Chapter 6, Problem 6.1P

Interpretation Introduction

Interpretation:

It should be shown that isobars in the vapor region of the Mollier diagram must have positive slope and positive curvature.

Concept Introduction:

Mollier diagram is graph between enthalpy and entropy. The slope of this diagram must be

  $\frac{dH}{dS}$

And for the isobars or constant pressure, the slope of this diagram would be

  ${(\frac{\partial H}{\partial S})}_P$

For the positive curvature isobars

  ${(\frac{{\partial }^{2}H}{\partial S^2})}_P$

Must be positive.

Answer to Problem 6.1P

The slope of the isobar in this diagram is:

  ${(\frac{\partial H}{\partial S})}_P=T$ And it is positive.

The curvature equation for the isobars is:

  ${(\frac{{\partial }^{2}H}{\partial S^2})}_P=\frac{T}{C_P}$

And it is positive

Explanation of Solution

Given information:

It is given that the fundamental equation $dH=TdS+VdP$ is applicable.

Differentiate given equation with respect to entropy keeping pressure constant

  $dH=TdS+VdP$

  ${(\frac{\partial H}{\partial S})}_P=T$

So, isobars in Mollier diagram has positive slope $T$ . Now for positive curvature, differentiate the above equation with respect to entropy,

  ${(\frac{{\partial }^{2}H}{\partial S^2})}_P={(\frac{\partial T}{\partial S})}_P$

As ${(\frac{\partial T}{\partial S})}_P=\frac{T}{C_P}$

So,

  ${(\frac{{\partial }^{2}H}{\partial S^2})}_P=\frac{T}{C_P}$

Which is positive

Conclusion

The isobars in Mollier diagram have straight lines because of constant pressure. This is the reason in superheated region with addition of heat at constant pressure, temperature keeps on increasing and if we draw them on enthalpy entropy curve it diverge from each other.