An open water tank in air springs a leak at position- 2 in...

An open water tank in air springs a leak at position- 2 in Fig. $14-2$, where the pressure due to the water at position- 1 is $500 \mathrm{kPa}$. What is the velocity of escape of the water through the hole? The pressure at position- 2 in the free jet is atmospheric. We use Bernoulli's Equation with $P_{1}-P_{2}=5.00 \times 10^{5}$ $\mathrm{N} / \mathrm{m}^{2}, h_{1}=h_{2}$, and the approximation $v_{1}=0 .$ Then $$\left(P_{1}-P_{2}\right)+\left(h_{1}-h_{2}\right) \rho g=\frac{1}{2} \rho v_{2}^{2}$$ whence $$v_{2}=\sqrt{\frac{2\left(P_{1}-P_{2}\right)}{\rho}}=\sqrt{\frac{2\left(5.00 \times 10^{5} \mathrm{~N} / \mathrm{m}^{2}\right)}{1000 \mathrm{~kg} / \mathrm{m}^{3}}}=31.6 \mathrm{~m} / \mathrm{s}. $$

An open water tank in air springs a leak at position- 2 in Fig. $14-2$, where the pressure due to the water at position- 1 is $500 \mathrm{kPa}$. What is the velocity of escape of the water through the hole? The pressure at position- 2 in the free jet is atmospheric. We use Bernoulli's Equation with $P_{1}-P_{2}=5.00 \times 10^{5}$ $\mathrm{N} / \mathrm{m}^{2}, h_{1}=h_{2}$, and the approximation $v_{1}=0 .$ Then $$\left(P_{1}-P_{2}\right)+\left(h_{1}-h_{2}\right) \rho g=\frac{1}{2} \rho v_{2}^{2}$$
whence
$$v_{2}=\sqrt{\frac{2\left(P_{1}-P_{2}\right)}{\rho}}=\sqrt{\frac{2\left(5.00 \times 10^{5} \mathrm{~N} / \mathrm{m}^{2}\right)}{1000 \mathrm{~kg} / \mathrm{m}^{3}}}=31.6 \mathrm{~m} / \mathrm{s}.
$$

Key Concepts

Bernoulli's Equation

Bernoulli’s Equation is a fundamental principle in fluid mechanics that represents the conservation of energy in a flowing fluid. It relates the pressure, velocity, and elevation (or potential energy) along a streamline, assuming the flow is steady, incompressible, and non-viscous. This equation is pivotal for predicting fluid behavior in systems where pressure differences result in fluid motion, such as in a tank with an orifice.

Pressure Difference

The pressure difference is the driving force in many fluid flow problems. It is the variation in pressure between two points in the fluid system, which causes the fluid to accelerate when moving from a region of higher pressure to a region of lower pressure. This concept is crucial in determining the exit velocity of a fluid emerging from an orifice in a tank.

Fluid Density

Fluid density, defined as the mass per unit volume of a fluid, plays a critical role in linking pressure energy to kinetic energy in fluid flow problems. When applying principles like Bernoulli’s Equation, the density of the fluid determines how effectively the pressure difference can be converted into the velocity of the fluid, influencing the overall dynamics of the flow.

Assumptions in Fluid Flow Analysis

Fluid flow problems often rely on simplifying assumptions to make analysis tractable. Common assumptions include considering the fluid as incompressible and non-viscous, assuming steady flow, and sometimes neglecting certain velocity components. These assumptions allow for the direct application of energy conservation principles, such as Bernoulli’s Equation, to evaluate fluid behavior, particularly in calculating exit velocities through apertures.

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