CN103277236B - A kind of low water head liquid gas energy conversion equipment and design method - Google Patents

Abstract

The present invention relates to a kind of low water head liquid gas energy conversion equipment and design method, described device comprises: the ascending tube being arranged on mill weir high water level side, ascending tube is connected with baffle pipe, baffle pipe is connected with the down tube of mill weir low water level side, down tube is arranged ventilation point, ventilation point arranges the sucking pipe connecting air turbine, and air turbine is connected with generator.Design method of the present invention exports with system and is design starting point to the maximum, sets each constraint conditio, releases two of syphon tube liquid-gas transfer device most basic parameters: the height of ventilation point and siphonal sectional area with this.Compared with the design method of traditional syphon tube liquid-gas transfer device, design method of the present invention is simpler, practical.The liquid gas shift efficiency calculation method proposed is more close with actual measurement, has promoted the practical application of syphon tube liquid-gas transfer device from theory.Theory shows, the efficiency of the liquid-gas transfer device designed according to design method of the present invention is maximum can reach 86%.

Description

A kind of low water head liquid gas energy conversion equipment and design method

Technical field

The present invention relates to a kind of low water head liquid gas energy conversion equipment and design method, is a kind of device and design method of transformation of energy, is the Optimization Design that a kind of flow energy by low water head is converted to device that gas can generate electricity or draw water and this device.

Background technique

Water can be at present in the world uniquely can the renewable energy sources of large-scale commercial applications exploitation.The expectation of the technical exploitable deposit of whole world hydroelectric resources is 15000TWh/, but only about half of, i.e. 6000-9000TWh/, and being considered at present can not economic development.Main cause is that the head of these hydroelectric resources is low, and power is little.Although utilize water turbine water can be changed into electric energy, system system efficiency often can only reach about 40%, and it is uneconomical that the very high investment costs expense of device specific power usually makes to build hydroelectric power plant, is not thus also utilized to generating so far.Waterpower air pump is the system that a kind of high efficiency, low cost utilizes micro-head and generates electricity.This system a kind ofly can utilize the low water head comprising ocean wave energy, or the renewable energy sources new technology of pole low head power generation.Waterpower air pump relies on syphon tube liquid gas energy transfer principle under certain condition, can convert water to gas energy, and air turbine then can be utilized gas to be converted to mechanical energy generating or to draw water.

Different from traditional water turbine power station, air turbine can arrange the place be arranged on away from river course, and the river that flood can be avoided to cause raises Flood Control Problem, and factory building construction investment is few; And because only have syphon tube in water, do not have moving element, maintenance cost is low.In addition, compared with water turbine, the air turbine rotating speed of same power is usually several times of water turbine, even tens times.On the other hand, the turbine volume of the air turbine that waterpower air pump uses is very little, and for given power stage, the cost of less high speed rotating machine transmitted power is lower.The most important thing is to only have the place of 0.5-2m at head, the waterpower air pump efficiency based on siphon can reach about 70%, and the efficiency of air turbine can reach more than 85%.In other words, can convert low water head water to gas energy, the total efficiency then utilizing air turbine to produce mechanical energy is more than 60%, and the efficiency of the conventional water turbine of corresponding low water head is between 60%-70%, and both efficiency closely.For these reasons, waterpower air pump system should have business market prospect widely.

In order to predict efficiency and the power of waterpower air pump system, French and Widden derives and obtains the analytic solutions of a waterpower air pump efficiency.In the process analyzed, they suppose that the evolution with distance of syphon tube section can ensure that the flow velocity of current is constants.Under steady flow condition, because hydraulic pressure in pipe can increase along with the decline of position height, thus cause the compression of section bubble volume, under the negligible condition of water body compressibility, in order to meet flow rate of water flow along the constant assumed condition of journey, then conduit section must reduce along with the minimizing of elevation.Should say, their method is very helpful at the aspect of performance of understanding waterpower air pump, but also may lead to misunderstanding, and people is thought and becomes can obtain higher efficiency like this by design of siphon pipes.Howey and Pullen has carried out preliminary test to such siphonic system, examines the difference of theoretical prediction and efficiency by inputoutput test, result show to use the efficiency of the model prediction of French and Widden and power higher.

Summary of the invention

In order to overcome the problem of prior art, the present invention proposes a kind of low water head liquid gas energy conversion equipment and design method.Described device is a kind of waterpower air pump of optimal design.Described design method is by mass conservation law, momentum conservation law and law of conservation of energy, establish waterpower air pump performance formula under conventional siphon condition, and the mathematical optimization models of waterpower air pump, existing waterpower air pump design is optimized, reduces the difference of theoretical prediction and efficiency by inputoutput test.

The object of the present invention is achieved like this: a kind of low water head liquid gas energy conversion equipment, described device comprises: the ascending tube being arranged on mill weir high water level side, described ascending tube is connected with baffle pipe, described baffle pipe is connected with the down tube of mill weir low water level side, described down tube is arranged ventilation point, described ventilation point is arranged the sucking pipe connecting air turbine, described air turbine is connected with generator, and the described position of ventilation point on down tube is:

$z_A=z_C+H+\frac{p_C-p_A}{\mathrm{\ρg}}-[L_A+{\left(\frac{n}{1-{\mathrm{\α}}_A}\right)}^2+(L_C-1){\left(\frac{n}{1-{\mathrm{\α}}_C}\right)}^2]\frac{{V_1}^2}{2g}$

Described siphonal sectional area is:

A ₁＝A＝Q/V ₁

In formula: z _athe position elevation of-ventilation point, z _cthe position elevation of-down tube outlet, H-acting head, p _athe pressure of-ventilation point, p _cthe pressure of-water outlet, L _a-ventilation point local loosening, α _athe voids at-ventilation point place, n-ventilation point upstream and downstream pipeline area ratio, when syphon tube basal area is along Cheng Xiangtong then n=1, L _c-water outlet local loosening, V ₁-ascending tube flow velocity, α _cthe voids of-water outlet, g-gravity accleration, the density of ρ-water, A ₁the syphon tube sectional area of-ventilation point, A-down tube sectional area, Q-flow.

