# Thermodynamics of Manufacturing Processes—The Workpiece and the Machinery

## Abstract

**:**

## 1. Introduction

#### 1.1. Least Dissipation Energy and Minimum Entropy Generation

#### 1.2. Exergy and System Irreversibilities

_{0}δS’ = δW

_{rev}− δW

_{out},

_{rev}and actual work δW

_{out}obtained from the system. T

_{0}, often 298 K, is the system temperature at the thermodynamic dead state: a fixed state of equilibrium with the extended surroundings assumed as a thermal reservoir (i.e., an idealized, unchanging source of energy) where T = T

_{0}, P = P

_{0}. δS’ is the entropy generation and δ indicates path dependence. The destruction of a system’s exergy or available energy during a process, Equation (1), is a direct measure of the system’s irreversibilities. As with entropy production, the destruction of a system’s exergy or available energy during a process, Equation (1), is a direct measure of the system’s irreversibilities, i.e., a system with significant irreversibilities will have high exergy destruction during the process interaction. Therefore, it is desired to keep system irreversibilities at a minimum to maximize available energy and minimize exergy destruction. However, it is noted that a system, such as a battery, could be at its lowest useful potential or exhaust its useful energy while remaining thermodynamically “alive”, far from the dead state. Also, the use of constant T

_{0}obtained far enough from the system where the surrounding temperature is truly steady includes external irreversibilities in the portion of the surroundings between the system and the location of T

_{0}establishment [8,9]. Moreover, if the temperature of the surroundings is significantly different from 298 K used in most availability/exergy analyses, the results are then less consistent with reality, possibly impacting the widespread usage of thermodynamics-based analysis in the degradation/performance modeling of non-thermal systems, such as electronics and electromechanical systems.

#### 1.3. Local Equilibrium

## 2. Product in Formation—The Workpiece

_{p}= δQ

_{p}+ δW

_{p}

_{p}is the net heat transfer between the component and the surroundings during manufacturing and δW

_{p}is the net work that transforms the material. Here, surroundings include machinery and the immediate vicinity of the composite system. Via the second law, the entropy change [4,5] during manufacturing is given as

_{p}= (δQ / T)

_{p}+ δS’

_{p}

_{p}is the component’s instantaneous absolute temperature (in K) and δS’

_{p}≥ 0 is the entropy produced or generated in the component. The principles of minimum entropy generation and least energy dissipation also assert that δS’

_{p}≥ 0 [3,16,17]. In Equation (3), the first right-hand side term is entropy transfer out of the component via heat. Equations (2) and (3) combine to give

_{p}= T

_{p}dS

_{p}− T

_{p}δS’

_{p}+ δW

_{p}

_{p}= (δW

_{p}− dA

_{p}− S

_{p}dT

_{p}) / T

_{p}

_{p}= dU

_{p}− T

_{p}dS

_{p}− S

_{p}dT

_{p}

_{p}dS

_{p}is the heat transfer in/out of the workpiece and S

_{p}dT

_{p}is the internal energy dissipation (within the workpiece via compositional change/diffusion [4,5], or induced at the workpiece-process interface via friction or other external heating). Some non-thermal processes that are not temperature-controlled may yield a workpiece temperature rise driven by the workpiece entropy content S

_{p}, the significance of which depends on the material properties, e.g., the coefficient of thermal expansion, melting point, and the nature of the manufacturing process. Manufacturing processes increase the workpiece’s Helmholtz energy—its utility or useful energy to do work—to a maximum (finished state), dA

_{p}≥ 0, while reducing its overall utility-based entropy, dS

_{p}≤ 0. Hence, the effect of the relatively low non-microstructure-changing temperature rise dT

_{p}is further minimized by the manufacturing process-reduced entropy content S

_{p}= S

_{0}+ dS

_{p}, where S

_{0}is the initial material entropy content. This indicates that during most non-thermal manufacture, the last term of Equation (6) can be neglected to simplify Equation (5) as

_{p}= (δW

_{p}− dA

_{p}) / T

_{p}≥ 0.

_{p}includes dissipative phenomena at the machine-workpiece interface that do not contribute to the product’s desired final form, e.g., using a non-ideal cutting tool for a machining task will increase δW

_{p}for a given dA

_{p}, and in turn, the entropy generation δS’

_{p}is increased. Therefore, it is easily inferred that a low δS’

_{p}is desirable. As mentioned previously, the second law prohibits negative entropy generation in all real systems/processes, i.e., δS’

_{p}≥ 0 or δW

_{p}≥ dA

_{p}, establishing δS’

_{p}= 0 or δW

_{p}= dA

_{p}as the limit of possibility or ideal (reversible) case. In other words, conforming with everyday experience, one cannot obtain from the product more than what has gone into its formation (efficiency ≥ 100% is unattainable), a corollary of the second law known as the Carnot limitation [10]. Equation (7) further indicates that the ideal case is only possible under perfectly isothermal conditions dT

_{p}= 0.

