The First Law for Open Systems

MEE1018: Thermodynamics - Lecture 8

Continuity equation

ṁ = ρ₁A₁V₁ = ρ₂A₂V₂

Volume equation

ṁ = ρAc

Volumetric flow equation

V˙ = Ac

Using ideal gas equation to find density

ρ = p/RT

If a closed system has a change in KE or PE

Q_1-2 - W_1-2 = m[ (½c₂² + gz₂ + u₂) - ( ½c₁² + gz₁ + u₁)]

Flow energy

For an open system the KE and the PE may change (as above), but in also, because there is mass transfer, there is also some additional energy or work required to cause the flow.

Flow into an open system

W_in = P₁A₁L₁ = P₁V₁ = mP₁v₁

The work that has to be done to add mass to the system is the flow energy.

A fluid coming in or going out will also bring its own internal energy and kinetic and potential energy with it.

The resistance force that prevents entry of mass at the inlet of an open system

F₁ = P₁A₁

The work done in moving mass along the inlet

W_in = F₁L₁ = P₁A₁L₁

Inlet volume

V₁ = A₁L₁

The rate at which energy flows in / out

Flow energy in = ṁ₁P₁v₁

Flow energy out = ṁ₂P₂v₂

Change in system energy

dE_s/dt = Q˙ - W˙

where Q˙ is the rate of heat transfer and W˙ is the rate of work

General equation for the first law for open systems

Q˙ - W˙ + ṁ₁ [h₁ + gz₁ + ½c₁²] - ṁ₂ [h₂ + gz₂ + ½c₂²] = dE_s/dt

Simplification to the First Law for Open Systems

1. No change in PE - is a fluid going in and out at the same level?

2. No work W = 0 - look for something moving or electricity added or generated

3. No heat transfer Q = 0

4. No change in KE (c₁ = c₂) - is air just flowing along a tube

5. Steady state dE_s/dt = 0 - occurs once a system is up and running at operating conditions. The internal energy will remain constant.

6. Stead flow mass in = mass out - in this case the continuity equation will apply and volumetric flow.

Final simplified equation

Q˙ + ṁ [h₁ + ½c₁²] = W˙ + ṁ [h₂ + ½c₂²]

Inflow only system

Where there is only inflow (e.g. filling a tank), there may be steady flow in, but not steady state as the internal system energy will change. There is also no exiting mass.

Q˙ - W˙ + ṁ [h₁ + ½c₁²] = dE_s/dt

Steady flow devices

Nozzle / diffuser

Turbine

Compressor / pump

Heat exchange

Nozzle / diffuser

Accelerates / decelerates flow

Turbine

Extracts energy from flows and turns it into useful work.

Compressor / Pump

Does work on flows to add energy to them.

Heat exchanger

Transfers heat between two fluid flows