A design method for above-mentioned low water head liquid gas energy conversion equipment, the step of described method is as follows:

Calculate the voids of ventilation point, the pressure of ventilation point and the step of ascending tube flow velocity: for maximizes water can be converted to gas can efficiency eta be voids, pressure and flow velocity according to calculating ventilation, according to maximum efficiency formula:

$\max \left\{\mathrm{\η}(p_A,V_1,{\mathrm{\α}}_A)\right\}=\max \left\{\frac{k}{k-1}\frac{p_A}{\mathrm{ng}\mathrm{\ρ}H}({\left(\frac{p_a}{p_A}\right)}^{(k-1)/k}-1){\mathrm{\α}}_A(\frac{n}{1-{\mathrm{\α}}_A}-\frac{V_r}{V_1})\right\}$

The constraint conditio maximizing η is:

$H=K\frac{{V_1}^2}{2g}+B---\left[1\right]$

$K=L_A+n^2(\frac{L_C}{{(1-{\mathrm{\α}}_C)}^2}+\frac{L_{\mathrm{AC}}}{{(1-{\mathrm{\α}}_A)}^2})---\left[2\right]$

$B=\frac{c_1}{\mathrm{\ρg}}[\frac{1}{V_r+c_2}\ln \frac{(c_2+V_r{\mathrm{\α}}_A)(1-{\mathrm{\α}}_C)}{(1-{\mathrm{\α}}_A)(c_2+V_r{\mathrm{\α}}_C)}-\frac{1}{c_2}\ln \frac{(c_2+V_r{\mathrm{\α}}_A){\mathrm{\α}}_C}{(c_2+V_r{\mathrm{\α}}_C){\mathrm{\α}}_A}-\frac{1}{c_2+V_r{\mathrm{\α}}_A}+\frac{1}{c_2+V_r{\mathrm{\α}}_C}]---\left[3\right]$

${\mathrm{\α}}_C=\frac{-(c_1+c_2{p}_C)+\sqrt{{(c_1+c_2{p}_C)}^2+4c_1{V}_r{p}_C}}{2V_r{p}_C}---\left[4\right]$

$c_1=p_A{\mathrm{\α}}_A(n\frac{V_1}{(1-{\mathrm{\α}}_A)}-V_r)---\left[5\right]$

c ₂＝nV ₁-V _r[6]

0＜α _A＜0.3[7]

p _A＞p _v[8]

In formula: α _athe voids at-ventilation point place, p _athe pressure at-ventilation point place, V ₁-ascending tube flow velocity, p _vthe vaporization pressure of-local liquid is known quantity, and H-acting head is known quantity, L _a-ventilation point local loosening is known quantity, L _c-water outlet local loosening is known quantity, L _aCpipeline waterhead fall between-ventilation point to water outlet is known quantity, p _a-local atmospheric pressure is known quantity, ρ _a-air density is known quantity, and k-gas adiabatic exponent is known quantity, g-gravity accleration, is known quantity, V _rthe speed difference of-current and air-flow is known quantity, n-ventilation point upstream and downstream pipeline area ratio, when syphon tube section area is along Cheng Xiangtong then n=1, p _cthe pressure of-water outlet, gets p in process of calculation analysis _c=p _afor known quantity, Two-level Optimization method is used to calculate the voids α of ventilation point A _a, pressure p _a, ascending tube current mean velocity in section V ₁;

Calculate the step of ventilation point height: for passing through formula:

$z_A=z_C+H+\frac{p_C-p_A}{\mathrm{\ρg}}-[L_A+{\left(\frac{n}{1-{\mathrm{\α}}_A}\right)}^2+(L_C-1){\left(\frac{n}{1-{\mathrm{\α}}_C}\right)}^2]\frac{{V_1}^2}{2g}$

Calculate the position height of ventilation point in syphon tube down tube;

Calculate the step of syphon tube sectional area: for passing through formula:

A ₁＝A＝Q/V ₁

Calculate siphonal sectional area.

The beneficial effect that the present invention produces is: design method of the present invention is design starting point to the maximum with system delivery efficiency, set each constraint conditio, release with this most basic parameter that syphon tube liquid-gas transfer device (waterpower air pump) two optimizes: the height of ventilation point and siphonal sectional area.Compared with the design method of traditional syphon tube liquid-gas transfer device, design method of the present invention is simpler, practical.The liquid gas shift efficiency calculation method proposed is more close with actual measurement, has promoted the practical application of waterpower air pump theoretically.Theoretical research shows, the efficiency of the liquid-gas transfer device designed according to design method of the present invention is maximum can reach 86%.

Accompanying drawing explanation

Below in conjunction with drawings and Examples, the invention will be further described.

Fig. 1 is the structural representation of device described in embodiments of the invention one;

Fig. 2 is z described in embodiments of the invention two _aowith α _awith the relation curve of H;

Fig. 3 is z described in embodiments of the invention three _aowith α _awith H relation curve.

Embodiment

Embodiment one:

The present embodiment is a kind of low water head liquid gas energy conversion equipment, as shown in Figure 1.Device described in the present embodiment comprises: the ascending tube 1 being arranged on mill weir 2 high water level side, described ascending tube is connected with baffle pipe 3, described baffle pipe is connected with the down tube 6 of mill weir low water level side, described down tube is arranged ventilation point 4, described ventilation point is arranged the sucking pipe connecting air turbine 5, described air turbine is connected with generator, and the described position of ventilation point on down tube is:

$z_A=z_C+H+\frac{p_C-p_A}{\mathrm{\ρg}}-[L_A+{\left(\frac{n}{1-{\mathrm{\α}}_A}\right)}^2+(L_C-1){\left(\frac{n}{1-{\mathrm{\α}}_C}\right)}^2]\frac{{V_1}^2}{2g}$

Described siphonal sectional area is:

A ₁＝A＝Q/V ₁

In formula: z _athe position elevation of-ventilation point, z _cthe position elevation of-down tube outlet, H-acting head (water-heads of mill weir both sides), p _athe pressure of-ventilation point, p _cthe pressure of-water outlet, L _a-ventilation point local loosening, α _athe voids at-ventilation point place, n-ventilation point upstream and downstream pipeline area compares n=A ₁/ A, when syphon tube basal area is along Cheng Xiangtong then n=1, L _c-water outlet local loosening, V ₁-ascending tube flow velocity, α _cthe voids of-water outlet, g-gravity accleration, the density of ρ-water, A ₁the syphon tube sectional area of-ventilation point, A-down tube sectional area, Q-flow.

Ascending tube described in the present embodiment, baffle pipe and down tube form complete syphon tube, use these nouns of line pipe, baffle pipe and down tube, just convenient in order to describe, and do not have other particular meanings.Water is drawn to low one end of mill weir water level 7 from one end 9 that mill weir water level is high by the syphon tube described in the present embodiment, and whole siphonal water pipe diameter is consistent, and namely the diameter dimension of three sections of pipes is identical.Reference level 8 is base level faces of position grid DEM, such as sea level.

