To tackle the problem of heat flow through a steam boiler made of copper, we will apply the principles of thermal conduction. The heat transfer through a material can be calculated using Fourier's law of heat conduction, which states that the rate of heat transfer (Q) through a material is proportional to the temperature difference across the material and the area, and inversely proportional to the thickness of the material. Let's break this down step by step.
Calculating Heat Flow Rate Across the Boiler
First, we need to calculate the rate of heat flow across 1 m² of the boiler. The formula for heat transfer through conduction is given by:
Q = (k * A * ΔT) / d
- Q = rate of heat transfer (W)
- k = thermal conductivity of the material (W/mK)
- A = area (m²)
- ΔT = temperature difference (K)
- d = thickness of the material (m)
(a) Heat Flow Rate Calculation
For the copper plate:
- k (copper) = 393 W/mK
- A = 1 m²
- ΔT = T_outer - T_inner = 350°C - 200°C = 150 K
- d = 7 mm = 0.007 m
Substituting these values into the formula:
Q = (393 W/mK * 1 m² * 150 K) / 0.007 m
Calculating this gives:
Q = (58950 W) / 0.007 = 8421428.57 W
Thus, the rate of heat flow across 1 m² of the boiler is approximately 8421.43 W.
Mass of Water Evaporated to Steam
Next, we need to determine how much water is evaporated per second per square meter. The latent heat of vaporization of water is approximately 2260 kJ/kg. To find the mass flow rate (m), we can use the formula:
m = Q / L
- Q = heat flow rate (W)
- L = latent heat of vaporization (J/kg)
Converting L to J/kg:
L = 2260 kJ/kg = 2260000 J/kg
Now substituting the values:
m = 8421.43 W / 2260000 J/kg
Calculating this gives:
m ≈ 0.00373 kg/s
Therefore, the mass of water evaporated to steam per second per square meter is approximately 3.73 g/s.
Impact of Lime Scale on Heat Flow Rate
Now, let's consider the effect of a lime scale layer that is 0.5 mm thick on the inside of the copper plate. The total thickness of the material will now be the sum of the copper thickness and the lime scale thickness:
d_total = d_copper + d_limescale = 0.007 m + 0.0005 m = 0.0075 m
For the lime scale:
- k (lime scale) = 1 W/mK
- ΔT remains the same at 150 K
(c) New Heat Flow Rate Calculation
We will use the same formula for heat transfer, but now with the new total thickness:
Q_new = (k_copper * A * ΔT) / d_total
Substituting the values:
Q_new = (393 W/mK * 1 m² * 150 K) / 0.0075 m
Calculating this gives:
Q_new = (58950 W) / 0.0075 = 7859999.99 W
Thus, the new rate of heat flow across 1 m² of the boiler with lime scale is approximately 7859.99 W.
New Mass of Water Evaporated to Steam
Now, we can calculate the new mass flow rate:
m_new = Q_new / L
Substituting the new heat flow rate:
m_new = 7859.99 W / 2260000 J/kg
Calculating this gives:
m_new ≈ 0.00347 kg/s
Therefore, the mass of water evaporated to steam per second per square meter with the lime scale is approximately 3.47 g/s.
Summary of Results
- Rate of heat flow without lime scale: 8421.43 W
- Mass of water evaporated without lime scale: 3.73 g/s
- Rate of heat flow with lime scale: 7859.99 W
- Mass of water evaporated with lime scale: 3.47 g/s
This analysis shows how the presence of lime scale affects both the heat transfer efficiency and the evaporation rate in a steam boiler system.
