Introduction

Abstract

This book is presenting in its first part two important heat pumping technologies (production of cooling and heating), introduced by the author lately. These are the coabsorbent technology (Chaps. 2–7) and the mecanical vapor compression refrigeration and heating COP increase, using the discharge gas superheat (Chap. 8). In the next chapters, the book is including author’s own researches concerning a non-equilibrium phenomenological two-point theory of mass and heat transfer in physical and chemical interactions (Chap. 9), a new wording of the Laplace equation and the variational numerical and analytical approach of the liquid capillary rise effect (Chap. 10). In the book last part, Marangoni convection basic mechanism explanation, pseudo-Marangoni cells model and the absorption-desorption mass and heat exchangers model application are presented (Chap. 11). In this first introductory chapter, selected topic of thermodynamics is attached, in order to complete the subjects elaborated and facilitate the book reading.

Appendix 1

• (i) In Eq. (1.120), $\ln \frac{{T_{2} }}{{T_{1} }}$ is written as:

$$ \ln \frac{{T_{2} }}{{T_{1} }} = \ln \frac{1 + x}{1 - x} = f\left( x \right) $$

(A1.1)

where

$$ x = \frac{{T_{2} - T_{1} }}{{T_{2} + T_{1} }},\;x \in \left[ {0,1} \right),\;x < < 1. $$

(A1.2)

The function f(x) of Eq. (A1.1), arranged in a different form:

$$ f\left( x \right) = \ln \left( {1 + x} \right) - \ln \left( {1 - x} \right) $$

(A1.3)

is derivated, in order to obtain:

$$ f^{{\prime }} \left( x \right) = \frac{1}{1 + x} + \frac{1}{1 - x} $$

(A1.4)

The ratios in Eq. (A1.4) are expressed using the infinite series of power:

$$ \frac{1}{1 \pm x} = 1 \pm \sum\limits_{i = 1}^{\infty } {\left( { - x} \right)}^{i} $$

(A1.5)

We introduce Eq. (A1.5) in Eq. (A1.4) and totalize. The same rank terms of Eq. (A1.5) are vanishing through summation, except the first ones, hence Eq. (A1.4) results as:

$$ f^{{\prime }} \left( x \right) = 2 $$

(A1.6)

Integrating Eq. (A1.6) between T ₁ and T ₂, the following result holds true (see Eq. A1.2):

$$ f\left( x \right) = 2x = \ln \frac{1 + x}{1 - x} $$

(A1.7)

Introducing Eqs. (A1.7) and (A1.2) in Eq. (A1.1), it results:

$$ \ln \frac{{T_{2} }}{{T_{1} }} = 2\frac{{T_{2} - T_{1} }}{{T_{2} + T_{1} }} $$

(A1.8)

and further, with it, Eq. (1.122) holds true:

$$ \theta \left( \xi \right) = 1 - \frac{{T_{0} }}{{\frac{{T_{1} + T_{2} }}{2}}} $$

(A1.9)

• (ii) The 2nd principle equation is written with intensive parameters and at equilibrium, as follows:

$$ Tds = du + pdv $$

(A1.10)

In Eq. (A1.10) the specific entropy, internal energy and volume are written in function of the extensive values as $s = \frac{S}{G}$, $u = \frac{U}{G}$ and $v = \frac{V}{G}$, respectively. Differentiating these expressions, it is obtained:

$$ \begin{aligned} ds &= \frac{GdS - SdG}{{G^{2} }} \\ du & = \frac{GdU - UdG}{{G^{2} }} \\ dv & = \frac{GdV - VdG}{{G^{2} }} \\ \end{aligned} $$

(A1.11)

Introducing Eq. (A1.11) in Eq. (A1.10), after arranging some terms, it results:

$$ TGdS = GdU - \left( {U - TS} \right)dG + p\left( {GdV - VdG} \right) $$

(A1.12)

In Eq. (A1.12):

$$ \left( {U - TS} \right) = F = Gf $$

(A1.13)

Introducing Eq. (A1.13) in Eq. (A1.12) and dividing the entire Eq. (A1.12) by G, it is obtained:

$$ TdS = dU - fdG + pdV - p\frac{V}{G}dG $$

(A1.14)

In Eq. (A1.14) we take into account that $\frac{V}{G} = v$. Further Eq. (A1.14) is partially derivated with respect to G for U = const. and V = const., wherefrom the following result holds true:

$$ T\left( {\frac{\partial S}{\partial G}} \right)_{U,V} = \left( {\frac{\partial U}{\partial G}} \right)_{U,V} - f + p\left( {\frac{\partial V}{\partial G}} \right)_{U,V} - pv $$

(A1.15)

In Eq. (A1.15) right side the members in the brackets are vanishing and bearing in mind that f + pv = φ, the needed entropy partial derivative is obtained:

$$ \left( {\frac{\partial S}{\partial G}} \right)_{U,V} = - \frac{\varphi }{T} $$

(A1.17)