THE ESTABLISHMENT OF THE NOVEL M-H EOS
The
basis of arithmetic solution of the Hou’s modified M-H EOS
The M-H EOS was originally
proposed
as an empirical equation by Martin and Hou in 1955 on the basis of a
large number of P-V-T data analysis about many kinds of substances. The
initial
formula can be expressed as follows9:
\(\mathrm{P=}\)\(\sum_{i=1}^{5}\frac{f_{i}\left(T\right)}{\left(V-b\right)^{i}}\)(1)
with\(\mathrm{f}_{\mathrm{i}}\left(\mathrm{T}\right)\mathrm{=}\mathrm{A}_{\mathrm{i}}\mathrm{+}\mathrm{B}_{\mathrm{i}}\mathrm{T+}\mathrm{C}_{\mathrm{i}}\mathrm{e}^{\mathrm{-k}\frac{\mathrm{T}}{\mathrm{T}_{\mathrm{c}}}}\)(2)
where k = 5.475, Ai, Bi,
Ci , and b are the characteristic constants for a
given
substance.
Based on the universal physical properties of substances,
there
are 10 initial value conditions for the EOS, however, the
unknown
characteristic constants are up to 169.
Fortunately,
not every characteristic constant is of equal importance or necessary,
which means those less important characteristic constants can be
omitted. Ref. (9) gave the detailed reasons for the
simplification
of the related unknown characteristic constants and the theory
foundations of the initial value conditions. In this work, we just
exhibit
the result of the simplified fi (T ) and
the initial value conditions of the M-H EOS.
The simplified fi (T ) can be expressed as
follows9:
\(\mathrm{f}_{\mathrm{1}}\left(\mathrm{T}\right)\mathrm{=}\mathrm{B}_{\mathrm{1}}\mathrm{T}\)(3)
\(\mathrm{f}_{\mathrm{2}}\left(\mathrm{T}\right)\mathrm{=}\mathrm{A}_{\mathrm{2}}\mathrm{+}\mathrm{B}_{\mathrm{2}}\mathrm{T+}\mathrm{C}_{\mathrm{2}}\mathrm{e}^{\mathrm{-5.475}\frac{\mathrm{T}}{\mathrm{T}_{\mathrm{c}}}}\)(4)
\(\mathrm{f}_{\mathrm{3}}\left(\mathrm{T}\right)\mathrm{=}\mathrm{A}_{\mathrm{3}}\mathrm{+}\mathrm{B}_{\mathrm{3}}\mathrm{T}\mathrm{+}\mathrm{C}_{\mathrm{3}}\mathrm{e}^{\mathrm{-5.475}\frac{\mathrm{T}}{\mathrm{T}_{\mathrm{c}}}}\)(5)
\(\mathrm{f}_{\mathrm{4}}\left(\mathrm{T}\right)\mathrm{=}\mathrm{A}_{\mathrm{4}}\)(6)
\(\mathrm{f}_{\mathrm{5}}\left(\mathrm{T}\right)\mathrm{=}\mathrm{B}_{\mathrm{5}}\mathrm{T}\)(7)
The
number of the unknown characteristic constants is 10:A 1, A 2,B 2, C 2,A 3, B 3,C 3, A 4,B 5, b .
