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Heat Convection

Basic component of Convection

Description

The Heat Convection component models the heat convection phenomenon, which is based on Newton's law of cooling.

The Convection type parameter selects the type of convection to use for this component. There are four options to select from: Constant, External input, Natural, and Forced.

Additionally, there are several built-in functions for Forced convection and Natural convection as references.

Refer to the following table for the implemented convection types.

Convection type	Use reference for Natural / Forced	Natural / Forced convection type	Use Correction input
Natural	false	-	false or true
	true	Vertical (Ra:10^4-10^13)	false or true
		Horizontal Upper Warm(Ra:10^4-10^11)	false or true
		Horizontal Upper Cold (Ra:10^5-10^11)	false or true
Forced	false	-	false or true
	true	Tube and Duct	false or true
		Over flat plates	false or true
		Over a cylinder	false or true
		Over a sphere	false or true
Constant	-	-	false or true
External input	-	-	false or true

Equations

Fundamental equation is :

$Q_{flow} = A \cdot h_act \cdot (T_{s} - T__f)$

The extended equation which is implemented in this library is:

$Q_{flow} = cor \cdot A \cdot h_act \cdot (T_{s} - T__f)$

when Use Correction input = true, $cor$ is specified by the input signal. If Use Correction input = false, $cor$ is the constant value "1".

Heat transfer coefficient $h_act$ is defined based on the selected option. The equation for each option is shown below.

Convection type: Natural

This type is for Natural convection, and there are 4 options including the default setting.

•	Default : Use references for Natural = false

If you select this option, the generalized equation is valid.

$Nu = {\begin{array}{c} C__lam \cdot {Ra}^{n__lam} & Ra < Threshold \\ C__tur \cdot {Ra}^{n__tur} & otherwise \end{array}$

$h_act = \overline{h} = Nu \cdot \frac{k}{X}$

The default value of the experimental parameters are for the vertical plate.

$Nu = {\begin{array}{c} 0.59 \cdot {Ra}^{\frac{1}{4}} & Ra < 10^{9} \\ 0.1 \cdot {Ra}^{\frac{1}{3}} & otherwise \end{array}$

$h_act = \overline{h} = Nu \cdot \frac{k}{X}$

And, several thermophysics properties are calculated with Modelica functions which call Maple's built-in library CoolProp.

For more information, see Fluid Properties Check.

$Gr = `HeatTransfer.Properties.Fluids.AirGrashof_pT` (p, T__s, T__f, g, X)$

$\Pr = `HeatTransfer.Properties.Fluids.AirPrandtl_pT` (p, T__s, T__f, X)$

$Ra = Gr \cdot \Pr$

$k = `HeatTransfer.Properties.Fluids.AirConductivity_pT` (p, \frac{T__s+T__f}{2})$

$g$ is gravity force and its value is 9.81.

•	Reference "Vertical (Ra:10^4-10^13)" : Use references for Natural = true and Natural Convection type = Vertical(Ra:10^4-10^13)

Parameter values for this case is referred from [1].

$Nu = {\begin{array}{c} 0.59 \cdot {Ra}^{\frac{1}{4}} & Ra < 10^{9} \\ 0.1 \cdot {Ra}^{\frac{1}{3}} & otherwise \end{array}$

$h_act = \overline{h} = Nu \cdot \frac{k}{X}$

The calculation for thermophysics properties is same as Default : Use references for Natural = false.

•	Reference "Horizontal Upper Warm (Ra:10^4-10^11)" : Use references for Natural = true and Natural Convection type = Horizontal_Upper_Warm(Ra:10^4-10^11)

Parameter values for this case is referred from [1].

$Nu = {\begin{array}{c} 0.54 \cdot {Ra}^{\frac{1}{4}} & Ra < 10^{7} \\ 0.15 \cdot {Ra}^{\frac{1}{3}} & otherwise \end{array}$

$h_act = \overline{h} = Nu \cdot \frac{k}{X}$

The calculation for thermophysics properties is same as Default : Use references for Natural = false.

•	Reference "Horizontal Upper Cold (Ra:10^5-10^11)" : Use references for Natural = true and Natural Convection type = Horizontal_Upper_Cold (Ra:10^5-10^11)

Parameter values for this case is referred from [1].

$Nu = {\begin{array}{c} 0.27 \cdot {Ra}^{\frac{1}{4}} & Ra < 10^{5} \\ 0.27 \cdot {Ra}^{\frac{1}{4}} & otherwise \end{array}$

$h_act = \overline{h} = Nu \cdot \frac{k}{X}$

The calculation for thermophysics properties is same as Default : Use references for Natural = false.

