# How Does a Steel Bar Sink Through Ice?

• grepecs
In summary, The problem involves a steel bar of rectangular cross section placed on a block of ice with a weight hanging from each end. As the ice melts beneath the bar and refreezes, heat is released and absorbed. The speed of the bar sinking through the ice can be approximated using the Clausius-Clapeyron equation and the equation for thermal conduction. The final expression for the speed includes the coefficient of thermal conductivity, the heat flux, the difference in densities between ice and water, and the densities and latent heat of fusion.
grepecs

## Homework Statement

Question no. 4 in this document (there's a helpful picture, too):

A steel bar of rectangular cross section (height a and width b) is placed on a
block of ice (width c) with its ends extending a triﬂe as shown in the ﬁgure. A weight of mass m is hung from each end of the bar. The entire system is at T = 0° C. As a result of the pressure exerted by the bar, the ice melts beneath the bar and refreezes above the
bar. Heat is therefore liberated above the bar, conducted through the metal, and then absorbed by the ice beneath the bar. (We assume that this is the most important way in which heat reaches the ice immediately beneath the bar in order to melt it.) Find an approximate expression for the speed with which the bar sinks through the ice. Take the latent heat of fusion per gram of ice to be l, and the densities of ice and water to be ρi and ρw, respectively.

(a) Let’s say the bar sinks a distance ∆z in time ∆t. Calculate ∆U, the amount of energy
required for this to happen.

(b) The energy calculated in part (a) must pass through the bar via the process called
thermal conduction, described by the following equation

$$F=-\kappa\frac{\delta \tau}{\delta z}\approx -\kappa\frac{∆\tau}{a}$$

Here F is the heat ﬂux, i.e. total energy crossing unit area per unit time, constant κ is
the coeﬃcient of thermal conductivity, and ∆τ is the diﬀerence between the temperatures
under and above the bar. Note that F is proportional to the negative of the gradient of
temperature (as expected, since heat ﬂows from high to low τ ). Use Eq. (1) and your result from part (a) to calculate the speed of the bar, v = dz/dt, as a function of ∆τ and other given quantities.

(c) Finally, eliminate ∆τ from your result in part (b) to obtain the ﬁnal expression for v:

$$v=\frac{2mg\kappa\tau}{abcl^2\rho_i}(\frac{1}{\rho_i}-\frac{1}{\rho_w})$$

## Homework Equations

The Clausius-Clapeyron equation:

$$\frac{\delta p}{\delta \tau}=\frac{l}{\tau ∆v},$$

where v is the volume per unit mass, i.e., the inverse of the density.

## The Attempt at a Solution

The answer to question a) is simply the change in potential energy of the bar divided by time, which is

$$\frac{∆U}{∆t}=-2mg\frac{∆z}{∆t}.$$

Since the rate of energy transport through the steel bar is

$$\frac{F}{bc}=\frac{Q}{\delta t},$$

where bc is the surface area of the bar, I think I should be able to set

$$\frac{Q}{\delta t}=\frac{∆U}{∆t}$$

so that I now have

$$2mg\frac{∆z}{∆t}=\kappa\frac{∆\tau}{a}$$

From here on I'm pretty lost, though. It seems to me that since ∆z/∆t is the speed, the factor 2mg will end up in the denominator instead of in the numerator. Also, I don't really know what to do with the Clausius-Clapeyron equation. I can see that the term ∆v will eventually give me the fractions of the densities that you can see in the final expression, but it also contains a pressure term that I'm not quite sure I understand. Would that be the difference in pressure between the ice and the water? If so, should I perhaps restate it in terms of energy and volume, i.e., ∆U/∆v?

No one?

Perhaps I should state my answer to b) explicitly: the speed with which the bar sinks is

$$v=\frac{∆z}{∆t}=\kappa\frac{∆\tau}{2mga}.$$

Is this correct?

I'd really need some help.

Substituting $$∆\tau$$ in the last expression for

$$\frac{\delta p ∆v\tau}{l}$$

(a rearrangement of the Clausius-Clapeyron equation, and using the fact that δT and $$∆\tau$$ are equal), I get

$$v=\frac{\kappa bc\delta p \tau}{2mgal}(\frac{1}{\rho_i}-\frac{1}{\rho_w})$$

It kind of resembles the correct answer, but as I said, the term 2mg is in the wrong place, and I don't know what to do with δp.

## What factors affect the speed of metal melting through ice?

The speed of metal melting through ice can be influenced by several factors, including the type and thickness of the ice, the type and temperature of the metal, and the surrounding environmental conditions such as air temperature and humidity. The chemical properties of the metal, such as its melting point and thermal conductivity, also play a role in determining the speed of melting.

## What is the process of metal melting through ice?

The process of metal melting through ice is a complex one that involves several physical and chemical changes. When a metal is placed on top of ice, the ice immediately begins to melt due to the weight and warmth of the metal. As the ice melts, it creates a layer of water between the metal and the remaining ice. The metal then transfers its heat to this water, causing it to warm up and melt the surrounding ice. This process continues until the metal has fully melted through the ice.

## Why does metal melt faster through ice than through other substances?

Metal melts faster through ice than through other substances because ice has a lower melting point and thermal conductivity than most other materials. This means that ice can absorb and transfer heat more quickly, allowing the metal to melt through it at a faster rate. Additionally, the layer of water created between the metal and the ice acts as a lubricant, making it easier for the metal to slide and melt through the ice.

## Can the speed of metal melting through ice be controlled?

While there are certain factors that can affect the speed of metal melting through ice, such as the ones mentioned above, the process itself is largely uncontrollable. Once the metal is placed on the ice, it will continue to melt through at its own pace until it has completely melted through. However, certain measures can be taken to speed up or slow down the process, such as adjusting the temperature of the metal or the environment.

## What real-world applications does the study of metal melting through ice have?

Studying the speed of metal melting through ice has several real-world applications, particularly in industries such as construction and transportation. Understanding the process can help engineers and designers develop more efficient and effective methods for melting ice, such as using heat-generating materials for de-icing roads and runways. It can also aid in the development of more durable materials that can withstand extreme temperatures and prevent ice buildup.

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