When current flow in syphon tube, can produce negative pressure in syphon tube, therefore, the present embodiment arranges ventilation point at the connection part of syphon tube baffle pipe and down tube.Represent the position of ventilation point in Fig. 1 with the A of letter capitalization, the A expressing syphon tube sectional area here in the A capitalized and each embodiment of the present invention is not a concept, in the footnote that capitalization the A here and capitalization C of water outlet occurs, and expression position, such as: α _arepresent ventilation point voids and A point voids; And α _cthen represent the voids of water outlet, i.e. C point voids.

Must be appropriate at the air that sucks of ventilation point, too much syphon tube just may be made to stop, mechanical efficiency is low at least.Described turbo machine can be axial flow type air turbo machine, also can be radial inflow air turbine.Namely air turbine described in the present embodiment is different from steam turbine and is also different from gas turbine.These two kinds of turbo machines are by concentrating the dynamic air-flow ejected to promote.Air turbine described in the present embodiment is then by the raw power of the air miscarriage sucked.Therefore, the air turbine described in the present embodiment and steam turbine and gas turbine have obvious difference in shape.The blade of the inlet end of steam turbine and gas turbine is less, and outlet side blade is larger.And air turbine described in the present embodiment is just in time contrary, the blade of inlet end is comparatively large, and the blade of outlet side is less.

Embodiment two:

The present embodiment is the improvement implementing embodiment one, is the refinement of embodiment one about air turbine.Air turbine described in the present embodiment is the one in axial flow turbine or radial-inward turbine.

Radial-inward turbine described in the present embodiment is for traditional radial outward flow turbine.In traditional radial outward flow turbine, the flowing of air is first along the axial flow at turbine center, then turns on turbine face, with centrifugal form, radial from the center of turbine along turbine, flows out from turbine edge.This type of flow be because in traditional turbo machine with the air stream of kinetic energy be from concentrate become diffusion.And be that to have dispersion to become concentrated with the air-flow direction of kinetic energy in air turbine described in the present embodiment.The flowing of its air on turbine face is just contrary: be enter turbine by turbine edge, turn, move along turbine central axis, therefore, be called radial-inward turbine after focusing on turbine center.

Research shows, in the waterpower air pump system described in the present embodiment, use radial inflow air turbine than the better effects if using axial flow type air turbo machine, efficiency is higher.

Embodiment three:

The present embodiment is the design method of low water head liquid gas energy conversion equipment described in above-described embodiment, and the step of described method is as follows:

Calculate the step of the voids of ventilation point, the mean velocity in section of pressure and ascending tube current: for maximizes water can be converted to gas can efficiency eta be voids, pressure and flow velocity according to calculating ventilation, according to maximum efficiency formula:

$\max \left\{\mathrm{\η}(p_A,V_1,{\mathrm{\α}}_A)\right\}=\max \left\{\frac{k}{k-1}\frac{p_A}{\mathrm{ng}{\mathrm{\ρ}}_{w}H}({\left(\frac{p_a}{p_A}\right)}^{(k-1)/k}-1){\mathrm{\α}}_A(\frac{n}{1-{\mathrm{\α}}_A}-\frac{V_r}{V_1})\right\}$

The constraint conditio maximizing η is:

$H=K\frac{{V_1}^2}{2g}+B---\left[1\right]$

$K=L_A+n^2(\frac{L_C}{{(1-{\mathrm{\α}}_C)}^2}+\frac{L_{\mathrm{AC}}}{{(1-{\mathrm{\α}}_A)}^2})---\left[2\right]$

$B=\frac{c_1}{\mathrm{\ρg}}[\frac{1}{V_r+c_2}\ln \frac{(c_2+V_r{\mathrm{\α}}_A)(1-{\mathrm{\α}}_C)}{(1-{\mathrm{\α}}_A)(c_2+V_r{\mathrm{\α}}_C)}-\frac{1}{c_2}\ln \frac{(c_2+V_r{\mathrm{\α}}_A){\mathrm{\α}}_C}{(c_2+V_r{\mathrm{\α}}_C){\mathrm{\α}}_A}-\frac{1}{c_2+V_r{\mathrm{\α}}_A}+\frac{1}{c_2+V_r{\mathrm{\α}}_C}]---\left[3\right]$

${\mathrm{\α}}_C=\frac{-(c_1+c_2{p}_C)+\sqrt{{(c_1+c_2{p}_C)}^2+4c_1{V}_r{p}_C}}{2V_r{p}_C}---\left[4\right]$

$c_1=p_A{\mathrm{\α}}_A(n\frac{V_1}{(1-{\mathrm{\α}}_A)}-V_r)---\left[5\right]$

c ₂＝nV ₁-V _r[6]

0＜α _A＜0.3[7]

p _A＞p _v[8]

In formula: α _athe voids at-ventilation point place, p _athe pressure at-ventilation point place, V ₁-ascending tube flow velocity, p _vthe vaporization pressure of-local liquid is known quantity, and H-acting head is known quantity, L _a-ventilation point local loosening is known quantity, L _c-water outlet local loosening is known quantity, L _aCpipeline waterhead fall between-ventilation point to water outlet is known quantity, p _a-local atmospheric pressure is known quantity, ρ _a-air density is known quantity, and k-gas adiabatic exponent is known quantity, g-gravity accleration, is known quantity, V _rthe speed difference of-current and air-flow is known quantity, n-ventilation point upstream and downstream pipeline area ratio, when syphon tube section area is along Cheng Xiangtong then n=1, p _cthe pressure of-water outlet, gets p in process of calculation analysis _c=p _afor known quantity, Two-level Optimization method is used to calculate the voids α of ventilation point A _a, pressure p _a, ascending tube current mean velocity in section V ₁;

Calculate the step of ventilation point height: for passing through formula:

$z_A=z_C+H+\frac{p_C-p_A}{\mathrm{\ρg}}-[L_A+{\left(\frac{n}{1-{\mathrm{\α}}_A}\right)}^2+(L_C-1){\left(\frac{n}{1-{\mathrm{\α}}_C}\right)}^2]\frac{{V_1}^2}{2g}$

Calculate the position height of ventilation point in syphon tube down tube;

Calculate the step of syphon tube sectional area: for passing through formula:

A ₁＝A＝Q/V ₁

Calculate siphonal sectional area.