Convection type: Forced

This type is for Forced convection, and there are 5 options including the default setting.

•	Default : Use references for Forced = false

If you select this option, the generalized equation is valid.

$Nu = C_{forced} \cdot ({Re}^{m_{forced}} - {offset}_{forced}) \cdot \Pr^{n_{forced}}$

$h_act = \overline{h} = Nu \cdot \frac{k}{X}$

The default value of the experimental parameters are for the flat plate with the laminar flow.

$Nu = 0.664 \cdot {Re}^{\frac{1}{2}} \cdot \Pr^{\frac{1}{3}}$

$h_act = \overline{h} = Nu \cdot \frac{k}{X}$

And, several thermophysics properties are calculated with Modelica functions which is to call Maple's built-in library CoolProp.

Please see more in Fluid Properties Check.

$Re = `HeatTransfer.Properties.Fluids.AirReynolds_pT` (p, T__s, T__f, X, v)$

$\Pr = `HeatTransfer.Properties.Fluids.AirPrandtl_pT` (p, T__s, T__f, X)$

$k = `HeatTransfer.Properties.Fluids.AirConductivity_pT` (p, \frac{T__s+T__f}{2})$

•	Reference "Tube and Duct" : Use references for Forced = true and Forced Convection type = Tube and Duct

For laminar flow, the constant Nusselt number is used in this option, and for turbulent, Dittus-Boetter's equation [2] is used.

$Nu =$ ${\begin{array}{c} 3.66 & Re < 2300 \\ {\begin{array}{c} 0.023 \cdot {Re}^{0.8} \cdot \Pr^{0.4} & T__s < T__f \\ 0.023 \cdot {Re}^{0.8} \cdot \Pr^{0.3} & otherwise \end{array} & otherwise \end{array}$

$h_act = \overline{h} = Nu \cdot \frac{k}{X}$

The calculation for thermophysics properties is same as Default : Use references for Forced = false.

•	Reference "Over flat plates (Laminar)" : Use references for Forced = true and Forced Convection type = Over flat plates

$Nu = 0.664 \cdot {Re}^{\frac{1}{2}} \cdot \Pr^{\frac{1}{3}}$

$h_act = \overline{h} = Nu \cdot \frac{k}{X}$

The calculation for thermophysics properties is same as Default : Use references for Forced = false.

•	Reference "Over a cylinder" : Use references for Forced = true and Forced Convection type = Over a cylinder

This equation is given by Churchill and Bernstein [3], which is valid for the range $10^{2} < Re < 10^{7}$ , $Re \cdot \Pr > 0.2$ .

$Nu = 0.3 + \frac{0.62 \cdot {Re}^{\frac{1}{2}} \cdot \Pr^{\frac{1}{3}}}{{(1 + {(0.4 \cdot \Pr)}^{\frac{2}{3}})}^{\frac{1}{4}}} \cdot {(1 + {(\frac{Re}{282000})}^{\frac{5}{8}})}^{\frac{4}{5}}$

$h_act = \overline{h} = Nu \cdot \frac{k}{X}$

The calculation for thermophysics properties is same as Default : Use references for Forced = false.

•	Reference "Over a sphere" : Use references for Forced = true and Forced Convection type = Over a sphere

This equation is developed by Whitaker [4], which is valid for the range $3.5 < Re < 8 \cdot 10^{4}$ , $0.7 < \Pr < 380$ .

$Nu = 2 + (0.4 \cdot {Re}^{\frac{1}{2}} + 0.06 \cdot {Re}^{\frac{2}{3}}) \cdot \Pr^{0.4} \cdot {(\frac{η__f}{η__s})}^{\frac{1}{4}}$

$h_act = \overline{h} = Nu \cdot \frac{k}{X}$

$η__f = `HeatTransfer.Properties.Fluid.AirViscosity_pT` (p, T__f)$

$η__s = `HeatTransfer.Properties.Fluid.AirViscosity_pT` (p, T__s)$

The calculation for the other thermophysics properties is same as Default : Use references for Forced = false.

	Convection type: Constant
	With this type, the heat transfer coefficient is specified by the value of parameter $h$ . $h_act = h$

	Convection type: External input
	If using this type, the heat transfer coefficient is specified by the signal input $h_{in}$ . $h_act = h__in$

References

[1] : J. P. Holman. "Heat Transfer Ninth Edition", McGraw-Hill Higher Education.

[2] : Dittus, F. W. and L. M. K. Boelter, Univ. Calif. (Berkeley) Pub. Eng. vol. 2, p.443, 1930.

