Question

Consider the cooking process of a roast beef in an oven. Would you consider this to be a steady or transient heat transfer problem? Also, would you consider this to be one-, two-, or three-dimensional? Explain.

Short answer

Answer: The heat transfer during the cooking process of roast beef in an oven is transient and occurs in three dimensions.

Step-by-step solution

Understanding Steady and Transient Heat Transfer

First, we need to understand the difference between steady and transient heat transfer. In steady heat transfer, the temperature at any point in the system remains constant over time. In transient heat transfer, the temperature at any point can change with time.

Understanding Dimensions of Heat Transfer

Next, we need to understand the dimensions of heat transfer. One-dimensional heat transfer assumes that heat only moves in one direction. Two-dimensional heat transfer occurs in two directions, and three-dimensional heat transfer occurs in all three directions.

Determining Steady or Transient Heat Transfer

Now, we need to determine if the cooking process of roast beef in an oven is a steady or transient heat transfer problem. We know that the temperature of the roast beef changes with time during the cooking process. Therefore, we can conclude that it’s a transient heat transfer problem since the temperature at any point in the roast beef does not remain constant over time.

Determining Dimensionality of Heat Transfer

Lastly, we need to determine the dimensionality of the heat transfer. During the cooking process, the heat is transferred from the oven to the roast beef in all directions. The heat not only moves from the outer surface to the inner parts of the roast beef but also throughout the roast beef in multiple directions. Considering that the roast beef has a non-uniform geometry, the heat transfer will likely occur in all three directions (length, width, and height), making it a three-dimensional heat transfer problem. In conclusion, the cooking process of a roast beef in an oven can be considered a transient and three-dimensional heat transfer problem.

Key concepts

Steady Heat Transfer

In steady heat transfer, the temperature within the system remains unchanged over time. Imagine a situation like maintaining the temperature in a room with a constant heater output. The key characteristic is that at any position or point in the system, the temperature does not vary as time progresses.
Steady heat transfer is typically studied under conditions where thermal equilibrium is achieved. This means that any heat entering a part of the system is immediately balanced by the heat leaving it.

• Temperatures remain constant over time.
• Occurs in situations of thermal equilibrium.
• Happens at a constant state or during stable operation.

When evaluating a heat transfer problem, if temperatures are seen to stabilize or reach a consistent value across the system after a certain period, it is likely classified as steady heat transfer.

Dimensionality of Heat Transfer

Dimensionality in heat transfer describes the directions in which heat moves throughout a system. Each dimension represents a potential path through which heat can be conducted. Understanding this helps in modeling and predicting how heat flows in various applications.
There are three types of dimensional analysis in heat transfer:

• One-Dimensional (1D): Heat moves in a single direction. It is often ideal for simple, straight paths like through a thin rod or a wall.
• Two-Dimensional (2D): Heat flows in two directions, like along the length and width of a flat plate.
• Three-Dimensional (3D): Heat moves in all directions - length, width, and height. It encompasses more complex shapes and volumes, like a solid sphere or a roast beef in an oven.

Understanding the dimensionality helps in simplifying complex heat transfer problems and aids in constructing mathematical models or simulations.

Three-Dimensional Heat Transfer

When heat transfer occurs in all three dimensions — length, width, and height — it is called three-dimensional heat transfer. This type of transfer is common in objects that have complex geometries or are exposed to different temperatures from multiple sides.
An example of 3D heat transfer is a piece of roast beef cooking in an oven. Heat from the oven enters not just from the top or bottom, but from all around, needing to be distributed throughout the entire mass.

• Heat can move freely in every direction.
• Common in objects with complex shapes.
• Requires more intricate calculations or computational analysis when compared to 1D or 2D.

Accurately modeling three-dimensional heat transfer is crucial for ensuring uniform cooking or thermal treatment of any material, ensuring no parts are undercooked or overheated.