3.5.1 Heat exchanger analysis of the double-pipe co-current
flow heat exchanger
The following assumptions are made concerning the analysis of the heat exchanger: • Steady state conditions are assumed to exist only during the 90 minute pe-
riod when flow rates are sampled on discrete days during the fouling tests. Over this period temperatures are sampled every 20 seconds and averaged every three minutes. The bulk inlet water temperature trends during each 90 minute sample period is observed to be less than about 0.1◦C/h. This
is less than the measurement uncertainty of the temperature probes and thus the steady state assumption is reasonable.
• The fluids are assumed to be incompressible which means constant spe- cific heat approximations are used.
• Thermophysical fluid properties are evaluated using pure water data for both the foulant and heated water, despite the presence of other species in the foulant. The error from this is considered small as the foulant is fresh water and also any small error introduced from this assumption is contained in the regression uncertainty of the annular fluid (the annular Nusselt number is experimentally determined using the same water as dis- cussed later).
Asserting these assumptions, the log mean temperature difference (LMTD) method (Incropera et al., 2007; Cengel & Ghajar, 2011) is applicable to each of the six heat exchangers (figure 3.10). For co-current (parallel) flow it can be shown that the total rate of heat transfer between the two fluids is
Q = U A∆TLM (3.1)
where U is overall heat transfer coefficient, and A is the surface area contacted by the heated water in the annulus3and∆TLMis the LTMD equal to
∆TLM=Tin,ann− Tin− Tout,ann+ Tout
lnTTout,ann−Toutin,ann−Tin (3.2) The overall heat transfer coefficient is equal to the reciprocal of the total ther- mal resistance between the foulant and the heated water in the annulus (Incr- opera et al., 2007)
(U A)−1= Rt (3.3)
CHAPTER 3. EXPERIMENTATION
Foulant enters
Tin
Foulant exits
Tout
Heated water enters annulus
Tin,ann
Heated water exits
Tout,ann
Figure 3.10:Schematic of a heat exchanger (not to scale)
The total thermal resistance is equal to the sum of the individual resistances
Rt= 1 hannAann+ ln(d3/d2) 2πktL + ln(d2/d1) 2πkPPFL + Rf A + 1 h A (3.4)
where kPPFis the PPF conductivity and Rf is the fouling factor based on the un-
fouled diameter. The convection coefficients h and hanndescribe the convection
on the inside and outside of the tube surfaces respectively. In fact they depend on the friction factor which has to determined first.
3.5.2 The analogy between friction factor and convection
Nusselt number
The Darcy friction factor fd, referred hereafter as friction factor, is a dimension-
less parameter relating the pressure drop along internal flows to the average ve- locity according to (Cengel & Ghajar, 2011)
∆p = fd
L∆p d
ρv2
2 (3.5)
Fundamentally the friction factor relates to the skin friction coefficient of the tube, and it is important to describe the pressure drop and hence the pumping power required for the heat exchanger. Further, the friction factor characterizes the surface which determines the turbulent boundary layer. From experiments with smooth tubes, Konakov, cited by Gnielinski (2009), found the fully-turbulent friction factor to be
fd=£1.8 log10(Re) − 1.5¤−2 (3.6)
Importantly the surface roughness tends to increase as the tube surface accu- mulates foulant (in addition to the obvious reduction in cross-sectional area). To account for this, the pressure drop is physically measured (see section 3.3.4) in order to determine the friction factor (using equation (3.5)).
CHAPTER 3. EXPERIMENTATION
The change in friction factor also plays a role in the convective heat transfer. Since the friction factor influences the turbulent boundary layer, and this dictates the convective heat transfer, there exists an analogy between the friction factor and the convective heat transfer known as the Chilton-Colburn analogy (Cengel
& Ghajar, 2011)
Nu = 0.125fdRePr
1
3 (3.7)
where Nu is the dimensionless Nusselt number, i.e the ratio of convective to con- ductive heat transfer across the tube surface, viz.
Nu =hdk
f (3.8)
The accuracy of equation (3.7) is generally improved through experimental data incorporated in various convection correlations.
