Cartridge heater - Heater power - Convection coefficients in air.xls

KNOWN: Dimensions of a cartridge heater. Heater power. Convection coefficients in air and water at a prescribed temperature.
FIND: Heater surface temperatures in water and air.
ASSUMPTIONS: 1) Steady-state conditions.
2) All of the electrical power is transferred to the fluid by convection.
3) Negligible heat transfer from ends.

Calculation Reference
Fundamentals of Heat and Mass Transfer - Frank P. Incropera

To calculate the annual cost of heat loss through pipes, considering the length, diameter, surface temperature, emissivity of the steam line, temperature, convection coefficient of the ambient air, efficiency, and fuel cost for the gas-fired furnace, you can follow these steps:

a) Rate of Heat loss:

1. Calculate the surface area of the steam line: The surface area of a pipe can be calculated using the formula:

   Surface Area = π * D * L

   Where D is the diameter of the steam line and L is the length of the steam line.

2. Calculate the radiative heat transfer: Use the Stefan-Boltzmann Law to calculate the radiative heat transfer (Q_rad) from the steam line to the surrounding walls:

   Q_rad = ε * σ * Surface Area * (T_surface^4 - T_surroundings^4)

   Where ε is the emissivity of the steam line, σ is the Stefan-Boltzmann constant, Surface Area is the surface area of the steam line, T_surface is the surface temperature of the steam line, and T_surroundings is the temperature of the surrounding walls.

3. Calculate the convective heat transfer: Calculate the convective heat transfer (Q_conv) from the steam line to the ambient air using the equation:

   Q_conv = h * Surface Area * (T_surface - T_ambient)

   Where h is the convection coefficient of the ambient air and T_ambient is the temperature of the ambient air.

4. Calculate the total heat loss: Sum the radiative and convective heat transfers to obtain the total heat loss from the steam line:

   Total Heat Loss = Q_rad + Q_conv

b) Annual cost of heat loss:

1. Calculate the total energy loss: Convert the total heat loss to energy units by dividing it by the efficiency of the gas-fired furnace:

   Total Energy Loss = Total Heat Loss / Efficiency

2. Calculate the annual fuel consumption: Divide the total energy loss by the fuel cost per unit of energy to obtain the annual fuel consumption:

   Annual Fuel Consumption = Total Energy Loss / Fuel Cost

3. Calculate the annual cost of heat loss: Multiply the annual fuel consumption by the fuel cost to determine the annual cost of heat loss:

   Annual Cost of Heat Loss = Annual Fuel Consumption * Fuel Cost

By following these steps, you can calculate the rate of heat loss from the steam line and determine the annual cost of heat loss through pipes. These calculations consider the relevant parameters such as surface area, temperature, emissivity, convection coefficient, efficiency, and fuel cost. The assumption that the net radiation transfer is between a small surface (steam line) and a large enclosure (plant walls) allows for simplified analysis and neglects other potential sources of heat loss.