Heat exchanger interface (P)#

This block allows for modeling heat exchange in a pneumatic network: pressure drop and temperature change of the medium as it passes through the heat exchanger. When combined with the NTU heat transfer block, it allows for modeling heat exchangers calculated using the NTU method.

Equation of conservation of mass

Compressible medium (1) :

(1)#\[\dot m_{вх} +\dot m_{вых}=(\frac{dp}{dt} \cdot \frac{1}{\beta}- \frac{dT}{dt}\cdot \alpha)\cdot \rho \cdot V\]

where:

p – pressure of the medium in the internal node, [Pa];

T – ambient temperature in the internal unit, [℃];

V – volume of the medium, [m\(^3\)];

\(\rho\) – density of the medium, [kg/m\(^3\)];

\(\beta\) – bulk modulus of elasticity, [Pa];

\(\alpha\) – coefficient of isobaric thermal expansion, [1/℃],

\(\dot m_{in},\dot m_{out}\) – mass flow rate at the inlet and outlet, [kg/s].

Equation of conservation of energy

Compressible medium (2) :

(2)#\[\phi_{вх}+\phi_{вых}-Q=((\frac{h}{\beta}-\frac{T \cdot \alpha}{\rho})\cdot \frac{dp}{dt}+(C_р - h\cdot \alpha) \cdot \frac{dT}{dt})\cdot \rho \cdot V\]

where:

h – enthalpy of the medium in the internal node, [J/kg];

\(\phi_{in},\phi_{out}\) – energy flow at the input and output, [W];

Q – heat flow, [W].

Equations of conservation of momentum

Compressible medium (3) :

(3)#\[\begin{split}\begin{cases}p_{вх}-p=\Delta p_{вх} \\p_{вых}-p=\Delta p_{вых} \end{cases}\end{split}\]

where:

\(p_{in},p_{out}\) – pressure at the inlet and outlet of the heat exchanger, [Pa];

\(\Delta p_{in}=p_{in}-p\) – pressure loss between the inlet and the internal unit, [Pa];

\(\Delta p_{out}=p_{out}-p\) – pressure loss between the outlet and the internal unit, [Pa];

p – pressure in the internal node, [Pa].

Calculation of pressure losses in a heat exchanger

The pressure loss between the block ports and the internal unit is calculated using the following formulas:

If the pressure loss calculation model selected is Constant pressure loss coefficient (4) :

(4)#\[\begin{split}p_{вх}^{ср}-p_{вн}^{ср}=\begin{cases}\dot m_{вх} \cdot \mu_{вх}\cdot K_{Потерь} \cdot Re_L \cdot \frac{1}{4\cdot d_г \rho_{вх}\cdot S_{min}}, Re_{вх}\leq Re_L\\K_{Потерь}\cdot \frac{\dot m_{вх}\cdot|\dot m_{вх}|}{4\cdot \rho_{вх}\cdot S_{min}^2}, Re_{вх}\geq Re_T\end{cases}\\ p_{вых}^{ср}-p_{вн}^{ср}=\begin{cases}\dot m_{вх} \cdot \mu_{вых}\cdot K_{Потерь} \cdot Re_L \cdot \frac{1}{4\cdot d_г \rho_{вых}\cdot S_{min}}, Re_{вых}\leq Re_L\\K_{Потерь}\cdot \frac{\dot m_{вых}\cdot|\dot m_{вых}|}{4\cdot \rho_{вых}\cdot S_{min}^2}, Re_{вых}\geq Re_T\end{cases}\end{split}\]

where:

\(p_{in}^{cp},p_{out}^{cp},p_{in}^{cp}\) – pressure of the medium at the inlet, outlet and in the internal unit, respectively, [Pa];

\(\dot m_{in},\dot m_{out}\) – mass flow rate of the medium at the inlet and outlet of the heat exchanger, [kg/s];

\(K_{Losses}\) – pressure loss coefficient;

\( ho_{in}, ho_{out}\) – density of the medium at the inlet and outlet, [kg/m\(^3\)];

\(S_{min}\) – minimum cross-sectional area of the flow section, [m\(^2\)];

\(Re_L\) – upper limit of the Reynolds number for laminar flow;

\(Re_T\) – lower limit of the Reynolds number for turbulent flow;

\(\mu_{in},\mu_{out}\) – dynamic viscosity of the medium at the inlet and outlet, [Pa\(\cdot\)s].

