# On Description of Acceleration of Spinless Electrons in Law of Heat Conduction a capite ad calcem in Temperature

## Abstract

**:**

## 1. Introduction

- (i)
- the microscopic theory of reversibility of Onsager [2] is violated;
- (ii)
- it neglects the time needed for the acceleration of heat flow by free electrons (Sharma, [9]);
- (iii)
- (iv)
- the development of Fourier’s law was from observations at steady state;
- (v)
- (vi)
- Landau and Lifshitz observed the contradiction of the infinite speed of propagation of heat with Einstein’s light speed barrier [23];
- (vii)
- Fourier’s law breaks down at the Casimir limit [24].

_{0}subject to a constant surface temperature boundary condition for times greater than zero. The hyperbolic partial differential equation (PDE) that forms the governing equation of heat conduction was solved by a new method called relativistic transformation of coordinates. The hyperbolic PDE is multiplied by e

^{τ/2}and transformed into another PDE in wave temperature. This PDE is converted to an ordinary differential equation (ODE) by a transformation using [10] a variable that is spatio-temporally symmetric. The resulting ODE is seen to be a generalized Bessel differential equation. The solution from this approach is within 12% of the exact solution obtained by Baumeister and Hamill using the method of Laplace transforms. There are no singularities in the solution. There are four regimes to the solution: (a) the inertial regime at shorter times compared with the lag time at the considered point; (b) the regime at larger distances from the surface further from the wave front X > τ characterized by the Bessel composite function of the zeroth order and first kind and; (c) the regime at shorter distances from the surface prior to the wave front at τ > X characterized by the modified Bessel composite function of the zeroth order and first kind and; (iv) the wave front regime at τ = X. Expressions for penetration length and inertial lag time are developed. The comparison between the solution from the method of relativistic transformation of coordinates and the method of Laplace transforms was made by use of Chebyshev polynomial approximation and numerical integration. In a similar manner, the exact solution to the hyperbolic PDE is solved by the method of relativistic transformation of coordinates for the infinite cylindrical and infinite spherical medium. For the case of heating a finite slab, the Taitel paradox problem is revisited. Taitel [34] found that when the hyperbolic PDE was solved the interior temperature in the slab was found to exceed the wall temperature of the slab. This is in violation of the second law of thermodynamics (Bai and Lavine [35], Zanchini [36]). By use of the final condition in time at steady state, the wave temperature was found to become zero at steady state. This condition, when mathematically posed as the fourth condition for the second-order PDE, leads to well-bounded solutions within the bounds of the second law of thermodynamics. For systems with large relaxation times, i.e., ${\mathsf{\tau}}_{r}>\frac{{\pi}^{2}}{{a}^{2}\alpha}$, subcritical damped oscillations can be seen in the temperature. In a similar manner, the transient temperature for a finite sphere and finite cylinder are also derived. Nanoscale effects in the time domain (Sharma [19]) are as important, if not more important than nanoscale phenomena in the space domain in a number of applications. When the advances in microprocessor speed are approaching the limits of physical laws on gate width and miniaturization, there is incentive to re-examine the physical laws at a level of scrutiny never done before. In this study, the application of free electron theory to derivation of a universal law of heat conduction is revisited. Acceleration of spinless electrons is not taken into account during the derivation of the law of heat conduction. When the applied force on the spinless electron by the applied temperature difference equals the drag force, the electron has attained its steady drift velocity. This can be seen later in the derivation that Fourier’s law of heat conduction will lead to. When the velocity of the electron is written in terms of heat flux, the Cattaneo and Vernotte equation results. The velocity of the electron gets eliminated in the derivation. Rather than eliminating the velocity, the acceleration of the spinless electron is expressed in terms of the accumulation of energy. This can lead to a de novo law of heat conduction. In an earlier study (Sharma, [10]), the accumulation at the surface of thermal energy transfer was shown to be causative in the need for an additional term in the law of heat conduction. The non-Fourier conduction equation that results is evaluated for use in the prediction of transient temperature in a finite slab.

## 2. Materials and Methods

#### 2.1. Free Electron Theory

#### 2.2. Derivation of Alternate Non-Fourier Conduction Equation

_{B}is the Boltzmann constant, m is the mass of the electron in (kg) and v

_{e}is the velocity of the free electron in (m s

^{−1}). The heat flux, q

_{z}, in (W m

^{−2}) is the rate of energy transfer across a cross-sectional area of the heat conduction path. The heat flux in terms of the properties of the moving electron can be written as follows:

_{r}where τ

_{r}is the relaxation time of the materials as defined by Cattaneo [25] and Vernotte [26].

_{v}, by definition is the energy needed to raise one mole of a substance by 1° K. For one molecule this would be (1.5 k

_{B}) as a corollary of the equi-partition energy theorem (Equation (1)). The electron cloud is assumed to be ideal gas. It can be shown that for an ideal gas (Sharma [31]):

_{r}is 0.3 τ, where τ is the collision time of the electron, Equation (16) becomes:

_{e}, is approximately equal to the 95% of the velocity of heat, v

_{h}. The velocity of heat is given in terms of the thermophysical properties of materials as given in Equation (17). The coefficient to the ballistic term in Equation (9) may be simplified as:

_{h}, Equation (20) reverts to Fourier’s law of heat conduction. In the asymptotic limit of the null transfer of heat, Equation (20) indicates a jump in temperature after an elapsed time at a considered point in the medium. This is at zero drag. This may not be achievable in practical applications.

