In previous post I shared how to estimate insulation thickness for both flat surface and cylindrical surface. In this post I want to share how to estimate optimum insulation thickness.

The total annual cost of insulation is the sum of the cost of heat energy lost and fixed cost. A plot of cost vs insulation thickness will determine the most economical insulation thickness. A dimensionless factor can be used to calculate insulation thickness, depending on the ratio of insulation thickness to pipe diameter in the following equation:

Where:

t = operating time in hours per year

C_{h} = annual cost of heat loss ($/million Btu)

C_{i} = installed cost of insulation per cubic foot

f = fraction of installed cost depreciated annually

D = outside diameter of pipe (in)

K = thermal conductivity of insulation material (Btu/hr.ft.^{o}F)

∆t_{o} = difference between surface temperature and ambient air temperature (F)

h = combined film coefficient of convection and radiation in the air film on the insulator (Btu/hr.ft^{2}.F)

To get h, we used the following Equation

Where:

τ_{1} = dimensionless factor depending on the surface (see Table 1)

τ_{2} = dimensionless factor (see Chart 1)

a = 1.2 Btu/ft^{2}.F.hr

b = 0.0048 Btu/ft^{2}.F.hr

The unknown surface temperature can be found in parallel with optimum insulation thickness by trial and error. The approach used to find optimum insulation thickness is to plot ∆t_{o }vs X (insulation thickness) from figure below and ∆t_{o }vs X (insulation thickness) from equation above and find the point of intersection.

**Procedures**

The recommended procedures to find optimum insulation thickness is as follows:

- Make a rough guess at the surface temperature and select two other points, say 25 degrees above and below this value.
- Find the corresponding insulation thickness for each of the three points by two ways: first using chart 1, chart 2, and chart 3 above and second using Equation 1.
- Plot these points with ∆t
_{o }for the X-axis and the insulation thickness for the Y-axis and connect them with two smooth curves.__The intersection of the two curves indicates the optimum thickness.__

**Example**

Let say we have vertical pipe with the following data.

Outside diameter : 4 in

Air temperature : 85 F

Fluid temperature : 650 F

Thermal conductivity of insulation material : 0.05 Btu/hr.ft.F (at 400 F)

Estimate the optimum insulation thickness.

**Answer**

As mentioned above, to find optimum insulation thickness, we need to make a plot between ∆t_{o }and X from Chart 1 and Chart 2, and ∆t_{o }and X from Equation 2 and Chart 3. The intersection is optimum insulation thickness.

I make a complete tabulation which we need to fill to get the plot. The complete tabulation is shown below.

This is only my approach. I use 8 steps to complete the tabulation and the plots. Let us get started.

**1 – Complete all the required data and make a rough guess of surface temperature**

Make a table and complete all the data we have, such as ambient air temperature, thermal conductivity of insulating material, and outside diameter of pipe. Also, make a rough guess of surface temperature. I used 160 F as middle value, 135 F (160 – 25) as upper value, and 185 (160 + 25) as above value. From these data, we can get ∆t_{o }(difference between surface temperature and ambient air temperature).

**2 – Find surface function (φ) from Chart 1**

Based on chart below, we can get surface function.

Complete the table with surface function.

**3 – Find insulation thickness (X) for each surface function (φ)**

Based on chart below, we get insulation thickness.

Complete the table with insulation thickness.

**4 – Find ****τ _{1}**

**and τ**

_{2}τ_{1 }is a function of surface. For cylindrical vertical surface, **τ _{1 }**is 1.187.

τ_{2 }can be found easily by using Chart 1. It is a function of ∆t_{o. }See Step 2 for the procedure.

Complete the table with τ_{1}, τ_{2, }a, and b.

**5 – Calculate h **

h is calculated by using equation 1. Complete the table with h.

**6 – Calculate ****τ**

τ is calculated by using equation 2. Complete the table with τ**. **

**7 – Find X/D (ratio of insulation thickness to pipe diameter) and X**

X/D is a function of τ. Use Chart 3 to find X/D and calculate X. Complete the table.

**8 – Make a Plot**

Plot ∆t_{o }and insulation thickness we found from Step 3 and Step 7. Connect them with two smooth curves.

From figure above, we found that at intersection of the two curves, ∆t_{o }is 58^{o}F and optimum thickness is 2.1 inches.

**References:**

Evans, F. L., *Equipment Design Handbook for Refineries and Chemical Plants, *Vol. 1, 2^{nd} Ed., Gulf Publishing Co., 1979.

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