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12.4: Thermodynamic Energy Conversion

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    19016
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    Four fundamental thermodynamic properties were introduced in Section 8.1: volume \(\mathbb{V}\), pressure \(\mathbb{P}\), temperature \(T\), and entropy \(S\). Many devices convert between some form of energy and either energy stored in a confined volume, energy stored in a material under pressure, energy in a temperature difference, or energy of a disordered system. We can describe energy conversion processes in these devices using the language of calculus of variations with one of these parameters, \(\mathbb{V}\), \(\mathbb{P}\), \(T\), or \(S\), as the generalized path and another as the generalized potential. Table \(\PageIndex{1}\) summarizes the results.

    Many sensors convert energy between electrical energy and energy stored in a volume, pressure, or temperature difference. A capacitive gauge can measure the volume of liquid fuel versus vapor in the tank of an aircraft. Strain gauges and Piranhi hot wire gauges (Sec. 10.4), for example, are sensors that can measure pressure on solids or in gases. Pyroelectric detectors (Sec. 3.1), thermoelectric detectors (Sec. 8.7), thermionic devices (Sec. 10.1), and resistance temperature devices (Sec. 10.4) can be used to sense temperature changes.

    Many other energy conversion devices convert between energy stored in a confined volume, energy stored in a material under pressure, or energy in a temperature difference and another form of energy without involving electricity. For example, if you tie a balloon to a toy car then release the air in the balloon, the toy car will move forward. Energy stored in the confined volume of the balloon, as well as in the stretched rubber of the balloon, is converted to kinetic energy of the toy car. An aerator or squirt bottle converts energy of a pressure difference to kinetic energy of a liquid. An eye dropper converts energy of a pressure difference to gravitational potential energy. An airfoil converts a pressure difference to kinetic energy in the form of lift. A piston converts energy of a gas under pressure to kinetic energy. As discussed in Sec. 10.5, a constricted pipe, or a weir, converts energy of a pressure difference in a flowing liquid to kinetic energy of the liquid. A baseball thrown as a curve ball converts the rotational energy of the rotating ball into a pressure differential to deflect the ball's path [162, p. 350]. A Sterling engine converts a temperature difference to kinetic energy.

    Calculus of variations can be used to gain insights into thermodynamic energy conversion processes in these devices. The first step in applying the ideas of calculus of variations is to identify an initial and final form of energy. The Lagrangian is the difference between these forms of energy as a function of time. Some authors choose the Lagrangian as an entropy instead of an energy [170] [171], but throughout this text Lagrangian is assumed to represent an energy as described in Ch. 11.

    Table \(\PageIndex{1}\): Describing thermodynamic systems in the language of calculus of variations.
    Energy storage device A balloon filled with air confined to a finite volume A compressed piston A cup of hot liquid (hot compared to the temp of the room) A container with two pure gases separated by a barrier
    Generalized Path Volume \(\mathbb{V}\) in \(m^3\) Pressure \(\mathbb{P}\) in Pa Temperature \(T\) in K Entropy \(S\) in \(\frac{J}{K}\)
    Generalized Potential Pressure \(\mathbb{P}\) in Pa = \(\frac{J}{m^3}\) Volume \(\mathbb{V}\) in \(m^3 = \frac{J}{Pa}\) Entropy \(S\) in \(\frac{J}{K}\) Temperature \(T\) in K
    Generalized Capacity \(\frac{\mathbb{V}}{\mathbb{B}} = -\frac{\partial \mathbb{V}}{\partial \mathbb{P}}\) in \(\frac{m^6}{J}\) \(\frac{\mathbb{B}}{\mathbb{V}} = -\frac{\partial \mathbb{P}}{\partial \mathbb{V}}\) in \(\frac{J}{m^6}\) \(\frac{T}{C_v} = \frac{\partial S}{\partial T}\) in \(\frac{g \cdot K^2}{J}\) \(\frac{C_v}{T} = \frac{\partial T}{\partial S}\) in \(\frac{J}{g \cdot K^2}\)
    Constitutive relationship \(\Delta \mathbb{V} = -\frac{\mathbb{V}}{\mathbb{B}}\Delta\mathbb{P}\) \(\Delta \mathbb{P} = -\frac{\mathbb{B}}{\mathbb{V}}\Delta\mathbb{V}\) \(\Delta T = \frac{T}{C_v}\Delta S\) \(\Delta S = \frac{C_v}{T}\Delta T\)

    Energy (int expression)

    Energy (const. potential)

    \(\int \mathbb{V} d \mathbb{P}\)

    \(\mathbb{V} \Delta \mathbb{P}\)

    \(\int \mathbb{P} d \mathbb{V}\)

    \(\mathbb{P} \Delta \mathbb{V}\)

    \(\int T d S\)

    \(T \Delta S\)

    \(\int S d T\)

    \(S \Delta T\)

    Law for potential Bernoulli's Equation Second Law of Thermodynamics
    This column assumes constant \(S, T\) constant \(S, T\) constant \(\mathbb{P}, \mathbb{V}\) constant \(\mathbb{P}, \mathbb{V}\)

    Assume that only one energy conversion process occurs in a device. Also assume that if we know three (not two) of the four thermodynamic parameters, we can calculate the fourth. Additionally, assume small amounts of energy are involved, and the energy conversion process occurs in the presence of a large external thermodynamic reservoir of energy.

