Basic thermal design theory for heat exchangers

CHAPTER 2

Basic thermal design theory for heat exchangers Wilfried Roetzela, Xing Luob, Dezhen Chenc a

Institute of Thermodynamics, Helmut Schmidt University/University of the Federal Armed Forces Hamburg, Hamburg, Germany Institute of Thermodynamics, Gottfried Wilhelm Leibniz University Hannover, Hannover, Germany c Institute of Thermal Energy and Environmental Engineering, Tongji University, Shanghai, China b

The basic theories of heat exchangers are based on the relationships between the amount of heat exchanged and the heat transfer surface area, mass flow rates, entrance and exit temperatures of fluids, etc. These relations are based on the energy balance, mass balance, momentum balance, and the exchange mechanism. Thus, the fundamental equations for all types of heat exchangers include the mass, momentum and energy conservation equations, and the equation describing the heat transfer rate. These equations are the starting point for the design of heat exchangers and also for the transient analysis of heat exchangers. Depending upon different specific cases, the general forms of these equations may be simplified.

2.1 Heat transfer fundamentals In most cases, recuperative heat exchangers are operated in a steady state under preset operation conditions. Although some small disturbances in environment, inlet fluid temperatures, and flow rates might happen, the mean values of the operation parameters can be well maintained by automatic control systems; therefore, the steady-state design methods can still be applied, and they are the basic background of the design and operation of heat exchangers and their networks not only for steady-state operation but also for transient operation such as startup, shutdown, and operation switching.

Design and Operation of Heat Exchangers and their Networks https://doi.org/10.1016/B978-0-12-817894-2.00002-9

© 2020 Elsevier Inc. All rights reserved.

13

14

Design and operation of heat exchangers and their networks

2.1.1 Heat transfer coefficient and overall heat transfer coefficient The conventional heat exchangers are of recuperative type. In a recuperative heat exchanger, the heat transfer occurs through a separating wall (e.g., tube wall, plate, or other interfaces) between the two fluids. It is convenient to express the heat transfer rate per unit area (heat flux q) in terms of the heat transfer coefficient α defined by the Newton’s law of cooling: q ¼ α ðt  tw Þ

(2.1)

in which tw is the wall temperature and t is the fluid temperature. For an external flow, t is the temperature in the main fluid stream outside the thermal boundary layer. For an internal flow, we usually use the fluid bulk temperature as t, which is defined as an equilibrium temperature after an adiabatic mixing of the fluid from a given cross section of the flow channel: Z Z ρuc p tdAc utdAc Ac Ac Z Z tb ¼  (2.2) ρuc p dAc udAc Ac

Ac

For concision, we will omit the subscript “b” for the fluid bulk temperature if it does not cause a confusion. The value of the heat transfer coefficient strongly depends on the flow and heat transfer patterns, wall geometry, fluid properties, and fluid velocity, and in most cases, they are correlated experimentally. For general applications, the heat transfer coefficient is represented in a dimensionless group, Nusselt number: Nu ¼ αl=λ

(2.3)

where l is the characteristic length and λ is the thermal conductivity of the fluid at its reference temperature. For internal flow, we often use the hydraulic diameter as the characteristic length: dh ¼ 4Ac =P

(2.4)

in which Ac is the cross-sectional area of flow passage and P is the wetted perimeter. For variable cross-sectional area along the flow passage, the minimum cross-sectional area of the flow passage can be used to define the hydraulic diameter: dh ¼ 4Ac, min =P

(2.5)

Basic thermal design theory for heat exchangers

15

For complicated geometry, especially for compact heat transfer surfaces, we can also define dh as dh ¼ 4V =A

(2.6)

where V is the fluid volume in the flow passage and A is the heat transfer area. The Nusselt number (Nu) strongly depends on the Reynolds number (Re), which is a ratio of inertial forces to frictional forces. According to its value, we can know whether a flow is laminar, or undergoes a transition to turbulent flow, or is fully turbulent. The transition from laminar flow to turbulent flow can be distinguished by the critical Reynolds number Recr. For the fluid flow in a straight circular tube, Recr ¼ 2300. For Re < Recr, the flow is laminar. If Re > Recr, the flow is in a transition region and may become turbulent. When Re > 104, the flow is fully turbulent. In the laminar flow region, the heat transfer and pressure drop will be influenced by the form of channels and heating or cooling boundary conditions. Two typical boundary conditions are as follows: uniform wall temperature denoted by the subscript “T” and uniform heat flux denoted by the subscript “H” for uniform heat flux in both flow direction and peripheral direction (thin-wall duct) and “H1” for constant heat flux in the flow direction and uniform peripheral wall temperature (thick-wall duct), respectively. The Nusselt number for fully developed laminar flow in a circular tube with uniform wall temperature can be analytically obtained as 3.6567935, and that under the uniform heat flux condition is 48/11 (Shah and London, 1978), 12% higher than the former. The real boundary condition in heat exchangers might lie between these two values. If we were not sure which one is more suitable, we would like to take the value for uniform wall temperature for a conservative design of the heat exchanger. The inlet conditions also affect the heat transfer and pressure drop characteristics. When a fluid enters the tube or when it is heated (or cooled) beginning from the inlet cross section, the velocity or temperature boundary layer will form and develop along the wall until the boundary layer fills the entire flow channel. Therefore, at the entrance, the local heat transfer coefficient and frictional pressure drop are high and then decrease with the increase of the boundary layer thickness. For short heat exchangers, the entrance effects should be taken into account. Three cases have been considered: (1) thermally developing and hydrodynamically developed laminar flow, (2) thermally and hydrodynamically developing laminar flow, and (3) thermally and hydrodynamically developed laminar flow.

16

Design and operation of heat exchangers and their networks

2.1.1.1 Fully developed laminar flow in straight circular tubes In a thermally and hydrodynamically developed laminar flow through a straight circular tube, the Nusselt number is a constant. For constant wall temperature, NuT ¼ 3:66 ð Re < 2200, Pr > 0:6, RePrdh =L < 10Þ

(2.7)

For constant heat flux, NuH ¼ 4:36 ð Re < 2200, Pr > 0:6, RePrdh =L < 10Þ

(2.8)

Example 2.1 Sizing an electrically heated tube The compressed air at 1.5 bar with a normal volumetric flow rate of 1.2 Nm3/h shall be heated from 20°C to 80°C by the heating wire uniformly wrapped around the tube as it flows through the tube. The tube outside diameter is 25 mm, tube wall thickness is 2 mm, and thermal conductivity of the tube material is 15 W/mK. The tube temperature shall not exceed 200°C. Determine the length of the tube heating section. Solution The mean temperature of air is tm ¼ ðtin + tout Þ=2 ¼ 50°C The properties of air are calculated with RefProp. At normal pressure pN ¼ 1.01325 bar and normal tN ¼ 0°C, the density ρN ¼ 1.293 kg/m3, resulting in the mass flow rate as m_ ¼ ρN VN ¼ 1:293  1:2=3600 ¼ 4:310  104 kg=s The mass velocity is G¼

m_ 4:310  104 ¼ ¼ 1:244 kg=m2 s πdi2 =4 π  0:0212 =4

At the mean temperature, we have μ ¼ 1.964  105 sPa, λ ¼ 0.02810 W/mK, cp ¼ 1008 J/kgK, and Pr ¼ 0.7047. Then, we have the Reynolds number Gd i 1:244  0:021 ¼ ¼ 1331 < 2200 μ 1:964  105 The heat power transferred through the tube wall to the air flow is evaluated as Re ¼

Q ¼ mc _ p ðtout  tin Þ ¼ 4:310  104  1008  ð80  20Þ ¼ 26:07W Because the uniform electric heating implies the constant heat flux boundary condition, we chose at first Eq. (2.8) for the Nusselt number, which yields

Basic thermal design theory for heat exchangers

17

NuH λ 4:36  0:0281 ¼ 5:833 W=m2 K ¼ di 0:021 The overall heat transfer coefficient between the tube outside surface and the air flow can be expressed as     1 ln ðdo =di Þ 1 1 ln ð0:025=0:021Þ 1 k¼ + 0:021  + di ¼ αH 2λw 5:833 2  15 ¼ 5:829 W=m2 K αH ¼

If we assume that the air properties are constant and the heat conduction in the tube wall and air flow along the tube length is negligible, then for the constant heat flux boundary condition and constant heat transfer coefficient, the temperature difference between the tube wall and the fluid is a constant, and the temperature distributions in the tube wall and air flow along the tube length are two parallel straight lines. Therefore, the maximum wall temperature happens near the end of the heating section, x ¼ L: Δt ¼ tw, o  t ¼ ðtw,o  t Þx¼L ¼ 200  80 ¼ 120 K The required tube length is determined by Q ¼ kA △ t, which yields Q 26:07 ¼ 0:5649 m ¼ πdi kΔt π  0:021  5:829  120 Finally, we will check the entrance length by calculating the parameter L¼

1330  0:7047  0:021 ¼ 34:86 0:5649 which is larger than 10. That means the heating section is still in the entrance region, and the design with L ¼ 0.57 m is a little conservative, but it would be safe. The detailed calculation can be found in the MatLab code for Example 2.1 in the appendix. RePrdi =L ¼

2.1.1.2 Thermally developing and hydrodynamically developed laminar flow in straight circular tubes For the thermally developing and hydrodynamically developed laminar flow (the Nusselt-Graetz problem), Gnielinski (1989) suggested an asymptotic equation of the mean Nusselt number for constant wall temperature as h i3 Nu3T ¼ 3:663 + 0:73 + 1:615ðRePrd=L Þ1=3  0:7 (2.9) Compared with the analytical results given by Shah and London (1978, Table 13), the maximum deviation of Eq. (2.9) in 0.1 < RePrd/L < 106 is

18

Design and operation of heat exchangers and their networks

0.98% at RePrd/L ¼ 2500. As is shown in Fig. 2.1, this equation offers us the best fitting with the analytical solution. An empirical equation of Hausen for the mean Nusselt number was given by Stephan (1959): NuT ¼ 3:65 +

0:0668RePrd=L

(2.10)

1 + 0:045ðRePrd=L Þ2=3

The maximum deviation of this equation in 0.1 < RePrd/L < 105 is +2.9% at RePrd/L ¼ 1600. However, if RePrd/L is larger than 2  104, Eq. (2.10) becomes lower than the analytical solution, and the relative deviation approaches to 8.1% when RePrd/L ! ∞. Another empirical equation of Hausen was presented by Stephan and Nesselmann (1961) as NuT ¼ 3:65 +

0:19ðRePrd=L Þ0:8 1 + 0:117ðRePrd=L Þ0:467

(2.11)

Its maximum deviation in 0.1 < RePrd/L < 106 is +9.6% at RePrd/L ¼ 15 but approaches to the analytical solution for RePrd/L ! ∞. For detailed simulation of heat exchangers, especially evaporators and condensers, local heat transfer coefficient might be required for the 103

NuT, Nux,T

102

NuT, analytical solution (Shah and London, 1978, Table 13) NuT, Eq. (2.9) (Gnielinski, 1989) NuT, Eq. (2.10) (Stephan, 1959) NuT, Eq. (2.11) (Stephan and Nesselmann, 1961) Nux,T, analytical solution (Shah and London, 1978, Table 13) Nux,T, Eq. (2.12) (Gnielinski, 2010a, 2013a)

10

1 0.1

1

10

103 102 RePrd/x

104

105

106

Fig. 2.1 Local and mean Nusselt number Nux, T and NuT for thermally developing and hydrodynamically developed laminar flow.

Basic thermal design theory for heat exchangers

19

determination of local heat flux and wall temperature distributions. An asymptote can be used for local Nusselt number (Gnielinski, 2010a, 2013a) h i3 Nu3x, T ¼ 3:663 + 0:73 + 1:077ðRePrd=xÞ1=3  0:7 (2.12) The maximum deviation of Eq. (2.12) in 0.1 < RePrd/x < 106 is +6.2% at RePrd/x ¼ 25. For constant heat flux boundary condition (H), the local Nusselt number can be expressed as (Gnielinski, 2010a, 2013a) h i3 1=3 3 3 Nux, H ¼ 4:354 + 1 + 1:302ðRePrd=xÞ  1 (2.13) The maximum deviation of Eq. (2.13) in 5 < RePrd/L < 106 is 4.0% at RePrd/L ¼ 200. Shah and London (1978) recommended a combination of their work and the approximate equation of Grigull and Tratz (1965) as follows: 8 x=ðdRePrÞ  0:00005 1:302ðRePrd=xÞ1=3  1, > > < 1=3 0:00005 < x=ðdRePrÞ < 0:0015 Nux, H ¼ 1:302ðRePrd=xÞ  0:5, 0:506 > x=ðdRePrÞ  0:0015 4:364 + 8:68 ð 0:001RePrd=x Þ > : e41=ðRePrd=xÞ , (2.14) which has the maximum deviation of 1.0% around RePrd/x ¼ 104. With a similar asymptote of Gnielinski, the mean Nusselt number can be calculated from (Gnielinski, 2010a, 2013a): h i3 Nu3H ¼ 4:3543 + 0:63 + 1:953ðRePrd=L Þ1=3  0:6 (2.15) of which the maximum deviation in 0.1 < RePrd/L < 106 is 0.89% at RePrd/L ¼ 50. A comparison among these correlations is shown in Fig. 2.2. It is interesting to notice that the experimental correlation of the mean Nusselt number of Sieder and Tate (1936) Nu ¼ 1:86ðRePrd=L Þ1=3 ðμ=μw Þ0:14 ðRe < 2200, 0:0044 < μ=μw < 9:75, and RePrdh =L > 10Þ

(2.16)

is close to the analytical solution for uniform heat flux boundary condition. Their experiments were carried out in a concentric-tube heat exchanger. The inner tube had a total length of 27.7 m including a calming section

20

Design and operation of heat exchangers and their networks

103

NuH, Nux,H

102

NuH, analytical solution (Shah and London, 1978, Table 18) NuH, Eq. (2.15) (Gnielinski, 2010a, 2013a) Nuexp, Eq. (2.16) (Sieder and Tate, 1936) Nux,H, analytical solution (Shah and London, 1978, Table 18) Nux,H, Eq. (2.13) (Gnielinski, 2010a) Nux,H, Eq. (2.14) (Shah and London, 1978)

10

1 0.1

1

10

102 103 RePrd/x

104

105

106

Fig. 2.2 Local and mean Nusselt number Nux,H and NuH for thermally developing and hydrodynamically developed laminar flow.

of 6.1 m and a mixing portion of 6.1 m. The effective length is 15.5 m. They used water flowing through the annular space to heat or cool the oil flowing inside the inner tube. According to their experimental data, they might have used counterflow arrangement, because in their test runs No. 7, 8, and 39, the wall temperatures were lower or equal to the outlet temperature of water that would never happen in a parallel-flow arrangement. In such a flow arrangement, the real boundary condition might more likely be the constant heat flux than the constant wall temperature, especially when the ratio of thermal capacity rates of the two fluids approaches to one. 2.1.1.3 Thermally and hydrodynamically developing laminar flow in straight circular tubes The asymptotic equations for both thermally and hydrodynamically developing laminar flow are given by Eqs. (2.17), (2.18) for constant wall temperature (Gnielinski, 1989, 2010a, 2013a): h i3 Nu3T ¼ 3:663 + 0:73 + 1:615ðRePrd=L Þ1=3  0:7 " #3 1=6 2 + ðRePrd=L Þ1=2 (2.17) 1 + 22 Pr

Basic thermal design theory for heat exchangers

h i3 Nu3x, T ¼ 3:663 + 0:73 + 1:077ðRePrd=xÞ1=3  0:7 "  #3 1=6 1 2 + ðRePrd=xÞ1=2 2 1 + 22 Pr

21

(2.18)

In Fig. 2.3, these equations are compared with the equation of Stephan (Baehr, 1960) NuT ¼ 3:66 +

0:0677ðRePrd=L Þ1:33 1 + 0:1 Prð Red=L Þ0:83

(2.19)

and the equation of Churchill and Ozoe (1973b) n o3=8 Nux, T ¼ 5:357 1 + ½ðπ=388ÞRePrd=x8=9 2

0

6 B 41 + @n

14=3 33=8 ðπ=284ÞRePrd=x C o1=2 n o3=4 A 1 + ð Pr=0:0468Þ2=3 1 + ½ðπ=388ÞRePrd=x8=9

1:7

7 5 (2.20)

The maximum deviation among them is 6.5% in 0.1 < RePrd/L < 106 and 0.7  Pr  10. 103 NuT, Eq. (2.17) (Gnielinski, 1989, 2010a, 2013a) NuT, Eq. (2.19) (Baehr, 1960) Nux,T, Eq. (2.18) (Gnielinski, 2010a, 2013a) Nux,T, Eq. (2.20) (Churchill and Ozoe, 1973)

102 NuT, Nux,T

Pr = 0.7 NuT Pr = 10

10 Nux,T

1 0.1

1

10

103 102 RePrd/x

104

105

106

Fig. 2.3 Local and mean Nusselt number Nux,T and NuT for thermally and hydrodynamically developing laminar flow.