The basic ideas of the present embodiment are the optimum efficiencies by studying conventional syphon tube liquid gas shift (waterpower air pump), obtain the optimised form of liquid-gas transfer device.Conventional siphonic system refers to that caliber is the siphon system of constant along journey, and as shown in Figure 1, wherein in down tube gas-liquid mixture, the mean velocity in section of current is variablees, instead of constant.

For the ease of analyzing, suppose: the flowing in syphon tube is steady flow; Gas is temperature-resistant in flow process; The compressibility of water body can be ignored.Analyze the mathematical model used:

1, the relation of voids and hydraulic pressure and flow velocity:

When representing voids with α, i.e. the volume of air and the ratio of control volume volume in control volume shown in Fig. 1, then the averag density of control volume gas-liquid mixture is

ρ _e＝αρ _a+ρ(1-α)≈ρ(1-α)(1)

In formula: ρ _ethe averag density of=gas-liquid mixture, kg/m ³; ρ _a=gas density, kg/m ³; The density of ρ=water, kg/m ³.The test of Wallis (1969) shows, as α > 0.3, bubble causes siphon flow disruption with regard to Hui Bian great, so the necessary condition maintaining normal gas-liquid two flowing is α < 0.3.

When gas is considered as perfect gas, then equation of state of gas can be expressed as

In formula: the pressure of p=fluid, Pa; m ³; The quality of m=gas, kg; R=universal gas constant; The kelvin temperature of T=gas, DEG C.

For steady flow, the throughput of down tube AC section two section air is

Q _a＝αA(V-V _r)＝α _AA(V _A-V _r)(3)

In formula: Q _a=throughput, m ³/ s; The mean velocity of V=section current, m/s; A=cross-sectional area, m ²; V _rthe speed difference of=current and air-flow, m/s; Subscript A=ventilation point A Outlet Section, i.e. AC section admission section.

The volume of gas in unit time the function relation that can be obtained voids and hydraulic pressure by formula (2) is

pαA(V-V _r)＝p _Aα _AA(V _A-V _r)(4)

The equation of continuity of down tube current can be described as

VA(1-α)＝V _AA(1-α _A)＝V ₁A ₁(5)

In formula: V ₁and A ₁the mean velocity of=ascending tube (ventilation point A upstream line) and area.

Can obtain from formula (5)

$V_A=\frac{V_1{A}_1}{A(1-{\mathrm{\&alpha;}}_A)}=n\frac{V_1}{(1-{\mathrm{\&alpha;}}_A)}---\left(6\right)$

$V=\frac{V_1{A}_1}{A(1-\mathrm{\&alpha;})}=n\frac{V_1}{(1-\mathrm{\&alpha;})}---\left(7\right)$

In formula: for constant, when siphonic system caliber is constant along journey, n=1.

Wushu (6) and formula (7) substitute into formula (4) and obtain

$p=\frac{p_A{\mathrm{\&alpha;}}_A(n\frac{V_1}{(1-{\mathrm{\&alpha;}}_A)}-V_r)}{\mathrm{\&alpha;}(n\frac{V_1}{(1-\mathrm{\&alpha;})}-V_r)}---\left(8\right)$

Arrange

$p=\frac{c_1(1-\mathrm{\&alpha;})}{\mathrm{\&alpha;}(c_2+V_r\mathrm{\&alpha;})}---\left(9\right)$

In formula: $c_1=p_A{\mathrm{\&alpha;}}_A(n\frac{V_1}{(1-{\mathrm{\&alpha;}}_A)}-V_r),$ c ₂＝nV ₁-V _r。

Formula (9) can be rewritten as

V _rα ²p+(c ₁+c ₂p)α-c ₁＝0

Solve

$\mathrm{\&alpha;}=\frac{-(c_1+c_{2}p)+\sqrt{{(c_1+c_{2}p)}^2+4c_1{V}_{r}p}}{2V_{r}p}---\left(10\right)$

For AC section, p _a, α _a, V ₁be import boundary conditions, can known quantity be thought.

The research of French and Widden (2001) shows, V _rchange very little, can constant be thought, V _r≈ 0.25m/s.Like this, can obtain formula (9) differential

$\mathrm{dp}=-c_1\{\frac{1}{\mathrm{\&alpha;}(c_2+V_r\mathrm{\&alpha;})}+\frac{(1-\mathrm{\&alpha;})(c_2+2V_r\mathrm{\&alpha;})}{{\left[\mathrm{\&alpha;}(c_2+V_r\mathrm{\&alpha;})\right]}^2}\}\mathrm{d\&alpha;}---\left(11\right)$

2, breather line momentum equation:

It is very little that the frictional loss of Fig. 1 Liquid Flow pipeline section is compared with gravity item, and reduction can be concentrated in the loss item of upstream and downstream.When reverse as a reference with flow direction, then the momentum equation of control volume can be described as

$\mathrm{pA}-(\mathrm{pA}+A\frac{\mathrm{dp}}{\mathrm{dz}}\mathrm{\&delta;z})-g{\mathrm{\&rho;}}_e\mathrm{A\&delta;z}={\mathrm{\&rho;}}_e\mathrm{AV}\frac{\mathrm{dV}}{\mathrm{dz}}\mathrm{\&delta;z}---\left(12\right)$

In formula: A=pipeline area, m ².The above formula equal sign left side: Section 1 is the water pressure acted on bottom control volume, Section 2 is the water pressure acting on control volume top, and Section 3 is the gravity acted on control volume.It is the knots modification of water body momentum on the right of equal sign.

