This article presents three different gas turbine phenomena and design cases. The sketch in the article shows a schematic of a combined cycle powerplant consisting of a Brayton cycle (gas turbine) whose exhaust provides energy to a Rankine cycle (steam turbine). Frequently, one can use simple but exact one-dimensional (1D) heat conduction solutions to estimate the heat loss or gain of gas turbine components under transient conditions. These easy-to-use solutions are found in most undergraduate heat transfer texts. The article suggests that those three widely different gas turbine phenomena and design cases all have the simple, nonlinear superposition form.

## Article

#33 March 2018

A Useful Equation for Gas Turbine Design

“The essence of mathematics is not to make simple things complicated, but to make complicated things simple.” [1]. Certainly we as engineers know this to be the case!

With this clear statement in mind, I would like to go over a simple mathematical equation that explains some gas turbine phenomena and design in a straightforward and elegant way. This equation is written in terms of the variables X1, X2 and X12, where 0≤X≤1.0 and X12 is given by
$X12=X1+X2−X1X2$
(1)

I have come to call (1) a nonlinear superposition expression, since it is composed of a sum of X1 and X2 (superpositioning) minus the nonlinear product X1X2. A plot of X12 yields a segment of the surface of a hyperbolic paraboloid, bounded by 0≤X≤1.0.

## Combined Cycle Powerplant Efficiency

One practical use of Eq. (1) results in an expression for the efficiency of a combined cycle gas turbine/steam powerplant. The sketch in Fig. 1 shows a schematic of a combined cycle powerplant consisting of a Brayton cycle (gas turbine) whose exhaust provides energy to a Rankine cycle (steam turbine).

Using just the First Law of Thermodynamics and the definition of thermal efficiency (useful energy output/costly energy input), the combined cycle efficiency, ηcc, becomes (Horlock [2])

$ηCC=ηB+ηR−ηBηR$
(2)

Where ηB = WB/Qin and ηR = WR/QBR are the thermal efficiencies of the Brayton and Rankine cycles respectively. Thus ηcc is directly analogous to the general nonlinear superposition equation given in (1).

Taking ηB = 40% and ηR = 30%, the sum minus the product in Eq. (2) yields ηcc = 58%, a value of combined cycle efficiency greater than either of the individual efficiencies -- which is the secret of success of this modern power plant.

## Film Cooling Effectiveness

Since at least the 1960s, film cooling, the use of air films bled from the compressor and directed onto the external surfaces of superalloy turbine airfoils and combustion liners, have been utilized by gas turbine designers. This has allowed thermal efficiency enhancing turbine temperatures to be dramatically increased, far above superalloy melting points.

The non-dimensional heat transfer variable e, (where 0≤e≤1.0) is called the film cooling effectiveness. It is used by gas turbine designers to define heat transfer coefficients for film cooling heat load calculations that are independent of gas temperature differences. In practice, values of e start at 1.0 at the injection site, and decrease rapidly downstream of coolant slots or rows of holes, due to the strong turbulent dispersion of the film.

Thus, multiple film cooling rows, introduced by slots or rows of holes mounted sequentially in a stream-wise direction may be required. An additive method, first developed by Sellers [3] has been widely used to account for values of e downstream of multiple cooling film injections, treating the coolant as confined stratified layers. Using the Sellers model, the resulting film cooling effectiveness, e12 of two rows of slots or holes (see Fig. 2) is given by

$e12=e1+e2−e1e2$
(3)

where e1 and e2 are the values of e for each row individually. The effect of a third row, e3, can be additively accounted by treating (3) as one row and obtaining a like expression for e123, and so on, for additional rows.

Thus, we see that (3), for gas turbine film cooling, has the same nonlinear superposition form as Equ.(1).

## Engine Component Heat Transfer Estimates

Frequently, one can use simple but exact one-dimensional (1-D) heat conduction solutions to estimate the heat loss or gain of gas turbine components under transient conditions. These easy-to-use solutions are found in most undergraduate heat transfer texts.

As an example, consider the case of estimating the amount of heat transfer per unit span, Q, that occurs to a high pressure compressor blade during an engine surge, relative to its stored internal energy Qo (based on the surge gas temperature, T. (A surge is the sudden reversal of flow in the compressor, accompanied by a sudden rise in compressor gas path temperatures [4].).

We consider a low cambered high compressor blade to be modeled as a rectangular bar, with a chord of 2L1 and an average thickness of 2L2 . The transient temperatures in this 2-D bar is then given by the product of the exact 1-D solutions [5] of transient conditions in two infinite plates of thickness 2L1 and 2L2.

It follows [5] that the per unit span heat transfer to the compressor blade, Q, is then given by
$QQ0=Q1Q01+Q2Q02−Q1Q01+Q2Q02$
(4)

where the subscripts 1 and 2 refer to the 1-D exact solutions for the two infinite plates. Again, (4) has the same form as (1).

## Conclusion

In summary, we see that three widely different gas turbine phenomena and design cases given by Equs. (2), (3) and (4), all have the simple, nonlinear superposition form, given by Equ. (1). The reader might try to find other situations that are described by Equ. (1), assuredly making complicated things simple, by mathematics!

## References

References
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Gudder
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Stanley
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Horlock
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J.H.
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