## 3. Manufacturing Equipment and Process—The Machinery

_{m/c}= δQ

_{m/c}+ δW

_{m/c}

_{m/c}= δW

_{in}− δW

_{p},

_{m/c}is always desired. δW

_{m/c}includes all the energy conversion losses, friction, Ohmic dissipation, corrosion, plasticity, and shaft misalignment effects, and should be resolved into appropriate constituents based on an order of magnitude analysis of the specific system or manufacturing process. Combined energy and entropy balance on the machinery (substituting entropy balance and Equation (9) into Equation (8)) yields

_{m/c}= T

_{m/c}(dS − δS’)

_{m/c}+ δW

_{in}− δW

_{p}.

_{m/c}= dU

_{m/c}− T

_{m/c}dS

_{m/c}− S

_{m/c}dT

_{m/c}, the entropy generation in the machinery is

_{m/c}= (δW

_{in}− δW

_{p}− dA

_{m/c}− S

_{m/c}dT

_{m/c}) / T

_{m/c}≥ 0.

_{m/c}is the differential ideal energy change, which can be specified via nominal machine/process specifications, or dropped if unknown, to give an instantaneous entropy generation for low-amplitude temperature changes (adequate for most pseudo-steady non-thermal processes)

_{m/c}= (δW

_{in}− δW

_{p}) / T

_{m/c}≥ 0,

_{m/c}dT

_{m/c}, a measure of the interfacial thermal fluctuations. Further breakdown of this term is given in Section 4 and references [18,19,20].

_{total}= δS’

_{p}+ δS’

_{m/c}= [(δW

_{p}− dA

_{p}) / T

_{p}] + [(δW

_{in}− δW

_{p}) / T

_{m/c}].

_{p}≈ T

_{m/c}, Equation (13) becomes

_{total}= (δW

_{in}− dA

_{p}) / T

_{p}.

_{total}, Equation (14), measures the efficiency of the entire system-process interaction and can be used as a first (overall system) analysis parameter, given its relative ease of evaluation: δW

_{in}is usually known/measurable and dA

_{p}is easily specified and typically standardized for a product and/or manufacturing task (e.g., drilling a hole in a thick steel plate or adding a thickener to grease in production)—a straight line joins the Helmholtz energy states before and after the task, an artifact of the thermodynamic state principle [4,5,8,9,15]. With a known/measured δW

_{p}, Equations (7) and (12) give the individual contributions from the workpiece and the machinery respectively. Further sub-system analyses can be performed as necessary to determine the significant sources of irreversibilities in the process.

## 4. Entropy Content S and Internal Free Energy Dissipation–SdT

_{A}= S(T,X,N) [15]: entropy of a system depends on the temperature T, generalized displacement $X$, and number of moles $N$, all of which are experimentally and instantaneously measurable. Via partial derivatives, the Helmholtz entropy change for a system with one reactive species is

## 5. Battery Charging and Other Example Processes

_{p}= ∫

_{t}VI dt (blue plot in Figure 2, left axis), can be determined via knowledge of the charging process, readily obtained from measurements. Similar to the Helmholtz free energy ∆A

_{p}for non-reactive and non-thermal systems, the minimum recharge energy required for a battery or other electrochemical energy device is its Gibbs free energy ∆G

_{p}= V

_{0}$\mathcal{C}$

_{rev}(green plot in Figure 2, left axis), determined via knowledge of the battery: V

_{0}is the battery’s standard potential and $\mathcal{C}$

_{rev}can be evaluated using Faraday’s first law [4,20]. In terms of chemical potential and number of moles or reaction affinity and extent [4,9], change in Gibbs potential ∆G

_{p}characterizes the chemical formation of products, a standardized form of which is the so-called Gibbs free energy of formation of certain pure substances. For a thermally dominant process, enthalpy ∆H

_{p}is the minimum thermal energy required for product formation, which has also been standardized as the enthalpy of formation.

_{p}(blue plot, left axis), Gibbs free energy ∆G

_{p}(green plot, left axis), and entropy generation S’

_{p}(red plot, right axis), via Equation (20), obtained from measured V and T with constant I of 3 A, verify the above formulations:

- W
_{p}≥ ∆G_{p}and S’_{p}≥ 0 for a product-forming or energy-adding process; and - as W
_{p}→ ∆G_{p}, S’_{p}→ 0 and ∆T → 0, limit of which is the reversible transformation, i.e., W_{p}= ∆G_{p}, S’_{p}= 0 and ∆T = 0.

_{p}= $\mathcal{V}$σdε where $\mathcal{V}$ is the volume, σ is stress, and ε is strain [18,21]; dA

_{p}= σ’

_{f}d(σ’

_{f}/E), a constant, where σ’

_{f}is the fatigue strength coefficient and E is Young’s modulus. Similarly, δW

_{p}for other processes can be expressed in terms of process-defining variables or characterized using the specific process energy as done in [22,23] for cutting and grinding processes and accompanying dA

_{p}specified in similar parameters as done above for battery charging and material deformation.

