Question

Consider heat loss from a \(200-\mathrm{L}\) cylindrical hot water tank in a house to the surrounding medium. Would you consider this to be a steady or transient heat transfer problem? Also, would you consider this heat transfer problem to be one-, two-, or three-dimensional? Explain.

Short answer

Answer: The heat loss from a 200 L cylindrical hot water tank is a transient heat transfer problem because the temperature distribution will change with time. It is also a three-dimensional heat transfer problem since the temperature distribution varies radially, axially, and circumferentially.

Step-by-step solution

1. Steady vs Transient Heat Transfer

Steady heat transfer implies that temperature distribution does not change with time, whereas transient heat transfer refers to the temperature distribution changing over time. In the case of a hot water tank in a house, the temperature of the water will decrease as heat is lost to the surrounding environment. This heat transfer will occur over a period of time until the temperature of the water reaches the temperature of the surrounding environment. Therefore, we can consider this problem as a transient heat transfer problem because the temperature distribution will change with time.

2. One-, Two- or Three-Dimensional Heat Transfer

One-dimensional heat transfer occurs when the temperature varies in only one direction. Two-dimensional heat transfer occurs when the temperature varies in two directions, and similarly, three-dimensional heat transfer occurs when the temperature varies in three directions. In the case of a cylindrical hot water tank, the heat loss can occur in all three physical dimensions: radial, axial, and circumferential dimensions. However, if the tank is well-insulated, the heat loss can be neglected in the axial and circumferential directions, simplifying the problem to one-dimensional heat transfer. Nonetheless, without having information about the insulation of the tank, we need to consider the realistic scenario involving heat transfer in all directions. Therefore, this heat transfer problem can be considered as a three-dimensional heat transfer problem, since the temperature distribution would vary radially, axially, and circumferentially.

Key concepts

Transient Heat Transfer

Transient heat transfer is a dynamic phenomenon where the temperature within a system changes over time. Unlike steady heat transfer, where temperatures remain constant, transient scenarios require consideration of time as a crucial variable. In the context of a hot water tank, as heat is steadily lost to the surroundings, the water temperature within the tank will gradually decrease. This scenario reflects the transient nature of the heat transfer process since the temperature distribution within the tank does not reach a stable state instantaneously but evolves until eventual equilibrium with the surroundings.

Understanding the factors that influence transient heat transfer, such as the properties of the water, the materials of the tank, and the surrounding environment, is key. Thermal conductivity, heat capacity, and convection coefficients play crucial roles in the rate at which the water loses heat. Using mathematical models, engineers can predict how temperature changes within the tank over time and design the tank to maintain its heat more effectively.

One-Dimensional Heat Transfer

One-dimensional heat transfer refers to the assumption that temperature varies in only one spatial dimension, simplifying the analysis significantly. For a long, slender object or for cases where variations in temperature are negligible along other dimensions, this assumption is fairly accurate. When we consider a well-insulated cylindrical hot water tank, for instance, the predominant heat transfer might take place predominantly in the radial direction, from the hot water to the tank walls.

Despite the cylindrical geometry, if the height of the tank is much larger than its diameter, and assuming insulation is effective at the top and bottom, the temperature gradient along the length (axial direction) and around the circumference (circumferential direction) can often be ignored. Hence, engineers can model the system as one-dimensional to simplify the calculations, which would primarily focus on the radial temperature gradient. This type of simplification is useful for initial designs and for understanding basic heat transfer behavior in elongated cylindrical shapes.

Three-Dimensional Heat Transfer

Three-dimensional heat transfer takes into account temperature variations in all three spatial dimensions, making the analysis much more complex but also more accurate for certain geometries and conditions. A cylindrical hot water tank, physically, can lose heat in the radial, axial, and circumferential directions. Realistically, to capture the complete essence of the heat transfer in a tank without assuming ideal insulation, engineers must consider it as a three-dimensional problem.

In practice, considering all three dimensions is crucial when the goal is precise modeling or when the conditions of the problem necessitate detailed analysis, for example, in tanks with significant height-to-diameter ratios or with varying insulation qualities along different surfaces. Three-dimensional heat transfer analysis helps in optimizing tank design, understanding the distribution of temperatures within the tank, and making informed decisions on how to minimize energy loss.

Steady Heat Transfer

Steady heat transfer implies that, unlike transient heat transfer, the temperature at any given point in the system does not change with time; it remains constant. Steady-state conditions are often an idealization used in engineering to represent systems that have had enough time to stabilize temperature profiles or are maintained by a balance of input and output heat flows.

In reality, truly steady conditions are rare, as most systems experience some form of transient phenomena before reaching an equilibrium. However, for theoretical and computational simplicity, steady assumptions are often made in cases where transient effects are deemed negligible or where a snapshot of a system at equilibrium is needed for analysis. Under this assumption, the heat transfer equations are time-independent, which means that engineers can focus solely on the spatial distribution of temperatures without considering their variation over time.