22

Design and operation of heat exchangers and their networks

The local and mean Nusselt numbers in circular tubes with uniform heat flux at the tube wall (H) can be expressed with similar asymptotes (Gnielinski, 2010a, 2013a); i3 h i3 h Nu3x, H ¼ 4:3543 + 1:302ðRePrd=xÞ1=3  1 + 0:462ð Red=xÞ1=2 Pr1=3 (2.21)

h i3 3 Nu3H ¼ 4:354 + 0:63 + 1:953ðRePrd=L Þ1=3  0:6 h i3 + 0:924ð Red=L Þ1=2 Pr1=3 ð Pr > 0:7Þ

(2.22)

Another available correlation was proposed by Churchill and Ozoe (1973a): n o3=10 Nux, H ¼ 5:364 1 + ½ðπ=220ÞRePrd=x10=9 2

0

15=3 33=10

6 B 41 + @h

ðπ=115:2ÞRePrd=x C i1=2 n o3=5 A 2=3 10=9 1 + ð Pr=0:0207Þ 1 + ½ðπ=220ÞRePrd=x

1

7 5 (2.23)

The curves of these equations for Pr ¼ 0.7 and Pr ¼ 10 are shown in Fig. 2.4. The maximum deviation between Eqs. (2.20) and (2.23) is 6.5% for Pr ¼ 0.7 and 3.8% for Pr ¼ 10. 103 NuH, Eq. (2.22) (Gnielinski, 2010a, 2013a) Nux,H, Eq. (2.21) (Gnielinski, 2010a, 2013a) Nux,H, Eq. (2.23) (Churchill and Ozoe, 1973)

Pr = 0.7 NuH, Nux,H

102 NuH Pr = 10 10

1 0.1

Nux,H

1

10

102 103 RePrd/x

104

105

106

Fig. 2.4 Local and mean Nusselt number Nux,H and NuH for thermally and hydrodynamically developing laminar flow.

Basic thermal design theory for heat exchangers

Example 2.2 Sizing an electrically heated tube (continued) The problem is the same as Example 2.1, which gives a safe design of L ¼ 0.57 m. For some reason, the tube should be shortened. A mixing disk with many small holes is assembled at the inlet of the tube so that the inlet velocity distribution could be uniform. Considering 5% uncertainty in the calculation of the heat transfer coefficient, evaluate the minimal acceptable length of the heating section. Solution Because of the use of a mixing disk in the front of the heating section, we can consider it as thermally and hydrodynamically developing laminar flow; therefore, Eq. (2.21) can be used for evaluating the local heat transfer coefficient. Since the local heat transfer coefficient in the entrance region decreases along the tube length, the use of its mean value will underestimate the temperature difference between the tube wall and air flow. The temperature distributions in the tube outside wall and air flow along the tube length are shown in Fig. 2.5, which indicates that the highest wall temperature appears near the end of the heating section, at which the local heat transfer coefficient is calculated with Eq. (2.21). We will at first take L ¼ 0.57 m as the initial value to calculate the local heat transfer coefficient at x ¼ L as follows: RePrdi =x ¼ 1330  0:7047  0:021=0:57 ¼ 34:54 250 200

t (°C)

tw 150 100 t 50 0 0

0.1

0.2

0.3

0.4

0.5

x (m) Fig. 2.5 Temperature distributions in tube wall and air flow along the tube length. Continued

23

24

Design and operation of heat exchangers and their networks

 i3 1=3 h i3 h Nux, H ¼ 4:3543 + 1:302ðRePrdi =xÞ1=3  1 + 0:462ðRePrdi =xÞ1=2 Pr1=6  i3 1=3 h i3 h ¼ 4:3543 + 1:302  34:541=3  1 + 0:462  34:541=2  0:70471=6 ¼ 5:198 αx, H ¼

Nux, H λ 5:198  0:0281 ¼ 6:954 W=m2 K ¼ di 0:021

By considering 5% uncertainty in αx,H, the overall heat transfer coefficient between the tube outside surface and the air flow at x ¼ L is   1 ln ðdo =di Þ 1 k¼ + di ð1  0:05ÞαH 2λw   1 ln ð0:025=0:021Þ 1 + 0:021  ¼ ¼ 6:601W=m2 K ð1  0:05Þ  6:954 2  15 from which we obtain the length of the heating section by Q 26:07 ¼ 0:4989 m ¼ πdi kΔt π  0:021  6:601  120 Using the newly calculated length and repeating the earlier calculations, we finally obtain L ¼ 0.4822m. That means that the heating section length should not be shorter than 0.4822 m, and L ¼ 0.5 m is recommended. The detailed calculation can be found in the MatLab code for Example 2.2 in the appendix. L¼

2.1.1.4 Laminar flow between parallel plates For fully developed laminar flow between parallel plates with the same constant wall temperature, the Nusselt number is a constant: NuT ¼ 7:5407

(2.24)

For fully developed laminar flow between parallel plates with constant but different heat flux at each wall, the Nusselt number are obtained as (Shah and London, 1978) NuH, p1 ¼

140 26  9qp2 =qp1

where p1 denotes one plate wall and p2 the other.

(2.25)

Basic thermal design theory for heat exchangers

25

In thermally developing laminar flow, we can use the approximate equations of Shah and London (1978): 8 L=ðdh RePrÞ  0:0005 < 1:849ðRePrdh =L Þ1=3 , 1=3 NuT ¼ 1:849ðRePrdh =L Þ + 0:6, 0:0005 < L=ðdh RePrÞ  0:006 : L=ðdh RePrÞ > 0:006 7:541 + 0:0235RePrdh =L, (2.26) 8 < 1:233ðRePrdh =xÞ1=3 + 0:4, x=ðdh RePrÞ  0:001 Nux, T ¼ 7:541 + 6:874ð0:001RePrdh =xÞ0:488 x=ðdh RePrÞ > 0:001 : e245=ðRePrdh =xÞ , (2.27) 8 L=ðdh RePrÞ  0:001 < 2:236ðRePrdh =L Þ1=3 , NuH ¼ 2:236ðRePrdh =L Þ1=3 + 0:9, 0:001 < L=ðdh RePrÞ < 0:01 : L=ðdh RePrÞ  0:01 8:235 + 0:0364RePrdh =L, (2.28) 8 x=ðdh RePrÞ  0:0002 1:490ðRePrdh =xÞ1=3 , > > < 1=3 0:0002 < x=ðdh RePrÞ  0:001 Nux, H ¼ 1:490ðRePrdh =xÞ  0:4 0:506 > ð =x Þ x=ðdh RePrÞ > 0:001 8:235 + 8:68 0:001RePrd > h : e164=ðRePrdh =xÞ , (2.29) For the thermally and hydrodynamically developing laminar flow, Stephan (1959) proposed the following correlation: 0:024ðRePrd=L Þ1:14 NuT ¼ 7:55 + 1 + 0:0358 Pr0:81 ð Red=L Þ0:64

(2.30)

2.1.1.5 Fully developed laminar flow in rectangular ducts The Nusselt number for fully developed laminar flow in rectangular channels was approximately expressed by Shah and London (1978) as ! 5 X n Nu ¼ a0 1 + an γ ðγ ¼ aspect ratio, 0  γ  1, Re < 2200, Pr > 0:6Þ n¼1

(2.31) For constant wall temperature (T): a0 ¼ 7.541, a1 ¼ 2.61, a2 ¼ 4.97, a3 ¼ 5.119, a4 ¼ 2.702, and a5 ¼ 0.548. For the boundary condition of constant heat flux in the flow direction and uniform peripheral wall

26

Design and operation of heat exchangers and their networks

temperature (H1): a0 ¼ 8.235, a1 ¼ 2.0421, a2 ¼ 3.0853, a3 ¼ 2.4765, a4 ¼ 1.0578, and a5 ¼ 0.1861. The maximum deviation of Eq. (2.31) was reported as 0.1% for NuT and 0.03% for NuH1. 2.1.1.6 Heat transfer in turbulent flow In the fully turbulent region, the velocity and temperature boundary layers are relative thin, and the form of the channel cross section has negligible influence on the heat transfer and pressure drop. Therefore, the correlations for a circular tube can be applied to other forms of channels except the ducts with sharp corners. A simple correlation for turbulent heat transfer is the Dittus-Boelter correlation (Dittus and Boelter, 1930):  Nu ¼ 0:023 Re0:8 Prn Re > 104 , 0:7 < Pr < 120, L=dh > 10 (2.32) where n ¼ 0.4 for heating and n ¼ 0.3 for cooling. For fine design calculation, the Gnielinski correlation is recommended (Gnielinski, 1975): "   # ðf =8ÞðRe  1000ÞPr dh 2=3 pffiffiffiffiffiffiffi Nu ¼ 1+ L 1 + 12:7 f =8 Pr2=3  1  K 2300 < Re < 106 , 0:6 < Pr < 105

(2.33)

f ¼ ½1:82lgð ReÞ  1:642

(2.34)

ð Pr= Prw Þ0:11 for liquid , 0:05 < Pr= Prw < 20 ðTb =Tw Þ0:45 for gas, 0:5 < Tb =Tw < 1:5

(2.35)

where  K¼

In the transition region, the heat transfer and pressure drop become very sensitive to the conditions of wall surface and incoming flow and have relative large uncertainties, which yielded large deviations among different experiments. A commonly used method for evaluating Nu in the transition region is the interpolation between the laminar and turbulent regions (Gnielinski, 1995): Nu ¼ Nulam, Re¼Recr + with Recr ¼ 2300.

Re  Recr  4  Nulam, Re¼Re (2.36) Nu tur , Re¼10 cr 104  Recr

Basic thermal design theory for heat exchangers

27

2.1.1.7 Heat transfer in concentric annular ducts For heat transfer in concentric annular ducts, three boundary conditions have often been met: (1) heat transfer through the inner tube, with the insulated outer tube; (2) heat transfer through the outer tube, with the insulated inner tube; and (3) heat transfer through both the inner and outer tubes having the same wall temperature. A typical example is double-pipe heat exchangers, which are usually treated as boundary condition (1). Stephan (1962) developed a set of correlations. For boundary conditions (1) and (2), the Nusselt number for turbulent flow (2300  Re  106) can be evaluated by the following equations, respectively: h i Nui ¼ 0:033ðdo =di Þ0:45 1 + ðdh =L Þ2=3 Re0:75  180 Pr0:42 ðμ=μw Þ0:14 (2.37) h i Nuo ¼ 0:037ð1  0:1di =do Þ 1 + ðdh =L Þ2=3 Re0:75  180 Pr0:42 ðμ=μw Þ0:14 (2.38) For boundary condition (3), Stephan suggested the following relation: Nu ¼

Nui di =do + Nuo di =do + 1

(2.39)

For hydrodynamically developed laminar flow in the thermal entrance region, Stephan expressed the Nusselt number as follows: 0:19ðRePrdh =L Þ0:8 Nu ¼ Nu∞ + f ðdi =do Þ 1 + 0:117ðRePrdh =L Þ0:467

(2.40)

in which Nu∞ is the Nusselt number for fully developed laminar flow under the corresponding boundary condition, and the function f (di/do) was given as 8 < 1 + 0:14ðdi =do Þ1=2 , heat transfer inner tube (2.41) f ðdi =do Þ ¼ 1 + 0:14ðdi =do Þ1=3 , heat transfer outer tube : 1 + 0:14ðdi =do Þ0:1 , heat transfer both tubes Martin’s expressions (Gnielinski, 2010b, 2013b) can be used for Nu∞: 8 > heat transfer inner tube 3:66 + 1:2ðdi =do Þ0:8 , > < heat transfer outer tube 3:66 + 1:2ðdi =do Þ0:5 , Nu∞ ¼ > 3:66 + ½4  0:102=ð0:02 + di =do Þ heat transfer both tubes > : ðdi =do Þ0:04 , (2.42)

