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Electronic Notes & Work Sheets

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 mn − . 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) 

Different blade setting angles β

Different blade setting angles β

Try to understand the meaning of the “Envelope of the optimum efficiency” under different geometrical conditions of a turbomachine.

 

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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: Sound speed______________________ Isentropic process_____________________ Stagnation enthalpy______________ __ Stagnation temperature_________________

Stagnation pressure________________ Stagnation density_____________________ Mach number_____________________ Rotational/blade Mach number___________

Specific speed vs. specific diameter. Redial type: low specific speeds with larger impeller diameters. Axial type: high specific speeds with smaller rotor diameters.

 

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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 Purposes:

• Small model data can be used to predict the performance of a full-scale prototype; • Predict the machine performance at other operating conditions.

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 efficiencies of the model and prototype are the same providing the similarity laws are satisfied.

 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).

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.

The textbook is wrong. Use the class notes.