Question

A 110 -volt electrical heater is used to warm \(0.3 \mathrm{m}^{3} / \mathrm{s}\) of air at \(100 \mathrm{kPa}\) and \(15^{\circ} \mathrm{C}\) to \(100 \mathrm{kPa}\) and \(30^{\circ} \mathrm{C}\). How much current in amperes must be supplied to this heater?

Short answer

Answer: Approximately 50.34 Amperes.

Step-by-step solution

Calculate the power required to heat the air

To find the power, we first need to determine the heat capacity of the air, which is given by the formula: \(C_p = nR \times \frac{C_v}{nR - C_v}\) Where n is the polytropic exponent (1.4 for air), \(C_v\) is the specific heat capacity of air at constant volume (approx. 718 \(J/kgK\)), and \(R\) is the specific gas constant for air (287 \(J/kgK\)). We can now plug these values into the formula and find \(C_p\): \(C_p = 1.4 \times \frac{718}{1.4 \times 287 - 718}\) \(C_p \approx 1005 \ J/kgK\) Now, we must calculate the mass flow rate (\(\dot{m}\)) of the air: \(\dot{m} = \rho \times \dot{V}\) We can find the density (\(\rho\)) using the ideal gas law: \(\rho = \frac{P}{RT}\) At the starting temperature (15°C), the specific air density is: \(\rho = \frac{100 \times 10^3}{287\times(15+273.15)}\) \(\rho \approx 1.225 \ kg/m^3\) Now, we can find the mass flow rate: \(\dot{m} = 1.225 \times 0.3\) \(\dot{m} \approx 0.3675 \ kg/s\) Now, we can calculate the power required (P) to heat the air: \(P = \dot{m} \times C_p \times \Delta T\) Where \(\Delta T = T_{final} - T_{initial} = 30 - 15 = 15K\). \(P = 0.3675 \times 1005 \times 15\) \(P \approx 5537.6 \ W\)

Calculate the current needed for the heater

Next, we must determine the current required to supply power to the heater. We can use the basic electrical power formula as follows: \(Power = Voltage \times Current\) Rearrange the formula to find the current: \(Current = \frac{Power}{Voltage}\) Now, we can plug in the power we calculated in step 1, and the voltage given (110V), to find the current needed: \(Current = \frac{5537.6}{110}\) \(Current \approx 50.34 \ A\) So, approximately 50.34 Amperes of current must be supplied to the 110-volt electrical heater to warm the given volume of air.

Key concepts

Specific Heat Capacity

Understanding the concept of specific heat capacity is essential when dealing with heat transfer problems. Specific heat capacity, symbolized as \(C_p\) when at constant pressure, refers to the amount of heat energy required to raise the temperature of a unit mass of a substance by one degree Celsius (or Kelvin). In mathematical terms, \(C_p\) is expressed in units of joules per kilogram per Kelvin (\(J/kgK\)).

Consider a situation where you need to determine how much energy is needed to heat a mass of air. Specific heat capacity is crucial here because it allows you to calculate the energy input needed based on the mass flow rate and the temperature change. Air, being a mixture of gases, has a specific heat capacity that varies slightly depending on the conditions, but for most practical purposes in thermodynamics, an average value can be used. Using the right specific heat capacity is crucial in our calculations to accurately determine the power required. It's the inefficiency and heat loss—ideas about which a good heading toward the concept of specific heat capacity will guide you to avoid—that can make or break a real-world engineering problem.

Ideal Gas Law

When tackling thermodynamics problems involving gases, the ideal gas law is a foundational equation to predict the behavior of the gas under various conditions. The law is given by the equation \(PV = nRT\), where \(P\) is the pressure, \(V\) is the volume, \(n\) is the number of moles, \(R\) is the ideal gas constant, and \(T\) is the temperature in Kelvin.

To use the ideal gas law in our heater problem, we first need to manipulate the equation to find the density \(\rho\) of the air at a specific state. Density, which is mass per unit volume, can be rearranged from the ideal gas law as \(\rho = \frac{P}{RT}\). The temperature must be in Kelvin, which is the Celsius temperature plus 273.15. This step is vital because we can then use the density to determine the mass flow rate, tying into the concept of mass-conserving flow systems and allowing for precise calculations in determining energy requirements for heating processes. Grasping the ideal gas law helps us accurately describe the state of an 'ideal' gas, serving as a stepping stone to understanding more complex gas behavior.

Mass Flow Rate

The mass flow rate is a measurement that indicates how much mass of a substance passes through a given surface per unit time. It's represented with the symbol \(\dot{m}\) and is typically expressed in kilograms per second (\(kg/s\)). This concept is frequently seen in problems involving fluid dynamics and thermodynamics, where we are interested in the amount of fluid or gas moving through a system.

In the context of our heating problem, after calculating the air's density using the ideal gas law, finding the mass flow rate allows us to determine how much air, by mass, flows past a certain point per second. This calculation is crucial because it factors directly into the equation used to calculate the power needed to heat the air, \(P = \dot{m} \times C_p \times \Delta T\). The mass flow rate impacts the total amount of energy needed for a heating process, as a higher mass flow rate requires more power to achieve the same temperature increase. Recognition of the relationship between mass flow rate, energy, and temperature change is critical for solving real-world thermodynamics problems.