If the pressure loss calculation model selected is Correlation for flow inside tubes (5) :

(5)#\[\begin{split}p_{вх}^{ср}-p_{вн}^{ср}=\begin{cases}\dot m_{вх} \cdot \mu_{вх}\cdot \lambda \cdot \frac{L+L_{доп}}{4\cdot d_г^2 \rho_{вх}\cdot S_{min}}, Re_{вх}\leq Re_L\\f_{вх}\cdot \frac{(L+L_{доп})\cdot\dot m_{вх}\cdot|\dot m_{вх}|}{4\cdot d_г \cdot \rho_{вх}\cdot S_{min}^2}, Re_{вх}\geq Re_T\end{cases}\\ p_{вых}^{ср}-p_{вн}^{ср}=\begin{cases}\dot m_{вых} \cdot \mu_{вых}\cdot \lambda \cdot \frac{L+L_{доп}}{4\cdot d_г^2 \rho_{вых}\cdot S_{min}}, Re_{вых}\leq Re_L\\f_{вых}\cdot \frac{(L+L_{доп})\cdot\dot m_{вых}\cdot|\dot m_{вых}|}{4\cdot d_г \cdot \rho_{вых}\cdot S_{min}^2}, Re_{вых}\geq Re_T\end{cases}\end{split}\]

where:

\(L\) – length of the flow path from the inlet to the outlet of the heat exchanger, [m];

\(L_{add}\) – total equivalent length to take into account additional pressure losses in local resistances, [m];

\(\lambda\) – form factor for calculating the Darcy friction coefficient in the laminar flow regime;

\(f_{in},f_{out}\) – Darcy friction coefficient at the input and output.

The Darcy friction coefficients at the inlet and outlet are calculated using the following formulas (6) :

(6)#\[\begin{split} f_{вх}=\frac{1}{(-1.8 \cdot log_{10} \cdot(\frac{6.9}{Re_{вх}}+(\frac{r}{3.7 \cdot d_г})^{1.11}))^2}\\ f_{вых}=\frac{1}{(-1.8 \cdot log_{10} \cdot(\frac{6.9}{Re_{вых}}+(\frac{r}{3.7 \cdot d_г})^{1.11}))^2} \end{split}\]

where \(r\) – is the absolute roughness of the inner surface, [m].

If the pressure loss calculation model is selected – Table - Darcy friction factor vs. Reynolds number (7) :

(7)#\[\begin{split}p_{вх}^{ср}-p_{вн}^{ср}=\begin{cases}\dot m_{вх} \cdot \mu_{вх}\cdot \lambda \cdot \frac{L}{4\cdot d_г^2 \rho_{вх}\cdot S_{min}}, Re_{вх}\leq Re_L\\f_{вх} (Re_{вх})\cdot \frac{L}{4\cdot d_г} \cdot\frac{\dot m_{вх}\cdot|\dot m_{вх}|}{\rho_{вх}\cdot S_{min}^2}, Re_{вх}\geq Re_T\end{cases}\\ p_{вых}^{ср}-p_{вн}^{ср}=\begin{cases}\dot m_{вых} \cdot \mu_{вых}\cdot \lambda \cdot \frac{L}{4\cdot d_г^2 \rho_{вых}\cdot S_{min}}, Re_{вых}\leq Re_L\\f_{вых}(Re_{вых})\cdot \frac{L}{4\cdot d_г} \cdot\frac{\dot m_{вых}\cdot|\dot m_{вых}|}{\rho_{вых}\cdot S_{min}^2}, Re_{вых}\geq Re_T\end{cases}\end{split}\]

where \(f_{in}(Re_{in}),f_{out}(Re_{out})\) are the Darcy friction coefficients at the inlet and outlet, calculated on the basis of the tabular data provided by the user, depending on the Reynolds number.

