Plate Heat Exchanger (TL-TL)

Block appearance in REPEAT VISION

 Block simulates a plate heat exchanger in which heat exchange occurs between two thermohydraulic systems.

 Heat flow is calculated using NTU method. User can add accounting of plates thermal resistance and deposits on their surfaces.

 The coolant movement (counterflow)[1] in the heat exchanger is shown in Fig. 1.

Fig. 1 General view of coolant movement in the heat exchanger (counterflow)

The geometry of plate corrugation[4] is shown in Fig. 2.

Fig. 2 - Corrugation geometry of plates
φ - corrugation angle;
2\(a\) is a distance between plates;
Λ - corrugation pitch;
L\(_h\) is a plate length;
W is a plate width.

  Total volume of medium in heat exchanger (1) :

(1)\[V_{общ}=(n_{пл}+1)\cdot b \cdot L_{пл}\cdot W_{пл}\]

where:

 \(n_{pl}\) – number of plates in the heat exchanger excluding extreme ones, [pcs];

 b – distance between plates, [m];

 \(L_{pl}\) – length of plates, [m];

 \(W_{pl}\) – width of plates, [m].

  Volume of one medium in the heat exchanger {eq} PlateHeatExchangerTH2 :

(2)\[V_{ср}=\frac{V_{общ}}{2}\]

 Cross-sectional area for one medium (3) :

(3)\[S_{ср}=\frac{(n_{пл}+1)\cdot b \cdot W_{пл}}{2}\]

 Total heat exchange surface area excluding plate corrugation (4) :

(4)\[A'_{общ}=2\cdot n_{пл} \cdot L_{пл} \cdot W_{пл} \]

 The heat exchange surface area for one medium is half of the total heat exchange surface without corrugation multiplied by the area increase factor for corrugation (5) :

(5)\[А_{ср}=\phi \cdot n_{пл} \cdot L_{пл} \cdot W_{пл}\]

where \(\phi\) is coefficient of increase of plate surface area.

 Coefficient \(\phi\) is the ratio of actual surface area to surface area excluding corrugation (6) :

(6)\[\phi = \frac{А_{действ}}{A'_{общ}}\]

where \(А_{actual}\) is the actual heat exchange surface area.

 In the block, the plate surface area increase factor can be set by the user or calculated using the following formula based on the specified corrugation geometry (if Corrugation geometry is selected as Specification of corrugation pattern)[2] (7) :

(7)\[\phi=\frac{1+\sqrt{1+(\frac{\pi \cdot b}{\lambda})^2}+4 \cdot \sqrt {1+(\frac{\pi \cdot b}{\sqrt{2}\cdot \lambda})^2}}{6}\]

where \(\lambda\) is corrugation pitch, [m].

  Hydraulic diameter (8) :

(8)\[d_г=\frac{2\cdot b}{\phi}\]

1.  Heat Exchange

NTU convective heat exchange (9) :

(9)\[Q_{конв}=\epsilon \cdot C_{мин}\cdot(T_{вх}^{ср1}-T_{вх}^{ср2})\]

where:

 \(\epsilon\) – heat exchanger efficiency factor;

 \(C_{min}\) – minimum values of \(\dot m_1 \cdot C_{p1}\) and \(\dot m_2 \cdot C_{p2}\), [\(\frac{W}{℃}\)];

 \(\dot m_1, \dot m_2\) – mass flow rate of medium 1 and 2, [\(\frac{kg}{s}\)];

 \(C_{p1}, C_{p2}\)– specific heat capacity of medium 1 and 2, [\(\frac{J}{kg\cdot ℃}\)];

 \(T_{in}^{aver1}, T_{in}^{aver2}\) – medium 1 and 2 temperature at heat exchanger input, [℃].

  Heat exchanger efficiency (10):

(10)\[\epsilon=\frac{Q}{Q_{макс}}\]

where:

  Q - actual heat flux, [W];

 Q\(_{max}\) is maximum possible heat flow, [W].

  Formula for calculating heat exchanger efficiency for different mutual motion of coolant [3]:

• Sump (11) :

(11)\[\epsilon=\frac{1-e^{-NTU \cdot (1+C_{отн})}}{1+C_{отн}}\]

where:

 NTU - number of heat transfer units;

 \(C_{rel}=\frac{C_{min}}{C_{max}}\).

