Can first law of thermodynamics defined for a closed system be applied to the steady flow energy equation? Why? I came across the derivation of $Vdp$ work and Every book applied the first law defined for closed system to steady flow energy equation.
Please clarify
OH , i get it now. You are right.We can't apply the first law if you don't consider the entire system and if the energy can leak in or out by other means ,say a more energetic fluid comes in. But in steady state flow ,no particular trait of a fluid changes with time.This means that the property of a fluid at a particular spatial point is constant in time(to be more mathematical,it's partial derivative with respect to time is zero).This means that the only way the energy of fluid can change in a constraint volume is by change in kinetic energy,change in potential energy or by work done.
For detailed explanantion,go to https://wiki.ucl.ac.uk/display/MechEngThermodyn/First+law+applied+to+flow+processes
It helps if you consider the first and second law expressed in terms of energy and entropy densities that depend explicitly on space and time, rather than differentials.
The first laws is: $$\frac{\partial e}{\partial t} + \nabla\cdot(e\boldsymbol{v}) = \boldsymbol{T}:\nabla\boldsymbol{v}+\nabla\cdot\boldsymbol{q} - Q,$$ where $e$ is the energy density; $\boldsymbol{v}$ the velocity of the fluid or material; $\boldsymbol{T}$ is the stress, e.g. the pressure or viscosity in a fluid; $\boldsymbol{q}$ is the surface flux of heat, and $Q$ is the volume sink of heat. The term $\boldsymbol{T}:\nabla\boldsymbol{v}$ represent the work being done on the fluid or material, and reduces to $p\partial V/\partial t$ in the case of a fluid with no viscosity. If you integrate this equation over a control volume you get a formula that says, in words, $$\frac{\mathrm{d}(\text{total internal energy within volume})}{\mathrm{d}t} + (\text{flux of energy through volume boundary}) = \text{power} + \text{heating}.$$ The "power" is the work done per unit time, and the "heating" is the heat transferred per unit time. The flux of energy through the boundary appears if the system is open, so matter that moves through the boundary is carrying its internal energy in or out.
The second law has a similar expression: $$\frac{\partial s}{\partial t} + \nabla\cdot(s\boldsymbol{v}) \geqslant \nabla\cdot\left(\frac{\boldsymbol{q}}{T}\right) - \frac{Q}{T},$$ where $s$ is the entropy density and $T$ the temperature. Again, if you integrate over a control volume you obtain $$\frac{\mathrm{d}(\text{total entropy within volume})}{\mathrm{d}t} + (\text{flux of entropy through volume boundary}) \geqslant \frac{\text{heating}}{\text{temperature}}.$$ The flux of entropy through the boundary appears if the system is open, so matter that moves through the boundary is carrying its entropy in or out.
I'm sorry if this looks complicated, but if you devote some time to it, even skipping the technical details, it can give you a better grasp of what the first and second laws say and how to use them in unfamiliar situations.
Check out C. A. Truesdell: Rational Thermodynamics (2nd ed., Springer 1984), or I. Samohýl & M. Pekař: The Thermodynamics of Linear Fluids and Fluid Mixtures (Springer 2014).
