How to calculate the thickness of polyurethane foam?

How to calculate the thickness of polyurethane foam?

In thermal insulation engineering, foam plastics are used in two types of situations: one as insulation material, and the other as refrigeration insulation material. Due to the different usage conditions, the calculation formulas are also different.

For the calculation of insulation thickness as used in cylindrical objects like pipelines, the following formula is recommended:

If used in constant-temperature flat surfaces, formula (11-39) is suitable for thickness calculation:

Where:

N = (1+N)m1/n(1+n)m

d1 — outer diameter of the insulation layer, m

d0 — inner diameter of the insulation layer, m

λ — thermal conductivity of the insulation layer, 1.163 W/(m·K)

a — surface heat transfer coefficient, 1.163 W/(m²·K)

b — cost of energy, yuan/4187J

α — construction cost of insulation material, 1000 yuan/m³

h — annual usage time, h

t0 — internal temperature, ℃

tt — ambient room temperature, ℃

n — annual interest rate

m — repayment years, a

μ — insulation layer thickness, m

The above formula includes human factors, such as insulation cost and investment payback period. Therefore, it is necessary to further understand the heat dissipation situation. For pipeline applications:

For flat surface applications:

In formula (11-40), the unit of Q refers to the total heat dissipation per meter per hour, i.e., 1.163 W/m; while in formula (11-41), the unit of Q is the total heat dissipation per square meter per hour, i.e., 1.163 J/(m²·h) or 1.163 W/m².

From a purely technical perspective, for pipeline systems:

For flat surfaces:

μ = λ/a(t0 - ts)/(ts - tt)   (11-43)

Where:

d1 — outer diameter of the insulation layer, m

d0 — inner diameter of the insulation layer, m

λ — thermal conductivity of the insulation material, 1.163 W/(m·K)

a — surface heat transfer coefficient of the insulation layer, W/(m²·K)

t0 — internal temperature, ℃

ts — surface temperature, ℃

tt — ambient temperature, ℃

μ — thickness of the insulation layer, m

It should be noted that a, the heat transfer coefficient, refers to the heat transfer between the insulation surface and the atmosphere, which is a combination of convection and radiation. Therefore, the heat transfer coefficient a should be:

a = at + ac    (11-44)

Where:

at — radiation heat transfer coefficient

ac — convection heat transfer coefficient

The value of at can be calculated from the following formula:

Where:

ε — emissivity of the insulation layer (0.84 when protected with silver paint)

C0 — blackbody radiation coefficient, taken as 5.7 J/m²

Tt — surface temperature, ℃

Tr — atmospheric temperature, ℃

ac is the convective heat transfer coefficient, which is related to the fluid flow rate, specific heat, thermal conductivity, etc., that is, related to the Nusselt number.

ac = Nu(λ / H)   (11-46)

Where:

Nu — Nusselt number

H — engineering height, m

Through multiple tests and comparisons in various countries, it has been considered that the a value in refrigeration applications is approximately 8.14 W/(m²·K), and in insulation applications, it is approximately 11.63 W/(m²·K), which closely match actual values.

【Example】

Using polyurethane rigid foam as the insulation material for a cold storage, with a thermal conductivity of 0.02326 W/(m·K), a design storage temperature of –20°C, and an external average annual temperature of 30°C, and relative humidity of 85%, the insulation thickness of foam plastic is calculated.

According to formula (11-43):

Given: λ = 0.02326 W/(m·K), tn = –20°C, tt = 30°C, a = 8.14 W/(m·K)

In refrigeration scenarios, the surface temperature is actually the dew point temperature. According to the saturated vapor pressure table, the saturated vapor pressure at 30°C is 4.23 kPa. Given 85% relative humidity, the actual vapor pressure is:

4.23 × 0.85 = 3.6 (kPa)

The dew point temperature refers to the saturated temperature at this pressure. From the saturated vapor pressure table, the saturated temperature at 3.6 kPa is 27.2°C, i.e., tn = 27.2°C.

The actual thermal conductivity λ of the cold storage insulation should be corrected using formula (11-37):

λ = λ0 + 0.00012T = 0.020 + 0.00012[(–20 + 27.2)/2]

= 0.020432 [kcal/(m·h·℃)]

= 0.02376 [W/(m·℃)]

μ = 0.02376 / 8.14 × [(–20 – 27.2)/(27.2 – 30)] = 0.049 (m)

The insulation thickness is 49 mm, approximately 50 mm. Therefore, subsequent calculations are based on 50 mm.

With 50 mm thickness, the heat transfer Q and surface temperature of the cold storage are calculated according to formula (11-41):

Q = (t0 – tt)/(1/a + a/λ) = (–20 – 30)/(1/8.14 + 0.05/0.02326) = –22 (W/m²)

The result is negative, indicating heat transfer from outside to inside. Surface temperature verification:

ts = Q/a + tt = (–22 W/m²)/(8.14 W/m²·℃) + 30 = 27.3 (℃)

ts = 27.3°C matches the value obtained from the saturated vapor pressure table. This shows that with 50 mm foam insulation, the design requirement is fully met.