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First Law of Thermodynamics

An introduction to the first law of thermodynamics, and discussing the general energy equation and its application to particular cases.

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Contents

Overview
The General Energy Equation
The Application Of The First Law To Particular Cases
Page Comments

Key Facts

Gyroscopic Couple: The rate of change of angular momentum ( $\inline \tau$ ) = $\inline I\omega\Omega$ (In the limit).

$\inline I$ = Moment of Inertia.
$\inline \omega$ = Angular velocity
$\inline \Omega$ = Angular velocity of precession.

Overview

Definitions

Potential energy (PE) is the energy possessed by a quantity of matter by virtue of its position above some specified datum.

Kinetic energy (KE) is the energy possessed by a quantity of matter by virtue of its translational velocity.

Internal energy is the sum of the molecular KE (translational and rotational) and the PE of individual molecules.

Heat engine is any machine that can convert heat into work, and vice versa.

Working substance (WS). The WS is used as the carrier for heat energy. The heat engine carries out the conversion process by a series of changes of state of the WS. The state of the WS is defined by the values of its properties, e.g. pressure, volume, temperature, internal energy, enthalpy. These properties are also sometimes called functions of state.

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Key facts

The first law of thermodynamics states that energy can neither be created nor destroyed, but only transformed from one form (e.g. work, heat, KE, PE, internal energy) to another.

The first law of thermodynamics can be used to develop the general energy equation, which can be written as:

$(E_1 + P_1 V_1) + KE_1 +$

$+ PE_1 + Q_S = (E_2 +$

$+ P_2 V_2) + KE_2 + PE_2 + W$

where $\inline E_1$ , $\inline KE_1$ , $\inline PE_1$ , and $\inline P_1$ , $\inline V_1$ , are the energies, and functions of state respectively characterizing the first state, $\inline E_2$ , $\inline KE_2$ , $\inline PE_2$ , and $\inline P_2$ , $\inline V_2$ , are the energies, and functions of state respectively characterizing the second state, $\inline Q_S$ is the heat supplied to the WS, and $\inline W$ the work done by the WS.

The general energy equation can also be expressed in terms of enthalpies as:

$H_1 + KE_1 + PE_1 + Q_S =$

$= H_2 + KE_2 + PE_2 + W$

where $\inline H_1$ is the enthalpy characterizing the first state, while $\inline H_2$ is the enthalpy characterizing the second state.

The form of the general energy equation can be adapted to particular cases. For example, for constant volume processes it can be reduced to:

$E_1 + Q_S = E_2$

for constant pressure non-flow processes:

$E_1 + Q_S = E_2 + W$

for constant pressure flow processes:

$H_1 + Q_S = H_2 + W$

for adiabatic non-flow processes:

$E_1 = E_2 + W$

for adiabatic flow processes:

$H_1 = H_2 + W$

while for throttling processes it states that there is no change in enthalpy:

$H_1 = H_2$

The first law of thermodynamics, also sometimes called the law of conservation of energy, states that energy can neither be created nor destroyed, but only transformed from one form to another. The different interchangeable forms of energy include, among others, work, heat, potential energy (PE), kinetic energy (KE), or internal energy.

The General Energy Equation

The first law of thermodynamics can be used to develop the so-called general energy equation for open systems. Imagine a heat engine taking in a working substance (WS) in an initial state characterized by the energies $\inline E_1$ , $\inline KE_1$ , and $\inline PE_1$ (see Figure 1).

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The WS passes through the machine having $\inline Q_S$ units of heat supplied to it and giving a work output of $\inline W$ units. The WS then leaves the machine with energies $\inline E_2$ , $\inline KE_2$ , and $\inline PE_2$ .

The work done to drive one unit mass of air through the system, this time diagramed as in Figure 2, is given by:

$Work \; done = F_2 l_2 - F_1 l_1$

(2)

which can also be written, by taking into account that the force acting on a piston is $\inline F = P \cdot A$ , as:

$Work \; done = P_2 A_2 l_2 - P_1 A_1 l_1$

(3)

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Equation (3) thus leads to:

$Work \; done = P_2 V_2 - P_1 V_1$

(4)

where $\inline P$ is the pressure and $\inline V$ the volume.

