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Detailed HX Water Solid

Detailed heat exchanger between Water and Solid

	Detailed heat exchanger between Water and Solid
	Description The Detailed HX Water Solid component models a heat exchanger between Fluid Water and Solid materials, which is for Laminar and Turbulent, for the lumped thermal fluid simulation of Water. This component calculates mainly pressure difference, mass flow rate, and heat flow rate.

Equations

The calculation is changed based on parameter values of Type of pipe, and Dynamics of mass in the Water Settings component.

The definition of Inner hydraulic diameter and Flow area and Geometrical coefficient for laminar flow, and the heat transfer coefficient calculation are explained in the following:

Type of pipe = General

Inner hydraulic diameter is defined with:

$D__h_act = D__h$

Flow area is defined with:

$A__act = A$

Surface area for Heat exchange is defined with:

$A__surface_act = A__surface$

Geometrical coefficient for laminar flow is defined with:

$Geo__act = \{\begin{array}{c} geo__in & External input of Geometrical coefficient = true \\ Geo & others \end{array}$

Heat transfer coefficient is calculated with:

$h__act = \frac{k}{D__h_act} \cdot C__general \cdot ({Re__h}^{m__general} - offset__general) \cdot \Pr^{n__general}$

Reynolds number for heat transfer coefficient is calculated with:

$Re__h = \max (\frac{\frac{(ρ__a + ρ__b)}{2} \cdot |v| \cdot D__h}{μ}, 0.1)$

Prandtl number is calculated with:

$\Pr = \frac{μ \cdot c__p}{k}$

Type of pipe = Circular

Inner hydraulic diameter is defined with:

$D__h_act = D__h$

Flow area is defined with:

$A__act = \frac{π \cdot {D__h}^{2}}{4}$

Surface area for Heat exchange is defined with:

$A__surface_act = π \cdot D__h \cdot L$

Geometrical coefficient for laminar flow is defined with:

$Geo__act = 1$

Heat transfer coefficient is calculated with:

$h__act = (1 - κ__h) \cdot h__lam + κ__h \cdot h__tur$

$h__lam = \frac{k}{D__h_act} \cdot 3.66$

$h__tur = {\begin{array}{c} \frac{k}{D__h_act} \cdot 0.023 \cdot {Re__h}^{0.8} \cdot \Pr^{0.4} & `solid.T` < \frac{`port_a.T` + `port_b.T`}{2} \\ \frac{k}{D__h_act} \cdot 0.023 \cdot {Re__h}^{0.8} \cdot \Pr^{0.3} & others \end{array}$

$κ__h = \frac{\tanh (\frac{IF__speed \cdot (Re__h - Re__CoT)}{2}) + 1}{2}$

Reynolds number for heat transfer coefficient is calculated with:

$Re__h_target = \max (\frac{\frac{ρ__a + ρ__b}{2} \cdot |v| \cdot D__h_act}{μ}, 0.1)$

$\frac{&DifferentialD; Re__h}{&DifferentialD; t} = \frac{(Re__h_target - Re__h)}{T__const}$

Prandtl number is calculated with:

$\Pr = \frac{μ \cdot c__p}{k}$

Type of pipe = Rectangular

Inner hydraulic diameter is defined with:

$D__h_act = \frac{2}{\frac{1}{a__rect} + \frac{1}{b__rect}}$

Flow area is defined with:

$A__act = a__rec \cdot b__rec$

Surface area for Heat exchange is defined with:

$A__surface_act = (a__rec + b__rec) \cdot 2 \cdot L$

Geometrical coefficient for laminar flow is defined with:

$Geo__act = MapleSim.Interpolate1D` (data, \frac{b__rect}{a__rect})$

(*) $`MapleSim.Interpolate1D`$ is the function of Lookup table of 1D.

(*) data is specified with:

- If data_source = inline, parameter $table__rect$ .

- If data_source = attachment, an attached file (.csv and .xls, .xlsx) is used

- If data_source = file, need to specify the path of file (.csv and .xls, .xlsx).