    As with the discussion of the previous tables, each column of Table \(\PageIndex{1}\) details the parameters of calculus of variations for a different choice of generalized path. In order, the columns can be used to describe energy storage in a gas confined to a finite volume, a material under pressure, a temperature differential, or an ordered system. The rows are labeled in the same way as in the previous tables of this chapter so that analogies between the systems can be drawn.

    Energy can be stored and released from a gas confined to a finite volume and a gas under pressure. These related energy conversion processes are detailed in the second and third columns of Table \(\PageIndex{1}\) respectively. The second column specifies parameters of calculus of variations with volume chosen as the generalized path and pressure as the generalized potential. The third column specifies parameters with pressure chosen as the generalized path and volume as the generalized potential. In reality, energy conversion processes involving changes in the pressure and volume of a gas are unlikely to occur without a change in temperature or entropy of the system simultaneously occurring. Resistive heating, friction, gravity, and all other energy conversion processes that could simultaneously occur are ignored. Temperature and entropy are explicitly assumed to remain fixed, and these assumptions are listed in the last row of the table for emphasis. These columns can apply to energy conversion in liquids and solids in addition to gases. Using the choice of variables in the second column, the capacity to store energy is given by \(\frac{\mathbb{V}}{\mathbb{B}}\) where \(\mathbb{B}\) is the bulk modulus in units pascals, and it is a measure of the ability of a compressed material to store energy [103]. Bulk modulus was introduced in Section 8.1. Using volume as the generalized path, the Euler-Lagrange equation can be set up and solved for the equation of motion. All terms of the resulting equation of motion have the units of pressure, and the equation of motion is a statement of Bernoulli's equation, an idea discussed in Section 10.5.

    The fourth and fifth columns of Table \(\PageIndex{1}\) specify parameters of calculus of variations with temperature and entropy chosen as the generalized path respectively. A cup of hot liquid stores energy. Similarly, a container with two pure gases separated by a barrier stores energy. The system is in a more ordered state before the barrier is removed than after, and it would take energy to restore the system to the ordered state. Both of these systems can be described by the language of calculus of variations. As detailed in the fourth column, temperature can be chosen as the generalized path and entropy can be chosen as the generalized potential. Alternatively as detailed in the fifth column, entropy can be chosen as the generalized path and temperature can be chosen as the generalized potential. Both of these columns assume that the pressure and volume remain constant. The quantity \(C_v\), which shows up in these columns, is the specific heat at constant volume in units \(\frac{J}{g \cdot K}\), and it was introduced in Sec. 8.2.

    The equation of motion that results when temperature is chosen as the path and entropy is chosen as the generalized potential is a statement of conservation of entropy, and each term of this equation has the units of entropy. This relationship is more commonly known as the second law of thermodynamics, and it shows up in the second to last row of Table \(\PageIndex{1}\). More commonly, the law is written for a closed system as [109, p. 236],

    \[\Delta S=\int \frac{\delta \mathbb{Q}}{T}+S_{p r o d u c e d}. \nonumber \]

    In words, it says the change in entropy within a control mass is equal to the sum of the entropy out of the control mass due to heat transfer plus the entropy produced by the system.

    \[\text{(change in entropy) = (entropy out due to heat) + (entropy produced)}\nonumber \]

    A system can become more organized or more disordered, so \(\Delta S\) may be positive or negative. If energy is supplied in or out, entropy can be transfered in or out of a system, so the quantity \(\int \frac{\delta \mathbb{Q}}{T}\) may be positive or negative.

    Energy is listed in the third to last row of Table \(\PageIndex{1}\) in two different forms. The first expression is an integral expression. For example, you can integrate the volume with respect to pressure to find the energy of a system.

    \[E=\int \mathbb{V} d \mathbb{P} \nonumber \]

    Alternatively, the second expression

    \[\Delta E=\mathbb{V} \Delta \mathbb{P} \nonumber \]

    can be used to find change in energy in the case when volume is not a strong function of pressure over a small element.


    This page titled 12.4: Thermodynamic Energy Conversion is shared under a CC BY-NC 4.0 license and was authored, remixed, and/or curated by Andrea M. Mitofsky via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.