Question

The polytropic or small stage efficiency of a compressor \(\eta_{\infty}, c\) is defined as the ratio of the actual differential work done on the fluid to the isentropic differential work done on the flowing through the compressor \(\eta_{\infty}, c=d h_{s} / d h\) Consider an ideal gas with constant specific heats as the working fluid undergoing a process in a compressor in which the polytropic efficiency is constant. Show that the temperature ratio across the compressor is related to the pressure ratio across the compressor by $$\frac{T_{2}}{T_{1}}=\left(\frac{P_{2}}{P_{1}}\right)^{\left(\frac{1}{\eta_{\infty, C}}\right)\left(\frac{R}{c_{p}}\right)}=\left(\frac{P_{2}}{P_{1}}\right)^{\left(\frac{1}{\eta_{\infty, C}}\right)\left(\frac{k-1}{k}\right)}$$

Short answer

Answer: The temperature ratio across the compressor is related to the pressure ratio across the compressor by the formula: $$ \frac{T_{2}}{T_{1}} = \left(\frac{P_{2}}{P_{1}}\right)^{\left(\frac{1}{\eta_{\infty, C}}\right)\left(\frac{k-1}{k}\right)} $$ where η∞,C is the constant polytropic efficiency, and k is the ratio of specific heats (cp/cv).

Step-by-step solution

Write down the given variables

We have the polytropic efficiency η∞,c and the working fluid as an ideal gas with constant specific heats, cp and R. The process in the compressor is undergoing a temperature and pressure change from T1, P1 to T2, P2 respectively.

Write down the isentropic relation

For an isentropic process for an ideal gas with constant specific heats, the temperature and pressure ratios are related as follows: $$ \left(\frac{T_{2}}{T_{1}}\right)^{\frac{c_{p}}{R}}=\left(\frac{P_{2}}{P_{1}}\right) $$

Calculate the isentropic exponent

The ratio of specific heats is defined as k, let's find the relation between cp and R: $$ k = \frac{c_{p}}{c_{v}} $$ Since \(c_p-c_v=R\), we find $$ c_v=c_p-R $$ and substituting, we get $$ k = \frac{c_p}{c_p - R} $$ Solve for cp and we get: $$ c_p= \frac{k}{k-1}R $$

Rewrite the isentropic relation with exponent

Substitute the cp expression into the isentropic relation from Step 2: $$ \left(\frac{T_{2}}{T_{1}}\right)^{\frac{k}{k-1}}=\left(\frac{P_{2}}{P_{1}}\right) $$

Write down the polytropic efficiency definition

We know that the polytropic efficiency is the ratio between the actual differential work done (enthalpy change) on the fluid and the isentropic differential work done: $$ \eta_{\infty, c}=\frac{d h_{s}}{d h} $$ For an ideal gas, enthalpy is a function of temperature, so the above relation can be written in terms of temperature change as: $$ \eta_{\infty, c}=\frac{dT_{s}}{dT} $$

Calculate temperature differential

We need to calculate the temperature differential \(dT_{s}\) from the isentropic relation in Step 4. Take the logarithm of both sides in that relation and differentiate: $$ \frac{k}{k-1}*\frac{dT_{s}}{T_{1}}= \frac{dT}{T_{1}} $$ Now, we can solve for \(dT_s\) to get: $$ dT_{s} = \frac{k}{k-1} * dT $$

Substitute temperature differential into polytropic efficiency definition

Replace \(dT_{s}\) in the efficiency equation from Step 5: $$ \eta_{\infty, c} = \frac{\frac{k}{k-1}*dT}{dT} $$

Solve for the temperature ratio

Rearrange the equation to get the desired expression for temperature ratio: $$ \frac{T_{2}}{T_{1}} = \left(\frac{P_{2}}{P_{1}}\right)^{\left(\frac{1}{\eta_{\infty, C}}\right)\left(\frac{k-1}{k}\right)} $$ Now, we've reached the conclusion that the temperature ratio across the compressor is related to the pressure ratio across the compressor as shown in the equation.

Key concepts

Compressor Thermodynamics

In the realm of thermodynamics, compressors are pivotal devices used in various applications such as refrigeration, air conditioning, and the oil and gas industry. A compressor increases the pressure of a gas by reducing its volume, which typically involves the rise in temperature of the gas due to the work done on it. The performance of a compressor can be defined by its polytropic efficiency, which is a measure of how effectively the compressor performs this process compared to an idealized, reversible process called an isentropic process.

The polytropic efficiency is critical as it impacts the overall energy efficiency of the system in which the compressor operates. Given that real-world compressors aren't perfectly efficient, the polytropic efficiency helps engineers calculate the actual work done and the resulting real temperature increase, taking into account the inefficiencies that occur due to factors such as friction, heat transfer to the surroundings, and internal turbulence within the compressor.

For an ideal gas, the relationship between pressure and temperature changes during compression can be determined by using the polytropic efficiency and the specific heat capacities of the gas. This relationship is crucial for designing and evaluating the compressor's performance. Sorting out this relationship allows one to predict how changes in operating conditions, like the pressure ratio across the compressor, will affect the final temperature of the gas, and subsequently helps to ensure the compressor operates within its intended thermal limits.

Isentropic Process

An isentropic process is an idealized thermodynamic process that is both adiabatic and reversible. The term 'isentropic' is derived from the Greek word 'isentropos', meaning 'equal entropy'. During an isentropic process, the entropy of the system remains constant. Entropy, a fundamental concept in thermodynamics, is a measure of the disorder within a system. A constant entropy implies that the process is operating without any entropy change, signifying no heat transfer and perfect reversibility.

This concept is especially important when studying the thermodynamics of compressors because it represents the 'gold standard' against which actual compressor performance is often compared. The work done to compress a gas in an isentropic process is the minimum possible for a given initial and final pressure. In reality, due to irreversibilities in the actual compression process, more work than this minimum is required. The relationship between the isentropic and the real process is captured by the isentropic efficiency of the compressor, allowing for a comparison between the ideal and actual scenarios.

Understanding how isentropic processes work can give insight into the maximum theoretical efficiency of thermodynamic cycles and helps gauge the potential improvements in actual processes. By examining isentropic processes, engineers can design systems that strive to achieve closer to this theoretical efficiency, thus optimizing energy usage.

Specific Heats

Specific heats, represented by the symbols, Cp and Cv, are foundational concepts in the study of thermodynamics, particularly when analyzing substances undergoing phase changes or temperature alterations due to heat transfer. In essence, the specific heat is the amount of heat per unit mass required to raise the temperature of a substance by one degree Celsius (or one Kelvin).

For an ideal gas—a theoretical gas that perfectly fits all the assumptions of the kinetic-molecular theory—the specific heat at constant pressure (Cp) and the specific heat at constant volume (Cv), provide insights into the energy needed to raise the temperature of the gas under different conditions. Specifically, at constant pressure, the gas is allowed to expand as it is heated, and at constant volume, the gas is contained and not allowed to expand.

A key relationship in thermodynamics involves the ratio of these two specific heats, represented as 'k' (Gamma or adiabatic index), which is utilized in various equations to describe the properties and behaviors of ideal gases, including adiabatic and isentropic processes commonly encountered in compressor operations. The value of k is particularly important in determining the isentropic relations between pressure and temperature during a compression process. In engineering and physics, understanding and applying the concepts of specific heats allow for precise energy calculations vital for the design, analysis, and optimization of thermal systems.