Thermal Decoupling: An Investigation

In recent years there have been a number of papers that have addressed various topics within the general subject area of thermal modeling in a pipeline. These have all been worthy papers and certainly present the PSIG membership with a reasonably comprehensive view of the subject. However, the approach to solving the temperature equation together with the hydraulic equations has only been briefly discussed and relatively few comparisons made between the alternative strategies. This paper investigates the differences between a fully coupled system, in which all three equations used to describe the flow of fluid in a pipe are solved simultaneously, and a decoupled system, in which the thermal equation is solved separately from the hydraulic equations. The advantages of such a decoupling are reduced complexity and improved computational speed. But what is the cost? Is the accuracy of a decoupled system compromised? Certainly if the decoupling is not undertaken carefully then the inaccuracies will render the methodology useless. However, can a properly constructed decoupled system produce solutions that are indistinguishable from those produced by a fully coupled system? To compare the different approaches a comprehensive set of test cases has been developed. As well as highlighting specific thermal modeling phenomena, the results of these tests demonstrate where differences in the solutions lie and the magnitude of such differences: ultimately the tests are used to determine the credence of decoupling the thermal solution from the hydraulic solution. INTRODUCTION Over the past 30 years there have been many papers presented at PSIG on the subject of thermal modeling ranging from tutorials on the physics and thermodynamics [1],[2] , to comparison of different solution methods [3],[4] , verification [5] and accuracy [6] and why thermal modeling is important in the real world [7],[8] . A number of these papers [3],[4],[8] also present investigations into various simplifications that can be made to the physics of the thermal model and to what extent these simplifications affect the accuracy. The general conclusion is that a rigorous thermal model, including an accurate description of the transient heat transfer through the pipe, is the most accurate approach, but under certain circumstances, and with suitable approximation, certain simplifications can be made. Modisette [4] provides some analysis on the different numerical approaches to solve the thermal equation in relation to the hydraulic equations. His investigation considers fully coupled, isothermal, sequential steady-states, pseudo-steady-state with moving knots and a leap-frog approach in which the hydraulic and thermal equations are decoupled. This latter approach involves the following steps: 1. Solve the hydraulic equations using previously computed pressures, velocities and temperatures. 2. Update the solution 3. Solve exactly the thermal equation using the updated pressures and velocities and the previously computed temperatures 4. Update the solution A key feature of this approach is that the temperature equation is solved exactly in each thermal step. However, a brief note in the paper indicates that some interpolation of previous temperature solutions is required to compute the density and viscosity in the hydraulic step to avoid large errors in the PSIG 1213 Thermal Decoupling: An investigation Jon Barley, Energy Solutions International, Ltd. 2 Jon Barley PSIG 1213 hydraulic solution. In the approach presented here we derive a decoupled solution which requires no such interpolation but solves the thermal solution using the same (approximate) numerical method used to solve the hydraulic equations. Furthermore, with this approach intermediate solutions are not required to be computed between hydraulic and thermal steps. As will be discussed in the paper, the advantages of decoupling are simplicity and computational efficiency. However, we need to ensure that, to be usable, a decoupled solution does not introduce significant errors into the solution. To this end various test cases are given and the results of the coupled and decoupled solutions for these cases are presented. Although the main thrust of this paper is consideration of the decoupling of the thermal equation, decoupling is often used in other places. For example the thermal mass of the pipe is often included in the temperature equation: the pipe and fluid are coupled. The justification for this is that for turbulent flow the pipe is in good thermal contact with the fluid and therefore, to a good approximation, both the pipe and fluid are at a uniform temperature. There are, however, circumstances when this is not the case and an example of such is given. Another area where decoupling is used is the pipe cross section area i.e. it is often re-evaluated after the hydraulic and thermal solutions have been computed. This “area decoupling” is almost taken for granted, however, it is not necessary and a fully coupled solution method which includes the pipe cross section area can be derived. Batch/composition tracking is another example. EQUATIONS OF MOTION The equations that describe the transient behavior of pressure, flow and temperature of a single phase fluid are: Mass Balance: 0 ) ( ) (   x t Av A   Momentum Balance: 0 2 1       D v fv gh P vv v x x x t 