New axial dispersion model for heat exchanger design

Wilfried Roetzel; Chakkrit Na Ranong; Georg Fieg

doi:10.1007/s00231-011-0847-z

Heat and Mass Transfer

August 2011, 47:1009 | Cite as

New axial dispersion model for heat exchanger design

Authors
Authors and affiliations

Wilfried RoetzelEmail author
Chakkrit Na Ranong
Georg Fieg

Original

First Online: 10 July 2011

Received: 30 April 2011

Accepted: 25 June 2011

232 Downloads
4 Citations

Abstract

The special case of unity dispersive Mach number of the hyperbolic axial dispersion model is investigated as the more realistic and simpler alternative to the parabolic model with zero Mach number. Simple corrections to the mean temperature difference or to the heat transfer coefficients are derived as functions of the dispersive Peclet numbers. As an example the model is applied to a cascade of stirred tanks in overall counterflow arrangement.

Keywords

Heat Transfer Coefficient Heat Exchanger Dispersion Model Peclet Number Parallel Flow

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

List of symbols

A: Area for heat transfer (m²)
A_c: Cross-sectional area (m²)
B: n × n matrix
b_ij: Elements of matrix B
C: Propagation velocity (m s⁻¹)
c_p: Specific isobaric heat capacity (J kg⁻¹ K⁻¹)
H: Operator
k: Overall heat transfer coefficient (W m⁻² K⁻¹)
k^*: Overall heat transfer coefficient (W m⁻² K⁻¹), Eq. 24a
L: Length of heat exchanger (m)
l: Space coordinate (m)
M: Dispersive Mach number, M = |w|/C
NTU: Number of transfer units, NTU = k A/(A _c wρc _p)
NTU^*: Number of transfer units, Eq. 24
NTU_α: Number of transfer units, NTU_α = αA/(A _c wρc _p)
$ {\text{NTU}}_{\alpha }^{*} $: Number of transfer units, Eq. 27a
n: Number of stirred tanks in series or number of fluid streams
P: Dimensionless temperature change, $ P_{1} = {{\left( {T_{1}^{\prime } - T_{1}^{\prime \prime } } \right)} \mathord{\left/ {\vphantom {{\left( {T_{1}^{\prime } - T_{1}^{\prime \prime } } \right)} {\left( {T_{1}^{\prime } - T_{2}^{\prime } } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {T_{1}^{\prime } - T_{2}^{\prime } } \right)}} $ and $ P_{2} = {{\left( {T_{2}^{\prime \prime } - T_{2}^{\prime } } \right)} \mathord{\left/ {\vphantom {{\left( {T_{2}^{\prime \prime } - T_{2}^{\prime } } \right)} {\left( {T_{1}^{\prime } - T_{2}^{\prime } } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {T_{1}^{\prime } - T_{2}^{\prime } } \right)}} $
Pe: Dispersive Peclet number, Pe = w L ρ c _p/λ ^*
$ \dot{Q} $: Heat transfer rate (W)
$ \dot{q}_{l} $: Heat flux (W m⁻²)
R_w: Wall resistance including fouling resistances (K W⁻¹)
T: Hypothetic temperature inside the heat exchanger and true fluid temperature outside the heat exchanger (K)
T: Vector of fluid temperatures (K)
ΔT_M: Non-dispersive mean temperature difference (K)
t: True fluid temperature inside the exchanger (K)
t: Vector of temperatures (K)
Δt_M: Dispersive mean temperature difference (K)
W₁: Capacity of residing fluid in the core of the heat exchanger (J K⁻¹)
W₂: Capacity of the core of the heat exchanger (J K⁻¹)
$ \dot{W} $: Heat capacity rate (W K⁻¹)
w: Flow velocity (m s⁻¹)
x: Dimensionless space coordinate (x = l/L)
y: Dimensionless space coordinate, perpendicular to x (cross-flow)
z: Dimensionless time coordinate (z = τ w/L)

Greek symbols

α: Heat transfer coefficient (W m⁻² K⁻¹)
α^*: Heat transfer coefficient (W m⁻² K⁻¹), Eq. 27
λ^*: Dispersive thermal conductivity (W m⁻¹ K⁻¹)
φ: Reduced axial dispersive heat flux (K)
ρ: Density (kg m⁻³)
τ: Time (s)
Θ: Dimensionless mean temperature difference, $ \Uptheta_{\text{dispersive}} = {{\Updelta t_{\text{M}} } \mathord{\left/ {\vphantom {{\Updelta t_{\text{M}} } {\left( {T_{1}^{\prime } - T_{2}^{\prime } } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {T_{1}^{\prime } - T_{2}^{\prime } } \right)}} $ and $ \Uptheta_{\text{plug}} = {{\Updelta T_{\text{M}} } \mathord{\left/ {\vphantom {{\Updelta T_{\text{M}} } {\left( {T_{1}^{\prime } - T_{2}^{\prime } } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {T_{1}^{\prime } - T_{2}^{\prime } } \right)}} $

