Compression ratio of a gasoline engine (ratio

The compression ratio of internal combustion (the ratio of the maximum volume of the working mixture to its minimum volume) is 8. It is necessary to find the ratio of the exhaust temperature to the combustion temperature. To solve this problem, we will consider the expansion to be adiabatic, and the working mixture (a mixture of air and gasoline vapor) to be a diatomic ideal gas.

Let us denote by V1 and V2 the volumes of the working mixture before and after compression, respectively. Then:

V2/V1 = 1/8

Consider the combustion process in an engine. Let us denote by Q1 the amount of heat released during the combustion of a unit mass of the working mixture. Then the thermal effect of combustion will be equal to Q = Q1 * m, where m is the mass of the working mixture.

After combustion, the heat will be converted into internal energy of the gas, increasing its temperature. Let us denote by Cv the specific heat capacity at constant volume and by Cp the specific heat capacity at constant pressure. Then, during adiabatic expansion of the gas:

Cv * (T2 - T1) = -Q

where T1 is the gas temperature before combustion, T2 is the gas temperature after combustion.

Let's consider the process of gas compression. Let us denote by P1 and P2 the gas pressures before and after compression, respectively. Then, during an adiabatic process:

P1 * V1^γ = P2 * V2^γ

where γ = Cp/Cv is the adiabatic exponent.

Using the ideal gas equation of state:

PV = mRT

where P is pressure, V is volume, m is mass of gas, R is universal gas constant, T is gas temperature, we get:

P1 * V1 = m * R * T1 P2 * V2 = m * R * T2

Dividing the last two equations, we get:

P2/P1 = V1/V2 * T2/T1

Replacing V2/V1 with the value obtained from the first equation, we get:

P2/P1 = 8 * T2/T1

Comparing this equation with the equation for an adiabatic process, we obtain:

(P2/P1)^(γ-1) = T2/T1

Considering that γ = Cp/Cv and that Cp - Cv = R, we get:

(P2/P1)^(R/Cp) = T2/T1

Let us express the temperature ratio in terms of known quantities:

T2/T1 = (P2/P1)^(R/Cp)

Let us substitute into this equation the value of the pressure ratio obtained from the equation for gas compression:

T2/T1 = (8^((γ-1)/γ))^(R/Cp) = 8^(R/Cp - 1)

Thus, the ratio of the exhaust temperature to the combustion temperature will be equal to 8^(R/Cp - 1).

Our digital product is a detailed solution to problem No. 20344 on thermodynamics related to the compression ratio of a gasoline engine and the ratio of exhaust temperature to combustion temperature. In our product you will find a brief recording of the conditions of the problem, formulas and laws used in the solution, the derivation of the calculation formula and the answer to the problem.

Our team of thermodynamics experts has prepared this solution using the latest advances in science and technology. We took into account all the features of the task and gave comprehensive answers to all her questions.

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This problem is related to the compression ratio of a gasoline engine and the ratio of exhaust temperature to combustion temperature. To solve it, it is necessary to take into account that gas expansion occurs adiabatically, and the working mixture is a diatomic ideal gas.

Let us denote by V1 and V2 the volumes of the working mixture before and after compression, respectively. Then the ratio of the volumes of the working mixture will be equal to V2/V1 = 1/8.

Consider the combustion process in an engine. Let us denote by Q1 the amount of heat released during the combustion of a unit mass of the working mixture. Then the thermal effect of combustion will be equal to Q = Q1 * m, where m is the mass of the working mixture.

After combustion, the heat will be converted into internal energy of the gas, increasing its temperature. Let us denote by Cv the specific heat capacity at constant volume and by Cp the specific heat capacity at constant pressure. Then, during adiabatic expansion of the gas:

Cv * (T2 - T1) = -Q,

where T1 is the gas temperature before combustion, T2 is the gas temperature after combustion.

Let's consider the process of gas compression. Let us denote by P1 and P2 the gas pressures before and after compression, respectively. Then, during an adiabatic process:

P1 * V1^γ = P2 * V2^γ,

where γ = Cp/Cv is the adiabatic exponent.

Using the equation of state of an ideal gas: PV = mRT, where P is pressure, V is volume, m is gas mass, R is the universal gas constant, T is gas temperature, we obtain:

P1 * V1 = m * R * T1 P2 * V2 = m * R * T2

Dividing the last two equations, we get:

P2/P1 = V1/V2 * T2/T1

Replacing V2/V1 with the value obtained from the first equation, we get:

P2/P1 = 8 * T2/T1

Comparing this equation with the equation for an adiabatic process, we obtain:

(P2/P1)^(γ-1) = T2/T1

Considering that γ = Cp/Cv and that Cp - Cv = R, we get:

(P2/P1)^(R/Cp) = T2/T1

Let us express the temperature ratio in terms of known quantities:

T2/T1 = (P2/P1)^(R/Cp)

Let us substitute into this equation the value of the volume ratio obtained from the first equation:

T2/T1 = (1/8)^(R/Cp)

Thus, the ratio of the exhaust temperature to the combustion temperature will be equal to 1/(1/8)^(R/Cp) = 8^(R/Cp - 1).

Answer: The ratio of the exhaust temperature to the combustion temperature is 8^(R/Cp - 1), where R is the universal gas constant, Cp is the specific heat at constant pressure.

