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New axial dispersion model for heat exchanger design

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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.

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Abbreviations

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

NTUα :

Number of transfer units, NTUα = 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)

W 1 :

Capacity of residing fluid in the core of the heat exchanger (J K−1)

W 2 :

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)

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 \( \Uptheta_{\text{plug}} = {{\Updelta T_{\text{M}} } \mathord{\left/ {\vphantom {{\Updelta T_{\text{M}} } {\left( {T_{1}^{\prime } - T_{2}^{\prime } } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {T_{1}^{\prime } - T_{2}^{\prime } } \right)}} \)

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

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Author information

Authors and Affiliations

  1. Institute of Thermodynamics, Helmut Schmidt University/University of the Federal Armed Forces, 22039, Hamburg, Germany

    Wilfried Roetzel

  2. Institute of Process and Plant Engineering, Hamburg University of Technology, 21071, Hamburg, Germany

    Chakkrit Na Ranong & Georg Fieg

Authors
  1. Wilfried Roetzel
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  2. Chakkrit Na Ranong
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  3. Georg Fieg
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Corresponding author

Correspondence to Wilfried Roetzel.

Additional information

Dedicated to Prof. Dr.-Ing. Dr.-Ing. E.h. mult. Franz Mayinger on the occasion of his 80th birthday.

Appendices

Appendix 1: Mixed-unmixed cross-flow

Stream 1 mixed, stream 2 unmixed.

1.1 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)
$$ {{\dot{W}_{2} } \mathord{\left/ {\vphantom {{\dot{W}_{2} } {\dot{W}_{1} }}} \right. \kern-\nulldelimiterspace} {\dot{W}_{1} }} = {{{\text{NTU}}_{1} } \mathord{\left/ {\vphantom {{{\text{NTU}}_{1} } {{\text{NTU}}_{2} }}} \right. \kern-\nulldelimiterspace} {{\text{NTU}}_{2} }} $$

1.2 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

2.1 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: \( \alpha_{i} = \alpha_{i} \left( {\bar{t}_{i} } \right) \).

2.2 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

3.1 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 \( \alpha_{i} = \alpha_{i} \left( {\bar{t}_{i} } \right) \)

3.2 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).

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Roetzel, W., Na Ranong, C. & Fieg, G. New axial dispersion model for heat exchanger design. Heat Mass Transfer 47, 1009–1017 (2011). https://doi.org/10.1007/s00231-011-0847-z

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