Lab #9: Dimensional Analysis and Similitude
Introduction
Dimensional analysis is one of the powerful tools in the determination of relationships between physical parameters. This technique involves manipulation and comparison of the units of physical features so as to achieve the desired relationship, whose units conform to the expectation of a quantity being derived. On the other hand, similitude refers to the a technique where a real life structure or component is reproduced by at a lower scale such that every dimension is drawn to the scale and at a constant proportion among all members of a structure. This technique assumes that a model behaves in a similar manner as a prototype and the results of a model can be scaled back to obtain the attributes of a real life structure (Gibbings 7). Using dimensional analysis and similitude, the discharge over a dam is expressed as
Q=KH32wg12
Whereas the relationship between the model and prototype is;
QmQp=LmLp52
Apparatus and Material
- Water channel
- Norris dam model
- Actual measurements of Norris dam
- A Ruler
Procedure
- Slope indicator was adjusted to zero mark
- After ensuring that all the dials were off, the apparatus was plugged in and levelled
- The tailgate was made horizontal by adjusting the knob at the control panel
- The width of the Norris dam model was obtained
- The model dam was placed on the channel and attached with screws
- The orifice knobs were opened slowly and pump turned on until when the largest possible difference in head was realized.
- The value of the head difference was recorded
- Steps 1 to 7 were repeated for four more trials.
- The obtained values were used to calculate the rate of discharge in the model, Qm and in the prototype, Qp.
Calculations and Analysis of Data
Dimensional Analysis
- Height of water above the dam, Hm
- Experimental values of K
Qm=bgyc312
Using the obtained values of Qm, the values of K were calculated using the relationship;
K=Qm(H)32w(g)12
- Flow over prototype, Qp using prototype characteristics
Qp=KH32wg12
Where;
Inserting the values in the equation,
Qp =1420.5 m3/s
- Flow over prototype, Qp using similitude method
QmQp=LmLp52
For the Norris dam, the ratio of Lm/Lp is 1/200; Replacing these values in the equation above;
QmQp=1.7678×10-6
Qp=Qm1.7678×10-6
Using the average value of Qm, Qp=0.00711.7678×10-6=4003.8 m3/s
- Comparison of flow over prototype, Qp values
Graph of Data
Qm vs. (Hm) 3/2w (g) 1/2
In the graph of Qm vs. (Hm) 3/2w (g) ½, the linear relationship was obtained;
y=0.3261x+0.0015
Discussion
The dimensional analysis and similitude carried out in this experiment managed to obtain varying values of flow rates in the prototype and in the model. The values of constant K were found to be 1.6880. Notably, there was a big difference in the values of flow rates obtained in the prototype and similitude methodology. This difference is attributed to the systematic errors associated with carried out the experiment. In the dimensional analysis techniques, the flow rate was found to be 1420.5m3/s whereas the similitude showed that the flow was 4003.8 m3/s. The huge difference noted in these two values of flow rate between the two methods is in contrary to the expectations of this experiment. It was anticipated that the values of flow rate in a prototype would be the same or close in both dimensional analysis and similitude.
When a graph of Qm vs. (Hm) 3/2w (g) ½ was plotted, a linear relationship whose gradient is 0.3261 was obtained. As compared to the value of K obtained in the dimensional analysis, 1.6880, there is a notable difference between the two values which were expected that they were going to be the same.
One of the possible errors that might have led to the difference in calculated values is failure to take accurate measurement in the dynamic upstream heights at respective intervals. Secondly, there is a big assumption that the model created in the laboratory to estimate the rate of flow in the Norris dam is exactly similar to the actual prototype in terms of critical measurements and in their proportions. This difference could not allow the behaviour of water flowing in the model to exhibit exactly similar behaviour when flowing through the prototype.
In conclusion, the experiment showed that there were fundamental errors which led to a massive disparity in the values of prototype flow rate in both similitude and in dimensional analysis methods. Furthermore, the value of a constant, K was too big, which indicate that the errors interfered immensely with the results of the experiment. Thus, a correction in the design of both design and construction of the prototype may help in maximization of the dimensional analysis and similitude techniques of comparing the actual flow rates. Notwithstanding the differences noted in the values of calculated flow rates, it is evident that the experiment was successful as it managed to obtain values.
Works Cited
Gibbings, John Cecil. Dimensional analysis. Springer, 2011. Print