Heat transfer coefficient is calculated with:

$h__act = (1 - κ__h) \cdot h__lam + κ__h \cdot h__tur$

$h__lam = \frac{k}{D__h_act} \cdot 3.66$

$κ__h = \frac{\tanh (\frac{IF__speed \cdot (Re__h - Re__CoT)}{2}) + 1}{2}$

Reynolds number for heat transfer coefficient is calculated with:

$Re__h_target = \max (\frac{\frac{ρ__a + ρ__b}{2} \cdot |v| \cdot D__h_act}{μ}, 0.1)$

$\frac{&DifferentialD; Re__h}{&DifferentialD; t} = \frac{(Re__h_target - Re__h)}{T__const}$

Prandtl number is calculated with:

$\Pr = \frac{μ \cdot c__p}{k}$

Reynolds number for Friction factor calculation is defined with:

$Re__target = \max (\frac{{\begin{array}{c} ρ__a & dp \geq 0 \\ ρ__b & others \end{array} \cdot |v| \cdot D__h_act}{{\begin{array}{c} μ__a & dp \geq 0 \\ μ__b & others \end{array}}, 0.1)$

$\frac{&DifferentialD; Re}{&DifferentialD; t} = \frac{(Re__target - Re)}{T__const}$

The friction factor of flow is calculated with:

$λ = `HeatTransfer.Functions.lambda_Re` (Re, roughness, D__h_act, Re__CoT, IF__speed, Geo__act)$

(*) The above function $`HeatTransfer.Functions.lambda_Re`$ is to calculated friction factor for Laminar and Turbulent flow.
The fundamental implementation is based on the following equations. Especially, the equation of Turbulent flow is Swamee and Jain's approximation[1] .

(Reference) Detailed implementation of Friction factor calculation

Friction factor of Laminar flow is calculated with:

$λ__lam = Geo__act \cdot \frac{64}{Re}$

And, Turbulent flow's friction factor is defined with (Swamee and Jain's approximation):

$λ__tur = \frac{0.25}{\log {(\frac{\frac{roughness}{D__h_act}}{3.7} + \frac{5.74}{{Re}^{0.9}})}^{2}}$

Intermittency is defined with:

$κ = \frac{\tanh (\frac{IF__speed \cdot (Re - Re__CoT)}{2}) + 1}{2}$

So, the friction factor is calculated with:

$λ = (1 - κ) \cdot λ__lam + κ \cdot λ__tur$

The following plot is Reynolds number vs Friction factor, and $\frac{roughness}{D__h_act} = 0.001$ , $IF__speed = 0.007$ , $Re__CoT = 3500$ , $Geo__act = 1$ .

The definition of Flow calculation is the following and:

	Dynamics of mass = Static
	Pressure difference is calculated with Darcy–Weisbach equation: $dp = \frac{1}{2} \cdot λ \cdot \frac{L}{D__h_act \cdot {A__act}^{2} \cdot {\begin{array}{c} ρ__a & dp \geq 0 \\ ρ__b & others \end{array}} \cdot {mflow}^{2} \cdot sign (mflow)$

Dynamics of mass = Dynamic

In theory, Mass flow rate is calculated with Darcy–Weisbach equation:

$mflow = \sqrt{\frac{2 \cdot D__h_act \cdot {A__act}^{2}}{λ \cdot L}} \cdot \sqrt{{\begin{array}{c} ρ__a & dp \geq 0 \\ ρ__b & others \end{array} \cdot |dp|} \cdot sign (dp)$

In the Heat Transfer Library, the following equation is used to resolve difficulties of the numerical calculation:

$mflow = \sqrt{\frac{2 \cdot D__h_act \cdot {A__act}^{2}}{λ \cdot L}} \cdot `HeatTransfer.Functions.regRoot2` (dp, dp_small, ρ__a, ρ__b, true, sharpness)$

(*) $`HeatTransfer.Functions.regRoot2`$ is the same function as $`Modelica.Fluid.Utilities.regRoot2`$ . To check the details of the package and view the original documentation, which includes author and copyright information, click here.

