## Theoretical Work

Theoretical work or compressor head is the heart and substance of compressor design. Some basic form of understanding must be devel oped even if involvement with compressors is less than that of design of the machine itself. Proper applications cannot be made if this understanding is absent. The following theoretical evaluations will be abbreviated as much as possible to reduce the length and still present the philosophy. For the reader with the ambition and desire, the presentation will be an outline to which the reader can fill in the spaces.

In deriving the head equation, the general energy Equation 2.41 will be used. The equation can be modified by regrouping and eliminating the z terms, as elevation differences are not significant with gas.

 (u V22^ f +vn it a I 2gJ — V 2gJ

The velocity term can be considered part of the enthalpy if the enthalpy is defined as the stagnation or total enthalpy. The equation can be simplified to

If the process is assumed to be adiabatic (no heat transfer), then

For the next step the enthalpy equation is written in differential form:

Recalling Equation 2.46, du = Tds - Pdv

and substituting Equation 2.46 into 2.49, dh = Tds + vdP

The process is assumed reversible. This defines entropy as constant and therefore ds = 0, making Tds = 0. The enthalpy equation is simplified to dh = vdP

For an isentropic, adiabatic process, Pvk = constant = C solving for P, P = Cv k

Taking the derivative of P with respect to v yields, dP = C(-k)v~ k ~ Mv

Substituting into the enthalpy Equation 2.51, dh = C(-k)v~ kdv

Integrating from state point 1 to 2 and assuming k is constant over the path yields, h I - h] =C

(k - l)/k Substitute C = P,v1k = P2v2k into Equation 2.55, which yields

Using the perfect gas Equation 2.1 and substituting into Equation 2.5ft yields h

As a check on the assumptions made, a comparison can be made to a different method of checking the derivation of the head. Enthalpy difference, as a function of temperature change, for an adiabatic process is expressed by h2-h,=cp (T2-T,

Specific heat cp can be calculated using specific gas constant R and specific heat ratio k.

Substitute Equation 2.61 into Equation 2.60 with the result, R(T2-T,)

This equation is identical with Equation 2.59 previously derived, giving a check on the method.

By regrouping Equation 2.59, substituting into Equation 2.48, and maintaining the adiabatic assumption Qh = 0, Equation 2.62 is developed.

RTik k-1

The -W signifies work done to the system, a driven machine, as contrasted to +W, which would indicate work done by the system as with a driver. If the adiabatic head is defined by the following equation:

and the term rp is introduced as the ratio of discharge pressure to inlet pressure.

Next, the temperature ratio relationship in Equation 2.65 will be used. This relationship is the result of combining Equations 2.6 and 2.57 as well as a half dozen algebraic steps:

When substituting Equation 2.65 into Equation 2.62, the result is the classical form of the adiabatic head equation.

An interesting note is that if in Equation 2.58, Equation 2.8 were used in place of 2.1, the result would be h,-h1 = R(T2Z2~TlZl) (2.67

Since the compressibility does not change the isentropic temperature rise, it should be factored out of the AT portion of the equation. To achieve this for moderate changes in compressibility, an assumption can be made as follows:

Javg

By replacing the values of Z2 and Zb with Zavg in Equation 2.67 and factoring, Equation 2.67 is rewritten as h2_hi3vgR(T2-Tl) (2.69)

Now with the same process used to obtain Equation 2.66, the final form of the head equation with compressibility is k , k -1

For a polytropic (reversible) process, the following definitions need to be considered;

n k rip where

Tjp = polytropic efficiency n = polytropic exponent

By regrouping Equation 2.71, a polytropic expression can be

T1p ~ k/(k - 1) By substituting n for k, the head equation becomes

One significant practical difference in use of polytropic head is that the temperature rise in the equation is the actual temperature rise when there is no jacket cooling. The other practical uses of the equation will be covered as they apply to each compressor in the later chapters.

### Real Gas Exponent

About the time it appears that there is some order to all the chaos of compressible flow, there comes another complication to worry about. It has been implied that k is constant over the compression path. The sad fact is that it is not really true. The k value has been defined in Equation

2.18 as cv

It has played a dual role, one in Equation 2.18 on specific heat ratio and the other as an isentropic exponent in Equation 2.53. In the previous calculation of the speed of sound, Equation 2.32, the k assumes the singular specific heat ratio value, such as at compressor suction conditions. When a non-perfect gas is being compressed from point 1 to point 2, as in the head Equation 2.66, k at 2 will not necessarily be the same as k at 1. Fortunately, in many practical conditions, the k doesn't change very much. But if one were inclined to be a bit more judicious about it and calculate a k at both state points, and if the values differed by a small amount, then one could average the two and never look back. This could not be done, however, with a gas near its critical pressure or one that's somewhat unruly like ethylene, where the k value change from point 1 to 2 is highly nonlinear. For a situation like this, the averaging approach is just not good enough and the following modification will be presented to help make the analysis more accurate.

To calculate a single compression exponent to represent the path from point I to point 2, the following equations will be used. Substitute y for k and Equation 2.64 into Equation 2.65

where y = compression path exponent.

The expression in Equation 2.52 can be modified to Equation 2.75 to show the basic relationship for the exponent.

To solve for y use the following equation:

To solve for the compression exponent, use a Mollier diagram to establish the T2 temperature value. By establishing a starting point at Pj, and T}, and taking a path of constant entropy to P2, the T2 value can be read from the diagram. For a gas mixture or gas with no convenient Mollier diagram available, the problem becomes more acute. There are two alternatives: one is to use an equation of state and the other is to use a method suggested by Edmister and McGarry [6]. The latter is somewhat tedious, making the equation of state the preferred method.

Power

Input shaft power is the head of the compressor multiplied by the weight flow and divided by an appropriate efficiency with the result being added to the mechanical losses. The head portion covers the fluid or thermodynamic portion of the cycle, whereas the mechanical losses cover items such as bearings and liquid seals that are not directly linked to the fluid process. The form shown here is generalized. Each compressor type has its own unique considerations and will be covered in the appropriate chapter. The adiabatic shaft work can be expressed as wH

For polytropic shaft work,

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