Evaluate the feasibility of heating the digester designed for Omega Three (Example 27-6) with the methane produced in the digester. The proposed digester design and temperatures are given in the sketch below. Ignore the slope of the bottom of the digester in calculating the area of the bottom. The

Question

Evaluate the feasibility of heating the digester designed for Omega Three (Example 27-6) with the methane produced in the digester. The proposed digester design and temperatures are given in the sketch below. Ignore the slope of the bottom of the digester in calculating the area of the bottom. The mean temperature in January is −8°C. The incoming sludge temperature is 10°C. Assume conservative values for U, a heat exchange efficiency of 80%, and a lower heating value for methane of 22.40 MJ/m³.

Accepted Answer

a. Compute the heat requirement to raise the sludge temperature to the digester temperature. From Example 27-3, the mass of sludge is 3,461 kg per pumping cycle and there are three pumping cycles per day.

[latex]q_{r}[/latex] = (3, 461 kg/cycle)(3 cycles/d)(4.186 kJ/kg ⋅ K)[(273.15 K + 35 K)
−(273.15 K + 10 K)]
[latex] = 1.09 × 10^{6} kJ/d = 1.09 × 10^{9} J/d[/latex]

b. Compute the areas for the walls, roof, and floor.

Wall area above ground = (π)(31 m)(3 m) = 292.2 m²
Wall area below ground = (π)(31 m)(4.5 m) = 438.3 m²
Roof area = [latex]\frac{(π)(31 m)^{2}}{4} [/latex] = 754.8 m²
Floor area = Roof area

c. Compute the heat loss by conduction using Equation 27-28.

[latex]q_{L} = (U)(A)(ΔT)[/latex] (27-28)

Wall above ground:
[latex]q_{L}[/latex] = (0.8 W/m² ⋅ K)(292.2 m²)(35°C + 8°C)(86, 400 s/d) = 8.68 × 10[latex]^{8}[/latex] J/d

Wall below ground:
[latex]q_{L}[/latex] = (1.4 W/m² ⋅ K)(438.3 m²)(35°C − 9°C)(86, 400 s/d) = 1.38 × 10[latex]^{9}[/latex] J/d

Roof:
[latex]q_{L}[/latex] = (1.0 W/m² ⋅ K)(754.8 m²)(35°C + 8°C)(86, 400 s/d) = 2.8 × 10[latex]^{9}[/latex] J/d

Floor:
[latex]q_{L}[/latex] = (2.85 W/m² ⋅ K)(754.8 m²)(35°C − 9°C)(86, 400 s/d) = 4.83 × 10[latex]^{9}[/latex] J/d

d. The total capacity of the heat exchanger required is

Capacity = [latex]1.09 × 10^{9} J/d + 8.68 × 10^{8} J/d + 1.38 × 10^{9} J/d + 2.8 × 10^{9} J/d + 4.83 × 10^{9} J/d[/latex]
[latex]= 1.10 × 10^{10} J/d[/latex]

e. With a heat exchanger efficiency of 80%, the lower heating value of the fuel must be
[latex]\frac{1.10 × 10^{10} J/d}{0.8} = 1.37 × 10^{10} J/d = 1.4 × 10^{4} MJ/d[/latex]

f. The methane produced by the digester has a lower heating value of
(395.3 m³/d)(22.40 MJ/m³) = 8.9 × 10³ MJ/d

The methane production will not be enough to maintain the digester temperature during January.

Comments:
1. Note that this assessment is the worst-case scenario, that is, January. For months with more moderate temperatures, the digester should be self-sustaining.

2. The options for design to overcome the need for more energy to maintain the temperature include auxiliary fuel, storing methane during the warmer months, and increasing the insulation to reduce the heat loss.

Question 27.7: Evaluate the feasibility of heating the digester designed fo...

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