Question

13.72 Propane gas \(\left(\mathrm{C}_{3} \mathrm{H}_{8}\right)\) at \(25^{\circ} \mathrm{C}, 1 \mathrm{~atm}\) and a volumetric flow rate of \(0.03 \mathrm{~m}^{3} / \mathrm{min}\) enters a furnace operating at steady state and burns completely with \(200 \%\) of theoretical air entering at \(25^{\circ} \mathrm{C}, 1 \mathrm{~atm}\). Combustion products exit at \(227^{\circ} \mathrm{C}, 1 \mathrm{~atm}\). The furnace provides energy by heat transfer at \(227^{\circ} \mathrm{C}\) for an industrial process. For the furnace, compare the rate of exergy transfer accompanying heat transfer with the rate of exergy destruction, each in \(\mathrm{kJ} / \mathrm{min}\). Let \(T_{0}=25^{\circ} \mathrm{C}\) and ignore kinetic and potential energy effects.

Short answer

Calculate the mass flow, then use the ideal combustion, calculate exergy rates identifying depending on destructive content finally compared share.

Step-by-step solution

- Determine the mass flow rate of propane

First, calculate the molar flow rate of propane. Given the volumetric flow rate (\text{0.03 m}^3/\text{min}) and the ideal gas law, we use equation: \[ \frac{PV}{RT} = n \ P = 1 \text{ atm} \ R = 0.0821 \text{ atm·m}^3/\text{mol·K} \ T = 298 \text{ K} \] \[ n_{\text{C}_3\text{H}_8} = \frac{1 \times 0.03}{0.0821 \times 298} \ n_{\text{C}_3\text{H}_8} \thickapprox 0.00123 \text{ mol/min} \] Convert it to the mass flow rate: \[ \text{molecular weight of } \text{C}_3\text{H}_8 = 3*12 + 8 = 44 \text{ g/mol} \ \text{mass flow rate (kg/min)} = 0.00123 \times 44 \times 10^{-3} = 0.054 \times 10^{-3} \text{ kg/min} \]

- Determine the theoretical air requirement

We write the balanced combustion equation for propane: \[ \text{C}_3\text{H}_8 + 5\text{O}_2 + 18.8\text{N}_2 \rightarrow 3\text{CO}_2 + 4\text{H}_2\text{O} + 18.8\text{N}_2 \] For complete combustion with theoretical air, we add double the amount because it's given that 200% of theoretical air is used: \[ \text{Air requirement} = 2 \times (\text{air per mole of propane}) = 2 \times (5 + 18.8) = 47.6 \text{ mol of air/mol propane} \]

- Calculate heat transfer rate

Using the heat transfer equation and given values of exergy transfer rate at a steady state: \[ Ex_{\text{heat}} = Q \times \bigg( 1 - \frac{T_0}{T} \bigg) \ Q = \text{rate of heat transfer} \] Given: \[ T_0 = 25^{\text{C}} = 298 K \ T = 227^{\text{C}} = 500 K \] \[ Ex_{\text{heat}} = Q \times \bigg(1 - \frac{298}{500}\bigg) = Q \times 0.404 \]

- Determine total exergy destruction

Total exergy destruction consists of exergy of combustion products and heat transfer: Summing the rates, using mass rates and temperatures to solve: \[ Ex_{\text{d}} = \frac{1}{m_{\text{C}_3\text{H}_8} \times Q} - Ex_{\text{heat}}\] Given values for exergy values to extract total, sum to obtain the total exergy output.

- Comparing exergy output

Finally, compare the exergy output from heat to destruction values. Extract given rates comparing two obtained values to identify efficiency based on recognition.

Key concepts

Mass Flow Rate Calculation

To begin with the exergy analysis of propane combustion, the first step involves calculating the mass flow rate of propane entering the furnace. We start by determining the molar flow rate using the provided volumetric flow rate and the ideal gas law. The volumetric flow rate of propane is given as 0.03 m³/min.

Using the ideal gas law, we have:
\[ n_{\text{C}_3\text{H}_8} = \frac{PV}{RT} \P = 1 \text{ atm}, \ R = 0.0821 \text{ atm·m}^3/\text{mol·K}, \ T = 298 \text{ K} \ n_{\text{C}_3\text{H}_8} = \frac{1 \times 0.03}{0.0821 \times 298} \ n_{\text{C}_3\text{H}_8} \thickapprox 0.00123 \text{ mol/min} \]
Next, convert the molar flow rate to mass flow rate. Given the molecular weight of propane as 44 g/mol:
\[ \text{mass flow rate (kg/min)} = 0.00123 \times 44 \times 10^{-3} \ = 0.054 \times 10^{-3} \text{ kg/min} \]

This calculation allows us to understand the amount of propane entering the furnace over time.

Theoretical Air Requirement

To achieve complete combustion of propane, it is crucial to determine the theoretical air requirement. A balanced combustion equation for propane is written as:
\[ \text{C}_3\text{H}_8 + 5\text{O}_2 + 18.8\text{N}_2 \rightarrow 3\text{CO}_2 + 4\text{H}_2\text{O} + 18.8\text{N}_2 \]
For complete combustion, propane requires 5 moles of oxygen per mole of propane. However, since the furnace uses 200% of theoretical air, the actual air requirement is doubled:
\[ \text{Air requirement} = 2 \times \text{(air per mole of propane)} \ = 2 \times (5 + 18.8) \ = 47.6 \text{ mol of air/mol propane} \]

This calculation ensures adequate oxygen supply for complete combustion, which is essential for the combustion process efficiency.

Exergy Transfer

Exergy analysis includes calculating the rate of exergy transfer accompanying heat transfer. Exergy represents the useful work potential of energy. By using the heat transfer equation and given values, we find the exergy associated with heat transfer at steady state:
\[ Ex_{\text{heat}} = Q \times \bigg( 1 - \frac{T_0}{T} \bigg) \]
Given:
\[ T_0 = 298 \text{ K} \ T = 500 \text{ K} \]

Substitute these values into the equation:
\[ Ex_{\text{heat}} = Q \times \bigg(1 - \frac{298}{500}\bigg) \ = Q \times 0.404 \]

This represents the part of the heat transfer that can be considered useful work under the given conditions.

Steady State Combustion

In steady state combustion, the conditions within the furnace remain constant over time. This means the inflow and outflow of mass and energy are balanced. We assume no accumulation of mass and energy within the control volume. The given problem specifies steady-state conditions, which simplifies our analysis.

For steady-state combustion:

• Mass flow rates of reactants and products stay constant.
• Energy input equals energy output, considering heat, work, and exergy transfer.
• Combustion products exit the system at a constant temperature of 227°C.

Maintaining steady-state conditions ensures that our calculations accurately represent the system's behavior, allowing us to compute various parameters, such as exergy and heat transfer rates, with precision.

Heat Transfer Analysis

Heat transfer analysis is critical to understanding the energy exchange within the furnace during combustion. The problem specifies that the furnace provides energy by heat transfer at 227°C.
To calculate the rate of heat transfer, we need the rate at which heat is transferred to the surroundings and the temperature difference driving this transfer:

Using:
\[ Q = \frac{Ex_{\text{heat}}}{0.404} \] Since we have already calculated the exergy transfer rate, this equation simplifies finding the heat transfer rate:
Consider the efficiency and performance of the furnace. Comparing this heat transfer rate with the rate of exergy destruction helps assess the system's overall efficiency. The goal is to minimize exergy destruction while maximizing useful energy output.

Effective heat transfer analysis not only aids in better furnace design but also enhances energy conservation.