Chapter 2. Dimensional Analysis of Turbomachinery 1. SI Units

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Chapter 2. Dimensional Analysis of Turbomachinery
1. SI Units
• There are 7 SI base/primary units in physics and engineering. Other quantities, called derived
quantities, are defined in terms of the seven base quantities via a system of quantity equations.
• In thermodynamics, we frequently use 4 units (marked with red colour).
• In this course, only three base/primary units (mass, lengthand and time) are used in analysis of a
“pure mechianical system” (in which the density, temperature and enthalpy of the working fluids
do not change). In dimensional analysis, we use “M”, “L”, and “t” to represent these three base
dimensions, repectively.
• In a thermal-mechanical system (e.g., where temperture varies, heat transfer is involved, and fluid
is compressible), the fourth base unit (temperature) needs to be considered. In dimensional
analysis, we use “T” to represent this base dimension.
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Note that there is an analogy between the momentum and the angular-momentum systems, indicated
using * and **, respectively:
mass ~ moment of inertia, velocity ~ angular velocity, force ~ torque,
acceleration ~ angular acceleration, momentum ~ angular momentum.
2. Buckingham Pi Theorem (Peng, 2008)
Theorem: the Buckingham pi theorem states that the number of independent nondimensional parameters
needs to correlate the variables for a given process is n − m .
Here, n is the number of relevant dimensional parameters involved and m is the number of primary
dimensions, i.e. (M, L, t or F, L, t) for a pure mechanical system; and (M, L, t, T or F, L, t, T) for a
thermal-mechanical system.
Procedure:
See page 19 in Peng (2008) for the 10-step method to perform a dimensional analysis based on the
Buckingham pi theorem. These 10 steps can be further summarized as:
1. Determine n and m, and the functional relation of the quantities: List the n physical quantities (Qn)
with dimensions and the m primary/base dimensions. There will be (n-m) π-terms.
2. Select the primary quantities: Select m number of these quantities, none dimensionless and no two
having the same dimensions. All primary dimensions must be included collectively in the quantities
selected.
3. Write down the π-terms: Express each π-term as the product of the selected quantities (each to an
unknown exponent) and one remaining (unused) quantity (to a known power usually taken as unity).
4. Group exponents: Group the exponents for the same primary dimension in each π-term.
5. Determine the exponents: Solve for the unknown exponents, and then report the final formulae of the
π-terms and the nondimentionalized functional relation. Try to analyze their physical meanings.
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3. Key Concepts in Dimensional Analysis for Turbomachinery (pp. 19-27,
Peng, 2008)
Write down the definitions of the following key concepts. Try to thoroughly understand the physical
meaning of these parameters, and their applications in turbomachinery.
Work____________________________ Specific work_____________________
Head____________________________ Efficiency (general)________________
Lift coefficient____________________ Drag coefficient___________________
Head coefficient___________________ Flow coefficient___________________
Reynolds number__________________ Power coefficient__________________
Speed ratio_______________________ Unit power_______________________
Specific speed_____________________ Specific diameter__________________
The Moody diagram shows that the drag coefficient is a function of the Reynolds number. It also
demonstrates the advantage to use non‐dimensional parameters in the analysis‐‐‐all experimental
results of different types of sand grain rough‐wall flows collapse.
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Variable Geometry Turbomachines (Dixon, 2005)
Try to understand
the meaning of the
“Envelope of the
optimum efficiency”
under different
geometrical
conditions of a
turbomachine.
Different
blade setting
angles β
Different
blade setting
angles β
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Efficiency of the turbomachinery vs. the specific speed
Specific speeds
characteristic of
different types of
turbomchinery.
For the fixed turbomachine, when the
Reynolds number effects are ignored,
the efficiency is approximately a
function of the specific speed.
There is an optimum specific speed
which corresponds to the highest
efficiency of a fixed turbomachine.
Efficiency vs.
specific speed.
Volume flow rates
vs. specific speed
for different types
of turbomachinery.
It shows the range
of specific speeds
and efficiencies for
a turbomachine.
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4. Compressible Fluid Analysis (pp.15-16, Dixon, 2005; Case 3 on p.22,
Peng, 2008)
4.1 Compressibility
Compressible = specific volume is changeable (or, density is changeable)
True or false?
If we put a huge pressure on a substance, it must become compressible.
Your answer is____
If the total volume of a certain amount of substance can change, then this substance is compressible.
Your answer is____
4.1 Stagnation properties:
Specific speed vs. specific
diameter.
Redial type: low specific speeds
with larger impeller diameters.
Axial type: high specific speeds
with smaller rotor diameters.
Sound speed______________________ Isentropic process_____________________
Stagnation enthalpy______________ __ Stagnation temperature_________________
Stagnation pressure________________ Stagnation density_____________________
Mach number_____________________ Rotational/blade Mach number___________
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The Mollier (h-s) diagram is useful in analyzing the performance of an adiabatic steady-flow
process, such as the flows in nozzles, diffusers, turbines and compressors.
Temperature-Entropy (T-S) Diagram
Water
Questions:
1. Show the constant p lines
(isobaric processes), constant
s lines (isentropic processes),
and constant enthalpy lines
(constant h processes).
2. Show where the two-phase
region is and where the
vapour region is.
3. Show the direction of
compression and expansion
between two pressure lines.
4. Can a gas turbine run in a
two-phase region? Why?
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5. Similitude
Similitude is a concept used in the testing of
engineering models. A model is said to have
similitude with the real application (full-size
prototype machine) if the two share geometric
similarity, kinematic similarity and dynamic
similarity. Similarity and similitude are
interchangeable in this context.
Purposes:
• Small model data can be used to predict the performance of a full-scale prototype;
• Predict the machine performance at other operating conditions.
Geometrical Similarity:
(L/D)m = (L/D)p
• Linear dimension ratios are the same
everywhere;
• Photographic enlargement.
Kinematic Similarity:
m = p;
(V1/V2)m=(V1/V2)p
• Same flow coefficients;
• Same fluid velocity ratios (triangles)
are the same.
Dynamic Similarity:
ψm = ψp,
(F1/F2)m=(F1/F2)p
• Same head/power/loading coefficients;
• Same force ratios (and force triangles).
The efficiencies of the model and
prototype are the same providing the
similarity laws are satisfied.
Example 1 (the same machine at different operating conditions. Example 2.1S, pp.29, Peng, 2008)
Example 2 (two machines: model and prototype machines. Example 1.1, pp.35, Gorla and Khan, 2003)
The textbook is
wrong. Use the
class notes.
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