Gus Omer Randy Hurd

Determining the most economic method for heating diesel fuel in a reservoir from 20 to 100 °C in under 90 minutes. PROBLEM

Three different power heaters are available from the chosen manufacturer: 1500 W, 2500 W and 3750 W. The 3750 W heater will obviously heat the fuel the fastest, but because our target heat-up time is 90 minutes it may make more sense economically to choose a smaller heater that will cost less to purchase and run long-term. INITIAL THOUGHT

Steel tank filled with diesel fuel Wall thickness = m k = 50 W/m∙K RESERVOIR m m m

Uniform heating Bottom of tank is insulated (q=0) Thin walled tank (fuel and reservoir at same temperature) Quiescent air (no forced convection) Lumped capacitance for natural convection ASSUMPTIONS

Mild Steel k = 50 W/m k C = 500 J/kg K ρ = 7,850 kg / m 3 Diesel Fuel C = 1,800 J/kg K ρ = 875 kg / m 3 Mathematic ModelMaterial Properties APPROACH

The coefficient h was calculated assuming all heat transfer through convection was due to natural convection in quiescent air. Separate Rayleigh, Nusselt, and h values were calculated for the top of the fuel reservoir and the sides. The weighted average is h ≈ 7 The total heat transfer q was then calculated: −q = A s h(T s – T ∞ ) CONVECTION

RADIATION Properties: − ε = 0.08 Heat Transfer: −q = A s εσ (T s 4 - T sur 4 ) The calculated heat transfer through radiation was comparatively small and could safely be considered insignificant in calculating temperature increase with respect to time.

RESULTS

Assuming the fuel reservoir heater was run for 16 hours a day, 300 days per year at a cost of 8.51¢/kWh, the annual cost to run one unit would be: −1500 W: $ −2500 W: $ −3750 W: $ In addition, smaller power heaters require a smaller initial investment. FINANCES

According to our calculation only the 2500W and 3750 W heaters were capable of accomplishing the proposed task. The initial cost of the heaters is relatively insignificant when compared to energy costs over long periods of time. When energy costs are considered the 2500W heater proves ideal because it performs the proposed task for the cheapest energy cost. CONCLUSION

QUESTIONS