Polytropic Process MCQ Quiz - Objective Question with Answer for Polytropic Process - Download Free PDF

Explanation:

• In many real processes, it is found that the states during an expansion or compression can be described approximately by a relation of the form Pvn = constant,
• where n is a constant called index of compression or expansion, P and v are the average value of pressure and specific volume for the system.
• Compressions and expansions of the form Pvn = constant are called polytropic process.
• For the reversible polytropic process, single values of P and v can truly define the state of a system, dW = -Pdv.
• The equation for the polytropic process:
• $P{v}^n=C\Rightarrow \frac{P}{{\rho }^n}=C\Rightarrow \frac{P_1}{P_2}={\left(\frac{{\rho }_1}{{\rho }_2}\right)}^n$

The general polytropic process is shown in the figure:

Additional Information 

Concept:

Work done in the Polytropic process is given by:

W1-2 = $\frac{P_1{V}_1-P_2{V}_2}{n-1}$

Calculation:

Given:

m = 9 kg, PV1.2= Constant, P₁ = 2 bar, V₁ = 0.5 m3, V₂ =  1 m3, du = - 6 kJ/kg.

W_1-2 = $\frac{P_1{V}_1-P_2{V}_2}{n-1}$

$\frac{P_2}{P_1}={\left(\frac{V_1}{V_2}\right)}^{1.2}\Rightarrow P_2=2{\left(\frac{0.5}{1}\right)}^{1.2}$

P₂ = 0.87 bar

$W=\frac{2\times 0.5-0.87\times 1}{0.2}\times {10}^2$

W = 64.724 KJ

From first law, dQ = dU+ dW

dQ = -6 × 9 + 64.724

dQ = 10.724 KJ

Concept:

A polytropic process is any general process in nature and is represented by the equation

PVn = C

Where, P = Pressure of system, V = Volume of system

n = Polytropic index  and 1 < n < γ 

Work done in polytropic process is,  $W=\frac{P_1{V}_1-P_2{V}_2}{n-1}=\frac{mR(T_1-T_2)}{n-1}$

Calculation:

Given:

P = 1 bar, T₁ = 27° C, T₂ = 327° C, n = 1.3

Gas constant will be, R = 0.287 kJ / kg-K

Compression work, $W=\frac{P_1{V}_1-P_2{V}_2}{n-1}=\frac{mR(T_1-T_2)}{n-1}$

⇒ W = $\frac{0.287\times (27-327)}{1.3-1}$

⇒ W = $\frac{0.287\times (-300)}{0.3}=-287kJ/kg$

∵ Here, the system is air, and work is done on the system, i.e. compression of air, That's why work done is negative

Concept:

Polytropic process is any general process in nature and represented by the equation

PVⁿ = C

Where, P = Pressure of system, V = Volume of system

n = Polytropic index  and 1 < n < $\gamma$

Work done in polytropic process is,  $W=\frac{P_1{V}_1^{}-P_2{V}_2^{}}{n-1}=\frac{mR(T_1-T_2)}{n-1}$

Calculation:

Given:

P₁ = 110 KPa, V₁ = 5 m³ , V₂ = 2.5 m³, P₂ = ?, W = ?

$P_1{V}_1^n=P_2{V}_2^n$

$110\times 5^{1.2}=P_2\times {2.5}^{1.2}$ $\Rightarrow$ P₂ = 252.7136 KPa

Work done during the process is 

$W=\frac{P_1{V}_1^{}-P_2{V}_2^{}}{n-1}=\frac{110\times 5^{}-252.7136\times {2.5}^{}}{1.2-1}$

W = - 408.92 kJ

The magnitude of Work done is 408.92 kJ

Additional Information

The general equation of any thermodynamic process is represented as

PV^k = C

Explanation:

According to First Law of Thermodynamics:

Q1-2 = ΔU + W1-2

Polytropic Process is given by:

PVn=C

Work-done for a polytropic process:

$W=\frac{P_2{V}_2-P_1{V}_1}{1-n}=\frac{mR(T_2-T_1)}{1-n}$

Cp - Cv = R and $\frac{C_p}{C_v}=\gamma$

$\therefore C_v=\frac{R}{\gamma -1}$

ΔU = mcv(T2 - T1)

$\mathrm{\Delta }U=m\frac{R}{\gamma -1}(T_2-T_1)=W\frac{(1-n)}{(\gamma -1)}$

$Q=\Delta U+W=W\left[1+\frac{(1-n)}{(\gamma -1)}\right]=W\frac{(\gamma -n)}{(\gamma -1)}$

$Q=\left(\frac{\gamma -n}{\gamma -1}\right)\times W$

Explanation:

According to First Law of Thermodynamics:

Q_1-2 = ΔU + W_1-2

Polytropic Process is given by:

PVn=C

Work-done for a polytropic process:

$W=\frac{P_2{V}_2-P_1{V}_1}{1-n}=\frac{mR(T_2-T_1)}{1-n}$

Cp - Cv = R and $\frac{C_p}{C_v}=\gamma$

$\therefore C_v=\frac{R}{\gamma -1}$

ΔU = mcv(T2 - T1)

$\mathrm{\Delta }U=m\frac{R}{\gamma -1}(T_2-T_1)=W\frac{(1-n)}{(\gamma -1)}$

$Q=\Delta U+W=W\left[1+\frac{(1-n)}{(\gamma -1)}\right]=W\frac{(\gamma -n)}{(\gamma -1)}$

$Q=\left(\frac{\gamma -n}{\gamma -1}\right)\times W$

Concept:

According to the Ideal gas equation for a polytropic process:

$P_1{V}_1^n=P_2{V}_2^n$

$\Rightarrow {\left(\frac{V_1}{V_2}\right)}^n=\frac{P_2}{P_1}$

Taking log on both sides

$n\ln \left(\frac{V_1}{V_2}\right)=\ln \left(\frac{P_2}{P_1}\right)$

$\Rightarrow n=\frac{\ln \left(\frac{P_2}{P_1}\right)}{\ln \left(\frac{V_1}{V_2}\right)}$

Work done for Polytropic process is, $W_{1-2}=\frac{P_1{V}_1-P_2{V}_2}{n-1}$

Heat transfer for Polytropic process is, $Q=W\left(\frac{\gamma -n}{\gamma -1}\right)$

Concept:

Polytropic Process is represented by:

PVn = C

• n = 0 ⇒ P = C ⇒ Constant Pressure Process (Isobaric Process)
• n = 1 ⇒ PV = C ⇒ Constant Temperature Process (Isothermal process)
• n = γ ⇒ PVγ = C ⇒ Adiabatic Process
• n = ∞ ⇒ V = C ⇒ Constant Volume Process (Isochoric process)

Concept:

A process in which no heat is supplied and rejected is known as an adiabatic process. But if entropy is not constant, then this process is called the polytropic process.

Hint: This question can be solved by eliminating the options.

Isothermal Process:

An isothermal process is a process in which temperature remains constant. i.e. T₁ = T₂ ⇒ ΔU = 0

• For the isothermal process δQ = -δW ≠ 0
• Isentropic Process: ΔS = 0
• Hyperbolic Process is the isothermal process

So the process cannot be isothermal/hyperbolic and isentropic. It can only be polytropic.

Explanation:

In many real processes, it is found that the states during an expansion or compression can be described approximately by a relation of the form Pvⁿ = constant,

where n is a constant called index of compression or expansion, P and v are the average value of pressure and specific volume for the system.

Compressions and expansions of the form Pvⁿ = constant are called polytropic process.