Definitions related to Mass flow rate and pressure:

$dp = `port_a.p` - `port_b.p`$

$v = \frac{mflow}{{\begin{array}{c} ρ__a & dp \geq 0 \\ ρ__b & others \end{array} \cdot A__act}$

$`port_a.mflow` = mflow$

$`port_b.mflow` = - mflow$

Definitions related to Heat flow rate:

$Q_flow = h__act \cdot A__surface_act \cdot (`solid.T` - \frac{inStream (`port_a.T`) + inStream (`port_b.T`)}{2})$

$q_flow = \frac{Q_flow}{\max (|mflow|, 0.00001)}$

If Dynamics of mass is Static, specific enthalpy is defined with:

$`port_a.hflow` = \{\begin{array}{c} inStream (`port_b.hflow`) & mflow \geq 0 \\ inStream (`port_b.hflow`) + q_flow & others \end{array}$

$`port_b.hflow` = \{\begin{array}{c} inStream (`port_a.hflow`) + q_flow & mflow \geq 0 \\ inStream (`port_a.hflow`) & others \end{array}$

If Dynamics of mass is Dynamic, specific enthalpy is defined with:

$`port_a.hflow` = \{\begin{array}{c} inStream (`port_b.hflow`) & dp \geq 0 \\ inStream (`port_b.hflow`) + q_flow & others \end{array}$

$`port_b.hflow` = \{\begin{array}{c} inStream (`port_a.hflow`) + q_flow & dp \geq 0 \\ inStream (`port_a.hflow`) & others \end{array}$

Density is calculated with:

$ρ__a = inStream (`port_a.rho`)$

$ρ__b = inStream (`port_b.rho`)$

If Fidelity of properties = Constant, properties $μ$ and $c__p$ and $k$ are constants and properties at each ports are:

$μ__a = μ$

$μ__b = μ$

(*) Regarding the value of properties for Constant, see more in Water Settings.

If Fidelity of properties = Liquid water (Lookup table of IAPWS/IF97), properties are calculated with:

$μ__a = LUT__μ__a (`port_a.rho`, inStream (`port_a.T`))$

$μ__b = LUT__μ__b (`port_b.rho`, inStream (`port_b.T`))$

$μ = LUT__μ__(\frac{`port_a.rho` + `port_b.p`}{2}, \frac{`solid.T` + (\frac{inStream (`port_a.T`) + inStream (`port_b.T`)}{2})}{2})$

$c__p = LUT__c__p (\frac{`port_a.p` + `port_b.p`}{2}, \frac{`solid.T` + (\frac{inStream (`port_a.T`) + inStream (`port_b.T`)}{2})}{2})$

$k = LUT__k__(\frac{`port_a.p` + `port_b.p`}{2}, \frac{`solid.T` + (\frac{inStream (`port_a.T`) + inStream (`port_b.T`)}{2})}{2})$

(*) The properties are defined with Liquid water (Lookup table of IAPWS/IF97), see more in Water Settings.

If Fidelity of properties = IAPWS/IF97 standard, properties are calculated with:

$μ__a = Function__μ__a (`port_a.rho`, inStream (`port_a.T`), `port_a.p`)$

$μ__b = Function__μ__b (`port_b.rho`, inStream (`port_b.T`), `port_b.p`)$

$μ = Function__μ__(\frac{inStream (`port_a.rho`) + inStream (`port_b.rho`)}{2}), \frac{`solid.T` + (\frac{inStream (`port_a.T`) + inStream (`port_b.T`)}{2})}{2}, \frac{`port_a.p` + `port_b.p`}{2})$

$c__p = Function__c__p (\frac{`port_a.p` + `port_b.p`}{2}, \frac{`solid.T` + (\frac{inStream (`port_a.T`) + inStream (`port_b.T`)}{2})}{2}, 0)$

$k = Function__k__(\frac{inStream (`port_a.rho`) + inStream (`port_b.rho`)}{2}, \frac{`solid.T` + (\frac{inStream (`port_a.T`) + inStream (`port_b.T`)}{2})}{2}, \frac{`port_a.p` + `port_b.p`}{2}, 0, true)$

(*) The properties are defined with IAPWS/IF97 standard, see more in Water Settings.

