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D=duct SD=distance between cable and duct E=earth 4.2 Single Layer of Insulation, Continuous Load The internal thermal circuit is shown in Figure 13-3 for a cable with continuous load. The conductor heat source passes through only one thermal resistance. This may be an insulation, covering, or a combination as long as they have similar thermal resistances. Note that these circuits stop at the surface of the cable. The remainder of the thermal circuit will be added in examples that follow.

Figure 13-3 This diagram shows a continuous load flowing through one layer of insulation. The heat does not travel beyond the surface of the cable in this example.

4.3 Cable Internal Thermal Circuit Covered By Two Dissimilar Materials, Continuous Load

Figure 13-4 In this example, the continuous load flows through two dissimilar materials, but the heat still stays at the surface of the last layer of insulation.

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4.4 Cable Thermal Circuit for Primary Cable with Metallic Shield and Jacket, Continuous Load

Figure 13-5 This thermal diagram shows a primary cable with its several heat sources and thermal resistances still with a constant load where p and (1-p) divide the thermal resistance to reflect Qi.

4.5 Same Cable as Example 3, but with Cyclic Load

Figure 13-6 This diagram shows the same cable as in Figure 13-5, but the cyclic load is accounted for with the capacitors that are parallel to the three heat sources.

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4.6 External Thermal Circuit, Cable in Duct, Continuous Load

Figure 13-7 In this diagram, the resistances that are external to the cable are shown.

4.7 External Thermal Circuit, Cable in Duct, Time Varying Load, External Heat Source

Figure 13-8

where HX=external Heat Source 4.8 External Thermal Circuit, Cable Buried in Earth, Load May Be Cyclic, External Heat Source May Be Present

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Figure 13-9

The depiction of possible cyclic load and external heat source are shown by dotted lines. 4.9 External Thermal Circuit, Cable in Air, Possible External Heat Source

Figure 13-10

The external thermal circuit is shown with the possible external heat source shown by dotted lines. 5.0 SAMPLE AMPACITY CALCULATION 5.1 General Methods to calculate the ampacity of operating cables continue to be a popular subject for technical papers. Fortunately, the portion of the work that had been done by slide-rule and copious quantities of notepaper has been replaced with computers. Manipulations were handled by assuming intermediate values of the various parameters prior to the advent of the computer. The hand calculations were laborious, but the user did achieve a feel for the concept. The availability of tables and computer programs could lead to quick, but possibly incorrect, answers. The Neher-McGrath paper [13-10] is the best reference to use before a hand calculation is attempted. As a matter of fact, you should read that paper even if you have decided to use any available tables or programs!

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The following simple example of a calculation is presented with the intent of giving insight into the process: The general equation that has been previously given: (13.7)

where I=current in amperes that can be carried (ampacity) TC=maximum allowable conductor temperature in °C TA=ambient temperature of ambient earth in °C ∆Td=temperature rise due to dielectric loss in °C Rel=electrical resistance of conductor in ohms/foot at TC Rth=thermal resistance from conductor to ambient in thermal ohm feet, assuming no other heat sources. Another form of this equation recognizes the other possible heat sources that have been indicated in the thermal circuit diagrams. Equation 13.1 expands to: (13.8)

where TD=temperature rise due to dielectric loss in °C =to account for thermal resistance of insulation and/or coverings between the conductor and the first heat source beyond the insulation. =is the thermal resistance to ambient adjusted to account for additional heat sources such as shield loss, armor loss, steam lines, etc. 6.0 AMPACITY TABLES AND COMPUTER PROGRAMS 6.1 Tables The IEEE Standard Power Cable Ampacity Tables, [13-11,13-12], IEEE Std. 835–1994, is a book (or electronic version) that contains over 3,000 tables in 3,086 pages. Voltages range from 5 kV to 138 kV. Although there are situations that are not covered by these tables, this is an excellent beginning point for anyone interested in cable ampacities. Manufacturers have also published catalogues that cover the more common situations [13-13,13-14]. 6.2 Computer Programs Most of the large cable manufacturers and architect/engineering firms have their own computer programs for ampacity determination. This is an excellent source of information when you are engineering a new cable system. These programs generally are not for sale.

