Question

Most correlations for the convection heat transfer coefficient use the dimensionless Nusselt number, which is defined as (a) \(h / k\) (b) \(k / h\) (c) \(h L_{c} / k\) (d) \(k L_{c} / h\) (e) \(k / \rho c_{p}\)

Short answer

a) \(h L_{c} \cdot k\) b) \(\frac{k}{h L_{c}}\) c) \(\frac{h L_{c}}{k}\) d) \(\frac{L_{c}}{h k}\) Answer: c) \(\frac{h L_{c}}{k}\)

Step-by-step solution

1. Recall the definition of Nusselt number

The Nusselt number (Nu) is a dimensionless number defined as the ratio of convective heat transfer to conductive heat transfer at a boundary. It is given by: Nu = \(\frac{h L_{c}}{k}\) Where: \(h\) - heat transfer coefficient (W/m²·K), \(L_c\) - characteristic length (m), \(k\) - thermal conductivity (W/m·K).

2. Compare the definition with the given options

Comparing the definition of the Nusselt number with the given options, we can see that the correct definition of the Nusselt number is: Nu = \(h L_{c} / k\) Hence, the correct answer is (c) \(h L_{c} / k\).

Key concepts

Convective Heat Transfer

Imagine a hot cup of coffee sitting on the table. After a while, the air around the cup gets warmer, and if you put your hand above the cup, you can feel the heat. This transfer of heat from the hot cup to the surrounding air is known as convective heat transfer. It's the process where heat moves through a fluid (which can be a liquid or a gas) due to the fluid's movement. The faster the fluid moves, the more heat it can carry away. This process can be natural, due to differences in density (think of our coffee example), or it can be forced through external means like a fan or pump.

In engineering and physics, we quantify this heat transfer using the dimensionless Nusselt number, which relates to how effective a fluid is at transferring heat by convection compared to conduction, which is another type of heat transfer.

Conductive Heat Transfer

Now, consider holding an ice cube in your hand. You can feel the cold because heat from your warm hand is being transferred to the colder ice. This direct transfer of heat through a material without the movement of the substance itself is conductive heat transfer. Imagine molecules in a substance as people in a line carrying buckets of water from one to another; the molecules don't move much, but they pass on the energy (like our water buckets) from one to the next. Materials vary widely in their ability to conduct heat; metals like copper are excellent conductors, while materials like wood are not. This is a key concept when we are looking at the Nusselt number since it helps us understand how heat transfer by conduction compares with that by convection.

Heat Transfer Coefficient

The heat transfer coefficient, denoted as 'h', is a measure that tells us how good a material or system is at transferring heat. Think of it as a rating that tells how quickly heat can move from one place to another. A higher value for 'h' means that the material or system is a better 'heat mover'. This is crucial in convective heat transfer, where the heat transfer coefficient depends on the fluid properties, surface conditions, and fluid velocity. When we solve problems involving the Nusselt number, the heat transfer coefficient is a key player since it represents how well convection is happening on the surface of an object.

Thermal Conductivity

Thermal conductivity, represented by 'k', is a physical property that tells us how well a material can conduct heat. If you think about placing a thin metal spoon in a hot soup, you'll notice that the handle also gets hot quickly because metal has a high thermal conductivity. However, if you use a wooden spoon, the handle won't heat up as much because wood has a lower thermal conductivity. In the realm of heat transfer, particularly when discussing the Nusselt number, thermal conductivity is used to compare how effective materials are at transferring heat by conduction. It's one-half of the relationship captured by the Nusselt number, which combines the effects of conduction and convection in heat transfer scenarios.