28

Design and operation of heat exchangers and their networks

For thermally and hydrodynamically developing laminar flow, the asymptote of Gnielinski (2010b, 2013b) can be used to calculate the mean Nusselt number: h i3 Nu3T ¼ Nu3∞ + 1:615f ðdi =do ÞðRePrd=L Þ1=3 " #3 1=6 2 1=2 ðRePrd=L Þ + (2.43) 1 + 22Pr 2.1.1.8 Heat transfer in curved tubes Curved tubes such as helically coiled tubes are widely used in industries as heat exchangers. Compared with the fluid flow and heat transfer in straight tubes, a higher heat transfer coefficient arises due to the centrifugal force generated by curvature of the tubes. A secondary flow is induced by the centrifugal force and enhances the heat transfer rate. Naphon and Wongwises (2006) presented a review of the work done on the characteristics of single-phase and two-phase heat transfer and flow in curved tubes. The pressure drop correlations for flow through helically coiled tubes were summarized by Ali (2001). A systematic review of heat transfer and pressure drop correlations for laminar flow in curved tubes including the correlations for critical Reynolds number was made by Ghobadi and Muzychka (2016). Based on the experimental data of air, water, and oil, Schmidt (1967) proposed the following correlations for single-phase flow and heat transfer in curved tubes in the laminar, transition, and turbulent regions. In his experiments, the fluids were heated by steam condensation on the tube outside; therefore, the boundary condition can be considered as the constant wall temperature. The ratio of the tube radius r to the radius of the tube curvature rc of the tested curved tubes covered the range of 0.12–0.2. For laminar flow (100 < Re < Recr),  0:194 Nu ¼ 3:65 + 0:08 1 + 0:8ðr=rc Þ0:9 Re0:5 + 0:2903ðr=rc Þ Pr1=3 (2.44) In the transition region (Recr < Re < 2.2  104), h i 0:1 Nu ¼ 0:023 1 + 14:8ð1 + r=rc Þðr=rc Þ1=3 Re0:80:22ðr=rc Þ Pr1=3

(2.45)

For turbulent flow (2  104 < Re < 1.5  105),  Nu ¼ 0:023 1 + 3:6ð1  r=rc Þðr=rc Þ0:8 Re0:8 Pr1=3

(2.46)

Basic thermal design theory for heat exchangers

29

In the curved tubes, the critical Reynolds number is larger than that in the straight tubes:  Recr ¼ 2300 1 + 8:6ðr=rc Þ0:45 (2.47) 2.1.1.9 Extended heat transfer surfaces For the extended heat transfer surfaces (fins), there are two parallel heat transfer processes. The one is the convective heat transfer from the unfinned surface to the fluid, and the other is the conductive heat transfer through the fins and then from the fin surface to the fluid by heat convection. The effect of the conductive thermal resistance on the heat transfer performance of fins can be expressed by the fin efficiency, defined as the ratio of the heat transferred from the fin to the heat that would be transferred by the fin if its thermal conductivity were infinite large (i.e., if the entire fin were at the same temperature as its base): ηf ¼

Qf , actual αf Af ðtw  tÞ

(2.48)

where tw is the wall temperature at the fin base, t is the fluid temperature, and αf is the heat transfer coefficient at the fin surface (usually we take αf ¼ α). Then, we can express the total heat transfer Q and the overall fin efficiency η0 as Q ¼ αðA  Af Þðtw  tÞ + ηf αf Af ðtw  t Þ ¼ η0 αAðtw  tÞ η0 ¼ 1  ð1  ηf αf =αÞAf =A

(2.49) (2.50)

The expression of fin efficiency depends on the fin profile. Some typical examples of the fin profiles are longitudinal fins of rectangular, trapezoidal, or parabolic profiles; radial fins of these profiles; and cylindrical, truncated conical, or truncated parabolic spines. Consider a fin on a plate wall. By assuming one-dimensional heat conduction along the fin height in x direction, for a given profile, we have the fin cross-sectional area Ac,f ¼ Ac,f (x) and fin wetted perimeter Pf ¼ Pf (x) at the position x. The heat conduction along the fin height x can be expressed as   d dtf λf Ac, f ðxÞ (2.51) ¼ αf Pf ðxÞðtf  tÞ dx dx x ¼ 0 : tf ¼ tw x ¼ h :  λf

dtf dtf ¼ αf ðtf  t Þ or ¼ 0 ðadiabatic at fin tipÞ dx dx

(2.52) (2.53)

30

Design and operation of heat exchangers and their networks

For the fin around a cylinder (annular finned tube), the heat conduction along the fin height r is described in the cylinder coordinates as   d dtf (2.54) λf Ac, f ðr Þ ¼ αf Pf ðr Þðtf  t Þ dr dr r ¼ R0 : tf ¼ tw

(2.55)

dtf dtf ¼ αf ðtf  tÞ or ¼ 0 ðadiabatic at fin tipÞ (2.56) dr dr For example, for an annular fin with constant fin thickness, Ac,f (r) ¼ 2πrδ, and Pf (r) ¼ 2πr. Eq. (2.54) then turns into   1 d dtf αf ðtf  tÞ (2.57) r ¼ r dr dr λf δ r ¼ R :  λf

The analytical solutions for several typical fin profiles are available in the literature (Kraus et al., 2001). For the fins with constant cross-sectional area, constant wetted perimeter, and adiabatic boundary condition at the fin tip, the fin efficiency is given by tanh ðmhÞ mh where m is referred to as the fin performance factor, sffiffiffiffiffiffiffiffiffiffiffiffi αf Pf m¼ λf Ac, f ηf ¼

(2.58)

(2.59)

If the heat convection at the fin tip should also be taken into account, we can still use Eq. (2.58) by extending the fin height with Δh ¼ Ac, f =Pf

(2.60)

2.1.1.10 Overall heat transfer coefficient The overall heat transfer coefficient k is principally based on the heat transfer coefficients at both sides of the wall separating the two fluids. It can be expressed as the sum of a series of thermal resistances: convective heat transfer resistances of the two fluids, conductive heat transfer resistance of the wall, and possible fouling resistances at hot and cold sides: 1 1 Rf , h δw Rf , c 1 + + + + ¼ kA αh Ah α c Ac Ah λw Am Ac

(2.61)

31

Basic thermal design theory for heat exchangers

Here, we should pay attention to kA, in which k is always related to its corresponding heat transfer area A. For example, if A is the outside area of a tube, then we should definitively express that k is based on the tube outside area. If k is based on the area of tube inside, A must be the tube inner area. The third term of the right side of Eq. (2.61) represents the thermal resistance of the wall, in which Am is the mean wall area perpendicular to conductive heat flux through the wall. For a tube wall, Am ¼

2πδw L lnðdo =di Þ

(2.62)

The conductive thermal resistance per unit area of tube inside can be expressed as R w, i ¼

δw di lnðdo =di Þ ¼ 2λw λw Am =ðπdi L Þ

(2.63)

For the heat exchangers with extended heat transfer surfaces (finned surfaces), Eq. (2.61) should be rewritten as 1 1 Rf , h δw Rf , c 1 + + + + ¼ kA η0, h αh Ah η0, h Ah λw Am η0, c Ac η0, c αc Ac

(2.64)

Example 2.3 Cooling of a printed circuit board A printed circuit board is cooled by blowing air through a heat sink as is shown in Fig. 2.6. The printed circuit board is 150 mm in length and 80 mm in width and has a heat duty of 100 W. The heat sink is made of aluminum and has 13 rectangular air flow channels with channel spacing

d hfs

sfs df

d

L

B

Fig. 2.6 Cooling of a printed circuit board. Continued

32

Design and operation of heat exchangers and their networks

of 25 mm in height, channel spacing of 3.5 mm in width, and wall thickness of 2.5 mm. The thermal conductivity of the heat sink material is 230 W/mK. The air flow rate is 18 m3/h at the inlet air temperature of 25°C. Assuming that the heat flux is uniform over the printed circuit board, evaluate its highest temperature. Solution The properties of air are calculated with RefProp. With the air density ρin ¼ 1.169 kg/m3 at 25°C, we have m_ ¼ ρin V ¼ 1:169  18=3600 ¼ 0:005844 kg=s m_ 0:005844 ¼ 5:138 kg=m2 s G¼ ¼ Nhfs sfs 13  0:025  0:0035 For the rectangular channel, the hydraulic diameter dh is given by dh ¼

2hfs sfs 2  0:025  0:0035 ¼ 0:00614 m ¼ 0:025 + 0:0035 hfs + sfs

The aspect ratio is equal to γ ¼ sfs =hfs ¼ 0:0035=0:025 ¼ 0:14 The wall thickness δf ¼ δ ¼ 0:0025 m We assume at first the air mean temperature of 25°C and obtain the specific isobaric thermal capacity cp ¼ 1007 J/kgK. The outlet air temperature is calculated as tout ¼ tin +

Q 100 ¼ 42°C ¼ 25 + m_cp 1007

and the new mean temperature is tm ¼ ðtin + tout Þ=2 ¼ ð25 + 42Þ=2 ¼ 33:5°C Since the thermal capacity at the new mean temperature is almost the same, no further iteration is needed. The air properties are given as ρ ¼ 1.136 kg/m3, cp ¼ 1007 J/kgK, μ ¼ 1.886  105 sPa, λ ¼ 0.02688 W/mK, and Pr ¼ 0.7062. Thus, we obtain the Reynolds number as Re ¼

Gd h 5:138  0:00614 ¼ ¼ 1673 μ 1:886  105

Because of the thick wall of the aluminum heat sink, the boundary condition of constant heat flux in the flow direction and uniform

Basic thermal design theory for heat exchangers

33

peripheral wall temperature (H1) would be reasonable. Therefore, the Nusselt number is evaluated with Eq. (2.31): NuH1 ¼ 8:235   1  2:0421γ + 3:0853γ 2  2:4765γ 3 + 1:0578γ 4  0:1861γ 5 ¼ 6:326 which yields the heat transfer coefficient as α¼

NuH1 λ 6:326  0:02688 ¼ 27:69 W=m2 K ¼ dh 0:00614

The fin efficiency can be determined by ηf ¼

m1 sinh ðm1 lf ,1 Þcosh ðm2 lf , 2 Þ + m2 cosh ðm1 lf ,1 Þsinh ðm2 lf , 2 Þ m1 ðlf , 1 + lf , 2 Þ½m1 cosh ðm1 lf ,1 Þcosh ðm2 lf ,2 Þ + m2 sinh ðm1 lf ,1 Þ sinh ðm2 lf , 2 Þ (2.65)

in which “1” and “2” denote the first and second fin sections, respectively, and the fin performance factor can be calculated with Eq. (2.59): sffiffiffiffiffiffiffiffiffiffiffiffi α f Pf m¼ λf Ac, f where Pf is the perimeter of the fin and Ac,f is its cross-sectional area. In this example, rffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2α 2  27:69 ¼ 9:814 m1 , lf ,1 ¼ hfs ¼ 0:025 m ¼ m1 ¼ λf δf 230  0:0025 rffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi α 27:69 ¼ 6:939 m1 , lf ,2 ¼ sfs =2 ¼ 0:0035=2 m2 ¼ ¼ λf δ 230  0:0025 ¼ 0:00175 m Substituting these values into Eq. (2.65), we get the value of the fin efficiency as ηf ¼ 0:947 The overall fin efficiency can be determined with Eq. (2.50) as Af 2hfs + sfs ¼ 1  ð1  ηf Þ A 2ðhfs + sfs Þ 2  0:025 + 0:0035 ¼ 0:9503 ¼ 1  ð1  0:947Þ 2  ð0:025 + 0:0035Þ Now, we can express the local temperature difference as η 0 ¼ 1  ð1  η f Þ

Continued

34

Design and operation of heat exchangers and their networks

  Q Q 1 δ ¼ + kA 2N ðhfs + sfs ÞL η0 α λf   100 1 0:0025 + ¼ ¼ 34:2K 2  13  ð0:025 + 0:0035Þ  0:15 0:9503  27:69 230

Δt ¼

The highest temperature of the printed circuit board appears near the air outlet: tmax ¼ tout + Δt ¼ 42:0 + 34:2 ¼ 76:2°C The detailed calculation can be found in the MatLab code for Example 2.3 in the appendix.

2.1.2 Basic equations for steady-state operations of heat exchangers The first law of thermodynamics should be satisfied in any heat exchanger both at macro- and microlevel. Taking the overall “macro” energy balance for a heat exchanger in a steady state, we have  Q ¼ C_ m, h th0  th00 (2.66)  00 0 (2.67) Q ¼ C_ m, c tc  tc The general heat exchange rate equation is given by Z A kðth  tc ÞdA ¼ km Δtm A Q¼

(2.68)

0

The mean thermal capacity rate of a fluid can be calculated according to the enthalpy change C_ m ¼ m_ ðh0  h00 Þ=ðt0  t00 Þ

(2.69)

or be calculated approximately according to the specific isobaric thermal capacity at its mean temperature: _ p, m ¼ mc _ p ½ðT 0 + T 00 Þ=2, p C_ m ¼ mc The mean temperature difference is defined by Z 1 A Δtm ¼ ðth  tc ÞdA A 0

(2.70)

(2.71)

Basic thermal design theory for heat exchangers

35

The substation of Eq. (2.71) into Eq. (2.68) yields the mean overall heat transfer coefficient as Z A kðth  tc ÞdA km ¼ Z0 A (2.72) ðth  tc ÞdA 0

Eqs. (2.66)–(2.68) are three basic equations for heat exchanger design. However, by using Eq. (2.68), the mean temperature difference Δtm should be known or needs to be determined. Fig. 2.7 shows the temperature variations in two typical heat exchangers: parallel-flow heat exchanger and counterflow heat exchanger. For both flow arrangements, the mean temperature difference is equal to the logarithmic mean temperature difference: Δtm ¼ ΔtLM ¼

Δt1  Δt2 lnðΔt1 =Δt2 Þ

(2.73)

in which Δt1 is the temperature difference at one end of the exchanger and Δt2 is that at the other end. A special case is Δt1 ¼ Δt2. It will happen in a counterflow heat exchanger if the thermal capacity rates of the two streams are the same, that is, C_ h ¼ C_ c . In such a case, Eq. (2.73) cannot be used. An expression was proposed by Chen (1987):  1=0:3275 (2.74) ΔtLM  Δt10:3275 + Δt20:3275 =2 which is a good approximation even for large values of Δt1/Δt2. A practical method is to use Eq. (2.75) for ΔtLM: 8 < Δt1  Δt2 , jΔt1  Δt2 j > 106 ΔtLM ¼ ln ðΔt1 =Δt2 Þ (2.75) : 6 ðΔt1 + Δt2 Þ=2, jΔt1  Δt2 j  10 Another special case is that the temperature of one fluid remains unchanged in the exchanger, t(z) ¼ t0 ¼ t00 . For example, in the two-phase heat transfer region of a condenser or an evaporator, the fluid temperature maintains at its saturation temperature. In such a case, Eq. (2.73) is valid not only for counterflow and parallel-flow but also for crossflow. This conclusion can be extended to other types of heat exchangers. A correction factor for the logarithmic mean temperature difference of counterflow can be introduced, which is defined by

36

Design and operation of heat exchangers and their networks

tc²

th¢

th²

.

Ch

th¢

Dt

tc²

.

tc¢

Cc

q

th² tc¢ z

(A)

0

L

.

tc¢

Cc

th¢

th²

.

Ch

th¢

tc²

Dt q

th² tc²

tc¢

(B)

z 0

L

Fig. 2.7 Counterflow (A) and parallel-flow (B) arrangements.

