The amount of air in cylinders depends on the volume of the cylinder, air pressure and its temperature. The relationship between air pressure and its volume at a constant temperature is determined by the relationship
where р1 and р2 are the initial and final absolute pressure, kgf/cm²;
V1 and V2 - initial and final volume of air, l. The relationship between air pressure and its temperature at a constant volume is determined by the relationship
where t1 and t2 are the initial and final air temperatures.
Using these dependencies, you can solve various problems that you encounter in the process of charging and operating air-breathing apparatus.
Example 4.1. The total capacity of the apparatus cylinders is 14 liters, the excess air pressure in them (according to the pressure gauge) is 200 kgf/cm². Determine the volume of free air, i.e. the volume reduced to normal (atmospheric) conditions.
Solution. Initial absolute atmospheric air pressure p1 = 1 kgf/cm². Final absolute pressure of compressed air p2 = 200 + 1 = 201 kgf/cm². The final volume of compressed air V 2 = 14 l. Volume of free air in cylinders according to (4.1)
Example 4.2. From a transport cylinder with a capacity of 40 liters with a pressure of 200 kgf/cm² (absolute pressure 201 kgf/cm²), air was transferred into the apparatus cylinders with a total capacity of 14 liters and a residual pressure of 30 kgf/cm² (absolute pressure 31 kgf/cm²). Determine the air pressure in the cylinders after air bypass.
Solution. Total volume of free air in the system of transport and equipment cylinders according to (4.1)
Total volume of compressed air in the cylinder system
Absolute pressure in the cylinder system after air bypass
excess pressure = 156 kgf/cm².
This example can be solved in one step by calculating the absolute pressure using the formula
Example 4.3. When measuring the air pressure in the apparatus cylinders in a room with a temperature of +17° C, the pressure gauge showed 200 kgf/cm². The device was taken outside, where a few hours later, during a working check, a pressure drop on the pressure gauge was discovered to 179 kgf/cm². The outside air temperature is -13° C. There is a suspicion of air leakage from the cylinders. Check the validity of this suspicion using calculations.
Solution. The initial absolute air pressure in the cylinders is p1 = 200 + 1 = 201 kgf/cm², the final absolute pressure p2 = 179 + 1 = 180 kgf/cm². Initial air temperature in cylinders t1 = + 17° C, final temperature t2 = - 13° C. Calculated final absolute air pressure in cylinders according to (4.2)
Suspicions are unfounded, since the actual and calculated pressures are equal.
Example 4.4. A submarine swimmer underwater consumes 30 l/min of air compressed to a pressure of a diving depth of 40 m. Determine the free air consumption, i.e., convert to atmospheric pressure.
Solution. Initial (atmospheric) absolute air pressure p1 = l kgf/cm². The final absolute pressure of compressed air according to (1.2) р2 =1 + 0.1*40 = 5 kgf/cm². Final compressed air flow V2 = 30 l/min. Free air flow according to (4.1)
The relationship between pressure, temperature, volume and number of moles of gas (the “mass” of gas). Universal (molar) gas constant R. Clayperon-Mendeleev equation = equation of state of an ideal gas.
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Limitations of practical applicability:
Within the range, the accuracy of the equation exceeds that of conventional modern engineering measuring instruments. It is important for the engineer to understand that significant dissociation or decomposition is possible for all gases as temperature increases. |
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Let's solve a couple of problems regarding gas volumetric and mass flow rates under the assumption that the composition of the gas does not change (the gas does not dissociate) - which is true for most of the gases in the above.
This task is relevant mainly, but not only, for applications and devices in which gas volume is directly measured.
V 1 And V 2, at temperatures, respectively, T 1 And T 2 let it go T 1< T 2. Then we know that:
Naturally, V 1< V 2
- The lower the temperature, the more significant the indicators of the volumetric gas meter are.
- it is profitable to supply “warm” gas
- it is profitable to buy “cold” gas
How to deal with this? At least simple temperature compensation is required, that is, information from an additional temperature sensor must be supplied to the counting device.
This task is relevant mainly, but not only, for applications and devices in which gas velocity is directly measured.
Let counter() at the delivery point give the volumetric accumulated costs V 1 And V 2, at pressures, respectively, P 1 And P2 let it go P 1< P2. Then we know that:
Naturally, V 1>V 2 for the same amounts of gas under given conditions. Let's try to formulate several practical conclusions for this case:
- The higher the pressure, the more significant the indicators of the gas volume meter are.
- It is profitable to supply low pressure gas
- profitable to buy high pressure gas
How to deal with this? At least simple pressure compensation is required, that is, information from an additional pressure sensor must be supplied to the counting device.
In conclusion, I would like to note that, theoretically, everyone gas meter must have both temperature compensation and pressure compensation. Practically......
Ideal gas equation of state determines the relationship between temperature, volume and pressure of bodies.
- Allows you to determine one quantity characterizing the state of a gas from two others (used in thermometers);
- Determine how processes proceed under certain external conditions;
- Determine how the state of the system changes if it does work or receives heat from external bodies.