Applying the first law of thermodynamics, which states that the energy that enters the system equals that which leaves it, and also taking into account (4), we can write that:

$E_1 + KE_1 + PE_1 + Q_S = E_2 + KE_2 + PE_2 + W + (P_2 V_2 - P_1 V_1)$

(5)

Equation (5) can also be written as:

$(E_1 + P_1 V_1) + KE_1 + PE_1 + Q_S = (E_2 + P_2 V_2) + KE_2 + PE_2 + W$

(6)

and is called the general energy equation.

The term $\inline (E+PV)$ is called enthalpy and is denoted with $\inline H$ . Thus, the general energy equation can also be written as:

$H_1 + KE_1 + PE_1 + Q_S = H_2 + KE_2 + PE_2 + W$

(7)

where $\inline H_1$ is the enthalpy characterizing the first state, while $\inline H_2$ is the enthalpy characterizing the second state.

The form of the general energy equation can also be adapted to particular applications. For example, for a petrol engine, the $\inline KE$ and $\inline PE$ terms are negligible. Thus equation (7) becomes:

$H_1 + Q_S = H_2 + W$

(8)

from which we have that:

$Q_S = W + (H_2 - H_1)$

(9)

Therefore, in this case, the heat supplied to the WS equals the work done by the WS plus the gain in enthalpy.

For a gas turbine, we can ignore only the $\inline PE$ terms, and thus equation (7) becomes:

$H_1 + KE_1 + Q_S = H_2 + KE_2 + W$

(10)

from which:

$Q_S = W + (KE_2 - KE_1) + (H_2 - H_1)$

(11)

Therefore, in this case, the heat supplied to the WS equals the work done by the WS plus the gain in kinetic energy plus the gain in enthalpy.

For a non-flow process, there are no $\inline PV$ , $\inline KE$ , or $\inline PE$ terms. Thus, in this case, equation (6) becomes:

$E_1 + Q_S = E_2 + W$

(12)

from which:

$Q_S = W + (E_2 - E_1)$

(13)

Hence, the heat supplied to the WS equals the work done by the WS plus the gain in internal energy.

The Application Of The First Law To Particular Cases

As we previously saw, the first law of thermodynamics can be expressed in terms of the general energy equation (see equations 6 and 7). Also as previously noted, the form of the general energy equation can be adapted to particular applications (see equations 8, 10, and 12). We can further apply the general energy equation to more comprehensive particular cases.

1) Constant Volume Processes

In this case, the work output is zero, and the processes are invariably non-flow.

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Nearly all the terms in the general energy equation (6) are eliminated, leaving:

$E_1 + Q_S = E_2$

(14)

or:

$Q_S = E_2 - E_1$

(15)

Therefore, for constant volume processes, all the heat supplied goes into increasing the internal energy of the WS.

2) Constant Pressure Processes

a) Non-Flow Processes

Consider that the WS experiences an isobaric change in volume as diagramed in Figure 4.

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As the force acting on the piston is given by:

$F = P \cdot A$

(16)

the work done:

$W = F \cdot l$

(17)

becomes:

$W = P \cdot A \cdot l$

(18)

or, furthermore:

$W = P(V_2 - V_1)$

(19)

In the general energy equation (6) most terms are eliminated, leaving:

$E_1 + Q_S = E_2 + W$

(20)

which leads to:

$Q_S = (E_2 - E_1) + W$

(21)

Taking into account (19), equation (21) becomes:

$Q_S = (E_2 - E_1) + P(V_2 - V_1)$

(22)

or, furthermore:

$Q_S = (E_2 + P_2 V_2) - (E_1 + P_1 V_1)$

(23)

Thus:

$Q_S = H_2 - H_1$

(24)

Therefore, for constant pressure non-flow processes, the heat supplied to the WS equals the gain in enthalpy.

b) Flow Processes

In this case, considering that all the changes to the kinetic and potential energies are negligible, the general energy equation (7) becomes:

$H_1 + Q_S = H_2 + W$

(25)

which leads to:

$Q_S = W + (H_2 - H_1)$

(26)

However, it is impossible to devise a machine that operates a constant pressure flow process giving a constant work output. Therefore, in all practical cases, the heat supplied to the WS equals the gain in enthalpy.