Port's variables are defined with:

$`port_a.rho` = inStream (`port_b.rho`)$

$`port_b.rho` = inStream (`port_a.rho`)$

$`port_a.T` = inStream (`port_b.T`)$

$`port_b.T` = inStream (`port_a.T`)$

	References
	[1] : Swamee P.K., Jain A.K. (1976): Explicit equations for pipe-flow problems. Proc. ASCE, J.Hydraul. Div., 102 (HY5), pp. 657-664.

Variables

Symbol	Units	Description	Modelica ID
$dp$	$Pa$	Pressure difference	p
$mflow$	$\frac{kg}{s}$	Mass flow rate	mflow
$v$	$\frac{m}{s}$	Velocity of flow	v
$D__h_act$	$m$	Inner hydraulic diameter used for Fluid simulation	Dh_act
$A__act$	$m^{2}$	Flow area used for Fluid simulation	A_act
$A__surface_act$	$m^{2}$	Surface area used for Heat exchange	A_surface_act
$Geo__act$	$-$	Geometrical coefficient used for Fluid simulation	Geo_act
$Re$	$-$	Reynolds number for Friction factor calculation	Re
$Re__target$	$-$	Targeted Reynolds number for Friction factor calculation	Re_target
$λ$	$-$	Friction factor	lambda
$λ__lam$	$-$	Friction factor for Laminar flow	lambda_lam
$λ__tur$	$-$	Friction factor for Turbulent flow	lambda_tur
$κ$	$-$	Intermittency factor to calculate Transition zone	kappa
$h__act$	$\frac{W}{m^{2} \cdot K}$	Heat transfer coefficient used for Fluid simulation	h_act
$Re__h$	$-$	Reynolds number for Heat transfer coefficient calculation	Re_h
$Re__h_target$	$-$	Targeted Reynolds number for Heat transfer coefficient calculation	Re_h_target
$\Pr$	$-$	Prandtl number	Pr
$κ__h$	$-$	Coefficient for Transition zone	kappa_h
$h__lam$	$\frac{W}{m^{2} \cdot K}$	Heat transfer coefficient for Laminar flow	h_lam
$h__tur$	$\frac{W}{m^{2} \cdot K}$	Heat transfer coefficient for Turbulent flow	h_tur
$Q_flow$	$W$	Heat flow rate between solid materials and fluid Water	Q_flow
$q_flow$	$\frac{W}{kg}$	Specific energy between solid materials and fluid Water	q_flow
$μ$	$Pa \cdot s$	Dynamic viscosity	vis
$c__p$	$\frac{J}{kg \cdot K}$	Specific heat capacity at the constant pressure	cp
$k$	$\frac{W}{m \cdot K}$	Thermal conductivity	k
$ρ__a$	$\frac{kg}{m^{3}}$	Density at port_a	rho_a
$ρ__b$	$\frac{kg}{m^{3}}$	Density at port_b	rho_b
$μ__a$	$Pa \cdot s$	Dynamic viscosity at port_a	vis_a
$μ__b$	$Pa \cdot s$	Dynamic viscosity at port_b	vis_b

Connections

Name	Condition	Description	Modelica ID
$port__a$		Water Port	$port_a$
$port__b$		Water Port	$port_b$
$solid$		Solid Port	$solid$
$geo_in$	if External input of Geometrical coefficient = false	Geometrical coefficient input	$geo_in$

