Adiabatic compression and expansion

Principle

To understand the concept of adiabatic phenomena, first Boyle’s law is applied to the simple set-up of Fig. 1. It is assumed that both compartments have constant volume and temperature.

Fig. 1

The amount of gas in both compartments together is proportional p_a·V_a + p_b·V_b which holds with the closed valve (at time zero) as well as after equilibration of the pressures (at time t) when the valve has been opened. So, it holds that :

p_{a, 0}·V_a + p_{b, 0}·V_{b =} p_t·(V_a + V_b). (0)

With given p_{a, 0}, p_{b, 0}, V_a and V_{b =,} p_t· can be calculated. However, this is only true when T in the whole system remains constant, i.e. isothermic compression and expansion holds. In daily life, a bicycle pump becomes warm when a tire is inflated. This is due to heat transfer from the heated, compressed air in the pump. When the heat produced by compression is not given off to the environment but “absorbed” by the gas it self, or when the energy needed for expansion is not provided by the environment but provided by the expanding gas itself, than Boyle’s law does not hold. So, a compressing gas heats and an expanding gas cools down. These processes are called adiabatic compression and expansion. The deviation from Boyle’s law can be very substantial (Fig. 2), as will be explained conceptually with the example of Fig. 1.

When the valve has a very narrow opening, the process of equilibration takes so much time (tens of minutes) that the whole process is isothermic. There is enough time for heat release and uptake between the system and the environment. When the equilibration is very fast (about a second; pipe and valve diameter large) the process is practically adiabatic and the changes of temperature can rather well be calculated.

For ideal gases under pure adiabatic conditions the p-V relation is:

p·V^γ = constant, (1)

with γ a numerical value greater than 1 and depending on the type of gas. γ is the so-called c_p/c_V ratio, the ratio of heat capacity of the gas (c_p) with constant p and the specific heat capacity of the gas (c_V) with constant V. It depends on the number of atoms in the gas molecule. For mono-atomic gases, e.g. the noble gases, γ is 5/3. For diatomic gases, for instance air, γ is 7/5 (= 1.4). More precisely, c_p/c_V =is :

c_p/c_V = c_p/(c_p-R), (2)

where R the universal gas constant (see Gas laws).

Fig.2 Isothermic and adiabatic p-V relation. The dashed curve gives the p-V relation according to Boyle’s law and the dashed straight horizontal line the temperature belonging to it. The solid curve represents the adiabatic p-V relation and the thick solid curve gives the adiabatic temperature. n = 0.1 kmol.

As follows from Eq (1), reducing a volume by a factor k¹ gives a pressure increase much higher than k (being k^1.4). Conceptually, one can say that a factor k is due to the volume decrease according to Boyle and that the remaining factor, k^0.4, is caused by the temperature increase.

An isothermal p-V curve according to Boyle and an adiabatic p-V curve are depicted in Fig. 2. Numerically, the resulting temperature effect can be very substantial. A further description of adiabatic processes can be found in More info. Since textbooks seldom give examples how to calculate the adiabatic effect, an example is given in More Info.

Application

Adiabatic effects are seldom applied and in contrary, are generally unwanted. So, prevention is required. (Semi)-artificial breathing apparatus (home care pulmonary diseases, pulmonology, IC, surgery theatre) should have valves, tubes etc. designed in such a way that valves are not blocked by freezing and the breathing gas is not cooled down by adiabatic effects. Adiabatic effects also play a role in the technology of (clinical) hyperbaric chambers, high pressure gas tanks and diving regulators.

In the human (and mammalian) body), compression and expansion of gas filed cavities, adiabatic effects play now role. Even when one dives in the water from some tens of meters altitude, the temperature rise in the cavities is physiologically irrelevant since the process is sufficiently isotherm.

A medical imaging method, Optoacoustic Imaging is based on adiabatic processes.

More Info

The exponent γ is the c_p/c_V ratio, the ratio of heat capacity of the gas (c_p) with constant p and the specific heat capacity of the gas (c_V) with constant V. The complicated derivation can be found in physical textbooks. Here, a conceptual explanation is given.

The process of compression always has the same energetic cost, so independent of the velocity of the compression. This external energy is added to the intrinsic energy of the gas, which is actually the kinetic energy of the particles. The kinetic energy of the particles is proportional with the temperature. When the external energy is supplied instantaneously, it is completely added to the kinetic energy of the particles. When the compression is slow, then during the time of compression, the particles have the time to transfer a part of their kinetic energy to the vessel wall. The result is that the temperature of the gas raises less and the temperature of the wall increases. When the compression is extremely slow, the gas remains its temperature since all external energy is (indirectly) transferred to the environment. Now, the compression is isothermic and Boyle's law holds.

For instantaneous compression, the increase can be calculated as follows. Suppose that volume V with pressure p is compressed to V’ yielding a new pressure p’. Applying Eq. (1) means that:

p·V^γ = p’(V’)^γ . (3).

Before compression and after compression the universal gas law (see Gas laws) holds, so that pV = nRT and p’V’ = nRT’ (T’ is the new temperature), and some more calculation yields:

ΔT = T’ – T = T((V/V’)^{(γ -1)} - 1) (4)

This relation also holds for volume expansion.

In a similar way, for a given change of pressure it can be derived that:

ΔT = T((p/p’)^{(γ-1)/ γ} - 1). (5)

Example of calculation

Suppose, everything ideal, that a gas volume at 300 K (27 ^oC) is instantaneously reduced in volume by a factor 2.5 and that there is no heat transport. According to Eq. (4) this yields ΔT = 300 x (2.5^0.4 - 1) = 133 K. However, due to the temperature increase, a much higher factor of pressure increase is obtained (being 2.5^1.4 = 3.6) rather than the factor of 2.5 according to Boyle’s law. Actually, a compression-factor of 1.92 (= 2.5^1/1.4) is required to obtain p = 2.5 bar. Then, the temperature increase becomes 300 x (1.92^0.4 – 1) = 90 K.

The equation p·V^γ = constant works also the other way around. Suppose that the pressure p is increased instantaneously to k·p by a factor of k. Then, according to Boyle, the volume V reduces with a factor k, but now, the reduction is only a factor k ^1/1.4. Again, in the pressure increase, a “Boyle factor” k^1/1.4 and a “temperature factor” k^0.4/1.4 are comprised. Of course, these two factors together have to yield k (= k^1/1.4·k^0.4/1.4). Now, with k = 2.5 and T = 300 K the temperature increase is again 90 K (2.5^0.4/1.4x300 K).

In practice, it appears that the processes of compression and expansion are seldom purely adiabatic or purely isothermic. When the process is in between the calculation is complicated and, moreover in practice valves may block by freezing, for instance the valve of Fig.1. This may result in oscillatory behavior between blocking and unblocking by freezing and unfreezing.