F¼

Δtm ΔtLM, cf

Z 1 A ðth  tc ÞdA A 0     1 (2.76) ¼  0 th  tc00  th00  tc0 = ln th0  tc00 = th00  tc0

Then, Eq. (2.68) can be expressed as 0  th  tc00  th00  tc0  Q ¼ FkAΔtLM, cf ¼ FkA  0 ln th  tc00 = th00  tc0

(2.77)

37

Basic thermal design theory for heat exchangers

The correction factor for the logarithmic mean temperature difference for various flow arrangements can be calculated with an approximate equation proposed by Roetzel and Spang (Spang and Roetzel, 1995; Roetzel and Spang, 2010, 2013, Table 1). In general, the steady-state fluid temperature distributions in a heat exchanger can be obtained by solving the “micro” energy equations: dQ ¼ m_ h dhh ¼ m_ c dhc dQ ¼ kðth  tc ÞdA th ¼ fh ðhh ph Þ, tc ¼ fc ðhc pc Þ

(2.78) (2.79) (2.80)

in which h is specific enthalpy and fh and fc are the equation of state for hot and cold fluids, respectively. Using the “micro” energy equations, we can solve the steady-state problems with variable overall heat transfer coefficient, variable heat transfer area along the flow direction, and even evaporation and condensation analytically or numerically. Example 2.4 Sizing a counterflow heat exchanger Consider a counterflow shell-and-tube heat exchanger with one shell pass and one tube pass. Hot water enters the tube at 100°C and leaves at 80°C. In the shell side, cold water is heated from 20°C to 70°C. The heat duty is expected to be 350 kW. There are totally 53 tubes with the inner diameter of 16 mm and wall thickness of 1 mm. The thermal conductivity of the tube wall is 40 W/mK. The shell-side heat transfer coefficient can be established as 1500 W/m2K. Calculate the tube length of the heat exchanger. Solution We use Eq. (2.68) to size the counterflow heat exchanger:   ln th0  tc00 = th00  tc0 Q  ðkAÞi ¼ ¼Q  0 ΔtLM, c th  tc00  th00  tc0 ln ½ð100  70Þ=ð80  20Þ ¼ 8087W=K ð100  70Þ  ð80  20Þ To get the heat exchanger area of tube inside Ai, we shall evaluate the overall heat transfer coefficient based on the area of tube inside. The properties of water to be used are calculated from the following equations. Specific isobaric thermal capacity of saturated liquid water (Popiel and Wojtkowiak, 1998) is as follows: ¼ 350  103 

cp, s ¼ 4:2174356  5:6181625  103 t + 1:2992528  103 t 1:5 1:1535353  104 t 2 6 2:5

10 t

+ 4:14964

ðkJ=kgKÞ ð0°C  t  150°CÞ

(2.81)

Continued

38

Design and operation of heat exchangers and their networks

Thermal conductivity of liquid water at 1 bar (Huber et al., 2012) is as follows: λ¼

4 X

ai

i¼1

½ðt + 273:15Þ=300bi

ðW=mKÞ ð0°C  t  110°CÞ

(2.82)

with ai ¼ 1.663, 1.7781, 1.1567, and 0.432115 and bi ¼ 1.15, 3.4, 6.0, and 7.6, respectively. Dynamic viscosity of liquid water at 1 bar (Pa´tek et al., 2009) is as follows: μ ¼ 106 

4 X

ai

i¼1

½ðt + 273:15Þ=300bi

ðsPaÞ ð20°C  t  110°CÞ (2.83)

with ai ¼ 280.68, 511.45, 61.131, and 0.45903 and bi ¼ 1.9, 7.7, 19.6, and 40, respectively. With these equations, we have the necessary fluid properties at the mean temperature of hot water, th,m ¼ (100 + 80)/2 ¼ 90°C as follows: cp,h ¼ 4:206 kJ=kgK, λh ¼ 0:6728 W=mK, μh ¼ 3:142  104 sPa, Prh ¼ cp,h μh =λh ¼ 1:964: According to the given heat duty, we can calculate the mass velocity inside the tubes by   m_ h Q= cp, h th0  th00 350=½4:206  ð100  80Þ ¼ Gh ¼ ¼ Ntube πdi2 =4 Ac,h 53  π  0:0162 =4 2 ¼ 390:5 kg=m s The tubeside Reynolds number is Reh ¼ Gh di =μh ¼ 390:5  0:016=3:142  104 ¼ 19;886 Since it is in the turbulent flow region, so we will use the Gnielinski correlation, Eq. (2.33), to calculate the Nusselt number and assume at first the correction term is equal to 1: f =8 ¼ ½1:82 lgð Reh Þ  1:642 =8 ¼ ½1:82 lgð19; 886Þ  1:642 =8 ¼ 0:003269 "   # ðf =8Þð Reh  1000Þ Prh d 2=3  1+ i Nuh ¼ ð Prh = Prw Þ0:11 pffiffiffiffiffiffiffi 2=3 L 1 + 12:7 f =8 Prh  1 "   # 0:003269  ð19, 886  1000Þ  1:964 0:016 2=3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   1+ ð1:964=Prw Þ0:11 ¼ L 1 + 12:7 0:003269  1:9642=3  1 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} ¼1 ¼ 85:84

Basic thermal design theory for heat exchangers

The heat transfer coefficient inside the tube can be established by αh ¼ Nuh λh =di ¼ 85:84  0:6728=0:016 ¼ 3610 W=m2 K The outside diameter of the tube do ¼ di + 2δw ¼ 0.016 + 2  0.001 ¼ 0.018 m. The conductive thermal resistance of the tube wall per unit inner area is calculated from Eq. (2.63): Rw,i ¼

di lnðdo =di Þ 0:016  ln ð0:018=0:016Þ ¼ 2:356  105 m2 K=W ¼ 2λw 2  40

Using Eq. (2.61), we have the expression of the overall thermal resistance as 1 1 Rw 1 1 Rw 1 + + ¼ + + ¼ ðkAÞi αh Ai Ai αc Ao αh Ntube πdi L Ntube πdi L αc Ntube πdo L

which yields

  ðkAÞi 1 di + Rw,i + Ntube πdi αh  αc do  8087 1 0:016 + 2:356  105 + ¼ ¼ 2:711m 53  π  0:016 3610 1500  0:018

L¼

and ðkAÞi 8087 ¼ 1120 W=m2 K ¼ Ntube πdi L 53  π  0:016  2:711 According to the energy equation, ki ¼

q ¼ αh ðth  th, w Þ ¼ ki ðth  tc Þ

we can express the mean wall temperature at the tube inside as th,w ¼ th  ki ðth  tc Þ=αh therefore, we have tc,m ¼ ð20 + 70Þ=2 ¼ 45°C th, w,m ¼ th,m  ki ðth, m  tc,m Þ=αh ¼ 90  1120  ð90  45Þ=3610 ¼ 76:04°C With the newly calculated tube length L and wall temperature at the tube inside th,w,m for the calculation of Prw, we can recalculate the Gnielinski correlation and repeat the earlier steps. After several iterations, the calculation converges to L ¼ 2.701 m. For a conservative design, we would like to enlarge the area by about 30% and set the tube length to be L ¼ 3.5 m. The detailed calculation can be found in the MatLab code for Example 2.4 in the appendix.

39

40

Design and operation of heat exchangers and their networks

Example 2.5 Rating a parallel-flow heat exchanger Consider a shell-and-tube heat exchanger designed in Example 2.4. The cold water flows through the shell side and should be heated from 20°C to 70°C by the hot water entering the tube at 100°C. The demanded heat duty is expected to be 350 kW. The shell-side heat transfer coefficient can be established as 1500 W/m2K. The exchanger is arranged in the parallel flow. Determine the mass flow rate of the hot water so that the outlet temperature of the cold water can be maintained at 70°C. Solution 00 At first, we assume the outlet temperature of the hot water th(0) ¼ 80°C. Then, we have th,m ¼ (100 + 80)/2 ¼ 90°C. Similar to the calculation procedure used in Example 2.4, we have cp,h ¼ 4:206 kJ=kgK, λh ¼ 0:6728 W=mK, μh ¼ 3:142  104 sPa, Prh ¼ cp,h μh =λh ¼ 1:964:   m_ h Q= cp, h th0  th00 350=½4:206  ð100  80Þ ¼ ¼ Gh ¼ Ntube πdi2 =4 Ac,h 53  π  0:0162 =4 2 ¼ 390:5 kg=m s Reh ¼ Gh di =μh ¼ 390:5  0:016=3:142  104 ¼ 19;886 f =8 ¼ ½1:82 lgð Reh Þ  1:642 =8 ¼ ½1:82 lgð19 886Þ  1:642 =8 ¼ 0:003269 "   # ðf =8Þð Reh  1000ÞPrh d 2=3  1+ i Nuh ¼ ð Pr=Prw Þ0:11 pffiffiffiffiffiffiffi 2=3 L 1 + 12:7 f =8 Prh  1 0:003269  ð19, 886  1000Þ  1:964 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  1"+ 12:7 0:003269# 1:9642=3  1     0:016 2=3 1:964 0:11 ¼ 88:21  1+ 3:5 Prw |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}

¼

¼1

αh ¼ Nuh λh =di ¼ 88:21  0:6728=0:016 ¼ 3709 W=m2 K di lnðdo =di Þ 0:016  ln ð0:018=0:016Þ ¼ 2:356  105 m2 K=W ¼ 2λw 2  40  1   1 di 1 1 0:016 5 + 2:356  10 + ki ¼ + Rw + ¼ αh 3709 1500  0:018 αc do 2 ¼ 1129 W=m K Rw, i ¼

With tc,m ¼ (20 + 70)/2 ¼ 45°C, we get

Basic thermal design theory for heat exchangers

th, w,m ¼ th,m  ki ðth, m  tc,m Þ=αh ¼ 90  1120  ð90  45Þ=3709 ¼ 76:30°C This temperature is used for the calculation of Prw appearing in the Gnielinski correlation. The logarithmic mean temperature difference for the parallel-flow arrangement is 0  t  t 0  t 00  t 00 ð100  20Þ  ð80  70Þ ¼ 33:66 K ΔtLM, p ¼ h 0 c 0 h00 c 00 ¼ ln ½ð100  20Þ=ð80  70Þ ln th  tc = th  tc Now, we shall use Eq. (2.68) to check the heat duty: Qð0Þ ¼ ki Ntube πdi LΔtLM, p ¼ 1129  103  53  π  0:016  3:5  33:66 ¼ 354:4 kW Because the calculated heat duty does not agree with its demanded value, we shall change the flow rate of the hot water by correcting the outlet temperature of the hot water:  Q ð nÞ 0 354:4 ðnÞ ð100  80Þ ¼ 79:75°C th  t 00 h ¼ 100  350 Q With the newly calculated hot water outlet temperature and wall temperature at the tube inside th,w,m for the calculation of Prw, we can repeat the earlier calculation. After several iterations, we finally obtain the outlet temperature and mass flow rate of the hot water: ðn + 1Þ

t 00 h

¼ th0 

th00 ¼ 79:87°C Q 350 ¼ ¼ 4:134 kg=s m_ h ¼  0 4:206  ð100  79:87Þ cp,h th  th00 It is interesting to compare the parallel-flow arrangement with the counterflow arrangement. If we connect the water streams to the heat exchanger as a counterflow heat exchanger, that is, 0  th  tc00  th00  tc0 ð100  70Þ  ð80  20Þ  00 ¼ ΔtLM, cf ¼  0 ¼ 43:28 K 00 0 ln ½ð100  70Þ=ð80  20Þ ln th  tc = th  tc 00

After several iterations, the calculation results in th ¼ 68.83°C and m_ h ¼ 2:673 kg/s, which are much lower than those in the parallel-flow heat exchanger. The detailed calculation can be found in the MatLab code for Example 2.5 in the appendix.

41

42

Design and operation of heat exchangers and their networks

2.1.3 Consideration of temperature-dependent heat transfer coefficients For highly temperature-dependent heat transfer coefficients, as may occur with viscous liquids and with radiation, the common calculation method with arithmetic mean of inlet and outlet temperatures as reference temperatures may lead to undesirable errors in design. For such cases, the two-point method (Roetzel, 1969; Roetzel and Luo, 2011; Roetzel and Spang, 2010, 2013) can be applied in which two special pairs of reference temperatures are used for the calculation of two overall heat transfer coefficients and their effective mean value. The method is valid for counterflow and parallel flow but can also be adapted to other flow arrangements. In the two-point method, the heat transfer coefficients are calculated at two reference points using two special pairs of reference temperatures for the fluids. At the reference points i ¼ 1 and i ¼ 2, the reference temperatures th,i and tc,i are determined for counterflow as follows:  (2.84) th, i ¼ th00 + ψ i th0  th00  (2.85) tc, i ¼ tc0 + ψ i tc00  tc0 where θ mi  1 (2.86) θ1 t0  t00 θ ¼ h00 c0 (2.87) th  tc 1 1 pffiffiffi 3 (2.88) m1, 2 ¼  2 6 For a balanced counterflow heat exchanger (C_ h ¼ C_ c ), we have ψ i ¼ mi. 0 00 For parallel-flow heat exchanger, the same equations can be used if tc and tc are exchanged. The heat transfer coefficients and overall heat transfer coefficients at the two reference points, that is, (kA)1 and (kA)2, are calculated in the usual way. The effective mean value (kA)m is calculated from   1 1 1 1 ¼ + (2.89) ðkAÞm 2 ðkAÞ1 ðkAÞ2 ψi ¼

Eqs. (2.84)–(2.86) are valid for constant heat capacities of the fluids. Variable heat capacities can be approximately replaced by constant mean values

Basic thermal design theory for heat exchangers

43

between the inlet and outlet temperatures. Strong variations can be taken into account by a refinement of the method (Roetzel, 1988). Example 2.6 Sizing the counterflow heat exchanger in Example 2.4 considering variable tubeside heat transfer coefficient The problem is the same as Example 2.4, however, with the consideration of variable tubeside heat transfer coefficient depending on the fluid temperature. Solution In Example 2.4, we have got do ¼ 0.018 m, (kA)i ¼ 8087 W/K, Gh ¼ 390.5 kg/m2 s, and Rw,i ¼ 2.356  105 K/W. The shell-side heat transfer coefficient is given as 1500 W/m2K. We will use Eq. (2.89) to calculate the mean overall heat transfer coefficient ki. According to Eqs. (2.86)–(2.88), we have   θ ¼ th0  tc00 = th00  tc0 ¼ ð100  70Þ=ð80  20Þ ¼ 0:5 pffiffiffi pffiffiffi m1 ¼ 1=2 + 3=6 ¼ 0:7887, m2 ¼ 1=2  3=6 ¼ 0:2113 θm1  1 0:50:7887  1 θm2  1 0:50:2113  1 ¼ ¼ 0:8422, ψ 2 ¼ ¼ θ1 0:5  1 θ1 0:5  1 ¼ 0:2725

