In this article, you will find everything you need to know about the **First Law of Thermodynamics**.

Prepared by: **Dr. Durga Bastakoti (****Ph.D. in Mechanical Engineering)**

The First Law of Thermodynamics is based on the principle of conservation of mass and energy. During the closed system, i.e. control mass, interaction occurs in the form of energy only so the first law of thermodynamic for control mass is explained by taking the reference of conservation of energy. During the open system, i.e. control volume, interaction occurs in the form of both mass and energy, so the first law of thermodynamic for control volume is explained by taking the reference of both conservation of mass and energy.

In general, the First law of thermodynamic states that “energy cannot be created or destroyed, but it can only change the forms.”

**First Law of Thermodynamics for control Mass**

**Conservation of mass for control mass:**

As for the control mass, i.e. a closed system, only energy transfer takes place in the system. Therefore the total mass of the control mass system always remains constant. Mathematically it can be written as:

dm=0 ........(1)

So for any process occurring between the state one and state 2, the above equation becomes:

m_2-m_1=0

∴ m_2=m_1 ……… (2)

Further, the equation (1) can also be expressed in terms of rate which is given below:

dm/dt=0 …….. (3)

**Conservation of energy for control mass:**

Let us consider a control mass undergoing a process during which the amount of supplied heat to the control mass, i.e. closed system is ‘*Del Q’ *which produces ‘*Del W’* amount of work.

When the amount of supplied heat is less than the produced work by the system, then the total amount of energy of the system reduces. At the same time, when the amount of supplied heat is greater than the produced work by the system, then the total amount of energy of the system increases.

Therefore in general conversation of energy for control mass undergoing process can be stated as “the variation in the total amount of energy of control mass is equal to the difference between the amount of heat supplied to the system and the work produced by the system.”

Mathematically it can be written as;

dE=δQ-δW ……. (4)

Above equation (4) can also be expressed in terms of the rate as given below;

δE/δt=Q-W . . . . . . (5)

So for any process occurring between state 1 and state 2, the above equation becomes as;

Where,

If we consider the kinetic energy, potential energy, and internal energy of the system then the total energy becomes the summation of all three energy so the equation (6) can be written as;

Further, the kinetic energy and potential energy can be expressed as;

Substituting the value of KE and PE in equation (7), we get,

Piston cylinder device is the most typical example of control mass, i.e. closed system. So for the piston-cylinder device, the change in PE and KE during the stationary position is negligible when contrasted to the change in internal energy of the device so the above equation (8) would change into the following equation as;

The above equation (9) can be rearranged to get the equation for the heat transfer between the state 1 and state 2 as;

**First law of Thermodynamics for control mass undergoing cyclic process:**

For the control mass undergoing the cyclic process, we should consider integral in the equation (4). When taking the integral of equation (4) then,

Since the initial and final state of the cyclic process is similar, so,

Then the above equation (11) becomes as;

Or this equation can also be expressed in terms of the equivalent forms as below;

This equation (13) can also be expressed as;

So the first law of thermodynamics for control mass undergoing cyclic process is defined as the amount of heat transferred to the closed system is equal to the net work done by that system.

Or it can also be defined as the amount of heat rejected by the control mass, i.e. closed system is equal to that net work done on that system, i.e. control mass.

Further, it can also be defined in terms of the principle of conservation of energy as incoming energy to the system is equal to the outgoing energy from that system.

**First Law of Thermodynamic for control volume**

**Conservation of mass for control volume:**

The control volume system is able to interact with the surrounding through mass transfer as well as energy transfer. So it is also said to be an open system where the mass of the system may remain constant or change according to specific conditions.

Conservation of mass for an open system is defined as the change in mass within the system is equal to the difference between the entering mass to the system and leaving mass from the system.

**Mathematically it is given as;**

Let us consider the open system, i.e. control volume where there are two inlets and three outlets for the mass. Then the total rate of mass flow for the system is given as;

**Mass flow rate expression:**

Let us assume fluid is flowing via a port which has ‘A’ as the cross-sectional area where the fluid crosses the distance of ‘*Delta L*‘ in ‘*Delta T’ *time interval. Then the total mass of the fluid crossing the ‘*Delta L*‘ length is;

Where,

As,

Then above equation (16) becomes as;

Further, the rate of mass flow is given as;

The above equation can also be expressed in specific volume as given below;

The equation (15) can be written as below in terms of specific volume as;

**Conservation of energy for control volume:**

The conservation of energy for control volume can be defined as the change in the total amount of energy of the system is equal to the net energy transferred by the working fluid to the system plus the heat transferred to the system minus the work done by the system.

So mass transfer also affects the total energy of the system in addition to the heat and work transfer. If the mass enters into the system, then there is an increase in the total energy of the system while if the mass leaves the system, then the total energy decreases.

Mathematically it can be written as;

So, the above equation becomes as;

Where,

Similarly, the heat transfer always occurs because of the temperature difference between the surrounding and the system whether the system id open or closed, i.e. control volume of control mass so,

However, the total work transfer in case of the control volume consists of various types of work transfer like compression or expansion work, flow work, shaft work and so on so the total work transfer can be written as;

**Expression for flow work:**

The amount of energy needed to get the flowing fluid into the system or work performed by the fluid coming out of the system is said to the flow work. During this process, flow work is taken as negative while the flow work at the outlet is taken as ** positive**.

