### Pressure Drop in Regenerative Heater

A regenerative heater is packed with a bed of 6 mm cubes. The cubes are poured into the cylindrical shell of the regenerator to a depth of 3.5 m such that the bed porosity was 0.44. If air flows through this bed entering at 25oC and 7 atm abs and leaving at 200oC, calculate the pressure drop across the bed when the flow rate is 500 kg/hr per square meter of empty bed cross section. Assume average viscosity as 0.025 cP and density as 6.8 kg/m3.

Data:

Mass flow rate of Air / unit area = 500 kg/(hr.m2) = 0.139 kg/(sec.m2)

Density of Air (ρ) = 6.8 kg/m3

Viscosity of Air (μ) = 0.025 cP = 0.025 x 10-3 kg/(m.sec)

Bed porosity (ϵ) = 0.44

Length of bed (L) = 3.5 m

Dia of particles (Dp)= 6 mm = 0.006 m

Formulae:

Sphericity (φs) = 6 vp / (DpSp)

vp = volume of particle = Dp3 (for cube)

Sp = surface area of particle = 6 x Dp2 (for cube)

NRePM = DpVoρ/(μ(1 - ϵ))

For laminar flow (i.e. NRePM < 10) pressure drop is given by Blake-Kozeny equation.

$$\displaystyle \frac{\Delta p \phi_s^2D_p^2\varepsilon^3}{LV_o\mu(1-\varepsilon)^2} = 150$$

For turbulent flow (i.e. NRePM > 1000) pressure drop is given by Burke-Plummer equation.

$$\displaystyle \frac{\Delta p \phi_sD_p\varepsilon^3}{L\rho V_o^2(1-\varepsilon)} = 1.75$$

For intermediate flows pressure drop is given by Ergun equation

$$\displaystyle \frac{\Delta p \phi_sD_p\varepsilon^3}{L\rho V_o^2(1-\varepsilon)} = \frac{150\mu(1-\varepsilon)}{\phi_sD_pV_o\rho} + 1.75$$

Superficial velocity Vo = Volumetric flow rate/ cross-sectional area of bed

Calculations:

Superficial velocity Vo = mass flow rate per unit area / density = 0.139 / 6.8 = 0.0204 m/sec

s) = 6 vp / (DpSp) = 6 x 0.0063 / (0.006 x 6 x 0.0062) = 1

NRePM= 0.006 x 0.0204 x 6.8 / (0.025 x 10-3 x (1-0.44)) = 59.45

We shall use Ergun equation to find the pressure drop.

$$\displaystyle \frac{\Delta p \phi_sD_p\varepsilon^3}{L\rho V_o^2(1-\varepsilon)} = \frac{150\mu(1-\varepsilon)}{\phi_sD_pV_o\rho} + 1.75$$

i.e. Δp x 0.006 x 0.443 / ( 3.5 x 6.8 x 0.02042 x ( 1 - 0.44 ) ) = 150 / 59.45 + 1.75

Δp x 0.0712= 4.273

Δp = 4.273 / 0.0712 = 60 N/m2