Wushu (1) substitutes into formula (12) and can obtain

$-\mathrm{dp}-\frac{\mathrm{\&alpha;}}{1-\mathrm{\&alpha;}}\mathrm{dp}-\mathrm{\&rho;gdz}=\mathrm{\&rho;VdV}---\left(13\right)$

Wushu (11) substitutes into (13) and obtains

$-\frac{\mathrm{dp}}{\mathrm{\&rho;g}}+\frac{c_1}{\mathrm{\&rho;g}}\{\frac{1}{(c_2+V_r\mathrm{\&alpha;})(1-\mathrm{\&alpha;})}+\frac{(c_2+2V_r\mathrm{\&alpha;})}{\mathrm{\&alpha;}{(c_2+V_r\mathrm{\&alpha;})}^2}\}\mathrm{d\&alpha;}-\mathrm{dz}=\frac{\mathrm{VdV}}{g}---\left(14\right)$

Formula (14) integration is obtained

$-\frac{1}{\mathrm{\&rho;g}}\underset{p_C}{\overset{p_A}{\&Integral;}}\mathrm{dp}+\frac{c_1}{\mathrm{\&rho;g}}\underset{{\mathrm{\&alpha;}}_C}{\overset{{\mathrm{\&alpha;}}_A}{\&Integral;}}\{\frac{1}{(c_2+V_r\mathrm{\&alpha;})(1-\mathrm{\&alpha;})}+\frac{(c_2+2V_r\mathrm{\&alpha;})}{\mathrm{\&alpha;}{(c_2+V_r\mathrm{\&alpha;})}^2}\}\mathrm{d\&alpha;}-\underset{z_C}{\overset{z_A}{\&Integral;}}\mathrm{dz}=\frac{1}{g}\underset{V_C}{\overset{V_A}{\&Integral;}}\mathrm{VdV}---\left(15\right)$

Can obtain

$-\frac{p_A-p_C}{\mathrm{\&rho;g}}+B-(z_A-z_C)=\frac{V_A^2-V_C^2}{2g}---\left(16\right)$

In formula:

$B=\frac{c_1}{\mathrm{\&rho;g}}\underset{{\mathrm{\&alpha;}}_C}{\overset{{\mathrm{\&alpha;}}_A}{\&Integral;}}\{\frac{1}{(c_2+V_r\mathrm{\&alpha;})(1-\mathrm{\&alpha;})}+\frac{(c_2+2V_r\mathrm{\&alpha;})}{\mathrm{\&alpha;}{(c_2+V_r\mathrm{\&alpha;})}^2}\}\mathrm{d\&alpha;}=$

$=\frac{c_1}{\mathrm{\&rho;g}}{[\frac{1}{V_r+c_2}\ln \frac{c_2+V_r\mathrm{\&alpha;}}{1-\mathrm{\&alpha;}}-\frac{1}{c_2}\ln \frac{c_2+V_r\mathrm{\&alpha;}}{\mathrm{\&alpha;}}-\frac{1}{c_2+V_r\mathrm{\&alpha;}}]}_{{\mathrm{\&alpha;}}_C}^{{\mathrm{\&alpha;}}_A}$

Arrange

$B=\frac{c_1}{\mathrm{\&rho;g}}[\frac{1}{V_r+c_2}\ln \frac{(c_2+V_r{\mathrm{\&alpha;}}_A)(1-{\mathrm{\&alpha;}}_C)}{(1-{\mathrm{\&alpha;}}_A)(c_2+V_r{\mathrm{\&alpha;}}_C)}-\frac{1}{c_2}\ln \frac{(c_2+V_r{\mathrm{\&alpha;}}_A){\mathrm{\&alpha;}}_C}{(c_2+V_r{\mathrm{\&alpha;}}_C){\mathrm{\&alpha;}}_A}-\frac{1}{c_2+V_r{\mathrm{\&alpha;}}_A}+\frac{1}{c_2+V_r{\mathrm{\&alpha;}}_C}]---\left(17\right)$

With French and Widden (2001) similarly, B is the head because the buoyancy of bubble produces, and is called buoyancy head.

Can be obtained by formula (10)

${\mathrm{\&alpha;}}_C=\frac{-(c_1+c_2{p}_C)+\sqrt{{(c_1+c_2{p}_C)}^2+4c_1{V}_r{p}_C}}{2V_r{p}_C}---\left(18\right)$

Formula (16) can be rewritten as

$(\frac{p_A}{\mathrm{\&rho;g}}+z_A+\frac{V_A^2}{2g})-(\frac{p_C}{\mathrm{\&rho;g}}+z_C+\frac{V_C^2}{2g})=B---\left(19\right)$

Above formula is exactly the momentum equation of ventilation syphon tube current.

When considering AC segment pipe loss of head, formula (19) can be rewritten as

$(\frac{p_A}{\mathrm{\&rho;g}}+z_A+\frac{V_A^2}{2g})-(\frac{p_C}{\mathrm{\&rho;g}}+z_C+\frac{V_C^2}{2g})=B+L_{\mathrm{AC}}\frac{V_A^2}{2g}---\left(20\right)$

In formula: L _aCthe waterhead fall of=AC section, local loosening and friction factor of head loss.

3, siphonic system momentum equation:

Syphon mouth does not have air in ventilation point A admission section pipeline section in FIG, can obtain according to bernoulli equation

$\frac{p_a}{\mathrm{\&rho;g}}+z_1=\frac{p_A}{\mathrm{\&rho;g}}+z_A+\frac{V_A^2}{2g}+L_A\frac{{V_1}^2}{2g}---\left(21\right)$

In formula: p _a=water surface atmospheric pressure, Pa; z ₁=weir upper pool elevation, m; L _a=syphon mouth, to the waterhead fall of ventilation point A Outlet Section pipeline section, comprises the local loosening of import local loosening, friction factor of head loss and ventilation point A micro-section.

C to the bernoulli equation of syphon tube outlet pipe section current is

$\frac{p_C}{\mathrm{\&rho;g}}+z_C+\frac{V_C^2}{2g}=\frac{p_a}{\mathrm{\&rho;g}}+z_2+L_C\frac{V_C^2}{2g}---\left(22\right)$

In formula: z ₂=weir the level of tail water, m; L _c=C, to the waterhead fall of syphon tube outlet pipe section, comprises outlet local and friction factor of head loss.

Can be obtained by formula (21) and formula (22)

$(\frac{p_A}{\mathrm{\&rho;g}}+z_A+\frac{V_A^2}{2g})-(\frac{p_C}{\mathrm{\&rho;g}}+z_C+\frac{V_C^2}{2g})=H-(L_A\frac{{V_1}^2}{2g}+L_C\frac{V_C^2}{2g})---\left(23\right)$

In formula: H=z ₁-z ₂for acting head, i.e. the water-head of weir upstream and downstream, m.

