Polytropic process

src: i.ytimg.com

Proses polytropic adalah proses termodinamika yang mematuhi hubungan:

${\ displaystyle pV ^ {\, n} = C}  ÂÂ$

where p is a pressure, n is a index polytropic i> is a constant. The polytropic process equation can describe some expansion and compression processes that include heat transfer.

Jika hukum gas ideal berlaku, proses adalah polytropic jika dan hanya jika rasio ( K ) dari transfer energi sebagai transfer panas ke energi sebagaimana bekerja pada setiap langkah yang sangat kecil dari proses dijaga konstan:

${\ displaystyle K = {\ frac {\ delta Q} {\ delta W}} = {\ text {constant}}}  ÂÂ$

Video Polytropic process

Certain cases

Beberapa nilai spesifik n sesuai dengan kasus-kasus tertentu:

• ${\ displaystyle n = 0}  ÂÂ$ adalah proses isobaric,
• ${\ displaystyle n = \ infty}  ÂÂ$ adalah proses isochoric.

Selain itu, ketika hukum gas ideal berlaku:

• ${\ displaystyle n = 1}  ÂÂ$ adalah proses isotermik,
• ${\ displaystyle n = \ gamma}  ÂÂ$ adalah proses adiabatik.

Maps Polytropic process

Kesetaraan antara koefisien polytropic dan rasio transfer energi

Consider the ideal gas in a closed system that undergoes a slow process with kinetic energy changes and potentially negligible potentials. For a very small time step, the first law of thermodynamics states that energy is added to the system as heat ? Q , minus the energy leaving the system working ? W , similar to the change of internal energy du of the system:

${\ displaystyle \ delta q- \ delta w = du}$ (Equation 1)

Tentukan rasio transfer energi,

${\ displaystyle K = {\ frac {\ delta q} {\ delta w}}}  ÂÂ$ atau ${\ displaystyle \ delta q = K \ delta w}  ÂÂ$ .

Transfer pekerjaan ke lingkungan dapat dinyatakan sebagai ${\ displaystyle {\ delta w} = p {dv}}  ÂÂ$ dan perubahan energi internal sebagai ${\ displaystyle du = nc _ {\, v} \, dT}  ÂÂ$

Dengan mengganti ekspresi di atas untuk ? W dan ? Q ke dalam hukum pertama:

${\ displaystyle (K-1) p \, dv = nc _ {\, v} \, dT}  ÂÂ$ (Persamaan 1)

Menulis hukum gas ideal dalam bentuk diferensial

$            {\displaystyle         p                d        v                v                d        p        =        n        R                d        T        ->                  \frac{              v                            d              p            }{              p                            d              v            }                        1        =                  \frac{              n              R                            d              T            }{              p                            d              v            }                =                  \frac{              n              R                            (              K              -              1              )              p                            d              T            }{              p                            n              c_{                  v                }              d              T            }                =                  \frac{              R                            (              K              -              1              )            }{c_{                v              }}              }        \{\backslash \; displaystyle\; p\; \backslash ,\; dv\; v\; \backslash ,\; dp\; =\; nR\; \backslash ,\; dT\; \backslash \; rightarrow\; \{v\; \backslash ,\; dp\; \backslash \; over\; p\; \backslash ,\; dv\}\; 1\; =\; \{nR\; \backslash ,\; dT\; \backslash \; over\; p\; \backslash ,\; dv\}\; =\; \{nR\; \backslash ,\; (K-1)\; p\; \backslash ,\; dT\; \backslash \; over\; p\; \backslash ,\; nc\_\; \{v\}\; dT\}\; =\; \{R\; \backslash ,\; (K-1)\; \backslash \; over\; c\_\; \{v\}\}\; \}\;  ÂÂ$

Dengan relasi Mayer, ini menjadi:

${\ displaystyle {v \, dp \ over p \, dv} = {(c_ {p} -c_ {v}) \, (K-1) \ over c_ {v}} - 1 = (K-1) \ gamma -K \ longrightarrow {dp \ over p} {((1- \ gamma) K \ gamma)} {dv \ over v} = 0}  ÂÂ$

dimana ? adalah rasio kapasitas panas. Dengan asumsi K (dan ? ) tetap konstan selama transformasi, seperti ${\ displaystyle {df \ over f} = d (log \, f)}  ÂÂ$ relasi ini dapat diintegrasikan sebagai

${\ displaystyle d \ left (log \, p {((1- \ gamma) K \ gamma)} log \, v \ right) = 0 \ longrightarrow pv ^ { (1- \ gamma) K \ gamma} = C}  ÂÂ$

where C is a constant.

Dengan demikian, prosesnya adalah polytropic, dengan koefisien ${\ displaystyle n = {(1- \ gamma) K \ gamma}}  ÂÂ$ .

This derivation can be expanded to include polytropic processes in open systems, including instances in which the kinetic energy (ie number of Mach) is significant. This can also be extended to include irreversible polytropic processes.

src: i.ytimg.com

Relationship to the ideal process

For certain values ​​of the polytropic index, the process will be identical to other general processes. Some examples of effects of various index values ​​are given in the table.

When the index n is between two previous values ​​(0, 1, ? , or?), That means the polytropic curve will cut (limited by) the curve of two indexes that leaping.

Untuk gas ideal, 1 & lt; Â ? Â & lt; Â 2, karena oleh relasi Mayer

${\ displaystyle \ gamma = {\ frac {c_ {p}} {c_ {v}}} = {\ frac {c_ {v} R} {c_ {v }}} = 1 {\ frac {R} {c_ {v}}} = {\ frac {c_ {p}} {c_ {p} -R}}}  ÂÂ$ .

src: i.ytimg.com

Lainnya

The solution to the Lane-Emden equation using polytropic liquids is known as polytrope.

src: i.ytimg.com

See also

• Polytrope
• Adiabatic Process
• Isentropic Process
• Isobaric Process
• Isochoric Process
• Isothermal process
• Vapor compression cooling
• Gas compressor
• Internal combustion engine
• quasistatic balance
• Thermodynamics

src: slideplayer.com

References

Source of the article : Wikipedia