• Counterflow (12) :

(12)\[\epsilon=\frac{1-e^{-NTU \cdot (1-C_{отн})}}{1- C_{отн}\cdot e^{-NTU \cdot (1-C_{отн})}}\]

 For \(C_{rel}\)=1 (13) :

(13)\[\epsilon=\frac{NTU}{1+NTU}\]

 Number of heat transfer units (14) :

(14)\[NTU=\frac{1}{C_{мин}\cdot R_{конв}}\]

where \(R_{conv}\) is the total thermal resistance between two liquids without taking into account the thermal conductivity of the liquids, [\(\frac{℃}{W}\)] (15) :

(15)\[R_{конв}=\frac{1}{\alpha_1 \cdot A_{ср1}}+\frac{R_{f1}}{A_{ср1}}+R_{пл}+\frac{R_{f2}}{A_{ср2}}+\frac{1}{\alpha_2 \cdot A_{ср2}}\]

where:

emsp;\(\alpha_1, \alpha_2\) - heat transfer coefficients for media 1 and 2, [\(\frac{W}{m^2 \cdot℃}]\);

 \(A_{aver1}, A_{aver2}\) – heat transfer surface area for medium 1 and 2, [m\(^2\)];

 \(R_{f1}, R_{f2}\) are surface contamination coefficients for media 1 and 2, [\(\frac{m^2 \cdot℃}{W}\)];

 \(R_{pl}=\frac{t}{\lambda_{pl}\cdot n_{pl}\cdot L_{pl}\cdot W_{pl}}\) – plate thermal resistance, [\(\frac{℃}{W}\)];

 \(\lambda_{pl}\) – plate thermal conductivity, [\(\frac{W}{m^2 \cdot℃}\)];

 t - plate thickness, [m].

• Heat transfer coefficient (16) :

(16)\[\alpha=\frac{\lambda \cdot Nu}{d_г}\]

where:

 \(\lambda\) – medium thermal conductivity, [\(\frac{W}{m^2 \cdot℃}\)];

 Nu is the Nusselt number.

  If Calculation type for the heat transfer coefficient property is seected (17):

(17)\[Nu=\alpha \cdot Re^b \cdot Pr^c\]

where:

 a,b,c are coefficients set by the user;

 Re=\(\frac{w \cdot d_г}{ u}\) – Reynolds number;

 w – medium speed, [\(\frac{m}{s}\)];

& emsp; \( u \) - kinematic viscosity of medium, [\(\frac{m^2}{s}\)];

 Pr is the Prandtl number.

  If Calculation type for the heat transfer coefficient property is selected - Correlation for flow between corrugated plates (without taking into account the correction for the temperature dependence of the medium viscosity)[2] (18) :

(18)\[Nu=c_1 \cdot (f\cdot Re^2 \cdot sin(2\cdot \beta))^{c_2}\cdot Pr^{c_3}\]

where:

 \(c_1, c_2, c_3\) – coefficients of dependence for flow between corrugated plates;

 \(f\) - Darcy friction factor;

 \(\beta\) - corrugation angle,°.

• Thermal flux of thermal conductivity (19) :

(19)\[Q_{тепл}=\frac{(T^{ср1}_{вых}-T^{ср2}_{вых})}{R_{тепл}}\]

where:

 \(T^{aver1}_{out}, T^{aver2}_{out}\) – medium 1 and 2 temperature at the heat exchanger output, [℃];

 \(R_{heat}\) – thermal resistance of thermal conductivity, \(\frac{℃}{W}\) (20):

(20)\[R_{тепл}=\frac{d_г}{\lambda_1 \cdot A_{ср1}}+\frac{R_{f1}}{A_{ср1}}+R_{пл}+\frac{R_{f2}}{A_{ср2}}+\frac{d_г}{\lambda_2 \cdot A_{ср2}}\]

where \(\lambda_1, \lambda_2\) are thermal conductivity coefficients for medium 1 and 2, [\(\frac{W}{m^2\cdot℃}\)].

