Apebble bed consists of uniform size spherical bed particles of diameter Dp = 2 cm. The void fraction has been experimentally determined to be ε = 0.36. Air at a temperature of Ta = 40 °C is blown through the bed at the rate of m⋅ = 0.5 kg/m^2s. Determine the pressure drop per unit bed height.

Question

Apebble bed consists of uniform size spherical bed particles of diameter [latex]D_{p}=2 cm[/latex]. The void fraction has been experimentally determined to be [latex]\epsilon=0.36[/latex]. Air at a temperature of [latex]T_{a}=40{ }^{\circ} C[/latex] is blown through the bed at the rate of [latex]\dot{m}=0.5 kg / m ^{2} s[/latex]. Determine the pressure drop per unit bed height. Discuss the contributions due to viscous and inertial effects.

Accepted Answer

Air properties at [latex]40^{\circ} C[/latex] are given below

[latex] ho=1.117 kg / m ^{3}[/latex] ; [latex]C_{p}=1005 J / kg ^{\circ} C[/latex]; [latex]k=0.0274 W / m ^{\circ} C[/latex] ; [latex] u=1.71 imes 10^{-5} m ^{2} / s[/latex]

Other data specified in the problem are:

[latex]D_{p}=2 cm =0.02 m[/latex] ; [latex]\dot{m}=0.5 kg / m ^{2} s[/latex] ; [latex]\epsilon=0.36[/latex]

Using the mass flux specified, the superficial velocity is calculated as

[latex]U_{s}=\frac{\dot{m}}{ ho}=\frac{0.5}{1.117}=0.448 m / s[/latex]

Reynolds number based on particle diameter of [latex]D_{p}=2 cm =0.02 m[/latex] is given by

[latex]R e_{p}=\frac{U_{s} D_{p}}{ u}=\frac{0.448 imes 0.02}{1.71 imes 10^{-5}}=523.5[/latex]

Contribution of viscous term to friction factor is given by Eq. 17.118 as

Kozeny–Carman equation : [latex]f=\frac{(1-\epsilon)^{2}}{\epsilon^{3}} \frac{150}{R e_{p}}[/latex] (17.118)

[latex]f_{v}=\frac{\left(1-0.36^{2} ight)}{0.36^{3}} imes \frac{150}{523.5}=2.515[/latex]

Contribution of inertial term to friction factor is given by Eq. 17.119 as

Blake–Plummer equation : [latex]f=\frac{1.75(1-\epsilon)}{\epsilon^{3}}[/latex] (17.119)

[latex]f_{i}=\frac{1.75(1-0.36)}{0.36^{3}}=24.005[/latex]

It is seen that the inertial contribution is dominant in this case. The friction factor using Ergun equation will then be [latex]f=f_{i}+f_{v}=2.515+24.005=26.520[/latex] . Based on the definition of friction factor (Eq. 17.116), we have

[latex]f=\frac{\Delta p}{ ho U_{s}^{2}} \frac{D_{p}}{H}[/latex] (17.116)

[latex]\frac{\Delta p}{H}=-\frac{f ho U_{s}^{2}}{D_{p}}=-\frac{26.520 imes 1.117 imes 0.448^{2}}{0.02}=-296.79 Pa / m[/latex]

Question 17.14: Apebble bed consists of uniform size spherical bed particles...

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