3) Adiabatic Processes

These are processes during which there is no heat transfer between the WS and the surroundings.

a) Non-Flow Processes

For adiabatic non-flow processes, the general energy equation (6) becomes:

$E_1 = E_2 + W$

(27)

Therefore, the work done is:

$W = E_1 - E_2$

(28)

and it thus equivalates the loss of internal energy.

b) Flow Processes

This category includes all rotary turbines and compressors. The general energy equation (7) can be written in this case as:

$H_1 = H_2 + W$

(29)

from which the work done becomes:

$W = H_1 - H_2$

(30)

and it thus equivalates the loss of enthalpy.

4) Throttling Processes

Consider two points, $\inline 1$ and $\inline 2$ , which are far enough from the orifice so that the $\inline KE$ terms are negligible (see Figure 5).

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Furthermore, consider that there is no heat supplied and there is no work output. Therefore, the general energy equation (7) becomes in this case:

$H_1 = H_2$

(31)

Hence, for throttling processes, there is no change in enthalpy.

Example:

[imperial]

Example - Horsepower of a gas turbine

Problem

Twenty pounds of gas flow per second through a gas turbine. The inlet pressure is $100\; psi$ ( $lb/in^2$ ), the specific volume is $3 \; ft^3/lb$ , and the inlet velocity is $500\; ft/s$ . In passing through the engine, the internal energy drops by $40\; BTU/lb$ , while $100\; BTU/s$ are lost through radiation. If the gas is discharged at $1000\; ft/s$ and at $50\; psi$ with a specific volume of $4\; ft^3/lb$ , find the horsepower ( $HP$ ) developed by the turbine.

Workings

From the general energy equation we can write that:

$E_1 + P_1 V_1 + KE_1 + Q_S = E_2 + P_2 V_2 + KE_2 + W$

(1)

which gives:

$W = (E_1 - E_2) + (P_1 V_1 - P_2 V_2) + (KE_1 - KE_2) + Q_S$

(2)

In order to calculate $W$ from (2) we have to analyze each individual term. From the hypothesis we know that:

$E_1 - E_2 = 40 \; \frac{BTU}{lb}$

(3)

Still from the hypothesis, and also considering that $1\; BTU = 778 \; ft-lb$ , and $1\; ft^2 = 144 \; in^2$ , the $(P_1 V_1 - P_2 V_2)$ term expressed in $BTU/lb$ can be calculated as:

$P_1 V_1 - P_2 V_2 = \frac{100 \cdot 144 \cdot 3}{778} - \frac{50\cdot 144 \cdot 4}{778}$

(4)

from which we obtain:

$P_1 V_1 - P_2 V_2 = 18.5 \; \frac{BTU}{lb}$

(5)

Following a similar reasoning, the $(KE_1 - KE_2)$ term can be expressed in $BTU/lb$ as:

$KE_1 - KE_2 = \frac{500^2}{2g\cdot 778} - \frac{1000^2}{2g\cdot 778}$

(6)

where $g$ is the standard gravity expressed in imperial:

$g = 32.174 \; \frac{ft}{s^2}$

(7)

From equation (6) we thus obtain that:

$KE_1 - KE_2 = -15 \; \frac{BTU}{lb}$

(8)

Considering that $100 \; BTU/s$ are lost through radiation, and that the flow rate of the turbine is $20 \; lb/s$ , the $Q_S$ term can be expressed in $BTU/lb$ as:

$Q_S = -\frac{100}{20}$

(9)

from which:

$Q_S = -5 \; \frac{BTU}{lb}$

(10)

By using (3), (5), (8), and (10) in equation (2), we can calculate $W$ as:

$W = 40 + 18.5 - 15 - 5$

(11)

from which we obtain:

$W = 38.5 \; \frac{BTU}{lb}$

(12)

Taking into account that the flow rate of the turbine is $20 \; lb/s$ , that $1\; BTU = 778 \; ft-lb$ , that $1\; HP = 550 \; ft-lb/s$ , and also considering (12), we get the horsepower output of the turbine as:

Solution

$\frac{38.5 \cdot 20 \cdot 778}{550} = 1090 \; HP$

(13)