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Engineering FIRST YEAR
Part IA: Dimensional Analysis
EXAMPLES PAPER 1
Voluntary/optional āwarm-upā questions are marked with a V.
Straightforward questions are marked with a ā , Tripos standard with a *.
V1. Using the unit conversion factors given in lectures and in the appendices of the āGuide to
Unitsā, complete the following: (a) 6 ft = .............................. m (b) 2 kg = .............................. lb (c) 70 mph = .............................. m/s (speed)
ā 2. Using the unit conversion factors given in lectures and in the appendices of the āGuide to
Unitsā, complete the following: (a) 1 year = .............................. s (b) 1 ft^3 /min = .............................. m^3 /hour (volumetric flow rate) (c) 1 kg/m^3 = .............................. lb/ft^3 (density) (d) 1 lbf/in^2 = .............................. N/m^2 (pressure or stress)
The lbf (pound force) is defined in Appendix D of the āGuide to Unitsā.
V3. Using the definitions given in the āGuide to Unitsā, find the dimensions (in the M-L-T-Ī
system) of the following quantities: (a) Energy (b) Power (c) Thermal conductivity
ā 4. Which of the following equations appear to be dimensionally inconsistent? In other words, in
which equations do the constants have dimensions?
(a) The formula used by heating contractors to determine the heating requirements of a room:
Q = 0.04 V + W + 0.33 A
Q = heat supply for room per Ā°F temperature difference between inside and outside expressed in Btu/hour Ā°F (a British Thermal Unit, Btu, is a measure of energy), V = volume of room expressed in ft^3 , W = area of windows expressed in ft^2 , A = area of external walls expressed in ft^2.
(b) The āWhiteā formula for the tension left in a straight weld joining two steel plates, on account of the shrinkage of the weld-metal:
T = 0.
v
Q
T = tension, Q = electrical power input to the welding arc, v = velocity of welding arc along the weld-line.
[Continued overleaf]
(c) The āChezyā formula for the mean flow velocity of water in a sloping pipe, the cross- section of which is not necessarily circular:
u = C L
AS
u = mean flow velocity, A = cross-sectional area of pipe, S = slope of pipe, L = āwetted perimeterā of cross-section, C = constant.
V5. You are considering a dimensional analysis problem that has N variables containing M
dimensions. What is the minimum number of dimensionless groups you can form if (a) N = 3, M = 2 (b) N = 5, M = 3 (c) N = 3, M = 3?
V6. Find as many dimensionless groups (that are independent of each other) as you can from the
following sets of variables: (a) Power P , mass m , speed V , length L (b) Pressure p , density Ļ, speed V , gravitational acceleration g , height h
- When a circular disc of material is rotating about a central axis that is perpendicular to the
plane of the disc, it will āburstā under the effects of its own inertia loading if a critical angular velocity Ļ c (rad/s) is exceeded. In tests carried out on gas turbine discs, it is found that the value of Ļ c is dependent only on the maximum stress Ļ (N/m^2 ) that the material can withstand, the density Ļ (kg/m^3 ) of the material and the radius R of the disc.
What is the form of the relationship governing Ļ c ,ļ¬ Ļ, Ļ and R? Use the elimination method to perform any dimensional analysis required.
- A construction company is required to build a long road bridge across a marsh. There are
many short spans but there is to be one long span in the middle. The figure below shows a stage of construction when all the short spans have been completed and the two arms of the long span are being extended so that they meet at the centre. The ground conditions are such that no supporting structure can be built.
The chief engineer is concerned that the individual arms of the incomplete long span may collapse under their own weight before they can be joined together, in the manner indicated by the broken lines above.
The bridge will be constructed using girders made from steel. The steel has a density Ļ of 7843 kg/m^3 and can withstand a maximum stress Ļ of 400 MN/m^2.
A simple model of one half of the long span is constructed out of aluminium. The half-span is 500 mm. The aluminium has a density Ļļ of 2720 kg/m^3 and can withstand a maximum stress Ļ of 70 MN/m^2. Using a centrifuge, it is found that the scale model collapses when the acceleration reaches 400 m/s^2. (The use of the centrifuge allows the acceleration due to gravity g applied to the model to be varied.)
