Transformation formula

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A fixed-mass system describes an amount of matter that can move, flow and interact with the surroundings, but the control volume approach depicts a region or volume of interest in a flow field, which is not unique and depends on the user. So conservation laws for a fixed-mass system need to be transformed to apply to a control volume. A mathematical relation that allows one to mathematically and physically link the conservation laws for a control volume with that of a fixed-mass system will be derived.

Figure 1 shows the flow field under consideration. At time t, the control volume shown by the solid line coincides with a single-phase fixed-mass system depicted by the dashed line. At time t + dt, a portion of the fixed-mass system moves outside of the boundaries of the control volume. It can be seen from Fig. 1 that region I is occupied by the system at time t only, region II is common to the system at both t and t + dt, and region III is occupied by the system at t + dt only. For a system with a fixed-mass, the change of the general property Φ, which has a specific value (per unit mass) φ, can be written as (Welty, 1978):


{\left. {\frac{{d\Phi }}{{dt}}} \right|_{system}} = \frac{{\Phi {{\left| {_{t + dt} - \Phi } \right|}_t}}}{{dt}}     \qquad \qquad(1)


Considering that the control mass occupies regions I and II at time t, one obtains {\left. \Phi  \right|_t} = {\left. \Phi  \right|_I} + {\left. \Phi  \right|_{II}}. At time t + dt, the fixed-mass system occupies both regions II and III, i.e., {\left. \Phi  \right|_{t + dt}} = {\left. \Phi  \right|_{II}} + {\left. \Phi  \right|_{III}}. Therefore, eq. (1) can be rewritten as


{\left. {\frac{{d\Phi }}{{dt}}} \right|_{system}} = \frac{{{\Phi _{II}}\left| {_{t + dt} - {\Phi _{II}}\left| {_t} \right.} \right.}}{{dt}} + \frac{{{\Phi _{III}}\left| {_{t + \Delta t}} \right.}}{{dt}} - \frac{{{\Phi _I}\left| {_t} \right.}}{{dt}}     \qquad \qquad(2)


Relation between a fixed mass system and a control volume.

The first term on the right-hand side of eq. (2) may be written as

\frac{{{\Phi _{II}}\left| {_{t + dt} - {\Phi _{II}}\left| {_t} \right.} \right.}}{{dt}} = \frac{{d\Phi {|_{II}}}}{{dt}}

and represents the rate of change of property Φ within the control volume, because region II becomes coincident with the control volume as dt \to 0. For the general case of variable Φ within a control volume, it is appropriate to write the time derivative of Φ as

\frac{{d\Phi {|_{II}}}}{{dt}} = {\left( {\frac{{d\Phi }}{{dt}}} \right)_{CV}} = \frac{\partial }{{\partial t}}\int_V {\rho \phi dV}

where V is the volume of the control volume.

The second and third terms on the right-hand side of eq. (2), respectively, represent the property Φ leaving and entering the control volume due to mass flow across its boundary. If the absolute velocity is V, and the reference frame moves with a constant velocity Vref, the relative velocity is {{\mathbf{V}}_{rel}} = {\mathbf{V}} - {{\mathbf{V}}_{ref}}. For the control volume’s entire surface area, A, the rate of movement of property Φ due to mass flow may be written as (see Fig. 2)

\frac{{{\Phi _{III}}\left| {_{t + \Delta t}} \right.}}{{dt}} - \frac{{{\Phi _I}\left| {_t} \right.}}{{dt}} = \int_A {\rho ({{\mathbf{V}}_{rel}} \cdot {\mathbf{n}})\phi dA}

where n is the normal direction of the control volume. Therefore, eq. (2) can be rewritten as

{\left. {\frac{{d\Phi }}{{dt}}} \right|_{system}} = \frac{\partial }{{\partial t}}\int_V {\rho \phi dV + \int_A {\rho ({{\mathbf{V}}_{rel}} \cdot {\mathbf{n}})\phi dA} }     \qquad \qquad(3)


Control volume in a flow field

which is the final form of the transformation formula that relates the change of property for a fixed-mass system to that of the control volume. It states that the rate of change of a property Φ for a fixed-mass system is equal to the rate of change of Φ in the control volume (the first term on the right-hand side) plus the net rate of efflux of Φ by mass flow into or out of the control volume (the second term on the right-hand side).

It should be pointed out that the control volume moves with the reference frame, which moves with a constant velocity, Vref. The coordinate system is attached to and moves with the reference frame. In other words, the coordinate system is stationary relative to the reference frame. The reference frame is inertial, so Newton’s second law is valid in the coordinate system that moves with the reference frame. Equation (3) will be used to obtain the macroscopic (integral) formulation of the basic laws for a control volume.

References

Welty, J.R., 1978, Engineering Heat Transfer, John Wiley & Sons, New York.

Further Reading

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