A Thermodynamic Approach of Mechanical Vapor Compression Refrigeration and Heating COP Increase

Abstract

The mechanical vapour compression (mvc) technology is laying down the basis of many important industrial, agricultural and household applications.

Appendix 1

• (i) First, we shall derive an equivalent expression of $\eta_{C} \left( \xi \right)$, Eq. (8.9), following the way covered in Appendix 1. In our case, Eqs. (A1.1) and (A1.2) become:

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

(A8.1)

and

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

(A8.2)

respectively. Bearing in mind Eq. (A1.7), Eq. (A1.8) becomes in this case:

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

(A8.3)

and Eq. (8.9) is rewritten in an equivalent form as follows:

$$ \eta_{C} \left( \xi \right) = 1 - \frac{{ln\frac{{T_{2a} }}{{T_{1} }}}}{{\frac{{T_{2a} }}{{T_{1} }} - 1}} \equiv \eta_{C,TFC} = 1 - \frac{{2T_{1} }}{{T_{2a} - T_{1} }}\frac{{T_{2a} - T_{1} }}{{T_{2a} + T_{1} }} $$

(A8.4)

or

$$ \eta_{C,TFC} = 1 - \frac{{T_{1} }}{{\frac{{T_{2a} + T_{1} }}{2}}} $$

(A8.5)

Equation (A8.5) can be arranged also as:

$$ \eta_{C,TFC} = \frac{{T_{2a} - T_{1} }}{{T_{2a} + T_{1} }} $$

(A8.6)

which is nothing else but the ideal thermal efficiency of the motor (direct) cycle named Trilateral Flash Cycle (TFC) (see Sect. 8.2.1).

• (ii) Second, let us consider the ideal TFC, which recovers from the heat source the discharge gas superheat $q_{a}$, rejects to the sink source the amount of heat $- \left| {q_{i} } \right| = - \left| {l_{i} } \right|$ (Popa and Vintila 1977, p. 373, Table 15.3) and produces the work $l_{a,r} = \left| {l_{a} } \right| - \left| {l_{i} } \right|$, Eq. (8.26). Combining Eq. (8.10), written as:

$$ l_{a,r} = \eta_{C,TFC} q_{a} $$

(A8.9)

with Eqs. (A8.5) and (8.26), it results:

$$ \left( {1 - \frac{{T_{1} }}{{\frac{{T_{2a} + T_{1} }}{2}}}} \right)q_{a} = \;\left| {l_{a} } \right| - \left| {l_{i} } \right| $$

(A8.10)

Taking into account that $\left| {l_{a} } \right| = \left| {q_{a} } \right|$, Eq. (8.25), and that $- \left| {q_{i} } \right| = - \left| {l_{i} } \right|$ (see above), Eq. (A8.10) becomes:

$$ \frac{{q_{a} }}{{\frac{{T_{1} + T_{2a} }}{2}}} - \frac{{\left| {q_{i} } \right|}}{{T_{1} }} = 0 $$

(A8.11)

which proves that TFC entropy variation is null when is completely covered, $\Delta s = s_{h} - s_{s} = 0$, therefore the considered TFC is reversible [see Eqs. (1.86) and (1.88) as well]. From Eq. (A8.11) it results also that the heat source delivers in an equivalent way the heat $q_{a}$ to the arithmetical mean temperature $\frac{{T_{1} + T_{2a} }}{2}$, as we expected.

• (iii) Third, in order to proof simpler the Theorem 4 of this chapter, Eq. (8.98) is rewritten in the new form with Eq. (A8.12) as:

$$ \frac{{COP_{c} }}{{COP_{c,TWRC} }} = \frac{{\left| {l_{a,TWRC} } \right|}}{{\left| {l_{a} } \right|}} = 1 - \varepsilon \eta_{C} \left( \xi \right) = 1 - \varepsilon \left( {1 - \frac{{T_{1} }}{{\frac{{T_{2a} + T_{1} }}{2}}}} \right) = f\left( {T_{1} ,T_{2a} } \right) $$

(A8.12)

and it suffices to show that $\frac{{\partial f\left( {T_{1} ,T_{2a} } \right)}}{{\partial T_{2a} }} = - \varepsilon \frac{{\partial \eta_{C} \left( {T_{1} ,T_{2a} } \right)}}{{\partial T_{2a} }} < 0$. Indeed, performing the last derivative, it results that:

$$ - \varepsilon \frac{{\partial \eta_{C} \left( {T_{1} ,T_{2a} } \right)}}{{\partial T_{2a} }} = \varepsilon \frac{\partial }{{\partial T_{2a} }}\left( {\frac{{2T_{1} }}{{T_{2a} + T_{1} }}} \right) = - \varepsilon \frac{{2T_{1} }}{{\left( {T_{2a} + T_{1} } \right)^{2} }} < 0\;{\text{q}}.{\text{e}}.{\text{d}}. $$

