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Chapter 5: Problem 94

Figure \(P 5.94\) shows the mixing of two streams. The shear stress between each fluid and its adjacent walls is negligible. Why can't Bernoulli's equation be applied between points in stream 1 and the mixed stream or between points in stream 2 and the mixed stream?

Short Answer

Expert verified
Bernoulli's equation cannot be applied between points in stream 1 or stream 2 and the mixed stream because the conditions for Bernoulli's equation to hold, specifically inviscid, steady, incompressible flow along a streamline, are not met in this situation of two streams mixing.

Step by step solution

01

Understand Bernoulli's Equation

Bernoulli's equation can be written as \(P1/(ρg) + V1^2/(2g) + Z1 = P2/(ρg) + V2^2/(2g) + Z2\), where P is the pressure, ρ is the fluid density, g is gravitational acceleration, V is velocity and Z is elevation head. The equation is derived from the principle of conservation of energy and it applies to a flow along a streamline during steady, inviscid and incompressible flow. No energy is wasted in viscous dissipation, no energy is added to or taken from the flow, and there are no changes in pressure or velocity from one line to another in the flow.
02

Understand Mixing of Streams

When two streams are mixing, their velocities change, thus energy conversion between kinetic energy and potential energy occurs. The mixing region is not a streamline flow and there will be viscous effects due to the interactions between the two streams. Therefore, the conditions under which Bernoulli's equation was derived aren't valid for the mixing of two streams.
03

Explaining the Inapplicability of Bernoulli's Equation

In considering points in either stream 1 or stream 2 and in the mixed stream, the flows are not along a streamline, nor are they inviscid or steady. Energy is being added to the system through the mixing process and the pressure and velocity are not constant. Therefore, Bernoulli's equation cannot be applied between points in stream 1 or stream 2 and the mixed stream because the conditions of the Bernoulli's equation are violated.

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