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November 4, 1999, 20:07 
time averaged heat transfer in oscillating flow

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I am working on the design of a shell & tube in which flue gas is used to heat oscillating helium to about 750 C. At the p,T and tube configuration, the peak Reynolds number for helium is about 6700  6800, rms about 4800. The amplitude of the oscillating helium is about equal to the tube length, which may affect turbulence. It is possible to estimate time averaged heat transfer coefficients from Pr and Re numbers and tube configuration, using St number: [N(St)*N(Pr)^0.67=f(L/D), U=Cp*mu*N(St)*N(Re)/D] However, I am not sure whether I should account for heat transfer enhancement by turbulence. I doubt that full turbulence will develop. That makes quite a difference in the heat transfer(factor 3  4). Do I need to sort this out in a CFD model, or can I just make a few assumptions ? I would be glad if anyone could suggest some literature on time average heat transfer in oscillating flow (in particular in the transition region) With thanks in advance and best regards, Matthieu 

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November 5, 1999, 11:31 
Re: time averaged heat transfer in oscillating flow

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The best answer that I can give you is that the usefulness of a CFD analysis will depend on your confidence in the heat transfer correlation that you use. All of the handbook heat transfer correlations that I have ever used have two important pieces of information that they carry with them: 1. The geometry that they are based on, and 2. The range of Reynolds numbers that they are applicable for.
If your heat transfer correlation were based on a scale model of the actual HX and was valid for a range of Reynold's numbers that encompassed yours (1000 > 15000 for example) then you would probably have a great deal of confidence in the correlation. In that case, I would recommend that you use the correlation and not worry about it. Odds are, though, that your correlation is based on a "similar" geometry, and you might be near the limit of the range of applicable Reynolds numbers. If this is the case, then you begin to wonder how good the analytical solution is going to be. Before deciding that you need a more accurate answer, you should look at the criticality and adjustability of the system. What is the overall consequence of the HX transferring more heat than predicted? Do you have a way to bypass flow around the HX and achieve the desired outlet temperature regardless of what happens inside the HX? If an accurate answer is critical, and the system is not designed to provide fine control over the flow, then a CFD analysis can help you. The analysis can be used to model the exact geometry of the heat exchanger, and the range of flow conditions that you expect. Most commercial CFD codes can model heat transfer as part of the flow solution, so you can get the temperature distribution along with the flow distribution. If the HX is large, a small portion of it can be modeled and an overall heat transfer coefficient can be calculated. The justification for performing a CFD analysis has to be based on your confidence in the correlations that you would otherwise use. The correlations also tend to work best under design conditions, and become less accurate in off design conditions (I've got some experience with that I'd be happy to share, if you don't mind a long story). If you are confident that your correlations are accurate for the HX you are designing, then there is no pressing need for a CFD analysis. 

November 5, 1999, 15:20 
Re: time averaged heat transfer in oscillating flow

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Hello Matthieu,
this is a good problem to be asking about. In general what you are talking about is the interaction of time and length scales (or alternatly time and velocity) in heat transfer and when the interaction is significant or can be neglected. 1. Back to basic assumptions! From dimensional analysis of the fundamental equations for a specified geometry at low speeds, neglecting effects of natural convection, IF the flow at the boundary is STEADY: Nu=f(Re,Pr) and we can buildup correlations from experimental results for geometrically similar geometries. This includes turbulent flows which enter steady at all length and time scales (ie. laminar) and go through transition in the domain. This does not include flows which enter as unsteady laminar or turbulent. For a flow which is unsteady at the boundary we SHOULD ensure that the nondimensional instantaneous velocity at the boundary is the same in the model and the device under consideration AT ALL TIMES AND AT ALL LOCATIONS......THIS IS OF COURSE A RIDUCULOUS AND UNACHIEVABLE REQUIREMENT! If we wish to have a solution that is correct in the statistical sense (time filtered Nu for some cutoff time scale...ie steady is a filter for all time scales) then IT appears the the most important factors to match are the statistical length and velocity scales of turbulence at the inlet! See Eckert and Drake 2. Experimental results show that when the length and velocity scales of turbulence and the physical scales are close there is a SIGNIFICANT interaction. One well documented example is in a shell and tube heat exchanger where the turbulent length scale relative to the tube diameter and pitch length scales. A maximum in heat transfer coeff. occurs when the two scales are about the sam (once described to me as a sort of resonance phenomena)! The difference between the "noninteracting scale assumption" and the "interacting" one is something of the order 70%........not negligible for most work! So the correlations that you use have to be now Nu=f(Re,Pr,Turb. Length Scale, Turb Velocity scale). A good reference to start with is Eckert and Drake's "Analysis of Heat and Mass Transfer". 3. If looking at a CFD solution, run through a full check on the assumptions especially looking at the turbulence treatment. We must not forget that we have to get both the turbulent Reynolds stresses AND turbulent heat flux terms right! Standard turbulence models and wall functions ARE BUILT AND TUNED on steady flows. For example, the loglaw wall treatment is an integration assuming a constant stress shear layer, turbulent kinetic energy production is in local equilibtium (at every instant in time) with dissipation and an assumption for turbulent length scale as a function of distance from the wall in the boundary layer. These are a lot of HUGE assumptions and this is only on the velocity boundary layer now we have to cary this over to the thermal B.L. remembering that pressure velocity fluctuations are significantly affect the Reynolds stress terms but NOT in the turbulent temperature fluxes how do we treat this...the old Reynolds analogy was also built on steady flows and I have seen some work that shows a phase difference between the temperature and momentum transfer (I don't have any references handy but this is JFM kind of stuff)! 4. One must remember at all times that in heat transfer the accurate determination of the thermal boundary layer is THE WHOLE PROBLEM and setting a code to run with a pile of assumptions which are not valid that you have not checked out is a waste of time and can be dangerous! 5. The two sides of the fluid problem are coupled in between with a transient thermal conduction problem...which of course has its own time and spacial scales! This is often the easy part and infact is the saviour for a lot of devices that the time scales (tau) in here are long. tau=h^2/alpha; tau [s] time scale h [m] lenth scale alpha [m^2/s] = k/(rho*Cp) thermal diffusivity The longer the time scale in between the more the coupling is "filtered" so large tau is like a low pass filter that averages out the fluctuations faster than tau and does not communicate them instantaneously to the other side! Aside from the wonderful high temperature properties of high alloys most of them have a lower k than, say carbon steel and hence a longer time scale which filters out more of the small time scale stuff! Pretty cool ...... eh! Best Regards, Duane Baker 

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