## 6. Product in Use

_{p}= (dA

_{p}− δW

_{out}) / T

_{p}≥ 0

_{p}and δW

_{out}are the product’s maximum and actual work outputs, respectively. More details on dissipative entropy generation during system/product operation can be found in [1,14,18,19,20,24].

## 7. Entropy: Generation or Change

_{p}= (dU

_{p}+ T

_{p}δS’

_{p}− δW

_{p}) / T

_{p}≤ 0,

_{p}+ T

_{p}δS’

_{p}) ≤ δW

_{p}, the input product formation work at the machine-workpiece interface δW

_{p}must be greater than or equal to the sum of the required energy content of the finished product dU

_{p}and the workpiece irreversibilities T

_{p}δS’

_{p}. However, according to the second law of thermodynamics, the process described by Equation (22) is not possible unless the total—workpiece and machine—entropy is monotonically non-decreasing, expressed as

_{total}= dS

_{p}+ dS

_{m/c}≥ 0.

_{m/c}(and that of the immediate vicinity considering the entropy transfer out via heat, see Equation (3)) will increase by an amount greater than or equal to the reduction in the component’s entropy via the manufacturing process.

## 8. Utility vs. Availability Analysis—Local Equilibrium vs. Thermodynamic Dead State

_{rev}~ dA

_{p}). However, it is noteworthy that Equations (1) and (20) will give different results under conditions where the component temperature T

_{p}is a variable on the “phenomenological” path, i.e., T

_{p}≠ T

_{0}, allowing for more accurate and consistent evaluation of internal irreversibilities within the system of interest only, while considering changes in the surroundings and the system’s thermal characteristic. The use of constant T

_{0}obtained far enough from the system where the temperature of the surroundings is truly steady can include significant external irreversibilities in the portion of the surroundings between the system and the location of T

_{0}establishment [8,9].

## 9. Fluctuations and Instabilities

_{p}= (δW

_{p}− dA

_{p}− S

_{p}dT

_{p}) / T

_{p}, and Equation (11): δS’

_{m/c}= (δW

_{in}− δW

_{p}− dA

_{m/c}− S

_{m/c}dT

_{m/c}) / T

_{m/c}≥ 0, give entropy generation in real systems far from equilibrium. Note that for an instantaneous application, dA

_{p}and dA

_{m/c}can be considered insignificant—both are elemental (differential) constants. For many real systems and processes (not merely existing at the “least dissipation” state), high-amplitude instantaneous fluctuations mean fast changing δW, S, and T, interactions included in Prigogine’s “local” potential formulation [4,5,6,7], developed for far-from-equilibrium transformations, which has been shown, via Equation (19), to be an equivalent expression of the –SdT term present in above formulations for real systems experiencing significant temperature changes during work interactions [18,19]. Unlike Equation (19) and Prigogine’s stability equation, Onsager’s reciprocity relations [3] do not apply to far-from-equilibrium interactions. With less fluctuations indicating stability, system optimization can be performed by minimizing –SdT, the limit of which leads to L.E.D. and M.E.G. Optimization strategies based on entropy generation minimization have been proposed by Bejan [25,26]. Equation (19) is central to the current work by the author on the degradation of all real systems [18,19,20,27].

## 10. Conclusions

## Funding

## Conflicts of Interest

## Abbreviations

Nomenclature | Name | Unit |

A | Helmholtz free energy | J |

G | Gibbs energy | J, Wh |

I | discharge/charge current or rate | A |

Q | heat | J |

S | entropy or entropy content | J/K, Wh/K |

S’ | entropy generation or production | J/K Wh/K |

t | time | sec |

T | temperature | degC or K |

U | internal energy | J |

V | voltage | V |

$\mathcal{V}$ | volume | m^{3} |

W | work | J |

Subscripts & acronyms | ||

0 | constant or initial reference | |

p | product, workpiece, component | |

m/c | machine, machinery | |

total | total | |

in | input | |

out | output | |

rev | reversible |

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**Figure 2.**Ohmic work W

_{p}= ∫

_{t}VI dt (blue plot, left axis), Gibbs free energy ∆G

_{p}= V

_{0}q

_{rev}(green plot, left axis), and entropy generation S’

_{p}from Equation (20) (red plot, right axis) during a 1.5 h constant-current charge of a 3.7 V, 11 Ah Li-ion battery of 3 A.

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**MDPI and ACS Style**

Osara, J.A.
Thermodynamics of Manufacturing Processes—The Workpiece and the Machinery. *Inventions* **2019**, *4*, 28.
https://doi.org/10.3390/inventions4020028

**AMA Style**

Osara JA.
Thermodynamics of Manufacturing Processes—The Workpiece and the Machinery. *Inventions*. 2019; 4(2):28.
https://doi.org/10.3390/inventions4020028

**Chicago/Turabian Style**

Osara, Jude A.
2019. "Thermodynamics of Manufacturing Processes—The Workpiece and the Machinery" *Inventions* 4, no. 2: 28.
https://doi.org/10.3390/inventions4020028