7.4.2: Linear Accelerated System


The acceleration can be employed in similar fashion as the gravity force. The linear acceleration "creates'' a conservative force of constant force and direction. The "potential'' of moving the mass in the field provides the energy. The Force due to the acceleration of the field can be broken into three coordinates. Thus, the element of the potential is

$\label{ene:eq:acceleration3C} d\,PE_{a} = \pmb{a} \cdot d\pmb{ll} \,dm$

The total potential for element material

$\label{ene:eq:elePE} PE_{a} = \int_{(0)}^{(1)} \pmb{a} \cdot d\pmb{ll} \,dm = \left( a_x \left( x_1 - x_0 \right) a_y \left( y_1 - y_0 \right) a_z \left( z_1 - z_0 \right) \right) \,dm$ At the origin (of the coordinates) $$x=0$$, $$y=0$$, and $$z=0$$. Using this trick the notion of the $$a_x \left( x_1 - x_0 \right)$$ can be replaced by $$a_x\,x$$. The same can be done for the other two coordinates. The potential of unit material is

$\label{ene:eq:PEtotal} {PE_a}_{total} = \int_{sys} \left( a_x\,x + a_y\,y + a_z\,z \right) \,\rho \,dV$ The change of the potential with time is

$\label{ene:eq:PEtotalDT} \dfrac{D}{Dt} {PE_a}_{total} = \dfrac{D}{Dt} \int_{sys} \left( a_x\,x + a_y\,y + a_z\,z \right) \,dm$ Equation can be added to the energy equation as

$\label{ene:eq:EneAccl} \dot{Q} - \dot{W} = \dfrac{D}{Dt} \int_{sys} \left[ E_u + \dfrac{U^2}{2\dfrac{}{}} + a_x\,x + a_y\, y + (a_z + g) z \right] \rho\,dV$
The Reynolds Transport Theorem is used to transferred the calculations to control volume as

Energy Equation in Linear Accelerated Coordinate

$\nonumber \dot{Q} - \dot{W} = \dfrac{d}{dt} \int_{cv} \left[ E_u + \dfrac{U^2}{2\dfrac{}{}} + a_x\,x + a_y\, y + (a_z + g) z \right] \rho\,dV \\ \label{ene:eq:ene:AccCV} + \int_{cv} \left( h + \dfrac{U^2}{2\dfrac{}{}} + a_x\,x + a_y\, y + (a_z + g) z \right) U_{rn}\, \rho\,dA\ \nonumber + \int_{cv} P\,U_{bn} \,dA$