Convective heat transfer with calculation of heat transfer coefficient

Block appearance in REPEAT VISION

 The Convective heat transfer with calculation of heat transfer coefficient block describes the heat transfer between air flows and the body surface.

 Heat flux in the block according to the Newton’s law of cooling (1) :

(1)\[ Q=\mathrm{k}\cdot \mathrm{F}\left(T_{1}-T_{2}\right) \]

where:

 \(k\) - heat transfer coefficient, [W/(m\(^2\) \(\cdot\) ℃)];

 \(F\) - surface area of the heat exchange, [m\(^2\)];

 \(T_1\) - air temperature, [℃];

 \(T_2\) - temperature of the body surface, [℃].

 Calculation of the heat transfer factor

 Calculation of heat transfer factor Convective heat transfer is described by the criterion equation as follows (2) :

(2)\[ \mathrm{N}\mathrm{u}=f\left(\mathrm{R}\mathrm{e},\mathrm{P}\mathrm{r},\mathrm{G}\mathrm{r}, Г \dots \right) \]

where:

 \(\mathrm{N}\mathrm{u}=\frac{k\cdot l}{\lambda }\) - Nusselt criterion;

 \(\mathrm{R}\mathrm{e}=\frac{w\cdot l}{\nu }\) - Reynolds criterion;

 \( \mathrm{P}\mathrm{r}=\frac{\nu }{\mathrm{а}}\) - Prandtl criterion;

 \( \mathrm{G}\mathrm{r}=\frac{g\cdot l^{3}}{\nu ^{2}}\beta ∆\mathrm{t}\) - Grashof criterion;

 \( Г=\frac{l_{i}}{l}\) - dimensionless geometric simplex;

 \(k\) = heat transfer factor, [\(\frac{\mathrm{В}}{\mathrm{m}^{2}\cdot \mathrm{℃}}\)];

 \(\nu\) - kinematic viscosity coefficient of the heat transfer medium, [\(\frac{m^2}{s}\)];

 \(\omega\) - heat transfer medium speed, [\(\frac{m}{s}\)];

 \(a\) - thermal diffusivity coefficient, [\(\frac{m^2}{s}\)];

 \(g\) - gravity factor, [\(\frac{m^2}{s}\)];

 \(l\) - defining dimension, [m];

 \(l_i\) - characteristic dimension, [m];

 \( \beta \) - thermal expansion coefficient, [\(\frac{1}{K}\)];

 \(\Delta t = t_{wall} - t_{f}\) - temperature difference between the wall and the heat transfer medium, [℃];

 \(\lambda\) - heat transfer coefficient of the heat transfer medium, [\(\frac{\mathrm{W}}{\mathrm{m}\cdot \mathrm{℃}}\)];

 The heat transfer coefficient is calculated using the following formula (3) :

(3)\[ \alpha =\frac{\mathrm{N}\mathrm{u}\cdot \lambda }{l} \]

 The Nusselt criterion for forced flow in straight pipes and channels is calculated using the following equations:

 Laminar flow of the heat transfer medium, \( \mathrm{R}\mathrm{e}\le 2320\) (4) :

(4)\[ \mathrm{N}\mathrm{u}=\mathrm{0,15}\cdot \mathrm{R}\mathrm{e}^{\mathrm{0,33}}\cdot \mathrm{P}\mathrm{r}^{\mathrm{0,43}}\cdot \mathrm{G}\mathrm{r}^{\mathrm{0,1}}\cdot \left(\frac{\mathrm{P}\mathrm{r}}{\mathrm{P}\mathrm{r}_{\mathrm{с}\mathrm{т}}}\right)^{\mathrm{0,25}}\cdot \epsilon _{l} \]

where:

 \( \Pr_{ст}\) - Prandtl criterion for the heat transfer medium with the wall temperature;

 \(\epsilon_l\) - – coefficient determined from the tables 1 and 2;

 Transient flow of the heat transfer medium, \( 2320\le \mathrm{R}\mathrm{e}<10000\) (5) :