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There are commercially available programs throughout North America. These are especially useful when you need to determine the precise ampacity of a cable, for instance, that is in a duct bank with other cables that are not fully loaded. The general cost of one of these programs is about $5,000 in US dollars. 7.0 REFERENCES [13-1] Power Cable Ampacities, AIEE Pub. No. S-135–1 and IPCEA Pub. No. p-46–426, 1962. [13-2] Kennelly, A.E., “On the Carrying Capacity of Electrical Cables…”, Minutes, Ninth Annual Meeting, Association of Edison Illuminating Companies, New York, NY, 1893. [13-3] Neher, J.H., “The Temperature Rise of Buried Cables and Pipes,” AIEE Paper No. 49–2, Winter General Meeting, New York, NY Jan. 31–Feb. 4, 1949. [13-4] Balaska, T.A., McKean, A.L., and Merrell, E.J. “Long Time Heat Runs on Underground Cables in a Sand Hill,” AIEE Paper No. 60–809, Summer General Meeting, June 19–24, 1960. [13-5] Schmill, J.V., “Variable Soil Thermal Resistivity—Steady State Analysis,” IEEE Paper No. 31 TP 66–14, Winter Power Meeting, New York, NY, Jan. 30–Feb. 4, 1966. [13-6] Insulated Conductors Committee Minutes, Appendices F-3, F-4, F-5, F6, F-7, and F-8, Nov. 1984. [13-7] “IEEE Guide for Soil Thermal Resistivity Measurements,” IEEE Std. 442–1979. [13-8] Black, W.Z. and Martin, M.A. Jr., “Practical Aspects of Applying Thermal Stability Measurements to the Rating of Underground Power Cables,” IEEE Paper No. 81 WM 050–4, Atlanta, GA, Feb. 1–8, 1981. [13-9] Simmons, D.M., “Calculation of the Electrical Problems of Underground Cables,” The Electrical Journal, East Pittsburgh, PA, May–Nov. 1932. [13-10] Neher, J.H. and McGrath, M.H., “The Calculation of the Temperature Rise and Load Capability of Cable Systems,” AIEE Transactions, Vol. 76, Pt.III , pp. 752–772, Oct. 1957. [13-11] IEEE Standard Power Cable Ampacity Tables, IEEE Std. 835–1994, (hard copy version). [13-12] IEEE Standard Power Cable Ampacity Tables, IEEE Std. 835–1994, (electronic version). [13-13] Engineering Data for Copper and Aluminum Conductror Electrical Cables, EHB-90, The Okonite Company, 1990. [13-14] Power Cable Manual, Second Edition, The Southwire Company, 1997.

