# Thermodynamics - Open Systems

## Conservation of Mass and Energy

$$\Sigma \dot{m}_{in} - \Sigma \dot{m}_{out} = \dot{m}_{CV}$$

Where $$\dot{m}_{CV}$$ is the rate of change of mass in the control volume.

Since the intake and output flows also have their associated energies, the general energy equation becomes

$$Q - W + \Sigma E_{in} - \Sigma E_{out} = \Delta E_{CV}$$

The work done to move the fluid from the intake to output through the control volume is

$$\dot{W} = \dot{W}_{flow,out} - \dot{W}_{flow,in} + \dot{W}_{CV} ; W_{flow} = PV$$

Where $$\dot{W}_{CV}$$ is the work for processes affecting the control volume.

Assumptions for a steady flow system are as follows:

$$\dot{m}_{in} = \dot{m}_{out}$$

$$\dot{E}_{in} = \dot{E}_{out}$$

$$V_{CV} = Constant$$

The general energy equation

$$Q - W + \Sigma E_{in} - \Sigma E_{out} = \Delta E_{CV}$$

can then be modified for steady flow to

$$\dot{Q} - ( \dot{W}_{CV} - \dot{W}_{flow} ) = \Sigma E_{out} - \Sigma E_{in}$$

Since enthalpy (H) also takes into account flow work,

$$H = PV + U ; \dot{W}_{flow} = PV$$

The general steady flow energy equation can be written as

$$\dot{Q} - \dot{W} = \Delta H + \Delta KE + \Delta PE$$

Where $$\Delta H = m C_p \Delta T ; H = PV+U ; \dot{W}_{flow} = PV$$

and $$KE = \frac{1}{2} m C^2 ; PE = mgz$$

## Work for Closed and Open systems

On a PV diagram, work in a closed system is the area under the graph:

$$W = \int P dV$$

For open systems, work done is the area to the left of the graph:

$$W = - \int V dP$$