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7.25.2 Heat Exchanger Model Theory

In FLUENT, the heat exchanger core is treated as a fluid zone with momentum and heat transfer. Pressure loss is modeled as a momentum sink in the momentum equation, and heat transfer is modeled as a heat source in the energy equation.

FLUENT provides two heat exchanger models: the default ntu-model and the simple-effectiveness-model. The simple-effectiveness-model interpolates the effectiveness from the velocity vs effectiveness curve that you provide. For the ntu-model, FLUENT calculates the effectiveness, $\epsilon$, from the NTU value that is calculated by FLUENT from the heat transfer data provided by the user in tabular format. FLUENT will automatically convert this heat transfer data to a primary fluid mass flow rate vs NTU curve (this curve will be piecewise linear). This curve will be used by FLUENT to calculate the NTU for macros based on their size and primary fluid flow rate.

The ntu-model provides the following features:

The simple-effectiveness-model provides the following features:



Streamwise Pressure Drop


In both heat exchanger models, pressure loss is modeled using the porous media model in FLUENT. The streamwise pressure drop can be expressed as


 \Delta p = \frac{1}{2} f \rho_{m} U^2_{A_{\rm min}} (7.25-1)


where      
  $\Delta p$ = streamwise pressure drop
  $f$ = streamwise pressure loss coefficient
  $\rho_{m}$ = mean gas density
  $U_{A_{\rm min}}$ = gas velocity at the minimum flow area

The pressure loss coefficient is computed from


 f = (K_c + 1 - \sigma^2) - (1 - \sigma^2 - K_e)\frac{\nu_e}... ...u_e}{\nu_i} - 1 \right) + f_c \frac{A}{A_c}\frac{\nu_m}{\nu_i} (7.25-2)


where      
  $\sigma$ = minimum flow to face area ratio
  $K_c$ = entrance loss coefficient
  $K_e$ = exit loss coefficient
  $A$ = gas-side surface area
  $A_c$ = minimum cross-sectional flow area
  $f_c$ = core friction factor
  $\nu_e$ = specific volume at the exit
  $\nu_i$ = specific volume at the inlet
  $\nu_m$ = mean specific volume $\equiv \frac{1}{2} (\nu_e + \nu_i)$

$K_c$ and $K_e$ are empirical quantities obtained from experimental data. You will need to specify these parameters based on graphs that are closest to the heat exchanger configuration that you are setting up [ 176], [ 174]. These parameters are used to set up large resistances in the two non-streamwise directions, effectively forcing the gas flow through the core to be unidirectional.

In Equation  7.25-2, the core friction factor is defined as


 f_c = a{\rm Re}^b_{\rm min} (7.25-3)


where      
  $a$ = core friction coefficient
  $b$ = core friction exponent
  ${\rm Re}_{\rm min}$ = Reynolds number for velocity at the minimum flow area

$a$ and $b$ are empirical quantities obtained from experimental data. You will need to specify the core friction coefficient and exponent based on graphs that are closest to the heat exchanger models that you set up [ 176], [ 174].

The Reynolds number in Equation  7.25-3 is defined as


 Re_{\rm min} = \frac{\rho_m U_{A_{\rm min}} D_h}{\mu_m} (7.25-4)


where      
  $\rho_m$ = mean gas density
  $\mu_m$ = mean gas viscosity
  $D_h$ = hydraulic diameter
  $U_{A_{\rm min}}$ = gas velocity at the minimum flow area

For a heat exchanger core, the hydraulic diameter can be defined as


 D_h = 4L \left( \frac{A_c}{A} \right) (7.25-5)

where $L$ is the flow length of the heat exchanger. If the tubes are normal to the primary fluid flow, then $L$ is the length in the primary fluid flow direction. Note that $U_{A_{\rm min}}$ can be calculated from


 U_{A_{\rm min}} = \frac{U} {\sigma} (7.25-6)

where $U$ is the gas velocity and $\sigma$ is the minimum flow to face area ratio.



Heat Transfer Effectiveness


For the simple-effectiveness-model, the heat-exchanger effectiveness, $\epsilon$, is defined as the ratio of actual rate of heat transfer from the hot to cold fluid to the maximum possible rate of heat transfer. The maximum possible heat transfer is given by


 q_{\rm max} = C_{\rm min} (T_{\rm in,hot} - T_{\rm in,cold}) (7.25-7)

where $T_{\rm in,hot}$ and $T_{\rm in,cold}$ are the inlet temperatures of the hot and cold fluids and


 C_{\rm min} = \min[(\dot{m} c_p)_{\rm hot}, (\dot{m} c_p)_{\rm cold}] (7.25-8)

Thus, the actual rate of heat transfer, $q$, is defined as


 q = \epsilon C_{\rm min} (T_{\rm in,hot} - T_{\rm in,cold}) (7.25-9)

The value of $\epsilon$ depends on the heat exchanger geometry and flow pattern (parallel flow, counter flow, cross flow, etc.). Though the effectiveness is defined for a complete heat exchanger, it can be applied to a small portion of the heat exchanger represented by a computational cell.

For the ntu-model, FLUENT calculates the effectiveness from the ratio of heat capacity and the number of transfer units using the relation


 \epsilon = 1 - \exp \left[-\frac{1}{C_r} N_{\rm tu}^{0.22} ({1-e^{-C_r N_{\rm tu}^{0.78} } } ) \right] (7.25-10)

where $C_r$ is the ratio of $C_{\rm min}$ to $C_{\rm max}$.

$N_{\rm tu}$ should be specified for various gas flow rates (for a single auxiliary fluid flow rate) as an input to the model. This $N_{\rm tu}$ for the heat exchanger is scaled for each macro in the ratio of their areas.

