Heat Exchanger (G-TL)
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Heat Exchangers
Description
The Heat Exchanger (G-TL) block models the complementary cooling and heating of fluids held briefly in thermal contact across a thin conductive wall. The wall can store heat in its bounds, adding to the heat transfer a slight transient delay that scales in proportion to its thermal mass. The fluids are single phase—pure gas on one side, pure liquid on the other. Neither fluid can switch phase and so, as latent heat is never released, the exchange is strictly one of sensible heat.
Sensible heat exchangers abound in machinery. The fuel heaters that in some jets keep ice from precipitating in fuel lines and from choking fuel strainers work by blasting bleed air still hot from the compressor over the fuel lines. The oil coolers that in some motorcycles keep lubricating oil from overheating work likewise by rushing ram air at ambient temperature over the oil lines. The bleed and ram air are gas flows and the fuel and oil are thermal liquid flows.
Block Variants
The heat transfer model depends on the choice of block variant. The block has two
variants: E-NTU Model
and Simple
Model
. Use the Modeling option
parameter to change the variant.
E-NTU Model
The default variant. Its heat transfer model derives from the Effectiveness-NTU method. Heat transfer in the steady state then proceeds at a fraction of the ideal rate which the flows, if kept each at its inlet temperature, and if cleared of every thermal resistance in between, could in theory support:
where QAct the actual heat transfer rate, QMax is the ideal heat transfer rate, and ε is the fraction of the ideal rate actually observed in a real heat exchanger encumbered with losses. The fraction is the heat exchanger effectiveness, and it is a function of the number of transfer units, or NTU, a measure of the ease with which heat moves between flows, relative to the ease with which the flows absorb that heat:
where the fraction is the overall thermal conductance between the flows and CMin is the smallest of the heat capacity rates from among the flows—that belonging to the flow least capable of absorbing heat. The heat capacity rate of a flow depends on the specific heat of the fluid (cp) and on its mass flow rate through the exchanger ():
The effectiveness depends also on the relative disposition of the flows, the number of passes between them, and the mixing condition for each. This dependence reflects in the effectiveness expression used, with different flow arrangements corresponding to different expressions. For a list of the effectiveness expressions, see the E-NTU Heat Transfer block.
Use the Flow arrangement block parameter to set how the flows meet in the heat exchanger. The flows can run parallel to each other, counter to each other, or across each other. They can also run in a pressurized shell, one through tubes enclosed in the shell, the other around those same tubes. The figure shows an example. The tube flow can make one pass through the shell flow (shown right) or, for greater exchanger effectiveness, multiple passes (left).
Other flow arrangements are possible through a generic parameterization based on tabulated effectiveness data and requiring little detail about the heat exchanger. Flow arrangement, mixing condition, and number of shell or tube passes, if relevant to the heat exchanger, are assumed to manifest in the tabulated data.
Use the Cross flow arrangement parameter to mix each of the flows, one of the flows, or none of the flows. Mixing in this context is the lateral movement of fluid in channels that have no internal barriers, normally guides, baffles, fins, or walls. Such movement serves to even out temperature variations in the transverse plane. Mixed flows have variable temperature in the longitudinal plane alone. Unmixed flows have variable temperature in both the transverse and longitudinal planes. The figure shows a mixed flow (i) and an unmixed flow (ii).
The distinction between mixed and unmixed flows is considered only in cross flow arrangements. There, longitudinal temperature variation in one fluid produces transverse temperature variation in the second fluid that mixing can even out. In counter and parallel flow arrangements, longitudinal temperature variation in one fluid produces longitudinal temperature variation in the second fluid and mixing, as it is of little effect here, is ignored.
Shell-and-tube exchangers with multiple passes (iv.b-e in the figure for 2, 3, and 4 passes) are most effective. Of exchangers with a single pass, those with counter flows (ii are most effective and those with parallel flows (i) are least.
Cross-flow exchangers are intermediate in effectiveness, with mixing condition playing a factor. They are most effective when both flows are unmixed (iii.a) and least effective when both flows are mixed (iii.b). Mixing just the flow with the smallest heat capacity rate (iii.c) lowers the effectiveness more than mixing just the flow with the largest heat capacity rate (iii.d).
The overall thermal resistance, R, is the sum of the local resistances lining the heat transfer path. The local resistances arise from convection at the surfaces of the wall, conduction through the wall, and, if the wall sides are fouled, conduction through the layers of fouling. Expressed in order from gas side to thermal liquid side:
where U is the convective heat transfer coefficient, F is the fouling factor, and ATh is the heat transfer surface area, each for the flow indicated in the subscript. RW is the thermal resistance of the wall.