If the pressure loss calculation model is selected – Table - Euler number vs. Reynolds number (8) :

(8)#\[\begin{split}p_{вх}^{ср}-p_{вн}^{ср}=\begin{cases}\dot m_{вх} \cdot \mu_{вх}\cdot Eu(Re_L) \cdot Re_L \cdot \frac{L}{4\cdot d_г \rho_{вх}\cdot S_{min}}, Re_{вх}\leq Re_L\\Eu(Re_{вх})\cdot\frac{\dot m_{вх}\cdot|\dot m_{вх}|}{4\cdot \rho_{вх}\cdot S_{min}^2}, Re_{вх}\geq Re_T\end{cases}\\ p_{вых}^{ср}-p_{вн}^{ср}=\begin{cases}\dot m_{вых} \cdot \mu_{вых}\cdot Eu(Re_L) \cdot Re_L \cdot \frac{L}{4\cdot d_г \rho_{вых}\cdot S_{min}}, Re_{вых}\leq Re_L\\Eu(Re_{вых})\cdot\frac{\dot m_{вых}\cdot|\dot m_{вых}|}{4\cdot \rho_{вых}\cdot S_{min}^2}, Re_{вых}\geq Re_T\end{cases}\end{split}\]

where \(Eu(Re_{in}),Eu(Re_{out})\) are the Euler numbers at the input and output, calculated on the basis of the tabular data provided by the user, depending on the Reynolds number.

Calculation of the heat transfer coefficient

The heat transfer coefficient is calculated using the following formulas:

If the heat transfer coefficient calculation model selected is Constant heat transfer coefficient (9) :

(9)#\[h_{ср-ст}=h_{const}\]

where:

\(h_{ср-ст}\) – heat transfer coefficient between the environment and the wall, [W/(m\(^2 \cdot \)℃)];

\(h_{const}\) – heat transfer coefficient from the environment to the wall, specified by the user in the block property, [W/(m\(^2 \cdot \)℃)].

For all other calculation models, the heat transfer coefficient is calculated as the arithmetic mean between the coefficients for the inlet and outlet ports (10) :

(10)#\[h_{ср-ст}=\frac{h_{вх}-h_{вых}}{2}\]

where (11):

(11)#\[\begin{split}h_{вх}=\frac{Nu_{вх}\cdot \lambda_{вх}}{d_{г,т/о}}\\h_{вых}=\frac{Nu_{вых}\cdot \lambda_{вых}}{d_{г,т/о}}\end{split}\]

where:

\(Nu_{in},Nu_{out}\) – Nusselt number for the input and output ports;

\(\lambda_{in},\lambda_{out}\) – thermal conductivity coefficients for the input and output ports, [W/(m\(\cdot\)℃)];

\(d_{g,t/o}\) – hydraulic diameter for heat exchange calculations, [m], (12):

(12)#\[d_{г,т/о}=\frac{4\cdot S_{min} \cdot L_{т/о}}{S_{т/о}}\]

where:

\(L_{h/e}\) – characteristic length for heat exchange calculation, [m];

\(S_{h/e}\) – heat exchange surface area, [m\(^2\)].