3.5.3 Literature correlation to calculate the inner convection
coefficient
For fully-developed convective heat transfer across the smooth tube surface, Dit- tus and Boelter, cited by Kröger (1998), found the Nusselt number to be a func- tion of Reynolds number and Prandtl number according to
Nu = 0.0265Re45Pr0.3 (3.9)
when the fluid is being cooled by the tube surface. When the fluid is heated by the surface they propose
Nu = 0.0243Re45Pr0.4 (3.10)
Compare this to the correlation found by Rabas & Cane (1983) (used by the ASME performance test code PTC12.2-2010):
Nu = 0.0158Re0.835Pr0.462 (3.11)
More recently, Gnielinski (2009) expanded on the work of Petukhov & Krillov (1958) to find Nu = ³f d 8 ´ RePr 1 + 12.7³fd 8 ´12 (Pr23− 1) (3.12)
valid for 2300 < Re < 106, 0.5 < Pr <104, and 0 < d1/L < 1. Taking into account
CHAPTER 3. EXPERIMENTATION
internal flow is fully developed by the time it enters the heat exchanger. Therefore equation (3.12) is applicable and the inner convection coefficient is
h =Nukd
1 (3.13)
with the fouling fluid thermal conductivity (k) evaluated at the mean bulk tem- perature, i.e.
Tm=1
2(Tin+ Tout) (3.14)
The effect of surface roughening is considered as the foulant alters the sur- face profile of the internal surface of the tube. The roughness Reynolds number (Lienhard & Lienhard, 2008) is
Re²= Re ² d1 µf d 8 ¶12 (3.15) whered²1 is the relative surface roughness. Although this cannot be measured di- rectly during the test, the relative surface roughness is estimated from the mea- sured friction factor, i.e. rewriting the correlation for the friction factor given by Kröger (1998), yields
² d1= e
((1.14−fd−0.5)/0.86) (3.16)
Provided the roughness Reynolds number is less than 5, equation (3.12) is used with the measured friction factor (Lienhard & Lienhard, 2008). If the rough- ness Reynolds number is between 5 and 70 the flow is transitionally rough and equation (3.12) is still used with the measured friction factor to estimate the Nus- selt number, albeit with less accuracy. When the roughness Reynolds number is above 70 the flow is termed fully rough and Bhatti & Shah (1987), cited by Lien- hard & Lienhard (2008), recommends the following correlation
Nu = ³f d 8 ´ RePr 1 +³fd 8 ´12 ¡ 4.5Re²0.2Pr0.5− 8.48¢ (3.17)
3.5.4 The annular convection coefficient
The annular convection analysis follows analogously to the inner tube, although slight differences arise from the velocity profile inside the annulus. The annular Reynolds number is based on the hydraulic diameter
Reann=ρannvann(d4− d3)
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Recently Dirker & Meyer (2005) showed the annular friction factor depends on the diameter ratio of the annulus. Gnielinski (2009) uses this in an effective Reynolds number for the annulus so that
fd,ann=¡1.8log10Re∗ ¢−2 (3.19) where Re∗= Reann µ 1 +³d4 d3 ´2¶ ln³d4 d3 ´ + µ 1 −³d4 d3 ´2¶ µ 1 −³d4 d3 ´2¶ ln³d4 d3 ´ (3.20)
Using this annular friction factor, Gnielinski (2009) found the following correla- tion with the annulus jacket insulated:
Nuann= ³f d,ann 8 ´ ReannPrann k1+ 12.7 ³f d,ann 8 ´0.5³ Prann 2 3− 1 ´ " 1 + µd h L ¶2/3# 0.75 µd 4 d3 ¶−0.17 (3.21) with k1= 1.07 + 900 Reann− 0.63 (1 + 10Prann) (3.22)
Equation (3.21) is the most comprehensive literature correlation describing the Nusselt number for fully developed annular convective flow. However, the heat exchangers in the apparatus have edge effects induced by the transition from circular to annular areas. Thence it is necessary to physically measure the actual annular Nusselt number for various Reynolds numbers before commenc- ing the fouling testing. Using the results from bare tube tests allows the outer convection coefficient to be determined, and the Nusselt number is regressed in terms of the Reynolds number in the form:
Nuann= ANuReannBNuPrann0.3 (3.23)
where ANuand BNuare the experimentally determined coefficients.