• The Reynolds transport theorem talks about how mass; momentum and energy are transported through a control volume.

• It provides equations that correlate control volume properties to the system properties.

• Reynolds transport theorem states that the rate of change of system property is equal to thesum of the rate of change of property within the control volume and the difference between property outflow and property inflow through the control surface.

Proof:

Consider any Extensive property of the system N. Therefore n is the property per unit mass (=N/m).

Rate of change of N for the system is $(\frac{dN}{dt})_{system}$

Consider a System bounded by a Control Surface (CS) at a time t as shown in the figure the system and the Control Volume (CV) coincides.

Due to motion of the fluid, the system occupies a new position, shown dotted, at a time t+dt.

Now, I, II and III denotes the areas bounded by the control surface.

$(\frac{dN}{dt})_system=lim┬(dt→0)⁡\frac{N_{t+dt}-N_{t}}{dt}$

=${\substack lim┬(dt→0)}⁡\frac{(N_{t+dt}-N_t)}{dt}$

=$lim┬(dt→0)⁡\frac{(N_{II}+N_{III})_{t+dt}-(N_{I}+N_{II})_{t})}{dt}$

=$lim┬(dt→0)⁡\frac{(∫^{\square}_{II}nρdV+∫^{\square}_{III}nρdV)_{t+dt}-(∫^{\square}_{In}ρdV+{∫^{\square} _II}nρdV)_{t}}{dt}$

where, ρ→Density of the fluid and V→Volume

Adding and subtracting $(∫^{\square}_InρdV)_{(t+dt)}$

=$lim┬(dt→0)⁡(\frac{∫^{\square}_InρdV+∫^{\square}_{II}nρdV)_{t+dt}-(∫^{\square}_{I}nρdV)_{t+dt}+(∫^{\square}_{III}nρdV)_{t+dt}-(∫^{\square}_{I}nρdV+∫^{\square}_{II}nρdV)_t}{dt}$

=$lim┬(dt→0)⁡[\frac{(∫^{\square}_{I}nρdV+∫^{\square}_{II}nρdV)_(t+dt)-(∫^{\square}_{I}nρdV+∫^{\square}_{II}nρdV)_t ]}{dt}$

$+lim┬(dt→0)⁡[\frac{(∫^{\square}_{III}nρdV)_{t+dt}-(∫^{\square}_{I}nρdV)_{t+dt}}{dt}]$

The first limitrepresents the change of property within the Control Volume (CV)and the second limit represents the difference between property outflow and inflow which can be represented as,

$\frac{dN}{dt}_system=\frac{∂}{∂t} [∫^{\square}_{CV}nρdV]+∫^{\square}_{CS}nρv ⃗∙(dA) ⃗$

where,v ⃗→ velocity vector and A ⃗→Area vector