v = 0 , in [ m / s ] {\displaystyle v=0{\mbox{, in}}\left[m/s\right]}
v ≠ 0 , in [ m / s ] {\displaystyle v\neq 0{\mbox{, in}}\left[m/s\right]}
∂ V ∂ t = 0 {\displaystyle {\frac {\partial V}{\partial t}}=0}
∂ V ∂ t ≠ 0 {\displaystyle {\frac {\partial V}{\partial t}}\neq 0}
∂ V ∂ x = 0 {\displaystyle {\frac {\partial V}{\partial x}}=0}
∂ V ∂ x ≠ 0 {\displaystyle {\frac {\partial V}{\partial x}}\neq 0}
p = ρ ⋅ g ⋅ h ⇒ h = p ρ ⋅ g {\displaystyle p=\rho \cdot g\cdot h\Rightarrow h={\frac {p}{\rho \cdot g}}}
q v = v ¯ ⋅ A , in [ m 3 / s ] {\displaystyle q_{v}={\bar {v}}\cdot A{\mbox{, in}}\left[m^{3}/s\right]}
F h y d r o s t a t i s c h = p g e m i d d e l d ⋅ A = 1 2 ⋅ ρ ⋅ g ⋅ h ∗ h ⋅ b = 1 2 ⋅ ρ ⋅ h 2 ⋅ b {\displaystyle F_{hydrostatisch}=p_{gemiddeld}\cdot A={\frac {1}{2}}\cdot \rho \cdot g\cdot \ h*h\cdot b={\frac {1}{2}}\cdot \rho \cdot h^{2}\cdot b}
F ⋅ a = ∑ p ⋅ b ⋅ d y ⋅ y F ⋅ a = ∫ p ⋅ b ⋅ y ⋅ d y F ⋅ a = ∫ ρ ⋅ g ( h − y ) ⋅ b ⋅ y ⋅ d y 1 2 ⋅ ρ ⋅ g ⋅ h 2 ⋅ b ⋅ a = ∫ ρ ⋅ g ⋅ ( h − y ) ⋅ b ⋅ y ⋅ d y 1 2 ⋅ h 2 ⋅ a = ∫ ( h − y ) ⋅ y ⋅ d y 1 2 h 2 ⋅ a = ∫ ( h ⋅ y − y 2 ) ⋅ d y 1 2 h 2 ⋅ a = ∫ y = 0 y = h ( h ⋅ y − y 2 ) ⋅ d y 1 2 h 2 ⋅ a = [ h ⋅ 1 2 y 2 − 1 3 y 3 ] y = o y = h 1 2 h 2 ⋅ a = [ h ⋅ 1 2 h 2 − 1 3 h 3 ] − [ 0 − 0 ] = 1 2 h 3 − 1 3 h 3 1 2 h 2 ⋅ a = 1 6 h 3 1 2 ⋅ a = 1 6 h a = 1 3 h {\displaystyle {\begin{array}{rcl}F\cdot a&=&\sum p\cdot b\cdot dy\cdot y\\F\cdot a&=&\int p\cdot b\cdot y\cdot dy\\F\cdot a&=&\int \rho \cdot g\left(h-y\right)\cdot b\cdot y\cdot dy\\{\frac {1}{2}}\cdot {\color {Red}\rho }\cdot {\color {Red}g}\cdot h^{2}\cdot {\color {Red}b}\cdot a&=&\int {\color {Red}\rho }\cdot {\color {Red}g}\cdot \left(h-y\right)\cdot {\color {Red}b}\cdot y\cdot dy\\{\frac {1}{2}}\cdot h^{2}\cdot a&=&\int \left(h-y\right)\cdot y\cdot dy\\{\frac {1}{2}}h^{2}\cdot a&=&\int \left(h\cdot y-y^{2}\right)\cdot dy\\{\frac {1}{2}}h^{2}\cdot a&=&\int _{y=0}^{y=h}\left(h\cdot y-y^{2}\right)\,\cdot dy\\{\frac {1}{2}}h^{2}\cdot a&=&\left[h\cdot {\frac {1}{2}}y^{2}-{\frac {1}{3}}y^{3}\right]_{y=o}^{y=h}\\{\frac {1}{2}}h^{2}\cdot a&=&\left[h\cdot {\frac {1}{2}}h^{2}-{\frac {1}{3}}h^{3}\right]-\left[0-0\right]={\frac {1}{2}}h^{3}-{\frac {1}{3}}h^{3}\\{\frac {1}{2}}h^{2}\cdot a&=&{\frac {1}{6}}h^{3}\\{\frac {1}{2}}\cdot a&=&{\frac {1}{6}}h\\a&=&{\frac {1}{3}}h\end{array}}}
z 1 + h 1 = z 2 + h 2 = z 3 + H 3 = constant {\displaystyle {\begin{matrix}z_{1}+h_{1}=z_{2}+h_{2}=z_{3}+H_{3}={\mbox{constant}}\end{matrix}}}