Calculation of the Nusselt criterion

If the heat transfer coefficient calculation model selected is Heat transfer coefficient for flow in a pipe (13):[1]

(13)#\[\begin{split}Nu_{вх} =\begin{cases}Nu_L,Re_{вх}\leq Re_L \\\frac{(\frac{f_{вх}}{8})\cdot(Re_{вх}-1000)\cdot Pr_{вх}}{1+12.7 \cdot (\frac{f_{вх}}{8})^{1/2} \cdot (Pr_{вх}^{2/3}-1)}, Re_{вх}\geq Re_T\end{cases}\\ Nu_{вых} =\begin{cases}Nu_L,Re_{вых}\leq Re_L \\\frac{(\frac{f_{вых}}{8})\cdot(Re_{вых}-1000)\cdot Pr_{вых}}{1+12.7 \cdot (\frac{f_{вых}}{8})^{1/2} \cdot (Pr_{вых}^{2/3}-1)}, Re_{вых}\geq Re_T\end{cases} \end{split}\]

where:

\(Nu_L\) – Nusselt number for calculating heat transfer in a laminar flow regime;

\(Pr_{in},Pr_{out}\) – Prandtl number for the input and output ports;

\(f_{in},f_{out}\) – friction coefficients calculated using the following formula (14):

(14)#\[f=(0.79 \cdot ln(Re) - 1.64)^{-2}\]

If the heat transfer coefficient calculation model is selected – Table - Colburn factor vs. Reynolds number (15):

(15)#\[\begin{split}Nu_{вх}=j(Re_{вх, т/о})\cdot Re_{вх, т/о} \cdot Pr_{вх}^{1/3}\\ Nu_{вых}=j(Re_{вых, т/о})\cdot Re_{вых, т/о} \cdot Pr_{вых}^{1/3} \end{split}\]

where:

\(Re_{input, h/e},Re_{output, h/e}\) – Reynolds numbers for the input and output ports, calculated based on the hydraulic diameter for heat transfer calculations;

\(j(Re_{input, h/e}),j(Re_{output, h/e})\) – Colburn numbers at the input and output calculated on the basis of tabular data provided by the user, depending on the Reynolds number (16):

(16)#\[\begin{split}Re_{вх, т/о}=\frac{\dot m_{вх}\cdot d_{г,т/о}}{S_{min} \mu_{вх}}\\Re_{вых, т/о}=\frac{\dot m_{вых}\cdot d_{г,т/о}}{S_{min} \mu_{вых}}\end{split}\]

If the heat transfer coefficient calculation model is selected – Table - Nusselt number vs. Reynolds number and Prandtl number (17):

(17)#\[\begin{split}Nu_{вх}=Nu(Re_{вх, т/о}, Pr_{вх})\\Nu_{вых}=Nu(Re_{вых, т/о}, Pr_{вых})\end{split}\]