ψ1 ¼

The two reference temperatures of hot water are obtained with Eq. (2.84):  th,1 ¼ th00 + ψ 1 th0  th00 ¼ 80 + 0:8422  ð100  80Þ ¼ 96:84°C  th,2 ¼ th00 + ψ 2 th0  th00 ¼ 80 + 0:2725  ð100  80Þ ¼ 85:45°C The two reference temperatures of cold water are obtained with Eq. (2.85):  tc,1 ¼ tc0 + ψ 1 tc00  tc0 ¼ 20 + 0:8422  ð70  20Þ ¼ 62:11°C  tc,2 ¼ tc0 + ψ 2 tc00  tc0 ¼ 20 + 0:2725  ð70  20Þ ¼ 33:63°C We assume initially the wall temperatures are equal to the reference temperatures of the hot water, th,w,1 ¼ th,1 and th,w,2 ¼ th,2, and take the tube length calculated in Example 2.4 as the initial tube length, L ¼ 2.701 m. With the same calculation steps described in Example 2.4, we can get the thermal properties and heat transfer coefficients as well as the overall heat transfer coefficients at these two reference temperatures as follows: cp,h, 1 ¼ 4:213 kJ=kgK, λh,1 ¼ 0:6760 W=mK, μh,1 ¼ 2:912  104 sPa, Prh, 1 ¼ 1:815 Continued

44

Design and operation of heat exchangers and their networks

cp,h, 2 ¼ 4:201 kJ=kgK,λh, 2 ¼ 0:6703 W=mK,μh,2 ¼ 3:313  104 sPa, Prh, 2 ¼ 2:076 At the first reference temperature, Reh,1 ¼ Gh di =μh, 1 ¼ 390:5  0:016=2:912  104 ¼ 21456 fh, 1 =8 ¼ ½1:82 lgð Reh, 1 Þ  1:642 =8 ¼ ½1:82 lgð21456Þ  1:642 =8 ¼ 0:003207 "   # ðfh, 1 =8Þð Reh,1  1000ÞPrh,1 di 2=3

 Nuh,1 ¼ 1 + ð Prh,1 =Prw,1 Þ0:11 pffiffiffiffiffiffiffiffiffiffiffi 2=3 L 1 + 12:7 fh, 1 =8 Prh,1  1 "   # 0:003207  ð21456  1000Þ  1:815 0:016 2=3  1+ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ¼ ð1:815=1:815Þ0:11 2:701 1 + 12:7 0:003207  1:8152=3  1 ¼ 91:01 αh,1 ¼ Nuh,1 λh, 1 =di ¼ 91:01  0:6760=0:016 ¼ 3845 W=m2 K   1 di 1 + Rw,i, 1 + ki, 1 ¼ α αc do  h,1 1 1 0:016 + 2:356  105 + ¼ ¼ 1141 W=m2 K 3845 1500  0:018 th, w,1 ¼ th,1  ki,1 ðth, 1  tc,1 Þ=αh,1 ¼ 96:84  1141  ð96:84  62:11Þ=3845 ¼ 86:54°C Similarly, at the second reference temperature, we can obtain Reh,2 ¼ 18,859, fh,2 ¼ 0.003314, Nuh,2 ¼ 87, αh,2 ¼ 3645 W/m2K, ki,2 ¼ 1123 W/m2K, and th,w,2 ¼ 69.48°C. The mean overall heat transfer coefficient is calculated by Eq. (2.89): ki,m ¼

2 2 ¼ 1132 W=m2 K ¼ 1=ki,1 + 1=ki,2 1=1141 + 1=1123

Since (kA)i ¼ ki,m πdiL, the tube length can be determined as ðkAÞi 8087 ¼ 2:681 m ¼ ki,m Ntube πdi 1132  53  π  0:016 With the newly calculated tube length L and wall temperatures at the tube inside th,w,1 and th,w,2 for the calculation of Prh,w,1 and Prh,w,2, we can recalculate the Gnielinski correlation and repeat the earlier steps. After several iterations, the calculation converges to L ¼ 2.697 m. The result shows that in this case, the common calculation method is sufficiently accurate. The detailed calculation can be found in the MatLab code for Example 2.6 in the appendix. L¼

Basic thermal design theory for heat exchangers

45

2.1.4 General energy equations for steady-state and dynamic analysis of heat exchangers For general flow processes, the energy equation can be expressed as (Whitaker, 1977)     ∂T ∂p ρcp + V rT ¼ rðλrT Þ + βT + V rp + μΦ + s (2.90) ∂τ ∂τ in which T is the thermodynamic temperature, s the volumetric heat source, Ф the dissipation function, and β the thermal expansion coefficient:   1 ∂v (2.91) β

v ∂T p Since the flow in a heat exchanger is usually a low-velocity flow (Mach number Ma < 0.3), the pressure variation would not be very large; therefore, even for gases, the pressure term in Eq. (2.90) can be omitted. Furthermore, if the Peclet number is not very high, the viscous dissipation in low-velocity flow can be neglected. Although the real velocity distribution in a heat exchanger is three-dimensional and could be very complicated, there is a main flow direction for each fluid. The main assumption to be used is that the fluid is completely mixed in the lateral direction but the axial mixing is negligible. Therefore, the flow velocity and temperature over the section perpendicular to the main flow direction are uniform. Such a flow pattern is called plug flow. By integrating Eq. (2.90) over the flow passage section perpendicular to the main flow direction and using the average values of the temperature and fluid properties over the cross-sectional area, the energy equation can be simplified to   Z ðð ∂t ∂t ∂ ∂t Ac ρcp + C_ ¼ Ac λ sdAc (2.92) + qdP + ∂τ ∂x ∂x ∂x P Ac Ð The term PqdP is the convective heat transfer from the solid wall of the exchanger to the fluid per unit length along the flow direction: Z Z qdP ¼ αðtw  t ÞdP (2.93) P

P

in which P is the wetted perimeter of the flow channel at x. If a flow channel consists of Nw walls and the wall temperature and convective heat transfer coefficient of each wall are uniformly distributed along its wetted perimeter, then the convective heat transfer term can be expressed as Z Nw X  qdP ¼ αj Pj tw, j  t (2.94) P

j¼1

46

Design and operation of heat exchangers and their networks

This expression can be applied to multichannel heat exchangers. For plate-fin heat exchangers, however, the temperature along the fin height is not constant, and therefore, Eq. (2.93) should be used. For the solid wall of the heat exchanger, we will assume that the heat conduction resistance in the direction perpendicular to its heat transfer surface is negligible, that is, the wall temperature across the wall thickness is uniform. Then, the partial differential equation for heat conduction in the wall can be written as     ∂tw ∂ ∂tw ∂ ∂tw ðδρc Þw ðδλÞw ðδλÞw ¼ + ∂y ∂τ ∂x ∂x ∂y Z Nf X A w, i α w, i ðtw  ti Þ + sw dz (2.95)  Axy δw i¼1 where Nf is the number of the fluid streams, δ is the wall thickness, and Aw,i/ Axy is the heat transfer area between the wall and the ith stream per unit area on the x-y plane. Eqs. (2.92), (2.95) are the fundamental equations of the distributed parameter model, which can well describe the steady-state or transient thermal performance of a heat exchanger if the flow in the exchanger is a plug flow. In practice, the heat conduction in the two fluids and solid wall is much smaller than the heat transfer between the two fluids through the wall, and therefore, the heat conduction terms in Eqs. (2.92), (2.95) can usually be neglected. For a two-stream heat exchanger without heat source in the fluids, they can be expressed as follows:  ∂th ∂th Ac ρcp h + C_ h ¼ ðαAÞh ðtw  th Þ, th jx¼0 ¼ th0 , th jτ¼0 ¼ th, 0 (2.96) ∂τ ∂x ∂tw (2.97) ¼ ðαAÞh ðth  tw Þ + ðαAÞc ðtc  tw Þ, tw jτ¼0 ¼ tw, 0 ðδρc Þw ∂τ  ∂tc ∂tc For parallel flow, Ac ρcp c + C_ c ¼ ðαAÞc ðtw  tc Þ, ∂τ ∂x tc jx¼0 ¼ tc0 , tc jτ¼0 ¼ tc, 0 For counterflow,

(2.98)

 ∂tc ∂tc Ac ρcp c  C_ c ¼ ðαAÞc ðtw  tc Þ, ∂τ ∂x tc jx¼L ¼ tc0 , tc jτ¼0 ¼ tc, 0

(2.99)

Basic thermal design theory for heat exchangers

For crossflow,



Ac ρcp

47

∂tc ∂tc + C_ c ¼ ðαAÞc ðtw  tc Þ, tc jy¼0 ¼ tc0 , c ∂τ ∂y (2.100) tc jτ¼0 ¼ tc, 0

If the heat exchanger operates at a steady state, Eqs. (2.96)–(2.100) further reduce to dth C_ h ¼ kAðtc  th Þ, th jx¼0 ¼ th0 dx dtc For parallel flow, C_ c ¼ kAðth  tc Þ, tc jx¼0 ¼ tc0 dx dtc For counterflow, C_ c ¼ kAðth  tc Þ, tc jx¼L ¼ tc0 dx dtc For crossflow , C_ c ¼ kAðth  tc Þ, tc jy¼0 ¼ tc0 dy

(2.101) (2.102) (2.103) (2.104)

In the dynamic analysis of heat exchangers, Eqs. (2.92), (2.95) have often been simplified by the lumped parameter model. In the lumped parameter model, it is assumed that each fluid in the whole heat exchanger has the same uniform temperature. The temperature is only a function of time. The same is for the solid wall. By integrating Eqs. (2.92), (2.95) over the whole volumes of the fluid and the solid wall, respectively, and expressing the temperatures and parameters with their volumetric average values, we obtain the energy equations for the hot and cold fluids and solid wall as Ci

Nw dti _  0 00 X ðαAÞij Δtm, ij + ðsV Þi ði ¼ 1, 2, …, Nf Þ ¼ C i ti  ti  dτ j¼1

C w, j

Nf dtw, j X ðαAÞij Δtm, ij + ðsV Þw, j ðj ¼ 1, 2, …, Nw Þ ¼ dτ i¼1

(2.105)

(2.106)

in which Nf is the number of fluid streams and Nw the number of walls. The mean temperature difference between the ith fluid and jth wall is calculated by integration over the heat exchanger area: Z  1 Δtm, ij ¼ ti  tw, j dA (2.107) Aij Aij Different definitions of the volumetric mean temperatures and mean temperature difference will yield different lumped parameter models.

48

Design and operation of heat exchangers and their networks

2.1.5 Axial dispersion models for design and rating of heat exchangers In heat exchangers having complex structure, there might exist severe maldistribution caused by bypassing, dead zones, recirculatings, and other nonuniformities of fluid flow. In such cases, the axial dispersion model can be used to take the influence of the nonuniform velocity distribution into account. The axial dispersion model was first proposed by Taylor (1954) for turbulent mass transfer in tubes. It assumes that the fluid flow is fully mixed in the lateral direction perpendicular to the flow direction and is also mixed to some extent in the flow direction. Mecklenburgh and Hartland (1975) introduced the axial dispersion term AcD(∂ t/∂ x) into the energy equation of the fluid and suggested that the axial dispersion model can be used for the thermal calculation of heat exchangers. The axial dispersion coefficient D has the same dimension as the thermal conductivity λ, but it is not a fluid property. The value of the axial dispersion coefficient depends on the flow pattern in the heat exchanger and should be determined experimentally. Diaz and Aguayo (1987) numerically investigated the effect of the axial dispersion on the steady-state thermal performance of heat exchangers. They found that the fluid flow in a heat exchanger can be considered as a plug flow if the axial dispersive Peclet number Pe ¼ uL/D > 100. If Pe < 20, the axial dispersion should be taken into account. Roetzel and his coworkers (Spang, 1991; Xuan, 1991; Lee, 1994; Luo, 1998; Balzereit, 1999; Roetzel, 1996) carried out a series of theoretical and experimental investigations on the axial dispersion in heat exchangers. The original parabolic dispersion model is applied in which the propagation velocity of thermal disturbances is assumed to be infinitely high, expressed by a zero dispersive Mach number, Ma ¼ 0. Later, the hyperbolic dispersion model has been investigated and further developed in which finite propagation velocities are considered (Luo and Roetzel, 1995; Roetzel and Das, 1995; Roetzel et al., 1998; Roetzel and Na Ranong, 1999; Sahoo and Roetzel, 2002; Das and Roetzel, 2004), which are more realistic for maldistribution effects. The type and degree of deviations from the plug flow are expressed with the dispersive Peclet number 0  Pe  ∞ and the dispersive Mach number 0  Ma  ∞. Their research shows that the axial dispersion model is suitable for the simulation of the complicated flow and heat transfer in the heat exchangers, especially for the dynamic simulation.

Basic thermal design theory for heat exchangers

49

For practical design and rating purposes, the model with two variable parameters, Pe and Ma, is not well suited. For that reason, the unity Mach number dispersion model has been developed in which the fixed mean value Ma ¼ 1 is applied together with the dispersive Peclet number (Roetzel, 2010; Roetzel et al., 2011; Na Ranong and Roetzel, 2012; Roetzel and Na Ranong, 2014, 2015, 2018a). The main advantage of this model over the parabolic model is that it leads merely to simple corrections of the mean temperature difference or the heat transfer coefficients, while the known steady-state design and rating methods can further be applied by using the true mean temperature difference for dispersive flow Δtm,d together with the true overall heat transfer coefficient in Eq. (2.68) as Q ¼ kAΔtm, d

(2.108)

or the hypothetic mean temperature difference Δtm (the mean temperature difference in an equivalent nondispersive plug-flow heat exchanger) and the apparent overall heat transfer coefficient k∗ as Q ¼ k∗ AΔtm

(2.109)

The mean temperature difference for dispersive flow is Δtm, d ¼ Δtm 

th0  th00 tc00  tc0  Peh Pec

(2.110)

with Peh and Pec as the dispersive Peclet numbers of the hot and cold fluid stream, respectively. The substitution of Eq. (2.111) Δtm  Δtm, d ðth, m  tc, m Þ  ðth, m, d  tc, m, d Þ ¼ th, m  tc, m Δtm

(2.111)

into Eq. (2.110) yields

0  th  th00 =Peh + tc00  tc0 =Pec ðth, m  tc, m Þ  ðth, m, d  tc, m, d Þ ¼ th, m  tc, m Δtm

(2.112)

From the theory of the dispersion model, it follows that for constant values of k or k∗, the earlier ratio of mean temperature differences is equal to the related local temperature differences. Thus, the indices “m” on the left-hand side of Eq. (2.112) can be omitted. Rearranging Eq. (2.112) yields 0  th  th00 =Peh + tc00  tc0 =Pec Δtd th, d  tc, d ¼ ¼1 (2.113) th  tc th  tc Δtm

50

Design and operation of heat exchangers and their networks

For rating problems, Eq. (2.109) can be used, in which the apparent overall heat transfer coefficient can be calculated from the apparent heat transfer coefficient αd: 1 1 1 1 1 1 ¼ + , ¼ + αd, h Ah αh Ah C_ h Peh αd, c Ac αc Ac C_ c Pec

(2.114)

and can be expressed as 1 1 1 1 + ¼ + k∗ A kA C_ h Peh C_ c Pec

(2.115)

