First Law of Thermodynamics – Control Volume

The concept of a control volume is fundamental in deriving the basic conservation equations, whether we’re using the Lagrangian or Eulerian approach. Regardless of the coordinate system chosen, there are two primary types of control volumes available.

The first type is a parallelepiped with sides of dx, dy, and dz. In this approach, various fluid properties like velocity or pressure are expressed using Taylor series expansions around the center of the control volume. This provides expressions for these properties at each face of the control volume. By invoking the conservation principle and allowing dx, dy, and dz to approach infinitesimally small values, we obtain differential equations representing the conservation principle. Often, simplifications are made by considering the control volume with sides of length dx, dy, and dz, and only retaining the first term of the Taylor series.

Control Volume types

The second type of control volume is arbitrary in shape, and each conservation principle is applied as an integral over this control volume. For instance, the mass within this control volume is calculated as the integral of the fluid density, represented as ∫∫∫V ρ dV, where ρ is the fluid density, and the integration covers the entire volume V containing the fluid within the control volume. This process yields an integro-differential equation of the form ∫∫∫V La dV = 0, where L is a differential operator, and a is a property of the fluid. To satisfy this equation, we set La = 0, resulting in the differential equation of the conservation law. If the integrand in the equation were not zero, we could redefine the control volume V in such a way that the integral of La would not be zero, contradicting the integro-differential equation.

Both types of control volumes have their merits, and in this book, both are utilized depending on which provides better insights into the physics of the discussed situation. The arbitrary control volume is typically used in deriving the fundamental conservation laws as it aligns well with the underlying principles. Importantly, it should be noted that the results obtained using these two methods are identical.