Question

The upstream and downstream pressures in a homogenizer during homogenization of milk are maintained at 250 bar and 10 bar, respectively. If density of milk is 1030 kg/m\(^3\), then the velocity at which milk comes out of the homogenizing valve in m/s is \underline{\hspace{3cm}} (rounded off to 3 decimal places).

Hint:

Homogenizers use very high pressure drops. The exit velocity can be estimated using Bernoulli's equation with pressure converted to velocity head.

Answer 1

Step 1: Use energy balance (Bernoulli's principle). Pressure drop is converted into velocity head: \[ \Delta P = \frac{1}{2} \rho v^2 \]

Step 2: Pressure difference. \[ \Delta P = (250 - 10) \, bar = 240 \, bar \] \[ = 240 \times 10^5 \, Pa = 2.4 \times 10^7 \, Pa \]

Step 3: Solve for velocity. \[ v = \sqrt{\frac{2 \Delta P}{\rho}} \] \[ = \sqrt{\frac{2 \times 2.4 \times 10^7}{1030}} \] \[ = \sqrt{46601.94} = 494.797 \, m/s \] \[ \boxed{494.797 \, m/s} \]