Question

Hot air at \(80^{\circ} \mathrm{C}\) is blown over a \(2-\mathrm{m} \times 4-\mathrm{m}\) flat surface at \(30^{\circ} \mathrm{C}\). If the convection heat transfer coefficient is \(55 \mathrm{W} / \mathrm{m}^{2} \cdot^{\circ} \mathrm{C},\) determine the rate of heat transfer from the air to the plate, in \(\mathrm{kW}\)

Short answer

Answer: The rate of heat transfer from the hot air to the flat surface is 22 kW.

Step-by-step solution

Find the area of the flat surface

To find the area of the flat surface, we need to multiply its length by its width. The given dimensions of the flat surface are \(2m\) and \(4m\). Therefore, the area can be calculated as: $$A = 2 \times 4 = 8 m^2$$

Calculate the temperature difference

The temperature difference between the hot air and the flat surface is given by the difference between their temperatures (in \(°C\)). We are given the temperatures as \(80°C\) and \(30°C\). Thus, we can find the temperature difference as: $$\Delta T = T_\text{air} - T_\text{surface} = 80°C - 30°C = 50°C$$

Calculate the rate of heat transfer

Now, we can use the convection heat transfer formula to find the rate of heat transfer: $$q = hA\Delta T$$ Plugging the given values for \(h\), \(A\), and \(\Delta T\): $$q = 55 \frac{W}{m^2\cdot °C} \cdot 8m^2 \cdot 50°C$$ $$q = 22000 W$$

Convert the rate of heat transfer to kW

To convert the rate of heat transfer from W to kW, we need to divide the value by 1000: $$q_\text{kW} = \frac{22000}{1000} kW = 22 kW$$ The rate of heat transfer from the air to the plate is \(22\text{ kW}\).

Key concepts

Understanding Convection Heat Transfer

Convection heat transfer is akin to a gentle dance of energy between a fluid (which can be a liquid or gas) and a solid surface. Imagine you have a hot cup of tea and you blow over it to cool it down. That process—where the heat from the tea is being carried away by the moving air—is an example of convection.

In the context of our problem, hot air is the fluid that transfers its energy to the cooler flat surface. This type of heat transfer is governed by both the fluid's movement (free or forced convection) and its properties, such as viscosity and thermal conductivity.

Forced convection, in particular, is when the fluid movement is caused by external means—like a fan or a pump. In the given exercise, we can infer that there is a movement of air, implying that the mode of heat transfer is forced convection. To quantify this heat exchange, we use the convection heat transfer coefficient which embodies the characteristics of both the fluid and the surface geometry.

The Role of Temperature Difference in Heat Transfer

In our daily lives, we inevitably experience the effects of temperature difference. For instance, feeling the warmth of sunlight on your face while standing in the cold. Temperature difference, often denoted as \(\Delta T\), is a driving force for heat transfer—it's the spark that sets heat in motion. It's based on a simple fact: heat always flows from warmer to cooler regions.

In the exercise, the temperature difference between the hot air and the flat surface is crucial. It dictates the rate at which heat will transfer from the air to the surface. A greater \(\Delta T\) would mean a faster heat transfer rate. It's essential to comprehend that even a small change in temperature can significantly affect the overall heat transfer rate, which is why accurate measurement and calculation of \(\Delta T\) is vital in solving such problems.

Decoding the Heat Transfer Coefficient

The heat transfer coefficient, symbolized by \(h\), is a measure that blends the thermal properties of the material and the dynamics of fluid flow into a single value. Think of it as a number that tells us how well the combination of the two can move heat from one place to another.

In technical terms, \(h\) has units of \(W/(m^2 \cdot^{\rc}C)\), indicating the rate of heat transfer per unit area per degree of temperature difference between the fluid and the surface. A higher coefficient means that the surface and fluid exchange heat more efficiently.

In our example, the given \(h\) is quite high, suggesting the hot air is quite effective at transferring its heat to the surface. It's this coefficient, multiplied by the surface area and temperature difference, that allows us to calculate the total heat transfer rate—bridging the gap between abstract numbers and the real-world phenomenon of a warming surface.