First Law of Thermodynamics for an open system

In this Article we will learn first law of thermodynamics in a practical way.

Applications of first law of Thermodynamics

First law of thermodynamics when it is applied to an open system has got tremendous applications all across industries. Using this law you can predict how much is the pressure drop across the nozzle, or how much is the energy required by the pump to pump the fluid out, or what is the heat transfer in heat exchanger, or what is the amount of work produced by the turbine.

In a nutshell first law simply means conservation of energy, or it states that energy is getting transformed from one form to another form.

First law for closed system

We will understand how first law is applied for a thermodynamic system by analyzing a simple example, an example of piston cylinder arrangement. Here the cylinder has got some gas inside it. Assume there is no air leakage to the surrounding. So this is an example of closed system where mass does not change. Assume the gas is absorbing some heat Q from the surrounding; also assume that this gas is able to push the piston upwards due to high pressure of gas. So the gas is doing some work on the piston with quantity W.

So there are 2 energy interactions to the gas, it will increase by a quantity Q, because it is absorbing energy. And it will decrease by a quantity W since it is losing energy by doing some work. So you can write change in energy (E) of gas as follows.

ΔE = Q - W

This is first law of thermodynamics for a closed system. Same equation you can write in differential form as follows. It is in form of rate of change of quantities per unit time.

dE / dt = Q - W

First law for an open system

Now we are going to open the system, or open the cylinder as shown below.

The system is no more closed now, it’s an open system. The mass is continuously varying. It can have an inlet mass flow rate at particular pressure and particular velocity. Similarly there will be exit flow rate of particular pressure and velocity. Here also our objective is the same. We want to find out energy change of the gas or the system. But here it is not possible to pin point a particular quantity of gas. The gas is continuously flowing. So before proceeding to the energy change calculation, we have to define a system first, a control volume where you will do energy balance.

Here the dotted line represents the control volume, or the space at which we will do energy balance. Here you can see there are 4 energy interactions to the system. 2 energy interactions which are coming to the system and another 2 energy interactions which leave the system. So if you want to find out energy change in system you should add energy transfer due to heat flow and inlet mass flow and subtract energy transfer due to work done and exit mass flow. So for an open system change in energy will be as follows.

Note that the flow stream has got 3 components of energy. Internal energy, kinetic energy and potential energy. Z represents the altitude of flow stream. This equation is the first law of thermodynamics for an open system.

Concept of flow work and enthalpy – More useful form of first law

But for an open system the term W, work done by the gas should be carefully examined. Here the gas is doing work to push the cylinder up, plus it is doing work to suck the fluid in or eject the fluid out. Or to maintain the flow gas has to do some work. This kind of work, the work which is required to maintain the flow is known as flow work. So the total work done by the system will be summation of visible work and flow work.

W = W _cv + Flow work

Wcv represents the visible work, in this case the work done by the gas on the piston. And we know flow work is the work required to eject the fluid out or suck the fluid out. The work required to eject the fluid out will be force at exit portion multiplied by velocity of this stream. Force is same as pressure at that portion times area. So we can represent flow work like this.

W = W_cv + (P₂A₂)V₂ - (P₁A₁)V₁

If you do some rearrangement to the equation by substituting volumetric flow rate as mass flow rate into specific volume, by representing u+Pv as a new property enthalpy,

h = u+P_v

the above equation will be simplified like this.

This is the final and most useful form of first law of thermodynamics for an open system.

One application of first law

We will work out one interesting example using firs law equation in this section. A pump problem, where fluid is getting pumped from point 1 to point 2.

We want to find out what’s the energy required by the pump to perform this action. To find out that we will use equation derived for first law of thermodynamics for an open system.We can assume the pump operation is in steady state. So energy of the pump does not change with time. So you can put first term in equation as zero. And usually there will not be any heat transfer to the pump. So you can put that term also as zero.

If cross sectional areas of point 1 and point 2 are equal, then velocities will be equal, so from this equation velocity part also get cancelled out. You can also assume height difference between inlet and outlets are negligible. So the altitude term also gets cancelled out. And finally what remains is this.

W_cv = m₁(h1₁ - h₂)

Work done by the control volume is mass flow rate times change in enthalpy. If you want work done on control volume or energy required by the pump, you have to just reverse the sign.

W_pump = m₁(h₁ - h₂)

Using the same approach you can solve lot of other energy transfer problems in industries.