Primeira Lei da Termodinâmica: Resolvendo problemas via métodos iterativos#

Alguns problemas em termodinâmica só podem ser resolvidos por métodos iterativos. Isto ocorre porque geralmente trabalhamos com equações de grau 3 ou superior, sistema de equações com mais variávies que equações possíveis e tipo de dependência entre variáveis do sistema. Vamos ilustrar como problemas deste tipo podem resolvidos utilizando um exemplo de cálculo de temperatura de chama adiabática.

Os seguintes módulos precisam ser carregados para resolução dos exercícios:

#bibliotecas necessárias para resolução de exercícios
import numpy as np
import sympy as sp

Exemplo 1: Cálculo da temperatura de chama adiabática, ou temperatura máxima teórica, da gasolina.#

Assumindo que a gasolina é composta majoritariamente por iso-octano, \(C_8 H_{18}\), e que este é inserido na câmara de combustão em fase gasosa: calcule a temperatura adiabática de chama para queima de 1,0 g gasolina em ar atmosférico, assumindo quantidades estequiométricas de oxigênio.

Dados:

\(C_8 H_{18}\)(g)

\(CO_2\)(g)

\(H_2O\)(g)

\(O_2\)(g)

\(\Delta_f Hº(kJ mol^{-1})\)

-224,0

-393,51

-241,82

-

\[\]
\[Cp(CO_2) = 23,585 + 0,044813T (J K^{-1} mol^{-1})\]
\[Cp(N_2) = 28,986 – 0,001854T (J K^{-1} mol^{-1})\]
\[Cp(H_2 O) = 30,092 + 0,00683T (J K{-1} mol^{-1})\]

Solução#

A solução é obtida resolvendo a seguinte equação: \(\int_{T_0}^{T} dH = \int_{T_0}^{T} Cp dT\).Onde é assumido que a combustão é completa e existem apenas os produtos da combustão e \(N_2\) do ar atmosférico ao fim da reação. Desta forma é preciso:

  • Fazer o cálculo estequiométrico para determinar número de mols de cada produto formado

  • Calcular \(\Delta Cp\)

  • Integrar simbolicamente a equação resultante

  • De forma iterativa calcular a temperatura da chama.

Reação de combustão:

\[ C_8 H_{18(l)} + 12.5O_{2(g)} \rightarrow 8CO_{2(g)} + 9H_2 O_{(g)}\]
#a) Estequiometria

# quantidade de matéria de gasolina (n)


n = 1 / 114.23     # mol   

n_co2 = 8*n        # mol
n_h2o = 9*n        # mol  
n_o2 = 12.25*n
n_n2 = 79 * n_o2/21
#b) calculando Cp dos produtos

cp_co2 = np.array([23.585,0.044813])
cp_n2 = np.array([28.986,-0.001854])
cp_h2o = np.array([30.092,0.00683])

cp = n_co2 *  cp_co2 + n_n2 * cp_n2 + n_h2o * cp_h2o
#c) Resolvendo a integral simbolicamente
# Define a variável simbólica 
T = sympy.Symbol('T')
def Cp (T): 
    return cp[0] + cp[1] * T

# Integração
res = sympy.integrate(Cp(T), T)
print("Cp dos produtos= ", res)
---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
Input In [4], in <cell line: 3>()
      1 #c) Resolvendo a integral simbolicamente
      2 # Define a variável simbólica 
----> 3 T = sympy.Symbol('T')
      4 def Cp (T): 
      5     return cp[0] + cp[1] * T

NameError: name 'sympy' is not defined
#d) calculando Tad
# Calculando Delta H de reação

DH = n_co2 * (-393.51) + n_h2o * (-241.82) - n * (-224.0) # em kJ/mol


# loop while para achar Tad

Q = -10 # J/mol
T = 1000 # Valor estimado de Tad deve ser bem acima da temperatura inicial

while Q > DH*1000: # multiplicado por 1000 para adequar as unidades de energia 
    
    T += 2
    Q = -(0.00146430666199772*(T**2-298.15**2) + 15.7163573492077*(T-298.15))
    print(T,Q)

print("Temperatura de chama adiabática= " + format(T , ' 5.2f'), "K")
    
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Temperatura de chama adiabática=  2546.00 K