Question

Consider a stainless steel spoon \(\left(k=8.7 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft} \cdot{ }^{\circ} \mathrm{F}\right)\) partially immersed in boiling water at \(200^{\circ} \mathrm{F}\) in a kitchen at \(75^{\circ} \mathrm{F}\). The handle of the spoon has a cross section of \(0.08\) in \(\times\) \(0.5\) in, and extends 7 in in the air from the free surface of the water. If the heat transfer coefficient at the exposed surfaces of the spoon handle is \(3 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft}^{2} \cdot{ }^{\circ} \mathrm{F}\), determine the temperature difference across the exposed surface of the spoon handle. State your assumptions. Answer: \(124.6^{\circ} \mathrm{F}\)

Short answer

Answer: The temperature difference across the exposed surface of the spoon handle is 124.6°F.

Step-by-step solution

Understand the problem

The problem essentially involves finding the temperature difference across the exposed surface of the spoon handle due to heat transfer through conduction and convection.

Gather necessary information

The following information is given in the problem: - Thermal conductivity of the stainless steel (k): 8.7 Btu/h·ft·°F - Boiling water temperature (T_boil): 200°F - Kitchen temperature (T_kitchen): 75°F - Cross-sectional dimensions of the handle: 0.08 in × 0.5 in - Length of the handle exposed in the air: 7 in - Heat transfer coefficient of the exposed surfaces of the handle (h): 3 Btu/h·ft²·°F Before we start solving the problem, we need to convert the dimensions of the handle to feet.

Convert dimensions to feet

To convert the dimensions of the handle to feet, divide the given dimensions by 12 (1 foot = 12 inches). Cross-sectional area (A): \((0.08 \times 0.5)/144 = 2.7778 \times 10^{-4} \,\text{ft}^2\) Length of the exposed handle (L): \(7/12 = 0.5833 \,\text{ft}\)

Find the heat transfer rate through conduction

To find the heat transfer rate (Q_dot) through conduction in the handle, use the following equation: $$ Q_\text{dot} = k \cdot \frac{A\cdot \Delta \text{T}}{L} $$ However, we first need to find the temperature difference (ΔT) across the handle.

Find the temperature difference across the handle

To find the temperature difference (ΔT) across the handle, we first need to find the conducted heat transfer rate. We know that the heat transfer rate is the same through conduction and convection. $$ Q_\text{dot} = h \cdot A \cdot\Delta \text{T}_\text{conv} $$ From step 4, we know: $$ Q_\text{dot} = k \cdot \frac{A\cdot \Delta \text{T}_\text{cond}}{L} $$ Equating the two expressions for Q_dot and solving for the temperature difference across the exposed surface of the spoon handle: $$ \Delta \text{T}_\text{conv} = \frac{k\cdot L}{h}\cdot \Delta \text{T}_\text{cond} $$ In order to find the temperature difference across the exposed surface of the spoon handle, we first need to find \(\Delta \text{T}_\text{cond}\): $$ \Delta \text{T}_\text{cond} = \frac{Q_\text{dot}\cdot L}{k\cdot A} $$ $$ \Delta \text{T}_\text{cond} = \frac{T_\text{boil} - T_\text{kitchen}}{\frac{k\cdot L}{h\cdot A} + 1} $$ Now plug in the given values and calculate: $$ \Delta \text{T}_\text{cond} = \frac{200 - 75}{\frac{8.7\cdot 0.5833}{3\cdot2.7778 \times 10^{-4}} + 1} $$ With this, we find the temperature difference across the stainless steel handle due to conduction: $$ \Delta \text{T}_\text{cond} = 69.4\,^{\circ}\text{F} $$ Now, we will find the temperature difference at the exposed surface using the above derived equation: $$ \Delta \text{T}_\text{conv} = \frac{8.7\cdot 0.5833}{3}\cdot 69.4 $$ $$ \Delta \text{T}_\text{conv} = 124.6\,^{\circ}\text{F} $$ The temperature difference across the exposed surface of the spoon handle is 124.6°F.

Key concepts

Thermal Conductivity

Thermal conductivity is a property of materials that measures their ability to conduct heat. It is denoted by the symbol 'k' and has units of Btu/h·ft·°F or W/m·K, depending on the system of units used. In the context of the stainless steel spoon in the exercise, thermal conductivity represents how well the material of the spoon can transfer heat from the boiling water it is partially immersed in to the other end. The higher the thermal conductivity, the more efficient a material is at transferring heat.

Understanding this concept is crucial because it allows us to predict how heat will flow through the spoon from the hot end to the cooler end. The value of thermal conductivity (8.7 Btu/h·ft·°F for the stainless steel spoon) directly impacts how much heat is transferred and consequently, the temperature distribution along the spoon handle. This is why materials with high thermal conductivity are used for cooking utensils, as they can quickly transfer heat from the cooking surface to the food.

Convection

Convection is the process of heat transfer through a fluid (which may be a liquid or a gas) that occurs due to the fluid's movement. This can be natural convection, caused by buoyancy forces when there is a temperature difference within the fluid, or forced convection, which occurs when an external force, such as a fan or a pump, impels the fluid to move.

In our exercise, heat transfer from the spoon handle to the surrounding air involves natural convection. The heat transfer coefficient 'h' (3 Btu/h·ft²·°F), also called the convective heat transfer coefficient, characterizes the efficiency of heat transfer from the solid surface to the surrounding air due to convection. The larger the value of 'h', the more efficient the convection process in dissipating heat. To fully understand the thermal behavior of the spoon in the air, it's important to consider how the cool air interacts with the spoon handle and how this interaction takes away heat through convection.

Temperature Difference

Temperature difference, denoted as \( \Delta T \), is the driving force for heat transfer. Heat naturally flows from regions of higher temperature to regions of lower temperature. The wider the temperature difference between two points, the greater the rate of heat transfer. In our exercise, there is a temperature difference between the boiling water and the ambient temperature of the kitchen, which creates the conditions necessary for heat transfer from the hot water to the cooler surroundings.

The calculated temperature difference across the exposed surface of the spoon handle (124.6°F) signifies a substantial gradient that leads to heat transfer through both conduction within the spoon and convection from the spoon handle to the air. It's vital to understand that without a temperature difference, there would be no heat transfer to drive the processes we observe or require, such as heating a room or cooking our food.