Engr Help

Finite Control Volume – Conservation of Energy (continued)

The figure below shows a general control volume (cv) with its control surface (cs). The velocity

profile of the fluid entering and leaving through the control surface may be nonuniform.

For conservation of energy:

∂/∂t ∫ e ρ dV + ∫ e ρ V . n dS = dQ/dt _{net in} + dW/dt_shaft _{net in}

cv cs

where ∂/∂t = the time rate of change

e = the energy per unit mass = u + p/ρ + ½ V² + gz (ft lb/slug or Nm/kg)

u = the internal energy per unit mass in the fluid, p is the fluid static pressure

ρ = the mass density of the fluid (slugs/ft³ or kg/m³)

dV = the element of volume within the control volume

dS = the element of area on the control surface

V = the absolute fluid velocity crossing the control surface (ft/sec or m/sec)

n = the unit outward normal to the control surface

V . n = the normal component of velocity crossing the control surface (dot product)

dQ/dt = net heat transfer into the control surface (ft lb/sec or Nm/sec)

dW/dt _shaft = the net shaft work into the control surface (pump is +, turbine is - )

(ft lb/sec or Nm/sec)