Integration
Definition
... the area under a curve or line, bounded by limits.
Where it occurs.
The reversible, heat transfer follows from the integral,
$$ \dot{Q} = \dot{m} \times \int_1^2 T ds $$
Special features
There are analytical solutions for isobaric, isothermal, isentropic and polytropic processes.
How to draw it
- Click the radio button by "vertical construction lines"
- Use the sliders to move the two construction lines to the start entropy and end entropy. (Note - the position of the construction line can be finely adjusted with the keyboard curser keys. )
- Identify a a point of interest > . Plot a curve .
- Press "integrate curve (power)". Values of heat transfer, change in enthalpy and heat transfer rate are reported immediately above the plot area.
- Alternatively press "integrate curve (thrust)". Values of thrust, exit velocity, change in enthalpy and heat transfer rate are reported immediately above the plot area.
A special case is when the curve is very steep and the temperature change is barely visible. Here one can use horizonal construction lines to describe limits in
terms of pressure.
The theory
The integral is found from the trapezium rule .
If one assumes an ideal gas then the change in enthalpy follows from
$$ \Delta \dot{H} = \dot{m} (h_2 - h_1) = \dot{m} c_p (T_2 - T_1) $$
In an open system, assuming no changes in kinetic energy and potential energy, the Steady-Flow Energy Equation is
manipulated to yield shaft power .
$$ \dot{W} = \Delta \dot{H} - \dot{W} $$
In an adiabatic system, with gas approaching a nozzle at velocity \(c_1\), the exit velocity follows from the Steady-Flow Energy Equation and is
$$ c_2 = ... $$
The thrust developed by the nozzle is
$$ F = \dot{m} (c_1 - c_2) $$
Exercises
... to follow.
Links
... to follow.