Example 2.7 Sizing a counterflow shell-and-tube heat exchanger with baffles Resizing the counterflow shell-and-tube heat exchanger of Example 2.4. The tubeside flow is assumed to be uniform; therefore, Peh ¼ ∞. The shell-side deviations from plug flow due to the baffles should be considered, with the baffle space ΔL  0.5 m. Calculate the tube length of the heat exchanger. Solution In this example, the deviation from shell-side plug flow can be expressed by a dispersive Peclet number Pec of the cold fluid stream, which depends on the construction of the baffled tube bundle. For the tube bundle with m baffles, the shell-side flow can be modeled with a cascade of n axially mixed zones (n ¼ m + 1), which yields the dispersive Peclet number (Roetzel et al., 2011): Pec ¼ 2n ¼ 2ðm + 1Þ The calculations presented in Example 2.4 result in the required tube length for plug flow L ¼ 2.701 m. Using this value as the assumed tube length, we have n  L=ΔL ¼ 2:701=0:5 ¼ 5:402 Therefore, we take n ¼ 6, which yields Pec ¼ 2n ¼ 12. The mean temperature difference for dispersive flow Δtm,d is calculated from Eq. (2.110): 0  th  tc00  th00  tc0 t 0  t 00 t 00  t 0  00  h h  c c Δtm,d ¼  0 Peh Pec ln th  tc00 = th  tc0 ¼

ð100  70Þ  ð80  20Þ 100  80 70  20   ¼ 39:11K ln ½ð100  70Þ=ð80  20Þ ∞ 12

Basic thermal design theory for heat exchangers

51

We use Eq. (2.108) to size the counterflow heat exchanger: 3

ðkAÞi ¼ ΔtQm, d ¼ 35010 39:11 ¼ 8948 W=K To get the heat exchanger area of tube inside Ai, we shall evaluate the overall heat transfer coefficient based on the area of tube inside. Using the same calculation procedure for Example 2.4, we have th,m ¼ ð100 + 80Þ=2 ¼ 90°C cp,h ¼ 4:206 kJ=kgK,λh ¼ 0:6728 W=mK, μh ¼ 3:142  104 sPa, Prh ¼ cp,h μh =λh ¼ 1:964:   m_ h Q= cp,h th0  th00 350=½4:206  ð100  80Þ ¼ ¼ Gh ¼ Ntube πdi2 =4 Ac,h 53  π  0:0162 =4 2 ¼ 390:5 kg=m s Reh ¼ Gh di =μh ¼ 390:5  0:016=3:142  104 ¼ 19; 886 f =8 ¼ ½1:82 lgð Reh Þ  1:642 =8 ¼ ½1:82 lgð19 886Þ  1:642 =8 ¼ 0:003269 "   # ðf =8Þð Reh  1000ÞPrh d 2=3  1+ i Nuh ¼ ð Pr= Prw Þ0:11 pffiffiffiffiffiffiffi 2=3 L 1 + 12:7 f =8 Prh  1 "   # 0:003269  ð19, 886  1000Þ  1:964 0:016 2=3  1+ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   ð1:964=Prw Þ0:11 ¼ 2:701 |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl} 1 + 12:7 0:003269  1:9642=3  1 ¼1

¼ 88:65 αh ¼ Nuh λh =di ¼ 88:65  0:6728=0:016 ¼ 3728 W=m2 K do ¼ di + 2δw ¼ 0:016 + 2  0:001 ¼ 0:018 m: di lnðdo =di Þ 0:016  ln ð0:018=0:016Þ ¼ 2:356  105 m2 K=W ¼ 2λw 2  40   ðkAÞi 1 di + Rw + L¼ Ntube πdi αh  α c do  8948 1 0:016 5 + 2:356  10 + ¼ 2:971m ¼ 53  π  0:016 3728 1500  0:018

R w, i ¼

and ðkAÞi 8948 ¼ 1131 W=m2 K ¼ Ntube πdi L 53  π  0:016  2:971 According to the energy equation q ¼ αh (th  th,w) ¼ ki Δtd, we have ki ¼

52

Design and operation of heat exchangers and their networks

th,w ¼ th  ki Δtd =αh where Δtd is the true temperature difference for dispersive flow at the reference point calculated from Eq. (2.113): 0  t  t 00 =Peh + tc00  tc0 =Pec Δtd h  00h  0  C¼ ¼ 1   0 th  tc th  tc00  th  tc0 = ln th  tc00 = th00  tc0 ð100  80Þ=∞ + ð70  20Þ=12 ¼ 0:9037 ½ð100  70Þ  ð80  20Þ= ln ½ð100  70Þ=ð80  20Þ The mean wall temperature is determined by ¼1 

th,w, m ¼ th, m  C ðth,m  tc,m Þki =αh   100 + 80 70 + 20  ¼ 90  0:9037   1131=3728 ¼ 77:66°C 2 2 With the newly calculated tube length L and wall temperature at the tube inside th,w,m for the calculation of Prw, we can recalculate the Gnielinski correlation and repeat the earlier steps. After several iterations, the calculation converges to L ¼ 2.99 m. The detailed calculation procedure can be found in the MatLab code for Example 2.7 in the appendix.

2.1.6 Application of the dispersion model to axial wall heat conduction The dispersion model can also be applied to the approximate consideration of axial wall heat conduction that has a similar negative effect on efficiency as fluid dispersion. One effective dispersive Peclet number Peeff ¼ Peh ¼ Pec ¼ Pe for both fluids is defined and has to be determined from the construction and heat transfer data at the operation point, with which the correct outlet temperatures can be calculated. The axial wall heat conduction in the separating wall and in the outer wall is taken into account under the assumption of adiabatic outside surface of the heat exchanger. Simple correlations are developed of the effective Peclet number for counterflow, parallel flow, and mixed-mixed crossflow (Roetzel and Na Ranong, 2018b; Roetzel and Spang, 2019). The heat conduction in the walls is expressed with wall Peclet numbers. For the separating wall, it is defined as Pew, h ¼ C_ h

Lh Lc , Pew, c ¼ C_ c λw Ac, w, h λw Ac, w, c

(2.116)

Basic thermal design theory for heat exchangers

53

where the subscript “w” indicates the separating wall, Lh and Lc are the heat exchanger length along the flow directions of hot and cold fluids, and Ac,w,h and Ac,w,c are the heat conduction cross-sectional areas of the separating walls for heat conduction along their flow directions. For counterflow and parallel flow, Lh/(λwAc,w,h) ¼ Lc/(λwAc,w,c). For crossflow, they should be determined individually. For the outer walls denoted with “wo,” we have similar expression: Pewo, h ¼ C_ h

Lh Lc , Pewo, c ¼ C_ c λwo, h Ac, wo, h λwo, c Ac, wo, c

(2.117)

If the hot fluid (or cold fluid) has no outer wall effect, then the corresponding heat conduction cross-sectional area Ac,wo,h (or Ac,wo,c) is zero, and we have Pewo,h (or Pewo,c) ¼ ∞. The number of transfer units N formed with heat transfer coefficient is defined as αh Ah α c Ac ,Nc ¼ (2.118) _ Ch C_ c for heat exchange between the hot and cold fluids. For heat exchange between the fluids and outer walls, N is denoted as Nwo and defined as Nh ¼

αwo, h Awo, h αwo, c Awo, c , Nwo, c ¼ (2.119) C_ h C_ c where αwo,h and αwo,c are the heat transfer coefficients of the hot and cold fluids at the fluid side surfaces of the outer walls Awo,h and Awo,c. Nwo, h ¼

2.1.6.1 Week conduction effect First, the limiting case of week conduction effect for high value of Pew, denoted with index “∞” 1 1 2a 1 1 1  +  ¼ 2+ + 2+ Pe∞ ψ h ψ h ψ c ψ c Pewo, h 1 + C_ h =C_ c Pewo, c 1 + C_ c =C_ h (2.120) where   qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffi αc Ac _ _ , ψ h ¼ Pew, h 1 + C h =C c 1 + α h Ah   qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffi αh Ah _ _ (2.121) ψ c ¼ Pew, c 1 + C c =C h 1 + α c Ac and a ¼ 1 for counterflow, a ¼ 1 for parallel flow, and a ¼ 0 for mixedmixed crossflow.

54

Design and operation of heat exchangers and their networks

2.1.6.2 Strong conduction effect The limiting case for infinite large thermal conductivity, that is, Pew ¼ 0, is denoted with index “0.” For this case, the dimensionless temperature change can be determined with 1 εh, 0

¼

Nwo, h 1 +  N Nh ðNh + Nwo, h Þ 1  e ð h + Nwo, h Þ   C_ h Nwo, c 1 + + C_ c Nc ðNc + Nwo, c Þ 1  eðNc + Nwo, c Þ C_ h εc, 0 ¼ εh, 0 C_ c

(2.122) (2.123)

Eq. (2.122) is valid for all one-pass flow arrangements. With Eqs. (2.122), (2.123), we can calculate the outlet temperatures of the hot and cold fluids and evaluate the mean temperature difference Δtm appearing in the expression of the limiting Peclet number Pe0: " # 1 1 Δtm 1 C_ h    (2.124) ¼ Pe0 1 + C_ h =C_ c εh, 0 th0  tc0 Nh C_ c Nc For parallel-flow and counterflow arrangements, Δtm is the logarithmic mean temperature difference. Eq. (2.124) is also exactly valid for pure crossflow and is approximate for other one-pass flow arrangements, if Δtm is replaced by the correct mean temperature difference for the flow arrangement under consideration. 2.1.6.3 Asymptotic equation for general cases For general cases 0  Pew  ∞, Roetzel and Na Ranong (2018b) proposed an asymptotic equation as follows:  1=0:87 Pe ¼ Pe0:87 + Pe0:87 (2.125) 0 ∞ Applying this Peclet number to both fluids according to the dispersion model gives the outlet fluid temperatures with consideration of axial wall heat conduction.

2.2 Pressure drop analysis In the thermal design of a heat exchanger, the total pressure drop (in some cases also the pressure distribution) analysis should be performed

Basic thermal design theory for heat exchangers

55

simultaneously, because it is directly related to the pumping power. Take a shell-and-tube heat exchanger as an example. We can decrease the inner diameter of tubes or the number of tubes to increase the fluid velocity, which will increase the heat transfer coefficient and reduce the size and weight of the heat exchanger. However, the frictional pressure drop will increase quadratically with the velocity, against which a much larger pump with much higher pumping power consumption might be required. The benefit of saving exchanger capital cost by increasing fluid velocities might be canceled by a more expensive pump or might be lost by increased operating costs in a short period. In some cases, the pressure drop will influence the heat transfer directly, especially in two-phase flow heat transfer. In such cases, the saturation temperature of a fluid decreases with the pressure decrease along the heat exchanger. The pressure drop analysis can help us judge a flow arrangement in a heat exchanger. For example, in an upward-flow evaporator, the boiling fluid flows upward with a pressure decrease in the flow direction, which results in a temperature decrease. Therefore, it is reasonable that the heating fluid also flows upward, so that the required mean temperature difference can be decreased slightly, which yields a little higher heat exchanger effectiveness. For the heat exchangers with parallel-flow passages, different flow and heat transfer conditions among the passages might introduce nonuniform flow rate distribution. A typical example is the flow rate distribution in a Z-type arrangement plate heat exchanger (Bassiouny and Martin, 1984). In the Z-type arrangement plate heat exchanger, the fluid velocity decreases in the intake conduit and increases in the exhaust conduit. According to the Bernoulli equation, it will make the pressure rise in the intake conduit and fall in the exhaust conduit. Such a pressure distribution might lead to a nonuniform flow in the channels. Whether this nonuniformity is significant should be estimated by the pressure drop analysis. The pressure drop of flows through heat exchangers is composed of three components: the frictional pressure drop, static pressure drop, and acceleration pressure drop; Δp ¼ Δpf + Δpg + Δpa

(2.126)

For flow through valves and pipeline fittings,

with ζ as the drag coefficient.

1 Δp ¼ ζ ρu2 2

(2.127)

56

Design and operation of heat exchangers and their networks

2.2.1 Frictional pressure drop For a pipe flow, the frictional pressure drop can be expressed as Δpf ¼ fD

L1 2 ρu dh 2

(2.128)

where fD is the Darcy friction factor. The frictional pressure drop can also be expressed by the shear stress near the wall with the Fanning friction factor defined by 1 τw ¼ f ρu2 2 Because of the force balance ΔpfAc ¼ τwPL, we have Δpf ¼ f

PL 1 2 ρu ¼ 2f ρu2 L=dh Ac 2

(2.129)

(2.130)

The Fanning friction factor should not be confused with the Darcy friction factor that is four times as large as the Fanning friction factor: fD ¼ 4f

(2.131)

2.2.1.1 Frictional pressure drop in circular tubes A set of equations are given in Table 2.1 for the calculation of Darcy friction factor for fully developed flow in a smooth circular tube in different ranges of Reynolds number. For laminar flow, the friction factor is independent of the surface roughness. However, in a fully developed turbulent flow, the friction factor depends solely on the roughness (Colebrook, 1939): 1 pffiffiffiffi ¼ 2 lgð3:7di =RÞ fD

(2.132)

In the transition zone, the friction factor not only depends on the surface roughness but also depends on the Reynolds number. As a general formula, the Colebrook-White equation (Colebrook, 1939)   1 2:51 R pffiffiffiffi ¼ 2 lg pffiffiffiffi + (2.133) fD Re fD 3:7di is recommended for Re > 4000. In the region of 2000 < Re < 4000, the value of the Darcy friction factor is subject to large uncertainties. In the absence of experimental data, a linear interpolation between the HagenPoiseuille equation and Colebrook-White equation can be used to calculate

Basic thermal design theory for heat exchangers

57

the Darcy friction factor. Eq. (2.133) is not explicit; therefore, an iteration with an initial value of fD ¼ 0.03 can be performed. Churchill (1977) suggested a single correlation of the Darcy friction factor for laminar, transitional, and turbulent flow: "  #1=12 64 12 1 fD ¼ + (2.134) Re ðA + BÞ3=2 where (

"

A ¼ 2 lg

#)16    7 0:9 R 13, 269 16 + , B¼ Re 3:7di Re

Churchill equation is very practical for the calculation of the Darcy friction factor for Re < 106. For Re > 15,800, we can use Eq. (2.134) to get an initial value for the calculation using the Colebrook-White equation (2.133) without further iteration, and the deviation is less than 0.2%. 2.2.1.2 Frictional pressure drop in laminar flow in rectangular ducts For fully developed laminar flow in a rectangular duct, the analytical solution of the Fanning friction factor can be expressed for the aspect ratio γ in the range of 0 < γ  1 as (See Shah and London, 1978, Eqs. (333) and (340)) f Re ¼

24

"

∞ X

192λ 1 ð2n + 1Þπ ð1 + γ Þ 1  5 5 tanh π n¼0 ð2n + 1Þ 2γ 2

#

(2.135)