Wushu (20) substitution formula (23) can obtain siphonic system momentum equation and be

$H=(L_A\frac{{V_1}^2}{2g}+L_C\frac{V_C^2}{2g}+L_{\mathrm{AC}}\frac{{V_A}^2}{2g})+B---\left(24\right)$

Can be obtained by formula (7)

$V_C=n\frac{V_1}{(1-{\mathrm{\&alpha;}}_C)}---\left(25\right)$

Wushu (6) and formula (25) substitute into formula (24) and can obtain

$H=K\frac{{V_1}^2}{2g}+B---\left(26\right)$

In formula:

$K=L_A+n^2(\frac{L_C}{{(1-{\mathrm{\&alpha;}}_C)}^2}+\frac{L_{\mathrm{AC}}}{{(1-{\mathrm{\&alpha;}}_A)}^2})---\left(27\right)$

Formula (26) the right Section 1 is total loss of head that water flow produces, and Section 2 B is the buoyancy head of bubble.So formula (26) shows acting head H, namely drives head, equal the algebraic sum of the total loss of head of syphon tube and buoyancy head.Increase Throughput and can increase buoyancy phase, but can water velocity V be reduced ₁.In addition, reduce siphonal resistance coefficient K and can increase flow rate of water flow.When not having gas in pipeline, buoyancy head B=0, formula (27) is reduced to the Bernoulli energy equation of current, and therefore, the size of B reflects the size that water can convert gas energy to.In other words, formula (20) describe water can gentle can function relation each other.

Can be obtained by formula (23)

$z_A=z_C+H+\frac{p_C-p_A}{\mathrm{\&rho;g}}-[L_A+{\left(\frac{n}{1-{\mathrm{\&alpha;}}_A}\right)}^2+(L_C-1){\left(\frac{n}{1-{\mathrm{\&alpha;}}_C}\right)}^2]\frac{{V_1}^2}{2g}---\left(28\right)$

4, the input power of air turbine and efficiency:

The working procedure of gas in air turbine, because gas and extraneous heat exchange are little, can be similar to and regard adiabatic process as.The unit mass flow power E that can obtain air turbine consumption according to law of conservation of energy is:

$E=\frac{k}{k-1}\frac{p_2}{{\mathrm{\&rho;}}_{a2}}({\left(\frac{p_1}{p_2}\right)}^{(k-1)/k}-1)+g(z_1-z_2)+\frac{C_1^2-C_2^2}{2}---\left(29\right)$

In formula: k=gas adiabatic exponent, general desirable k=1.4; P=pressure, Pa:C=gas mean velocity, m/s; Z=elevation, m; Subscript 1 and 2 represents the inlet/outlet section of air turbine respectively.

In the ordinary course of things, have

$\frac{k}{k-1}\frac{p_2}{{\mathrm{\&rho;}}_{a2}}({\left(\frac{p_1}{p_2}\right)}^{(k-1)/k}-1)>>g(z_1-z_2)+\frac{C_1^2-C_2^2}{2}$

So E can be expressed as

$E=\frac{k}{k-1}\frac{p_2}{{\mathrm{\&rho;}}_{a2}}({\left(\frac{p_1}{p_2}\right)}^{(k-1)/k}-1)---\left(30\right)$

When not considering the loss due to duct friction between air turbine to siphonic system, then p ₁≈ p _a(atmospheric pressure) and p ₂=p _a, like this, the input power of air turbine is

$P={\mathrm{\&rho;}}_{a2}{Q}_{a}E=\frac{k}{k-1}{p}_A({\left(\frac{p_a}{p_A}\right)}^{(k-1)/k}-1)Q_a---\left(31\right)$

In formula: the input power of P=air turbine, W; Q _a=enter the throughput that A is put in syphon tube ventilation.

Can be obtained by formula (3)

$Q_a={\mathrm{\&alpha;}}_A{A}_A(V_A-V_r)={\mathrm{\&alpha;}}_A{A}_A(\frac{n}{1-{\mathrm{\&alpha;}}_A}{V}_1-V_r)$

So

$P=\frac{k}{k-1}{p}_A({\left(\frac{p_a}{p_A}\right)}^{(k-1)/k}-1){\mathrm{\&alpha;}}_A{A}_A(\frac{n}{1-{\mathrm{\&alpha;}}_A}{V}_1-V_r)---\left(32\right)$

Because the power of current is

P _w＝gρ _wHQ _w＝gρ _wHA ₁V ₁(33)

So the efficiency that water can be converted to gas energy is

$\mathrm{\&eta;}=\frac{P}{P_w}=\frac{k}{k-1}\frac{p_A}{\mathrm{ng}{\mathrm{\&rho;}}_{w}H}({\left(\frac{p_a}{p_A}\right)}^{(k-1)k}-1){\mathrm{\&alpha;}}_A(\frac{n}{1-{\mathrm{\&alpha;}}_A}-\frac{V_r}{V_1})---\left(34\right)$

5, the optimal design of waterpower air pump:

Waterpower air pump optimal design target be water can be converted to gas can efficiency eta maximum, namely

$\max \left\{\mathrm{\&eta;}(p_A,V_1,{\mathrm{\&alpha;}}_A)\right\}=\max \left\{\frac{k}{k-1}\frac{p_A}{\mathrm{ng}{\mathrm{\&rho;}}_{w}H}({\left(\frac{p_a}{p_A}\right)}^{(k-1)/k}-1){\mathrm{\&alpha;}}_A(\frac{n}{1-{\mathrm{\&alpha;}}_A}-\frac{V_r}{V_1})\right\}---\left(35\right)$

Constraint conditio

$H=K\frac{{V_1}^2}{2g}+B---\left(36\right)$

$K=L_A+n^2(\frac{L_C}{{(1-{\mathrm{\&alpha;}}_C)}^2}+\frac{L_{\mathrm{AC}}}{{(1-{\mathrm{\&alpha;}}_A)}^2})---\left(37\right)$

$B=\frac{c_1}{\mathrm{\&rho;g}}[\frac{1}{V_r+c_2}\ln \frac{(c_2+V_r{\mathrm{\&alpha;}}_A)(1-{\mathrm{\&alpha;}}_C)}{(1-{\mathrm{\&alpha;}}_A)(c_2+V_r{\mathrm{\&alpha;}}_C)}-\frac{1}{c_2}\ln \frac{(c_2+V_r{\mathrm{\&alpha;}}_A){\mathrm{\&alpha;}}_C}{(c_2+V_r{\mathrm{\&alpha;}}_C){\mathrm{\&alpha;}}_A}-\frac{1}{c_2+V_r{\mathrm{\&alpha;}}_A}+\frac{1}{c_2+V_r{\mathrm{\&alpha;}}_C}]---\left(38\right)$

${\mathrm{\&alpha;}}_C=\frac{-(c_1+c_2{p}_C)+\sqrt{{(c_1+c_2{p}_C)}^2+4c_1{V}_r{p}_C}}{2V_r{p}_C}---\left(39\right)$

$c_1=p_A{\mathrm{\&alpha;}}_A(n\frac{V_1}{(1-{\mathrm{\&alpha;}}_A)}-V_r)---\left(40\right)$

c ₂＝nV ₁-V _r(41)

0＜α _A＜0.3(42)

p _A＞p _v(43)

In formula: p _vthe vaporization pressure of=local liquid, Pa.In the process of optimal design, controlled variable or design parameter are: the voids α of down tube section A _a, pressure p _a, ascending tube current mean velocity in section V ₁.Known quantity is: acting head H, pipeline waterhead fall L _a, L _c, L _aC, local atmospheric pressure p _a, air density ρ _a, gas adiabatic exponent k, gravity acceleration g, the speed difference V of current and air-flow _r, some A upstream and downstream pipeline area compares n.Desirable p in process of calculation analysis _c=p _afor known quantity.When syphon tube basal area along journey mutually simultaneously, n=1.

