Heat Loss From An Insulated Pipe

This Excel spreadsheet models heat loss from an insulated pipe. This is a very common system in the process industries - insulated pipes are everywhere, and engineers need a sound grasp of heat transfer principles to model their effects. Although the model in the spreadsheet is simplified to aid understanding, complexity can be easily added.

Liquid flows through the pipe, with heat exchanged with the insulation. Heat is lost from the insulation to the environment via convection (no radiation losses are considered). The thermal effects of the pipe wall are ignored (although this can be easily implemented).

Cross-Section of Insulated Pipe

These equations are used in the spreadsheet to define the heat transfer process.

  • q is the heat flowrate through the pipe and insulation (W m-1)
  • Ts is the temperature at the surface of the insulation (K)
  • Ta is the ambient air temperature (K)
  • Tf is the fluid temperature inside the pipe (K)
  • DO is the pipe diameter (m)
  • DS is the outside diameter of the insulated pipe (i.e. the pipe diameter plus two times the insulation thickness) (m)
  • k is the insulation thermal conductivity (W m-1 K-1)
  • ΔT is the temperature difference between the insulation surface and ambient air Ts-T(K)
  • hs is the insulation-to-air heat surface heat transfer coefficient (W m2 K-1)
The equation for the surface heat transfer hs coefficient is a correlation; any other valid relationship can be substituted.

The equations are implicit - the heat transfer coefficient is a function of the surface temperature Ts, but the surface temperature is a function of the heat transfer coefficient. 

Hence the equations need to be solved iteratively with Goal Seek in Excel. Simply 
  • break the loop by estimating a value of Ts
  • use this to calculate all other properties (including the heat transfer rate)
  • use the heat transfer rate to backcalculate Ts
  • use Goal Seek to make the two values of Ts equal by varying the estimated value of Ts (or any other parameter
You can easily modify the heat transfer equations to include more complex effects, such as effect of fouling on the pipe surface, multiple layers of different insulation, radiative losses, thick large pipe walls (which act as a heat sink) etc.

Calculate d-Exponent to Predict Pore Pressure Trends

This Excel spreadsheet calculates the d-exponent, a parameter used by drilling engineers to investigate ppore pressure trends when drilling into over-pressurized zones.

Normally, formation density increases with the drilling depth. But if the formation contains sand below the surface, then the drilling rate may increases with drilling depth. The d-exponent is used in several calculations to investigate these effects.

Calculate D-Exponent in Excel

Jorden and Shirley (1966) suggested the d-exponent method, basing it on the Bingham equation. A later adjustment to the equation by Rehm and Mcledon (1971) included the effect of mud weight

The modified equation to calculate the d-exponent is

d exponent
  • c is the shale compactibility coefficient
  • ρn is the mud weight equivalent (lb gal-1)
  • ρm is the mud weight used (lb gal-1)
  • D is drill bit diameter (in)
  • W is the weight on the bit (x 1000 lb)
  • R is the penetration rate (rpm)
  • N is the rotary speed.
The d-exponent is often used in Eaton's equation

Eaton's equation
  • P is the formation pore pressure (psi)
  • Pob is the overburden pressure (ft)
  • Z is the vertical depth (ft)
  • d is the d exponent
  • dn is the normal compaction trend of the d exponent
  • Pp is the formation pore pressure (psi)
  • Pp/Z is the normal pore pressure gradient (psi ft-1)