(A8.13)

• (iv) The classic compression saves input work, practicing staged compression with interstage cooling of the discharge gas (Popa and Vintila 1977). In Fig. 8.30 we plotted a compression process benefiting of two stages, with intermediary cooling, instead of a single one. The intermediary pressure $p_{x} ,\;p_{1} < p_{x} < p_{2}$ is established according to the known relation:

  Fig. 8.30

  

  Single-stage versus two-stage compression

$$ p_{x} = \sqrt {p_{1} p_{2} } , $$

(A8.14)

which ensures the minimum work consumption in two-stage compression (Popa and Vintila 1977). The single-stage compression is the adiabatic transformation, represented in Fig. 8.30 by the adiabat $T_{1} - T_{2a}$. In case of the two-stage compression, the first and second stage compressions are the adiabatic transformations $T_{1} - T_{ax}$ and $T_{ix} - T_{2ax}$, respectively. The work savings is that corresponding to the $T_{ax} - T_{ix} - T_{2ax} - T_{2a} - T_{ax}$ area comprised between the two adiabats $T_{ax} - T_{{T_{2a} }}$ and $T_{ix} - T_{2ax}$ and the two isobar $p_{x}$ and $p_{2}$ in the $p - V$ diagram. The relationship (A8.14) is valid for the compression practiced so far, without TWRC. The next lemma will prove that Eq. (A8.14) remains valid even in case of TWRC.

Lemma 2

A two-stage adiabatic mechanical vapor compression process, provided with TWRC and operating between the initial and final pressures $p_{1}$ and $p_{2}$ , respectively, has the same value of the intermediary pressure ensuring the minimum work consumption as that of the compression process without TWRC, that is $p_{x} = \sqrt {p_{1} p_{2} } ,\;p_{1} < p_{x} < p_{2}$.

Proof

From considering Eqs. (8.27)–(8.30), the work of the single-stage compression provided with TWRC results in:

$$ \left| {l_{a,TWRC} } \right| = \left| {l_{a} } \right|\left[ {1 - \varepsilon \left( {1 - \frac{{\ln \frac{{T_{2a} }}{{T_{1} }}}}{{\frac{{T_{2a} }}{{T_{1} }} - 1}}} \right)} \right] $$

(A8.15)

where $\left| {l_{a} } \right|$ is given by Eq. (8.14):

$$ \left| {l_{a} } \right| = \frac{k}{k - 1}p_{1} V_{1} \left[ {\left( {\frac{{p_{2a} }}{{p_{1} }}} \right)^{{\frac{k - 1}{k}}} - 1} \right] $$

(A8.16)

Combining the two Eqs. (A8.15) and (A8.16), it is obtained:

$$ \left| {l_{a,TWRC} } \right| = \frac{k}{k - 1}p_{1} V_{1} \left[ {\left( {\frac{{p_{2a} }}{{p_{1} }}} \right)^{{\frac{k - 1}{k}}} - 1} \right]\left[ {1 - \varepsilon \left( {1 - \frac{{\ln \frac{{T_{2a} }}{{T_{1} }}}}{{\frac{{T_{2a} }}{{T_{1} }} - 1}}} \right)} \right] $$

(A8.17)

The single-stage TWRC provided compression work of Eq. (A8.17) is written in case of the first and second TWRC provided stages works as follows:

$$ \left| {l_{a,TWRC,I} } \right| = \frac{k}{k - 1}p_{1} V_{1} \left[ {\left( {\frac{{p_{x} }}{{p_{1} }}} \right)^{{\frac{k - 1}{k}}} - 1} \right]\left[ {1 - \varepsilon \left( {1 - \frac{{ln\frac{{T_{ax} }}{{T_{1} }}}}{{\frac{{T_{ax} }}{{T_{1} }} - 1}}} \right)} \right] $$

(A8.18)

and

$$ \left| {l_{a,TWRC,II} } \right| = \frac{k}{k - 1}p_{1} V_{1} \left[ {\left( {\frac{{p_{2a} }}{{p_{x} }}} \right)^{{\frac{k - 1}{k}}} - 1} \right]\left[ {1 - \varepsilon \left( {1 - \frac{{ln\frac{{T_{ax} }}{{T_{ix} }}}}{{\frac{{T_{ax} }}{{T_{ix} }} - 1}}} \right)} \right] $$