The
initial value conditions of the M-H EOS are expressed as
follows9:
\(\mathrm{P}_{\mathrm{c}}\mathrm{=}\mathrm{P}\left(\mathrm{T}_{\mathrm{c}}\mathrm{,}\mathrm{V}_{\mathrm{c}}\right)\)(8)
\(\mathrm{PV=RT}\mathrm{\text{\ as\ }}\mathrm{P}\mathrm{\rightarrow 0}\)(9)
\(\left(\mathrm{\text{dP}}\mathrm{/}\mathrm{\text{dV}}\right)_{\mathrm{T}_{\mathrm{c}}}\mathrm{=0}\)(10)
\(\left(\frac{\mathrm{d}^{\mathrm{2}}\mathrm{P}}{\mathrm{d}\mathrm{V}^{\mathrm{2}}}\right)_{\mathrm{T}_{\mathrm{c}}}\mathrm{=0}\)(11)
\(\left(\frac{\mathrm{d}^{\mathrm{3}}\mathrm{P}}{\mathrm{d}\mathrm{V}^{\mathrm{3}}}\right)_{\mathrm{T}_{\mathrm{c}}}\mathrm{=0}\)(12)
\(\left(\frac{\mathrm{d}^{\mathrm{4}}\mathrm{P}}{\mathrm{d}\mathrm{V}^{\mathrm{4}}}\right)_{\mathrm{T}_{\mathrm{c}}}\mathrm{=0}\)(13)
\(\left[\left(\frac{\mathrm{\text{dZ}}}{\mathrm{d}\mathrm{P}_{\mathrm{r}}}\right)_{\mathrm{T}_{\mathrm{r}}}\right]_{\mathrm{P}_{\mathrm{r}}\mathrm{=0}}\mathrm{=\ -}\left(\mathrm{1-}\mathrm{Z}_{\mathrm{c}}\right)\mathrm{\text{\ \ \ \ \ \ }}\mathrm{\text{at}}\mathrm{\text{\ \ \ \ }}\mathrm{T}^{\mathrm{{}^{\prime}}}\mathrm{\cong}\mathrm{\ 0.8}\mathrm{T}_{\mathrm{c}}\)(14)
\(\left[\left(\frac{\mathrm{\text{dZ}}}{\mathrm{d}\mathrm{P}_{\mathrm{r}}}\right)_{\mathrm{T}_{\mathrm{r}}}\right]_{\mathrm{P}_{\mathrm{r}}\mathrm{=0}}\mathrm{=0}\mathrm{\text{\ \ \ \ \ }}\mathrm{at\ \ \ \ Boyle-Point\ Temperature}\mathrm{\ }\mathrm{T}_{\mathrm{B}}\)(15)
\(\left(\mathrm{\text{dP}}\mathrm{/}\mathrm{\text{dT}}\right)_{\mathrm{V}}\mathrm{=}\mathrm{m}\mathrm{=\ -}\mathrm{M}\frac{\mathrm{P}_{\mathrm{c}}}{\mathrm{T}_{\mathrm{c}}}\mathrm{\text{\ \ \ \ \ }}\mathrm{\text{at}}\mathrm{\text{\ \ \ V}}\mathrm{=}\mathrm{V}_{\mathrm{c}}\)(16)
\(\left(\frac{\mathrm{d}^{\mathrm{2}}\mathrm{P}}{\mathrm{d}\mathrm{T}^{\mathrm{2}}}\right)_{\mathrm{V}}\mathrm{=0}\mathrm{\text{\ \ \ \ \ \ }}\mathrm{\text{at\ }}\mathrm{\text{\ V}}\mathrm{=}\mathrm{V}_{\mathrm{c}}\)(17)
Here,
the number of the initial value conditions of the M-H EOS is also
10.
Although the arithmetic solutions of the unknown characteristic
constants can be obtained by using the initial value conditions above,
there still is a major defect that the precision calculated by the M-H
EOS in liquid-phase state is too low in practical
applications9,12. In fact, the original EOS was mostly
suitable for the calculation in gas-phase state.
After the original M-H EOS was put forward,
Martin10,11 and Hou12 independently
revised the original M-H EOS in order to extend application range to
liquid-phase state. Among them, the new formula modified by Hou is more
prominent12. According to the constraint condition of
gas-liquid equilibrium, Hou introduced a new initial value condition and
a new characteristic constant B 4 into the
original M-H EOS. Hou’s modified M-H EOS
expands
the range of applications to liquid-phase state12.
The
formula is similar as the original M-H EOS except that
f4 (T) has a change which adds a new
characteristic
constant B4. The new f4 (T) is expressed
as follows12: