Chapter 4
Pressure Drop in Channels and Heat Exchangers Learning Objectives At the end of this chapter the student should:
• Recognise the importance of pressure drop in heat transfer system design • Know the three mechanisms governing pressure drop, i.e. gravity, fluid acceleration and friction.
• Be able to use correlations to determine the magnitude of the pressure drop in single phase flows.
• Be aware of the strategy used in determining pressure drop in two-phase flows.
4.1
Introduction As a fluid flows through a heat exchanger there will normally be a pressure drop in the direction of the flow (in some special situations where the fluid velocity decreases there may be an increase in pressure). Pressure drops occur in the flow channels, nozzles, manifolds and turning regions in the headers of heat exchangers and each of these pressure drops must be evaluated, unless experience suggests that one or more may be neglected.
As we have seen in section 3, when deriving Reynolds Analogy, there is a relationship between heat transfer coefficient and frictional pressure gradient. The designer will therefore find that measures to increase heat transfer coefficients tend to also increase the frictional pressure gradient. This does not necessarily have an adverse effect on overall pressure drop - clever design may mean that an increased pressure gradient is outweighed by a decrease in required passage length so the overall pressure drop remains acceptable.
The designer should aim to “use” all or most of the allowable or available pressure drop. Having a lower than permissible pressure drop across the heat exchanger implies that improvements could be made:, depending upon the application e.g. a longer heat exchanger would have a higher effectiveness
4.1
allowing greater heat recovery; changes in the heat exchanger geometry (for example, use of smaller tubes) would result in a more compact design for a particular heat load.
Determination of the maximum allowable pressure drop involves practical and thermodynamic considerations. If starting from first principles, it is possible to pursue various strategies to minimise entropy generation within a heat exchanger, remembering that entropy is generated by irreversible pressure drops in the heat exchanger and by heat transfer through a finite temperature
difference.
From
purely
thermodynamic
considerations,
frictional pressure drops should be minimised since they are irreversible, while in principle the work done in accelerating and raising the fluid may be recovered. An entropy analysis is particularly relevant if considering heat exchangers in power or refrigeration plant. For many heat exchangers the inlet and outlet temperatures and flow rate are fixed (e.g. an oil cooler) and the allowable pressure drop is a function of the pumps and fans used. There may, of course, be a trade off in this situation; for example, it may be decided to uprate a fan to permit the use of a more compact or cheaper heat exchanger. In applications involving natural convective circulation the pressure drop and flow rate are determined by the geometry of the convection loop (including the heat exchanger) the properties of the fluids involved and the rate of heat transfer.
For the purposes of the design examples in this module, and many practical cases, it is assumed that the maximum allowable pressure drops are given to the designer. The designer must then predict the pressure drop for candidate heat exchanger designs. If the pressure drop on one or both sides of the heat exchanger is excessive, then the design is unacceptable and an alternative has to be developed. If the pressure drops are significantly below the permissible level then the designer may wish to attempt to reduce the size and cost of the heat exchanger while “using” the available pressure drops.
4.2
Pressure Drop in Channels
4.2
The pressure gradient for a fluid flowing in the z direction along a channel is given by: dp dz
=
dp f dz
+
dp a dz
+
dp h
(4.1)
dz
where: dp dz dp f dz
dp a dz
= Pressure gradient at position z in the channel
= frictional pressure gradient at position z in the channel = Pressure gradient due to the momentum change at position z
in the channel dp h dz
=Hydrostatic pressure gradient at position z in the channel
and z is the coordinate in the flow direction along the channel
4.2.1 Single Phase Pressure Drop in Channels
In most single phase flows in channels (the exception being gases undergoing significant temperature change) the pressure gradient due to momentum change may be neglected.
With reference to Fig 4.1, the hydrostatic pressure gradient is given by: dp h dz
= −ρg sin θ
(4.2)
z
θ Flow Direction
l
Figure 4.1 Nomenclature used in defining pressure drop
4.3
For θ = 0, i.e. a horizontal channel, then the hydrostatic pressure gradient is zero.
For constant fluid density, or where the density change is small and a representative mean density may be used, equation 4.1 may be integrated over the length of the channel, and arranging the signs so that ∆p, the pressure drop, is positive, this gives:
∆ p h = ρgl sin θ
(4.3)
The frictional pressure gradient may be determined from: dp f dz
ρV 2
= −2c f
d e
1
ρV 2
2
d e
= − f
(4.4)
where c f and f are the Fanning skin friction coefficient and Darcy friction factor, as defined in equation 2.59, respectively. The hydraulic diameter, de, is defined in equation 2.60. the negative sign in equation 4.4 indicating that the pressure decreases in the direction of flow. If fluid properties may be reg arded as constant over a length l then equation 4.4. may be integrated, again with the pressure drop regarded as positive:
∆ p f = 2c f
l d e
ρV 2 =
1 2
f
l d e
ρV 2
(4.5)
Clearly, application of equation 4.4 or 4.5 requires knowledge of the appropriate value of the factor c f or f. Since f =4c f , by definition, there is little to choose between the two forms of equation 4.4. The student must, however, be sure which factor is given by a particular data source.
For the remainder of this section the Darcy friction factor, f , will be used (also, to add to the potential confusion, the value τ o
1 2
ρV 2 is referred to
in some texts as the Fanning friction factor and given the symbol f !)
The value of f is a function of the flow Reynolds nu mber, the rou ghness of the channel surface and the channel geometry. It will not surprise the reader
4.4
to learn that there are numerous correlations which may be used in the estimation of f . As with heat transfer, the pressure drop characteristics differ greatly depending whether the flow is laminar or turbulent, with transition occurring at a Reynolds number of 2000-10000.
For laminar flow f is independent of surface roughness and inversely proportional to the Reynolds number. Values of the constant of proportionality for a range of channel shapes are given in Table 2.2.
For round tubes: f =
64
(4.6)
Re
The simplest expression for friction factor f in turbulent flow, which is applicable to smooth pipes, is that due to Blasius: f =
0.3164 Re
(3000 < Re < 105 )
0.25
(4.7a)
which may be extended to higher Reynolds numbers f = 0.0032 +
0.221 Re
(10 5 < Re < 3 x 10 6 )
0.237
(4.7b)
an alternative expression for commercial pipe or slightly corroded tubes: f = 0.014 +
1.056
(3380
Re 0.42
(4.7c)
The variation of f is traditionally (at least since 1944!) presented on a Moody Diagram, as reproduced in Figure 4.2.1 Roughness values for a range of pipe materials and conditions are given in Table 4.1 .
1
ASHRAE Handbook of Fundamentals, ASHRAE, 1997
4.5
Table 4.1 Roughness value
Many heat exchanger tubes are drawn copper and therefore have a representative roughness of some 0.0025mm; for a 19mm diameter tube this implies a relative roughness of 0.00014, suggesting, in conjunction with figure 4.2, that equation 4.6 will give reasonable results within its range of applicability. Steel tubes having a representative roughness of 0.025mm, for a 19mm tube this gives a relative roughness of 0.0014, rising by a further factor of 10 when a coating of light rust forms. If dealing with initially rough tubes, tubes which are roughened by corrosion, or high Reynolds number flow, then the roughness must be taken into account. When carrying out hand calculations involving rough tubes or pipes then the quickest method of estimating f is to use a Moody Diagram.
4.6
For calculations using a computer it is necessary to put this data in numerical form. Moody produced a correlation (equation 4.8) which matched his diagram to within 5% for Reynolds numbers between 4000 and 107 and for values of ε /d /d of less than 0.01. 0.33 6 ⎡ ⎛ ⎤ ε 10 ⎞ f = 0.005496 ⎢1 + ⎜ 20000 + ⎟ ⎥ d Re ⎠ ⎥ ⎢⎣ ⎝ ⎦
(4.8)
Alternative correlations are available, for example ASHRAE ASHRAE2 recommend that for complete turbulence, where, as can be seen from fi gure 4.2, the friction factor becomes independent of Reynolds number, the follow following ing is used: used:
⎛ d ⎞ log10 ⎜ ⎟ = 1.14 + 2 lo f ⎝ ε ⎠
1
In general,
2
ASHRAE Handbook of Fundamentals, ASHRAE, 1997
4.7
(4.9)
1
⎛ ⎞ 9.3 ⎛ d ⎞ = 1.14 + 2 log10 ⎜ ⎟ − 2 log10 ⎜1 + ⎟ ⎜ ε f d ε Re ⎝ ⎠ ( ) ⎟⎠ f ⎝
(4.10)
The difficulty with this equation being t hat it must be solved iteratively.
The frictional pressure gradient or drop in long, non-circular channels (including annuli) may be estimated by using equation 4.4 or 4.5 and evaluating f for laminar flow using table 2.2. or for turbulent flow using one of equations 4.5-4.10.
Entry effects are significant when the channels are short, for example in many compact heat exchangers or in heat sinks for cooling electronic devices, then this must be taken taken into account. Values of f or c f are available for many plate-fin surfaces. Examples are given in figure 2.16. Correlations are available for friction factors which are appropriate to plate-heat exchangers, however the general procedure for calculating the pressure drop remains the same and is based upon equation 4.4.
4.8
4.2.2 Single phase pressure drop across tube bundles
The pressure drop across a tube bundle depends on the geometry of the bundle, fluid properties and flow rate. For plain tube bundles, assuming constant fluid properties, the pressure drop (excluding the gravitational component) is given by
∆ p = Eu
ρ V max
2
2
N r
(4.11)
where Eu is the Euler number and Nr is the number of tube rows. Eu is analogous to the friction factor in internal flows (and is referred to as a friction factor in some texts). V max is the maximum velocity between the tubes, as determined from figure .3.12 and equation 3.84 or 3.85.
Eu = f ( Re d , a, b, Nr )
where (using the nomenclature of fig 3.12), a =
(4.12)
ST d
, b=
SL d
. For many tube
rows, Eu is independent of Nr .
Curves showing Eu/k1 for inline and staggered tube banks having a large number of rows with (k1 being unity for a=b) together with values of k1 for a ≠ b are reproduced here as figures 4.3(a) and (b)3. Equations have been
fitted to these curves, However because of the complexity of the relationships these are not reproduced here. They are available in 4, where 2 and 3 Re ranges, for each value of a, with appropriate equations to correlate Eu and k1 for in-line and staggered tube banks respectively.
The pressure drop in the first few (3 or 4) rows differs from that predicted from fig 4.3. It may be higher or lower than the average value, depending upon
3 4
geometry
and
Re.
Correction
factors
may
be
Handbook of Heat Exchanger Design, Ed Hewitt, G.F., Begell House, New York, 1992 Handbook of Heat Exchanger Design, Ed Hewitt, G.F., Begell House, New York, 1992
4.9
defined
c z= Eu z/ Eu and C z= Eu z/ Eu where Eu is the Euler number from figure *
4.3, Eu z* is the Euler number determined from the pressure drop over tube row z and Eu z is the Euler number relating to the pressure drop for a bank having z rows. Table 4.2 gives values of c z and C z for bundles having less than 10 rows.
Figure 4.3a
Figure 4.3b
Pressure drop of in-line banks as referred to the relative longitudinal longitudinal pitch b
Pressure drop of staggered banks as referred to the relative transverse pitcha pitch a
4.10
Table 4.2 Correction factors for for row-to-row variations variations
4.2.3 Property variations
In general, fluid properties should be evaluated at the mean bulk temperature When the fluid properties vary due to heat transfer (or in the case of a gas, due to pressure drop) this may require the introduction of correction factors, or in the case of properties varying in the flow direction, division of the heat exchanger into several sections over which the fluid properties may be regarded as constant.
If the temperature difference between the wall and the bulk of the fluid is large, the viscosity may vary significantly between the bulk of the fluid and the n
⎛ µ ⎞ fluid close to the wall. Typically, a correction factor of the form ⎜ w ⎟ is ⎝ η bulk ⎠ used.
4.2.4 Pressure drop in nozzles and headers.
The pressure drop in headers and nozzles (and in pipe fittings in general) is usually expressed in terms of velocity heads. The appropriate velocities and typical values of K , the number of velocity heads lost, are given below.
4.11
Channel (i.e. tube-side) inlet and outlet nozzles:
⎛G2 ⎞ ⎛ ρ V 2 ⎞ ∆ pnt = Knt ⎜ = Knt ⎜ ⎜ 2 ρ ⎟⎟ ⎜ 2 ⎟⎟ ⎝ ⎠ ⎝ ⎠ n
n
(4.13)
based on mass flux or velocity in nozzles. K nt = 1.1 (inlet nozzle)
= 0.7 (outlet nozzle)
Headers:
⎛G2 ⎞ ⎛ ρ V 2t ⎞ N = Knt ⎜ ∆ ph = Kh ⎜ ⎟ N p ⎜ 2 ρ ⎟⎟ p 2 ⎝ ⎠ ⎝ ⎠ t
(4.14)
based on mass flux or velocity in tubes, Np= number of tube side passes. K k = 0.9 (one tube-side pass)
= 1.6 (two or more tube side passes)
Shell- side inlet and outlet nozzles:
⎛G2 ⎞ ⎛ ρ V 2 ⎞ ∆ pns = Kns ⎜ = Kns ⎜ ⎜ 2 ρ ⎟⎟ ⎜ 2 ⎟⎟ ⎝ ⎠ ⎝ ⎠ n
n
(4.15)
based on mass flux or velocity in nozzles. With impingement plate
⎛ An ⎞ ⎟ A e ⎝ ⎠
K nsi = 1.0 + ⎜
⎛ ⎞ 1 ⎟ Without impingement plate : K nsi = 1.0 + ⎜ ⎜ {( Ae An ) + 0.6 ( S − do S)}2 ⎟ ⎝ ⎠ Where An= flow area of nozzle Ae= escape are at nozzle = (perimeter of nozzle x distance from
nozzle to impingement plate or closest tubes) d o= tube outside diameter S=tube pitch
4.12
Nozzle and header losses must be added to the pressure drops calculated for the core of the heat exchanger.
4.3
Two-phase pressure drop (vapour + liquid)
The evaluation of the pressure drop during two-phase flow in a heat exchanger is generally complex. Each component of the pressure gradient is a function of, amongst other parameters, the quality, x, defined as the ratio of vapour mass flow to total mass flow. Each component of pressure gradient listed in equation 4.1 must be evaluated independently and applied to a short flow length, using a mean value of x for that length. The pressure drop for that length is then determined. The mean quality for the next increment of flow length is then calculated and the process repeated. The pressure drop is then the sum of the pressure gradient x section length for each of the sections. A comprehensive guide to the correlations which may be used is given in5
5
Handbook of Heat Exchanger Design, Chapter 2.3, Ed Hewitt, G.F., Begell House, New York, 1992
4.13
Summary Points •
The acceptable pressure drop in a heat exchanger is usually specified.
•
The designer should ensure the allowable pressure drop is not exceeded, but in optimising a design should attempt to 'use' all of the available pressure drop.
•
Correlations are available which allow the pressure drop in the heat exchanger core and attachments to be calculated.
•
Pressure drops in single-phase, incompressible flows are relatively easy to calculate. For two-phase flows numerical integration is required, a computer package or programme is required for all but the simplest of cases.
4.14