Super- and subscripts

c: Cascade of stirred tanks
i: Stirred tank
i, j: Fluid stream i, j
plug: Plug flow
w: Wall
1: Fluid 1
2: Fluid 2
′: Inlet
″: Outlet
-: Mean value
τ: Transposition

This is a preview of subscription content, to check access.

Appendix 1: Mixed-unmixed cross-flow

Stream 1 mixed, stream 2 unmixed.

Individual solution for dispersion model

$$ P_{1} = 1 - \exp \left[ { - \left( {\frac{{{{\dot{W}_{1} } \mathord{\left/ {\vphantom {{\dot{W}_{1} } {\dot{W}_{2} }}} \right. \kern-\nulldelimiterspace} {\dot{W}_{2} }}}}{{1 - \exp \left( { - {{{\text{NTU}}_{2} } \mathord{\left/ {\vphantom {{{\text{NTU}}_{2} } {\left( {1 + \frac{{{\text{NTU}}_{2} }}{{Pe_{2} }}} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {1 + \frac{{{\text{NTU}}_{2} }}{{Pe_{2} }}} \right)}}} \right)}} + \frac{1}{{Pe_{1} }}} \right)^{ - 1} } \right] $$

(36)

$$ {{\dot{W}_{2} } \mathord{\left/ {\vphantom {{\dot{W}_{2} } {\dot{W}_{1} }}} \right. \kern-\nulldelimiterspace} {\dot{W}_{1} }} = {{{\text{NTU}}_{1} } \mathord{\left/ {\vphantom {{{\text{NTU}}_{1} } {{\text{NTU}}_{2} }}} \right. \kern-\nulldelimiterspace} {{\text{NTU}}_{2} }} $$

General approximation

$$ P_{1} = 1 - \exp \left\{ { - \frac{{\dot{W}_{2} }}{{\dot{W}_{1} }}\left[ {1 - \exp \left( { - {\text{NTU}}_{2}^{*} } \right)} \right]} \right\} $$

(37)

NTU ₂ ^* from Eq. 24

$$ {\text{NTU}}_{2}^{*} = \frac{{{\text{NTU}}_{2} }}{{1 + \frac{{{\text{NTU}}_{1} }}{{Pe_{1} }} + \frac{{{\text{NTU}}_{2} }}{{Pe_{2} }}}} $$

(24)

For Pe ₁ = ∞ the Eqs. 36 and 37 become identical.

Appendix 2: Two-fluid mixed–mixed cross-flow

Individual solution for dispersion model

i = 1, 2.

$$ \begin{gathered} F_{i} = \frac{1}{{{\text{NTU}}_{i} }}\left( {1 - \exp \left[ { - \frac{{{\text{NTU}}_{i} }}{{1 + \frac{{{\text{NTU}}_{i} }}{{Pe_{i} }}}}} \right]} \right) \hfill \\ \bar{t}_{1} = \frac{{\frac{{T_{1}^{\prime } }}{{F_{2} }} + \frac{{T_{2}^{\prime } }}{{F_{1} }} - T_{2}^{\prime } }}{{\frac{1}{{F_{2} }} + \frac{1}{{F_{1} }} - 1}} \hfill \\ \bar{t}_{2} = \frac{{\frac{{T_{1}^{\prime } }}{{F_{2} }} + \frac{{T_{2}^{\prime } }}{{F_{1} }} - T_{1}^{\prime } }}{{\frac{1}{{F_{2} }} + \frac{1}{{F_{1} }} - 1}} \hfill \\ T_{1}^{\prime \prime } = T_{1}^{\prime } - {\text{NTU}}_{1} \left( {\bar{t}_{1} - \bar{t}_{2} } \right) \hfill \\ T_{2}^{\prime \prime } = T_{2}^{\prime } + {\text{NTU}}_{2} \left( {\bar{t}_{1} - \bar{t}_{2} } \right) \hfill \\ \end{gathered} $$

(38)

The true mean temperature can serve as reference temperature: $ \alpha_{i} = \alpha_{i} \left( {\bar{t}_{i} } \right) $.