[3] : Churchill, S. W., and M. Bernstein. "A Correlating Equation for Forced Convection from Gases and Liquids to a Circular Cylinder in Crossflow",

J. Heat Transfer, vol.99, pp.300-306, 1977.

[4] : Whitake, S. "Forced Convection Heat-Transfer Correlations for Flow in Pipes, Past Flat Plates, Single Cylinders, Single Spheres, and Flow

in Packed Bids and Tube Bundles", AIChE J., vol.18 p361, 1972.

Variables

Symbol	Units	Description	Modelica ID
$Q__flow$	$W$	Heat flow rate from port a to solid port	Q_flow
$T__s$	$K$	Temperature of solid port
$T__f$	$K$	Temperature of fluid port
$h_act$	$\frac{W}{m^{2} \cdot K}$	Actual heat transfer coefficient	h_act
$\overline{h}$	$\frac{W}{m^{2} \cdot K}$	The average heat transfer coefficient
$Nu$		Nusselt number
$Re$		Reynolds number
$\Pr$		Prandtl number
$Gr$		Grashof number
$Ra$		Rayleigh number
$k$	$\frac{W}{m \cdot K}$	Thermal conductivity
$η__s$	$Pa \cdot s$	Viscosity calculated with temperature of solid port
$η__f$	$Pa \cdot s$	Viscosity calculated with temperature of fluid port

Connections

Name	Units	Condition	Description	Modelica ID
$solid$	-	-	Thermal port of the solid side	solid
$fluid$	-	-	Thermal port of the fluid side	fluid
$h in$	$\frac{W}{m^{2} \cdot K}$	if Convection type is External input.	Input signal of the heat transfer coefficient	h_in
$v$	$\frac{m}{s}$	if Convection type is Forced.	Input signal of Wind speed for Forced convection	v
$cor$	-	if Use correction input is true.	Input signal of the correction factor for $Q_{flow}$	cor

Parameters

Symbol	Default	Units	Description	Modelica ID
$Convection Type$	$Natural$		Select Type of Convection Natural : Natural convection Forced : Force convection Constant : Constant heat transfer coefficient External input : Heat transfer coefficient given by input	TypeOfMedium
$Use reference for Natural$	$false$		If true, all parameters are defined by references	use_reference_natural
$Natural Convection Type$	$Vertical (Ra &colon; 10 &Hat; 4 - 10 &Hat; 13)$		Geometry type of Natural convection as references	NaturalConvecType
$Use reference for Forced$	$false$		If true, all parameters are defined by references	use_reference_forced
$Forced Convection Type$	$Tube and Duct$		Geometry type of Natural convection as references	ForcedConvecType
$A$	$1.0$	$m^{2}$	Area of flow	A
$X$	$1.0$	$m$	Streamwise length	X
$p$	$101325$	$Pa$	Air pressure	p
$C_{forced}$	$0.664$		Gain parameter for Reynolds number in the generalized experimental equation of Forced convection generalized equation	C_forced
$m_{forced}$	$\frac{1}{2}$		Exponent parameter for Reynolds number in the generalized experimental equation of Forced convection generalized equation	m_forced
${offset}_{forced}$	$0$		Offset parameter for Reynolds number in the generalized experimental equation of Forced convection generalized equation	offset_forced
$n_{forced}$	$\frac{1}{3}$		Exponent parameter for Prandtl number in the generalized experimental equation of Forced convection generalized equation	n_forced
$C_{lam}$	$0.59$		Gain parameter for Reynolds number in the generalized experimental equation of Natural convection generalized equation when Fluid is laminar	C_lam
$n_{lam}$	$\frac{1}{4}$		Exponent parameter for Reynolds number in the generalized experimental equation of Natural convection generalized equation when Fluid is laminar	n_lam
$C_{tur}$	$0.1$		Gain parameter for Reynolds number in the generalized experimental equation of Natural convection generalized equation when Fluid is Turbulent	C_tur
$n_{tur}$	$\frac{1}{3}$		Exponent parameter for Reynolds number in the generalized experimental equation of Natural convection generalized equation when Fluid is Turbulent	n_tur
$Threshold$	$10^{9}$		Threshold value for Reynolds number to define Laminar or Turbulent	TH
$h$	$1$	$\frac{W}{m^{2} \cdot K}$	Constant heat transfer coefficient	h
$Use Correction input$	$false$		If true, input of correction for h is valid	use_correction

	See Also
	Heat Transfer Library Overview Basic Overview

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