Chapter 7, Problem 7.18P

Interpretation Introduction

Interpretation:

The gas discharge temperature and the work output of the turbine per mole of the gas under the given conditions need to be calculated

Concept Introduction:

• For an ideal gas under isentropic adiabatic conditions, the temperature and pressure are related as:

$\frac{{\text{T}}_{\text{2}}}{{\text{T}}_{\text{1}}}\text{=}{\left(\frac{{\text{P}}_{\text{2}}}{{\text{P}}_{\text{1}}}\right)}^{\text{γ-1/γ}}---(1)$

${\text{where, γ = C}}_{\text{p}}{\text{/C}}_{\text{v}}$

• The isentropic efficiency of a turbine is given as:

$\text{η}=\frac{\text{Actual work done by turbine}}{\text{ Adiabatic Work }}=\frac{{\text{W<mo>˙</mo>}}_{\text{a}}}{{\text{W<mo>˙</mo>}}_{\text{s}}}\text{ ----(3)}$

$\text{where, the rate of work done is given as:}$

$\text{W<mo>˙</mo> = n<mo>˙</mo>}\Delta \text{H -----(4)}$

$\text{n<mo>˙</mo> = molar flow rate}$

$\Delta \text{H = change in enthalpy}$

$\text{Thus, turbine efficiency is given as:}$

$\text{η =}\frac{{\text{W<mo>˙</mo>}}_{\text{a}}}{{\text{W<mo>˙</mo>}}_{\text{s}}}=\frac{{\text{H}}_{\text{2a}}{\text{-H}}_{\text{1}}}{{\text{H}}_{\text{2s}}{\text{-H}}_{\text{1}}}$

$\text{(or) η }=\frac{{\text{ΔH}}_{\text{actual}}}{{\text{ΔH}}_{\text{adiabatic}}}\text{ ----(5)}$

The gas discharge temperature = $593.8\text{C}$

Work output of turbine = -8.78 kJ/mol

Given Information:

Inlet pressure, P₁ = 10 bar

Inlet Temperature, T₁= $950\text{C}$

Outlet pressure, P₂ = 1.5 bar

Heat capacity, C_p = 32 J.mol-1.K-1

Turbine efficiency, ? = 77% = 0.77

Explanation:

In this case, the combustion products have been assumed to behave as an ideal gas. The discharge temperature can be deduced from equation (1) and the work output can be deduced from equation (4).

The heat capacity ratio for gas turbine, i.e. ? = 1.33

Calculation:

Step 1:

Calculate the discharge/final temperature, T₂

Based on equation (1) we have:

$\frac{{\text{T}}_{\text{2}}}{{\text{T}}_{\text{1}}}\text{=}{\left(\frac{{\text{P}}_{\text{2}}}{{\text{P}}_{\text{1}}}\right)}^{\text{γ-1/γ}}$

$\frac{{\text{T}}_{\text{2}}}{950}={\left(\frac{1.5}{10}\right)}^{\text{1}\text{.33-1/1}\text{.33 }}=0.625$

${\mathrm{T}}_2=593.8C$

Step 2:

Calculate the actual enthalpy change

The adiabatic enthalpy change is related to the change in temperature through the specific heat capacity, C_p

${\text{ΔH}}_{\text{adiabatic}}{\text{ = C}}_{\text{p}}{\text{(T}}_{\text{2}}{\text{-T}}_{\text{1}}\text{) = 32(593}\text{.8-950) = -11}\text{.4 kJ/mol}$

Based on equation (5) the actual enthalpy change is:-

${\text{ ΔH}}_{\text{actual}}\text{ = η}\times {\text{ΔH}}_{\text{adiabatic}}\text{ = 0}\text{.77}\times \text{(-11}\text{.4) = -8}\text{.78 kJ/mol}$

Step 3:

Calculate the work output of the turbine

Based on equation (4)

$\text{ W<mo>˙</mo> = n<mo>˙</mo>}\Delta {\text{H}}_{\text{actual}}$

$\frac{\text{W<mo>˙</mo>}}{\text{n<mo>˙</mo>}}=\text{ work output = }\Delta {\text{H}}_{\text{actual}}\text{ = -8}\text{.78 kJ/mol}$

Thus, the gas discharge temperature = $593.8\text{C}$

Work output of turbine = -8.78 kJ/mol