Question

An automobile engine uses methyl alcohol (CH \(_{3} \mathrm{OH}\) ) as fuel with 200 percent excess air. Air enters this engine at 1 atm and \(25^{\circ} \mathrm{C}\). Liquid fuel at \(25^{\circ} \mathrm{C}\) is mixed with this air before combustion. The exhaust products leave the exhaust system at 1 atm and \(77^{\circ} \mathrm{C}\). What is the maximum amount of work, in \(\mathrm{kJ} / \mathrm{kg}\) fuel, that can be produced by this engine? Take \(T_{0}=25^{\circ} \mathrm{C} .\)

Short answer

Answer: To determine the maximum amount of work produced by the engine, first write down the balanced combustion equation and determine the quantities of exhaust products. Next, note the inlet and outlet conditions, and use these conditions to find the change of specific Gibbs free energy (\(\Delta G\)) between products and reactants. Finally, the maximum work output is given by \(-\Delta G\), which can then be expressed in kJ/kg fuel.

Step-by-step solution

Write down the balanced combustion equation

First, we need to write down the balanced combustion equation for the reaction of methyl alcohol and air (oxygen and nitrogen). For each mole of methyl alcohol (\(CH_3OH\)), we will consider 1 mole of oxygen (\(O_2\)) as stoichiometric, and \(3.76\) moles of nitrogen (\(N_2\)) as inert gas: $$ CH_3OH + (1 + 2)O_2 + 3.76(1+2)N_2 \rightarrow x_1CO_2 + x_2H_2O + x_3N_2 $$

Determine the quantities of products in the combustion process

Based on the balanced equation obtained in the previous step, we can determine the quantities of exhaust products as follows: $$ x_1 = 1, \quad x_2= 2, \quad x_3 = 3.76(1+2) $$

Determine the inlet and outlet conditions of the engine

From the exercise, we are given that air enters the engine at 1 atm and \(25^\circ C\) and leaves the engine after the combustion process at 1 atm and \(77^\circ C\). Inlet conditions: \(P_1 = 1\,atm, \ T_1 = 25^\circ C\) Outlet conditions: \(P_2 = 1\,atm, \ T_2 = 77^\circ C\)

Determine the change of specific Gibbs free energy for the engine

Using the conditions we noted in the previous step, we will determine the change of specific Gibbs free energy (\(\Delta G\)): $$ \Delta G = G_{products}(T_2, P_2) - G_{reactants}(T_1, P_1) $$ We need to find the values for \({G}_{products}\) and \({G}_{reactants}\) using the Janaf tables or thermodynamics tables.

Determine the maximum work output of the engine

The maximum work that can be produced by the engine is given by the change of specific Gibbs free energy (\(\Delta G\)) determined in Step 4: $$ W_{max} = -\Delta G $$ Now that we have determined the maximum work output, we can express it in kJ/kg fuel.