# 7. First Law of Thermodynamics - Open Systems#

In the previous two chapter’s Section 5 First Law of Thermodynamics - Introduction to Closed Systems and Section 6 First Law of Thermodynamics - Boundary Work and Specific Heats, we were focused on applying the energy conservation equation to closed systems, where there was no mass exchange with the surroundings. However, there are many thermodynamic systems where accounting for the mass transfer, in addition to the energy transfer, is critical, such as turbines, mixing chambers, and so on. Recall, these are considered open systems. To evaluate open systems, we will have to revise our energy conservation equation to account for the energy content of the mass entering and/or leaving the system. In addition, we will introduce the concept of flow work, which is related to the work done by or to the fluid entering or leaving the system, and is the force behind it’s movement.

## 7.1. Flow Work#

The work required for matter to flow through a control volume is called flow work. Flow work is the work required to push a fluid into the control volume (recall, a control volume is an open system), and work done by the system in the form of flow work to push fluid out. Consider the control volume below in Figure 7.1 - the work required to push fluid into and out of the control volume can be evaluated by imagining a piston cylinder that the fluid is acting against. Fig. 7.1 Exemplary open system with 1 inlet and 1 exit. The flow work is represented at the inlet side as an imaginary piston-cylinder. The flow work is the work required to push the fluid into the system and the work required to push the fluid out of the system.#

Recall, that the force ($$F$$) acting over an area ($$A$$) is pressure ($$p$$) and if the force is constant, the flow work ($$W$$flow) is equal to the force times distance, or

(7.1)#$W_{\rm flow} = F \Delta x$

Thus, for a piston with area $$A$$, $$W$$flow is related to the pressure and volume ($$V$$) by,

(7.2)#$W_{\rm flow} = pA \Delta x = pV$

and in terms of mass specific properties

(7.3)#$w_{\rm flow} = pv$

## 7.2. Energy Conservation for Open Systems#

In addition to heat transfer and boundary work interactions that were considered in the closed system energy conservation equation, flow work and energy content of the mass entering or leaving the system need to also be accounted for in open systems. We have already defined the flow work above. The energy $$E$$ of the mass entering is it’s specific energy content, $$e$$, scaled by the mass, $$m$$. And further, this can be broken down into the individual components, internal energy ($$U$$), kinetic energy (KE) and potential energy (PE). These terms should not be confused with the system’s energy which can also be broken down similarly. Thus, we can say the energy content entering ($$E$$i) or exiting ($$E$$e) the control volume is the following.

(7.4)#$E_{\rm i} = U_{\rm i} + KE_{\rm i} + PE_{\rm i} E_{\rm e} = U_{\rm e} + KE_{\rm e} + PE_{\rm e}$

Now, we can add these terms to the closed system energy conservation equation. Recall, for a stationary system this was $$\Delta U = Q_{\rm net}-W_{\rm net}$$. For a system with one inlet and one exit, adding the inlet and exit flow work and energy of the entering and exiting masses, the energy equation is thus.

(7.5)#$\Delta U = Q_{\rm net}-W_{\rm net} + W_{\rm flow,i}-W_{\rm flow,e} + E_{\rm i} - E_{\rm e}$

and

(7.6)#$\Delta U = Q_{\rm net}-W_{\rm net} + p_{\rm i}V_{\rm i}-p_{\rm e}V_{\rm e} + (U_{\rm i}+\frac{1}{2}m_{\rm i}V_{\rm i}^2+m_{\rm i}gh_{\rm i}) - (U_{\rm e}+\frac{1}{2}m_{\rm e}V_{\rm e}^2+m_{\rm e}gh_{\rm e})$