(A8.19)

respectively. In Eq. (A8.19), the product $p_{1} V_{1}$ was used instead of $p_{x} V_{ix}$ because the points $\left( {T_{1} ,p_{1} } \right)$ and $\left( {T_{ix} ,p_{x} } \right)$ belong to the same isothermal curve $T_{1} = T_{ix} = T_{2i} = const.$, and along of which we have $p_{1} V_{1} = p_{x} V_{ix}$, Fig. 8.30. The ratios of temperatures in Eqs. (A8.18) and (A8.19) are written with the help of Eq. (8.18), valid along of an adiabat:

$$ \frac{{T_{ax} }}{{T_{1} }} = \left( {\frac{{p_{x} }}{{p_{1} }}} \right)^{{\frac{k - 1}{k}}} $$

(A8.20)

and

$$ \frac{{T_{2ax} }}{{T_{ix} }} = \left( {\frac{{p_{2a} }}{{p_{x} }}} \right)^{{\frac{k - 1}{k}}} $$

(A8.21)

respectively. Introducing Eqs. (A8.20) and (A8.21) in Eqs. (A8.18) and (A8.19), it is obtained;

$$ \left| {l_{a,TWRC,I} } \right| = \frac{{p_{1} V_{1} }}{y}\left[ {\left( {\frac{{p_{x} }}{{p_{1} }}} \right)^{y} - 1} \right]\left[ {1 - \varepsilon \left( {1 - \frac{{yln\left( {\frac{{p_{x} }}{{p_{1} }}} \right)}}{{\left( {\frac{{p_{x} }}{{p_{1} }}} \right)^{y} - 1}}} \right)} \right] $$

(A8.22)

and

$$ \left| {l_{a,TWRC,II} } \right| = \frac{{p_{1} V_{1} }}{y}\left[ {\left( {\frac{{p_{2a} }}{{p_{x} }}} \right)^{y} - 1} \right]\left[ {1 - \varepsilon \left( {1 - \frac{{yln\left( {\frac{{p_{2a} }}{{p_{x} }}} \right)}}{{\left( {\frac{{p_{2a} }}{{p_{x} }}} \right)^{y} - 1}}} \right)} \right] $$

(A8.23)

where $\frac{k - 1}{k} = y$. Further, Eqs. (A8.22) and (A8.23) are added term by term, resulting:

$$ \left| {l_{a,TWRC,I + II} } \right| = \left| {l_{a,TWRC,I} } \right| + \left| {l_{a,TWRC,II} } \right| = \frac{{p_{1} V_{1} }}{y}\left\{ {\left( {1 - \varepsilon } \right)\left[ {\left( {\frac{{p_{x} }}{{p_{1} }}} \right)^{y} + \left( {\frac{{p_{2a} }}{{p_{x} }}} \right)^{y} } \right] + \varepsilon yln\frac{{p_{2a} }}{{p_{1} }}} \right\} $$

(A8.23)

The searched value of $p_{x}$ is found solving equation below:

$$ \frac{d}{{dp_{x} }}\left| {l_{a,TWRC,I + II} } \right| = 0 $$

(A8.24)

or,

$$ \frac{d}{{dp_{x} }}\left| {l_{a,TWRC,I + II} } \right| = \frac{d}{{dp_{x} }}\left[ {\left( {\frac{{p_{x} }}{{p_{1} }}} \right)^{y} + \left( {\frac{{p_{2a} }}{{p_{x} }}} \right)^{y} } \right] = 0 $$

(A8.25)

Performing the derivation in Eq. (A8.25), it results:

$$ y\left( {\frac{{p_{x} }}{{p_{1} }}} \right)^{y - 1} \frac{1}{{p_{1} }} - y\left( {\frac{{p_{2a} }}{{p_{x} }}} \right)^{y - 1} \frac{{p_{2a} }}{{p_{x}^{2} }} = 0 $$

(A8.26)

wherefrom the lemma is proved, $\left( {p_{2a} = p_{2} } \right)$:

$$ p_{x} = \sqrt {p_{1} p_{2} } ,\;{\text{q}}.{\text{e}}.{\text{d}}. $$

(A8.27)

Bearing in mind that both mechanical vapor compression processes, without and with TTRC method application, have the same work input [see Eq. (8.3)], and taking into account the lemma 2 result, it is concluded that:

Corollary 4

Both TWRC and TTRC methods preserve the same relationship of the intermediary pressure calculus, in two-stage mechanical vapor compression, Eq. (A8.27), in order to minimize the work input.