Homework (for 27 July 2015)

1. What is the dimensionless -group gH/N2D2 in Equation (2.64) called?

2. How is the flow velocity V related to rotational speed N and size D in

Equation (2.92)? 3. What is the relationship between the power coefficient and torque

coefficient of a turbomachine? 4. What is the relationship between torque and pressure ito pump- and

fluid parameters? 5. Which additional dimensionless number is required for dimensional

analysis of aircraft propellers? Does this apply to small model aircraft as well?

Answers to questions asked in week 2 “Dimensional analysis of Turbomachines” Q1. What is the dimensionless -group gH/N2D2 in Equation (2.64)

called? A1. Head coefficient

Q2. How is the flow velocity V related to rotational speed N and size D in Equation (2.92)?

A2.

The Buckingham – theorem may be used to obtain the required relationship (next three slides)

Buckingham – theorem

cba VDN

Obtain parameter dimensions in [MLT]-system from table 2.1

1][ TN

LD ][

1][ LTV

Dimensions of -group must be zero

cbaLTLTTL 1100 ][

Exponent equations

cacb 0;0 Solutions of exponent equations

arbitrary; ccba

c

ccc

ND

VVDN

For c = 1

NDV

V/ND is the dimensionless flow coefficient Why flow coefficient? Multiply both the numerator and denominator with D2. The numerator becomes VD2 (with dimensions of velocity times area) and the denominator becomes ND3. Both have dimensions of volume flow – see also Equation (2.74).

Q3. What is the relationship between the power coefficient and torque coefficient of a turbomachine?

A3. Divide the power coefficient by the torque coefficient (alternatively obtain the inverse)

W

τN

τN

W

DNDN

W

or

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Comment

For aircraft propellers the ratio of power coefficient to torque

coefficient is 2 [Asselin, M., “An Introduction to Aircraft Performance,” AIAA, 1997]

Q4. What is the relationship between torque and pressure ito the pump- and fluid parameters?

A4. Buckingham - theorem

edcba PDN

edcbaTMLTMLLTMLTLM 222113000

Exponent equations

edaM 0:

edcaL 230:

edbT 220:

Solutions of exponent equations

edcedbeda 52;22;

with d and e arbitrary

ed

edededed

DNDN

PPDN

5222

5222

For d = -1, e = 1

PDDN

P

DNP

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Q5. Which additional dimensionless number is required for dimensional analysis of aircraft propellers? Does this apply to small model aircraft as well?

A5. The Mach number M: Propeller tip speeds are high enough to be

in the compressible speed range and may approach sonic speed.

M depends on the size D and speed N of the propeller, and the and static temperature of the air. On small aircraft propeller tip speeds may vary widely. If the tip Mach number is below 0,3 it is unnecessary to consider compressible flow effects. However, at high tip speeds usability suffers as propeller efficiency decreases with speed (due to the so-called “wave drag”). In order to improve efficiency, propfans, which are more efficient at high speeds, are becoming popular on large aircraft. See the next slide (also shown in Week 1 “Introduction”, Figure 1.12).

Airbus A400M propfans