Chapter 7, Problem 7.21P

Interpretation Introduction

Interpretation:

The discharge temperature and work produced per mole of carbon dioxide gas under the given conditions needs to be calculated

Concept Introduction:

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

$\begin{array}{l}\frac{{\text{T}}_{\text{2}}}{{\text{T}}_{\text{1}}}\text{=}{\left(\frac{{\text{P}}_{\text{2}}}{{\text{P}}_{\text{1}}}\right)}^{\text{γ-1/γ}}---(1)\\ {\text{where, γ = C}}_{\text{p}}{\text{/C}}_{\text{v}}\end{array}$

• The work done under isentropic conditions is:

$\text{W = }\frac{{\text{nR(T}}_{\text{1}}{\text{-T}}_{\text{2}}\text{)}}{\text{γ-1}}\text{ -----(2)}$

The initial temperature = $247.6\text{C}$

Work done/mole of nitrogen = 4223.5 J

Given Information:

Inlet pressure of steam, P₁ = 8. bar

Inlet Temperature of steam T₁= $400\text{C}$

Outlet pressure of steam, P₂ = 1.0 bar

Explanation:

In this case, carbon dioxide has been assumed to behave as an ideal gas. The discharge temperature can be deduced from equation (1) and the work done per mole of carbon dioxide can be calculated from equation (2).

The heat capacity ratio for CO₂, i.e. C_p/C_v = 1.3

Calculation:

Step 1:

Calculate the discharge/final temperature, T₂

Based on equation (1) we have:

$\begin{array}{l}\frac{{\text{T}}_{\text{2}}}{{\text{T}}_{\text{1}}}\text{=}{\left(\frac{{\text{P}}_{\text{2}}}{{\text{P}}_{\text{1}}}\right)}^{\text{γ-1/γ}}\\ \frac{{\text{T}}_{\text{2}}}{\text{400}}={\left(\frac{1}{8}\right)}^{\text{1}\text{.3-1/1}\text{.3 }}=0.619\\ {\mathrm{T}}_1=247.6C\end{array}$

Step 2:

Calculate the work done

Based on equation (2) for 1 mole of nitrogen we have:

$\text{W = }\frac{\text{1}\times \text{8}\text{.314}\times \text{(400-247}\text{.6)}}{\text{1}\text{.3-1}}\text{ = 4223}\text{.5 J}$

Thus, the initial temperature = $247.6\text{C}$

Work done/mole of nitrogen = 4223.5 J