Before there were computers we let nature be the computer. This was performed by using dimensionless numbers and developing correlations between them, particularly when designing complicated flow or heat transfer processes. The essence of the procedure was to take the differential equation applicable to the process at hand and make it dimensionless. The variables were all made dimensionless with process parameters such that they were all of order one in dimensionless form. How this allows nature to be the computer will become evident below.
Consider the velocity gradient dvz/dr for pressure driven flow of water in a pipe, see Fig. C.1. The appropriate velocity component of interest is that in the zdirection, vz for cylindrical coordinates. The radial position is given by r. We know there is a gradient since the fluid velocity at the wall will (normally) be zero and gradually rise to a maximum at the centerline (r = 0). In fact, for a Newtonian fluid of constant viscosity the velocity profile is a parabola.
By convention, the radial position r is made dimensionless with the pipe diameter Dp, while vz is made dimensionless with the average velocity in the pipe, Vw (the subscript w stands for water since we are primarily concerned with water here). The average velocity is the ratio of the volumetric flow rate to the pipe area and determined from the water mass flow rate m by









where pw is the density of water and the symbol [=] means ‘has dimensions of.’ One can now make the velocity gradient dimensionless to arrive at
where the r indicates a dimensionless number. In the above equation all dimensionless numbers are of order one, despite the value of Dp or Vw. The key to dimensional analysis is this, all the variables and derivatives of variables should be of order one and, as the reader will see below, the
physics will be confined to the dimensionless numbers made from the numerical prefactors.
The steadystate momentum equation for pressure driven flow of a Newtonian fluid in a pipe at constant temperature is given by
dP _ 1 d / 6vz
dz r dr V dr )
where P is the pressure at any given axial position z and pw is the viscosity of water. Physically this equation states that the pressure drop or gradient along the pipe is caused by momentum transfer within the fluid (i. e. dissipation) from the centerline to the pipe wall.
If the upstream and downstream pressures, Pin and Pout, respectively, are kept constant one expects dP/dz to also be constant along the pipe axis since there is no physical reason for it to vary other than linearly as long as the fluid temperature is constant (we have not begun to discuss heat transfer yet which will make the viscosity vary along the pipe length!). From a more mathematical point of view, since we are concerned with steadystate flow, pressure is expected to only be a function of z and the velocity only a function of r; since one side of the equation in a function of z and the other side r they must be equal to a constant. Now, one can discern why vz is a parabolic function in r since its second derivative must be a constant. Regardless, let’s make eqn (C.2) dimensionless to arrive at
ДР dP _ pwVw 1 (r—)
Lp dz Dp r df – V dr )
where ДР = Pin – Pout and has been used to make the pressure dimensionless while z was made dimensionless with the pipe length Lp, see Fig. C.1. Because of convention and many years (well over a hundred) of studying flow in a pipe, the dimensionless equation is written
where f and Re are the friction factor and Reynolds number, respectively. The factor of 2 in the friction factor is used to define the Fanning friction factor, while the Moody friction factor has a value of 1/2 rather than 2; we use the Fanning friction factor here.
So, one might do a series of experiments and measure the pressure drop as a function of mass flow rate for a variety of pipes with various lengths and diameters. Equation (C.3) can be used to show that if you plot f as a function of Re all the data should collapse onto one master curve; as is found. Although one can solve eqn (C.2) analytically for laminar flow it is not possible to do so for turbulent flow and this is where the dimensionless numbers are of use and how nature becomes a computer.1 The fRe correlation should collapse all the data for both laminar and turbulent flows allowing the designer to find the pressure drop for his or her circumstance.
