Heat Convection - MapleSim Help

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 :

The extended equation which is implemented in this library is:

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

Heat transfer coefficient $\mathrm{h_act}$ 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.

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

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

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

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

For more information, see Fluid Properties Check.

$\mathrm{Ra}=\mathrm{Gr}\cdot \mathrm{Pr}$

$g$ 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].

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

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].

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

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].

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

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.

$\mathrm{Nu}={C}_{\mathrm{forced}}\cdot \left({\mathrm{Re}}^{{m}_{\mathrm{forced}}}-{\mathrm{offset}}_{\mathrm{forced}}\right)\cdot {\mathrm{Pr}}^{{n}_{\mathrm{forced}}}$

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

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

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.

 • 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.

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

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

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

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 ${\mathit{10}}^{\mathit{2}}\mathit{<}\mathrm{Re}\mathit{<}{\mathit{10}}^{\mathit{7}}$, $\mathrm{Re}\cdot \mathrm{Pr}>0.2$.

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

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 $3.5<\mathrm{Re}<8\cdot {10}^{4}$, $0.7<\mathrm{Pr}<380$.

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

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 $h$. $\mathrm{h_act}=h$
 Convection type: External input If using this type, the heat transfer coefficient is specified by the signal input ${h}_{\mathit{in}}$. $\mathrm{h_act}=\mathrm{h__in}$
 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 $\mathrm{Q__flow}$ $W$ Heat flow rate from port a to solid port Q_flow $K$ Temperature of solid port $K$ Temperature of fluid port $\mathrm{h_act}$ $\frac{W}{{m}^{2}\cdot K}$ Actual heat transfer coefficient h_act $\stackrel{‾}{h}$ $\frac{W}{{m}^{2}\cdot K}$ The average heat transfer coefficient $\mathrm{Nu}$  Nusselt number $\mathrm{Re}$  Reynolds number $\mathrm{Pr}$  Prandtl number $\mathrm{Gr}$  Grashof number $\mathrm{Ra}$  Rayleigh number $k$ $\frac{W}{m\cdot K}$ Thermal conductivity $\mathrm{η__s}$ $\mathrm{Pa}\cdot s$ Viscosity calculated with temperature of solid port $\mathrm{η__f}$ $\mathrm{Pa}\cdot s$ Viscosity calculated with temperature of fluid port

Connections

 Name Units Condition Description Modelica ID $\mathrm{solid}$ - - Thermal port of the solid side solid $\mathrm{fluid}$ - - Thermal port of the fluid side fluid $h\mathrm{in}$ $\frac{W}{{m}^{2}\cdot K}$ 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 $\mathrm{cor}$ - if Use correction input is true. Input signal of the correction factor for ${Q}_{\mathrm{flow}}$ cor

Parameters

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