A piston/cylinder contains water at 500^∘ C, 3 MPa It is cooled in a polytropic process to 200^∘

A piston/cylinder contains water at 500^∘ C, 3 MPa It is...

Key Concepts

Thermodynamic State Properties

Thermodynamic state properties, such as pressure, temperature, and specific volume, define the condition of a system at a given moment. In the context of a piston-cylinder arrangement, these properties allow the determination of the working fluid’s behavior during a process and are fundamental for evaluating the work transfer, energy changes, and for establishing relations such as the polytropic relation.

Specific Work in a Polytropic Process

The specific work done during a polytropic process can be calculated by integrating the pressure with respect to volume. For a process following PV^n = constant, the work per unit mass is given by the formula W = (p2v2 - p1v1) / (1 - n) when n is not equal to 1. This formula arises from applying the first law of thermodynamics, connecting the work transfer with changes in the state properties.

Polytropic Process

A polytropic process is a thermodynamic process that follows the relation PV^n = constant, where P is the pressure, V is the specific volume, and n (the polytropic exponent) characterizes the process. This process model is versatile and can represent various specific processes by adjusting n, such as adiabatic (n = specific heat ratio) or isothermal (n = 1) processes.

Polytropic Exponent

The polytropic exponent, n, is a key parameter in polytropic processes that defines how pressure and volume change relative to each other. It is typically determined by applying the condition that PV^n remains constant between two states, often using logarithmic relationships to relate initial and final pressures and specific volumes.

Transcript

00:01 So here on this problem we are given a piston cylinder arrangement in which water is contained at temperature t1 we have 500 degrees celsius and pressure p1 we have three megapaascal now then water is cooled by a polytropic process to temperature t2 becomes 200 degrees celsius and the pressure p2 becomes one megapaascal now we have to calculate two things first of all the polytropic exponent in and next we have to calculate the work done during the process so for that we'll be referring the table b .1 .3 because these two states would be in the superheated region so corresponding to these value we can calculate a specific volume from the table b .1 .3 that is given in the book so v1 from here we'll get this value as 0 .11619 and similarly specific volume at state 2 v2 it will would be 0 .20596 now we have to calculate the polytropic index for that we know that for polytropic process p1 v1 to the power n is equals to p2 v2 to the power in so from here the polytropic index n would be we can take this term here and this is v1 so v1 over v2 it would be p1 over p2 and this is to the power n so from here we can write that we can take natural log to both the side so natural log n would come here and this is v1 over v2 is equals to natural log p2 over p1 now we're going to solve it for n so n would be equals to natural lock p2 over p1 divide by natural lock v1 over v2 now we have all the village we can plug it here so natural log p2 we have this much this is one megapascal and p1 we have three divided by natural log now v1 we have this much so 0 .11619 divide by v2 we have this much 0 .2056 now we're going to simplify this by using a calculator and we'll get the value as nearly 1 .919 so this is the exponent n for the polytropic process.

02:40 Now next we'll be calculating the work done during the process.

02:44 Now the work done during polytopic process is given by p2 v2 minus p1v1 divide by 1 minus n.

02:55 So p2 we have this is 1 megapascal so we'll be taking the values in kilo pascal so that we can get the answer in kilojoule.

03:03 So this is kilo pascal times v2...

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