MrsDrPoe: The First Law of Thermodynamics, Part 3

Once again Thesis Tuesday is upon us, and today we'll continue our discussion of the First Law of Thermodynamics.

If we examine the integral form of the first law that we derived at the beginning of our discussion and make the assumptions of steady, laminar, and incompressible flow with uniform properties, pressure and constant average velocity at our inlets and outlets, our equation transforms as follows:

rho*int((u + (p/rho) + 0.5*|V|*|V| + g*z)*|V|)dAout - rho*int((u + (p/rho) + 0.5*|V|*|V| + g*z)*|V|)dAin = Qdot,net_in + Wdot,shaft_in

rho*(u + (p/rho) + 0.5*|V|*|V| + g*z)*|V|int(1)dAout - rho*(u + (p/rho) + 0.5*|V|*|V| + g*z)*|V|int(1)dAin = Qdot,net_in + Wdot,shaft_in

rho*(u + (p/rho) + 0.5*|V|*|V| + g*z)*|V|*Aout - rho*(u + (p/rho) + 0.5*|V|*|V| + g*z)*|V|*Ain = Qdot,net_in + Wdot,shaft_in 

mdot*(u + (p/rho) + 0.5*|V|*|V| + g*z)out - mdot*(u + (p/rho) + 0.5*|V|*|V| + g*z)in = Qdot,net_in + Wdot,shaft_in  

Furthermore, if we consider this equation per unit mass:

((p/rho) + 0.5*|V|*|V| + g*z)out - ((p/rho) + 0.5*|V|*|V| + g*z)in = qdot,net_in + wdot,shaft_in

Since the flow is steady, we can combine uout - uin - qnet,in into a losses term so that:

((p/rho) + 0.5*|V|*|V| + g*z)out = ((p/rho) + 0.5*|V|*|V| + g*z)in + wdot,shaft_in - losses

This form of the equation is known as the mechanical energy equation or the extended Bernoulli equation.  Each of the terms in this equation are of the form energy per unit mass.


If we divide the mechanical energy equation by the gravitational constant, g, we obtain:

((p/gamma) + 0.5*|V|*|V|/g + z)out = ((p/gamma) + 0.5*|V|*|V|/g + z)in + wdot,shaft_in/g - losses/g

The term wdot,shaft_in/g has the dimensions of energy per unit weight, which simplifies to the unit of height or length and can thus be expressed as hs (shaft head).  This term becomes hT (turbine head) if a turbine is present in the system or hP (pump head) if a pump is present in the system.  The term losses/g has the same units and can be written as hL.  So our final equation is:

((p/gamma) + 0.5*|V|*|V|/g + z)out = ((p/gamma) + 0.5*|V|*|V|/g + z)in + hs - hL

which is the head form of the energy equation.  Each term has units of head (length).  This particular form is used extensively in pipe flow applications, which we'll look at briefly next week.  Until then, happy studying!