Mendeleev-Clapeyron equation (ideal gas equation of state)
- universal gas constant, R = kN A
Clapeyron's equation (united gas law)
Special cases of the equation are gas laws that describe isoprocesses in ideal gases, i.e. processes in which one of the macroparameters (T, P, V) in a closed isolated system is constant.
Quantitative relationships between two parameters of a gas of the same mass with a constant value of the third parameter are called gas laws.
Boyle's Law - Mariotte
The first gas law was discovered by the English scientist R. Boyle (1627-1691) in 1660. Boyle’s work was called “New Experiments Concerning an Air Spring.” Indeed, gas behaves like a compressed spring; this can be verified by compressing air in a regular bicycle pump.
Boyle studied the change in gas pressure as a function of volume at constant temperature. The process of changing the state of a thermodynamic system at a constant temperature is called isothermal (from Greek words isos - equal, therme - heat).
Independently of Boyle, somewhat later, the French scientist E. Marriott (1620-1684) came to the same conclusions. Therefore, the found law was called the Boyle-Mariotte law.
The product of the pressure of a gas of a given mass and its volume is constant if the temperature does not change
pV = const
Gay-Lussac's Law
The discovery of another gas law was published only in 1802, almost 150 years after the discovery of the Boyle-Mariotte law. The law defining the dependence of gas volume on temperature at constant pressure (and constant mass) was established by the French scientist Gay-Lussac (1778-1850).
The relative change in the volume of a gas of a given mass at constant pressure is directly proportional to the change in temperature
V = V 0 αT
Charles's Law
The dependence of gas pressure on temperature at constant volume was experimentally established by the French physicist J. Charles (1746-1823) in 1787.
J. Charles in 1787, i.e., earlier than Gay-Lussac, established the dependence of volume on temperature at constant pressure, but he did not publish his works in a timely manner.
The pressure of a given mass of gas at constant volume is directly proportional to the absolute temperature.
p = p 0 γT
| Name | Formulation | Charts |
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Boyle-Mariotte Law – isothermal process |
For a given mass of gas, the product of pressure and volume is constant if the temperature does not change
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Gay-Lussac's Law – isobaric process |
Since P is constant during an isobaric process, after reduction by P the formula takes the form
V 1 /T 1 =V 2 /T 2,
V 1 /V 2 =T 1 /T 2.
The formula is a mathematical expression of Gay-Lussac's law: at a constant gas mass and constant pressure, the volume of the gas is directly proportional to its absolute temperature.
Isothermal process
A process in a gas that occurs at a constant temperature is called isothermal. The isothermal process in gas was studied by the English scientist R. Boyle and the French scientist E. Mariot. The connection they established experimentally is obtained directly from the formula by reducing it to T:
p 1 V 1 =p 2 V 2 ,
p 1 /p 2 =V 1 /V 2.
The formula is a mathematical expression Boyle-Mariota law: At a constant mass of gas and a constant temperature, the pressure of the gas is inversely proportional to its volume. In other words, under these conditions, the product of the gas volume and the corresponding pressure is a constant:
The graph of p versus V during an isothermal process in a gas is a hyperbola and is called an isotherm. Figure 3 shows isotherms for the same mass of gas, but at different temperatures T. During an isothermal process, the gas density changes in direct proportion to the pressure:
ρ 1 /ρ 2= p 1 /p 2
Dependence of gas pressure on temperature at constant volume
Let's consider how gas pressure depends on temperature when its mass and volume remain constant. Let's take a closed vessel with gas and heat it (Figure 4). We will determine the gas temperature t using a thermometer, and the pressure using a pressure gauge M.
First, we will place the vessel in melting snow and denote the gas pressure at 0 0 C as p 0, and then we will gradually heat the outer vessel and record the values of p and t for the gas.
It turns out that the graph of p and t, constructed on the basis of such an experiment, looks like a straight line (Figure 5).
If we continue this graph to the left, it will intersect with the x-axis at point A, corresponding to zero gas pressure. From the similarity of triangles in Figure 5, a can be written:
P 0 /OA=Δp/Δt,
l/OA=Δp/(p 0 Δt).
If we denote the constant l/OA through α, we get
α = Δp//(p 0 Δt),
Δp= α p 0 Δt.
In essence, the proportionality coefficient α in the experiments described should express the dependence of the change in gas pressure on its type.
Magnitude γ, characterizing the dependence of the change in gas pressure on its type in the process of changing temperature at a constant volume and constant mass of gas is called the temperature coefficient of pressure. The temperature coefficient of pressure shows by what part of the pressure of a gas taken at 0 0 C changes when heated by 1 0 C. Let us derive the unit of temperature coefficient α in SI:
α =l ΠA/(l ΠA*l 0 C)=l 0 C -1
In this case, the length of the segment OA is equal to 273 0 C. Thus, for all cases, the temperature at which the gas pressure should go to zero is the same and equal to – 273 0 C, and the temperature coefficient of pressure α = 1/OA = (1/273 ) 0 C -1 .
