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A spherical ice cube (an "ice sphere") that is \(6 \mathrm{cm}\) in diameter is removed from a \(0^{\circ} \mathrm{C}\) freezer and placed on a mesh screen at room temperature \(T_{a}=20^{\circ} \mathrm{C} .\) What will be the diameter of the ice cube as a function of time out of the freezer (assuming that all the water that has melted immediately drips through the screen)? The heat transfer coefficient \(h\) for a sphere in a still room is about \(3 \mathrm{W} /\left(\mathrm{m}^{2} \cdot \mathrm{K}\right) .\) The heat flux from the ice sphere to the air is given by $$\mathrm{Flux}=\frac{q}{A}=h\left(T_{\mathrm{a}}-T\right)$$ where \(q=\) heat and \(A=\) surface area of the sphere. Use a numerical method to make your calculation. Note that the latent heat of fusion is \(333 \mathrm{kJ} / \mathrm{kg}\) and the density of ice is approximately \(0.917 \mathrm{kg} / \mathrm{m}^{3}\).

Step by step solution

01

Calculate initial surface area and volume of the ice sphere

Given the diameter of the ice sphere is \(6 \mathrm{cm}\), we can compute the radius \(r\) of the sphere as: \[r = \frac{6}{2} \mathrm{cm} = 3 \mathrm{cm}\] The surface area \(A\) and volume \(V\) are given by: \[A = 4\pi r^2\] \[V = \frac{4}{3}\pi r^3\]

02

Write an equation for the heat flux and find the heat rate

Heat flux is given by: \[\mathrm{Flux}=\frac{q}{A}=h\left(T_{a}-T\right)\] The heat transfer rate \(\dot{q}\) is the amount of heat transferred per unit time; this can be written as: \[\dot{q} = h A\left(T_{a}-T\right)\]

03

Write an equation for melting the mass of ice

The mass \(m\) of the ice sphere is given by: \[m = \rho V\] where \(蟻\) is the density of the ice. The energy required to melt the mass of ice is given by: \[Q = mL\] where \(L\) is the latent heat of fusion, and \(Q\) is the energy needed for melting the ice. From the heat transfer rate, we get: \[Q = \dot{q}t\] where \(t\) is time. So, mass equation: \[mL = \dot{q}t\]

04

Relate the melting mass of ice to time and volume

From the mass equation, we can solve for time: \[t = \frac{mL}{\dot{q}}\] Substituting the previous expression for the heat transfer rate: \[t = \frac{mL}{hA(T_{a} - T)}\] Now, let \(V_m\) be the volume of the melted ice. The volume and mass of the melted ice can be related by: \[V_m = \frac{m}{蟻}\] From the relationship between time and mass, we can write: \[t = \frac{V_mL蟻}{hA(T_a - T)}\]

05

Calculate the change in diameter over time

We need to find the relationship between the change in the diameter of the ice sphere and time. Let the new diameter be \(x\), then the new volume \(V'\) is: \[V' = V - V_{m}\] \[V' = \frac{4}{3}蟺{\left(\frac{x}{2}\right)}^3\] Having the relationship between \(V_m\) and time we can find the diameter \(x\) as a function of time: \[x(t) = 2{\left[3\frac{V - V_{m}}{4蟺}\right]}^{\frac{1}{3}}\] This equation gives us the diameter as a function of time.

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Key Concepts

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

Latent Heat of Fusion

The latent heat of fusion is the amount of heat required to change a solid into a liquid at its melting point without changing its temperature. This concept is essential in understanding how ice melts. When a spherical ice cube is placed in a warmer environment, it absorbs heat until it begins to melt. The energy needed to break the bonds holding the ice molecule structures together is called latent heat of fusion.

For ice, this value is approximately 333 kJ/kg. This means that to melt 1 kilogram of ice at 0掳C into 1 kilogram of water at 0掳C, 333 kilojoules of energy are required. This absorbed energy doesn't raise the temperature of the ice but alters its state.

In our context, knowing the latent heat of fusion helps estimate the time needed to melt the ice sphere completely as we factor this energy into our calculations.

Numerical Methods

Numerical methods are techniques aimed at solving mathematical problems numerically rather than by symbolic manipulation. They're especially useful when dealing with complex equations or simulations that cannot be easily solved by hand.

When determining the diameter of an ice sphere over time, numerical methods can be applied to solve the heat transfer equations. Since the melting rate depends on the heat transfer rate and the latent heat, computations are required to iterate and find precise changes over small time increments.

These methods often involve calculating repeated iterations until the outcome stabilizes at an approximate solution. Numerical techniques provide a practical way to explore heat transfer problems without getting bogged down by unrealistic simplifications.

Density of Ice

The density of a material is its mass per unit volume. For ice, its density is approximately 0.917 kg/m鲁, meaning a cubic meter of ice weighs about 917 kilograms. Compared to liquid water, ice is less dense, which explains why it floats.

In the problem of the melting ice sphere, the density of ice is a critical parameter because it relates the mass of the ice to its volume. By knowing the initial volume of the sphere and using the density, we can calculate the mass of the ice sphere.

Understanding this, we can find the energy needed to melt the ice, which is proportional to the mass involved. This knowledge helps us link the heat transfer rate with the rate of volume decrease, which ultimately affects the sphere's diameter over time.

Spherical Geometry

Spherical geometry involves the properties and measurements of spheres. It's important for solving problems involving spherical objects, like an ice sphere.

The geometry provides formulas for calculating surface area and volume. The surface area of a sphere is given by the formula \(A = 4\pi r^2\), and the volume is \(V = \frac{4}{3}\pi r^3\). These formulas are essential in our problem because the heat transfer rate depends on the surface area.

As the ice melts, the sphere's surface and volume shrink, impacting heat absorption. The decrease in volume pertains directly to changes in diameter. Thus, understanding spherical geometry allows us to comprehend how distribution and magnitude change over time, aiding precise measurement and analysis of the sphere's transformation.