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A 0.80 m deep bed of sand filter (length 4m and width 3m) is made of uniform particles (diameter = 0.40 mm, specific gravity = 2.65, shape factor = 0.85) with bed porosity of 0.4. the bed has to be backwashed at a flow rate of 3.60 m3/min. During backwashing, if the terminal settling velocity of sand particles is 0.05 m/s, the expanded bed depth (in m, round off to 2 decimal places) is ______
Correct answer is '1.2075'. Can you explain this answer?
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Calculation of Minimum Fluidization Velocity:
The first step in determining the expanded bed depth is to calculate the minimum fluidization velocity (Umf) of the sand particles. This can be done using the Ergun equation:
ΔP/L = (150(1-ε)ρu + 1.75ρu2)/ε3d2
Where:
ΔP/L = Pressure drop per unit bed length (Pa/m)
ε = Bed porosity
ρ = Density of fluid (kg/m3)
u = Fluid velocity (m/s)
d = Diameter of sand particles (m)
Rearranging the equation to solve for Umf, we get:
Umf = (4ε3d2ΔP/L/150ρ)1/2
Substituting the given values, we get:
Umf = (4(0.4)3(0.0004)2(0.8)(9.81)/150(2650))1/2
Umf = 0.067 m/s
Calculation of Expanded Bed Depth:
The next step is to determine the expanded bed depth, which can be calculated using the Richardson-Zaki equation:
V/Vmf = (h/hmf)α
Where:
V = Fluid velocity (m/s)
Vmf = Minimum fluidization velocity (m/s)
h = Bed depth (m)
hmf = Expanded bed depth (m)
α = Richardson-Zaki exponent (assumed to be 3)
Rearranging the equation to solve for hmf, we get:
hmf = h(V/Vmf)1/α
Substituting the given values, we get:
V = 3.60 m3/min = 0.06 m3/s
h = 0.80 m
Vmf = 0.067 m/s
α = 3
Converting the volumetric flow rate to fluid velocity, we get:
V/A = Q/Av = 0.06/(4*3) = 0.005 m/s
Substituting this value into the equation, we get:
hmf = 0.80(0.005/0.067)1/3
hmf = 1.2075 m (rounded to 2 decimal places)
The first step in determining the expanded bed depth is to calculate the minimum fluidization velocity (Umf) of the sand particles. This can be done using the Ergun equation:
ΔP/L = (150(1-ε)ρu + 1.75ρu2)/ε3d2
Where:
ΔP/L = Pressure drop per unit bed length (Pa/m)
ε = Bed porosity
ρ = Density of fluid (kg/m3)
u = Fluid velocity (m/s)
d = Diameter of sand particles (m)
Rearranging the equation to solve for Umf, we get:
Umf = (4ε3d2ΔP/L/150ρ)1/2
Substituting the given values, we get:
Umf = (4(0.4)3(0.0004)2(0.8)(9.81)/150(2650))1/2
Umf = 0.067 m/s
Calculation of Expanded Bed Depth:
The next step is to determine the expanded bed depth, which can be calculated using the Richardson-Zaki equation:
V/Vmf = (h/hmf)α
Where:
V = Fluid velocity (m/s)
Vmf = Minimum fluidization velocity (m/s)
h = Bed depth (m)
hmf = Expanded bed depth (m)
α = Richardson-Zaki exponent (assumed to be 3)
Rearranging the equation to solve for hmf, we get:
hmf = h(V/Vmf)1/α
Substituting the given values, we get:
V = 3.60 m3/min = 0.06 m3/s
h = 0.80 m
Vmf = 0.067 m/s
α = 3
Converting the volumetric flow rate to fluid velocity, we get:
V/A = Q/Av = 0.06/(4*3) = 0.005 m/s
Substituting this value into the equation, we get:
hmf = 0.80(0.005/0.067)1/3
hmf = 1.2075 m (rounded to 2 decimal places)
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A 0.80 m deep bed of sand filter (length 4m and width 3m) is made of uniform particles (diameter = 0.40 mm, specific gravity = 2.65, shape factor = 0.85) with bed porosity of 0.4. the bed has to be backwashed at a flow rate of 3.60 m3/min. During backwashing, if the terminal settling velocity of sand particles is 0.05 m/s, the expanded bed depth (in m, round off to 2 decimal places) is ______Correct answer is '1.2075'. Can you explain this answer?
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