Turbulent flow relative roughness

Since Reynolds number is greater than 4000, the flow is turbulent. The relative roughness of the smooth pipe is... 

For laminar flow (Re < 2000), generally found only in circuits handling heavy oils or other viscous fluids, / = 16/Re. For turbulent flow, the friction factor is dependent on the relative roughness of the pipe and on the Reynolds number. An approximation of the Fanning friction factor for turbulent flow in smooth pipes, reasonably good up to Re = 150,000, is given by / = (0.079)/(4i e ). 

For turbulent flow, Rmjpit is almost independent of velocity although it is a function of the surface roughness of the channel. Thus the resistance force is proportional to the square of the velocity. Rm/pu2 is found experimentally to be proportional to the one-third power of the relative roughness of the channel surface and may be conveniently written as ... 

The behavior of the flow in micro-channels, at least down to 50 pm in diameter, shows no difference with macro-scale flow. For smooth and rough micro-channels with relative roughness 0.32% turbulent flow occurs between 1,800 < Recr < 2,200, in full agreement with flow visualization and flow resistance data. In the articles used for the present study there was no evidence of transition below these results. 

A constant value of the friction factor f = 0.009 is assumed, for fully developed turbulent flow and a relative pipe roughness e = 0.01. The assumed constancy of f, however, depends upon the magnitude of the discharge Reynolds number which is checked during the program. The program also uses the data values given by Szekely and Themelis (1971), but converted to SI. 

All models for turbulent flows are semiempirical in nature, so it is necessary to rely upon empirical observations (e.g., data) for a quantitative description of friction loss in such flows. For Newtonian fluids in long tubes, we have shown from dimensional analysis that the friction factor should be a unique function of the Reynolds number and the relative roughness of the tube wall. This result has been used to correlate a wide range of measurements for a range of tube sizes, with a variety of fluids, and for a wide range of flow rates in terms of a generalized plot of/ versus /VRe- with e/D as a parameter. This correlation, shown in Fig. 6-4, is called a Moody diagram. 

Although Eq. (9-17) appears to be explicit for G, it is actually implicit because the friction factor depends on the Reynolds number, which depends on G. However, the Reynolds number under choked flow conditions is often high enough that fully turbulent flow prevails, in which case the friction factor depends only on the relative pipe roughness ... 

For turbulent flow of a Newtonian fluid, / decreases gradually with Re, which must be the case in view of the fact that the pressure drop varies with flow rate to a power slightly lower than 2.0. It is also found with turbulent flow that the value of / depends on the relative roughness of the pipe wall. The relative roughness is equal to eld, where e is the absolute roughness and d, the internal diameter of the pipe. Values of absolute roughness for various kinds of pipes and ducts are given in Table 2.1. 

Turbulent flow of Newtonian fluids is described in terms of the Fanning friction factor, which is correlated against the Reynolds number with the relative roughness of the pipe wall as a parameter. The same approach is adopted for non-Newtonian flow but the generalized Reynolds number is used. 

As discussed before, the transition from laminar to turbulent flow in the PPR channels already occurs at relatively low Reynolds number as a consequence of the roughness of the channel walls. Under typical operating conditions in practice, flow through the channels is quite turbulent, in contrast to the situation generally prevailing in monoliths as used in exhaust convertors, where due to the much smaller channel diameter and smoothness of the wall, flow is generally laminar. Therefore, in a PPR mass transfer in the gas inside the channel is generally relatively fast. 

In Section III.B, theoretical and empirical mass transfer and heat transfer relations arc discussed. The theoretical relations pertain to laminar flow, because for that flow regime the governing equations can be relatively easily solved numerically. The empirical relations pertain to turbulent flow in smooth rod assemblies. In this section the implications of the nonstandard boundary conditions in a BSR are also discussed. In Section III.C, the validity of the presented relations will be illustrated using experimental data obtained for turbulent flow in a lab-scale BSR with hydraulically rough strings of beads. 

For completely turbulent flow, it is assumed that fp approaches a constant value for all packed beds with the same relative roughness. The constant is found by experiment to be 1.75 [26]. 

For turbulent flows in noncircular conduits, the Moody diagram or ChurchiU equation can be used with good results, if the relative roughness is taken as e/Hj. 

Moody s [58] plot, shown in Fig. 5.9, gives the friction factor for laminar and turbulent flow in both smooth and rough circular ducts. Relative roughness el Dk is used as a parameter for... 

It should be noted that the horizontal portions of the curves to the right of the broken line are represented by Nikuradse s [60] correlation, which is presented in Table 5.9. The downward-sloping line for the smooth turbulent flow is represented by the PKN correlation shown in Table 5.8. The downward-sloping line for laminar flow is represented by Eq. 5.17. Relative roughness e can be obtained from Table 5.10 for a variety of commercial pipes. 

From Fig. 6.10 we see that for this value of the relative roughness the possible range of/for turbulent flow is 0.0056 to about 0.008. As our first guess, let us try /= 0.0057. Then from Eq. 6.20, rearranged to solve for F, we have... 

Check the assumed friction factor in Example 6.17. For the value of relative roughness shown, the range of possible friction factors in turbulent flow is 0.004 to 0.01. How much would the economic diameter differ at/= 0.01 ... 

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