For the reversible polytropic process, single values of P and v can truly define the state of a system, dW = -Pdv.
The equation for the polytropic process: 
$P{v}^n=C\Rightarrow \frac{P}{{\rho }^n}=C\Rightarrow \frac{P_1}{P_2}={\left(\frac{{\rho }_1}{{\rho }_2}\right)}^n$

Additional Information

Concept:

Polytropic process is any general process in nature and represented by the equation

PVⁿ = C

Where, P = Pressure of system, V = Volume of system

n = Polytropic index  and 1 < n < $\gamma$

Work done in polytropic process is,  $W=\frac{P_1{V}_1^{}-P_2{V}_2^{}}{n-1}=\frac{mR(T_1-T_2)}{n-1}$

Calculation:

Given:

P₁ = 110 KPa, V₁ = 5 m³ , V₂ = 2.5 m³, P₂ = ?, W = ?

$P_1{V}_1^n=P_2{V}_2^n$

$110\times 5^{1.2}=P_2\times {2.5}^{1.2}$ $\Rightarrow$ P₂ = 252.7136 KPa

Work done during the process is 

$W=\frac{P_1{V}_1^{}-P_2{V}_2^{}}{n-1}=\frac{110\times 5^{}-252.7136\times {2.5}^{}}{1.2-1}$

W = - 408.92 kJ

The magnitude of Work done is 408.92 kJ

Additional Information

The general equation of any thermodynamic process is represented as

PV^k = C

Explanation:

In many real processes, it is found that the states during an expansion or compression can be described approximately by a relation of the form Pvn = constant,

where n is a constant called index of compression or expansion, P and v are the average value of pressure and specific volume for the system.

Compressions and expansions of the form Pvn = constant are called polytropic process.

For the reversible polytropic process, single values of P and v can truly define the state of a system, dW = -Pdv.
The equation for the polytropic process: 
$P{v}^n=C\Rightarrow \frac{P}{{\rho }^n}=C\Rightarrow \frac{P_1}{P_2}={\left(\frac{{\rho }_1}{{\rho }_2}\right)}^n$

or ${\mathrm{P}}_1{\mathrm{V}}_1^{\mathrm{n}}={\mathrm{P}}_2{\mathrm{V}}_2^{\mathrm{n}}$

The general polytropic process is shown in the figure:

Additional Information

Concept:

Given device is piston-cylinder, hence it is closed system.

In closed system work done by the system in a polytropic process (PVⁿ = constant) is given by,

$W_{1-3}=\frac{P_1\times V_1-P_3\times V_3}{n-1}$

Calculation:

Given P₁ = 100 kPa, V₁ = 0.1 m³, P₂ = 37.9 kPa, V₂ = 0.2 m³, P₃ = 14.4 kPa, V₃ = 0.4 m³;

Work done during process from 1 to 3

$W_{1-3}=\frac{P_1\times V_1-P_3\times V_3}{n-1}$

Concept:

Polytropic Process: A process which follows

PV^γ = Constant 

⇒ P₁V₁ⁿ = P₂V₂ⁿ 

⇒ $\frac{P_1}{P_2}={\left(\frac{V_2}{V_1}\right)}^n$

ln 2 = 0.693  & ln 5 = 1.62

Calculation:

Given:

$\frac{P_1}{P_2}=8$, $\frac{V_2}{V_1}=5$ & n = ?

Using $\frac{P_1}{P_2}={\left(\frac{V_2}{V_1}\right)}^n$

8 = 5ⁿ 

Taking log on both sides

ln 8 = n ln 5

n = $\frac{\ln 8}{\ln 5}=\frac{3\ln 2}{\ln 5}$

n = $\frac{3\times 0.693}{1.62}$

n = 1.292

Explanation:

Polytropic Process is represented by

PVⁿ = C

• n = 0 ⇒ P = C ⇒ Constant Pressure Process (Isobaric Process)
• n = 1 ⇒ PV = C ⇒ Constant Temperature Process (Isothermal process)
• n = γ ⇒ PV^γ = C ⇒ Adiabatic Process
• n = ∞ ⇒ V = C ⇒ Constant Volume Process (Isochoric process)

For n = 0, it is a constant pressure process and it is a horizontal line.

For n = ∞, it is a constant volume process and it is a vertical line.

So, as the value of n increases, the process line will come closer to the y-axis.