the water, m is the coefficient of the side slope, i is the slope of the longitudinal slope of the bottom of the channel, n_bIs the roughness of the trench, n_iThe roughness of the ice cover, Q the water delivery flow of the open channel under the ice cover, Q_iWater flow, Q, for the upper ice cover affected zone_bWater flow of the lower-layer canal bed influence area, A is the area of the hydraulic section of the water delivery channel of the open channel under the ice cover, V_iThe average flow velocity, V, of the water flow in the upper ice cover affected zone_bThe average flow velocity of the water flow in the lower layer channel bed influence area.

Detailed Description

In order to clearly explain the technical features of the present invention, the following detailed description of the present invention is provided with reference to the accompanying drawings. The following disclosure provides many different embodiments, or examples, for implementing different features of the invention. To simplify the disclosure of the present invention, the components and arrangements of specific examples are described below. Furthermore, the present invention may repeat reference numerals and/or letters in the various examples. This repetition is for the purpose of simplicity and clarity and does not in itself dictate a relationship between the various embodiments and/or configurations discussed. It should be noted that the components illustrated in the figures are not necessarily drawn to scale. Descriptions of well-known components and processing techniques and procedures are omitted so as to not unnecessarily limit the invention.

As shown in fig. 1, the open channel water delivery channel under ice cover of the invention comprises a channel bottom 1, a channel left side slope 2, a channel right side slope 3, a left bank top 4 and a right bank top 5, wherein the left side and the right side of the channel bottom 1 are respectively connected with the lower ends of the channel left side slope 2 and the channel right side slope 3, the upper end of the channel left side slope 2 is connected with the left bank top 4, and the upper end of the channel right side slope 3 is connected with the right bank top 5; the section of the water delivery channel of the open channel under the ice cover is a trapezoidal section, and when water is delivered under the ice cover under the condition that the average flow rate of water flow in the upper ice cover influence area is the same as the average flow rate of water flow in the lower channel bed influence area, the width-depth ratio of the hydraulic optimal section of the water delivery channel of the open channel under the ice cover is as follows:depth of water ofThe width of the trench bottom isWherein b is the width of the bottom of the channel, h is the depth of the water, m is the coefficient of the side slope, i is the gradient of the longitudinal slope of the bottom of the channel, n_bIs the roughness of the trench, n_iIs the ice cover roughness.

The width-depth ratio of the hydraulic optimal section of the ice cover lower open channel water delivery channel is solved according to the following optimization model of the hydraulic optimal section of the ice cover lower open channel water delivery channel:

the objective function is to minimize the cross-sectional area:

A＝(b+hm)h (1)

the constraint condition is that the specified flow Q needs to be passed under the flow area:

wherein Q is the water delivery flow of the open channel under the ice cover, b is the width of the channel bottom, h is the water depth, m is the coefficient of the side slope, i is the gradient of the longitudinal slope of the channel bottom, n_bIs the roughness of the trench, n_iFor ice cover roughness, Φ is the constraint function.

The water delivery channel of the open channel under the ice cover comprises an excavation channel, a filling channel or a filling excavation channel.

The invention also provides a method for solving the hydraulic optimal section of the water delivery channel of the open channel under the ice cover.

The method for solving the hydraulic optimal section of the water delivery channel of the open channel under the ice cover comprises the following steps of:

firstly, establishing an optimization model of the hydraulic optimal section of an open channel water delivery channel under an ice cover:

the objective function is to minimize the cross-sectional area:

A＝(b+hm)h (1)

the constraint condition is that the specified flow Q needs to be passed under the flow area:

wherein Q is the water delivery flow of the open channel under the ice cover, b is the width of the channel bottom, h is the water depth, m is the coefficient of the side slope, i is the gradient of the longitudinal slope of the channel bottom, n_bIs the roughness of the trench, n_iFor ice cover roughness, Φ is the constraint function.

Obtaining a differential equation for solving the hydraulic optimal section of the water delivery channel of the open channel under the ice cover by using a Lagrange multiplier method, wherein the equation is shown in a formula (3):

thirdly, as shown in fig. 2, the water flow under the ice cover is divided into an upper ice cover influence area and a lower canal bed influence area, and the water flow under the ice cover is divided into the following areas according to the mass conservation theorem:

Q＝Q_i+Q_b

(4)

wherein Q is the water delivery flow of the open channel under the ice cover, Q_iWater flow, Q, for the upper ice cover affected zone_bWater flow of the lower-layer channel bed influence area;

setting dimensionless parametersV_iThe average flow velocity, V, of the water flow in the upper ice cover affected zone_bObtaining an expression of the comprehensive roughness rate for the average flow velocity of the water flow in the lower-layer trench bed influence area according to a theory of talent ability formula, wherein the expression is shown in formula (5):

in the formula, P_iWet cycle, P, for ice cover_bWet cycles caused by the bed;

fourthly, setting the width-depth ratio dimensionless parameterAnd solving the width-depth ratio of the hydraulic optimal section of the water delivery channel of the open channel under the ice cover by solving the equation after the derivative of A and phi in the formula (1) and the formula (2) is solved and the formula (3) is replaced.

The steps are divided into the following two cases:

(1)V_b≈V_i(Or ∈ ≈ 1)

(1-1) the study showed that (Chow Ven Te. (1959), Open Channel Hydraulics, McGraw-Hill, New York.) when water is delivered under the ice cover, the average flow rates of the water flow in the upper ice cover affected zone and the lower bed affected zone are both approximately equal to the average flow rate of the water flow over the entire cross-section, i.e., V_b＝V_iAt this time, ∈ ≈ 1(∈ ═ V)_i/V_b) Then the corresponding overall roughness is expressed as:

(1-2) setting a non-dimensional parameter of width-depth ratioAnd the derivative of A and phi is calculated, and the optimization model of the hydraulic optimal section can be finally converted into the following equation after the formula (3) is substituted:

(1-3) solving an equation shown in the formula (7), and finally obtaining the width-depth ratio (the only feasible solution) of the section with the maximum flow capacity under the ice cover as follows:

and (3) substituting the formula (8) into a constraint condition formula (2) of the hydraulic optimal section optimization model of the water delivery channel of the open channel under the ice cover to obtain:

(1-4) solving equation (9) to obtain water depth:

the width of the trench bottom is

In the formula, Q is the water delivery flow of the open channel under the ice cover, b is the width of the channel bottom, h is the water depth, m is the coefficient of the side slope, i is the gradient of the longitudinal slope of the channel bottom, n_bIs the roughness of the trench, n_iIs the ice cover roughness.

(1-5) when the section of the water delivery channel of the open channel under the ice cover is a rectangular section, namely m is 0 (the trapezoidal section becomes the rectangular section), replacing m with 0 for obtaining the width-depth ratio of the hydraulic optimal section of the rectangular section in the formula (8):

at this time, the water depth isThe width of the trench bottom is

Comparing the formula (11a) with the formula (8), the hydraulic optimal section of the trapezoidal section under the ice cover provided by the invention has the optimal width-depth ratio which is not only related to the slope coefficient m, but also related to the ice cover roughness n_iRoughness n of trench_bRelated (to the relative relationship between ice cover roughness and trench roughnessRelated) whose base width b is the depth of water hDoubling; whereas the conventional hydraulic optimum profile is independent of roughness (Zhao Zheng xing, He Jian Jing, hydraulics, Qinghua university Press);

substituting m-0 into formula (10) to obtain the water depth of the hydraulic optimal section of the rectangular section:

in the formula, Q is the water delivery flow of the open channel under the ice cover, b is the width of the channel bottom, h is the water depth, and m isCoefficient of side slope, i is the slope of longitudinal slope at bottom of channel, n_bIs the roughness of the trench, n_iIs the ice cover roughness.

(2) When V is_b≠V_i(or ∈ ≠ 1)

(2-1) when the average flow velocity of the water flow in the upper ice cover influence area is not equal to the average flow velocity of the water flow in the lower canal bed influence area in the ice cover lower water delivery process, namely V_b≠V_iAt this time, ∈ ≠ 1(∈ ═ V)_i/V_b) Then, the optimization model of the hydraulic optimal section is converted into the following equation by the same method:

(2-1) solving an equation shown in the formula (12) to obtain the width-depth ratio of the hydraulic optimal section of the water delivery channel of the open channel under the ice cover as follows:

wherein,

Z₁＝n_i ^3/2(∈^5/2+∈^3/2)n_b ^3/2+∈⁴n_i ³+n_b ³

wherein b is the width of the bottom of the channel, h is the depth of the water, m is the coefficient of the side slope, i is the gradient of the longitudinal slope of the bottom of the channel, n_bIs the roughness of the trench, n_iIs the ice cover roughness.

The invention obtains the optimal water delivery section for delivering water under the ice cover in winter, and the section can ensure that the channel has the maximum flow capacity (or the minimum flow area under the condition of certain flow) under the condition of certain flow area when delivering water under the ice cover. The optimal width-depth ratio obtained under the condition without the ice cover is the same as the width-depth ratio of the conventional hydraulic optimal section, and the hydraulic optimal section calculation formula provided by the patent has a wider application range, is suitable for solving the hydraulic optimal section under the ice cover and can also be used for solving the hydraulic optimal section without the ice cover. The conventional hydraulic optimal section is only a calculation formula provided by the patent at n_iA special case when 0 (no ice cover). The hydraulic optimal section of water delivery under the ice cover is a narrow and deep section, which is beneficial to geothermal energy and water temperature maintenance, thereby reducing ice damage. Under the condition of the hydraulic optimal section, the water surface width is always smaller when freezing is carried out than when not freezing, so that the heat dissipation of the water surface (or ice surface) is favorably reduced, the water temperature is kept, and the ice damage is reduced.

The technical principle adopted by the invention is as follows.

(1) Optimum hydraulic section model for water delivery under ice cover

As shown in FIG. 1, when water is delivered under the ice cover, the cross-sectional area A can be expressed as:

A＝(b+hm)h (14)

different from non-ice period, water surface width B, and wet period P caused by ice cover_iWet circumference P caused by channel_bAnd the total wet week P can be expressed as

P_i＝B＝b+2mh (15)

And setting the optimization variables h and b for the trapezoidal section. According to the definition of the optimal section (the hydraulic optimal section refers to the maximum flow rate under the condition of a certain area (or the minimum water delivery section area under the condition of a certain flow rate)), the hydraulic optimal section of the water delivery under the ice cover can be expressed as the following optimization problem:

and (3) minimizing: a ═ A (h, b) (18)

Constraint conditions are as follows:

in the formula, n_tFor combined roughness including ice cover and bottom

With the lagrange multiplier method, the above optimization problem can be expressed as:

λ is the lagrange multiplier.

The joint solution of the equations (20) and (21) can obtain a differential equation for solving the hydraulic optimal section under the ice cover:

in the above formula, the function Φ includes the integrated roughness n_tComprehensive roughnessn_tRoughness n of trench_bWet periphery P of trench bed_bIce cover roughness n_iAnd ice cover wet cycle P_iIt is related.

(2) Comprehensive roughness of optimum hydraulic section for water delivery under ice cover

As shown in FIG. 2, the water flow under the ice cover can be divided into an upper layer and a lower layer, and the integrated roughness of the water delivery under the ice cover can be expressed by the following formula (Chow,1959)

Because R, R_iAnd R_bAre all unknown, so dimensionless parameters are assumed∈ can be expressed as Schoenbergia's formula (hydraulics, Zhao Zheng xing, Qing Hua university Press)

In the formula, V_b,V_iThe average flow velocity of the water flow in the lower and upper layers of the canal (see FIG. 2)

Represented by the formula (24), R_iAvailable as ∈, R_b,n_iAnd n_bRepresents:

r is to be_iSubstituting the formula (25) into the formula (25), the expression of the comprehensive roughness of the water delivery under the ice cover can be obtained

The following discussion is divided into two cases, one being V_i＝V_bI.e. ∈ ═ 1, another is V_i≠V_bI.e. ∈ ≠ 1.