General approximation

$$ \begin{gathered} F_{i} = \frac{1}{{{\text{NTU}}_{i}^{*} }}\left( {1 - \exp \left[ { - {\text{NTU}}_{i}^{*} } \right]} \right) \hfill \\ T_{1}^{\prime \prime } = T_{1}^{\prime } - {\text{NTU}}_{1}^{*} \left( {\bar{t}_{1} - \bar{t}_{2} } \right) \hfill \\ T_{2}^{\prime \prime } = T_{2}^{\prime } + {\text{NTU}}_{2}^{*} \left( {\bar{t}_{1} - \bar{t}_{2} } \right) \hfill \\ \end{gathered} $$

(39)

NTU^* according to Eq. 24, $ \bar{t}_{1} $ and $ \bar{t}_{2} $ from Eq. 38. These temperatures are now only approximations of the true mean temperatures.

Appendix 3: Multi-fluid mixed–mixed cross-flow

Individual solution for dispersion model

Inlet temperature: $ {\mathbf{T}}^{\prime } = \left( {T_{1}^{\prime } ,T_{2}^{\prime } , \ldots ,T_{n}^{\prime } } \right)^{\tau } $

True mean temperature: $ {\bar{\mathbf{t}}} = \left( {\bar{t}_{1} ,\bar{t}_{2} , \ldots ,\bar{t}_{n} } \right)^{\tau } $

Using the n × n matrix $ {\mathbf{B}} = \left( {b_{ij} } \right) $,

$$ {\bar{\mathbf{t}}} = {\mathbf{B}}^{ - 1} \cdot {\mathbf{T}}^{\prime } . $$

(40)

$$ b_{ij} = \left\{ {\begin{array}{*{20}l} {\frac{{N_{i} }}{{1 - \exp \left( { - {{N_{i} } \mathord{\left/ {\vphantom {{N_{i} } {\left( {1 + \frac{{N_{i} }}{{Pe_{i} }}} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {1 + \frac{{N_{i} }}{{Pe_{i} }}} \right)}}} \right)}}} & {\left( {i = j} \right)} \\ {\frac{{\left( {Ak} \right)_{ij} }}{{\sum\nolimits_{\begin{subarray}{l} j^{\prime } = 1 \\ j^{\prime } \ne i \end{subarray} }^{n} {\left( {Ak} \right)_{{ij^{\prime } }} } }}\left[ {1 - \frac{{N_{i} }}{{1 - \exp \left( { - {{N_{i} } \mathord{\left/ {\vphantom {{N_{i} } {\left( {1 + \frac{{N_{i} }}{{Pe_{i} }}} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {1 + \frac{{N_{i} }}{{Pe_{i} }}} \right)}}} \right)}}} \right]} & {\left( {i \ne j} \right)} \\ \end{array} } \right. $$

(41)

$$ N_{i} = \frac{1}{{\dot{W}_{i} }}\sum\limits_{\begin{subarray}{l} j = 1 \\ j \ne i \end{subarray} }^{n} {\left( {Ak} \right)_{ij} } $$

(42)

$$ T_{i}^{\prime \prime } = g_{i} T_{i}^{\prime } + \left( {1 - g_{i} } \right)\;\bar{t}_{i} $$

(43)

$$ g_{i} = \frac{{\left( {1 + N_{i} } \right)\exp \left( { - \frac{{N_{i} }}{{1 + {{N_{i} } \mathord{\left/ {\vphantom {{N_{i} } {Pe_{i} }}} \right. \kern-\nulldelimiterspace} {Pe_{i} }}}}} \right) - 1}}{{N_{i} + \exp \left( { - \frac{{N_{i} }}{{1 + {{N_{i} } \mathord{\left/ {\vphantom {{N_{i} } {Pe_{i} }}} \right. \kern-\nulldelimiterspace} {Pe_{i} }}}}} \right) - 1}} $$

(44)

Reference temperature $ \alpha_{i} = \alpha_{i} \left( {\bar{t}_{i} } \right) $

General approximation

Equation 40 remains unchanged. Equations 41 and 42 are replaced by Eqs. 45 and 46

$$ b_{ij} = \left\{ {\begin{array}{*{20}l} {\frac{{N_{i}^{*} }}{{1 - \exp \left( { - N_{i}^{*} } \right)}}} & {\left( {i = j} \right)} \\ {\frac{{\left( {Ak^{*} } \right)_{ij} }}{{\sum\nolimits_{\begin{subarray}{l} j^{\prime } = 1 \\ j^{\prime } \ne i \end{subarray} }^{n} {\left( {Ak^{*} } \right)_{{ij^{\prime } }} } }}\left[ {1 - \frac{{N_{i}^{*} }}{{1 - \exp \left( { - N_{i}^{*} } \right)}}} \right]} & {\left( {i \ne j} \right)} \\ \end{array} } \right. $$

(45)

$$ N_{i}^{*} = \frac{1}{{\dot{W}_{i} }}\sum\limits_{\begin{subarray}{l} j = 1 \\ j \ne i \end{subarray} }^{n} {\left( {Ak^{*} } \right)_{ij} } $$

(46)

with $ \alpha_{i}^{*} $ from Eq. 27.