CHAPTER 14 THERMAL RESISTIVITY OF CONCRETE William A.Thue Consultant, Hendersonville, North Carolina, U.S.A. 1.0 INTRODUCTION The backfill material that surrounds a buried cable system is an extremely critical element in achieving optimum heat dissipation of the system. When soils are used for backfill for power cables, moisture migration can change a soil from about 60 rho (thermal resistivity in °C-cm/watt) to 300–400 rho as the moisture content is reduced from 10% to 3%. [14-1] Analysis of the thermal resistivity of soils by Fink [14-2], Sinclair [14-3], Adams and Baljet [14-4], Black and Martin [14-5], and others have clearly shown that migration of moisture from the backfill soil is a critical element in the ampacity of cable systems. Concrete has been used as backfill around buried cables in Europe for many years. The possibility of using concrete instead of soil was considered but initially rejected when it was seen that the previous Ampacity Tables [14-6] used a value of 85 for the thermal resistivity of concrete. This was contradicted in reports of the construction of concrete dams in the United States [14-7]. It should be noted that the value used in the present IEEE Ampacity Tables is a rho 90 for concrete in duct banks [14-8]. 2.0 EARLY FIELD EXPEREINCE A search for a local source of an ideally graded, natural soil often proves fruitless. Native soils frequently have a rather uniform size and hence dry out very quickly when a heat source is applied. A typical silica sand has 88% of the sand passing through a number 40 sieve (0.42 mm or smaller) and with 0.6% silt. Dry densities of only about 100 pounds per cubic foot, mostly made of quartz crystals that have a thermal resistivity of about 11, are attainable with normal construction techniques. The thermal resistivity of these sands in their native state is about 60 to 80 rho as long as they have about 3% by weight moisture. Typically, a load of about 30 watts per foot from a cable or duct bank are sufficient to dry such sand to near zero percent moisture where the rho can rise to 350 to 400. Thermal runaway conditions of native soils have been experienced by many utilities involving transmission cables as well as single-circuit, direct buried distribution feeder cables. In one situation, the sand had less than one-half percent moisture adjacent to a direct buried feeder cable exiting a substation even though lawn irrigation sprinklers were

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only two feet above. With four hours of watering every night, the sand could not retain enough moisture to prevent thermal runaway of one buried three conductor 500 kcmil copper feeder cable. The pattern of eccentric circles of dry sand was found to be in complete agreement with the paper of Balaska, Merrell, and McKean [14-9] that described a simulated transmission cable in a sand hill in New York. In another example, the native soil around a pipe cable was found to be baked completely dry and adhering to the pipe’s outer surface even though the cable was 30 feet under the surface of a bay. The material in this situation had a clay-like composition (marl) with a high amount of organic fillers. Similar reports have been given regarding high thermal resistivity of soils around cables under waterways in Denmark, England [14-10] and the deepest water in Lake Champlain [14-11]. In the situation described in Denmark, it was stated that “Cooling conditions for a submarine cable are normally assumed to be very good and the ampacity is based on a low value of thermal resistivity of the seabed.” Since the land section had a thermal resistivity of 43 to 54°C-cm/watt, it was assumed that the value in the seabed was equally as low. After two joints failed in service, it was discovered in laboratory investigations that the seabed material contained high organic levels and that the thermal resistivity was 105°C-cm/watt. Needle probes into the seabed discovered a rho of 94°C-cm/watt. In the London investigation [14-10], they found the silt in the bottom of canals to be as high as 118°C-cm/watt and that even higher values could be reached in the presence of heated cables. The Lake Champlain 115 kV cables [14-11] were installed in 1958 and failed in 1969 at a depth of about 300 feet. A sample of the soil near the failure was sent to a laboratory for analysis. They found the silt to have an average value of rho of 90 to 100 even though the silt “was not tested in the condition that it was in the lake bottom…” The new cable rating was based on the lakebed silt to have a rho of 140°C-cm/watt. The lessons to be learned here is that moisture can migrate from soils even in deep waterways. To maintain a low thermal resistivity soil in a seabed, water must be free to move through a porous or granular environment and have a limited level of organic material. It should also be pointed out that readings taken with thermal probes along a proposed route may give optimistically low values of thermal resistivity if the heat source is not left on long enough to detect moisture migration in that soil. 3.0 CONCRETE FOR CABLE BACKFILL Cement bound sand (weak concrete) has been used as a backfill material around direct buried cables in Europe for many years. A typical material in use there is a 12:1 sand/cement mix. Although this provides for some cable movement and relative ease for removal, these mixes do attain a rather high thermal resistivity when a load is applied for many months. A resistivity of about 105°C-cm/watt is typical. Lower values of rho may be attained by decreasing the amount of sand. In other words, by making the concrete more structurally sound. Greebler and Barnett [14-12] reported that concrete around a laboratory installed duct bank had a rho of 85. This paper was the source of the 85 rho that may be found in the