The wall thermal resistance and fouling factors are simple constants obtained
from block parameters. The heat transfer coefficients are elaborate functions of
fluid properties, flow geometry, and wall friction, and derive from standard
empirical correlations between Reynolds, Nusselt, and Prandtl numbers. The
correlations depend on flow arrangement and mixing condition, and are detailed
for each in the E-NTU Heat Transfer block on which the E-NTU
Model
variant is based.
The wall is more than a thermal resistance for heat to pass through. It is also a thermal mass and, like the flows it divides, it can store heat in its bounds. The storage slows the transition between steady states so that a thermal perturbation on one side does not promptly manifest on the side across. The lag persists for the short time that it takes the heat flow rates from the two sides to balance each other. That time interval scales with the thermal mass of the wall:
where is the cp,W is the specific
heat capacity and MW the inertial mass
of the wall. Their product gives the energy required to raise wall temperature
by one degree. Use the Wall thermal mass block parameter to
specify that product. The parameter is active when the Wall thermal
dynamics setting is On
.
Thermal mass is often negligible in low-pressure systems. Low pressure affords a thin wall with a transient response so fast that, on the time scale of the heat transfer, it is virtually instantaneous. The same is not true of high-pressure systems, common in the production of ammonia by the Haber process, where pressure can break 200 atmospheres. To withstand the high pressure, the wall is often thicker, and, as its thermal mass is heftier, so its transient response is slower.
Set the Wall thermal dynamics parameter to
Off
to ignore the transient lag, cut the
differential variables that produce it, and, in reducing calculations, speed up
the rate of simulation. Leave it On
to capture the
transient lag where it has a measurable effect. Experiment with the setting if
necessary to determine whether to account for thermal mass. If simulation
results differ to a considerable degree, and if simulation speed is not a
factor, keep the setting On
.
The wall, if modeled with thermal mass, is considered in halves. One half sits on the gas side and the other half sits on the thermal liquid side. The gas side is denoted side 1 and the thermal liquid side is denoted side 2. This notation is used in the calculations for heat transfer. The thermal mass divides evenly between the pair:
Energy is conserved in the wall. In the simple case of a wall half at steady state, heat gained from the fluid equals heat lost to the second half. The heat flows at the rate predicted by the E-NTU method for a wall without thermal mass. The rate is positive for heat flows directed from side 1 of the heat exchanger to side 2:
In the transient state, the wall is in the course of storing or losing heat, and heat gained by one half no longer equals that lost to the second half. The difference in the heat flow rates varies over time in proportion to the rate at which the wall stores or loses heat. For side 1 of the heat exchanger:
where is the rate of change in temperature in the wall half. Its product with the thermal mass of the wall half gives the rate at which heat accumulates there. That rate is positive when temperature rises and negative when it drops. The closer the rate is to zero the closer the wall is to steady state. For side 2 of the heat exchanger:
The E-NTU Model
variant is a composite component
built from simpler blocks. A Heat Exchanger Interface
(G) block models the gas flow, a Heat Exchanger Interface
(TL) block models the thermal liquid flow, and an
E-NTU Heat Transfer block
models the heat exchanged across the wall between the flows. The figure shows
the block connections for the E-NTU Model
block
variant.
Simple Model
The alternative variant. Its heat transfer model depends on the concept of specific dissipation, a measure of the heat transfer rate observed when gas and thermal liquid inlet temperatures differ by one degree. Its product with the inlet temperature difference gives the expected heat transfer rate:
where ξ is specific dissipation and
TIn is inlet temperature for gas
(subscript G
) or thermal liquid (subscript
TL
). The specific dissipation is a tabulated function of the mass
flow rates into the exchanger through the gas and thermal liquid inlets:
To accommodate reverse flows, the tabulated data can extend over positive and negative flow rates, in which case the inlets can also be thought of as outlets. The data normally derives from measurement of heat transfer rate against temperature in a real prototype:
The heat transfer model, as it relies almost entirely on tabulated data, and as that data normally derives from experiment, requires little detail about the exchanger. Flow arrangement, mixing condition, and number of shell or tube passes, if relevant to the heat exchanger modeled, are assumed to manifest entirely in the tabulated data.
See the Specific Dissipation Heat Transfer block for more detail on the heat transfer calculations.
The Simple Model
variant is a composite component.
A Simple Heat
Exchanger Interface (G) block models the gas flow and a
Simple Heat
Exchanger Interface (TL) block models the thermal liquid
flow. Mass, momentum, and energy conservation in the flow channels derive from
the corresponding interface blocks. A Specific
Dissipation Heat Transfer block captures the heat exchanged
across the wall between the flows.
Examples
Ports
Conserving
Parameters
Extended Capabilities
Version History
Introduced in R2019a