Let us consider a fluid is flowing via the inlet section having ‘A’ as the cross-section area which crosses the distance in the time interval as shown in the figure.

The amount of energy needed to displace the fluid is given as;

Where F= forcing acting on the particles of the fluids which is equal to the product of the pressure and the cross section area’ A’. Then the above equation (25) becomes as;

Similarly, the specific flow work is defined as the flow work per unit mass of the flowing fluid and is given as;

Then the rate of flow work is equal to;

Further, the shaft work is defined as the work produced by the shaft by consuming the energy carried by the fluid or work consumed by the shaft for increasing the fluid energy.

Substituting the value of *Wflow* in the equation (24) we get;

Then substituting the equation (23) and (29) in the equation (21) we get,

Similarly replacing *h = u + Pv,* we get,

This equation (31) is the generalized energy equation for the control volume, i.e. an open system where,

gives the energy carried by the fluid and known as the flow energy.

**Control volume analysis:**

The control volume or open system is analyzed in two different way. One being regarding time and others being regarding space. If we analyzed the control volume regarding the time, then it can be steady-state or unsteady state system. If the system’s properties at the specific point do not change with respect to time, then it is said to be steady-state system whereas if it changes with respect to time, then the system is said to be unsteady state system.

Similarly, the control volume with reference to space of coordinate can also be categorized into two systems, i.e. uniform and non-uniform. If the system’s properties do not change with space at the specific time instant, then the system is said to be uniform system whereas if it changes with respect to space at specific time instant, then the system is said to be non-uniform.

### i) **Steady-state analysis of control volume**

In order to be in a steady-state system, properties of the system should not be varied with the time. So for the steady-state system, mathematically control volume can be written as below;

Substituting the equation (32) in equation (15) we get,

Similarly substituting the equation (33) in equation (31) we get,

The above equation (34) and (35) illustrates that the incoming mass should be equal to the outgoing mass, and the incoming energy should be equal to outgoing energy for the steady-state system. Turbines, nozzles, compressors etc. are the typical examples of steady-state conditions.

**ii)** **Unsteady state analysis of control volume:**

In order to be in the unsteady state system, the properties of the system should be varied with time. So for the unsteady state system, mathematically control volume can be written as below;

The above equation (36) and (3&) can be integrated in order to determine the generalized mass and energy conservation equation for the unsteady state devices.

**Control Volume Application**

Control volume devices performance is based on the mass and energy conservation equations. So based on the performance and operation of the devices, control volume application can be grouped into four types which are explained below:

**i) Steady-state work application:**

Those devices which operate under the condition of steady-state and produce or consume works are considered as the steady-state work application. Turbine, fan, pumps, etc. are common examples. Since the turbine is a device having one inlet and one outlet, so the energy equation for the turbine is;

As the energy is lost from the turbine in the form of heat transfer so if we insulated the surface of the turbine then the loss of heat transfer is negligible so the process is said to be adiabatic turbine and the energy equation for the adiabatic turbine is given as;

Not only for the turbine, above equations (38) and (39) are valid for compressor, pump and fan.

**ii) Unsteady state work application:**

The above-mentioned devices operate at unsteady state conditions during the starting and shutting down period. Therefore the mass and energy conservation equations for these devices are;

For the piston-cylinder device the KE and PE energy are very less in comparison to internal energy so above equation (41) becomes as;

**ii) Steady-state flow application:**

Those devices which operate under the condition of steady-state and do not produce or consume works are considered as the steady-state flow application. Nozzle, evaporators, diffuser, condenser, heat exchanger, throttling valve etc. are the common examples.

Nozzle increases the velocity of fluid by decreasing the cross-section area, whereas the diffuser decreases the fluid velocity by increasing the cross-section area. Both these devices have one inlet and one outlet, so the energy equation for these devices are given as;

For the adiabatic nozzle or diffuser, the energy equation is given as;

**iv) Unsteady state flow application:**

A cooking gas cylinder is an example of an unsteady-state flow application where there is a continuous decrease in the mass of the system which does not produce any boundary work. So the mass and energy equation for the unsteady state flow application is given as;

Further, the PE and the KE of the system is negligible in this case in comparison to the change in internal energy. Therefore the equation (46) becomes as;

**Other statements of the first law**

**First law of thermodynamics for an isolated system:**

If the control mass, i.e. closed system is isolated from the surrounding then there would be no interaction between the system and the surroundings so then the energy equation would become as;

Therefore for the isolated system, the total energy of the system remains always constant.

**ii) First law of thermodynamics for control mass undergoing an adiabatic process:**

When the boundary of the closed system is insulated, and the heat transfer is zero then the energy equation would become as;

Hence, for an adiabatic process, the total energy of the closed system is equal to the work done on the system.

**iii) Perpetual motion machine of the first kind (PMM-1) is not possible:**

As friction is present in any running machine due to which the continuously running machine is not possible. Therefore, it is not possible to take the continuous useful output effect without a corresponding supply of input energy.