First Law- Control Mass (Closed) Systems Upto Monday 9 March

First Law for Controlled Mass (closed) Systems Work

Work done at a moving boundary (simple compressible systems) Constant Volume Process (Isochoric) Constant Pressure Process (Isobaric) Constant Temperature (Isothermal) Polytropic Processes

Heat Transfer Revisited (Conduction, Convection, Radiation)

Internal Energy and Enthalpy

Specific Heats

Internal Energy, Enthalpy and Specific Heat (for gases, solids and liquids)

Conservation of energy principle (1st law) for a closed system Revisited

Problem Solving technique 1

Work

Consider the two systems labeled A and B.

In system A, a gas is stirred by a paddle wheel: the paddle

wheel does work on the gas.

In principle, the work could

be evaluated in terms of the

forces and the motions at the

boundary between the paddle

wheel and the gas.

By contrast, consider system B, which includes only the battery.

At the boundary of system B, forces and motions are not evident. Rather, there is

an electric current i driven by an electrical potential difference existing across the

terminals a and b. That this type of interaction at the boundary can be classified

as work follows from the thermodynamic definition of work given previously:

We can imagine the current is supplied to a hypothetical electric motor that lifts a

weight in the surroundings. 2

Work Revisited

A thermodynamic definition of work states:

Work is done by a system on its surroundings if the sole effect on everything external to the system could have been the raising of a weight.

In other words, work is the energy transfer (across a boundary) associated with a force acting through a distance resulting (usually) in a movement of the boundary.

Mathematically:

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Forms of Work

dVp

V

V

2

1

Expansion/Compression Work (Moving Boundary Work)

Elongation of a solid bar

Stretching of a Liquid Film

Rotating Shaft

Electric

Others: Polarization

Magnetization

Surface tension

Spring work

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Moving Boundary Work This is the work associated with the expansion or

compression of a gas in a pistoncylinder device. During this process, part of the boundary (the inner face of the piston) moves back and forth. Therefore, the expansion and compression work is often called moving boundary work, or simply boundary work. Some call it the P dV work.

Moving boundary work is the primary form of work involved in automobile engines. During their expansion, the combustion gases force the piston to move, which in turn forces the crankshaft to rotate.

The moving boundary work associated with real engines or compressors cannot be determined exactly from a thermodynamic analysis alone because the piston usually moves at very high speeds, making it difficult for the gas inside to maintain equilibrium.

Then the states through which the system passes during the process cannot be specified, and no process path can be drawn. Work, being a path function, cannot be determined analytically without a knowledge of the path. Therefore, the boundary work in real engines or compressors is determined by direct measurements.

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Moving Boundary Work We analyze the moving boundary work for a quasi-

equilibrium process, a process during which the system remains nearly in equilibrium at all times. A quasi-equilibrium process, also called a quasi-static process, is closely approximated by real engines, especially when the piston moves at low velocities.

Consider the gas enclosed in the pistoncylinder device shown. The initial pressure of the gas is P, the total volume is V, and the cross-sectional area of the piston is A. If the piston is allowed to move a distance ds in a quasi-equilibrium manner, the differential work done during this process is

The total boundary work done during the entire process as the piston moves is

obtained by adding all the differential works from the initial state to the final state:

Strictly speaking, the pressure P in the above equation is the pressure at the inner

surface of the piston. It becomes equal to the pressure of the gas in the cylinder only

if the process is quasi-equilibrium and thus the entire gas in the cylinder is at the

same pressure at any given time. 6

Moving Boundary Work An idealized type of process is called a quasi-equilibrium

process; this a process in which all states through which the system passes may be considered equilibrium states. The intensive properties of the system are assumed uniform throughout the system.

To consider how a gas (or liquid) might be expanded or compressed in a quasi-equilibrium fashion, refer to the figure on the right which shows a system consisting of a gas initially at an equilibrium state. The gas pressure is maintained uniform throughout by a number of small masses resting on the freely moving piston.

Imagine that one of the masses is removed, allowing the piston to move upward as the gas expands slightly.

During such an expansion the state of the gas would depart only slightly from equilibrium. The system would eventually come to a new equilibrium state, where the pressure and all other intensive properties would again be uniform in value.

Moreover, were the mass replaced, the gas would be restored to its initial state, while again the departure from equilibrium would be slight.

If several of the masses were removed one after another, the gas would pass through a sequence of equilibrium states without ever being far from equilibrium. In the limit as the increments of mass are made vanishingly small, the gas would undergo a quasi-equilibrium expansion process. 7

Moving Boundary Work Remember that the limits on the integral

mean from state 1 to state 2 and cannot be interpreted as the values of work at these states. The notion of work at a state has no meaning as the differential of work, W, is inexact because, in general, the integral cannot be evaluated without specifying the details of the process; i.e. the integral can be evaluated only if we know the functional relationship between P and V during the process.