What is the half-span of the largest steel bridge that can be constructed in this way using this particular design? Use the elimination method to perform any dimensional analysis required.
(c) Assuming that the four variables listed above represent a complete description of the problem (i.e. they are the only relevant independent variables), use Buckinghamās Pi Theorem to show that there must be at least one more independent dimensionless group.
Several forms are possible for this remaining group. The most useful form is when the remaining group involves only the geometry of the estuary. Suggest a suitable form for this group.
(d) Reynolds found that there was, in fact, no need to have the same horizontal and vertical scales in his model. What can now be concluded about the factors on which the formation of sandbanks in estuaries depends?
(e) To investigate in detail the tidal motion within the Mersey Estuary, Reynolds built a model with horizontal scale of 2 inches to the mile (1:31680) and vertical scale of 1 inch to 80 feet (1:960). Determine the tidal period required for the model to simulate correctly the development of the sandbanks due to tidal motion.
*11. An oil tanker is damaged and spills oil of density! o = 800 kg/m^3 onto the ocean surface; the
sea density! s is 1025 kg/m^3. Following the damage the spilt oil emerges from the tanker at a
steady rate Q of 1 m^3 /s and spreads over the ocean surface in a radially expanding surface layer of pure oil.
Model experiments are to be performed in a centrifuge at 1:1000 scale to establish the time
taken t for pollution to spread over a certain distance D. The sea water is modelled by oil of
density 900 kg/m^3 and the spilt oil by liquid of a specially selected density. The centrifuge creates an artificial gravity of 100 times that at the sea surface.
(a) Explain why the time t is the dependent variable in this problem.
(b) Under what conditions will the model test M have dynamical similarity to the real case R?
(c) In order to model the spilt oil, what density and flow rate should the fluid have in the scaled experiment?
(d) The pollution is observed in the model to travel a certain distance DM in 1 s after the
vessel ruptures. What duration does this correspond to in the full-scale case?
Suitable Past Tripos Questions:
Realistically, you should not attempt these questions until you have covered the Engineering topics
with which they are concerned in the relevant lecture course.
IA 2007, Paper 1, Q5(e); IA 2006, Paper 1, Q6(b)(i); IA 2005, Paper 1, Q4;
IA 2004, Paper 1, Q4; IA 2003, Paper 2, Q6(b), Q10(a)
Older questions that will be harder to find:
IA 2002, Paper 2, Q3; IA 2001, Paper 1, Q2(d); IA 2000, Paper 1, Q10;
IA 1998, P1, Q
ANSWERS
- (a) 1.829 (b) 4.409 (c) 31.
- (a) 31.5Ć 106 (b) 1.70 (c) 0.0624 (d) 6894
- (a) ML^2 Tā^2 (b) ML^2 Tā^3 (c) MLTā^3 Īā^1
- (a) and (c) have constants with dimensions
- (a) 1 (b) 2 (c) 0
- (a) 3 mV
PL
(b) 2 V
p
!
and 2 V
gh (other equivalent results are possible)
- Ļ cR " /! = constant
- 40.4 m (dimensional analysis gives " gs! = constant)
- 2 dimensionless groups (a) 288 m/s (b) 2.9 MN
(dimensional analysis gives v A
F
2 !
depends on v
c or other equivalent results)
- (b) the number of times waves travel up and down the estuary during a tidal period
(c) d / l (d) sandbank formation depends on { t gd l } only (e) 40 s
- Dimensional analysis gives 3 D
tQ depends on
!
2
5 , Q
gD
s
o
'
(or other equivalent results)
(b) For DS s M
o
s R
o !
!!^ =
and
R Q M
gD
Q
gD
!
2
5
2
5
(c) 702.4 kg/m^3 and 3.16Ć 10 ā^7 m^3 /s
(d) 316 s
G.T. Parks & R.S. Cant September 2007
