Overview

Topic 1 - Radial compressors

Topic 1 - Application

Difference Between Axial Flow & Centrifugal Flow Air Compressors (Youtube, 4:30)

Also, air density changes due to variation in flow area as radius increases.  So there are multiple factors which truly come into play.

Topic 2 - Axial compressors

Let us now consider axial compressors (sometimes referred to as a “straight-through” compressor because the air goes straight-through the compressor, rather than turning from axial flow to radial flow as with an impeller).  As the constant mass flow of air is pushed back into a decreasing cross-sectional area, the pressure increases. As you go back in the compressor, the blades are smaller, because the area is smaller……which allows more of them, further allowing you to increase pressure.

The centrifugal effects on these rotating blades will create a centrifugal stress at blade root, which can be found from:

${\text{σ}}_{\text{m}\text{a}\text{x}}=\text{ρ}{\text{ω}}^2/{\text{a}}_{\text{r}}{\int }_{\text{r}}^{\text{t}}\text{a}\left(\text{R}\right) \text{R}\text{d}\text{R}$

Where

$\text{ρ}$ = mass density of blade material

$\text{ω}$ = rotational velocity

${\text{a}}_{\text{r}}$ = area of blade at root

$\text{r}$ = root radius

$\text{t}$ = tip radius

$\text{a}\left(\text{R}\right)$ = area of blade as function of radial distance from blade root

$\text{R}$ = radius

The pressure change through each individual stage of the compressor can be found from:

${\text{R}}_{\text{s}}={\text{p}}_{\text{e}\text{x}\text{i}\text{t}}/{\text{p}}_{\text{e}\text{n}\text{t}\text{e}\text{r}}={\left[1+{\text{η}}_{\text{s}}\left({\text{T}}_{\text{e}\text{x}\text{i}\text{t}}-{\text{T}}_{\text{e}\text{n}\text{t}\text{e}\text{r}}\right)/{\text{T}}_{\text{e}\text{n}\text{t}\text{e}\text{r}}\right]}^{\text{γ}\left(\text{γ-1}\right)}\mathrm{}$

Topic 2 - Application

You are designing an axial compressor blade – is it strong enough?

First,

$\text{ρ}{\text{ω}}^2/{\text{a}}_{\text{r}}=\left[\left(490 \text{l}\text{b}/\text{f}{\text{t}}^3\right){\left(12 \text{i}\text{n}/\text{f}\text{t}\right)}^3/\left(386 \text{i}\text{n}/\text{s}\text{e}{\text{c}}^2\right)\right]$ $\left[\left(15000\text{r}\text{e}\text{v}/\text{m}\text{i}\text{n}\right)\left(2\text{π} \text{rad}/\text{rev}\right)\left(\text{m}\text{i}\text{n}\text{/}60\text{s}\text{e}\text{c}\right)\right]/\left(0.5\text{i}{\text{n}}^2\right)=3622$

So

${\text{σ}}_{\text{m}\text{a}\text{x}}=\text{ρ}{\text{ω}}^2/{\text{a}}_{\text{r} }{\int }_{\text{r}}^{\text{t}}\text{a}\left(\text{R}\right) \text{R}\text{d}\text{R} =$$3622{\int }_3^6\text{a}\left(\text{R}\right) \text{R}\text{d}\text{R}$

$=3622{\int }_3^6\left[0.5-0.25\text{R}/3+0.25\left(3/3\right)\right]\text{R}\text{d}\text{R}$

$=3622{\int }_3^6\left[0.75\right]\text{R}\text{d}\text{R} -{\int }_3^6\left[0.083\right]{\text{R}}^2\text{d}\text{R }}$

$=36673-18939 = 1733\text{p}\text{s}\text{i} =17.7 \text{k}\text{s}\text{i}$

This value is the continuous stress that will be occurring at the blade root when it is operating at this condition.  By continuous, we sometimes mean that this stress is not varying.   So long as the engine runs at this condition, this stress remains constant.

However, there is the potential for another type of stress, due to vibration.  If the blade encounters a vibratory mode, there will be stresses created by that vibration which will be in addition to the stress that we just found due to the centrifugal loading.  This stress related to vibration is referred to as “dynamic” stress, because it is changing, as in, not constant.  If we refer to this as dynamics stress, then it makes sense that we sometimes refer to the earlier, continuous stress, as “static” stress.

