What is a non-polytropic process

Polytropic change of state

Is it apolytropic change of state this means that the product $ pV ^ n $ remains constant:

$ pV ^ n = const $.

The exponent becomes $ n $Polytropic exponentcalled.

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The simple changes of state dealt with in the previous sections represent special cases of the polytropic change of state.

Special cases of the polytropic change of state

exponent $ n $Thermal equation of stateChange of state
$ n = 0 $$ pV ^ 0 = const $Isobar
$ n = 1 $$ pV ^ 1 = const $Isothermal
$ n \ to \ infty $$ pV ^ {\ infty} = const $Isochore
$ n = \ kappa = \ frac {c_p} {c_v} $$ pV ^ {\ kappa} = const $Isentrop

p, v diagram

The polytropes can be shown in the p-v diagram. The graphic illustration of the isobars, isochores, isotherms and isentropes has already been made from the previous chapters. Three more polytropes are considered. Namely the polytropes between the isotherms and the isentropes with $ 1

Polytropic change of state in the p-v diagram

Of particular interest is the area between the isentropes and the isotherms, i.e. the polytropes with the polytropic exponent $ 1

Thermal equation of state

The thermal equation of state applies to all ideal gases and is generally given by

$ pV = m \; R_i \; T $ or $ pV = n \; R \; T $.


Since the product of $ pV ^ n $ is constant, the following applies:

$ p_1V_1 ^ n = p_2V_2 ^ n $.

The following context has been drawn from the previous sectionIsentropic change of state accepted and set $ \ kappa = n $:

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$ \ frac {T_1} {T_2} = (\ frac {V_2} {V_1}) ^ {n-1} = (\ frac {p_2} {p_1}) ^ {\ frac {1-n} {n}} $.

The polytropic exponent can be determined if the initial and final state are given with:

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$ n = \ frac {\ ln \ frac {p_2} {p_1}} {\ ln \ frac {p_2} {p_1} - \ ln \ frac {T_2} {T_1}} = \ frac {\ ln \ frac {p_2 } {p_1}} {\ ln \ frac {V_1} {V_2}} $.

Volume change work

The work of volume change for a closed system can be determined with $ pV ^ n = const $ using the following equations (the equations were taken from the previous section and set $ \ kappa = n $):

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$ W_V = \ frac {p_1V_1} {n-1} [(\ frac {V_1} {V_2}) ^ {n-1} - 1] $.

With the above connection $ \ frac {T_1} {T_2} = (\ frac {V_2} {V_1}) ^ {n-1} $ results:

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$ W_V = \ frac {p_1V_1} {n-1} [\ frac {T_2} {T_1} - 1] $.

With the relation $ (\ frac {V_2} {V_1}) ^ {n-1} = (\ frac {p_2} {p_1}) ^ {\ frac {n-1} {n}} $ we get:

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$ W_V = \ frac {p_1V_1} {n-1} [(\ frac {p_2} {p_1}) ^ {\ frac {n-1} {n}} - 1] $.

By inserting the thermal equation of state $ p_1V_1 = m \; R_i \; T_1 $ results in:

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$ W_V = \ frac {m \; R_i} {n-1} \; (T_2 - T_1) $.

Replacing $ R_i = c_ {vm} | _ {T_1} ^ {T_2} (\ kappa -1) $ results in:

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$ W_V = m \; c_ {vm} | _ {T_1} ^ {T_2} \ frac {\ kappa -1} {n-1} (T_2 - T_1) $.

All 5 equations are relevant for calculating the work of volume change depending on which state variables are given.

The volume change work can - as already shown in the previous chapters - be shown in the p, V diagram and shows the area under the polytropes to the V axis.

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Let $ n = 0 $ (isobaric change of state) be given. That means $ p = const $. Which of the above equations can be used to determine the work of volume change in the isobaric change of state?

All equations can be used (depending on which state variables are given) except the one which contains $ p_2 $, since the pressure remains constant and thus $ p_1 = p_2 = p $.

Reversible engineering work (pressure change work)

The reversible technical work results for the polytropic change of state with

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$ W_t ^ {rev} = n \ cdot W_V $.

The above equations for the volume change work $ W_V $ can be adopted. In order to determine the reversible technical work (work of pressure change) from this, this must be multiplied by $ n $.

