First Law of Thermodynamics
Statement:
When a system is taken through a cyclic change, the net heat to or from the system is equal to the
net work from or to the system.
Qnet = QH + (- QL)
= QH - QL
Wnet = Wout - Win
Mathematically, ∑ 𝑄 = ∑ 𝑊
=> ∑ 𝑑𝑄 = ∑ 𝑑𝑊
∮(𝑑𝑄 − 𝑑𝑊) = 0
Heat and work must be in the same unit.
Corollary-1:
There exist a property of a closed system such that a change in it’s a value is equal to the
difference between the heat supplied and work done during any change of state.
From 1st law of thermodynamics,
For cycle 1A2B1, ∮(𝑑𝑄 − 𝑑𝑊) = 0
=> ∫ (𝑑𝑄 − 𝑑𝑊) + ∫ (𝑑𝑄 − 𝑑𝑊) = 0 − − − − − (1)
1𝐴2
2𝐵1
For cycle 1A2C1, ∮(𝑑𝑄 − 𝑑𝑊) = 0
=> ∫ (𝑑𝑄 − 𝑑𝑊) + ∫ (𝑑𝑄 − 𝑑𝑊) = 0 − − − − − (2)
1𝐴2
2𝐶1
From (1) & (2), => ∫2𝐵1(𝑑𝑄 − 𝑑𝑊) = ∫2𝐶1(𝑑𝑄 − 𝑑𝑊)
Thus we can say, ∫(𝑑𝑄 − 𝑑𝑊) is independent of the path rather depends on state 1 and 2. Since
it depends on points, we can easily say that it is a property.
As heat and work is energy, so this property is to be termed as Internal energy. So for a process 1
2
to 2, ∫(𝑑𝑄 − 𝑑𝑊) = ∫1 𝑑𝑈
First law can be expressed as,
𝑄1−2 = ∆𝑈 + 𝑊1−2
Corollary-2:
The internal energy of a closed system will remain unchanged when the system is isolated from
the (its) surroundings.
From corollary-1,
𝑄1−2 = ∆𝑈 + 𝑊1−2
If the system is isolated, then 𝑄1−2 = 0 𝑎𝑛𝑑 𝑊1−2 = 0
So
∆𝑈 = 0
=> 𝑈2 − 𝑈1 = 0
∴ 𝑈2 = 𝑈1
Corollary-3:
A perpetual motion machine of first kind is impossible.
PMM1: It is such a machine which will produce work continuously without absorbing heat from
surroundings and or without work being done on it. It does not consume any energy.
The first law states the general principle of the conservation of energy. Energy is neither
created nor destroyed, but only gets transformed from one form to another. There can
be no machine which would continuously supply mechanical work without some other form
of energy disappearing simultaneously. Such a fictitious machine is called a perpetual
motion machine of the first kind, or in brief, PMM1. A PMMI is thus impossible.
Limitations of the First Law
The First law deals with the amounts of energy of various forms transferred between the
system and its surroundings and with changes in the energy stored in the system. It treats
work and heat interactions as equivalent forms of energy in transit and does not indicate the
possibility of a spontaneous process proceeding in a certain direction. It is the second law
which assigns a quality to different forms of energy, and also indicates the direction of any
spontaneous process.
Steady Flow Energy Equation (SFEE)
Steady flow: The flow in which various properties don’t change with respect to time.
Assumption:
Following assumption should be made in order to derive SFEE.
1. Mass flow rate is constant with time; mass flow rate at inlet is equal to the mass flow rate
at outlet.
2. The properties at any time within the open system do not vary with time.
3. The properties are constant over the cross section of flow at inlet and outlet.
4. Any heat and work crossing the control surface does so at a uniform rate.
5. The effects of capillary, electricity or magnetism are neglected.
Derivation:
Since there is no accumulation of mass or energy
∂m1 e1 + ∂m1 p1 V1 + Q = ∂m2 e2 + ∂m2 p2 V2 + Wsf -------------(1)
e = I. E + K. E + P. E [Total stored energy per unit mass]
1
=> 𝑒 = 𝑈 + v 2 + gz
2
From equation (1),
Q = Wsf + (∂m2 p2 V2 − ∂m1 p1 V1 ) + (∂m2 e2 − ∂m1 e1 )
Considering ∂m2 = ∂m1 = ∂m
1
=> 𝑄 = Wsf + ∂m(p2 V2 − p1 V1 ) + ∂m(U2 − U1 ) + 2 ∂m(v22 − v12 ) + ∂mg(z2 − z1 )
1
=> 𝑄 = Wsf + ∂m∆pV + ∂m∆U + 2 ∂m(v22 − v12 ) + ∂mg(z2 − z1 )
1
=> 𝑄 = Wsf + ∂m(∆pV + ∆U) + 2 ∂m(v22 − v12 ) + ∂mg(z2 − z1 )
=> 𝑄 = Wsf + ∆H + ∆K. E + ∆P. E
𝐐 = 𝐖𝐬𝐟 + ∆𝐇 + ∆𝐊. 𝐄 + ∆𝐏. 𝐄
This is the required SFEE.
Application of SFEE:
Nozzle:
It is a device used for the increase of velocity at the cost of pressure drop.
 Nozzles are thermally insulated.
 No work is done by the nozzle.
 The change in the elevation between entry and exit are negligible.
From SFEE,
Q = Wsf + ∆H + ∆KE + ∆PE
Here, ∆PE = 0, Q = 0 and Wsf = 0
=> ∆𝐾𝐸 = −∆𝐻
1
=> 2 (v22 − v12 ) = H1 − H2
=> (v22 − v12 ) = 2(H1 − H2 )
If entry velocity is negligible then v1 = 0
=> 𝐯𝟐 = √𝟐(𝐇𝟏 − 𝐇𝟐 )
Boiler:
It is a device to produce steam at a predetermined condition of pressure and temperature by
burning of fuel.
 Velocity change is not appreciable.
 Change in potential energy is also negligible.
 Steady flow work is zero.
From SFEE,
Q = Wsf + ∆H + ∆KE + ∆PE
Here, ∆PE = 0, ∆KE = 0 and Wsf = 0
=> Q = ∆H
=> 𝑄 = 𝐇𝟐 − 𝐇𝟏
Turbine:
Its objective is to produce work. It is a device used to obtain work from a steady flowing fluid
expanding from high pressure to low pressure.
 Turbines are thermally insulated.
 Velocity change is negligible.
 Change in potential energy is negligible since they are horizontally mounted.
From SFEE,
Q = Wsf + ∆H + ∆KE + ∆PE
Here, ∆PE = 0, ∆KE = 0 and Q = 0
=> Wsf + ∆H = 0
=> Wsf = −∆H = −(H2 − H1 )
=> 𝑊𝐬𝐟 = 𝐇𝟏 − 𝐇𝟐
Compressor:
It is reciprocal or inverse to turbine. Actually it’s a work consuming device.
=> 𝑊𝐬𝐟 = 𝐇𝟐 − 𝐇𝟏