Lumped Adiabatic Method - Detailed Discussion
When this Arrow Steady Solution Method is used, the following simplifying assumptions apply:
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No Heat Transfer - All heat transfer data entered for individual pipes is neglected such that they are adiabatic. Any entered data is kept in the background in case the user switches back to one of the marching methods.
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No Elevation Changes - All elevation data is neglected. The output will still show the elevation as entered by the user, and the data is kept in the background in case the user switches back to one of the marching methods. Since elevation effects are usually not very important in gas systems, neglecting elevation changes is frequently an acceptable approximation. One application where this is definitely not the case is in Rotating Systems.
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Calorically Perfect Gas - This solution method assumes a calorically perfect gas law, meaning that the specific heat capacity is constant, and therefore the specific heat ratio (gamma) is also constant. However, note that this only applies to the mathematical derivation of the equation below. In the final solution output, the specific heat may still change due to other settings in the model such as which fluid library is used, what equation of state model is selected, or what enthalpy model is selected. For example, using the REFPROP fluid library may result in values of Cp that change down a pipe.
An analytical solution to the Fundamental Equations for adiabatic flow neglecting elevation changes is given by the following (e.g., see Saad, 1993, page 209):
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(42) |
This equation is valid for real gases as long as the following approximation is accepted:
using a suitably averaged value of cp. Integrating from 0 to L obtains:
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(43) |
Equation 43 is then solved for each pipe, using an average value of friction factor and g for each pipe. The solution of Equation 43 typically is iterative where the unknown is M2.
Note: In Saad 1993 on page 209 equation 5.27 (Equation 43 above) the square brackets are not included. They have been added above to clarify that the natural log applies to all of the encapsulated terms. Furthermore, the left-hand side in Saad contains a multiplier of 4 since the Fanning friction factor was used, whereas Equation 43 above used in Datacor Pipe Flow Modeling software uses the Darcy friction factor.