#### 2.3. Entropy Production Term

_{h}can be seen as:

_{xx}are needed in addition to the applied wall shear stress, τ

_{xy}, in order to completely describe and characterize the flow of these fluids. The viscosity changes exponentially with change in the volume fraction of the polymer in the solution. The change of viscosity with temperature is exponential as well. Master curves have been developed by industrial technologists and rheologists and used in the industry. Rheology is the study of deformation and flow of non-Hookean solids and non-Newtonian liquids. The Newton Law of Viscosity can be written as follows:

_{xy}is the tangential shear stress and µ is the viscosity of the liquid and v

_{y}is the velocity of the fluid in the y direction. The gradient of the velocity is in the x direction.

_{mom}is the relaxation time of the momentum. Sharma [9] has shown in analogous heat transfer problems that this parameter is a measure of the acceleration time of the free electron that can be used to describe Fourier conduction before it reaches steady state. It was found to be about one-third of the collision time of the free electron and an obstacle. Equation (30) has been used to characterize viscoelastic fluids where µ is the zero shear viscosity. The effects of Equation (30) exist for all transient flows, Newtonian or otherwise. However, these effects are only seen in a pronounced manner in some materials. For instance, “silly putty” is considered viscoelastic. The material flows readily when squeezed slowly using the palms of one’s hands and may be considered to be in the viscous Newtonian state. The material can be rolled into a ball and the ball will rebound when dropped onto a hard surface. It can be expected that the stresses change rapidly and the material can be seen to behave analogous to an elastic solid. In some cases, Equation (30) is simplified to include only the shear rate term and the accumulation of the momentum term. This may be applicable when the changes are rapid. Sharma [9] discussed how poor use of the initial conditions can result in model solutions that may be in dissonance with the second law of thermodynamics using Equation (30). Damped wave transport and relaxation were studied. The results from an in-depth study of the Cattaneo and Vernotte non-Fourier heat conduction answered a few questions. One issue is that the entropy production becomes negative when the momentum flux is high and the momentum rate is in the opposite direction. This can lead to a violation of the second law of thermodynamics locally. The “overshoot” phenomena were shown to be a mathematical artifact and, when physically reasonable, the final time condition or the higher accumulation of the temperature condition is used and the overshoot was found to disappear. An equation to describe real mass diffusion is derived from Gibbs’ chemical potential formulation for a non-relativistic solute particle. The acceleration term eliminated between the equation of motion for the spinless particle and the accumulation of chemical potential formulation leads to an equation for mass diffusion that is a capite ad calcem in concentration (Equation (39)). This entropy production for this term can be seen to be positive for real mass transfer events. The dC/dt, the time derivative of concentration and mass flux, can either be “both positive” and “both negative” but never one positive and one negative for spontaneous mass diffusion events. So for all extemporaneous and practical purposes, the second law of thermodynamics is obeyed.

_{2}can is a retardation time.

#### 2.4. Transport Parameters

^{−2}), D

_{AB}is the binary diffusivity, C

_{A}is the concentration of the diffusing solute and z is the spatial direction of the solute transfer. The solute transfer is considered to be one-dimensional. When motion in other dimensions becomes important, terms can be added for each ordinate. The diffusion coefficient D

_{AB}is obtained from the Stokes–Einstein formulation as follows:

_{B}is the Boltzmann constant with units of J molec

^{−1}K

^{−1}, T is the absolute temperature, f is the molecular drag coefficient, µ is the viscosity of the surrounding medium, and R

_{0}is the radius of the solute molecule. Rigid spheres are assumed. Equations (32) and (33) are derived as follows. The chemical potential of an ideal solution of solute A in solvent B can be written as:

_{A}can be approximated for dilute solutions as $\left(\frac{{C}_{A}}{{C}_{B}}\right)$. At steady state, when the solute is in motion caused by the chemical potential gradient, the driving force and drag forces will be equal to each other and:

_{A}v

_{A}and:

_{0}. The diffusion coefficient recovered from Equation (36) by comparing Equation (36) and Equation (32) can be seen to be the same as given for the diffusion coefficient in Equation (33). Equation (36) is at steady state. Often times, during the electrophoretic measurements there exists a time period between the start of the experiment to the time when the fragment motion can be considered to be at steady state. During this transient regime the solute molecules can be expected to undergo translational acceleration. The Newtonian acceleration effects are not accounted for in Fick’s law of diffusion. The use of the Cattaneo and Vernotte equation in order to account for transient diffusion effects was discussed in Sharma [41]. The acceleration motion of the diffusing solute may be modeled by looking at the accumulation of the chemical potential. Thus, Equation (35) can be written including the accelerating term and:

_{m}can be taken as the velocity of mass v

_{mass}, (Sharma [34]) it is characterized by a relaxation time τ

_{mr}such that:

_{mr}is the mass relaxation time. It is a characteristic measure of the acceleration time of the solute from the instant of application of the driving force which causes the flow to the steady state regime. It can be seen to be:

## 3. Results and Discussion

_{0}for times less than 0. At time 0 the surface of the semi-infinite medium is raised to a solute concentration C

_{As}(C

_{As}> C

_{0}) and maintained constant at C

_{s}for all times t > 0. The initial time condition and the boundary conditions can be written as follows:

_{A}= C

_{0}

_{A}= C

_{As}

_{A}= C

_{0}

_{A}in one dimension is obtained by combining Equations (39) and (45) and can be written as follows:

_{1}can be seen to be zero.

