## 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, , 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 model can be used to check heat capacity for both the gas and the auxiliary fluid and takes the lesser of the two for the calculation of heat transfer.

• The model can be used to model heat transfer to the gas from the auxiliary fluid and vice versa.

• The model can be used to model gas-side reverse flow.

• The model can be used with variable density of gas.

• The model can be used in either the serial or parallel FLUENT solvers.

The simple-effectiveness-model provides the following features:

• The model can be used to model heat transfer from the auxiliary fluid to the fluid.

• The auxiliary fluid properties can be a function of pressure and temperature, thus allowing phase change of the auxiliary fluid.

• The model can be used by serial as well as parallel solvers.

• The model can be used to make a network of heat exchangers using a heat exchanger group (Section  7.25.4).

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

 (7.25-1)

 where = streamwise pressure drop = streamwise pressure loss coefficient = mean gas density = gas velocity at the minimum flow area

The pressure loss coefficient is computed from

 (7.25-2)

 where = minimum flow to face area ratio = entrance loss coefficient = exit loss coefficient = gas-side surface area = minimum cross-sectional flow area = core friction factor = specific volume at the exit = specific volume at the inlet = mean specific volume

and 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

 (7.25-3)

 where = core friction coefficient = core friction exponent = Reynolds number for velocity at the minimum flow area

and 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

 (7.25-4)

 where = mean gas density = mean gas viscosity = hydraulic diameter = gas velocity at the minimum flow area

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

 (7.25-5)

where is the flow length of the heat exchanger. If the tubes are normal to the primary fluid flow, then is the length in the primary fluid flow direction. Note that can be calculated from

 (7.25-6)

where is the gas velocity and is the minimum flow to face area ratio.

Heat Transfer Effectiveness

For the simple-effectiveness-model, the heat-exchanger effectiveness, , 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

 (7.25-7)

where and are the inlet temperatures of the hot and cold fluids and

 (7.25-8)

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

 (7.25-9)

The value of 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

 (7.25-10)

where is the ratio of to .

should be specified for various gas flow rates (for a single auxiliary fluid flow rate) as an input to the model. This 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

 (7.25-11)

 where = heat exchanger effectiveness = gas capacity rate (flow rate specific heat) = auxiliary fluid inlet temperature of macro containing the 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

 (7.25-12)

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

 (7.25-13)

 where = macro effectiveness = macro auxiliary fluid inlet temperature = macro gas inlet temperature

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

 (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:

 (7.25-15)

The auxiliary fluid inlet temperature to each macro ( 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,

 (7.25-16)

where and are the inlet and outlet enthalpies of the auxiliary fluid in the macro. The auxiliary fluid outlet temperature from the macro is calculated as

 (7.25-17)

 where = user-defined function = auxiliary fluid pressure

The values of and 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:

 (7.25-18)

where is the total auxiliary mass flow rate for the heat exchanger group. refers to the volume of the th finite volume cell within the 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:

 (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

 (7.25-20)

where is the absolute pressure and 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

 (7.25-21)

 where = macro row index = inlet auxiliary fluid pressure = overall pressure drop across a heat exchanger group = number of rows per pass number of passes.

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