H = z + p ρ ⋅ g + v 2 2 g = constant, in [ m ] {\displaystyle H=z+{\frac {p}{\rho \cdot g}}+{\frac {v^{2}}{2g}}={\mbox{constant, in}}\left[m\right]}
q v = constant ⇒ v 1 ⋅ A 1 = v 2 ⋅ A 2 = v 3 ⋅ A 3 {\displaystyle q_{v}={\mbox{constant}}\Rightarrow v_{1}\cdot A_{1}=v_{2}\cdot A_{2}=v_{3}\cdot A_{3}}
H = z 1 + p 1 ρ ⋅ g + v 1 2 2 g = z 2 + p 2 ρ ⋅ g + v 2 2 2 g = z 3 + p 3 ρ ⋅ g + v 3 2 2 g {\displaystyle H=z_{1}+{\frac {p_{1}}{\rho \cdot g}}+{\frac {v_{1}^{2}}{2g}}=z_{2}+{\frac {p_{2}}{\rho \cdot g}}+{\frac {v_{2}^{2}}{2g}}=z_{3}+{\frac {p_{3}}{\rho \cdot g}}+{\frac {v_{3}^{2}}{2g}}}
pi e ¨ tzometrisch niveau (p.n.) = z + p ρ ⋅ g {\displaystyle {\mbox{pi}}\mathrm {\ddot {e}} {\mbox{tzometrisch niveau (p.n.)}}=z+{\frac {p}{\rho \cdot g}}}
H = z 1 + p 1 ρ ⋅ g + v 1 2 2 g = z 2 + p 2 ρ ⋅ g + v 2 2 2 g + Δ H 1 − 2 {\displaystyle H=z_{1}+{\frac {p_{1}}{\rho \cdot g}}+{\frac {v_{1}^{2}}{2g}}=z_{2}+{\frac {p_{2}}{\rho \cdot g}}+{\frac {v_{2}^{2}}{2g}}+\Delta H_{1-2}}
E t o t a a l = E p o t e n t i e e l + E k i n e t i s c h , in [ N m ] E t o t a a l = m ⋅ g ⋅ ( z + h ) + 1 2 ⋅ m ⋅ v 2 E t o t a a l m ⋅ g = z + h + v 2 2 g = H E t o t a a l m ⋅ g = H E t o t a a l m ⋅ g ⇒ E t o t a a l = m ⋅ g ⋅ H ; m = ρ ⋅ V ⇒ E t o t a a l = ρ ⋅ V ⋅ g ⋅ H {\displaystyle {\begin{array}{rclcl}E_{totaal}&=&E_{potentieel}+E_{kinetisch}{\mbox{, in}}\left[Nm\right]\\E_{totaal}&=&m\cdot g\cdot \left(z+h\right)+{\frac {1}{2}}\cdot m\cdot v^{2}\\{\frac {E_{totaal}}{m\cdot g}}&=&z+h+{\frac {v^{2}}{2g}}&=&H\\{\frac {E_{totaal}}{m\cdot g}}&=&H\\{\frac {E_{totaal}}{m\cdot g}}&\Rightarrow &E_{totaal}=m\cdot g\cdot H;\ \ m=\rho \cdot V&\Rightarrow &E_{totaal}=\rho \cdot V\cdot g\cdot H\\\end{array}}}
Δ E = ρ ⋅ Δ V ⋅ g ⋅ H d E = ρ ⋅ d V ⋅ g ⋅ H {\displaystyle {\begin{array}{ccc}\Delta E&=&\rho \cdot \Delta V\cdot g\cdot H\\dE&=&\rho \cdot dV\cdot g\cdot H\end{array}}}
P ≡ d E d t = ρ ⋅ d V d t ⋅ g ⋅ H ; q v = d V d t ⇒ P = ρ ⋅ g ⋅ q v ⋅ H , in [ N m / s ] ; [ J / s ] ; [ W ] {\displaystyle P\equiv {\frac {dE}{dt}}=\rho \cdot {\frac {dV}{dt}}\cdot g\cdot H;\ \ q_{v}={\frac {dV}{dt}}\Rightarrow P=\rho \cdot g\cdot q_{v}\cdot H{\mbox{, in}}\left[Nm/s\right];\left[J/s\right];\left[W\right]}
Δ P 1 − 2 = ρ ⋅ q ⋅ q v ⋅ Δ H 1 − 2 {\displaystyle \Delta P_{1-2}=\rho \cdot q\cdot q_{v}\cdot \Delta H_{1-2}}
![{\displaystyle v=0{\mbox{, in}}\left[m/s\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b548b50e1e52c5409d904bbfe0208cf3e52439e2)