For each macro, the gas inlet temperature is calculated using the mass average of the incoming gas temperatures at the boundaries. This automatically takes into account any reverse flow of the gas at the boundaries.



Heat Rejection


Heat rejection is computed for each cell within a macro and added as a source term to the energy equation for the gas flow. Note that heat rejection from the auxiliary fluid to gas can be either positive or negative.

For the simple-effectiveness-model, the heat transfer for a given cell is computed from


 q_{\rm cell} = \epsilon (\dot{m} c_p)_{\rm g} (T_{\rm in,auxiliary fluid} - T_{\rm cell}) (7.25-11)


where      
  $ \epsilon$ = heat exchanger effectiveness
  $(\dot{m} c_p)_{\rm g}$ = gas capacity rate (flow rate $\times$ specific heat)
  $T_{\rm in,auxiliary fluid}$ = auxiliary fluid inlet temperature of macro containing the cell
  $T_{\rm cell}$ = cell temperature
       

For the simple-effectiveness-model, the heat rejection from a macro is calculated by summing the heat transfer of all the cells contained within the macro


 q_{\rm macro} \; \; = \; \; \sum_{\rm all \; cells \; in \; macro \;} q_{\rm cell} (7.25-12)

For the ntu-model, the heat transfer for a macro is calculated from


 q_{\rm macro} = \epsilon C_{\rm min} (T_{\rm in,auxiliary fluid}-T_{\rm in,gas}) (7.25-13)


where      
  $ \epsilon$ = macro effectiveness
  $T_{\rm in,auxiliary fluid}$ = macro auxiliary fluid inlet temperature
  $T_{\rm in,gas}$ = macro gas inlet temperature
       

For the ntu-model, the heat transfer for a given cell is computed from


 q_{\rm cell} = q_{\rm macro} \frac{V_{\rm cell}}{V_{\rm macro}} (7.25-14)

For both heat exchanger models, the total heat rejection from the heat exchanger core is computed as the sum of the heat rejection from all the macros:


 q_{\rm total} \; \; = \; \; \sum_{\rm all \; \rm macros} q_{\rm macro} (7.25-15)

The auxiliary fluid inlet temperature to each macro ( $T_{\rm in,auxiliary fluid}$ in Equations  7.25-11 and 7.25-13) is computed based on the energy balance of the auxiliary fluid flow. For a given macro,


 q_{\rm macro} = (\dot{m})_{\rm auxiliary fluid} (h_{\rm out} - h_{\rm in}) (7.25-16)

where $h_{\rm in}$ and $h_{\rm out}$ are the inlet and outlet enthalpies of the auxiliary fluid in the macro. The auxiliary fluid outlet temperature from the macro is calculated as


 T_{\rm out} = \left\{ \begin{array}{cl} \frac{h_{\rm out}}{... ...\ f(h_{\rm out}, p) & {\rm UDF \; method} \end{array} \right. (7.25-17)


where      
  $f$ = user-defined function
  $p$ = auxiliary fluid pressure
       

The values of $h_{_{\rm out}}$ and $T_{_{\rm out}}$ then become the inlet conditions to the next macro.

The first row of macros (Macros 0, 1, and 2 in Figure  7.25.1) are assumed to be where the auxiliary fluid enters the heat exchanger core. When the fixed total heat rejection from the heat exchanger core is specified, the inlet temperature to the first row of macros is iteratively computed, so that all of the equations are satisfied simultaneously. When a fixed auxiliary fluid inlet temperature is specified, the heat transfer for the first row of macros are used to calculate their exit enthalpy, which becomes the inlet condition for the next row macros. At the end of each pass, the outlet enthalpy of each macro (in the last row) is mass averaged to obtain the inlet condition for the next pass macros.



Heat Exchanger Group Connectivity


If the optional heat exchanger group is used, a single heat exchanger may be comprised of multiple fluid zones. In this case, the auxiliary fluid is assumed to flow through these zones in parallel. Thus, after taking into account any auxiliary stream effects, the auxiliary fluid inlet mass flow rate is automatically apportioned to each zone in the group as follows:


 \dot{m}_i = \left(\frac{\sum_k V_{i,k}}{\sum_i \sum_k V_{i,k}} \right)\dot{m} (7.25-18)

where $\dot{m}_i$ is the total auxiliary mass flow rate for the heat exchanger group. $V_{i,k}$ refers to the volume of the $k$th finite volume cell within the $i$th fluid zone. Within each zone, the auxiliary fluid flows through each macro in series as usual.

At the outlet end of the group, the parallel auxiliary fluid streams through the individual zones are recombined, and the outlet auxiliary fluid enthalpy is calculated on a mass-averaged basis:


 \bar{h} = \left(\frac{\sum_i{\dot{m}_i h_i}}{\sum_i{\dot{m}_i}} \right) (7.25-19)

With user-defined functions, the simple-effectiveness-model allows you to simulate two-phase auxiliary fluid flows and other complex auxiliary fluid enthalpy relationships of the form


 h = h(T,p,x) (7.25-20)

where $p$ is the absolute pressure and $x$ is the quality (mass fraction of vapor) of a two-phase vapor-liquid mixture. When pressure-dependent auxiliary fluid properties are used, the mean pressure within each macro is calculated and passed to the user-defined function as


 \bar{p}_j = p_{\rm in} + \left(j + \frac{1}{2} \right) \frac{\Delta p}{N} (7.25-21)


where      
  $j$ = macro row index
  $p_{\rm in}$ = inlet auxiliary fluid pressure
  $\Delta p$ = overall pressure drop across a heat exchanger group
  $N$ = number of rows per pass $\times$ number of passes.


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Up: 7.25 Heat Exchanger Models
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© Fluent Inc. 2006-09-20