Chapter 3, Problem 3.26P

Interpretation Introduction

Interpretation:

It is to derive an equation for final temperature of gas remaining in the tank contains gas at some given temperatureinitially after tank has been bled from an initial pressure to final temperature at given initial and final pressure $P_1$ and $P_2$ .

Concept Introduction:

Since temperature is involved in this problem so energy balance is required. Since assumption is that gas is ideal with constant properties, so energy balance shall be

  $\stackrel{•}{Q}=\stackrel{•}{n^{\text{'}}}{H}^{\text{'}}+\frac{d(nU)}{dt}$

Where $n^{\text{'}}$ and $H^{\text{'}}$ are the molar rate and enthalpy of exit streams so energy balance has positive sign.

This is a throttling process since stream bled from the tank is throttled. So, $\begin{array}{l}\Delta H=0\\ H=\text{constant}\end{array}$

Or $H_1=H^{\text{'}}$.....(a)

In terms of moles or material balance is

  $\frac{dn}{dt}+\Delta n=0$

Where $\frac{dn}{dt}$ is the rate of change of moles of air inside the room and $\Delta n$ is difference between exit and entrance of number of moles of air.

For steady state $\frac{dn}{dt}$ that implies $\Delta n=0$

Or $d{n}_1=-d{n}^{\text{'}}$.....(b)

To solve the problem, it is needed to find a relation which relates heat transfer of the tank to quantity of air supplies to it and its temperaturechange. Hence,

  $n_1=\frac{P_1{V}^t}{RT}$

And

  $n_2=\frac{P_2{V}^t}{RT}$

Where $V^t$ is the total volume of the tank, $n_1$ and $n_2$ are the number of moles at different pressure and T is constant temperature.

So, the quantity of air supplies to tank is

  $n^{\text{'}}=n_2-n_1=\frac{(P_2-P_1)V^t}{RT}$

For non adiabatic, no shaft work steady state neglecting kinetic and potential terms, the resultant energy balance shall be

  $\stackrel{•}{Q}=-n^{\text{'}}{H}^{\text{'}}+\frac{d(nU)}{dt}$....(1)

Multiplying by $dt$ and integrating with respect to time implies

  $\begin{array}{l}Q=n_2{U}_2-n_1{U}_1-n^{\text{'}}{H}^{\text{'}}\\ Q=n_2(U_2-H^{\text{'}})-n_1(U_1-H^{\text{'}})\\ Q=n_2(H_2-RT-H^{\text{'}})-n_1(H_1-RT-H^{\text{'}})\\ Q=n_2(C_P(T-T^{\text{'}})-RT)-n_1(C_P(T-T^{\text{'}})-RT)\end{array}$

Hence

  $Q=n^{\text{'}}(C_P(T-T^{\text{'}})-RT)$....(2)