This digital product is a detailed solution to problem No. 20344 on thermodynamics related to the compression ratio of a gasoline engine and the ratio of exhaust temperature to combustion temperature.

To solve the problem we use the following formulas and laws:

• The compression ratio of internal combustion (the ratio of the maximum volume of the working mixture to its minimum volume) is 8.
• We consider the working mixture (a mixture of air and gasoline vapor) to be a diatomic ideal gas.
• We consider the expansion of the gas in the cylinder to be adiabatic.
• Let us denote by V1 and V2 the volumes of the working mixture before and after compression, respectively. Then: V2/V1 = 1/8.
• Let us denote by Q1 the amount of heat released during the combustion of a unit mass of the working mixture. Then the thermal effect of combustion will be equal to Q = Q1 * m, where m is the mass of the working mixture.
• Let us denote by Cv the specific heat capacity at constant volume and by Cp the specific heat capacity at constant pressure. Then, during adiabatic expansion of gas: Cv * (T2 - T1) = -Q, where T1 is the gas temperature before combustion, T2 is the gas temperature after combustion.
• Let's consider the process of gas compression. Let us denote by P1 and P2 the gas pressures before and after compression, respectively. Then, for an adiabatic process: P1 * V1^γ = P2 * V2^γ, where γ = Cp/Cv is the adiabatic exponent.
• Using the equation of state of an ideal gas: PV = mRT, where P is pressure, V is volume, m is gas mass, R is the universal gas constant, T is gas temperature, we obtain: P1 * V1 = m * R * T1, P2 * V2 = m * R * T2.
• Dividing the last two equations, we get: P2/P1 = V1/V2 * T2/T1.
• Replacing V2/V1 with the value obtained from the first equation, we get: P2/P1 = 8 * T2/T1.
• Comparing this equation with the equation for the adiabatic process, we obtain: (P2/P1)^(γ-1) = T2/T1.
• Considering that γ = Cp/Cv and that Cp - Cv = R, we obtain: (P2/P1)^(R/Cp) = T2/T1.
• Let's express the temperature ratio through known quantities: T2/T1 = (P2/P1)^(R/Cp).
• Let us substitute into this equation the value of the pressure ratio obtained from the equation for gas compression: T2/T1 = 8^(R/Cp - 1).
• Thus, the ratio of the exhaust temperature to the combustion temperature will be equal to 8^(R/Cp - 1).

These formulas and laws make it possible to solve the problem related to the compression ratio of a gasoline engine and the ratio of the exhaust temperature to the combustion temperature. However, to solve this problem it is necessary to know the specific heat capacity at constant volume (Cv) and specific heat capacity at constant pressure (Cp) for the working mixture of a gasoline engine. These values ​​depend on the composition of the working mixture and may be different for different types of fuel and fuel additives.

Therefore, in order to solve this problem, you need to know not only the formulas and laws of thermodynamics, but also the specific values ​​of the specific heat capacity at constant volume and at constant pressure for a given working mixture. If these values ​​are unknown, then additional data or assumptions must be used to determine them.

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This product is a description of the solution to problem No. 20344, related to determining the ratio of the exhaust temperature to the combustion temperature in a gasoline engine. In the problem statement, it is known that the compression ratio of the engine is 8, and the expansion is considered adiabatic. It is also assumed that the working mixture is a diatomic ideal gas.

To solve the problem, you must use the following laws and formulas:

1. Boyle-Mariotte law: pV = const, where p is pressure, V is volume.

2. Law of adiabatic expansion: pV^γ = const, where γ is the adiabatic exponent.

3. Gay-Lussac's law: V/T = const, where T is temperature.

4. Equation of state of an ideal gas: pV = nRT, where n is the amount of substance, R is the universal gas constant.

5. Adiabatic index for a diatomic gas: γ = 1.4.

Based on the conditions of the problem, we can write formulas for the volumes of the working mixture at different stages of engine operation:

V1 is the volume of the working mixture at the inlet into the cylinder, V2 is the volume of the working mixture when compressed to the maximum compression ratio, V3 is the volume of the working mixture at the end of combustion and the beginning of expansion, V4 is the volume of the working mixture at the exhaust.

Using the ideal gas equation of state and the Boyle-Mariotte law, we can write the following relationships:

p1V1 = nRT1, p2V2 = nRT2, p3V3 = nRT3, p4V4 = nRT4.

Also, given that the compression ratio is 8, we can write the relationship between the volumes of the working mixture:

V2/V1 = 1/8.

Next, using the law of adiabatic expansion, we can write the relationship between pressures and volumes at different stages of engine operation:

p1V1^γ = p2V2^γ, p3V3^γ = p4V4^γ.

Also, given that the expansion is considered adiabatic, we can write the relationship between temperatures and volumes at different stages of engine operation using Gay-Lussac's law:

V1/T1 = V2/T2, V3/T3 = V4/T4.

Based on these relationships, we can express the ratio of the exhaust temperature to the combustion temperature:

T4/T3 = (V3/V4)^(γ-1) = (V1/V2)^(γ-1) = (1/8)^(γ-1) = 0.16.

Thus, the ratio of the exhaust temperature to the combustion temperature in this case is 0.16.

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