(5)\[ \mathrm{N}\mathrm{u}=\mathrm{0,008}\cdot \mathrm{R}\mathrm{e}^{\mathrm{0,9}}\cdot \mathrm{P}\mathrm{r}^{\mathrm{0,43}} \]

 Turbulent flow of the heat transfer medium, \(\mathrm{R}\mathrm{e}\ge 10000\) (6) :

(6)\[ \mathrm{N}\mathrm{u}=\mathrm{0,021}\cdot \mathrm{R}\mathrm{e}^{\mathrm{0,8}}\cdot \mathrm{P}\mathrm{r}^{\mathrm{0,43}}\left(\frac{\mathrm{P}\mathrm{r}}{\mathrm{P}\mathrm{r}_{\mathrm{с}\mathrm{т}}}\right)^{\mathrm{0,25}}\cdot \epsilon _{l} \]

 The defining temperature in the equations presented above is the mean temperature of the fluid, the defining dimension is the equivalent diameter of the flow cross-section (7) :

(7)\[ d_{\mathrm{э}\mathrm{к}\mathrm{в}}=\frac{4\cdot S}{\mathrm{П}} \]

where:

 \(S\) - cross-sectional area of liquid flow, [m\(^2\)];

 П - wetted perimeter, [m].

Table 1 - \(\epsilon _1\) coefficient value if \(Re < 10000\)

\(\frac{l}{d_{equ}}\)	1	2	5	10	15	20	30	40	50
\( \epsilon_l \)	1,9	1,7	1,44	1,26	1,18	1,13	1,05	1,02	1

Table 2 - \(\epsilon _1\) coefficient value if \(Re > 10000\)

Re	\(\frac{l}{d_{equ}}\)
	10	20	30	40	50
\(1 \cdot 10^4\)	1.23	1.13	1.07	1.03	1
\(2 \cdot 10^4\)	1.18	1.1	1.05	1.02	1
\(5 \cdot 10^4\)	1.13	1.08	1.04	1.02	1
\(1 \cdot 10^5\)	1.1	1.06	1.03	1.02	1
\(1 \cdot 10^6\)	1.05	1.03	1.02	1.01	1

 For free convection, the Nusselt criterion is calculated using the following formula (8):

(8)\[ \mathrm{N}\mathrm{u}=C\cdot \left(\mathrm{G}\mathrm{r}\cdot \mathrm{P}\mathrm{r}\right)^{n}\cdot \left(\frac{\mathrm{P}\mathrm{r}}{\mathrm{P}\mathrm{r}_{\mathrm{с}\mathrm{т}}}\right)^{\mathrm{0,25}} \]

where C and n depend on the mode of heat transfer medium flow and are determined according to the Table 3.

Table 3 - Value of coefficient C and the exponent n

Mode	\(Gr \cdot Pr\)	\(C\)	\(n\)
Laminar	\( 1\cdot 10^{-3}\div 5\cdot 10^{2}\)	1,18	0,125
Transient	\( 5\cdot 10^{2}\div 2\cdot 10^{7}\)	0,54	0,25
Turbulent	\( 2\cdot 10^{7}\div 1\cdot 10^{12}\)	0,185	0,33

 The defining geometric dimension is the height of the vertical heat exchange surface, in case of horizontal pipes it is the diameter. The defining temperature is the average temperature of the heat transfer medium.

 For plate flow the Nusselt criterion is calculated by formulas:

• laminar flow of coolant \(Re < 5 \cdot 10^5\) (9):

(9)\[ Nu=\mathrm{0,66}\cdot \mathrm{R}\mathrm{e}^{\mathrm{0,5}}\cdot \mathrm{P}\mathrm{r}^{\mathrm{0,33}}\cdot \left(\frac{\mathrm{P}\mathrm{r}}{\mathrm{P}\mathrm{r}_{\mathrm{с}\mathrm{т}}}\right)^{\mathrm{0,25}} \]

• turbulent mode of coolant flow (10):

(10)\[ Nu=\mathrm{0,037}\cdot \mathrm{R}\mathrm{e}^{\mathrm{0,8}}\cdot \mathrm{P}\mathrm{r}^{\mathrm{0,43}}\cdot \left(\frac{\mathrm{P}\mathrm{r}}{\mathrm{P}\mathrm{r}_{\mathrm{с}\mathrm{т}}}\right)^{\mathrm{0,25}} \]

 Calculation of the Nusselt criterion for natural and forced convection of a horizontal cylinder

 The determining temperature: t=\(\frac{(t_{ст}+t_ж)}{2}\), the determining size is the diameter.

