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(See The Wide World of Fluids article titled "Ice Engineering." Section \(7.9 .3 .)\) A model study is to be developed to determine the force exerted on bridge piers due to floating chuaks of ice in a river. The piers of interest have square cross sections. Assume that the force, \(R\), is a function of the pier width, \(b\), the thickness of the ice, \(d\), the velocity of the ice, \(V\), the acceleration of gravity, \(g,\) the density of the ice, \(\rho_{i},\) and a measure of the strength of the ice, \(E_{i},\) where \(E_{i}\) has the dimensions \(F L^{-2}\) (a) Based on these variables determine a suitable set of dimensionless variables for this problem. (b) The prototype conditions of interest include an ice thickness of 12 in. and an ice velocity of \(6 \mathrm{ft} / \mathrm{s}\). What model ice thickness and velocity would be required if the length scale is to be \(1 / 10 ?(\mathrm{c})\) If the model and prototype ice have the same density, can the model ice have the same strength properties as that of the prototype ice? Explain.

Step by step solution

01

Identify the Dimensionless Parameters

To form the dimensionless parameters, we need to apply the Buckingham Pi theorem. This problem includes 7 variables (\(R, b, d, V, g, \rho_{i}, E_{i}\)) and 3 fundamental dimensions (mass \(M\), length \(L\), time \(T\)), so we can obtain 4 dimensionless parameters. To simplify the process, we can choose \(b, g,\) and \(\rho_{i}\) as recurring variables.

02

Calculate Model Ice Thickness and Velocity

If the scale of the model is 1/10th, the length ratio \(L_{r}\) is \(1/10\). Therefore, the model ice thickness \(d_{m} = d_{p} / L_{r} = 12 \, \text{inches} / 10 = 1.2 \, \text{inches}\). The velocity ratio is equal to \(\sqrt{L_{r}}\) , so the model ice velocity \(V_{m} = V_{p} / \sqrt{L_{r}} = 6 \, \text{ft}/s / \sqrt{10} = 1.9 \, \text{ft}/s\) approximately.

03

Analyze the Strength Properties of the Model Ice

If the model ice has the same density as the prototype ice, the model ice cannot have the same strength properties as that of the prototype. This is because the strength of the ice, \(E_{i}\) is a property that also depends on ice's microstructure, its impurities, and other factors. Therefore, it wouldn't be correct to assume that two ice samples with the same density have the same strength as well.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Buckingham Pi theorem

The Buckingham Pi theorem is an essential principle in dimensional analysis. It helps us determine dimensionless parameters, which simplify complex physical problems. The theorem states that you can describe a system with fewer dimensionless variables than the original variables by identifying repeating or fundamental dimensions. For example, if a problem involves 7 variables and 3 fundamental dimensions like mass \(M\), length \(L\), and time \(T\), you can derive 4 dimensionless groups or parameters.
In the case of the ice engineering problem, the force exerted on the bridge piers \(R\) depends on dimensions such as the width of the pier, the ice thickness, the velocity, gravity, density, and ice strength. By applying the Buckingham Pi theorem, we select recurring variables (e.g., \(b, g, \ ho_i\), simplifying our task of finding dimensionless parameters.

Dimensionless Parameters

Dimensionless parameters are a vital tool in engineering and science, allowing the comparison of different systems on a simplified scale. They are formed by aggregating variables such that all units cancel out, resulting in pure numbers with no physical units. This simplification can be crucial for understanding how different systems behave under similar conditions.
In our problem, after applying the Buckingham Pi theorem, we formulated 4 dimensionless numbers which describe the system fully even if the individual variables have different scales or units. These dimensionless groups are crucial for drawing parallels between model studies and real-world scenarios, ensuring our models behave like their full-scale counterparts when subjected to similar dimensionless conditions.

Model Scaling

Model scaling involves creating a smaller representation of something physical to study how it behaves. It is particularly useful in civil engineering and structural analysis for experimenting with miniature versions of full-scale systems or constructions. Scaling factors, such as \(1/10\), ensure that physical phenomena like force or velocity scale correctly so that we can safely apply the model’s results to the real system.
In the ice engineering problem, we calculated the model's thickness and velocity based on the prototype by using length and velocity ratios. The length ratio \(L_r = 1/10\) and the velocity ratio \((V_m = V_p / \sqrt{L_r}\))\ allowed us to maintain the system's dynamic similarity, a crucial aspect when validating models against real-world counterparts.

Strength of Materials

The strength of materials involves understanding how different materials behave under various forces and impacts. It's a critical field that examines properties such as elasticity, plasticity, and toughness, to predict how materials respond to stresses. This concept is fundamental in predicting the failure points and ensuring safety in engineering designs.
In our exercise, we analyzed whether the model ice could have similar strength properties as the prototype. We observed that even if ice densities are the same, the strength \(E_i\) largely depends on its microstructure, making it impossible to assume identical strength merely based on density. Therefore, while density might match, microscopic and compositional factors mean strength properties between model and prototype do not necessarily align.