Question

A \(1000-\mathrm{W}\) iron is left on the ironing board with its base exposed to the air at \(20^{\circ} \mathrm{C}\). If the temperature of the base of the iron is \(150^{\circ} \mathrm{C}\), determine the rate of exergy destruction for this process due to heat transfer, in steady operation.

Short answer

Answer: R = hA(130) (0.3075)

Step-by-step solution

Convert the given temperatures to Kelvin

To work with temperatures in thermodynamic equations, we need to convert them to the Kelvin scale. To do this, we add 273.15 to the given Celsius temperatures. T0 (air temperature) = 20°C + 273.15 = 293.15 K T (iron base temperature) = 150°C + 273.15 = 423.15 K

Calculate the heat transfer rate (Q)

The heat transfer rate (Q) is not provided in the exercise, but we can estimate it using the Newton's law of cooling which states that the rate of heat transfer (Q) is proportional to the temperature difference between the iron base and the surrounding air (ΔT). Therefore, we can write Q as: Q = hAΔT Where h is the heat transfer coefficient, A is the surface area of the iron base, and ΔT is the temperature difference between the iron base and the surrounding air. In this exercise, we don't have enough information to find the exact value of Q. But since we are only interested in the rate of exergy destruction, we can rewrite the formula for the exergy destruction rate (R) in terms of Q: R = hAΔT (1 - T0 / T)

Calculate the rate of exergy destruction (R)

Now, using the formula derived in the previous step, we can calculate the rate of exergy destruction (R): R = hAΔT (1 - T0 / T) = hA(423.15 - 293.15) (1 - 293.15 / 423.15) R = hA(130) (1 - 0.6925) = hA(130) (0.3075) We couldn't find the exact value of exergy destruction rate (R) as it depends on h (heat transfer coefficient) and A (surface area). However, the expression for the exergy destruction rate in terms of h and A has been derived using the given temperatures and steady operation assumption. Once we know the values of h and A, we can substitute them into the expression to find the exergy destruction rate.

Key concepts

Understanding Thermodynamics

Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, radiation, and physical properties of matter. The fundamental principles of thermodynamics are expressed in four laws. These laws describe how thermal energy is converted to and from other forms of energy and how it affects matter.

The first law, also known as the law of energy conservation, states that energy cannot be created or destroyed in an isolated system. The second law of thermodynamics is particularly pertinent to our exercise as it introduces the concept of entropy and explains that total entropy can only increase over time in an isolated system, meaning that processes are irreversible. In our problem, exergy destruction is directly tied to the second law, as it is a measure of the irreversibility of the process and the lost potential to do work due to heat transfer.

The Kelvin Temperature Scale

The Kelvin temperature scale is an absolute temperature scale that is a fundamental part of thermodynamics. It starts at absolute zero, the theoretical point where molecular motion ceases. Unlike the Celsius and Fahrenheit scales, where temperatures can be negative, the Kelvin scale sets absolute zero as 0 K, which is equivalent to -273.15°C or -459.67°F.

In thermodynamic calculations, it is essential to use the Kelvin scale to ensure accuracy in mathematical formulations, as seen when converting the temperatures for our exergy destruction rate problem. To convert Celsius to Kelvin, as shown in the solution, you add 273.15 to the Celsius temperature.

Newton's Law of Cooling

Newton's law of cooling is an empirical relationship that describes the rate at which an exposed body changes temperature through radiation. It asserts that the rate of heat loss of a body is proportional to the difference in temperatures between the body and its surroundings. The law makes it easier to calculate heat transfer rates, especially in situations where we know the temperature difference and the heat transfer coefficient.

The importance of Newton's law of cooling in our problem is evident, as it provides a basis for estimating the heat transfer rate (Q) through the relationship Q = hAΔT, connecting the temperature difference and the heat transfer characteristics of the iron's surface and the surrounding air.

Heat Transfer Coefficient

The heat transfer coefficient, h, is a parameter that describes the convective heat transfer properties of a fluid in contact with a solid. It is a measure of the convective heat transfer capability from the solid surface to the fluid and vice versa, which depends on the properties of the fluid, the flow characteristics, and the surface geometry. The higher the coefficient, the more efficient is the heat transfer.

In practical applications, such as calculating the rate of exergy destruction for the iron, the heat transfer coefficient is crucial. The exact rate of exergy destruction can't be determined without the value of h, along with the surface area A. These coefficients are often determined experimentally or by following guidelines provided for similar physical systems and materials.