The work of pressure change can - as already shown in the previous chapters - be shown in the p, V diagram and shows the area next to the polytropes to the p-axis.

warmth

The heat is calculated from the polytropic change of state

$ U_2 - U_1 = Q + W_V + W_ {diss} $.


Solved for $ Q $ results in:

$ Q = U_2 - U_1 - W_V - W_ {diss} $.

It becomes the final equation for the volume change work $ W_V $

$ W_V = m \; c_ {vm} | _ {T_1} ^ {T_2} \ frac {\ kappa -1} {n-1} (T_2 - T_1) $ inserted:

$ Q = U_2 - U_1 - m \; c_ {vm} | _ {T_1} ^ {T_2} \ frac {\ kappa -1} {n-1} (T_2 - T_1) - W_ {diss} $.


For the change in internal energy becomes the equation

$ U_2 - U_1 = m \; c_ {vm} | _ {T_1} ^ {T_2} (T_2 - T_1) $ inserted:

$ Q = m \; c_ {vm} | _ {T_1} ^ {T_2} (T_2 - T_1) - m \; c_ {vm} | _ {T_1} ^ {T_2} \ frac {\ kappa -1} {n-1} (T_2 - T_1) - W_ {diss} $.

After summarizing the terms it results:

$ Q = m \; c_ {vm} | _ {T_1} ^ {T_2} (T_2 - T_1) (1- \ frac {\ kappa -1} {n-1}) - W_ {diss} $.

Combining $ (1- \ frac {\ kappa -1} {n-1}) $ to $ \ frac {n - \ kappa} {n-1} $ results in:

$ Q = m \; c_ {vm} | _ {T_1} ^ {T_2} (T_2 - T_1) \ frac {n - \ kappa} {n-1} - W_ {diss} $.

For oneirreversible process the result for the heat is:

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$ Q = m \; c_ {vm} | _ {T_1} ^ {T_2} (T_2 - T_1) \ frac {n - \ kappa} {n-1} - W_ {diss} $.


For onereversible process with $ W_ {diss} = 0 $ we get:

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$ Q = m \; c_ {vm} | _ {T_1} ^ {T_2} (T_2 - T_1) \ frac {n - \ kappa} {n-1} $.

Replacing $ c_ {vm} | _ {T_1} ^ {T_2} = \ frac {R_i} {\ kappa -1} $ results in:

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$ Q = m \; \ frac {R_i} {\ kappa - 1} (T_2 - T_1) \ frac {n - \ kappa} {n-1} $.

entropy

The change in entropy can be determined from the following equations:

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$ S_2 - S_1 = m \; c_ {vm} | _ {T_1} ^ {T_2} \ frac {n - \ kappa} {n - 1} \ ln \ frac {T_2} {T_1} $

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$ S_2 - S_1 = m \; c_ {pm} | _ {T_1} ^ {T_2} \ ln \ frac {T_2} {T_1} - m \; R_i \ ln \ frac {p_2} {p_1} $

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$ S_2 - S_1 = m \; c_ {vm} | _ {T_1} ^ {T_2} \ ln \ frac {T_2} {T_1} + m \; R_i \ ln \ frac {V_2} {V_1} $.

The entropy can be represented in a T, S diagram. Entropy can also be written as

$ \ int T \; dS = Q + W_ {diss} $.

In general, the area under the curve (polytropes) to the $ S $ axis is the sum of heat $ Q $ and dissipation work $ W_ {diss} $.

In the case of the polytropic change of state (where the polytropic is considered with the exponent $ 1 Isochores the change in internal energy $ U_1 - U_2 $ can also be displayed. This corresponds to the area under the isochores (see also section isochore change of state). The total area (area under the polytropes + area under the isochores) corresponds to the work of volume change $ W_V $.

Polytropic change of state with isochore (volume change work)


If you take the isochore instead of the Isobars the change in enthalpy $ H_1 - H_2 $ can also be displayed. This corresponds to the area under the isobars (see also section isobaric change of state). The total area (area under the isobars + area under the polytropes) corresponds to the technical reversible work (pressure change work) $ W_t ^ {rev} $.

Polytropic change of state with isobars (pressure change work)