(3) When V is_i＝V_bOr ∈ -1 h water conveying hydraulic optimum section under ice cover

Many researchers have shown that the average flow rate of the upper and lower streams is approximately equal to the average flow rate of the entire cross-section when delivering water under the ice cap (Chow, 1959; Gary,1987, WEI, 2002; Nzhikhovski, 1964; Uzuner, 1975; Pratte, 1979). I.e. V_b＝V_i∈ ≈ 1, the corresponding overall roughness may be expressed as:

substituting equations (14), (17), and (27) into equation (19), the function Φ (h, b) can be expressed as:

from equation (14), the derivative of the flow area a (h, b) with respect to h and b can be obtained:

from equation (28), the derivative of Φ (h, b) with respect to h can be obtained:

in the formula,

in the same way, the derivative of Φ (h, b) with respect to b can be obtained:

substituting the expressions (29), (30) and (31) into the expression (27) can obtain an equation for solving the hydraulic optimal section under the ice cover:

simplifying the above equation can result:

the above formula has two unknowns, and the dimensionless parameter eta of the width-depth ratio is set as

η and A, P_i、P_bSubstituting the expressions of P and W into the expression (33) to obtain the simplified expression

Solving the above formula can obtain only feasible solution

When n is_iWhen the water surface is not frozen, equation (36) will become:

η in the above formula_freeThe water surface icing-free water surface optimum cross section is the known width-depth ratio of the water surface icing-free water surface optimum cross section and is the conventional open channel water surface optimum cross section in the teaching material. It can be seen that formula (36) is a general expression, and the conventional hydraulic optimum section on the textbook is only that formula (36) is in n_iA special case when 0.

(4) Characteristic of water delivery section under ice cover (E ═ 1)

The width-depth ratio of the hydraulic optimal section of the water delivery under the ice cover is always smaller than that of the conventional (non-icing time).

Comparison η_ic(formula (36)) and η_free(equation (37)), the following equation always holds:

as can be seen from the formula (38), the width-depth ratio of the hydraulic optimal section of the water delivery under the ice cover is always smaller than that of the conventional (non-icing) hydraulic optimal section

η_ic＜η_free(39)

This means that the hydraulic optimum section for water delivery under the ice cover is a narrow deep section which is beneficial to geothermal energy to maintain water temperature and reduce ice damage.

Secondly, the water surface width of the hydraulic optimal section for water delivery under the ice cover is always smaller than that of the conventional (non-icing) hydraulic optimal section.

η will be mixed_ic(equation (36)) substituting equations (14), (15) and (16) can obtain A, P under the ice cover directly_i，P_bAnd Q:

substituting (40), (41) and (42) into (28) can obtain the formula of direct ice cover Q:

from the conventional hydraulic optimum profile, or from equation (41), when there is no ice cover (n)_i0), the water surface width can be expressed as:

in the formula B_freeThe water surface width of the conventional hydraulic optimal section is obtained.

Comparing B (formula (41)) with B_free(equation (44)), the following inequality holds:

i.e. the width B of the water surface (ice surface) under the ice cover when the hydraulic power is optimal_icIs always smaller than the water surface width B without ice cover_freeNamely:

B_ic＜B_free(46)

formula (46) shows that the optimal hydraulic section under the ice cover can reduce the heat dissipation from the water surface or the ice surface in winter, is beneficial to keeping the water temperature, postponing the icing time, reducing the thickness of the ice cover and reducing the damage of ice damage in winter.

③ if n_i＝n_bFrom the formula (36), the optimum aspect ratio and hydraulic radius of

When n is compared with the width-depth ratio (no ice cover) of the conventional hydraulic optimal section on the teaching material_i＝n_bThe optimal aspect ratio is half of the conventional optimal aspect ratio, and the shape is shown in fig. 3. The bottom, the edge and the top of the hydraulic optimal section can be obtained to be tangent and round (radius)h)

(4) Calculation of normal and critical water depth of hydraulic optimum section (under ice cover)

Solving equation (43) can obtain an explicit expression of normal water depth:

the general expression for calculating the critical water depth is:

wherein g is the acceleration of gravity; α is a cross-sectional energy correction coefficient (generally, α is 1.0).

Substituting equations (40) and (41) for equation (50), equation (50) may be written as:

solving the formula (51), wherein the expression for explicitly solving the critical water depth of the hydraulic optimal section under the ice cover is as follows:

in the formula,

(5) the conventional hydraulic optimal section is not optimal when water is delivered under the ice cover

Substituting h in formula (49) into formula (40) to obtain

Solving the above equation, Q can be represented as A

In the formula,

the formula shows the flow capacity under a certain flow area, which is obtained by the width-depth ratio of the hydraulic optimal section under the ice cover.

