I
A. B. METZNER University of Delaware, Newark, Del.
Non-Newtonian Fluid Flow Relationships between Recent Pressure-Drop Correlations Applicability of the generalized Reynolds number to Bingham plastic and power-law non-Newtonians is shown in detail
WITHIN
3 months two similar publications concerning the flow of nonNewtonian fluids inside round tubes appeared in the technical literature (4, 7). Each publication had the same purposes in mind: quantitatively establishing the end of the stable laminar flow region for non-Newtonian fluids, and correlating pressure drop data for these materials in both laminar and nonlaminar flow regions. Furthermore, parts of the theoretical methods of approach of both authors were identical. In spite of these similarities, the results achieved were considerably different. Metzner and Reed (4) stated their correlation to be rigorously applicable to all of the usual fluids, including pseudoplastic, dilatant, and Bingham plastic non-Newtonians as well as Newtonian fluids. Weltmann (7) described two correlations : Hedstrom’s (2) earlier correlation for Bingham plastics and one applicable to fluids that follow the empirical power-law relationship between shear stress and shear rate. This latter category includes Newtonian fluids and is frequently a satisfactory approximation to the true behavior of non-Newtonians over limited ranges of shear rate. Thus both approaches described by Weltmann are dependent on the availability of an analytical expression between shear stress and shear rate, while Metzner and Reed needed to make no such assumptions. The present report points out more clearly the rigorous nature of the Metzner-Reed correlation, and incidentally presents relationships which show how this correlation reduces to that of Weltmann for each of the special cases treated by her.
Development This discussion is restricted to flow of fluids under laminar conditions, as the theoretical developments of both articles (4, 7) are of proved rigor only in this region. For fluids which obey the “power-law” relationship between shear stress and shear rate: r =
k( -du/dr)n
(1)
hereafter called power-law fluids, the shear rate at the wall of a round tube when the flow is laminar may be related to the average velocity of flow rate as follows :
Equation 2 may be derived by a double integration of Equation 1 (3, pp. 96-7). The minus sign within the shear rate parentheses in Equations 1 and 2 is used to emphasize that the shear rate terms, for flow inside round tubes, are negative if distance r is measured from the center line of the tube-as is usually the case. (Most authors omit the sign, as frequently it is not necessary to consider it.) Equation 2 is superficially very similar to a rearrangement of the Rabinowitsch equation given by Metzner and Reed (4):
In the case, n’ is defined as (4) The complete development of Equation 3 is given by Mooney (6) and Metz-
ner and Reed ( 4 ) . By comparison with Equation 4, if one takes logarithms of Equation 1 one obtains: l o g r = log K
+ n log ( - d u / d r )
(5)
Because in writing Equation 1, n and IC are assumed to be constant over a t least a differential range of shear rates and stresses, differentiation of Equation 5 gives, within this range:
Comparison of Equations 4 and 6 clearly brings out distinction between n and n‘: n is the slope of a logarithmic plot of shear stress r us. the corresponding shear rate (-u’u/dr), O n the other hand, n’ is the slope of a logarithmic plot of shear stress at the wall of a round tube, rW,us. 8V/D. The shear rate a t the wall, at any given value of the wall shear-stress is not, in general, equal to 8V/D, as inspection of Equation 3 shows. Therefore, these latter plots from which n’ is determined are not actually shear stress-shear rate curves and should not be termed such, except for the special case of Newtonian behavior (n‘ = 1.00). The shear stress, T ~ at , the wall of a round tube is equal to DAP/4L (3). As n’ is the slope of the logarithmic rU,us. 8V/D curve, the equation of the straight line which is the tangent to the curke at a chosen value of rtomay be written: rUl= DAp -=
4L
K’
x
(8V/D)“’
(7)
where IC‘ corresponds to a particular value of shear stress, r W . If the slope VOL. 49, NO. 9
SEPTEMBER 1957
1429
of the logarithmic curve happens to be constant over a range of shear stresses (as is frequently the case), Equation 7 also represents the equation of this straight line. If the slope of the logarithmic T , us. SV/D plots is not a straight line (and the slope, n’, of the tangent to the curve varies with shear stress in this case), it is necessary to evaluate n’ and K‘ at the appropriate value of shear stress. The physical significance of the fluid property parameters, n‘-Le., the flow behavior index of the fluid at given shear stress-and K’ (the consistency index) has been discussed (3, 4 ) . A comparison of Equations 1 and 7 illustrates the practical importance of Equation 7 over Equarion 1. Equation 7 clearly is a direct relationship between the pressure drop, Ab, and the flow rate (or average velocity) of the fluid, V, in terms of tube dimensions and parameters K’ and n’. As such, it may be used for a rigorous pipeline design, provided n‘ and K’ are known at the particular values of 8V/D (or T ~ )of interest. O n the other hand, Equations 1 and 6 relate shear stress only to shear rate and not to flow rate. To introduce the average velocity, V, Equation 1 must be integrated twice, over the entire range of shear stresses encountered in a tube-Le., from 7 = 0 to T = T ~ .Unless one is willing to carry out multiple integrations, this requires that exponent n be a constant over this entire rangea condition satisfied rigorously only for the special case of Newtonian behavior. Fortunately, even appreciable errors in description of the fluid behavior at low values of shear stress ( T