That is, P = f (V) should be available either as a mathematical relationship or from experimental data. Note that

P = f (V) is simply the equation of the process path on a P-V diagram.

The quasi-equilibrium expansion process described is shown on a P-V diagram opposite. The differential area dA is equal to P dV, which is the differential work. The total area A under the process curve 12 is obtained by adding these differential areas: 8

Moving Boundary Work A gas can follow several different paths as it

expands from state 1 to state 2. In general, each path will have a different area underneath it, and since this area represents the magnitude of the work; the work done will be different for each process (as shown). This is expected, since work is a path function (i.e., it depends on the path followed as well as the end states).

If work were not a path function, no cyclic devices (car engines, power plants) could operate as work-producing devices. The work produced by these devices during one part of the cycle would have to be consumed during another part, and there would be no net work output. The cycle shown on the right produces a net work output because the work done by the system during the expansion process (area under path A) is greater than the work done on the system during the compression part of the cycle (area under path B), and the difference between these two is the net work done during the cycle (the colored area).

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Moving Boundary Work

Remember that work is a mechanism for energy interaction between a system and its surroundings, and Wb represents the amount of energy transferred from the system during an expansion process (or to the system during a compression process). Therefore, it has to appear somewhere else and we must be able to account for it since energy is conserved.

In a car engine, for example, the boundary work done by the expanding hot gases is used to overcome friction between the piston and the cylinder, to push atmospheric air out of the way, and to rotate the crankshaft. Mathematically:

Of course the work used to overcome friction appears as frictional heat and the

energy transmitted through the crankshaft is transmitted to other components

(such as the wheels) to perform certain functions. But note that the energy

transferred by the system as work must equal the energy received by the

crankshaft, the atmosphere, and the energy used to overcome friction.

The use of the boundary work relation is not limited to the quasi-equilibrium

processes of gases only. It can also be used for solids and liquids.

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Moving Boundary Work As mentioned earlier, to perform the integral

requires a relationship between the gas pressure at the moving boundary and the system

volume.

This relationship may be difficult, or even impossible, to obtain for actual compressions

and expansions.

In the cylinder of an automobile engine, for example, combustion and other non-

equilibrium effects give rise to non-uniformities throughout the cylinder. Scatter might

exist in the pressurevolume data, as illustrated below. Still, performing the integral of

the above equation based on a curve fitted to the data could give a plausible estimate of

the work.

Note that the relationship between

pressure and volume during an expansion

or compression process is often described

analytically (coming up).

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Moving Boundary Work (Isochoric process)

Consider a rigid tank containing air at 500 kPa and 150C. As a result of heat transfer to the surroundings, the temperature and pressure inside the tank drop to 65C and 400 kPa, respectively. Let us determine the boundary work done during this process.

Is there a change in volume?

Note that in an isochoric process, heat is added (or removed) to change the temperature

without changing the volume.

Since a rigid tank has a constant volume,

i.e. dV = 0. Therefore:

i.e. there is no boundary work done

during this process. The boundary

work done during a constant-

volume process is always zero. This

is also evident from the P-V

diagram of the process (the area

under the process curve is zero).

Constant Volume Process

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Moving Boundary Work Constant Pressure Process (Isobaric process) Consider a frictionless pistoncylinder device which contains 10 lbm of steam at 60

psia and 320oF. Heat is now transferred to the steam until the temperature reaches 400oF. If the piston is not attached to a shaft and its mass is constant, determine the work done by the steam during this process.

Usually the isobaric expansion is accomplished by adding heat to a system. A consequence

of the process is that a total quantity of heat Q is added and the temperature increases.

The expansion process is quasi-

equilibrium. Even though it is not

explicitly stated, the pressure of

the steam within the cylinder

remains constant during this

process since both the

atmospheric pressure and the

weight of the piston remain

constant. Therefore, this is a

constant-pressure process, and,

from

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Moving Boundary Work Constant Temperature Process (Isothermal process)

Consider a pistoncylinder device which initially contains 0.4 m3 of air at 100 kPa and 80C. The air is now compressed to 0.1 m3 in such a way that the temperature inside the cylinder remains constant. Determine the work done during this process.

The compression process is quasi-

equilibrium. At the specified

conditions, air can be considered to

be an ideal gas since it is at a high

temperature and low pressure relative

to its critical-point values. For an

ideal gas the relationship is given by:

Substituting this into

we have:

PV = NRT = C or

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Moving Boundary Work As mentioned earlier, the relationship between pressure and volume during an

expansion or compression process can also be described analytically. Pressure and volume are often related by PVn = C, where n and C are constants. A process of this kind is called a Polytropic process. The pressure for a polytropic process can be expressed as:

Substituting this into

We have:

Since

. Note the following:

The equation for work, Wb, above can be written in terms of temperatures, n and R For the case of n=1 the above equation cannot be used!

For the case when n=1 and for an ideal gas equations of state assumption, the

equations used in the previous slide for an isothermal process can be used 15