Detailed heat transfer model between two general fluids
Fluid Network Interfaces/Heat Exchangers/Fundamental Components
The E-NTU Heat Transfer block models the heat exchange between two general fluids based on the standard Effectiveness-NTU method. The fluid thermal properties are specified explicitly through Simscape™ physical signals. Combine with the Heat Exchanger Interface (TL) block to model the pressure drop and temperature change between the inlet and outlet of a heat exchanger.
The block dialog box provides a choice of common heat exchanger configurations. These include concentric-pipe with parallel and counter flows, shell-and-tube with one or more shell passes, and cross-flow with mixed and unmixed flows. A generic configuration lets you model other heat exchangers based on tabular effectiveness data.
Heat Exchanger Configurations
The E-NTU model defines the heat transfer rate between fluids 1 and 2 in terms of an effectiveness parameter ε:
$$\begin{array}{cc}{Q}_{1}=-{Q}_{2}=\u03f5{Q}_{Max},& 0<\epsilon <1\end{array},$$
where:
Q_{1} and Q_{2} are the heat transfer rates into fluid 1 and fluid 2.
Q_{Max} is the maximum possible heat transfer rate between fluid 1 and fluid 2 at a given set of operating conditions.
ε is the effectiveness parameter.
The maximum possible heat transfer rate between the two fluids is
$${Q}_{Max}={C}_{Min}\left({T}_{1,In}-{T}_{2,In}\right),$$
where:
C_{Min} is the minimum value of the thermal capacity rate:
$${C}_{Min}=min\left({\dot{m}}_{1}{c}_{p,1},{\dot{m}}_{2}{c}_{p,2}\right)$$
T_{1,In} and T_{2,In} are the inlet temperatures of fluid 1 and fluid 2.
$${\dot{m}}_{1}$$ and $${\dot{m}}_{2}$$ are the mass flow rates of fluid 1 and fluid 2 into the heat exchanger volume through the inlet.
c_{p,1} and c_{p,2} are the specific heat coefficients at constant pressure of fluid 1 and fluid 2. The Minimum fluid-wall heat transfer coefficient parameter in the block dialog box sets a lower bound on the allowed values of the heat transfer coefficients.
The heat exchanger effectiveness calculations depend on the flow arrangement type selected in
the block dialog box. For all but Generic — effectiveness
table
, the block computes the thermal exchange effectiveness
through analytical expressions written in terms of the number of transfer units
(NTU) and thermal capacity ratio. The number of transfer units is defined as
$$NTU=\frac{{U}_{Overall}{A}_{Heat}}{{C}_{Min}}=\frac{1}{{C}_{Min}{R}_{Overall}},$$
where:
NTU is the number of transfer units.
U_{Overall} is the overall heat transfer coefficient between fluid 1 and fluid 2.
R_{Overall} is the overall thermal resistance between fluid 1 and fluid 2.
A_{Heat} is aggregate area of the primary and secondary, or finned, heat transfer surfaces.
The thermal capacity ratio is defined as
$${C}_{rel}=\frac{{C}_{Min}}{{C}_{Max}}$$
where:
C_{rel} is the thermal capacity ratio.
The overall heat transfer coefficient and thermal resistance used in the NTU calculation are functions of the heat transfer mechanisms at work. These mechanisms include convective heat transfer between the fluids and the heat exchanger interface and conduction through the interface wall [2]:
$${R}_{Overall}=\frac{1}{{U}_{Overall}{A}_{Heat}}=\frac{1}{{h}_{1}{A}_{Heat,1}}+{R}_{Foul,1}+{R}_{Wall}+{R}_{Foul,2}+\frac{1}{{h}_{2}{A}_{Heat,2}},$$
where:
h_{1} and h_{2} are the heat transfer coefficients between fluid 1 and the interface wall and between fluid 2 and the interface wall.
A_{Heat,1} and A_{Heat,2} are the heat transfer surface areas on the fluid-1 and fluid-2 sides.
R_{Foul,1} and R_{Foul,2} are the fouling resistances on the fluid-1 and fluid-2 sides.
R_{Wall} is the interface wall thermal resistance.
Heat Transfer From Fluid 1 to Fluid 2
The tables show some of the analytical expressions used to compute the heat exchange effectiveness [1]. The parameter N refers to the number of shell passes and the parameter ε_{1} to the effectiveness for a single shell pass.
