MrsDrPoe: The First Law of Thermodynamics, Part I

It's another Thesis Tuesday here at the blog, and today we begin discussion of the third governing equation of fluid mechanics- the first law of thermodynamics.  (In case you've forgotten, the other governing equations are the conservation of mass equation and the conservation of momentum equation.)

The first law of thermodynamics (also called the first law and the conservation of energy equation) simply tracks the energy in a system- just like mass and momentum, energy is conserved and every bit of it in every system/process can be accounted for at any instant in time.  In mathematical speak, the first law states that, "the time rate of increase of the total stored energy of the system is equal to the sum of the net time rate of energy addition by heat transfer into the system, and the net time rate of energy addition by work transfer into the system" or:

D/Dt(int(E*rho)dVsys) = (sum(Qdot)in - sum(Qdot)out)sys + (sum(Wdot)in - sum(Wdot)out)sys

This form of the equation is useful for Lagrangian viewpoints; recall that in order to transform it into a form useful for Eulerian viewpoints, we must employ the Reynolds Transport Theorem to the term on the left side:

D/Dt(int(E*rho)dVsys) = d/dt(int(e*rho)dV) + int(e*rho*(V.n))dA

meaning, "the time rate of increase of the total stored energy of the system is equal to the sum of the time rate of increase of the total stored energy of the contents of the control volume and the net rate of flow of the total stored energy out of the control volume across the control surface.  Our first law equation then becomes:

d/dt(int(e*rho)dV) + int(e*rho*(V.n))dA = Qdot,net in + Wdot, net in

In general, we will denote both work and heat coming into a system as positive and going out of a system as negative.

In the equations above, e is the sum of the potential, kinetic, and internal energies in the fluid (e = u + (|V|*|V|)/2 + g*z).  The heat transfer rate, Qdot, denotes all the ways energy is exchanged between the control volume and its surroundings due to a temperature difference.  You may recall from other classes that making the adiabatic assumption means that there is no heat transfer to or from the system.  This net rate of heat transfer is also zero if the heat transfer into the control volume is equal to the heat transfer out of the control volume.  

The work rate (or power), Wdot, denotes the work done on the control volume by the surroundings (when positive).  This work can be done by a shaft at the control surface, such as in a piston/cylinder arrangement, which calculated by multiplying the torque of the shaft by the shaft's angular velocity.  The work also occurs due to the normal stresses present on the surfaces of the control volume due to pressure and tangential stresses due to shearing.  Typically, the control volume is chosen in such a way that the tangential stress work is zero; the normal stress work is calculated by:

Wdot, norm = int(-p*(V.n))dA

If we put our expanded work terms into the energy equation derived earlier, we are left with:

d/dt(int((u + (|V|*|V|)/2 + g*z)*rho)dV) + int((u + (|V|*|V|)/2 + g*z)*rho*(V.n))dA = Qdot,net in + Wdot, net in - int(p*(V.n))dA

or:

d/dt(int((u + (|V|*|V|)/2 + g*z)*rho)dV) + int((u + (|V|*|V|)/2 + g*z)*rho*(V.n))dA = Qdot,net in + Wdot, shaft in

And that's the integral form of the first law.  Next week we'll look at a brief example of its use, but until then, happy studying!