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When solving problems, they usually use an approximate value of α equal to α =1/OA=(1/273) 0 C -1 . From experiments, the value of α was first determined by the French physicist J. Charles, who in 1787. installed next law: the temperature coefficient of pressure does not depend on the type of gas and is equal to (1/273.15) 0 C -1. Note that this is only true for gases having low densities and for small changes in temperature; at high pressures or low temperaturesα depends on the type of gas. Only an ideal gas strictly obeys Charles's law. Let's find out how we can determine the pressure of any gas p at an arbitrary temperature t.
Substituting these values Δр and Δt into the formula, we get
p 1 -p 0 =αp 0 t,
p 1 =p 0 (1+αt).
Since α~273 0 C, when solving problems the formula can be used in the following form:
p 1 =p 0
The combined gas law is applicable to any isoprocess, taking into account that one of the parameters remains constant. In an isochoric process, the volume V remains constant, the formula after reduction by V takes the form
At the core physical properties gases and the laws of the gas state lies the molecular kinetic theory of gases. Most of the laws of the gas state were derived for an ideal gas, the molecular forces of which are zero, and the volume of the molecules themselves is infinitely small compared to the volume of intermolecular space.
Molecules of real gases in addition to energy rectilinear motion possess the energy of rotation and vibration. They occupy a certain volume, that is, they have finite dimensions. The laws for real gases are somewhat different from the laws for ideal gases. This deviation is greater the higher the pressure of the gases and the lower their temperature; it is taken into account by introducing a compressibility correction factor into the corresponding equations.
When transporting gases through pipelines under high pressure, the compressibility coefficient is of great importance.
At gas pressures in gas networks up to 1 MPa, the laws of gas state for an ideal gas quite accurately reflect the properties natural gas. With more high pressures or at low temperatures, equations are used that take into account the volume occupied by molecules and the forces of interaction between them, or correction factors are introduced into the equations for an ideal gas - gas compressibility coefficients.
Boyle's Law - Mariotte.
Numerous experiments have established that if you take a certain amount of gas and expose it different pressures, then the volume of this gas will change in inverse proportion to the pressure. This relationship between pressure and gas volume at constant temperature is expressed by the following formula:
p 1 /p 2 = V 2 /V 1, or V 2 = p 1 V 1 /p 2,
Where p 1 And V 1- initial absolute pressure and volume of gas; p2 And V 2 - pressure and volume of gas after the change.
From this formula we can obtain the following mathematical expression:
V 2 p 2 = V 1 p 1 = const.
That is, the product of the gas volume by the gas pressure corresponding to this volume will be a constant value at a constant temperature. This law has practical use in the gas industry. It allows you to determine the volume of a gas when its pressure changes and the gas pressure when its volume changes, provided that the gas temperature remains constant. The more the volume of a gas increases at a constant temperature, the lower its density becomes.
The relationship between volume and density is expressed by the formula:
V 1/V 2 = ρ 2 /ρ 1 ,
Where V 1 And V 2- volumes occupied by gas; ρ 1 And ρ 2 - gas densities corresponding to these volumes.
If the ratio of gas volumes is replaced by the ratio of their densities, then we can obtain:
ρ 2 /ρ 1 = p 2 /p 1 or ρ 2 = p 2 ρ 1 /p 1.
We can conclude that at the same temperature the densities of gases are directly proportional to the pressures under which these gases are located, that is, the density of a gas (at a constant temperature) will be greater, the greater its pressure.
Example. Volume of gas at a pressure of 760 mm Hg. Art. and a temperature of 0 °C is 300 m 3. What volume will this gas occupy at a pressure of 1520 mm Hg? Art. and at the same temperature?
760 mmHg Art. = 101329 Pa = 101.3 kPa;
1520 mmHg Art. = 202658 Pa = 202.6 kPa.
Substituting given values V, p 1, p 2 into the formula, we get m 3:
V 2= 101, 3-300/202,6 = 150.
Gay-Lussac's law.
At constant pressure, with increasing temperature, the volume of gases increases, and with decreasing temperature, it decreases, that is, at constant pressure, the volumes of the same amount of gas are directly proportional to their absolute temperatures. Mathematically, this relationship between the volume and temperature of a gas at constant pressure is written as follows:
V 2 /V 1 = T 2 /T 1
where V is the volume of gas; T - absolute temperature.
From the formula it follows that if a certain volume of gas is heated at constant pressure, then it will change as many times as its absolute temperature changes.
It has been established that when a gas is heated by 1 °C at constant pressure, its volume increases by a constant amount equal to 1/273.2 of the original volume. This quantity is called the thermal expansion coefficient and is denoted p. Taking this into account, Gay-Lussac's law can be formulated as follows: the volume of a given mass of gas at constant pressure is a linear function of temperature:
V t = V 0 (1 + βt or V t = V 0 T/273.
Charles's law.
At constant volume, the absolute pressure of a constant amount of gas is directly proportional to its absolute temperatures. Charles's law is expressed by the following formula:
p 2 / p 1 = T 2 / T 1 or p 2 = p 1 T 2 / T 1
Where p 1 And p 2 - absolute pressures; T 1 And T 2— absolute gas temperatures.
From the formula we can conclude that at a constant volume, the pressure of a gas when heated increases as many times as its absolute temperature increases.