Question

Consider a hot dog being cooked in boiling water in a pan. Would you model the heat transfer to the hot dog 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

Answer: The heat transfer in a cooking hot dog can be modelled as a one-dimensional transient heat transfer. The Cartesian coordinate system is appropriate for this process, with the origin placed at one end of the hot dog.

Step-by-step solution

Determine the heat transfer dimensionality

In a hot dog cooking process, boiling water surrounds the hot dog, and heat transfers from water to the hot dog mostly through convection. Since the hot dog has a long cylindrical shape, its heat transfer can be considered as primarily one-dimensional, along the length of the hot dog. In the actual application, some deviations take place due to ends of the hot dog and unevenness of the boiling water, but for modelling purposes, we can simplify it to a one-dimensional heat transfer.

Identify if the heat transfer is steady or transient

In this context, steady heat transfer means that the temperature distribution in the hot dog is constant and doesn't change over time. On the other hand, transient heat transfer means that the temperature distribution changes with time. Since the hot dog is cooked from an initial temperature towards the cooking temperature over time and the temperature distribution changes, we can consider the heat transfer to the hot dog as a transient process.

Choose the appropriate coordinate system

Since we assumed the heat transfer is mainly one-dimensional for simplification purposes, we can use the Cartesian (rectangular) coordinate system to describe the problem. This system has an x-axis along the length of the hot dog, and considering its cylindrical shape, we don't need to consider the radial coordinates (r, theta, or z) for this case.

Determine the origin placement

The most convenient point to place the origin (x=0) would be one end of the hot dog where it is initially in contact with the boiling water. The x-axis would then extend along the length of the hot dog. In summary, the heat transfer in a cooking hot dog can be modelled as one-dimensional transient heat transfer in the Cartesian coordinate system, with the origin placed at one end of the hot dog.

Key concepts

One-Dimensional Heat Transfer

One-dimensional heat transfer is a simplification of thermal energy movement that occurs along a single spatial dimension. Imagine a ruler that only measures length without worrying about width or height; that's what we do when we look at one-dimensional heat transfer.

In the case of cooking a hot dog, we see this concept in action. A hot dog has a long shape and, therefore, heat primarily travels along its length, from one end to the other. This situation is perfect for one-dimensional analysis because the temperature does not change drastically across the hot dog's width or depth. To capture this phenomenon in a model, we set up what's known as a temperature gradient along the length, often described by the marine Fourier's law of heat conduction.

However, it's important to note that this is a simplified model. Real-life objects rarely experience pure one-dimensional heat transfer due to factors like end effects or variations in material composition. Nonetheless, for educational and practical purposes, assuming one-dimensional heat transfer in certain scenarios like cooking a hot dog allows for easier analysis and grasp of the fundamental concepts of thermal energy distribution.

Transient Heat Transfer

Transient heat transfer, unlike its steady counterpart, describes a situation where temperature changes over time. Think of transient heat transfer like watching a weather time-lapse, where temperatures swing between day and night rather than staying at a constant midday heat.

When a hot dog is dropped into boiling water, it starts at room temperature and gradually warms up. This change in temperature isn't instant; it occurs over time as heat from the water flows into the hot dog. We observe this as the hot dog cooks, describing a dynamic process where the temperature at any point in the hot dog is a function of time.

Mathematically, this type of problem is often characterized by partial differential equations (PDEs) such as the heat equation. For students modeling transient heat transfer, understanding its time-dependent nature requires grasping how boundary conditions and initial temperatures contribute to shaping the temperature profile at any given moment.

Cartesian Coordinate System

The Cartesian coordinate system is a pillar in geometry and physics, providing a framework to describe the position of points in space using perpendicular axes labeled x, y, and z. These axes form a three-dimensional space allowing us to pinpoint locations in a way similar to using a map and coordinates to find a destination.

In our hot dog scenario, we're primarily interested in the x-axis, which aligns with the length of the hot dog. By placing the origin at one end, we can describe the temperature at any point along the hot dog with just one number—the distance from the origin. This simplifies our calculations and helps us focus solely on how heat moves along the hot dog's length.

Within this Cartesian framework, we don't need to worry about the y-axis or z-axis due to our one-dimensional assumption. Yet, it's vital to recognize a Cartesian coordinate system's versatility would let us easily extend our model into two or three dimensions, should we need a more complex analysis for objects with significant heat transfer in multiple directions.