E-NTU Heat Transfer

Detailed heat transfer model between two general fluids

Library

Fluid Network Interfaces/Heat Exchangers/Fundamental Components

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Description

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

Heat Transfer Rate

The E-NTU model defines the heat transfer rate between fluids 1 and 2 in terms of an effectiveness parameter ε:

$$\begin{array}{cc}{Q}_1=-Q_2=ϵQ_{Max},& 0<\epsilon <1\end{array},$$

where:

• Q₁ and Q₂ 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.

Heat Exchanger Effectiveness

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:

• h₁ and h₂ 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 ε₁ to the effectiveness for a single shell pass.

Concentric Tubes
Counter Flow	$$\epsilon =\{\begin{array}{cc}\frac{1-\exp \left[-NTU\left(1-C_{rel}\right)\right]}{1-C_{rel}\exp \left[-NTU\left(1-C_{rel}\right)\right]},& \text{if }{C}_{rel}<1\\ \frac{NTU}{1+NTU},& \text{if }{C}_{rel}=1\end{array}$$
Parallel Flow	$$\epsilon =\frac{1-\exp \left[-NTU\left(1+C_{rel}\right)\right]}{1+C_{rel}}$$
Shell and Tube
One shell pass and two, four, or six tube passes	$${\epsilon }_1=\frac{2}{1+C_{rel}+\sqrt{1+C_{rel}{}^2}\frac{1+\exp \left(-NTU\sqrt{1+C_{rel}{}^2}\right)}{1-\exp \left(-NTU\sqrt{1+C_{rel}{}^2}\right)}}$$
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	$$\epsilon =1-\exp \left(\frac{\exp \left(-C_{rel}NT{U}^{0.78}\right)-1}{C_{rel}NT{U}^{-0.22}}\right)$$
Both Fluids Mixed	$$\epsilon =\frac{1}{\frac{1}{1-exp\left(-NTU\right)}+\frac{C_{rel}}{1-\exp \left(-C_{rel}NTU\right)}-\frac{1}{NTU}}$$
CMax mixed, CMin unmixed	$$\epsilon =\frac{1}{C_{rel}}\left(1-\exp \left(-C_{rel}\left(1-\exp \left(-NTU\right)\right)\right)\right)$$
CMax unmixed, CMin mixed	$$\epsilon =1-\exp \left(-\frac{1}{C_{rel}}\left(1-\exp \left(-C_{rel}NTU\right)\right)\right)$$

Assumptions and Limitations

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.

Parameters

Common Tab

Flow arrangement

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 shell passes

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.

Cross flow type

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.

Number of heat transfer units vector, NTU

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].

Thermal capacity ratio vector, CR

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].

Effectiveness table, E(NTU, CR)

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.

Wall thermal resistance

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.

Controlled Fluid 1 Tab | Controlled Fluid 2 Tab

Heat transfer surface area

Aggregate surface area for heat transfer between the cold and hot fluids. The default value is 0.01 m^2.

Fouling factor

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.

Minimum fluid-wall heat transfer coefficient

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.

Ports

• 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

References

[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.

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.