Energy balance equation for the open system

So far, the system was closed. Then, let’s consider the first law in open systems. Now the material may enter or leave the system so we must consider the energy balance and mass balance together. Mass balance is the material accumulation inside the system. And it is the difference between the amount of materials entering the system and the amount of materials leaving the system. In formulating the energy balance equation, we must consider the energy accompanied with the materials transferring. Besides heat and work, materials in and out carries energies. The materials definitely bring in or out their inherent internal energy. But for the materials in transit, we must consider the flow work.

The flow work is the work done on the system when entering material is pushed into it or the work done by the system when the system pushes the material out of the system. In other words, the flow work is the energy required to make a room or space for the material by pushing environment and establishing its volume and pressure. So the work flow in is integration of PidVi involved with the materials entering in. Here the values with underbar are specific values, the quantities per unit mass. So the work flow in is Pi times specific volume Vi underbar times mi, the mass the goes in. Likewise, the work flow out is PoVomo involved with the materials leaving out.

Internal energy of materials is the specific internal energy times the mass. So the materials mi that enter in, accompany the internal energy Ui and the flow work PiVi. Likewise, the material Mo that go out accompany Uo and PoVo. Therefore, the energy change inside the system is the energy accumulation related with the materials transfer and the change in heat and work. So the first law for the open system is summation of the first law for the closed system and the energy change related with the materials entering and leaving the system. The complete version equation expressing the 1st law of thermodynamics is as follows. The energy change inside the system is the difference between energy going in and energy going out.

It is categorized into two part. The first part is energy change related to the material exchange and the second part is the energy change related to energy in transit, the heat and work. The full version formulation includes potential and kinetic energies. So this is the complete first law equation for the open system. Here, U+PV can be combined to define the enthalpy. So the enthalpy represents the energy of a material in transit so it is the summation of internal energy and the flow work. Then the first law equation becomes like this. The change in internal, potential and kinetic energy of the system equals enthalpy goes in minus enthalpy goes out and delta Q and W.

If we express the enthalpy and internal energy in differential form, dH equals dU+PdV+VdP and dU is delta Q-PdV. Insert dU into dH equation, then dH equals delta Q+VdP.