Insulation and Pipe Heat Loss: Method

Insulation and Pipe Heat Loss

The tool finds the steady heat flow from a fluid in a pipe or vessel to the surrounding air through the wall, up to three layers of insulation and a jacket, and from it the temperatures, the thickness needed for safety or for condensation control, and the thickness that costs least over the life of the plant. The same calculation serves hot and cold service: below the ambient temperature the heat flow reverses and is reported as a gain.

Resistances in series

For a pipe every quantity is per unit length. A cylindrical layer between radii r₁ and r₂ with conductivity k has the resistance R = ln(r₂/r₁)/(2πk); a film of coefficient h on a surface of radius r has R = 1/(2πrh). For a flat wall every quantity is per unit area, R = t/k for a layer of thickness t and R = 1/h for a film. The heat flow is q = (T_f − T_a)/ΣR and the temperature falls across each element by qR. The inside film is neglected unless a coefficient is entered, which puts the inside of the wall at the fluid temperature; for liquids and condensing steam the error is negligible, for a gas it is not, and a coefficient should then be entered.

Conductivity varying with temperature

Insulation conducts better when it is hot. Each layer follows k = a + bT, and the tool uses the value at the mean of the layer's two face temperatures. For a straight-line law this is exact, not an approximation, in a slab and in a cylinder alike: the conduction equation integrates to q = (shape factor) × ∫k dT, and the integral of a straight line between two temperatures is its mid-point value times the difference. The test script confirms it against a numerical integration of the conduction equation. Because the face temperatures depend on the heat flow, the conductivities and the heat flow are iterated together until they stop changing.

The material list gives a and b for seven materials as straight lines fitted to the conductivity polynomials that ship with the U.S. Department of Energy's MEASUR software, each over the temperature range shown in the notes of the results when it is exceeded; the lines follow the polynomials within 2 to 6 percent. Three further entries are the single values of Table 12-2 of Army TM 5-852-5. All are representative values. A design should use the curve of the product to be installed, entered as a and b.

The outer film and the surface temperature

The outer coefficient is the sum of a convection and a radiation part and both depend on the surface temperature, which is itself the unknown. The surface temperature is the root of the energy balance (T_f − T_s)/ΣR_conduction = h(T_s) × perimeter × (T_s − T_a), found by bisection between the ambient and fluid temperatures to a residual far below 10⁻⁸ of the heat flow.

• Free convection. Horizontal pipe: Churchill and Chu's relation for the horizontal cylinder, Nu = {0.60 + 0.387 Ra^1/6/[1 + (0.559/Pr)^9/16]^8/27}², with the outside diameter as the length. Vertical wall: their relation for the vertical plate, with 0.825 and 0.492 in place of 0.60 and 0.559 and the height as the length. Horizontal surfaces: McAdams' relations, 0.54 Ra^1/4 or 0.15 Ra^1/3 (above Ra = 10⁷) for a warm surface facing up or a cold one facing down, 0.27 Ra^1/4 for the opposite cases, with area divided by perimeter as the length.
• Forced convection. Wind across a pipe: Churchill and Bernstein, Nu = 0.3 + 0.62 Re^1/2Pr^1/3[1 + (0.4/Pr)^2/3]^−1/4[1 + (Re/282,000)^5/8]^4/5. Wind along a wall: the flat-plate averages 0.664 Re^1/2Pr^1/3 up to Re = 5 × 10⁵ and (0.037 Re^4/5 − 871)Pr^1/3 beyond.
• Combination. Nu = (Nu_forcedⁿ + Nu_freeⁿ)^1/n, with n = 4 for the pipe (the value used by MEASUR) and n = 3 for the wall.
• Radiation. h_rd = εσ(T_s² + T_a²)(T_s + T_a), a grey surface seeing surroundings at the air temperature. The emissivity list is that of the MEASUR pipe-insulation calculator.

Air properties are taken at the film temperature from the equations of the 1976 U.S. Standard Atmosphere at sea-level pressure. A coefficient entered in the Outside Film field replaces all of this.

A bright metal jacket radiates little. It reduces the heat loss slightly and raises the surface temperature noticeably, which matters where the surface can be touched. The tool compares the surface with the entered limit and finds, by bisection on the thickness of layer 1, the thickness that just meets it. The default limit of 60 °C (140 °F) is the customary five-second contact value associated with ASTM C1055; the Department of Energy's Steam Tip Sheet 2 advises insulating any surface above 120 °F.

Critical radius

Insulation on a cylinder adds conduction resistance but also enlarges the surface that loses heat. With a constant outer coefficient the total ln(r/r₁)/(2πk) + 1/(2πrh) is least at r_c = k/h. A pipe or tube smaller than r_c loses more heat with thin insulation than bare, until the outer radius passes r_c. For industrial pipe r_c is about 10 mm and the effect is absent; it matters for small tubing and for electrical conductors. Figure 3 of the results is computed with the real, varying film coefficient and shows the turn of the curve when there is one.

