Chapter 6, Problem 6.10P

Interpretation Introduction

Interpretation:

The force exerted on the given device in an upward direction by water and air is to be calculated. Also, explain why Bernoulli’s equation is not used in the calculation of the force.

Concept Introduction:

The general energy balance equation according to the first law of thermodynamics written for a control volume is:

  $\frac{\delta Q}{dt}-\frac{\delta W}{dt}={\displaystyle {\iint }_{C.S}\left(e+\frac{P}{\rho }\right)\rho \left(v\cdot n\right)dA}+\frac{\partial }{\partial t}{\displaystyle {\iiint }_{C.V}e\rho dV}+\frac{\delta W_{\mu }}{dt}$   ...... (1)

The general equation written for the overall linear momentum balance for a controlled volume is:

  ${\displaystyle \sum F}={\displaystyle {\iint }_{C.S}v\rho \left(v\cdot n\right)dA}+\frac{\partial }{\partial t}{\displaystyle {\iiint }_{C.V}\rho vdV}$   ...... (2)

Here, C.S is the control surface over which the integral dA is taken, C.V is the control volume over which the integral dV is taken, $\rho$ is the density of the fluid, $v$ is the velocity vector, $n$ is the direction of the vector $v$ , $\delta Q/dt$ is the rate of heat added (positive) or removed (negative) from the system, $\delta W/dt$ is the rate of work done by (positive) or on (negative) the system, and $\delta W_{\mu }/dt$ is the rate of work done to overcome the viscous effect at the control surface. The product $v\cdot n$ is scalar defined as:

  $v\cdot n=\left|v\right|\left|n\right|\cos \theta$