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Integrating the 1st Law

Closed
System

ΔEsys

This diagram shows a closed system with a system boundary enclosing it and sign convention arrows showing that heat transfer is positive when heat flow into the system and work is positive when it is done by the system.

Multiply the 1st Law equation, above, by dt and integrate both sides of the equation from t1 to t2:

These three terms are relatively easy to evaluate.

In order to solve these two integrals we need to know the inlet and outlet flow rates and properties as functions of time.

Lesson
5E Blog

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Lesson 5E: page 3 of 9

The next step is to multiply the enthalpy form of the 1st Law by the differential dt.
Then, we can integrate both sides of the equation.
The LHS is pretty easy to integrate and it turns out to be the change in the energy of the system during the process.
When we integrate the heat transfer RATE over the time interval, we just get the total amount of heat transferred during the process.
Similarly, when we integrate the RATE at which work is done by the system over the time interval, we just get the total amount of work done by the system during the process.
Notice that there are no longer dots over the Q & W. Q and W have units such as kJ or BTU, but not W because we have already integrated with respect to time.
The other two terms on the RHS are not as easy to integrate.
The mass flow rate, the specific enthalpy, or the velocity of the inlet and outlet streams may be functions of time.
It seems less likely that the altitude of the inlet and outlet streams would be functions of time, but it is not inconceivable.
In any case, it is time to apply our assumptions and see if they can help us get out of this mess.
Flip the page and see how it turns out.

Integrating the 1st Law

Ch 5, Lesson E, Page 3 - Integrating the 1st Law