We can consider the likelihood of dynamic stress being an issue by examining the frequencies of the blade modes (predicted by Finite Element Analysis, or from dynamic rig test data) compared against the likely excitation sources.  These likely excitation sources are the rotational frequency of the compressor rotor (referred to as engine-order or EO), and the multiple of this frequency times any blockages in the flow path, like the number of inlet struts, or the number of airfoils in the vane row just in front of the blades.  Let us assume that we have the following chart for our engine.  As the engine operates in its normal running range between idle and max speed, the number of vanes times engine order intersects the first vibrational mode of the blade near operating range, which is a concern.

Because this concern has been raised, we should probably acquire test data on the vibratory (dynamics) stresses that the blade experiences at this operating point.  We can do this by applying strain gages to the blade and running an engine test.  Typical data is shown here, indicating that at the max end of the operating range, a dynamic stress of 32 ksi was measured.

With this information, we can now go to a Goodman Diagram.  This diagram plots the allowable static stress that can be combined with a simultaneously applied dynamic stress.   The line defines the combination of stresses which would be acceptable and still provide infinite life (usually defined as 10⁶) cycles.  So let’s take the 32 ksi from the data above and go to the Goodman Diagram.  This indicates that 20 ksi static stress is allowed.  We previously determined that we had 17.7 ksi static stress, which is less than 20.  Therefore, we would anticipate that this component would not fail due to the combination of stresses up to the life limit for which the Goodman Diagram was constructed.

Had this not proven to be acceptable, we might have tried to solve the problem, by raising the frequency of the mode, thus shifting, to the right, the hump in the yellow line on the stress versus rpm plot above, so as to reduce the 32 ksi stress.  Some things that we might try to accomplish this are:

• Stiffer blade cross-section
• Stiffer blade material
• Change boundary conditions (how the blade is attached to the wheel)

Topic 3 - Blades and vanes

The flowpath of a compressor module, or turbine module, consists of both rotating blades and static vanes which are attached to the engine case. 

The vanes redirect the air flow between stages of blades, so that the flow is impacting the blades in the most effective manner.

Some additional turbine equations, relevant to this flow, are:

Efficiency equation: $∆{\text{T}}_{\text{s}\text{t}\text{a}\text{g}\text{e}}={\text{η}}_{\text{s}\text{t}\text{a}\text{g}\text{e}}{\text{T}}_{\text{s}\text{t}\text{a}\text{g}\text{e}\text{i}\text{n}\text{l}\text{e}\text{t}}\left[1-{\left({\text{p}}_3/{\text{p}}_1\right)}^{\left(\text{γ}-1\right)/\text{γ}}\right]$

Flow Coefficient: $\text{ϕ} = {\text{C}}_{\text{a}}/\text{U}$

Blade Loading Coefficient: $\text{ψ}=\left(2{\text{C}}_{\text{a}}/\text{U}\right)\left(\text{tan}{\text{β}}_2 + \text{tan}{\text{β}}_3\right)$

Reaction: $∆=\left({\text{T}}_2-{\text{T}}_3\right)/\left({\text{T}}_1-{\text{T}}_3\right)$

Many designers believe that when developing a two spool gas turbine engine, there is an advantage to designing a Vaneless Counter-Rotating Turbine.  Ponder for a moment how this would work, and what advantage it might present.

We have already discussed the occasional need for turbine blade cooling in the previous section.  Typical this cooling involves 1.5 to 2.0% of air mass flow with the cooling distribution something like : 40% blade, 30% vanes, 20% walls, 10% disk 

In consideration of this, we need to be aware of the

Blade Relative Temperature Parameter =

$\frac{{\text{T}}_{\text{b}\text{l}\text{a}\text{d}\text{e}}-{\text{T}}_{\text{c}\text{o}\text{o}\text{l}\text{a}\text{n}\text{t}}}{{\text{T}}_{\text{g}\text{a}\text{s}}-{\text{T}}_{\text{c}\text{o}\text{o}\text{l}\text{a}\text{n}\text{t}}}$

And the heat transfer relationship:

${\text{h}}_{\text{g}\text{a}\text{s}}{\text{S}}_{\text{g}\text{a}\text{s}}\left({\text{T}}_{\text{g}\text{a}\text{s}}-{\text{T}}_{\text{b}\text{l}\text{a}\text{d}\text{e}}\right)={\text{h}}_{\text{c}\text{o}\text{o}\text{l}\text{a}\text{n}t}{\text{S}}_{\text{c}\text{o}\text{o}\text{l}\text{a}\text{n}\text{t}}\left({\text{T}}_{\text{b}\text{l}\text{a}\text{d}\text{e}}-{\text{T}}_{\text{c}\text{o}\text{o}\text{l}\text{a}\text{n}\text{t}}\right)$

where: 

h is the heat transfer coefficient between the blade and either gas or coolant

S is the wetted perimeter of either the blade profile or coolant passage

And the temperature differential along blade length, L:

$\frac{\text{d}{\text{T}}_{\text{c}\text{o}\text{o}\text{l}\text{a}\text{n}\text{t}}}{\text{d}\text{L}}=\frac{\left(1 + {\text{h}}_{\text{g}\text{a}\text{s}}{\text{S}}_{\text{g}\text{a}\text{s}}/{\text{h}}_{\text{c}\text{o}\text{o}\text{l}\text{a}\text{n}\text{t}}{\text{S}}_{\text{c}\text{o}\text{o}\text{l}\text{a}\text{n}\text{t}}\right)\text{d}{\text{T}}_{\text{b}\text{l}\text{a}\text{d}\text{e}}}{\text{d}\text{L}}$

When blades are relatively short, they are usually unshrouded, that is, they are like simple cantilever beams, extending outward from the turbine wheel.   However, as blades get taller, they have an increased tendency to have their bending mode at a lower and lower frequency, possibly dropping into the operating range, which we have already shown can be problematic.  When this occurs, we can raise the frequency of this mode, by designing a shroud at the top of the blade, such that the shrouds from adjacent blades contact, constraining the blade at both top and bottom rather than being like a simple cantilever.  When even more constraint is needed, the shroud may be configured in a Z shape, rather than a straight cut edge.

We have already discussed why it is necessary to be able to assess a blade’s natural frequency.  There are a few ways to do this, usually involving a shaker rig, which excites the blade at a range of frequencies.  The actual blade response can be measured by Laser Holography.

A hologram is made of the object in its free, unstressed condition. Then the object is stressed and a new hologram made. The stressed hologram is viewed through the original unstressed hologram, and the superposition provides the interference fringe pattern that would have been produced by a double exposure.

By such means, time variations can be studied.  When hologram interferometry is applied to the examination of vibrations set up in a rapidly rotating turbine blade, stroboscopic techniques aid the analysis.

The laser light is stroboscopically interrupted at the same frequency as the rotation of the turbine blade, and, with the blade thus apparently at rest, a hologram is produced.

Consequently, a holographic interference pattern is created for the blade whose motion is stopped by stroboscopic action. By slightly altering the frequency of the stroboscope arrangement, a slow scan can be made over the complete vibrational stress pattern to which the blade is subjected. Much information about stresses in turbine blades and other rotating or vibrating objects can be obtained from such holograms.

Material Selection is Critical for Turbine Blade Life.  The blades can be made from metal alloys or high temperature ceramics.  Their external surface characteristics can be altered by the addition of blade coatings.  

Metal blades can also be grown from metallic crystals.  We do not normally think of metals as being crystalline structures…..but they are.  They can be grown as equi-directional (or random directional) crystal structures, as uni-directional crystal structures (all crystals form in the same direction), or as one large single crystal.  These forms are progressively difficult to achieve, but they also produce increasingly better material properties, so can be worth the trouble.

The Evolution of Jet Engine Turbine Blades (Youtube, 5:44)

Task

Reference

Langston, L. & Opdyke, G. (1997) “Introduction to Gas Turbines for Non-Engineers,” Global Gas Turbine News, Vol 37, No. 2.

Langston, L. (2013). “Powering Out of Trouble”  Mechanical Engineering, Vol 135, No. 12, doi: 10.1115/1.2013-DEC-3.  

Saravanamutto, H., Rogers, G., Cohen, H. & Strznicky, P. (2009).  Gas Turbine Theory.  Essex, UK: Pearson.

Sforza, P. (2011). Theory of Aerospace Propulsion. New York: Butterworth-Heinemann.

Wilson, D & Korakianitis, T. (2014).  The Design of High-Efficiency Turbomachinery and Gas Turbines.  Cambridge, USA:  MIT Press.