Concentric Tubes | |
Counter Flow |
$$\epsilon =\{\begin{array}{cc}\frac{1-\mathrm{exp}\left[-NTU\left(1-{C}_{rel}\right)\right]}{1-{C}_{rel}\mathrm{exp}\left[-NTU\left(1-{C}_{rel}\right)\right]},& \text{if}{C}_{rel}1\\ \frac{NTU}{1+NTU},& \text{if}{C}_{rel}=1\end{array}$$ |
Parallel Flow |
$$\epsilon =\frac{1-\mathrm{exp}\left[-NTU\left(1+{C}_{rel}\right)\right]}{1+{C}_{rel}}$$ |
Shell and Tube | |
One shell pass and two, four, or six tube passes |
$${\epsilon}_{1}=\frac{2}{1+{C}_{rel}+\sqrt{1+{C}_{rel}{}^{2}}\frac{1+\mathrm{exp}\left(-NTU\sqrt{1+{C}_{rel}{}^{2}}\right)}{1-\mathrm{exp}\left(-NTU\sqrt{1+{C}_{rel}{}^{2}}\right)}}$$ |
N Shell Passes and 2N, 4N, or 6N Tube Passes |
$$\epsilon =\frac{{\left[\left(1-{\epsilon}_{1}{C}_{rel}\right)/\left(1-{\epsilon}_{1}\right)\right]}^{N}-1}{{\left[\left(1-{\epsilon}_{1}{C}_{rel}\right)/\left(1-{\epsilon}_{1}\right)\right]}^{N}-{C}_{rel}}$$ |
Cross Flow (Single Pass) | |
Both Fluids Unmixed |
$$\epsilon =1-\mathrm{exp}\left(\frac{\mathrm{exp}\left(-{C}_{rel}NT{U}^{0.78}\right)-1}{{C}_{rel}NT{U}^{-0.22}}\right)$$ |
Both Fluids Mixed |
$$\epsilon =\frac{1}{\frac{1}{1-exp\left(-NTU\right)}+\frac{{C}_{rel}}{1-\mathrm{exp}\left(-{C}_{rel}NTU\right)}-\frac{1}{NTU}}$$ |
C_{Max} mixed, C_{Min} unmixed |
$$\epsilon =\frac{1}{{C}_{rel}}\left(1-\mathrm{exp}\left(-{C}_{rel}\left(1-\mathrm{exp}\left(-NTU\right)\right)\right)\right)$$ |
C_{Max} unmixed, C_{Min} mixed |
$$\epsilon =1-\mathrm{exp}\left(-\frac{1}{{C}_{rel}}\left(1-\mathrm{exp}\left(-{C}_{rel}NTU\right)\right)\right)$$ |
The flows are single-phase. The heat transfer is strictly one of sensible heat. The transfer is limited to interior of the exchanger, with the environment neither gaining heat from nor providing heat to the flows—the heat exchanger is an adiabatic component.
Heat exchanger geometry. Common geometries that you can select
include Parallel or counter flow
, Shell
and tube
, and Cross flow
.
Select Generic — effectiveness table
to
model other heat exchanger geometries based on tabular effectiveness
data.
In the Parallel or counter flow
configuration,
the relative flow directions of fluids 1 and 2 determine whether the
heat exchanger is based on parallel or counter flows. The flow directions
depend on the remainder of the Simscape
Fluids™ model.
Number of times the flow traverses the shell before exiting.
This parameter is visible only when the Flow arrangement
parameter is set to Shell and tube
. The default value
is 1
, corresponding to a single shell pass.
Fluid mixing configuration. The fluids can be mixed or unmixed.
The block uses the mixing configuration to determine which empirical
heat transfer correlations to use. This parameter is visible only
when the Flow arrangement parameter is set to Cross
flow
. The default setting is Both fluids
mixed
.
M-element vector of NTU values at which to specify the effectiveness tabular data. The number of transfer units (NTU) is a dimensionless parameter defined as
$$NTU=\frac{{A}_{s}U}{{C}_{min}},$$
where:
A_{S} is the heat transfer surface area.
U is the overall heat transfer coefficient.
C_{min} is the smallest of the thermal capacity rates for the hot and cold fluids.
This parameter is visible only when the Flow Arrangement parameter
is set to Generic — effectiveness table
.
The default vector is [0.5, 1.0, 2.0, 3.0, 4.0]
.
N-element vector of thermal capacity ratios at which to specify the effectiveness tabular data. The thermal capacity ratio is the fraction
$${C}_{r}=\frac{{C}_{min}}{{C}_{max}},$$
where C_{min} and C_{max} are
the minimum and maximum thermal capacity rates. This parameter is
visible only when the Flow arrangement parameter
is set to Generic — effectiveness table
.
The default vector is [0.0, 0.25, 0.5, 0.75, 1.0]
.
M-by-N matrix with the heat exchanger effectiveness values. The matrix rows correspond to the different values specified in the Number of heat transfer units vector, NTU parameter. The matrix columns correspond to the values specified in the Thermal capacity ratio vector, CR parameter.
This parameter is visible only when the Flow arrangement parameter is set
to Generic — effectiveness table
. The default
table is a 6-by-5 matrix ranging in value from 0.30
to
0.99
.
Thermal resistance of the interface wall separating the two
heat exchanger fluids. The block uses this parameter to compute the
rate of heat transfer between the fluids. The default value is 1.6e-4
k/W.
Aggregate surface area for heat transfer between the cold and
hot fluids. The default value is 0.01
m^2.
Empirical parameter used to quantify the increased thermal resistance
due to dirt deposits on the heat transfer surface. The default value
is 1e-4
m^2*K/W.
Smallest allowed value of the heat transfer coefficient. The
heat transfer coefficients specified through physical signal ports
HC1 and HC2 saturate at this value. The default value is 5
W/(m^2*K).
The block uses the heat transfer coefficient to calculate the heat transfer rate between fluids 1 and 2 as described in Heat Transfer Rate.
H1 — Thermal conserving port associated with the inlet temperature of fluid 1
H2 — Thermal conserving port associated with the inlet temperature of fluid 2
C1 — Physical signal input port for the thermal capacity rate of fluid 1
C2 — Physical signal input port for the thermal capacity rate of fluid 2
HC1 — Physical signal input port for the heat transfer coefficient between fluid 1 and the interface wall
HC2 — Physical signal input port for the heat transfer coefficient between fluid 2 and the interface wall
[1] Holman, J. P. Heat Transfer. 9th ed. New York, NY: McGraw Hill, 2002.
[2] Shah, R. K. and D. P. Sekulic. Fundamentals of Heat Exchanger Design. Hoboken, NJ: John Wiley & Sons, 2003.