Equation 44 is replaced by

$$ g_{i} = \frac{{\left( {1 + N_{i}^{*} } \right)\exp \left( { - N_{i}^{*} } \right) - 1}}{{N_{i}^{*} + \exp \left( { - N_{i}^{*} } \right) - 1}} . $$

(47)

For n = 2 the multi-fluid solution turns to the two-fluid solutions (Eqs. 38 and 39).

References

1.
(2006) VDI-Wärmeatlas, 10. Auflage. Springer, BerlinGoogle Scholar
2.
(2010) Heat Atlas, 2nd edn. Springer, BerlinGoogle Scholar
3.
Hewitt GF, Shires GL, Bott TR (1994) Process heat transfer. CRC Press, Boca RatonGoogle Scholar
4.
Das SK, Roetzel W (2004) The axial dispersion model for heat transfer equipment—a review. Int J Transp Phenom 6:23–49Google Scholar
5.
Spang B (1991) Über das thermische Verhalten von Rohr-bündelwärmeübertragern mit Segmentumlenkblechen. VDI-Fortschrittsberichte, Reihe 19, Nr. 48, VDI-Verlag, DüsseldorfGoogle Scholar
6.
Xuan Y (1991) Thermische Modellierung mehrgängiger Rohrbündelwärmeübertrager mit Umlenkblechen und geteiltem Mantelstrom. VDI-Fortschrittsberichte, Reihe 19, Nr. 52, VDI-Verlag, DüsseldorfGoogle Scholar
7.
Luo X (1998) Das axiale Dispersionsmodell für Kreuzstrom-wärmeübertrager. VDI-Fortschrittsberichte, Reihe 19, Nr. 109, VDI-Verlag, DüsseldorfGoogle Scholar
8.
Roetzel W, Das SK (1995) Hyperbolic axial dispersion model: concept and its application to a plate heat exchanger. Int J Heat Mass Transf 38:3065–3076CrossRefGoogle Scholar
9.
Roetzel W, Na Ranong C (2000) Axial dispersion models for heat exchangers. Int J Heat Technol 18:7–17MATHGoogle Scholar
10.
Sahoo RK, Roetzel W (2002) Hyperbolic axial dispersion model for heat exchangers. Int J Heat Mass Transf 45:1261–1270MATHCrossRefGoogle Scholar
11.
Na Ranong C, Höhne M, Franzen J, Hapke J, Fieg G, Dornheim M, Bellosta von Colbe M, Eigen N, Metz O (2009) Concept, design and manufacture of a prototype hydrogen storage tank based on sodium alanate. Chem Eng Technol 32(8):1154–1163CrossRefGoogle Scholar
12.
Chester M (1963) Second sound in solids. Phys Rev 131:2013–2015CrossRefGoogle Scholar
13.
Roetzel W, Spang B, Luo X, Das SK (1998) Propagation of the third sound wave in fluid: hypothesis and theoretical foundation. Int J Heat Mass Transf 41:2769–2780MATHCrossRefGoogle Scholar
14.
Roetzel W, Luo X (2010) Thermal design of multi-fluid mixed–mixed cross-flow heat exchangers. Heat Mass Transf 46(10):1077–1085CrossRefGoogle Scholar
15.
Roetzel W, Na Ranong C (2003) On the application of the Wilson plot technique. Int J Heat Technol 21:125–130Google Scholar

Copyright information

Authors and Affiliations

Wilfried Roetzel
- 1
Email author
Chakkrit Na Ranong
- 2
Georg Fieg
- 2

1.Institute of ThermodynamicsHelmut Schmidt University/University of the Federal Armed ForcesHamburgGermany
2.Institute of Process and Plant EngineeringHamburg University of TechnologyHamburgGermany

New axial dispersion model for heat exchanger design

Abstract

Keywords

List of symbols

Greek symbols

Super- and subscripts

References

Copyright information

Authors and Affiliations

Personalised recommendations

Cite article