In the same way, the relationship of the aspect ratio of the optimum section with conventional (ice-free) hydraulics can be obtained:by substituting formulae (14), (15) and (16) and further substituting formula (28), the conventional hydraulic optimum cross section under ice cover Q can be obtained₂The calculation formula of (2) is as follows:

solving the above equation can obtain

Will be provided withSubstituting h in the above formula to obtain

Solving the above equation, Q can be obtained₂The relationship represented as A is

The above formula shows the flow capacity at a certain flow area, obtained with the optimal aspect ratio of the conventional (under no ice cover) hydraulic optimum section.

But A is^*＞0,m＞0,n_b＞0,n_iWhen the value is more than 0, Q is more than Q by verification₂。

Optimum aspect ratio for hydraulic optimum profile using conventional (no ice under cover)The hydraulic power obtained is not "optimal" when delivering water under the ice cover.

(6) Hydraulic optimal section algorithm for rectangular section under ice cover

When m is equal to 0, the trapezoidal section becomes a rectangular section, and according to the formula (36), the width-depth ratio of the hydraulic optimal section of the rectangular section is obtained as follows:

when the water surface is not frozen, n_i(iii) β from formula (61): 0_ic2. The width-depth ratio of the hydraulic optimal section is the same as that of the rectangular section on the teaching material.

From equation (43), a flow calculation equation of a rectangular cross section can be obtained:

from equations (49) and (52), the normal water depth and the critical water depth can be obtained:

(7) when V is_i≠V_bOr ∈ ≠ 1 hydraulic characteristics of hydraulic optimum fracture

When V is_i≠V_bOr ∈ ≠ 1, a more general hydraulic optimum fracture surface solving equation can be obtained in the same manner (equations (28) to (35)):

solving the above equation can obtain a more general optimal aspect ratio solving equation:

in the formula,

Z₁＝n_i ^3/2(∈^5/2+∈^3/2)n_b ^3/2+∈⁴n_i ³+n_b ³

η derived from formula (66) when ∈ is 1_icη obtained by using the value and equation (36)_icThe values are identical.

The foregoing is only a preferred embodiment of the present invention, and it will be apparent to those skilled in the art that various modifications and improvements can be made without departing from the principle of the invention, and such modifications and improvements are also considered to be within the scope of the invention.

Claims (8)

1. An open channel water delivery channel under an ice cover is characterized by comprising a channel bottom (1), a channel left side slope (2), a channel right side slope (3), a left bank top (4) and a right bank top (5), wherein the left side and the right side of the channel bottom (1) are respectively connected with the lower ends of the channel left side slope (2) and the channel right side slope (3), the upper end of the channel left side slope (2) is connected with the left bank top (4), and the upper end of the channel right side slope (3) is connected with the right bank top (5); the section of the water delivery channel of the open channel under the ice cover is a trapezoidal section, and the average flow velocity V of water flow in the influence area of the ice cover on the upper layer_iAnd the average flow velocity V of the water flow in the lower layer channel bed influence area_bWhen water is delivered under the ice cover under the same condition, the width-depth ratio of the hydraulic optimal section of the water delivery channel of the open channel under the ice cover is as follows:average flow velocity V of water flow in upper ice cover influence area_iAnd the average flow velocity V of the water flow in the lower layer channel bed influence area_bWhen carrying out under the ice cover water delivery under the inequality circumstances, the width-depth ratio of the optimum section of open channel water delivery channel water conservancy under the ice cover is:wherein b is the width of the bottom of the channel, h is the depth of the water, m is the coefficient of the side slope, i is the gradient of the longitudinal slope of the bottom of the channel, n_bIs the roughness of the trench, n_iIn order to obtain the roughness of the ice cover,

Z₁＝n_i ^3/2(∈^5/2+∈^3/2)n_b ^3/2+∈⁴n_i ³+n_b ³，

$\begin{array}{c}{Z}_2=-2({n_i}^{3/2}({\∈}^{5/2}+{\∈}^{3/2}){n_b}^{9/2}-{n_i}^{9/2}({\∈}^{11/2}+{\∈}^{13/2}){n_b}^{3/2}+2{n_b}^3({n_b}^3+15/2({\∈}^2-\frac{32\∈}{15}+1){\∈}^3{n_i}^3))\×\\ m\sqrt{m^2+1}+4{n_i}^{3/2}(m^2+1)({\∈}^{5/2}+{\∈}^{3/2}){n_b}^{9/2}\end{array},$