Now consider heat transfer by having a cold fluid entering a heated pipe, as one may encounter in a flat plate solar energy collector. The equation of energy with constant transport properties (i. e. the heat capacity Cpw and thermal conductivity kw of water are assumed to be constant) for steadystate, nonisothermal flow in a pipe, is
^ dT , Г1 d ( 8T 3T1
pw Cpw vz^, — kw _ yr" I r "y 1 + "y
dz L r dr dr / dz _
where T is temperature. This equation can be written in terms of dimensionless variables as
PwCPwVw r dT _ k Гdj dL
Lp z dz w Dp f df / df j Lp dz
where the dimensionless temperature T was written as [T – Tin]/[Tout – Tin]; one can subtract Tin from T in the derivatives since it is a constant and will naturally drop out when the derivative is taken. The subtraction was done so T is of order one at all times. This equation can be divided by pw on both sides and rewritten as
Equation (C.4) can be used to demonstrate the importance of the Reynolds and Prandtl (Pr) numbers in heat transfer and any correlation must be developed with them in mind since T will be a function of them. This equation can be solved for laminar flow, however, we need to develop a correlation that relates the heat input from the hot pipe
wall to the cold fluid under any flow condition, laminar or turbulent. 2 2This being said eqn (C.4) is an adEquation (C.4) will allow calculation of the temperature rise for a venture in арр1Ы even
cold fluid flowing in a hot pipe. Yet, details of the heat transfer from in laminar flow.
the hot pipe to the fluid have not been discussed. Any time there is an interface between a liquid or gas and a solid there is not good thermal contact between the two materials, at least not as good as within a solid, homogenous material alone.
Consider simple heat transfer from the heated pipe to liquid water, this is usually written as
qw _ hw [Tp – Toe][_]—2 (C.5)
similar to that written in Section 8.1 and shown in eqn (8.10). This is Newton’s law of cooling and relates the rate of heat transfer per unit area to the difference between the pipe temperature Tp (see Fig. C.1) and the temperature far from it, TTC (this is an ambiguous temperature and is included to relate this heat transfer relation to thermal conduction, suffice to say it is ‘far’ from the pipe wall). The heat transfer coefficient hw can only be found by correlations and is not solely a material property. It is dependent on the material’s properties as well as process conditions.
The type of heat transfer considered in eqn (C.4) is only conduction from the pipe wall to the water, via knowledge of the thermal conductivity kw. This can be written




by applying Fourier’s law right at the pipe wall. The temperature of the fluid just at the pipe wall T is unknown. The second equation results by making the distance x dimensionless with the pipe diameter Dp. Of course, this is an imprecise discussion since radial coordinates are not being used; the mathematical details are simplified to promote the gist of the discussion.
These two rates of heat transfer should be equal when describing the same system and using eqns (C.5) and (C.6) one arrives at
д(Г – Г») і
Nu = hwDp =————— дх——— (C.7)
k T – T y
wp
So, the Nusselt number represents a dimensionless temperature gradient. The utility of the Nusselt number is this: assuming one could solve eqn (C.4) the boundary condition of water having a temperature Tp at the pipe wall would have to be used. One does not know the temperature of the water at this position since heat transfer from the pipe at temperature Tp to the water in contact with it is unknown. There will be a finite temperature drop at the interface. The experimentally determined Nusselt number corrects for this.
Thus, correlations must be determined experimentally and the Nu correlated with Re and Pr. This would be accomplished by measuring the temperature rise ToutTin for a given pipe maintained at Tp, allowing one to determine qw, the rate of heat transfer per unit area of pipe, via
qw = mCpw [Tout – Tin] (C.8)
The heat capacity of water Cpw will be determined at the mean temperature, see eqn (8.11). This is just the sensible heat rise of the water. A similar equation exists in eqn (8.10), written in slightly different form here,
?w = hw AT, m (C.9)
since this is the rate of heat transfer per unit area based on the log mean temperature difference. Equating eqns C.8 and C.9 allows one to determine hw,
m Cpw[Tout Tin ]
AT,
One would calculate hw from this equation and then determine Nu and a correlation with Re and Pr is made.
A simple correlation is the DittusBoelter correlation which is of limited utility,
Nu = 0.023Re08Prn
where n = 0.3 if the fluid is cooled and n = 0.4 if the fluid is heated. Equation (8.13) is more complicated, and accurate, as it was developed over many years and includes the friction factor f.
Below is a summary of the dimensionless numbers used in fluid flow and heat transfer in this monograph. Also included is the absorber number, introduced in eqn (9.19), that is a useful grouping of variables and is a modified Stanton number. The dimensionless numbers are formulated as a ratio of forces like the ratio of inertial to viscous, for Re, or rates.
Table C.1 Dimensionless numbers used in fluid flow and heat transfer in this monograph.
*As discussed in Section 9.3 and shown in eqn (9.20), the absorber number is related to the Stanton number through, Ab = 4LS’t, where
Dp
hNsnDpLp
Ab =———– t—kL, Ap is the internal pipe area.
m oCp
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