_{2}is obtained from the constant wall temperature BC as given in Equation (43) and is seen to be given by (1/s). The solution for the dimensionless temperature in the Laplace domain may be written as follows:

_{0}(y) occurs at y = 2.4048. Thus, the penetration distance Z

_{pen}can be estimated for a given instant of time τ as follows:

**Figure 2.**Comparison of transient concentration from Fick model, damped wave transport model and ballistic transport model.

## 4. Conclusions

## Acknowledgments

## Conflicts of Interest

## Abbreviations

A_{N} | Avagadro Number (6.023 × 10 ^{23} molecules/mole) |

A | cross-sectional area across which transport occurs (m ^{2}) |

C_{A} | concentration of species A (mol/m ^{3}) |

C_{B} | concentration of species B (mol/m ^{3}) |

C_{p} | heat capacity of material at constant pressure (J/mole/K) |

C_{v} | heat capacity of material at constant volume (J/mole/K) |

D_{AB} | binary diffusion coefficient of species A in B (m ^{2}/s) |

erf(z) | error function of z. $erf(z)=\frac{2}{\sqrt{\pi}}{\displaystyle \underset{0}{\overset{z}{\int}}{e}^{-{z}^{2}dz}}$ |

f | molecular drag coefficient (kg/s/molecule) |

k | thermal conductivity of the material (W/m/K) |

k_{B} | Boltzmann Constant (J/molecule/K) |

H | enthalpy (J/mole) |

J″ | area averaged molar flux (mole/m ^{2}/s) |

J_{p}(x) | Bessel function of the p ^{th} order and first kind |

I_{p}(x) | modified Bessel function of the p ^{th} order and first kind |

t | time (s) |

T | Temperature ( °K) |

m | mass of the molecule (kg) |

N | molecular weight of oligonucleotide (kg/mole) |

q_{z} | heat flux (area averaged) (W/m ^{2}) |

n | electron density (# of electrons/m ^{3}) |

R | universal molar gas constant (J/mole/K) |

R_{0} | radius of solute molecule (m) |

S | entropy (J/mole/K) |

u | dimensionless concentration, $u=\left(\frac{{C}_{A}-{C}_{0}}{{C}_{As}-{C}_{0}}\right)$ |

u(s) | temperature in Laplace domain |

v_{e} | velocity of electron (m/s) |

v_{A} | velocity of solute molecule (m/s) |

v_{y} | velocity of fluid in y cartesian direction (m/s) |

v_{m} | velocity of mass (m/s) |

z | z Cartesian distance (m) $Z=\left(\frac{x}{\sqrt{{D}_{AB}{\mathsf{\tau}}_{mr}}}\right)$ |

Z | dimensionless distance |

v_{h} | velocity of heat (m/s) |

Z_{pen} | dimensionless penetration distance |

## Greek

σ | Entropy production term (W/m ^{3}/K) |

α | thermal diffusivity of material (m ^{2}/s) |

t | collision time of the electron and obstacle (seconds) |

τ | Dimensionless time in governing equation $\mathsf{\tau}=\left(\frac{t}{{\mathsf{\tau}}_{r}}\right)$ |

τ_{r} | relaxation time (heat) of material (s) |

τ_{mr} | relaxtion time (mass) of material (s) |

ρ | density of material (kg/m ^{3}) |

τ_{xy} | tangential shear stress (N/m ^{2}) |

τ_{mom} | relaxtion time (momentum) (s) |

µ_{A} | chemical potential (J/molecule) |

µ | viscosity (kg/m/s) |

λ_{2} | retardation time (s) |

x_{A} | mole fraction of species A |

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Sharma, K.R.
On Description of Acceleration of Spinless Electrons in Law of Heat Conduction *a capite ad calcem* in Temperature. *C* **2016**, *2*, 1.
https://doi.org/10.3390/c2010001

**AMA Style**

Sharma KR.
On Description of Acceleration of Spinless Electrons in Law of Heat Conduction *a capite ad calcem* in Temperature. *C*. 2016; 2(1):1.
https://doi.org/10.3390/c2010001

**Chicago/Turabian Style**

Sharma, Kal Renganathan.
2016. "On Description of Acceleration of Spinless Electrons in Law of Heat Conduction *a capite ad calcem* in Temperature" *C* 2, no. 1: 1.
https://doi.org/10.3390/c2010001