$Z_3=(5{n_b}^6+31{\∈}^3({\∈}^2-\frac{60\∈}{31}+1){n_i}^3{n_b}^3+{\∈}^8{n_i}^6)m^2+4{n_b}^3({n_b}^3+4{\∈}^3({\∈}^2-7/4\∈+1){n_i}^3),$

$Z_4=(4{\∈}^{5/2}-4{\∈}^{3/2}){n_i}^{3/2}{n_b}^{3/2}\sqrt{m^2+1}+(-5m{\∈}^{5/2}+3m{\∈}^{3/2}){n_i}^{3/2}{n_b}^{3/2}+(-{\∈}^4{n_i}^3-{n_b}^3)m,$

∈＝V_i/V_b。

2. the under-ice-cover open channel water delivery channel of claim 1, wherein the width-to-depth ratio of the hydraulically optimal cross section of the under-ice-cover open channel water delivery channel is solved according to the following optimization model of the hydraulically optimal cross section of the under-ice-cover open channel water delivery channel:

minimizing the objective function:

A＝(b+hm)h (1)

constraint conditions are as follows:

$\mathrm{\Φ}(h,b)=Q-\frac{{\left(\right(hm+b\left)h\right)}^{5/3}{i}^{0.5}}{{(2b+2\sqrt{m^2+1}h+2hm)}^{2/3}}{\left(\frac{(b+2\sqrt{m^2+1}h){n_b}^{3/2}+(2hm+b){n_i}^{3/2}}{2b+2\sqrt{m^2+1}h+2hm}\right)}^{-2/3}=0---\left(2\right)$

wherein Q is the water delivery flow of the open channel under the ice cover, b is the width of the channel bottom, h is the water depth, m is the coefficient of the side slope, i is the gradient of the longitudinal slope of the channel bottom, n_bIs the roughness of the trench, n_iFor ice cover roughness, Φ is the constraint function.

3. The under ice cover open channel water delivery channel of claim 1 or 2, wherein the under ice cover open channel water delivery channel comprises a cut channel, a fill channel or a fill channel.

4. A method for solving the hydraulic optimal section of the water delivery channel of the open channel under the ice cover is characterized by firstly establishing a solution model of the hydraulic optimal section of the water delivery channel of the open channel under the ice cover, then obtaining a differential equation for solving the hydraulic optimal section of the water delivery channel of the open channel under the ice cover by utilizing a Lagrange multiplier method, and finally calculating the width-depth ratio of the hydraulic optimal section of the water delivery channel of the open channel under the ice cover according to the differential equation.

5. The method for solving the hydraulic optimal section of the water delivery channel of the open channel under the ice cover as claimed in claim 4, wherein the method for solving the hydraulic optimal section of the water delivery channel of the open channel under the ice cover comprises the following steps:

the method comprises the following steps: establishing an optimization model of the hydraulic optimal section of the water delivery channel of the open channel under the ice cover:

A＝(b+hm)h (1)

$\mathrm{\Φ}(h,b)=Q-\frac{{\left(\right(hm+b\left)h\right)}^{5/3}{i}^{0.5}}{{(2b+2\sqrt{m^2+1}h+2hm)}^{2/3}}{\left(\frac{(b+2\sqrt{m^2+1}h){n_b}^{3/2}+(2hm+b){n_i}^{3/2}}{2b+2\sqrt{m^2+1}h+2hm}\right)}^{-2/3}=0---\left(2\right)$

in the formula, Q is the water delivery flow of the open channel under the ice cover, b is the width of the channel bottom, h is the water depth, m is the coefficient of the side slope, i is the gradient of the longitudinal slope of the channel bottom, n_bIs the roughness of the trench, n_iThe ice cover roughness;

step two: a differential equation for solving the hydraulic optimal section of the water delivery channel of the open channel under the ice cover is obtained by utilizing a Lagrange multiplier method, and the equation is shown as the formula (3):

$\frac{\∂A(h,b)}{\∂h}\frac{\∂\mathrm{\Φ}(h,b)}{\∂b}=\frac{\∂\mathrm{\Φ}(h,b)}{\∂h}\frac{\∂A(h,b)}{\∂b}---\left(3\right)$

step three: divide into the rivers under the ice cover in upper strata ice cover influence district and lower floor's bed influence district, have according to the conservation of mass theorem:

Q＝Q_i+Q_b(4)

wherein Q is the water delivery flow of the open channel under the ice cover, Q_iWater flow, Q, for the upper ice cover affected zone_bWater flow of the lower-layer channel bed influence area;

setting dimensionless parametersV_iThe average flow velocity, V, of the water flow in the upper ice cover affected zone_bObtaining an expression of the comprehensive roughness rate for the average flow velocity of the water flow in the lower-layer trench bed influence area according to a theory of talent ability formula, wherein the expression is shown in formula (5):

$n_t=\frac{{(P_i{\∈}^{3/2}{n_i}^{3/2}+P_b{n_b}^{3/2})}^{5/3}}{{(P_i+P_b)}^{2/3}({\∈}^{5/2}{n_i}^{3/2}{P}_i+P_b{n_b}^{3/2})}---\left(5\right)$

in the formula, P_iWet cycle, P, for ice cover_bWet cycles caused by the bed;

step four: dimensionless parameter with width-depth ratioAnd solving the width-depth ratio of the hydraulic optimal section of the water delivery channel of the open channel under the ice cover by solving the equation after the derivative of A and phi in the formula (1) and the formula (2) is solved and the formula (3) is replaced.

6. The method for solving the hydraulic optimal section of the water delivery channel of the open channel under the ice cover as claimed in claim 5, wherein in the fourth step, if the average flow velocity of the water flow in the upper layer of the ice cover influence area and the average flow velocity of the water flow in the lower layer of the canal bed influence area are both equal to the average flow velocity of the water flow of the whole section, i.e. V_b＝V_iWhen ∈ ≈ 1, the corresponding rough combinationThe expression for the ratio is:

$n_t={\left(\frac{P_b{n_b}^{3/2}+P_i{n_i}^{3/2}}{P_b+P_i}\right)}^{2/3}---\left(6\right)$

the equation of the optimization model conversion of the hydraulic optimal section is as follows:

$h^4(\mathrm{\η}+m)\[(2-{\mathrm{\η}}^2+(\sqrt{m^2+1}-3m\left)\mathrm{\η}\right){n_b}^{3/2}-{n_i}^{3/2}\mathrm{\η}(\sqrt{m^2+1}+m+\mathrm{\η})\]=0---\left(7\right)$

solving the equation shown in the formula (7) to obtain the width-depth ratio of the hydraulic optimal section of the water delivery channel of the open channel under the ice cover as follows:

${\mathrm{\η}}_{ic}=\frac{b}{h}=\frac{2{n_b}^{3/2}(\sqrt{m^2+1}-m)}{{n_b}^{3/2}+{n_i}^{3/2}}---\left(8\right)$

and (3) substituting the formula (8) into a constraint condition formula (2) of the hydraulic optimal section optimization model of the water delivery channel of the open channel under the ice cover to obtain:

$Q=1/2\frac{(2{n_b}^{3/2}\sqrt{m^2+1}-{n_b}^{3/2}m+{n_i}^{3/2}m)h^{8/3}\sqrt{i}\sqrt[3]{2}}{{({n_b}^{3/2}+{n_i}^{3/2})}^{5/3}}---\left(9\right)$

solving equation (9) to obtain water depth h:

$h=\sqrt[4]{2}{\left(\frac{Q{({n_b}^{3/2}+{n_i}^{3/2})}^{5/3}}{(2{n_b}^{3/2}\sqrt{m^2+1}-{n_b}^{3/2}m+{n_i}^{3/2}m)\sqrt{i}}\right)}^{3/8}---\left(10\right)$

in the formula, Q is the water delivery flow of the open channel under the ice cover, b is the width of the channel bottom, h is the water depth, m is the coefficient of the side slope, i is the gradient of the longitudinal slope of the channel bottom, n_bIs the roughness of the trench, n_iIs the ice cover roughness.