 The resulting heat flux between the coolants is the sum of the convective and conductive component (21) :

(21)\[Q=Q_{конв}+Q_{тепл}\]

Pressure loss in the heat exchanger

 The pressure loss between the block ports and the internal unit is calculated using the following formulas (22) :

(22)\[\begin{split}\begin{cases}p_{вх}^{ср} - p_{вн}^{ср} = \frac{\xi_{порт}\cdot \dot m_{вх} \cdot |\dot m_{вх}|}{4 \cdot \rho \cdot S^2_{вх.порт}}+\frac{f \cdot L_{пл}\cdot \dot m_{вх}\cdot |\dot m_{вх}|}{4 \cdot \rho \cdot S^2_{ср} \cdot d_г}\\p_{вых}^{ср} - p_{вн}^{ср} = \frac{\xi_{порт}\cdot \dot m_{вых} \cdot |\dot m_{вых}|}{4 \cdot \rho \cdot S^2_{вых.порт}}+\frac{f \cdot L_{пл}\cdot \dot m_{вых}\cdot |\dot m_{вых}|}{4 \cdot \rho \cdot S^2_{ср} \cdot d_г} \end{cases}\end{split}\]

where:

 \(p^{aver}_{input}, r^{aver}_{output}, p^{aver}_{input}\) - medium pressure at input, output and in internal unit, respectively, [Pa];

 \(\dot m_{input},\dot m_{output}\) – mass flow rate of medium at input and output of heat exchanger, [\(\frac{kg}{s}\)];

 \(\xi\) – input and output pressure loss factor;

 \( ho\) – medium density, [\(\frac{kg}{m^2}\)];

 \(S_{input port}, S_{output port}\) – cross-sectional area of input and output port, [m\(^2\)].

• If Pressure loss calculation type is selected - Constant Darcy friction factor, then the loss ratio is equal to the one set by the user.

• If Pressure loss calculation type is selected - Correlation for flow between corrugated plates, then the loss factor is calculated from the following formula [2]:

  • For Re<2000 (23) :

  (23)\[\frac{1}{\sqrt{f}}=\frac{cos \beta}{\sqrt{0.18 \cdot tg \beta +0.36 \cdot sin \beta + \frac{64}{Re \cdot cos \beta}}}+\frac{1-cos \beta}{\sqrt{3.8 \cdot (\frac{597}{Re}+3.85)}}\]

  • For Re\(\geq\) 2000 (24) :

  (24)\[\frac{1}{\sqrt{f}}=\frac{cos \beta}{\sqrt{0.18 \cdot tg \beta +0.36 \cdot sin \beta + \frac{1}{(1.8 \cdot lgRe - 1.5)^2 \cdot cos \beta}}}+\frac{1-cos \beta}{\sqrt{\frac{3.8\cdot 39}{Re^{0.289}} }}\]

Mass conservation equation (25):

(25)\[\frac{dM}{dt}=\dot m^{ср}_{вх}+ \dot m^{ср}_{вых}\]

where M – mass of medium, [kg].

Energy conservation equation (26):

(26)\[\phi^{ср}_{вх}+\phi^{ср}_{вых}-Q=((\frac{h}{\beta}-\frac{T_{вн}\cdot \alpha}{\rho})\cdot \frac{dp^{ср}_{вн}}{dt}+(С_р - h \cdot \alpha)\cdot \frac{dT_{вн}}{dt})\cdot \rho \cdot V\]

where:

 \(\phi^{aver}_{in}, \phi^{aver}_{out}\) – energy flow at input and output, [W];

 Q – heat flux, [W];

 h – enthalpy of medium in internal unit, [\(\frac {J}{kg}\)];

 \(\alpha\) – coefficient of isobaric thermal expansion, \(\frac {1}{K}\);

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

 \(T_{in}\) – medium temperature in internal unit, [℃];

 V – volume of medium in the heat exchanger, [m\(^3\)].