Table 2.1 Darcy friction factor for fully developed flow in a smooth circular tube (Kast, 2010, 2013). Correlation

Hagen-Poiseuille equation Blasius equation Konakov equation Hermann equation Prandtl-von Karman equation Filonenko equation

Valid range

fD ¼ 64/Re

Re < 2320

fD ¼ 0.3164Re0.25 fD ¼ [1.8 lg(Re)  1.5]2 0.3 fD ¼ 0.0054 + 0.3964Re pffiffiffiffi 1 pffiffiffi ¼ 2 lgð Re fD =2:51Þ

3000 < Re < 105 104 < Re < 106 2  104 < Re < 2  106 Re > 106

p1ffiffiffi ¼ 1:819 lgð ReÞ  1:64

105 < Re < 5  107

fD fD

58

Design and operation of heat exchangers and their networks

For γ ¼ 0 (parallel plates), Eq. (2.135) approaches to f Re ¼ 24

(2.136)

A polynomial correlation was provided by Shah and London (1978, Eq. (341)) as  f Re ¼ 24 1  1:3553γ + 1:9467γ 2  1:7012γ 3 + 0:9564γ 4  0:2537γ 5 (2.137) with the maximum error of 0.05%. 2.2.1.3 Frictional pressure drop in laminar flow in isosceles triangular ducts For fully developed laminar flow in isosceles triangular ducts, the Fanning friction factor was obtained in a closed form by Migay (See Shah and London, 1978, Eq. (365)) as f Re ¼

12ðB + 2Þð1  tan 2 θÞ ðB  2Þð tan θ + sec θÞ2

(2.138)

where θ is half of the apex angle of the isosceles triangle and B is given by Eq. (2.139):   1=2 5 1 B¼ 4+ (2.139) 1 2 tan 2 θ For a equilateral triangle, Eq. (2.138) yields f Re ¼ 131⁄3

(2.140)

2.2.1.4 Frictional pressure drop in laminar flow in concentric annular ducts For fully developed laminar flow in a concentric annular duct, the velocity distribution can be expressed as (Lundberg et al., 1963)    lnr 2 1  r 2 + r 2i  1 u lnr i ¼ (2.141)  lnr um 1 + r 2i  r 2i  1 lnr i where r ¼ r=ro and r i ¼ ri =ro . According to the definition of the Fanning friction factor,  du 1 τi ¼ μ  ¼ fi ρu2m (2.142) dr r¼ri 2

Basic thermal design theory for heat exchangers

 du 1 τo ¼ μ  ¼ fo ρu2m dr r¼ro 2 τ¼

τi ri + τo ro 1 ¼ f ρu2m 2 ri + ro

59

(2.143) (2.144)

Eq. (2.141) yields the following Fanning friction factors: 2  ri  1 2 16ð1  r i Þ r 2 lnr i i   (2.145) fi Re ¼ r 2i  1 2 ri 1 + ri   lnr2 i  r 1 16ð1  r i Þ r 2o  i 2 lnr i   fo Re ¼ (2.146) 2 ri  1 2 ro 1 + ri  lnr i 16ð1  r i Þ2 f Re ¼ (2.147) r 2i  1 2 1 + ri  ln r i where the Reynolds number is defined with the hydraulic diameter dh: Re ¼ um dh =μ ¼ um ðdo  di Þ=μ

(2.148)

2.2.1.5 Frictional pressure drop in two-phase flow The frictional pressure drop in two-phase flow is much more complicated than that in single-phase flow. The two-phase pressure drop is linked to the momentum exchange between the liquid phase and vapor (or gas) phase and can be characterized by the two-phase flow pattern. For example, for flow boiling in a horizontal tube, the flow pattern changes from single-phase liquid flow to bubbly flow, stratified flow, wavy flow, slug flow, annular flow, mist flow, and finally single-phase vapor flow. For upward flow boiling in a vertical tube, the flow pattern can be bubbly flow, slug flow, churn flow, wispy-annular flow, annular flow, and mist flow. More detailed descriptions about the flow patterns and flow pattern maps can be found in the book of Collier and Thome (1994). We can use the method of Lockhart and Martinelli (1949) for approximate prediction of the frictional pressure drop for fully developed, incompressible horizontal gas/liquid flow. The two-phase frictional pressure drop is evaluated by the single-phase frictional pressure drop multiplied with a two-phase multiplier ϕ2:

60

Design and operation of heat exchangers and their networks



dpf dz



 ¼ ϕ2l tp

dpf dz

 ¼ ϕ2g l

  dpf dz g

(2.149)

where the two frictional pressure drops are calculated for each of the two phases as if it flows alone in single-phase flow. By defining the LockhartMartinelli parameter X as the ratio of these pressure drops, ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s    dpf dpf X¼ = (2.150) dz l dz g the two-phase multiplier can be correlated as (Collier and Thome, 1994) ϕ2l ¼ 1 + C=X + 1=X 2

(2.151)

ϕ2g ¼ 1 + CX + X 2

(2.152)

where C has the following values: Liquid turbulent viscous turbulent viscous

Gas  turbulent ðttÞ  turbulent ðvtÞ  viscous ðtvÞ  viscous ðvvÞ

C 20 12 19 5

Whether the liquid pressure drop or gas pressure drop is used in Eq. (2.149) depends on the values of corresponding two-phase multiplier: 8   2 dpf 2 2 > >   < ϕl dz , ϕl  ϕg dpf  l ¼ (2.153) dpf dz tp > 2 2 2 > ϕ , ϕ > ϕ : g l g dz g 2.2.1.6 Frictional pressure drop in curved tubes A correlation for frictional pressure drop in curved tubes was proposed by Schmidt (1967) as 8 0:312 < 1 + 0:14ðr=rc Þ0:97 Re10:644ðr=rc Þ , ð100 < Re < Recr Þ fD, c =fD, s ¼ 1 + 2:88  104 ðr=rc Þ0:62 = Re, ð Recr < Re < 2:2  104 Þ : 1 + 0:0823ð1 + r=rc Þðr=rc Þ0:53 Re0:25 , 2  104 < Re < 1:5  105 (2.154)

Basic thermal design theory for heat exchangers

with

61



64= Re, Re < Recr 0:3164= Re0:25 , Re > Recr  Recr ¼ 2300 1 + 8:6ðr=rc Þ0:45

fD, s ¼

(2.155) (2.156)

For two-phase frictional pressure drop, Colombo et al. (2015) proposed a correlation, in which corrective parameters are included into the two-phase multiplier of Lockhart and Martinelli to account for the effect of the centrifugal force: ϕ2l ¼ 0:13ϕ2l, s De0:15 ðρm =ρl Þ0:37 l

(2.157)

h i1 _ g + ð1  x_ Þ=ρl ρm ¼ x=ρ

(2.158)

where

ϕ2l,s is the two-phase multiplier for straight tube, calculated from Eq. (2.151) with C ¼ 20 for turbulent flow of both liquid and vapor phases and the Lockhart-Martinelli parameter X as !0:1     1  x_ 0:9 ρg 0:5 μl X¼ (2.159) x_ ρl μg

2.2.2 Static pressure drop The static pressure drop (hydrostatic pressure) is the difference in pressure at two points within a fluid column, due to the weight of the fluid: Δpg ¼ ρm gΔH

(2.160)

in which g is acceleration due to gravity, H the elevation, and ρm the mean density of the fluid column. In a pipe flow, the static pressure drop can be expressed as. dpg ¼ ρ gsin θdz

(2.161)

where θ denotes the angle of inclination of the tube. For upward flow, θ > 0; therefore, the static pressure drop is positive. For downward flow, the static pressure drop is negative. ρ is the local fluid density. The total static pressure drop can be calculated by integration of the local pressure drop gradient over the whole pipe length: Z L Δpg ¼ g ρ sinθdz (2.162) 0

62

Design and operation of heat exchangers and their networks

For two-phase flow, the local fluid density can be expressed as ρ ¼ aρg + ð1  aÞ ρl

(2.163)

in which a is the void fraction defined as the space- and time-averaged fraction of the channel volume (or channel cross-sectional area) that is occupied by the gas. Unlike the vapor mass fraction, the void fraction cannot be evaluated by the mass balance or energy balance. The relationship between the void fraction and vapor mass fraction depends on the ratio of the gas phase velocity to the liquid phase velocity (known as the slip ratio s): s¼

ug x_ ð1  aÞρl ¼ ul ð1  x_ Þaρg

(2.164)

If the two phases are well mixed or the property differences between the two phases are small, then we can use the homogeneous model with s ¼ 1. It yields a¼

x=ρ _ g x=ρ _ g + ð1  x_ Þ=ρl

(2.165)

For annular flow, it can be evaluated by a simple relationship with the two-phase multiplier ϕ2l calculated from Eq. (2.151) (Hewitt and HallTaylor, 1970, Eq. (5.10)):  1=2 a ¼ 1  ϕ2l (2.166)

2.2.3 Acceleration pressure drop The acceleration pressure drop arises from the momentum change of fluid flowing from one cross section to another. For single-phase flow, dpa ¼ Gdu ¼ GdðG=ρÞ

(2.167)

If the density ρ is constant, after integration from section 1 to section 2, Eq. (2.167) becomes Δpa, 12 ¼

ρ 1 2 G2  G12 ¼ u22  u21 2ρ 2

(2.168)

In a straight channel, the cross-sectional area is constant; therefore, G also keeps constant. Eq. (2.167) can be easily integrated, which yields Δpa, 12 ¼ Gðu2  u1 Þ ¼ G2 ð1=ρ2  1=ρ1 Þ

(2.169)

Basic thermal design theory for heat exchangers

63

In this case, the acceleration pressure drop is usually negligible compared with the total pressure drop, except in the case of flow boiling or flow condensation. For two-phase in a straight channel, the acceleration pressure drop is often expressed as " # " # Gg2 Gg2 Gl2 Gl2 Δpa, 12 ¼ +  + (2.170) aρg ð1  aÞρl aρg ð1  aÞρl 2

1

For variable cross-sectional area, Eq. (2.171) can be used for the calculation of the acceleration pressure drop: " !# Gg2 1 1  Gl2 dpa ¼ d m_ g ug + m_ l ul ¼ d Ac (2.171) + Ac Ac aρg ð1  aÞρl

2.3 Heat exchanger dynamics Heat exchanger dynamics is very important for the design of automatic control systems dealing with heat exchangers and their networks. The task of the dynamic analysis of a heat exchanger is to obtain the dynamic response of the outlet fluid temperatures to the variations of various operating conditions. The linearization method and Laplace transform are the useful tools for the dynamic analysis of heat exchangers.

2.3.1 Linearization of nonlinear problems with small disturbances If the properties of fluids and wall materials depend on temperature, or thermal flow rates of fluids and heat transfer coefficients between the fluids and heat transfer surfaces vary with time, the dynamic thermal analysis of heat exchangers is a nonlinear problem. To simplify the problem, one can use average values of properties, thermal flow rates, and heat transfer coefficients in the real operation region of the heat exchanger to get a linear mathematical model. With this method, the transient temperature responses of heat exchangers to the disturbances in inlet fluid temperatures can be obtained analytically. However, if the disturbances to be investigated are thermal flow rates or heat transfer coefficients or the properties strongly depend on the temperatures, this method cannot be used. Another linearization method is the method for small disturbances. Assume that the properties of the fluids and wall materials in the heat

64

Design and operation of heat exchangers and their networks

exchanger are constant; there is no phase change in the heat exchanger; and the disturbances in thermal flow rates, heat transfer coefficients, and inlet fluid temperatures are small. Then, the nonlinear terms in the mathematical model for dynamic thermal analysis of heat exchangers can be linearized. Let us consider a nonlinear term y(τ) ¼ f (τ)g(τ). The functions f (τ) and g(τ) vary with time around their average values f and g under the new steadystate operating condition or in the new operating period. The disturbances Δf ðτÞ ¼ f ðτÞ  f and ΔgðτÞ ¼ gðτÞ  g are small. Then, this term can be expressed as  yðτÞ ¼ f + Δf ðτÞ ½g + ΔgðτÞ ¼ f gðτÞ + gΔf ðτÞ + Δf ðτÞΔgðτÞ (2.172) The first three terms at the right side of Eq. (2.172) are linear, and the fourth term is nonlinear. For small disturbances, Δf (τ) and Δg(τ) are small; therefore, their product Δf(τ)Δg(τ) can be neglected, which yields a linear expression as follows: yðτÞ ¼ f ðτÞgðτÞ  f gðτÞ + g Δf ðτÞ

(2.173)

Such a treatment is usually reasonable because for a heat exchanger system running at a normal operating condition, the disturbances in the mass flow rates and heat transfer coefficients are usually not large. If the disturbances in mass flow rates are less than 20%, the linearization of the nonlinear terms would not yield a large deviation.

2.3.2 Real-time solutions of heat exchanger dynamics The early research on heat exchanger dynamics was mainly restricted in the solutions of temperature responses in the Laplace domain, that is, the transfer functions of outlet fluid temperatures to disturbances in inlet fluid temperatures and mass flow rates. With the development of the modern control theory, more and more interest has been put on the real-time solutions of heat exchanger dynamics. There are several ways to get the real-time solutions. Some of them will be described briefly as follows. For the lumped parameter model, after the linearization, we obtain the following ordinary differential equation system: dT ¼ AT + BðτÞ dτ τ ¼ 0 : T ¼ T0

(2.174) (2.175)

Basic thermal design theory for heat exchangers

65

The temperature vector T consists of M temperatures of fluids and walls. A is the M  M coefficient matrix of the governing equations. The solution of Eq. (2.174) with its initial condition (2.175) reads Z τ 0 Rτ 1 T ¼ He H T0 + HeRðττ Þ H1 Bðτ0 Þdτ0 (2.176) 0 Rτ

Rτ

in which e is a diagonal matrix, e ¼ diag {er1τ, er2τ, ⋯, erMτ}, ri (i ¼ 1, 2, …, M) are the eigenvalues of the coefficient matrix A, and H is the eigenvector matrix whose columns are the eigenvectors of the corresponding eigenvalues. For the distributed parameter model, the Laplace transform can be applied to eliminate the time variable. Then, the analytical solution in the Laplace domain can be obtained. The real-time temperature responses can be calculated by means of numerical inverse algorithm. It is suggested to use the fast Fourier transform (FFT) algorithm to inverse the solution into the real-time domain. The formula of Ichikawa and Kishima (1972) " # M 1 X eaτn 1 f ðτ n Þ ¼ feða + ikπ=τÞe2iπnk=M  feðaÞ (2.177) Re 2 τ k¼0 can be adopted. With this algorithm, the temperature variation at all time points τn ¼ 2nτ/M in the time interval [0, τ] can be obtained simultaneously. The value of a in Eq. (2.177) is taken as 4 < aτ < 5. M is an exponent of 2, usually M ¼ 211 ¼ 2048. The FFT algorithm has no special requirements for the Laplace transforms to be inversed. However, if the function in the realtime domain τ > 0 has a discontinuity, the FFT algorithm will add an additional oscillation near this discontinuous point. If the value of M is large enough, the oscillation becomes an additional sharp pulse. The maximum value of the pulse amplitude is 8.949% of the step change at the discontinuous point (Anon., 1979). In some cases, the inverse Laplace transform can be obtained analytically by means of the residuum theorem (Anon., 1979): Z σ + i∞ ∞ h  i h i X 1 L 1 feðsÞ ¼ (2.178) res fe sj esj τ feðsÞesτ ds ¼ 2π i σi∞ j¼0 in which sj is the jth singularity of function feðsÞ. For double Laplace transform, we can get the analytical solution with Eq. (2.178) and then use numerical inverse algorithm to get the real-time solution. This method has been successfully used for predicting the transient behavior of crossflow heat exchangers (Luo, 1998).

