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:
t = operating time in hours per year
Ch = annual cost of heat loss ($/million Btu)
Ci = 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.oF)
∆to = 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.ft2.F)
To get h, we used the following Equation
τ1 = dimensionless factor depending on the surface (see Table 1)
τ2 = dimensionless factor (see Chart 1)
a = 1.2 Btu/ft2.F.hr
b = 0.0048 Btu/ft2.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 ∆to vs X (insulation thickness) from figure below and ∆to vs X (insulation thickness) from equation above and find the point of intersection.
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 ∆to 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.
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.
As mentioned above, to find optimum insulation thickness, we need to make a plot between ∆to and X from Chart 1 and Chart 2, and ∆to 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 ∆to (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 τ1 and τ2
τ1 is a function of surface. For cylindrical vertical surface, τ1 is 1.187.
τ2 can be found easily by using Chart 1. It is a function of ∆to. See Step 2 for the procedure.
Complete the table with τ1, τ2, 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 ∆to 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, ∆to is 58oF and optimum thickness is 2.1 inches.
Evans, F. L., Equipment Design Handbook for Refineries and Chemical Plants, Vol. 1, 2nd Ed., Gulf Publishing Co., 1979.