Question

13.55 Liquid ethanol \(\left(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}\right)\) at \(25^{\circ} \mathrm{C}, 1\) atm enters a reactor operating at steady state and burns completely with \(130 \%\) of theoretical air entering in a separate stream at \(25^{\circ} \mathrm{C}\), 1 atm. Combustion products exit at \(227^{\circ} \mathrm{C}, 1 \mathrm{~atm}\). Heat transfer from the reactor takes place at an average surface temperature \(T_{\mathrm{b}}\). For \(T_{\mathrm{b}}\) ranging from 25 to \(200^{\circ} \mathrm{C}\), determine the rate of exergy destruction within the reactor, in \(\mathrm{kJ}\) per kmol of fuel. Kinetic and potential energy effects are negligible. Let \(T_{0}=25^{\circ} \mathrm{C}\).

Short answer

The rate of exergy destruction is calculated using the entropy change of the system and the environmental temperature, which involves integrating the entropy changes over the specified range.

Step-by-step solution

Write the combustion reaction

Ethanol combusts completely according to the reaction: \[ \text{C}_2 \text{H}_5 \text{OH} + 3\text{O}_2 \rightarrow 2\text{CO}_2 + 3\text{H}_2 \text{O} \] Since 130% of theoretical air is used, adjust for excess air: \[ \text{C}_2 \text{H}_5 \text{OH} + 3(1.3)\text{O}_2 \rightarrow 2\text{CO}_2 + 3\text{H}_2 \text{O} + 0.3\text{O}_2 \] So, the balanced equation is: \[ \text{C}_2 \text{H}_5 \text{OH} + 3.9\text{O}_2 \rightarrow 2\text{CO}_2 + 3\text{H}_2 \text{O} + 0.9\text{O}_2 \]

Calculate the exit gas stream temperature

Given: Initial temperature = 25°C, Final temperature = 227°C. This step considers the heat value at these temperatures.

Determine the heat transfer (\text{\text{Q}})

Using the energy balance for the reactor, considering negligible kinetic and potential energy effects, \[ \text{Q} = n_{fuel} (\text{h}_{products} - \text{h}_{reactants}) \] where \(\text{h}\) represents the specific enthalpies at the given temperatures.

Calculate the exergy destruction

Exergy destruction can be determined with \[ \text{Exergy destruction} = \text{T}_0 \times \text{Δs}_{gen} \] where \[ \text{Δs}_{gen} = n_{fuel} \times (\text{s}_{products} - \text{s}_{reactants}) \] using the specific entropy values at the given temperatures.

Integrate over the temperature range

Assess exergy destruction over a temperature range from 25°C to 200°C by determining the above value for each temperature within the range and integrating.

Key concepts

combustion reaction

In a combustion reaction, a fuel reacts with an oxidizer, like oxygen, to produce heat and combustion products. For ethanol (\text{C}_2\text{H}_5\text{OH}), the complete combustion reaction with theoretical air can be written as
\[ \text{C}_2\text{H}_5\text{OH} + 3\text{O}_2 \rightarrow 2\text{CO}_2 + 3\text{H}_2\text{O} \] The phrase '130% of theoretical air' means we are using 30% more air than required. This is adjusted in the combustion equation as:
\[ \text{C}_2\text{H}_5\text{OH} + 3(1.3)\text{O}_2 \rightarrow 2\text{CO}_2 + 3\text{H}_2\text{O} + 0.3\text{O}_2 \] The new balanced equation becomes:
\[ \text{C}_2\text{H}_5\text{OH} + 3.9\text{O}_2 \rightarrow 2\text{CO}_2 + 3\text{H}_2\text{O} + 0.9\text{O}_2 \]

energy balance

The energy balance is a fundamental principle in thermodynamics. For a steady-state reactor, the energy balance helps us evaluate the heat transfer and changes in internal energy of the system. To perform an energy balance for a combustion process:
\[ Q = n_{fuel} \times (h_{products} - h_{reactants}) \] Where:

• Q is the heat transfer
• n_{fuel} is the moles of fuel
• h_{products} and h_{reactants} are the specific enthalpies of the products and reactants

Since kinetic and potential energy effects are negligible, the heat transfer can be directly related to changes in specific enthalpies.

specific enthalpies

Specific enthalpy (\text{h}) measures the total heat content per unit mass of a substance. To appraise this for the combustion reaction, we look at the specific enthalpies of the reactants (ethanol and air) and products (carbon dioxide and water) at given temperatures. This can be represented as:
\[ \text{h}_{ethanol} + 3.9\text{h}_{O_2} \rightarrow 2\text{h}_{CO_2} + 3\text{h}_{H_2O} + 0.9\text{h}_{O_2} \] We use standard enthalpy tables to find these values at different temperatures. The enthalpy change for the reaction is determined by subtracting the enthalpies of reactants from products.

specific entropy

Specific entropy (\text{s}) is a thermodynamic property that reflects the disorder or randomness in a system. For exergy destruction, we need to determine the change in entropy (\text{Δs}) which occurs during the combustion process. This is given by:
\[ \text{Δs}_{gen} = n_{fuel} \times (\text{s}_{products} - \text{s}_{reactants}) \] Like enthalpy, we derive specific entropy values from standard tables. The entropy of reactants and products at various temperatures helps us find the net entropy change, crucial for exergy analysis.

theoretical air

Theoretical air is the exact amount of air needed for complete combustion of a fuel without any excess. Complete combustion of ethanol requires 3 moles of oxygen, known as stoichiometric air. However, practical applications often use excess air to ensure complete combustion. In our case, 130% of the theoretical air means we use 1.3 times the stoichiometric amount:
\[ 3 \times 1.3 = 3.9 \text{ moles of O}_2 \] This excess air influences the total mass and energy calculations, impacts the heat transfer, and alters the specific enthalpy and entropy values.

exergy analysis

Exergy analysis assesses the efficiency of energy conversion processes, accounting for irreversibilities. Exergy destruction in a combustion process indicates energy lost during conversion due to irreversibility. The exergy destruction rate is determined by:
\[ \text{Exergy destruction} = \text{T}_0 \times \text{Δs}_{gen} \] Where \text{T}_0 is the surrounding temperature (25°C or 298K), and \text{Δs}_{gen} is the entropy generation change. By calculating \text{Δs}_{gen} and integrating over the temperature range of interest, we predict the extent of exergy destruction, which helps improve reactor efficiency.