Chapter 10, Problem 10.1P

Show that, for water at 1-atm pressure with $T_s-T_{sat}=10°C$ , the Jakob number is much less than unity. What is the physical significance of this result? Verify that this conclusion applies to other fluids.

To determine

The physical significance of Jakob number for water and verify if this conclusion applies to other fluids and to show that Jakob number is unity for water.

Answer to Problem 10.1P

The Jakob number is less than unity for water and this applies to all the fluids which signify that Jakob number is always to be much less than unity.

Explanation of Solution

Given:

Pressure of water is $1\text{ atm}$ .

The temperature difference $T_s-T_{sat}$ is $10°\text{C}$ .

Concept used:

Refer to table A-6 “Thermo physical properties of saturated water at pressure of 1.0133 bar” to obtain the value of heat of vaporization as $2257\text{ kJ/kg}$ and specific heat of water as $4.217\text{kJ/kg}\cdot \text{K}$ corresponding to 1 atm pressure.

Write the expression for Jakob number of water.

  $Ja=\frac{c_p\left(T_s-T_{sat}\right)}{h_{fg}}$ …… (1)

Here, $Ja$ is the Jakob number, $c_p$ is the specific heat of water, $T_s$ is the surface temperature, $h_{fg}$ is the heat of vaporization and $T_{sat}$ is the saturation temperature.

Calculation:

Substitute $10°\text{C}$ for $T_s-T_{sat}$ , $2257\text{ kJ/kg}$ for $h_{fg}$ and $4.217\text{kJ/kg}\cdot \text{K}$ for $c_p$ in equation (1).

  $\begin{array}{c}Ja=\frac{4.217\left(10\right)}{2257\text{}}\\ =0.01868\end{array}$

The value of Jakob number is $0.01868$ which less than unity therefore the statement is correct for water.

As the Jakob number is very less for water, it implies that the vapors absorbing latent heat is more than the vapors absorbing the sensible heat during the boiling phase change.

Check the value of Jakob number for other fluids like Ethylene glycol and mercury.

Refer to table A-5 “Thermo physical properties of saturated fluids at pressure of 1 atm” to obtain the value of heat of vaporization for ethylene glycol as $812\text{ kJ/kg}$ and specific heat of ethylene glycolat temperature of $\text{373K}$ as $2.742\text{kJ/kg}\cdot \text{K}$ corresponding to 1 atm pressure.

Substitute $10°\text{C}$ for $T_s-T_{sat}$ , $812\text{ kJ/kg}$ for $h_{fg}$ and $2.742\text{kJ/kg}\cdot \text{K}$ for $c_p$ in equation (1).

  $\begin{array}{c}Ja=\frac{2.742\left(10\right)}{812\text{}}\\ =0.0338\end{array}$

Thus, the value of Jakob number for ethyl glycol is also than unity.

Refer to table A-5 “Thermo physical properties of saturated fluids at pressure of 1 atm” to obtain the value of heat of vaporization for mercury as $301\text{ kJ/kg}$ and specific heat of mercury at temperature of $\text{630K}$ as $0.1355\text{kJ/kg}\cdot \text{K}$ corresponding to 1 atm pressure.

Substitute $10°\text{C}$ for $T_s-T_{sat}$ , $301\text{ kJ/kg}$ for $h_{fg}$ and $0.1355\text{kJ/kg}\cdot \text{K}$ for $c_p$ in equation (1).

  $\begin{array}{c}Ja=\frac{0.1355\left(10\right)}{301\text{}}\\ =0.0045\end{array}$

Thus, the value of Jakob number for mercury is also than unity.

Conclusion:

Thus, the Jakob number is less than unity for water and this applies to all the fluids which signify that Jakob number is always to be much less than unity.