下射式閘門 作用力及流量( Undershoot gate ,force and flow)

 

Google Play 商店 連結網址:

 明渠水理計算_曼寧公式

https://play.google.com/store/apps/details?id=com.yuchihung.manning

例題

已知條件：y1=3m ， y2=0.3m , B=10m

連續方程式(C.E.)：q=V1*y1=V2*y2

動量方程式(M.E.)：1/2*r*y1^2-1/2*r*y2^2-f=ρ*q*(V2-V1)

==> 1/2*r*(y1^2-y2^2)-f=ρ*q^2*(1/y2-1/y1)

==> f=1/2*r*(y1-y2)/(y1*y2)*[y1*y2*(y1+y2)-2*q^2/g]........(1)

上式中：V1=q/y1 ， V2=q/y2 ， r=ρ*g 

                 q=Q/B(單位寬度流量)

                 f=F/B(單位寬度作用力)

能量方程式(E.E.)：y1+V1^2/(2*g)=y2+V2^2/(2*g)

==> y1-y2=1/(2*g)*(q^2/y2^2-q^2/y1^2)

                 =q^2/(2*g)*(y1-y2)*(y1+y2)/(y1^2*y2^2)

==> q^2/g=2*y1^2*y2^2/(y1+y2) .........................(2)

將(2)式代入(1)式：

f=1/2*r*(y1-y2)/(y1*y2)*[y1*y2*(y1+y2)-4*y1^2*y2^2/(y1+y2)]

==> f=1/2*r*(y1-y2)^3/(y1+y2) .............................(3)

將已知條件代入(3)式：

F=f*B=1/2*(9.81*998)*(3-0.3)^3/(3+0.3)*10=291975.8(N)=291.9758(KN)

依據(2)式：

q=SQRT[2*y1^2*y2^2/(y1+y2)*g]=SQRT[2*3^2*0.3^2/(3+0.3)*9.81]=2.19450(cms/m)

Q=q*B=2.19450*10=21.9450(cms)

V1=q/y1=2.19450/3=0.7315(m/s)

V2=q/y2=2.19450/0.3=7.3150(m/s)

Google Play Store link URL:

 Open channel hydraulic calculation_Manning formula

https://play.google.com/store/apps/details?id=com.yuchihung.manning

example

Known conditions: y1=3m, y2=0.3m, B=10m

Continuity equation (CE): q=V1*y1=V2*y2

Momentum equation (ME): 1/2*r*y1^2-1/2*r*y2^2-f=ρ*q*(V2-V1)

==> 1/2*r*(y1^2-y2^2)-f=ρ*q^2*(1/y2-1/y1)

==> f=1/2*r*(y1-y2)/(y1*y2)*[y1*y2*(y1+y2)-2*q^2/g]........ (1)

In the above formula: V1=q/y1, V2=q/y2, r=ρ*g 

                 q=Q/B (flow rate per unit width)

                 f=F/B (force per unit width)

Energy Equation (EE): y1+V1^2/(2*g)=y2+V2^2/(2*g)

==> y1-y2=1/(2*g)*(q^2/y2^2-q^2/y1^2)

                 =q^2/(2*g)*(y1-y2)*(y1+y2)/(y1^2*y2^2)

==> q^2/g=2*y1^2*y2^2/(y1+y2) ...................( 2)

Substituting equation (2) into equation (1):

f=1/2*r*(y1-y2)/(y1*y2)*[y1*y2*(y1+y2)-4*y1^2*y2^2/(y1+y2)]

==> f=1/2*r*(y1-y2)^3/(y1+y2) .......................... ...(3)

Substitute the known conditions into equation (3):

F=f*B=1/2*(9.81*998)*(3-0.3)^3/(3+0.3)*10=291975.8(N)=291.9758(KN)

According to formula (2):

q=SQRT[2*y1^2*y2^2/(y1+y2)*g]=SQRT[2*3^2*0.3^2/(3+0.3)*9.81]=2.19450(cms/m)

Q=q*B=2.19450*10=21.9450(cms)

V1=q/y1=2.19450/3=0.7315(m/s)

V2=q/y2=2.19450/0.3=7.3150(m/s)