Heat-Up Considerations

As per IEEE 515

Relationship between heat-up time and the heating device input for a pipe:-

t = H X Ln {(Qc – U X (Ti – Ta)) / (Qc – U X (Tf– Ta))} + {(P1 X Vc1 X hf) / (Qc – U X (Tsc– Ta))}

Where: -

t = Desired Heat-Up Time (s)

H = Thermal Time Constant (s)

Qc = Heating Cable Output (W/M)

U = Heat Loss per Unit Length of Pipe per Degree of Temperature (W/M·°C)

Ti = initial temperature of the pipe (°C)

Tf = final temperature of the fluid and the pipe (°C)

Ta = ambient temperature (°C)

P1 = density of product in the pipe (kg/m3)

Vc1 = internal volume of the pipe (m3/m)

hf = Latent Heat Of Fusion For The Product (J/Kg)

Tsc = Temperature at Which Phase Change Occurs (°C)

 

We calculate heat loss: -

 

U = 1 / ((1 / (Pie X D1 X hi)) + (Ln (D2 / D1) / (2 X Pie X k1)) + (Ln (D3 / D2) / (2 X Pie X k2)) + (1 / (Pie X D3 X hco)) + (1 / (Pie X D3 X ho)))

Where: -

U = Heat Loss per Unit Length of Pipe per Degree of Temperature (W/M·°C)

D1= Inside Diameter of the Insulation Layer (M)

D2 = Outside Diameter of the Inner Insulation Layer (M)

D3 = Outside Diameter of the Outer Insulation Layer (M)

k1 = Thermal Conductivity of the Inner Insulation Evaluated At Its Mean Temperature (W/M·°C)

k2 = Thermal Conductivity of the Outer Insulation Evaluated At Its Mean Temperature (W/M·°C)

hco = Inside Air Contact Coefficient of the Weather Barrier (W/M·°C)

hi = Inside Air Contact Coefficient From The Pipe To The Inside Insulation Surface (W/M·°C)

ho = Outside Air Contact Coefficient of the Weather Barrier To The Ambient Temperature (W/M2·°C)

 

We calculate Thermal Time Constant: -

H = ((P1 X Vc1 X Cp1) + (P2 X Vc2 X Cp2) + (0.5 x P3 X Vc3 X Cp3)) / U

Where:-

H = Thermal Time Constant (s)

P1 = density of product in the pipe (kg/m3)

Cp1 = specific heat of the product (J/kg·°C)

Vc1 = internal volume of the pipe (m3/m)

P2 = density of the pipe (kg/m3)

Cp2 = specific heat of the pipe (J/kg·°C)

Vc2 = pipe wall volume (m3//m)

P3 = density of the insulation (kg/m3)

Cp3 = specific heat of the insulation (J/kg·°C)

Vc3 = insulation wall volume (m3/m)

U = Heat Loss per Unit Length of Pipe per Degree of Temperature (W/M·°C)

 

If we consider some assumption:-

1.    Single layer insulation.

2.    (1 / (Pie X D2 X hco)) value is negligible due to mastic weather barrier &

3.    (1 / (Pie X D2 X ho))) Term may be omitted to give a conservative (high)       heat loss.

4.    (Pie X D1 X hi) Term is also negligible.

 

After that we find out result:-

U = 1 / (Ln (D2 / D1) / (2 X Pie X k))

Where: -

U = Heat Loss per Unit Length of Pipe per Degree of Temperature (W/M·°C)

D1= Inside Diameter of the Insulation Layer (M)

D2 = Outside Diameter of the Insulation Layer (M)

k= Thermal Conductivity of the Insulation Evaluated at Its Mean Temp. (W/M·°C)