0
14kviews
Froude's Model law
0
394views

1) It is the law in which the models are based on froude's number which means for dynamic similarity between model and prototype the froude's number for both of them should be equal .

2) When the gravity force is only predominant force which controls the flow in addition to the force of inertia froude's model law is applicable.

Fraude's model law is applied in the following fluid flow problems:-

1) Free surface flows such as flow over spillways, weirs, sluices channels etc.

2) Flow of jet from an orifice or nozzle.

3) Where waves are likely to be formed on surface.

4) Where fluids of different densities flow over one another.

Let $V_m$=Velocity of fluid in model

$L_m$ = liner or length of model

$g_m$= Acceleration due to gravity at a place where model is tested

and similarly

$V_p$= corresponding values of velocity in prototype

$L_p$= corresponding values of velocity in length

$g_p$= corresponding values of velocity of gravity for prototype

(Fe) model=(F) Prototype

$\frac{V_m}{\sqrt{g_m L_m}}=\frac{V_p}{\sqrt{g_p L_p}}$....(1)

If tests on model are performed on the same place where prototype is to operate then

$[g_m=g_p]$

eqn (1) becomes

$\frac{g_m}{\sqrt{L_m}}=\frac{V_p}{\sqrt{L_p}}$...(2)

$\frac{V_m}{Vp}\times \frac{1}{\sqrt{\frac{L_m}{L_p}}}= 1$

$\frac{Vp}{Vm}=\sqrt{\frac{L_p}{L_m}}$

$=\sqrt{L_r}$

[$\frac{L_p}{L_m}=Lr]$

when $L_r$= scale ratio for length

$\frac{V_p}{V_m}$=$V_r$= scale ratio for velocity

$\therefore \frac{V_p}{v_m}=V_r=\sqrt{L_r}$....(1)

a) Scale ratio for time:-

$Time=\frac{length}{velocity}$

then ratio of time for prototype and model is

$T_r=\frac{T_p}{T_m}$

=$\frac{(\frac{L}{V})_{p}}{(\frac{L}{v})_{m}}$

=$\frac{\frac{L_p}{V_p}}{\frac{L_m}{V_m}}$

=$\frac{L_p}{L_m}\times \frac{V_m}{V_p}$

=$L_r\times \frac{1}{\sqrt{L_r}} \ ------- since\ \frac{V_p}{V_m}=\sqrt{L_r}$

=$\sqrt{L_r}$

b) Scale ratio for acceleration

Acceleration=$\frac{V}{T}$

$ar=\frac{ap}{am}$

=$\frac{(V/T)_{p}}{(V/T)_{m}}$

=$\frac{V_p}{T_p}\times\frac{T_m}{V_m}$

=$\frac{V_p}{V_m}\times \frac{T_m}{T_p}$

=$\sqrt{L_r}\times\frac{1}{\sqrt{L_r}}$

[ar=1]

c) Scale ratio for discharge

$Q=A\times V$

=$L^{2}\times \frac{L}{T}$

=$\frac{L^{3}}{T}$

$Q_{r}=\frac{Q_p}{Q_m}=\frac{(\frac{L^{3}}{T})_{p}}{(\frac{L^{3}}{T})_{m}}$

$(\frac{L_p}{L_m})^{3}\times(\frac{T_m}{T_p})$

=$L_r^{3}\times\frac{1}{\sqrt{L_r}}$

$[Q_r=L_r^{2.5}]$

d) Scale ratio for force

Force = Mass $\times$ Acceleration

=$\rho L^{3}\times \frac{V}{T}$

=$\rho L^{2}.\frac{L}{T}.V$

=$\rho L^{2}V^{2}$

$F_r=\frac{F_p}{F_m}$

=$\frac{\rho_p L_p^{2} V_p^{2}}{\rho_m L_m^{2} V_m^{2}}$

[Fr=$\frac{\rho_p}{\rho_m}\times(\frac{L_p}{L_m})^{2}\times (\frac{V_p}{V_m})^{2}$]

If the fluid used in model and prototype is same then

$\frac{\rho_p}{\rho_m}$=1 or $\rho_m=\rho_p$

$F_r=(\frac{L_p}{L_m})^{2}\times(\frac{V_p}{V_m})^{2}$

=$L_r^{2}\times(\sqrt{L_r})^{2}$

=$L_r{2}.Lr$

[$F_r=Lr^{3}]$

e) Scale ratio in pressure intensity

P=$\frac{Force}{Area}$

=$\frac{\rho L^{2} V^{2}}{L^{2}}$

=$\rho V^{2}$

$P_r=\frac{\rho_p}{\rho_m}$

=$\frac{\rho_p V_p^{2}}{\rho_m V_m^{2}}$

If fluid is same then

[$\rho_p=\rho_m$]

$P_r=\frac{V_p^{2}}{V_m^{2}}$

=$(\frac{V_p}{V_m})^{2}$

[$P_r=L_r$]

f) Scale ratio for work, energy torque moment etc

Torque =Force $\times$ Distance

=F$\times$ L

$T_r=\frac{T_p}{T_m}$

=$\frac{(F\times L)_p}{(F\times L)_m}$

$T_r=F_r\times Lr$

$L_r^{3}\times L_r$

=$L_r^{4}$

g) Scale ratio for Power:-

Power=work per unit time

=$\frac{F\times L}{T}$

Power ratio

$P_r=\frac{P_p}{P_m}$

=$\frac{(\frac{F_p\times L_p}{T_p})}{(\frac{F_m\times L_m}{T_m}}$

=$\frac{F_p}{F_m}\times \frac{L_p}{L_m}\times \frac{1}{T_p/T_m}$

=$F_r.L_r.\frac{1}{T_r}$

=$L_r^{3}.L_r.\frac{1}{\sqrt{L_r}}$

[$P_r=L_r^{3.5}]$