![{\displaystyle v\neq 0{\mbox{, in}}\left[m/s\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/121d3b04c9cba43f164919e4395aca826a1ab1f2)
![{\displaystyle q_{v}={\bar {v}}\cdot A{\mbox{, in}}\left[m^{3}/s\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8e2f5fa1ff29512a394aaf1b1fa23aa47e071fb4)
![{\displaystyle {\begin{array}{rcl}F\cdot a&=&\sum p\cdot b\cdot dy\cdot y\\F\cdot a&=&\int p\cdot b\cdot y\cdot dy\\F\cdot a&=&\int \rho \cdot g\left(h-y\right)\cdot b\cdot y\cdot dy\\{\frac {1}{2}}\cdot {\color {Red}\rho }\cdot {\color {Red}g}\cdot h^{2}\cdot {\color {Red}b}\cdot a&=&\int {\color {Red}\rho }\cdot {\color {Red}g}\cdot \left(h-y\right)\cdot {\color {Red}b}\cdot y\cdot dy\\{\frac {1}{2}}\cdot h^{2}\cdot a&=&\int \left(h-y\right)\cdot y\cdot dy\\{\frac {1}{2}}h^{2}\cdot a&=&\int \left(h\cdot y-y^{2}\right)\cdot dy\\{\frac {1}{2}}h^{2}\cdot a&=&\int _{y=0}^{y=h}\left(h\cdot y-y^{2}\right)\,\cdot dy\\{\frac {1}{2}}h^{2}\cdot a&=&\left[h\cdot {\frac {1}{2}}y^{2}-{\frac {1}{3}}y^{3}\right]_{y=o}^{y=h}\\{\frac {1}{2}}h^{2}\cdot a&=&\left[h\cdot {\frac {1}{2}}h^{2}-{\frac {1}{3}}h^{3}\right]-\left[0-0\right]={\frac {1}{2}}h^{3}-{\frac {1}{3}}h^{3}\\{\frac {1}{2}}h^{2}\cdot a&=&{\frac {1}{6}}h^{3}\\{\frac {1}{2}}\cdot a&=&{\frac {1}{6}}h\\a&=&{\frac {1}{3}}h\end{array}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fbdaa440295dbe76c6109bd240ddc5b284961325)
![{\displaystyle H=z+{\frac {p}{\rho \cdot g}}+{\frac {v^{2}}{2g}}={\mbox{constant, in}}\left[m\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2a20364600922b7221b3f8ab34eea6252ac25c88)
![{\displaystyle {\begin{array}{rclcl}E_{totaal}&=&E_{potentieel}+E_{kinetisch}{\mbox{, in}}\left[Nm\right]\\E_{totaal}&=&m\cdot g\cdot \left(z+h\right)+{\frac {1}{2}}\cdot m\cdot v^{2}\\{\frac {E_{totaal}}{m\cdot g}}&=&z+h+{\frac {v^{2}}{2g}}&=&H\\{\frac {E_{totaal}}{m\cdot g}}&=&H\\{\frac {E_{totaal}}{m\cdot g}}&\Rightarrow &E_{totaal}=m\cdot g\cdot H;\ \ m=\rho \cdot V&\Rightarrow &E_{totaal}=\rho \cdot V\cdot g\cdot H\\\end{array}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3629d2af4fb39815abeff123c697af27da3cd3f)
![{\displaystyle P\equiv {\frac {dE}{dt}}=\rho \cdot {\frac {dV}{dt}}\cdot g\cdot H;\ \ q_{v}={\frac {dV}{dt}}\Rightarrow P=\rho \cdot g\cdot q_{v}\cdot H{\mbox{, in}}\left[Nm/s\right];\left[J/s\right];\left[W\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/54f041a1766dc993a4fbad01b507b8032a36289b)