2.  Properties

Property	Default Value	Description	Units of measurement
Flow arrangement:<brCounter flow; Parallel flow	Counter flow	-	-
Number of plates	27	-	pieces
Plate length	1.2	-	m
Plate width	0.6	-	m
Spacing between plates	0.006	-	m
Specification of corrugation pattern: Surface area enlargement factor; Corrugation geometry	Surface area enlargement factor	-	-
Surface area enlargement factor, available when selecting Specification of corrugation pattern-Surface area enlargement factor	1.25	-	-
Corrugation angle, available when selecting Specification of corrugation pattern-Corrugation geometry	60.0	-	degrees
Corrugation depth to pitch ratio, available when selecting Specification of corrugation pattern-Corrugation geometry	0.28	-	-
Consider the thermal resistance of the plates	выкл	-	-
Plate thickness, available when selecting Consider the thermal resistance of the plates-true	5.5E-4	-	m
Plate thermal conductivity, available when selecting Consider the thermal resistance of the plates-true	13.0	-	W/(m\(\cdot\)K)
Inlet cross-sectional area (liquid 1)	0.1	-	m\(^2\)
Inlet cross-sectional area (liquid 2)	0.1	-	m\(^2\)
Outlet cross-sectional area (liquid 1)	0.1	-	m\(^2\)
Outlet cross-sectional area (liquid 2)	0.1	-	m\(^2\)
Pressure loss calculation type (liquid 1): Constant Darcy friction factor; Correlation for flow between corrugated plates	Constant Darcy friction factor	-	-
Darcy friction factor for corrugated plates (liquid 1), available when selecting Pressure loss calculation type (liquid 1) - Constant Darcy friction factor	1.8	-	-
Inlet and outlet loss coefficient (liquid 1)	1.5	-	-
Calculation type for the heat transfer coefficient (liquid 1): Nu = \(\alpha \cdot Re^b \cdot Pr^c\); Correlation for flow between corrugated plates	Nu = \(\alpha \cdot Re^b \cdot Pr^c\)	-	-
Coefficients for equation \(\alpha \cdot Re^b \cdot Pr^c\) (liquid 1), available when selecting Calculation type for the heat transfer coefficient (liquid 1) - Nu = \(\alpha \cdot Re^b \cdot Pr^c\)	0.141; 0.748; 0.333;	-	-
Correlation coefficients for flow between corrugated plates (liquid 1), available when selecting Calculation type for the heat transfer coefficient (liquid 1) - Correlation for flow between corrugated plates	0.122; 0.374; 0.333;	-	-
Fouling factor (liquid 1)	0.1	-	K\(\cdot m^2\)/kW
Initial liquid pressure (liquid 1)	0.0	-	MPa
Initial liquid temperature (liquid 1)	20.0	-	℃
Pressure loss calculation type (liquid 2): Constant Darcy friction factor; Correlation for flow between corrugated plates	Constant Darcy friction factor	-	-
Darcy friction factor for corrugated plates (liquid 2), available when selecting Pressure loss calculation type (liquid 2) - [Constant Darcy friction factor]	1.8	-	-
Inlet and outlet loss coefficient (liquid 2)	1.5	-	-
Calculation type for the heat transfer coefficient (liquid 2): Nu = \(\alpha \cdot Re^b \cdot Pr^c\); Correlation for flow between corrugated plates	Nu = \(\alpha \cdot Re^b \cdot Pr^c\)	-	-
Coefficients for equation \(\alpha \cdot Re^b \cdot Pr^c\) (liquid 2), available when selecting Calculation type for the heat transfer coefficient (liquid 2) - Nu = \(\alpha \cdot Re^b \cdot Pr^c\)	0.141; 0.748; 0.333;	-	-
Correlation coefficients for flow between corrugated plates (liquid 2), available when selecting Calculation type for the heat transfer coefficient (liquid 2) - Correlation for flow between corrugated plates	0.122; 0.374; 0.333;	-	-
Fouling factor (liquid 2)	0.1	-	K\(\cdot m^2\)/kW
Initial liquid pressure (liquid 2)	0.0	-	MPa
Initial liquid temperature (liquid 2)	20.0	-	℃

3.  Inputs and outputs

Parameters	Input/Output	Description	Units of measurement	Port type
Liquid 1 inlet	Input	-	-	Thermohydraulics
Liquid 2 inlet	Input	-	-	Thermohydraulics
Liquid 1 outlet	Output	-	-	Thermohydraulics
Liquid 2 outlet	Output	-	-	Thermohydraulics

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

4.  Examples of use

 Examples of using this block can be found in the following sections:

• How-to guides;

• Tutorials;

• Description of Demo.

How-to guides

Section	Guide name
Energy	Development of a model of the industrial circuit system of oil coolers of NPP with WCWR

5.  Sources

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[1]

Sadik Kakaç, Hongtan Liu, Anchasa Pramuanjaroenkij, HEAT EXCHANGERS Selection, Rating, and Thermal Design

[2] (1,2,3)

Martin, Holger. “ Pressure Drop and Heat Transfer in Plate Heat Exchangers”, 2010

[3]

F. P. INCROPERA, D. P. DEWITT: “Fundamentals of Heat and Mass Transfer”, Fourth Edition, John Wiley & Sons Inc, 2007

[4]

Shah, R. K. Fundamentals of heat exchanger design / Ramesh K. Shah, Dusˇan P. Sekulic´.