Observation type (36)-Shi (41), at given α _aand p _acondition under, buoyancy head B and α in formula (38) and formula (39) _cjust controlled variable V ₁function, and formula (37) in K be α _cfunction, so only have a unknown quantity V in formula (36) ₁, can be calculated by iterative numerical and solve V ₁.

Once determine the controlled variable α making maximizing efficiency _a, p _a, V ₁optimum value, then can be determined the position elevation z of siphonic system suction port by formula (28) _a, and by A ₁=A=Q/V ₁determine siphonal sectional area, thus determine the optimum design of siphonic system.

Obviously, waterpower air pump Optimized model belongs to nonlinear programming problem, has a variety of mode, such as: Bao Weier (Powell) algorithm and the variation quantity algorithm etc. utilizing difference coefficient, can adopt the method for numerical calculation to obtain its optimal solution.

In the process of design, generally desirable atmospheric pressure p _a=100000Pa, air density ρ _a=1.2kg/m ³, gas adiabatic exponent k=1.4, gravity acceleration g=9.8m/s ², the speed difference V of current and air-flow _r=0.25m/s.As shown in Figure 1, when hypothesis syphon tube inlet/outlet respectively arranges a butterfly valve, then ascending tube local resistance item is made up of import, a butterfly valve and two elbows, desirable L _a=1.3; Down tube resistance to flow output coefficient L _c=1; If L _aC=0.3.

Fig. 2 and Fig. 3 respectively illustrate under these conditions given acting head H value time optimum efficiency η _oand z _aowith α _aone-to-one relationship.Observe Fig. 2 and Fig. 3, at 0 < α _ain the interval of < 0.3, following conclusion can be drawn:

(1) as H≤1.5m, then there is a α _avalue and z _aomake the efficiency of waterpower air pump maximum.Maximal efficiency excursion is 70%≤η _max≤ 86%, and acting head H is large, then η _maxgreatly.

(2) as H > 1.7m, then optimum efficiency η _oalong with voids α _aincrease and monotone increasing.In other words, larger α is adopted in this case _athe efficiency of waterpower air pump can be improved.

(3) as H>=2.0m, for given α _avalue, along with the increase of acting head H, optimum efficiency η _oto significantly reduce.

By above analysis, following conclusion can be drawn:

(1) as H≤1.5m, the maximal efficiency excursion of waterpower air pump is 70%≤η _max≤ 86%, and acting head H is large, then η _maxgreatly.

(2) as H > 1.7m, then optimum efficiency η _oalong with voids α _aincrease and monotone increasing.

(3) as H>=2.0m, for given α _avalue, along with the increase of acting head H, optimum efficiency η _oto significantly reduce.

Embodiment four:

The present embodiment is the improvement of embodiment three, the refinement of embodiment three about " calculating the voids of ventilation point, the pressure of ventilation point and the step of ascending tube flow velocity ", the present embodiment calculates voids, pressure and ascending tube flow velocity and uses two layers of loop optimization numeration, and described numeration comprises following sub-step:

1) input initial data, and make efficiency eta _o=0; For maximal efficiency;

2) first layer circulation starts: make controlled variable α _a=α _{a, i}=α _{a, 1}for initial value, wherein subscript i is cyclic variable;

3) second layer circulation starts: make controlled variable p _a=p _{a, j}=p _{a, 1}for just establishing value, wherein subscript j is cyclic variable;

4) controlled variable V is solved ₁, iterative numerical calculates V ₁program be:

1. V is made ₁=V ₁₀;

2. by formula

$c_1=p_A{\mathrm{\&alpha;}}_A(n\frac{V_1}{(1-{\mathrm{\&alpha;}}_A)}-V_r)$ And formula

c ₂＝nV ₁-V _r

Calculate c respectively ₁and c ₂, then by formula

${\mathrm{\&alpha;}}_C=\frac{-(c_1+c_2{p}_C)+\sqrt{{(c_1+c_2{p}_C)}^2+4c_1{V}_r{p}_C}}{2V_r{p}_C}$

Calculate α _c;

3. by formula

${\mathrm{\&alpha;}}_C=\frac{-(c_1+c_2{p}_C)+\sqrt{{(c_1+c_2{p}_C)}^2+4c_1{V}_r{p}_C}}{2V_r{p}_C}$

And formula

$B=\frac{c_1}{\mathrm{\&rho;g}}[\frac{1}{V_r+c_2}\ln \frac{(c_2+V_r{\mathrm{\&alpha;}}_A)(1-{\mathrm{\&alpha;}}_C)}{(1-{\mathrm{\&alpha;}}_A)(c_2+V_r{\mathrm{\&alpha;}}_C)}-\frac{1}{c_2}\ln \frac{(c_2+V_r{\mathrm{\&alpha;}}_A){\mathrm{\&alpha;}}_C}{(c_2+V_r{\mathrm{\&alpha;}}_C){\mathrm{\&alpha;}}_A}-\frac{1}{c_2+V_r{\mathrm{\&alpha;}}_A}+\frac{1}{c_2+V_r{\mathrm{\&alpha;}}_C}]$

Calculate B and K respectively; If H < is B, then jump out the second circulation, namely turn step 7); Otherwise, by formula

$H=K\frac{{V_1}^2}{2g}+B$

Solve unknown quantity V ₁;

4. judge | V ₁-V ₁₀|≤10 ^-3if set up, then V ₁be exactly given α _{a, i}and p _{a, j}solution under condition, then turns next step; Otherwise, get V ₁₀=0.5 (V ₁+ V ₁₀), repeat to walk 2.-4.;