1. Properties#

Property	Default Value	Description	Units of measurement
Minimum free flow area	0.01	-	m\(^2\)
Hydraulic diameter for pressure loss	0.15	-	m
Thermal liquid volume	0.01	-	m\(^3\)
Laminar flow upper Reynolds number limit	2000.0	-	-
Turbulent flow lower Reynolds number limit	4000.0	-	-
Pressure loss model: Pressure loss coefficient; Correlation for flow inside tubes; Table - Darcy friction factor vs. Reynolds number; Table - Euler number vs. Reynolds number	Pressure loss coefficient	-	-
Pressure loss coefficient, available when selecting `Pressure loss model` - `Pressure loss coefficient`	0.1	-	-
Length of flow path from inlet to outlet, available when selecting `Pressure loss model` - `Correlation for flow inside tubes` or `Table - Darcy friction factor vs. Reynolds number`	1.0	-	m
Aggregate equivalent length of local resistance, available when selecting `Pressure loss model` - `Correlation for flow inside tubes`	0.1	-	m
Internal surface absolute roughness, available when selecting `Pressure loss model` - `Correlation for flow inside tubes`	1.5E-5	-	m
Reynolds number vector for Darcy friction factor, available when selecting `Pressure loss model` - `Table - Darcy friction factor vs. Reynolds number`	400.0;1000.0;1500.0;3000.0;4000.0;6000.0; 10000.0;20000.0;40000.0;60000.0;100000.0;1.0E8;	-	-
Darcy friction factor vector, available when selecting `Pressure loss model` - `Table - Darcy friction factor vs. Reynolds number`	0.264;0.112;0.071;0.0417;0.0387;0.0268; 0.025;0.0232;0.0226;0.022;0.0214;0.0214;	-	-
Laminar friction constant for Darcy friction factor, available when selecting `Pressure loss model` - `Correlation for flow inside tubes` or `Table - Darcy friction factor vs. Reynolds number`	64.0	-	-
Reynolds number vector for Euler number, available when selecting `Pressure loss model` - `Table - Euler number vs. Reynolds number`	400.0;1000.0;1500.0;3000.0;4000.0;6000.0; 10000.0;20000.0;40000.0;60000.0;100000.0;1.0E8;	-	-
Euler number vector, available when selecting `Pressure loss model` - `Table - Euler number vs. Reynolds number`	4.4505;0.6864;0.4791;0.3755;0.36; 0.32;0.3;0.28;0.26;0.24;0.22;0.2;	-	-
Heat transfer coefficient model: Constant heat transfer coefficient; Correlation for flow inside tubes; Table - Colburn factor vs. Reynolds number; Table - Nusselt number vs. Reynolds number and Prandtl number	Constant heat transfer coefficient	-	-
Heat transfer surface area, available when selecting `Heat transfer coefficient model` - `Correlation for flow inside tubes` or `Table - Colburn factor vs. Reynolds number`, or `Table - Nusselt number vs. Reynolds number and Prandtl number`	0.5	-	m\(^2\)
Liquid wall heat transfer coefficient, available when selecting `Heat transfer coefficient model` - `Constant heat transfer coefficient`	100.0	-	W/(m\(^2 \cdot\)°С)
Length of flow path for heat transfer, available when selecting `Heat transfer coefficient model` - `Correlation for flow inside tubes` or `Table - Colburn factor vs. Reynolds number`, or `Table - Nusselt number vs. Reynolds number and Prandtl number`	1.0	-	m
Nusselt number for laminar flow heat transfer, available when selecting `Heat transfer coefficient model` - `Correlation for flow inside tubes`	3.66	-	-
Reynolds number vector for Colburn factor, available when selecting `Heat transfer coefficient model` - `Table - Colburn factor vs. Reynolds number`	100.0;150.0;1000.0;	-	-
Colburn factor vector, available when selecting `Heat transfer coefficient model` - `Table - Colburn factor vs. Reynolds number`	0.019;0.013;0.002;	-	-
Reynolds number vector for Nusselt number, available when selecting `Heat transfer coefficient model` - `Table - Nusselt number vs. Reynolds number and Prandtl number`	100.0;150.0;1000.0;	-	-
Prandtl number vector for Nusselt number, available when selecting `Heat transfer coefficient model` - `Table - Nusselt number vs. Reynolds number and Prandtl number`	1.0;10.0;	-	-
Nusselt number table, available when selecting `Heat transfer coefficient model` - `Table - Nusselt number vs. Reynolds number and Prandtl number`	\u003C3.72;4.21\u003E_\u003C3.75;4.44 \u003E_\u003C4.21;7.15\u003E	-	-
Initial medium temperature	20.0	-	°С
Initial medium gauge pressure	0.0	-	MPa

2. Inputs and outputs#

Parameters	Input/Output	Description	Units of measurement	Port type
Medium inlet	Input	-	-	Pneumatics
Medium outlet	Output	-	-	Pneumatics
Heat port	Output	-	-	Heat exchange
Thermal capacity rate	Output	-	W/°С	Automation
Heat transfer coefficient	Output	-	W/(m\(^2 \cdot\)°С)	Automation

In order to see the connection capabilities of the block, you need to go to page Description of block ports

Heat exchanger interface (P)#

1. Properties#

2. Inputs and outputs#

3. Sources#