66

Design and operation of heat exchangers and their networks

However, the Laplace transform can only be used for linear problems. For nonlinear problems with small disturbances, we should first linearize the problem and then solve it with Laplace transform. If the linearization is no longer allowed, a numerical method has to be used to calculate the temperature dynamics.

References Ali, S., 2001. Pressure drop correlations for flow through regular helical coil tubes. Fluid Dyn. Res. 28 (4), 295–310. Anon., 1979. Handbook of Mathematics. People’s Education Publishing House (in Chinese). Baehr, H.D., 1960. Gleichungen f€ ur den W€arme€ ubergang bei hydrodynamisch nicht ausgebildeter Laminarstr€ omung in Rohren. Chem. Ing. Tech. 32 (2), 89–90. Balzereit, F., 1999. Bestimmung von axialen Dispersionskoeffizienten in W€arme€ ubertragern aus Verweilzeitmessungen. Fortschritt-Berichte VDI, Reihe 19, Nr. 120, VDI Verlag, D€ usseldorf. Bassiouny, M.K., Martin, H., 1984. Flow distribution and pressure drop in plate heat exchangers—II Z-type arrangement. Chem. Eng. Sci. 39 (4), 701–704. Chen, J.J.J., 1987. Comments on improvements on a replacement for the logarithmic mean. Chem. Eng. Sci. 42 (10), 2488–2489. Churchill, S.W., 1977. Friction-factor equation spans all fluid-flow regimes. Chem. Eng. 84 (24), 91–92. Churchill, S.W., Ozoe, H., 1973a. Correlations for laminar forced convection with uniform heating in flow over a plate and in developing and fully developed flow in a tube. J. Heat Transf. 95 (1), 78–84. Churchill, S.W., Ozoe, H., 1973b. Correlations for laminar forced convection in flow over an isothermal flat plate and in developing and fully developed flow in an isothermal tube. J. Heat Transf. 95 (3), 416–419. Colebrook, C.F., 1939. Turbulent flow in pipes, with particular reference to the transition region between the smooth and rough pipe laws. J. Inst. Civ. Eng. 11 (4), 133–156. Collier, J.G., Thome, J.R., 1994. Convective Boiling and Condensation, third ed. Oxford University Press, Oxford. Colombo, M., Colombo, L.P.M., Cammi, A., Ricotti, M.E., 2015. A scheme of correlation for frictional pressure drop in steam-water two-phase flow in helicoidal tubes. Chem. Eng. Sci. 123, 460–473. Das, S.K., Roetzel, W., 2004. The axial dispersion model for heat transfer equipment—a review. Int. J. Transp. Phenom. 6 (1), 23–49. Diaz, M., Aguayo, A.T., 1987. How flow dispersion affects exchanger performance. Hydrocarb. Process. 66 (4), 57–60. Dittus, F.W., Boelter, L.M.K., 1930. Heat transfer in automobile radiators of the tubular type. In: University of California Publications in Engineering. vol. 2(13), pp. 443–461. University of California Press, Berkeley, CA. Reprinted in: International Communications in Heat and Mass transfer, 12(1):3–22, 1985. Ghobadi, M., Muzychka, Y.S., 2016. A review of heat transfer and pressure drop correlations for laminar flow in curved circular ducts. Heat Transfer Eng. 37 (10), 815–839. Gnielinski, V., 1975. Neue Gleichungen f€ ur den W€arme- und den Stoff€ ubergang in turbulent durchstr€ omten Rohren und Kan€alen. Forsch. Ingenieurwes. 41 (1), 8–16. Gnielinski, V., 1989. Zur W€arme€ ubertragung bei laminarer Rohrstr€ omung und konstanter Wandtemperatur. Chem. Ing. Tech. 61 (2), 160–161.

Basic thermal design theory for heat exchangers

67

Gnielinski, V., 1995. Ein neues Berechnungsverfahren f€ ur die W€arme€ ubertragung im € Ubergangsbereich zwischen laminarer und turbulenter Rohrstr€ omung. Forsch. Ingenieurwes. 61 (9), 240–248. Gnielinski, V., 2010a. G1 Heat transfer in pipe flow. In: 5825 Atlas, second ed. Springer, D€ usseldorf. Gnielinski, V., 2010b. G2 Heat transfer in concentric annular and parallel plate ducts. In: VDI Heat Atlas, second ed. Springer, D€ usseldorf. Gnielinski, V., 2013a. G1 Durchstr€ omte Rohre. In: VDI W€armeatlas, eleventh ed. Springer, D€ usseldorf. Gnielinski, V., 2013b. G2 W€arme€ ubertragung im konzentrischen Ringspalt und im ebenen Spalt. In: VDI W€armeatlas, eleventh ed. Springer, D€ usseldorf. Grigull, U., Tratz, H., 1965. Thermischer einlauf in ausgebildeter laminarer rohrstr€ omung. Int. J. Heat Mass Transf. 8 (5), 669–678. Hewitt, G.F., Hall-Taylor, N.S., 1970. Annular Two-Phase Flow. Pergamon Press, Oxford. Huber, M.L., Perkins, R.A., Friend, D.G., Sengers, J.V., Assael, M.J., Metaxa, I.N., Miyagawa, K., Hellmann, R., Vogel, E., 2012. New international formulation for the thermal conductivity of H2O. J. Phys. Chem. Ref. Data 41 (3), 033102. Ichikawa, S., Kishima, A., 1972. Applications of Fourier series technique to inverse Laplace transform. Kyoto University Memories 34 (part 1), 53–67. Kast, W., 2010. L1.2 Pressure drop in flow through pipes. In: VDI Heat Atlas, second ed. Springer, D€ usseldorf. Kast, W., 2013. L1.2 Druckverlust in durchstr€ omten Rohren. In: VDI W€armeatlas, eleventh ed. Springer, D€ usseldorf. Kraus, A.D., Aziz, A., Welty, J., 2001. Extended Surface Heat Transfer. John Wiley & Sons, New York. Lee, D.-Y., 1994. Thermisches Verfahren von Rohrb€ undelw€arme€ ubertragern. FortschrittBerichte VDI, Reihe 19, Nr. 78, VDI Verlag, D€ usseldorf. Lockhart, R.W., Martinelli, R.C., 1949. Proposed correlation of data for isothermal twophase two-component flow in a pipe. Chem. Eng. Prog. 45 (1), 39–48. Lundberg, R.E., McCuen, P.A., Reynolds, W.C., 1963. Heat transfer in annular passages. Hydrodynamically developed laminar flow with arbitrarily prescribed wall temperatures or heat fluxes. Int. J. Heat Mass Transf. 6 (6), 495–529. Luo, X., 1998. Das axiale Dispersionsmodell f€ ur Kreuzstromw€arme€ ubertrager. Dissertation, University of the Federal Armed Forces Hamburg, Germany. Also in: FortschrittBerichte VDI, Reihe 19, Nr. 109, VDI Verlag, D€ usseldorf. Luo, X., Roetzel, W., 1995. Extended axial dispersion model for transient analysis of heat exchangers. In: Proceedings of the 4th UK National Conference on Heat Transfer, 26-27 September 1995. IMechE, London, pp. 411–416. Mecklenburgh, J.C., Hartland, S., 1975. The Theory of Backmixing: The Design of Continuous Flow Chemical Plant With Backmixing Cover. John Wiley and Sons. Na Ranong, C., Roetzel, W., 2012. Unity Mach number axial dispersion model for heat exchanger design. J. Phys. Conf. Ser. 395, 012052. Naphon, P., Wongwises, S., 2006. A review of flow and heat transfer characteristics in curved tubes. Renew. Sust. Energ. Rev. 10 (5), 463–490. Pa´tek, J., Hruby´, J., Klomfar, J., Souckova´, M., 2009. Reference correlations for thermophysical properties of liquid water at 0.1MPa. J. Phys. Chem. Ref. Data 38 (1), 21–29. Popiel, C.O., Wojtkowiak, J., 1998. Simple formulas for thermophysical properties of liquid water for heat transfer calculations (from 0°C to 150°C). Chem. Ing. Tech. 19 (3), 87–101. Roetzel, W., 1969. Ber€ ucksichtigung ver€anderlicher W€arme€ ubergangskoeffizienten und W€armekapazit€aten bei der Bemessung von W€armeaustauschern. W€arme Stoff€ ubertragung 2, 163–170.

68

Design and operation of heat exchangers and their networks

Roetzel, W., 1988. Analytische Berechnung von W€arme€ ubertragern mit nachtr€aglicher Ber€ ucksichtigung temperaturabh€angiger W€armekapazit€aten. W€arme Stoff€ ubertragung 23, 175–177. Roetzel, W., 1996. Transient analysis in heat exchangers. In: Afgan, N. et al., (Eds.), New Developments in Heat Exchangers, Gordon and Breach Publishers, Amsterdam, pp. 547–575. Roetzel, W., 2010. New axial dispersion model for heat exchanger design. In: Stachel, A.A., Mikielewicz, D. (Eds.), Proceedings of the 13th International Symposium on Heat Transfer and Renewable Sources of Energy. Wydawnistwo Uczelnianie ZUT w Szczecinie, pp. 567–568. Roetzel, W., Das, S.K., 1995. Hyperbolic axial dispersion model: concept and its application to a plate heat exchanger. Int. J. Heat Mass Transf. 38 (16), 3065–3076. Roetzel, W., Luo, X., 2011. Mean overall heat transfer coefficient in heat exchangers allowing for temperature-dependent fluid properties. Heat Transfer Eng. 32 (2), 141–150. Roetzel, W., Na Ranong, C., 1999. Consideration of maldistribution in heat exchangers using the hyperbolic dispersion model. Chem. Eng. Process. 38, 675–681, Also in: Progress in Engineering Heat Transfer, B. Grochal, J. Mikielewicz and B. Sunden (eds.), Institute of Fluid-Flow Machinery Publishers, 569–580. Roetzel, W., Na Ranong, C., 2014. Evaluation of residence time measurements on heat exchangers for the determination of dispersive Peclet numbers. Arch. Thermodyn. 35 (2), 103–115. Roetzel, W., Na Ranong, C., 2015. Evaluation method of single blow experiment for the determination of heat transfer coefficient and dispersive Peclet number. Arch. Thermodyn. 36 (4), 3–24. Roetzel, W., Na Ranong, C., 2018a. Evaluation of temperature oscillation experiment for the determination of heat transfer coefficient and dispersive Peclet number. Arch. Thermodyn. 39 (1), 91–110. Roetzel, W., Na Ranong, C., 2018b. Thermal calculation of heat exchangers with simplified consideration of axial wall heat conduction. In: Proceedings of the 17th International Conference Heat Transfer and Renewable Sources of Energy (HTRSE-2018), E3S Web of Conferences. vol. 70. (02013). Roetzel, W., Spang, B., 2010. C1 Thermal design of heat exchangers. In: VDI Heat Atlas, second ed. Springer, D€ usseldorf. Roetzel, W., Spang, B., 2013. C1 Berechnung von W€arme€ ubertragern. In: VDI W€armeatlas, eleventh ed. Springer, D€ usseldorf. Roetzel, W., Spang, B., 2019. W€arme€ ubertrager: W€armedurchgang und W€armedurchgangskoeffizienten. In: Stephan, P., Kabelac, S., Kind, M., Mewes, D., Schaber, K., Wetzel, T. (Eds.), VDI W€armeatlas, twelfth ed. Springer, Berlin, D€ usseldorf. Roetzel, W., Spang, B., Luo, X., Das, S.K., 1998. Propagation of the third sound wave in fluid: hypothesis and theoretical foundation. Int. J. Heat Mass Transf. 41, 2769–2780. Roetzel, W., Na Ranong, C., Fieg, G., 2011. New axial dispersion model for heat exchanger design. Heat Mass Transf. 47, 1009–1017. Sahoo, R.K., Roetzel, W., 2002. Hyperbolic axial dispersion model for heat exchangers. Int. J. Heat Mass Transf. 45 (6), 1261–1270. Schmidt, E.F., 1967. W€arme€ ubergang und Druckverlust in Rohrschlangen. Chem. Ing. Tech. 39 (13), 781–789. Shah, R.K., London, A.L., 1978. Laminar Flow Forced Convection in Ducts. Academic Press, New York. Sieder, E.N., Tate, G.E., 1936. Heat transfer and pressure drop of liquids in tubes. Ind. Eng. Chem. 28 (12), 1429–1435.

Basic thermal design theory for heat exchangers

69

€ Spang, B., 1991. Uber das thermische Verhalten von Rohrb€ undelw€arme€ ubertragern mit Segmentumlenkblechen. Fortschritt-Berichte VDI, Reihe 19, Nr. 48, VDI Verlag, D€ usseldorf. Spang, B., Roetzel, W., 1995. Neue N€aherungsgleichung zur einheitlichen Berechnung von W€arme€ ubertragern. W€arme Stoff€ ubertragung 30, 417–422. Stephan, K., 1959. W€arme€ ubertragung und Druckabfall bei nicht ausgebildeter Laminarstr€ omung in Rohren und in ebenen Spalten. Chem. Ing. Tech. 31 (12), 773–778. Stephan, K., 1962. W€arme€ ubergang bei turbulenter und bei laminarer Str€ omung in Ringspalten. Chem. Ing. Tech. 34 (3), 207–212. Stephan, K., Nesselmann, K., 1961. Gleichungen f€ ur den W€arme€ ubergang laminar str€ omender Stoffe in ringf€ ormigen Querschnitten. Chem. Ing. Tech. 33 (5), 338–343. Taylor, S.G., 1954. The dispersion of matter in turbulent flow through a pipe. Proc. R. Soc. Lond. A Math. Phys. Sci. 223 (1155), 446–468. Whitaker, S., 1977. Fundamental Principles of Heat Transfer. Pergamon Press, New York. Xuan, Y., 1991. Thermische Modellierung mehrg€angiger Rohrb€ undelw€arme€ ubertrager mit Umlenkblechen und geteiltem Mantelstrom. Fortschritt-Berichte VDI, Reihe 19, Nr. 52, VDI Verlag, D€ usseldorf.