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## Homework Statement

A mechanical servo-mechanism comprising of a movable piston-cylinder within a vertical cylinder operates based on a venturi contraction in a horizontal 350mm diameter pipe that delivers a fluid of relative density 0.95. The upper end of the 100mm diameter vertical cylinder is connected by a pipe to the throat of the venturi, while the lower end of the cylinder is connected by a pipe to the pipe inlet. The piston in the cylinder is to be lifted vertically when the flow through the venturi exceeds 0.15m

^{3}/s, such that it will activate a controller to throttle back the flow rate. Note that the piston rod is known to have a diameter of 20mm and that it passes through both ends of the vertical cylinder. Neglecting friction, calculate the diameter required of the venturi throat if the gross effective load on the piston rod is 180N.

Ans: 0.162m

## Homework Equations

Benoulli equation & conservation of mass

## The Attempt at a Solution

I tried solving using Bernoulli's equation and conservation of mass.

$$0.15=0.175^2\pi* v_p=r_t^2*\pi*v_t$$

where the subscript p represents location at pipe inlet while t represents location at throat

Area of cylinder is,$$(0.05^2-0.01^2)\pi = 0.00024\pi$$

Pressure difference in area of cylinder is,

$$P_p -P_t=180/(0.0024\pi)=23873$$

$$P_p + 0.5\rho v_p^2 = P_t + 0.5\rho v_t^2 +\rho *g(0.175-r_t)$$

$$\rho=950, \ \ g=9.81, \ \ v_p=1.56$$

$$r_t=\sqrt{\frac{0.15}{\pi*v_t}}$$

After subbing in, $$25028.96=475*v_t^2+9319.5(0.175-\sqrt{\frac{0.15}{\pi*v_t}})$$

I can only solved after I can get velocity at throat. But I am stuck at the above part.