Chapter 9, Problem 9.38P

Interpretation Introduction

(a)

Interpretation:

The dendritic fraction of iron metal at ${10}^0\text{C}$ undercooling needs to be determined.

Concept Introduction:

Dendritic solidification is the process in which a liquid having lack of nucleating properties are undercooled to solidify the liquid.

A dendritic fraction is given by the following relation:

  $F=\frac{C\Delta T}{\Delta Hf}$

Answer to Problem 9.38P

Dendritic fraction of iron metal at ${10}^0\text{C}$ is 0.0333.

Explanation of Solution

Given Information:

Undercooling temperature ${10}^0\text{C}$, the specific heat of the iron $5.785{\text{/cm}}^{\text{3}}{}^{\text{0}}\text{C}$

Calculation:

A dendritic fraction is given by the following relation:

  $F=\frac{c\Delta T}{\Delta HF}$ ----------------------- (1)

Here,

  $\begin{array}{l}F=\mathrm{Dendritic}fraction\hfill \\ C=\text{Specific heat}\hfill \\ \Delta T=\text{Undercooling temperature}\hfill \\ \Delta HF=\text{Latent heat of fusion}\hfill \end{array}$

But

  $\begin{array}{l}C=5.78{\text{ s/cm}}^{\text{3}\text{0}}\text{C}\\ \Delta HF=17{\text{37 s/cm}}^{\text{3}}\end{array}$ (From latent heat table)

  $\Delta T={10}^0\text{C}$

Equation (1)becomes,

  $\begin{array}{l}F=\frac{5.78\times 10}{1737}\\ F=0.0333\end{array}$

Therefore dendritic fraction of iron nucleates at ${10}^0\text{C}$ is 0.0333.

Interpretation Introduction

(b)

Interpretation:

Thedendritic fraction of iron nucleates at ${100}^{\text{0}}\text{C}$ undercooling needs to be determined.

Concept Introduction:

Dendritic solidification is the process in which a liquid having lack of nucleating properties are undercooled to solidify the liquid.

A dendritic fraction is given by the following relation:

  $F=\frac{C\Delta T}{\Delta Hf}$

Answer to Problem 9.38P

The dendritic fraction of iron nucleates at ${100}^0\text{C}$ solidification is 0.333.

Explanation of Solution

Given Information:

Undercooling temperature ${\text{100}}^{\text{0}}\text{C}$

Specific heat of iron = $5.78{\text{s/cm}}^{\text{3}\text{o}}\text{C}$

Calculation:

A dendritic fraction is given by the following relation:

  $F=\frac{c\Delta T}{\Delta HF}$

Here,

  $\begin{array}{l}F=\text{Dendritic fraction}\hfill \\ C=\text{Specific heat}\hfill \\ \Delta T=\mathrm{Undercooling}temperature\hfill \\ \Delta HF=\text{Latent heat of fusion}\hfill \end{array}$

But latent of fusion as per general latent heat of fusion for all metals is $173{\text{7 J/cm}}^{\text{3}}$

  $\begin{array}{l}C=5.7{\text{8 s/cm}}^{\text{3}\text{0}}\text{C}\\ \Delta T=1{\text{00}}^{\text{0}}\text{C}\end{array}$

Equation (1) becomes,

  $\begin{array}{l}F=\frac{5.78\times 100}{1737}\\ F=0.333\end{array}$

Thus, the dendritic fraction of iron nucleates is 0.333.

Interpretation Introduction

(c)

Interpretation:

The dendritic fraction of homogenous iron nucleatewith specific heat of iron metal equals to $\text{5}{\text{.78 J/cm}}^{\text{3}\text{0}}\text{C}$ needs to be calculated.

Concept Introduction:

Homogeneous iron nucleate is the subcontains that are added in iron nucleates such as silicon tungsten, cobalt, zirconium is an equal proportion.

Answer to Problem 9.38P

The dendritic fraction of iron nucleates for homogeneous nucleation is 1.4, as dendritic fraction becomes greater than one, it indicates the pure solidification.

Explanation of Solution

Given Information:

Specific of iron nucleates = $5.78J/c{m}^{30}C$

Calculation:

Equation for dendritic fraction is as follow:

  $F=\frac{c\Delta T}{\Delta HF}$ --------------------- (1)

Here,

  $\begin{array}{l}\text{F=Dendritic fraction}\hfill \\ \text{C=Specific heat}\hfill \\ \text{ΔT=Undercooling temperature}\hfill \\ \text{ΔHF=Latent heat of fusion}\hfill \end{array}$

But at homogeneous nucleation of iron nucleates,

  $\begin{array}{l}\Delta T={420}^0\text{C}\\ C=5.7{\text{8 J/cm}}^{\text{3}\text{0}}\text{C}\\ \Delta HF=173{\text{7 J/cm}}^{\text{3}}\end{array}$

(From latent heat table)

Equation (1) becomes,

  $\begin{array}{l}F=\frac{5.78\times 100}{1737}\\ F=1.4\end{array}$

The dendritic fraction of iron nucleates at homogeneity is 1.4.

As F>1, pure solidification of iron nucleates formed.