# Control volume

Thermodynamics :
The classical Carnot heat engine :
Branches
Classical
Statistical
Chemical
Quantum thermodynamics
Equilibrium / Non-equilibrium :
Laws
Zeroth
First
Second
Third :
Systems
Closed system
Open system
Isolated system
State
Equation of state
Ideal gas
Real gas
State of matter
Phase (matter)
Equilibrium
Control volume
Instruments
Processes
Isobaric
Isochoric
Isothermal
Adiabatic
Isentropic
Isenthalpic
Quasistatic
Polytropic
Free expansion
Reversibility
Irreversibility
Endoreversibility
Cycles
Heat engines
Heat pumps
Thermal efficiency :
System propertiesNote: Conjugate variables in italics
Property diagrams
Intensive and extensive properties
Process functions
Work
Heat
Functions of state
Temperature / Entropy (introduction)
Pressure / Volume
Chemical potential / Particle number
Vapor quality
Reduced properties :
Material properties
Property databases
Specific heat capacity 
$c=$
$T$$\partial S$$N$$\partial T$
Compressibility 
$\beta =-$
$1$$\partial V$$V$$\partial p$
Thermal expansion 
$\alpha =$
$1$$\partial V$$V$$\partial T$ :
Equations
Carnot's theorem
Clausius theorem
Fundamental relation
Ideal gas law
Maxwell relations
Onsager reciprocal relations
Bridgman's equations
Table of thermodynamic equations :
Potentials
Free energy
Free entropy
Internal energy $U(S,V)$Enthalpy $H(S,p)=U+pV$Helmholtz free energy $A(T,V)=U-TS$Gibbs free energy $G(T,p)=H-TS$ :
HistoryCulture
History
General
Entropy
Gas laws
"Perpetual motion" machines
Philosophy
Entropy and time
Entropy and life
Brownian ratchet
Maxwell's demon
Heat death paradox
Loschmidt's paradox
Synergetics
Theories
Caloric theory
Vis viva ("living force")
Mechanical equivalent of heat
Motive power
Key publications
An Inquiry Concerning the Source ... Friction
On the Equilibrium of Heterogeneous Substances
Reflections on the Motive Power of Fire
Timelines
Thermodynamics
Heat engines
ArtEducation
Maxwell's thermodynamic surface
Entropy as energy dispersal :
Scientists
Bernoulli
Boltzmann
Bridgman
Carathéodory
Carnot
Clapeyron
Clausius
de Donder
Duhem
Gibbs
von Helmholtz
Joule
Kelvin
Lewis
Massieu
Maxwell
von Mayer
Nernst
Onsager
Planck
Rankine
Smeaton
Stahl
Tait
Thompson
van der Waals
Waterston :
Other
Nucleation
Self-assembly
Self-organization
Order and disorder :
Category :
vte :
In continuum mechanics and thermodynamics, a control volume (CV) is a mathematical abstraction employed in the process of creating mathematical models of physical processes. In an inertial frame of reference, it is a fictitious region of a given volume fixed in space or moving with constant flow velocity through which the continuuum (a continuous medium such as gas, liquid or solid) flows. The closed surface enclosing the region is referred to as the control surface.
At steady state, a control volume can be thought of as an arbitrary volume in which the mass of the continuum remains constant. As a continuum moves through the control volume, the mass entering the control volume is equal to the mass leaving the control volume. At steady state, and in the absence of work and heat transfer, the energy within the control volume remains constant. It is analogous to the classical mechanics concept of the free body diagram.

## Overview

Typically, to understand how a given physical law applies to the system under consideration, one first begins by considering how it applies to a small, control volume, or "representative volume". There is nothing special about a particular control volume, it simply represents a small part of the system to which physical laws can be easily applied. This gives rise to what is termed a volumetric, or volume-wise formulation of the mathematical model.
One can then argue that since the physical laws behave in a certain way on a particular control volume, they behave the same way on all such volumes, since that particular control volume was not special in any way. In this way, the corresponding point-wise formulation of the mathematical model can be developed so it can describe the physical behaviour of an entire (and maybe more complex) system.
In continuum mechanics the conservation equations (for instance, the Navier-Stokes equations) are in integral form. They therefore apply on volumes. Finding forms of the equation that are independent of the control volumes allows simplification of the integral signs. The control volumes can be stationary or they can move with an arbitrary velocity.

## Substantive derivative

Computations in continuum mechanics often require that the regular time derivation operator $d/dt\;$ is replaced by the substantive derivative operator $D/Dt$. This can be seen as follows.
Consider a bug that is moving through a volume where there is some scalar, e.g. pressure, that varies with time and position: $p=p(t,x,y,z)\;$.
If the bug during the time interval from $t\;$ to $t+dt\;$ moves from $(x,y,z)\;$ to $(x+dx,y+dy,z+dz),\;$ then the bug experiences a change $dp\;$ in the scalar value,
    $dp={\frac {\partial p}{\partial t}}dt+{\frac {\partial p}{\partial x}}dx+{\frac {\partial p}{\partial y}}dy+{\frac {\partial p}{\partial z}}dz$
(the total differential). If the bug is moving with a velocity $\mathbf {v} =(v_{x},v_{y},v_{z}),$ the change in particle position is $\mathbf {v} dt=(v_{x}dt,v_{y}dt,v_{z}dt),$ and we may write
    ${\begin{alignedat}{2}dp&={\frac {\partial p}{\partial t}}dt+{\frac {\partial p}{\partial x}}v_{x}dt+{\frac {\partial p}{\partial y}}v_{y}dt+{\frac {\partial p}{\partial z}}v_{z}dt\\&=\left({\frac {\partial p}{\partial t}}+{\frac {\partial p}{\partial x}}v_{x}+{\frac {\partial p}{\partial y}}v_{y}+{\frac {\partial p}{\partial z}}v_{z}\right)dt\\&=\left({\frac {\partial p}{\partial t}}+\mathbf {v} \cdot \nabla p\right)dt.\\\end{alignedat}}$
where $\nabla p$ is the gradient of the scalar field p. So:
    ${\frac {d}{dt}}={\frac {\partial }{\partial t}}+\mathbf {v} \cdot \nabla .$
If the bug is just moving with the flow, the same formula applies, but now the velocity vector,v, is that of the flow, u. The last parenthesized expression is the substantive derivative of the scalar pressure. Since the pressure p in this computation is an arbitrary scalar field, we may abstract it and write the substantive derivative operator as
    ${\frac {D}{Dt}}={\frac {\partial }{\partial t}}+\mathbf {u} \cdot \nabla .$