5) elevation z is calculated respectively _awith efficiency eta=η (p _{a, j}, V _{1, i, j}, α _{a, i}), if η > is η _o, then make: η _o=η, p _ao=p _{a, j}, α _ao=α _{a, i}, V _o=V ₁, z _ao=z _a;

6) second layer circulation is continued: make j=j+1 and p _{a, j}=p _{a, j-1}-δ, wherein δ > 0 is fractional increments.If cyclic variable j is less than prespecified CLV ceiling limit value, then turn step 4); Otherwise, turn next step;

7) first layer circulation is continued: make i=i+1, and α _{a, i}=α _{a, i-1}+ ε, wherein ε > 0 is fractional increments.If cyclic variable i is less than prespecified CLV ceiling limit value, or α _{a, i}≤ 0.3, then turn step 4); Otherwise, turn next step;

8) optimal solution: η is exported _o=η, p _ao=p _{a, j}, α _ao=α _{a, i}, V _o=V ₁, z _ao=z _a.Calculate and terminate;

Wherein: subscript o represents optimum value optimal.As: η _orepresent the optimum value etc. of η.

Finally it should be noted that, below only in order to technological scheme of the present invention to be described and unrestricted, although with reference to preferred arrangement scheme to invention has been detailed description, those of ordinary skill in the art is to be understood that, can to technological scheme of the present invention (the such as priority of siphonal form, computational process, the formula etc. used) modify or equivalent replacement, and do not depart from the spirit and scope of technical solution of the present invention.

Claims (2)

1. the design method of a low water head liquid gas energy conversion equipment, described device comprises: the ascending tube being arranged on mill weir high water level side, described ascending tube is connected with baffle pipe, described baffle pipe is connected with the down tube of mill weir low water level side, described down tube is arranged ventilation point, described ventilation point is arranged the sucking pipe connecting air turbine, described air turbine is connected with generator, it is characterized in that, the step of described method is as follows:

Calculate the voids of ventilation point, the pressure of ventilation point and the step of ascending tube flow velocity: for maximizes water can be converted to gas can efficiency eta be voids, pressure and flow velocity according to calculating ventilation, according to maximum efficiency formula:

The constraint conditio maximizing η is:

c ₂＝nV ₁-V _r[6]

0＜α _A＜0.3[7]

p _A＞p _v[8]

In formula: α _athe voids at-ventilation point place, p _athe pressure at-ventilation point place, V ₁-ascending tube flow velocity, p _vthe vaporization pressure of-local liquid is known quantity, and H-acting head is known quantity, L _a-ventilation point local loosening is known quantity, L _c-water outlet local loosening is known quantity, L _aClocal loosening between-ventilation point to water outlet is known quantity, p _a-local atmospheric pressure is known quantity, ρ _a-air density is known quantity, and k-gas adiabatic exponent is known quantity, g-gravity accleration, is known quantity, V _rthe speed difference of-current and air-flow is known quantity, n-ventilation point upstream and downstream pipeline area ratio, when syphon tube section area is along Cheng Xiangtong then n=1, p _cthe pressure of-water outlet, gets p in process of calculation analysis _c=p _afor known quantity, Two-level Optimization method is used to calculate the voids α of ventilation point A _a, pressure p _a, ascending tube current mean velocity in section V ₁;

Calculate the step of ventilation point height: for passing through formula:

Calculate the position height of ventilation point in syphon tube down tube;

Calculate the step of syphon tube sectional area: for passing through formula:

A ₁＝A＝Q/V ₁

Calculate siphonal sectional area,

In formula: z _athe position elevation of-ventilation point, z _cthe position elevation of-down tube outlet, H-acting head, p _athe pressure of-ventilation point, p _cthe pressure of-water outlet, L _a-ventilation point local loosening, α _athe voids at-ventilation point place, n-ventilation point upstream and downstream pipeline area ratio, when syphon tube basal area is along Cheng Xiangtong then n=1, L _c-water outlet local loosening, V ₁-ascending tube flow velocity, α _cthe voids of-water outlet, g-gravity accleration, the density of ρ-water, A ₁the syphon tube sectional area of-ventilation point, A-down tube sectional area, Q-flow.

2. design method according to claim 1, it is characterized in that, in described " calculating the voids of ventilation point, the pressure of ventilation point and the step of ascending tube flow velocity ", calculating voids, pressure and ascending tube flow velocity is two layers of loop optimization numeration, and described numeration comprises following sub-step:

1) input initial data, and make efficiency eta _o=0; For maximal efficiency;

2) first layer circulation starts: make controlled variable α _a=α _a,i=α _{a, 1}for initial value, wherein subscript i is cyclic variable;

3) second layer circulation starts: make controlled variable p _a=p _a,j=p _{a, 1}for just establishing value, wherein subscript j is cyclic variable;

4) controlled variable V is solved ₁, iterative numerical calculates V ₁program be:

1. V is made ₁=V ₁₀;

2. by formula

and formula

c ₂＝nV ₁-V _r

Calculate c respectively ₁and c ₂, then by formula

Calculate α _c;

3. by formula

And formula

Calculate B and K respectively; If H < is B, then jump out the second circulation, namely turn step 7); Otherwise, by formula

Solve unknown quantity V ₁;

4. judge | V ₁-V ₁₀|≤10 ^-3if set up, then V ₁be exactly given α _a,iand p _a,jsolution under condition, then turns next step; Otherwise, get V ₁₀=0.5 (V ₁+ V ₁₀), repeat to walk 2.-4.;

5) elevation z is calculated respectively _awith efficiency eta=η (p _a,j, V _{1, i, j}, α _a,i), if η > is η _o, then make: η _o=η, p _ao=p _a,j, α _ao=α _a,i, V _o=V ₁, z _ao=z _a;

6) second layer circulation is continued: make j=j+1 and p _a,j=p _{a, j-1}-δ, wherein δ > 0 is fractional increments, if cyclic variable j is less than prespecified CLV ceiling limit value, then turns step 4); Otherwise, turn next step;

7) first layer circulation is continued: make i=i+1, and α _a,i=α _{a, i-1}+ ε, wherein ε > 0 is fractional increments, if cyclic variable i is less than prespecified CLV ceiling limit value, or α _a,i≤ 0.3, then turn step 4); Otherwise, turn next step;

8) optimal solution: η is exported _o=η, p _ao=p _a,j, α _ao=α _a,i, V _o=V ₁, z _ao=z _a.Calculate and terminate;

Wherein: footnote o represents optimum value optimal.