Real thermodynamic systems

In a real system there are inevitably losses such that the con­version process is less than 100% efficient.

The Second Law of Thermodynamics therefore states that:

It is impossible for a system to produce net work in a ther­modynamic cycle if it only exchanges heat with sources /sinks at a single fixed temperature.

This Law is based on a principle proposed by Clausius. He stated that heat flows unaided from hot to cold but cannot flow, unassisted, from cold to hot. Lord Kelvin used the proposal to show that work may be completely transformed into heat. How­ever, only a proportion of heat could be transformed into work. If a gas is heated at constant volume there will be no work done but the energy level of the gas will be increased thus:

Q = mcv(T2 — T^ Equ 3.2

= m(U2-U1) where

Q = heat transferred (kJ)

M = mass of gas (kJ)

Cv = specific heat capacity at

Constant volume (kJ/kg. k)

T2 = final absolute temperature (k)

Ti = initial absolute temperature (k)

U2 = final specific internal energy (kj/kg)

Ui = initial specific internal energy (kJ/kg)

Note: There is no degree symbol associated with the abso­lute temperature. Absolute temperatures in Kelvin can be converted to degrees Celsius by subtracting 273.15.

Specific heat capacity is normally abbreviated to specific heat. It is easy to see that specific internal energy, Ui is equal to the product cv and the absolute temperature, internal energy is an intrinsic property of a gas and is dependent upon the tempera­ture and pressure. In this case it would have been possible to use degrees Celsius to obtain the same result.

However it is worthwhile working in absolute temperatures con­sistently to avoid problems with rations. If a gas is restrained and applied at constant pressure there will be work done, thus:

Q = mcv(T2 — T,) + W Equ 3.3

= mcp(T2 — T,)

= m(h2-h1) so that:

W = m[(h2-U2)-(h1-U1)]

Also

W=p(V2-V1)

And:

H = U + pv where:

W	= work done (kJ)
Cp	= specific heat capacity at constant pressure (kJ/(kg. K))
H2	= specific enthalpy (kJ/kg)
Hi	= specific enthalpy (kj/kg)
P	= absolute gas pressure (kPa)
V2	= final gas volume (m3)
V — i	= initial gas volume (m3)
V	= gas specific volume (m3/kg)

Absolute pressures are gauge pressures plus 101.325 kPa. The International Standard Atmosphere, at sea level, is

101.325 kPa. The actual local sea level atmospheric pressure is not constant and will vary with the weather by +/- 4%. some locations which experience severe weather conditions may ex­perience larger variations. The atmospheric pressure will re­duce at altitudes above sea level.

Enthalpy is an intrinsic property of a gas and is dependent upon the temperature, pressure and volume. The total enthalpy in a system, H, is the product of gas mass, m, and the specific enthalpy, h. Equation 3.3 can be rewritten as shown in

Where: Q K A Th To

Equ 3.7

Equation 3.4 when it is known as the Non-flow energy equation. U is the product of m and u.

Note: The specific heat capacities, cv and cp1 are variables not constants. The values for dry air, not real air, at at­mospheric pressure and 275 K are 0.7167 and 1.0028; at 1000 K the values increase to 0.854 and 1.411.

Q = (U2-U1) + W Equ 3.4

For heat to be transferred into or out of a system a temperature differential must exist. The general equation for heat transfer by conduction is thus:

3 Ka(Th-Tc)

Equ 3.5

L

= energy transfer (kW)

= thermal conductivity (kWm/(m2K)) = area (m2)

= hot absolute temperature (K)

= cold absolute temperature (K)

L = length of conductive path (m)

The thermal conductivity, k, will not be a simple value based on the boundary material. The conductivity value used must take account of the inside and outside boundary layer films and, if necessary, an allowance made for the reduction in conductivity due to surfaces being coated with deposits or modified by corrosion.

It will be appreciated that the rate of heat transfer due to con­duction is proportional to the temperature differential. If the heat source cools as transfer proceeds it will take an infinite length of time to transfer all the heat available providing there are no losses. Energy losses usually occur via convection and radiation and by heating the system as well as the gas. Perfect systems are massless; only the mass of the working fluid is considered.

Entropy is another intrinsic property of gases. Entropy is very unusual when compared to other gas properties; entropy only changes when heat transfer occurs. Entropy is not dependent upon temperature, pressure or volume. A change in entropy is defined as: due to change in height above some datum:

-pg фs sin 0 SA = — wфHфA Rate of change of momentum in direction of flow = p5Av(v+ Sv) — pSAv2 = pфAvфv

Thus

-фpфA = pSHSA = pSAvфv and rearranging

V&v + — + g5H = 0 P

Which in the limit becomes

Vdv + — + gdH = 0

P

On integration, this gives

-2 ………

— gH = constant

2 J p

H is measured from any arbitrary datum, and any change of da­tum results in a change in H and an equal change in the con­stant of integration. If the air is considered as incompressible, which is acceptable for fan pressure below about 2.0 kPa, then equation 3.7 reduces to

2

— + — + H = constant, known as Total Head Equ 3.8 2g pg

Although strictly only applicable to flow along a stream tube of an ideal frictionless fluid, equation 3.8 is often used to relate conditions between two sections in a practical system of flow through a duct. If the mean total head is measured at the two sections, it will be found that the value at the downstream sec­tion is less than that at the upstream section. This is due to re­sistance to flow between the sections and the difference in head is known as loss of total head. When making measure­ment however, it is customary to use gauge pressure, i. e. pres­sures greater or less than atmospheric pressure.

Considering two sections, subscript 1 referring to the upstream section and subscript 2 referring to the downstream section, then

2g

P9

Pg

Pat1 + Pl + H1 = + Pal2 +-P2 + H2 + AH Equ 3.9

2g

. dQ ds = — T

Equ 3.6

Where Ds

= change in entropy (kJ)

DQ = heat transfer (kJ)

T = absolute temperature (K)

The units for specific entropy, s, are kJ/(kg. K). Values of intrin­sic properties: ui hM s; are quoted in gas tables and appear on the axes of gas charts. It is very important to verify the base temperature of printed data before starting calculations. Some gases use 0 °C and some, like refrigerants, use — 40°C.

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