Cold service and condensation

When the fluid is colder than the air, moisture condenses on any surface colder than the dew point of the air. The dew point comes from the Magnus relation for the saturation pressure of water vapour, p_ws = 610.94 exp[17.625 t/(t + 243.04)] Pa with t in °C: the vapour pressure is the relative humidity times p_ws at the air temperature, and the dew point is the temperature at which that pressure saturates. Against the IAPWS-IF97 saturation line the dew point agrees within 0.03 °C from 5 to 60 °C. The tool reports the thickness of layer 1 that keeps the surface at the dew point; a design adds a margin and needs a continuous vapour retarder, because a surface that stays dry does not stop vapour from diffusing into the insulation. At very high humidity the required thickness grows without limit.

Economic thickness

More insulation costs more to install and saves more energy. For each candidate thickness the annual energy is the heat flow times the operating hours, divided by the efficiency of the boiler or heater that makes it up, and its cost is that energy times the price. The present worth of n years of this cost, with the price escalating at the rate e and money discounted at the rate i, is the annual cost times

PWF = x(1 − xⁿ)/(1 − x), x = (1 + e)/(1 + i),

which reduces to [(1 + i)ⁿ − 1]/[i(1 + i)ⁿ] without escalation, the factor of TM 5-852-5. The economic thickness is the candidate with the least sum of installed cost and present worth of energy. Installed costs are entered as a table, since they depend on the market and on standard thicknesses, or as a straight line, for which the continuous minimum is also found by golden-section search. The simple payback is the installed cost divided by the first-year saving against the bare surface, and the carbon dioxide avoided is the fuel energy saved times the entered emission factor. Maintenance, the residual value and taxes are not included.

Temperature drop, freeze-up and heat tracing

A single-phase fluid flowing at the mass rate ṁ with specific heat c_p through a line of total resistance R per unit length approaches the ambient temperature exponentially, T₂ = T_a + (T₁ − T_a) exp(−L/D) with D = ṁc_pR, the resistance taken as constant along the run. Saturated steam does not follow this law: it stays at its saturation temperature and the loss appears as condensate.

When flow stops in freezing weather the water first cools to 0 °C and then freezes. Treating the water as one lump, neglecting the heat capacity of the pipe and insulation and the resistance of the ice as it forms (TM 5-852-5 Figure 12-4), the time to reach the freezing point is t = πr_w²RC ln[(T₁ − T_a)/(T₀ − T_a)] and the further time to freeze the fraction f of the water is t = fπr_w²RL/(T₀ − T_a), with C the volumetric heat capacity and L the volumetric latent heat of water. The manual recommends the first as the design time. The resistance is solved again for the cold ambient. Heat tracing must replace the loss at the temperature to be maintained; the tool multiplies that loss by the entered factor.

Buried pipe

For a pipe below ground the outer film is replaced by the resistance of the soil between the pipe and the surface, R = arccosh(2z/D)/(2πk_s), where z is the depth of the centreline and D the outside diameter of the insulated pipe; this is the conduction shape factor S = 2π/arccosh(2z/D) of a cylinder below an isothermal plane. The soil is uniform, the surface is at the entered ground temperature and the solution is steady. For a deep pipe TM 5-852-5 suggests using the undisturbed ground temperature at the pipe depth in place of the surface temperature.

Checks

The test script reproduces the worked examples of TM 5-852-5 chapter 12: the insulated pipe above ground (12.7 Btu/h per foot), the same pipe buried (5.79), the design, safety-factor and complete freezing times (5.6, 9.5 and 107 hours) and the table of the economical-thickness example with its 3 in result. The manual's own film coefficient is entered for these, since it uses a different convection formula. The convection and radiation relations are checked against the unit-test values of MEASUR, and the whole solution against an independent numpy and scipy model for the default case, eleven variants, twenty random pipes and twenty random walls.

The bare-pipe heat losses are 2 to 16 percent above the table of Steam Tip Sheet 2 (2 to 6 percent for 1 and 2 in lines, rising with diameter), whose emissivity and convection formula are not stated; the tool does not reproduce that table to its printed digits.

Limits

• Steady state, uniform temperatures along the pipe, one-dimensional conduction. Supports, valves, flanges and wet or compressed insulation are not modelled and often dominate the loss of a real line.
• The pipe relations are for a horizontal pipe; a vertical pipe differs little once insulated.
• Radiation is to surroundings at the air temperature. Solar gain and a cold night sky are not included.
• Air at sea-level pressure; film relations within their published ranges (the results note when they are exceeded).
• The freeze-up times neglect the heat capacity of the pipe wall and insulation, which lengthens the real time, and assume still water with no supercooling.