Parameters

Symbol	Default	Units	Description	Modelica ID
$Water simulation settings$	$WaterSettings1$	$-$	Specify a component of Water simulation settings	Settings
$Type of pipe$	$Circular$	$-$	Select pipe type - General - Circular pipe - Rectangular pipe	TypeOfPipe
$L$	$0.5$	$m$	Pipe length	L
$D__h$	$0.01$	$m$	Internal hydraulic diameter if Type of pipe is General or Circular.	Dh
$a__rect$	$0.02$	$m$	Horizontal length only if Type of pipe = Rectangular.	a_rec
$b__rect$	$0.01$	$m$	Vertical length only if Type of pipe = Rectangular.	b_rec
$A$	$\frac{1}{4} \cdot Pi__\cdot {D__h}^{2}$	$m^{2}$	Flow area only if Type of pipe = General.	A
$A__surface$	$Pi \cdot D__h \cdot L$	$m^{2}$	Surface area for Heat exchange if Type of pipe = General.	A_surface
$roughness$	$0.000025$	$m$	Absolute roughness of pipe, with a default for a smooth steel pipe	roughness
$External input of Geometrical coefficient$	$false$	$-$	If true, Geometrical coefficient is defined by the input. And, if Type of pipe = Rectangular, this parameter is valid.	Geo_ext
$Geo$	$1$	$-$	Geometrical coefficient for Laminar flow only if Type of pipe = General and External input of Geometrical coefficient = false.	Geo
$C__general$	$0.664$	$m$	Gain parameter for Reynolds number in the generalized experimental equation of Internal flow convection generalized equation, only if Type of pipe = General.	C_forced
$m__general$	$0.5$	$m$	Exponent parameter for Reynolds number in the generalized experimental equation of Internal flow convection generalized equation, only if Type of pipe = General.	m_forced
$offset__general$	$0$	$m$	Offset parameter for Reynolds number in the generalized experimental equation of Internal flow convection generalized equation, only if Type of pipe = General.	offset_forced
$n__general$	$\frac{1}{3}$	$m$	Exponent parameter for Prandtl number in the generalized experimental equation of Internal flow convection generalized equation, only if Type of pipe = General.	n_forced
$data source__rect$	inline	-	See Data Source Options section above.	DSM_geo_rec
$table__rect$	$[\begin{array}{c} 0 & 1.5 \\ 0.1 & 1.323 \\ 0.2 & 1.192 \\ 0.3 & 1.094 \\ 0.4 & 1.023 \\ 0.5 & 0.9716 \\ 0.6 & 0.9360 \\ 0.7 & 0.9120 \\ 0.8 & 0.8983 \\ 0.9 & 0.8909 \\ 1.0 & 0.8887 \end{array}]$	$-$	Geometrical coefficient for Rectangular pipe, if $data source__rect$ = inline. [1] :Volume flow rate [2] :Pressure difference	table_geo_rec
$data__rect$	$2$	-	Geometrical coefficient for Rectangular pipe, if $data source__rect$ =file or attachment. You can specify data by using an attached file or specifying the path of file (.csv and .xls, .xlsx)	data_geo_rec
$columns__rect$	$[2]$	-	Determines which columns of the data table will be used to interpolate. For example, in an Excel spreadsheet, column A corresponds with 1, column B corresponds with 2, and so on.	columns_geo_rec
$skip rows__rect$	0	-	Number of rows that are skipped from the top of the data table.	skiprows_geo_rec
$smoothness__rect$	Table points are linearly interpolated	-	Determines whether the data points will be interpolated linearly or with a cubic spline.	smoothness_geo_rec
$dp__small$	$0.1$	$Pa$	Approximation of function for \|dp\| <= dp_small	dp_small
$sharpness$	$1.0$	$-$	Sharpness of approximation for sqrt(dp) and sqrt(rho * dp)	sharpness
$T__const$	$0.001$	$s$	Time constant for Reynolds number calculation	T_const
$Re__CoT$	$3500$	$-$	Reynolds number of the center of Transition zone	Re_CoT
$Spread of Intermittency factor$	$0.007$	$-$	Changing rate of Intermittency factor	IF_spread

	See Also
	Heat Transfer Library Overview Water Overview Water Shapes Overview

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