First Law and Energy
Q=\Delta U+ \Delta KE + \Delta PE +W
For a steady-state control volume system the first law is as follows:\dot{Q}=\Delta \dot{H}+ \Delta \dot{KE} + \Delta \dot{PE} +\dot{W}
- Q is heat input, energy transfer due to a temperature difference
- W is work output, energy transfer due to a force acting over a distance
- \Delta U is internal energy change of a substance
- \Delta H is enthalpy change of a substance
- \Delta KE is kinetic energy change
- \Delta PE is potential energy change
Energy Equations
Boundary Work
The work done by a piston cylinder can be described in terms of differentials:
\partial W=Fdy=PAdy=PdV
We can therefore express total work as:
W={\displaystyle\int_{V_1}^{V_2}PdV}
Likewise, we can express specific work, w=\frac{W}{m}, as:
w={\displaystyle\int_{\nu_1}^{\nu_2}Pd\nu}
Using the area interpretation of integrals we can see that specific work can be thought of as the area underneath a P–\nu curve.
A polytropic process is a process where P and V are related to a constant C by an exponent n.
PV^n=C
Substituting for P into the equation for total work:
\boxed{W=\dfrac{P_2V_2-P_1V_1}{1-n}\text{, }n\ne 1}
\boxed{W=P_2V_2\ln \left(\dfrac{V_2}{V_1}\right)\text{, }n=1}
Work for a piston cylinder pressing against a spring:
\boxed{W=\frac{1}{2}(P_2+P_1)(V_2-V_1)}
Note: For ideal gases we can use ideal gas law to substitute P_iV_i for mRT_i into the work equations. For an ideal gas, when n=1 the process is isothermal (constant temperature).
Internal Energy
Internal energy is the energy required to keep a substance or system in a certain state. Typically internal energy is used to describe the energy of a substance at a specific state point. Internal energy is defined as the product of the substance mass and substance specific internal energy:
U=mu
Values for specific internal energy must be looked up in tables as it is a unique value based on the substance, temperature, pressure, and specific volume. The change in system internal energy will be the difference between final state internal energies and initial state internal energies.
\Delta U=\sum m_eu_e -\sum m_iu_i
For ideal gases the change in internal energy is found to be proportional to the change in temperature of the substance:
\Delta u=C_{V_0}\Delta T
Likewise, for solids the change in internal energy can be found by:
\Delta u=\Delta h=C\Delta T
Enthalpy
Enthalpy is defined as the sum of internal energy and work flow:
H=U+W_{flow}
Work flow is the energy required to maintain fluid flow through a control volume.
W_{flow}=PV
Enthalpy can be described in terms of specific enthalpy, just as specific internal energy is used to describe internal energy.
h=u+P\nu
H=mh
\Delta H=\sum m_eh_e-\sum m_ih_i
\Delta \dot{H}=\sum \dot{m}_eh_e-\sum \dot{m}_ih_i
For ideal gases the change in enthalpy is found to be proportional to the change in temperature of the substance:
\Delta h=C_{P_0}\Delta T
Likewise, the change in enthalpy can be found for solids by:
\Delta u=\Delta h=C\Delta T
Kinetic Energy
Kinetic energy is the energy associated with motion.
\Delta KE=\frac{1}{2}\sum m_ev_e^2-\frac{1}{2}\sum m_iv_i^2
Potential Energy
Typically the only potential energy that has to be considered is gravitational potential. Gravitational potential energy is the energy associated with displacing an object a vertical distance, z within a gravitational field.
\Delta PE=\sum m_egz_e-\sum m_igz_i
Power and Heat Rate
Power and heat rate are concepts typically used for control volumes.
- Power: \dot{W}=\dfrac{dW}{dt}
- Heat Rate: \dot{Q}=\dfrac{dQ}{dt}
The first law can be rewritten for power and heat rate by changing mass to mass flow rate. The following is the first law for a steady-state system, where \dot{m_i} and \dot{m_e} are the input and output flow rates respectively.
\dot{F}_e = \sum{\dot{m}_e}\left(h_e+\dfrac{v_e^2}{2}+gz_e\right)
\dot{F}_i = \sum{\dot{m}_i}\left(h_i+\dfrac{v_i^2}{2}+gz_i\right)
\boxed{\dot{Q}-\dot{W}=\dot{F}_e-\dot{F}_i}
Conservation of Mass
The mass in a control volume must be conserved, that is, like energy, mass can not spontaneously be created or destroyed. Therefore, the rate that mass changes in a system must be the same as the net flow into the system.
\sum\dot{m_i}-\sum\dot{m_e}=\dfrac{dm_{cv}}{dt}
By definition, for a steady-state process the net mass change of the system is:
\dfrac{dm_{cv}}{dt}=0
Therefore this leads to the conclusion that flow in to the system must equal flow out of the system.
\boxed{\sum\dot{m_i}=\sum\dot{m_e}}
Transient Flow Processes
Transient processes are not steady-state, and have a finite duration. These processes are transitional in terms of system mass, for example, filling a tank with fluid. The first law for a transient flow process is dependent on the mass that will flow in to the system \sum{m_i}, the mass that will flow out of the system \sum{m_e}, the mass in the system at the end of the process m_2, and the mass that was in the system at the start of the process m_1.
F_e = \sum{m_e}\left(h_e+\dfrac{v_e^2}{2}+gz_e\right)
F_i = \sum{m_i}\left(h_i+\dfrac{v_i^2}{2}+gz_i\right)
E_2={m_2}\left(u_2+\dfrac{v_2^2}{2}+gz_2\right)
E_1={m_1}\left(u_1+\dfrac{v_1^2}{2}+gz_1\right)
\boxed{Q_{cv}-W_{cv}=F_e-F_i+E_2-E_1}