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6.1.2 Macro Heat Exchanger Model Theory

In ANSYS 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.

ANSYS FLUENT provides two heat transfer 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, ANSYS FLUENT calculates the effectiveness, $\epsilon$, from the NTU value that is calculated by ANSYS FLUENT from the heat transfer data provided by the user in tabular format. ANSYS 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 ANSYS 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 transfer models, pressure loss is modeled using the porous media model in ANSYS FLUENT. For the dual cell model (Section  6.2), pressure loss is used for both streams, while for the macro model, it is used only for the primary side.

The loss coefficients of the porous media model are computed using curve fitting of the pressure-versus-flow rate data outside of ANSYS FLUENT, which you will specify for the cell zone conditions. However, in some cases, the data for curve-fitting is not available. The macro model provides an additional means of getting the coefficients if the data is not available. The coefficients can also be automatically computed (and updated) using a known pressure loss coefficient as a function of some geometric parameters, the theory of which is defined below:


\Delta p = \frac{1}{2} f \rho_{m} U^2_{A_{\rm min}} (6.1-1)


where      
  $\Delta p$ = streamwise pressure drop
  $f$ = streamwise pressure loss coefficient
  $\rho_{m}$ = mean primary fluid density
  $U_{A_{\rm min}}$ = primary fluid 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} (6.1-2)


where      
  $\sigma$ = minimum flow to face area ratio
  $K_c$ = entrance loss coefficient
  $K_e$ = exit loss coefficient
  $A$ = primary fluid-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 [ 160], [ 158]. These parameters are used to set up large resistances in the two non-streamwise directions, effectively forcing the primary fluid flow through the core to be unidirectional.

In Equation  6.1-2, the core friction factor is defined as


f_c = a{\rm Re}^b_{\rm min} (6.1-3)


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 [ 160], [ 158].

The Reynolds number in Equation  6.1-3 is defined as


Re_{\rm min} = \frac{\rho_m U_{A_{\rm min}} D_h}{\mu_m} (6.1-4)


where      
  $\rho_m$ = mean primary fluid density
  $\mu_m$ = mean primary fluid viscosity
  $D_h$ = hydraulic diameter
  $U_{A_{\rm min}}$ = primary fluid 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) (6.1-5)

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} (6.1-6)

where $U$ is the primary fluid 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}) (6.1-7)

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}] (6.1-8)

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


q = \epsilon C_{\rm min} (T_{\rm in,hot} - T_{\rm in,cold}) (6.1-9)

The value of $\epsilon$ depends on the heat exchanger geometry and flow pattern (parallel flow, counter flow, cross flow, etc.). Even though the effectiveness of the primary fluid is computed using uniform conditions on the entire heat exchanger core, it is being applied to a small portion of the core represented by a computational cell. This can make it less accurate for some heat exchanger cores, where there is a strong variation in the primary flow. For a core with a strong primary flow variation, the NTU model must be used.

For the ntu-model, given the heat exchanger performance data (total heat rejection versus primary flow rate) based on uniform test conditions, ANSYS FLUENT calculates the effectiveness of the entire heat exchanger 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] (6.1-10)

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

The heat exchanger performance data should be specified for a number of auxiliary flow rates so that ANSYS FLUENT can compute the number of transfer units versus the primary fluid flow rate for a number of auxiliary fluid flow rates. This NTU, which is based on the full heat exchanger and uniform conditions, is scaled for each macro using the ratio of their volumes and minimum heat capacities.

For each macro, the primary fluid inlet temperature is calculated using the mass average of the incoming primary fluid temperatures at the boundaries. This automatically takes into account any reverse flow of the primary fluid 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 primary fluid flow. Note that heat rejection from the auxiliary fluid to primary fluid 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} - T_{\rm cell}) (6.1-11)


where      
  $ \epsilon$ = heat exchanger effectiveness
  $(\dot{m} c_p)_{\rm g}$ = primary fluid capacity rate (flow rate $\times$ specific heat)
  $T_{\rm in,auxiliary}$ = 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} (6.1-12)

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


q_{\rm macro} = \epsilon C_{\rm min} (T_{\rm in,auxiliary}-T_{\rm in,primary}) (6.1-13)


where      
  $ \epsilon$ = macro effectiveness
  $T_{\rm in,auxiliary}$ = macro auxiliary fluid inlet temperature
  $T_{\rm in,primary}$ = macro primary fluid 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}} (6.1-14)

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} (6.1-15)

The auxiliary fluid inlet temperature to each macro ( $T_{\rm in,auxiliary}$ in Equations  6.1-11 and 6.1-13) is computed based on the energy balance of the auxiliary fluid at a previous macro computation. For a given macro,


q_{\rm macro} = (\dot{m})_{\rm auxiliary} (h_{\rm out} - h_{\rm in}) (6.1-16)

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. (6.1-17)


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  6.1.1) 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.



Macro Heat Exchanger Group Connectivity

If the optional macro heat exchanger group is used, a single heat exchanger may be consist 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} (6.1-18)

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) (6.1-19)

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) (6.1-20)

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} (6.1-21)


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.

To learn how to use the macro heat exchanger models, refer to this section and this section in the separate User's Guide.

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Up: 6.1 The Macro Heat
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