Question

Consider a round potato being baked in an oven. Would you model the heat transfer to the potato as one-, two-, or three-dimensional? Would the heat transfer be steady or transient? Also, which coordinate system would you use to solve this problem, and where would you place the origin? Explain.

Short answer

2) Is the heat transfer steady or transient? 3) Which coordinate system should be used for solving this problem? 4) Where should the origin be placed in the potato?

Step-by-step solution

Dimension of Heat Transfer

The heat transfer in a potato while baking can be considered as a three-dimensional model, as heat is transferred in all three directions (x, y, and z-axis). However, in most cases, it can be simplified to a one-dimensional problem, assuming that heat transfer is primarily radial in nature, with temperature gradients being most significant in the radial direction in the potato.

Steady or Transient Heat Transfer

The heat transfer in this case is transient. This is because the temperature in the potato changes over time until it reaches a uniform temperature, depending on the oven's set temperature.

Coordinate System Selection

As the potato is round, the most appropriate coordinate system to use would be the spherical coordinate system. This allows for easier modeling of the problem as it takes the potato's shape into account, enabling the calculation of temperature changes across the spherical potato as a function of radius, polar angle, and azimuthal angle.

Placement of the Origin

The origin in the spherical coordinate system should be placed at the center of the potato. This simplifies the problem as it allows us to analyze heat transfer from the oven to the potato by using radial distance from the origin, which is the center of the potato.

Key concepts

Three-Dimensional Heat Transfer

When we talk about three-dimensional heat transfer, it involves the flow of heat in all three spatial dimensions: length, width, and height. This is crucial to understand when considering objects like a potato in an oven. The heat must move through the entire shape in three different directions: front to back, side to side, and top to bottom.

However, in many real-world applications like with the potato, although the heat is three-dimensional, we often simplify the scenario. In cases where an object has symmetry or specific geometric characteristics, one-dimensional models such as radial heat flow might suffice. But it's always essential to understand that three-dimensional analysis provides a complete picture. It considers how heat is truly distributed throughout the object, leading to more accurate predictions of cooking times or required energy input.

Transient Heat Transfer

Transient heat transfer is all about the temporal evolution of temperature in a material. This means we're looking at how temperatures change over time within an object until they reach equilibrium.

In our potato baking example, as the potato is exposed to the oven's heat, its internal temperature gradually changes and rises. Initially, the heat is absorbed at the surface and then starts migrating inward as time progresses.

In transient scenarios, the heat transfer never stays constant. It's a process that happens over a period as opposed to being static. This is important for predicting points at which the potato is adequately baked all the way through or to prevent overheating.

Spherical Coordinate System

The spherical coordinate system is a natural choice whenever dealing with objects that are globe-like in shape, such as our round potato in the oven. This system uses three variables: radius, polar angle, and azimuthal angle. They help locate any given point inside or on the surface of the sphere.

This coordinate system simplifies calculations for such objects by aligning well with their inherent geometry.

• **Radius (r):** measures how far from the center a point is. For the potato, the center is the starting point of measurement.
• **Polar Angle (θ):** defines the angle from a reference vertical direction.
• **Azimuthal Angle (φ):** is the angle around the object, similar to how longitude is measured on Earth.

The beauty of using this system lies in its ability to model spherical objects and analyze heat distribution precisely and uniformly, ensuring that calculations align perfectly with the physical properties and dimensions of the object at hand.