7. The method for solving the hydraulic optimal section of the ice cover under-channel water delivery channel according to claim 6, wherein when the section of the ice cover under-channel water delivery channel is a rectangular section, that is, when m is 0, the width-depth ratio of the hydraulic optimal section of the rectangular section is obtained in the formula (8) by substituting m-0:

${\mathrm{\β}}_{ic}=\frac{b}{h}=2\frac{{n_b}^{3/2}}{{n_b}^{3/2}+{n_i}^{3/2}}---\left(11a\right)$

substituting m-0 into formula (10) to obtain the water depth of the hydraulic optimal section of the rectangular section:

$h=\frac{1}{2}{2}^{5/8}{\left(\frac{Q{2}^{2/3}{({n_b}^{3/2}+{n_i}^{3/2})}^{5/3}}{{n_b}^{3/2}\sqrt{i}}\right)}^{3/8}---\left(11b\right)$

in the formula, Q is the water delivery flow of the open channel under the ice cover, b is the width of the channel bottom, h is the water depth, m is the coefficient of the side slope, i is the gradient of the longitudinal slope of the channel bottom, n_bIs the roughness of the trench, n_iIs the ice cover roughness.

8. The method for solving the hydraulic optimal section of the water delivery channel of the open channel under the ice cover as claimed in claim 5, wherein in the fourth step, the average flow velocity V of the water flow in the influence area of the ice cover on the upper layer in the water delivery process under the ice cover_iAverage flow velocity V of water flow in the lower layer channel bed influence area_bWhen not equal, i.e. V_b≠V_iAt this time, ∈ ≠ 1, the equation for the optimization model transformation of the hydraulic optimal section is as follows:

$\begin{array}{c}(-2{n_i}^{3/2}{\∈}^{3/2}(4\∈\mathrm{\η}+5\∈m-4\mathrm{\η}-3m){n_b}^{3/2}+4{n_b}^{3}m)\sqrt{m^2+1}+\\ {\∈}^{3/2}(\∈{\mathrm{\η}}^2+10\∈\mathrm{\η}m+10\∈m^2+{\mathrm{\η}}^2-6\mathrm{\η}m-6m^2){n_i}^{3/2}{n_b}^{3/2}+\\ ({\∈}^4{n_i}^3+{n_b}^3){\mathrm{\η}}^2+2m({\∈}^4{n_i}^3+{n_b}^3)\mathrm{\η}-4{n_b}^3(m^2+1)=0\end{array}---\left(12\right)$

solving the equation shown in the formula (12) to obtain the width-depth ratio of the hydraulic optimal section of the water delivery channel of the open channel under the ice cover as follows:

${\mathrm{\η}}_{ic}=\frac{b}{h}=\frac{\sqrt{Z_2+Z_3}+Z_4}{Z_1}---\left(13\right)$

wherein,

Z₁＝n_i ^3/2(∈^5/2+∈^3/2)n_b ^3/2+∈⁴n_i ³+n_b ³

$\begin{array}{c}{Z}_2=-2({n_i}^{3/2}({\∈}^{5/2}+{\∈}^{3/2}){n_b}^{9/2}-{n_i}^{9/2}({\∈}^{11/2}+{\∈}^{13/2}){n_b}^{3/2}+2{n_b}^3({n_b}^3+15/2({\∈}^2-\frac{32\∈}{15}+1){\∈}^3{n_i}^3))\×\\ m\sqrt{m^2+1}+4{n_i}^{3/2}(m^2+1)({\∈}^{5/2}+{\∈}^{3/2}){n_b}^{9/2}\end{array}$

$Z_3=(5{n_b}^6+31{\∈}^3({\∈}^2-\frac{60\∈}{31}+1){n_i}^3{n_b}^3+{\∈}^8{n_i}^6)m^2+4{n_b}^3({n_b}^3+4{\∈}^3({\∈}^2-7/4\∈+1){n_i}^3)$

$Z_4=(4{\∈}^{5/2}-4{\∈}^{3/2}){n_i}^{3/2}{n_b}^{3/2}\sqrt{m^2+1}+(-5m{\∈}^{5/2}+3m{\∈}^{3/2}){n_i}^{3/2}{n_b}^{3/2}+(-{\∈}^4{n_i}^3-{n_b}^3)m$

∈＝V_i/V_b

wherein b is the width of the bottom of the channel, h is the depth of the water, m is the coefficient of the side slope, i is the gradient of the longitudinal slope of the bottom of the channel, n_bIs the roughness of the trench, n_iIs the ice cover roughness.