Natural convection: (11):

(11)\[Nu=(0,6+\frac{0.387 \cdot Ra^{\frac{1}{6}}}{(1+(\frac{0.559}{Pr})^{\frac{9}{16}})^{\frac{8}{27}}})^2\]

Forced convection: (12):

(12)\[Nu=C\cdot Re^m\cdot Pr^\frac{1}{3}\]

Table 4 - The value of the coefficient C and the exponent m

Re	\(C\)	\(m\)
< 4	0,989	0,33
4 - 40	0,911	0,385
4 - 4000	0,683	0,466
4000 - 40000	0,193	0,618
> 40000	0,027	0,805

 Calculation of the Nusselt criterion for natural and forced convection of a plate

 Determining temperature: t=\(\frac{(t_{ст}+t_ж)}{2}\), determining size:

Table 5 - Value of coefficient C and degree index m

Plate inclination angle	Defining size
0 or 180	\(\frac{(L\cdot W)}{2\cdot (L+W)}\)
0 < Plate angle < 180	L

where:

 L - plate length, [m];

 W - plate width, [m].

Natural convection

Cold plate - hot coolant

• Tilt angle = 0 (13):

(13)\[Nu=0,27 \cdot Ra^{\frac{1}{4}}\]

• 0 < Inclination angle < 30 (14):

(14)\[Nu=Nu(0)+\frac{Nu(30)-Nu(0)}{0,5}\cdot sin(\alpha)\]

where \(\alpha\) is the plate inclination angle, [deg].

• 30 ≤ Inclination angle < 90 (15):

(15)\[Nu=(0,825+\frac{0.387 \cdot (Ra\cdot sin(\alpha))^{\frac{1}{6}}}{(1+(\frac{0,492}{Pr})^{\frac{9}{16}})^{\frac{8}{27}}})^2\]

where \(\alpha\) is the plate inclination angle, [deg].

• Inclination angle = 90 (16):

(16)\[Nu=(0,825+\frac{0.387 \cdot Ra^{\frac{1}{6}}}{(1+(\frac{0,492}{Pr})^{\frac{9}{16}})^{\frac{8}{27}}})^2\]

• 90 < Inclination angle < 180 (17):

(17)\[Nu=Nu(180)+(Nu(90)-Nu(180))\cdot sin(\alpha)\]

where \(\alpha\) is the plate inclination angle, [deg].

• Inclination angle = 180

Ra < 4.74E6 (18):

(18)\[Nu=0,54 \cdot Ra^{\frac{1}{4}}\]

Ra > 4.74E6 (19):

(19)\[Nu=0,15 \cdot Ra^{\frac{1}{3}}\]

Hot plate - cold coolant

• Inclination angle = 0

Ra < 4.74E6 (20):

(20)\[Nu=0,54 \cdot Ra^{\frac{1}{4}}\]

Ra > 4.74E6 (21):

(21)\[Nu=0,15 \cdot Ra^{\frac{1}{3}}\]

• 0 < Inclination angle < 90 (22):

(22)\[Nu=Nu(0)+(Nu(90)-Nu(0))\cdot sin(\alpha)\]

where \(\alpha\) is the plate inclination angle, [deg].

• Inclination angle = 90 (23):

(23)\[Nu=(0,825+\frac{0.387 \cdot Ra^{\frac{1}{6}}}{(1+(\frac{0,492}{Pr})^{\frac{9}{16}})^{\frac{8}{27}}})^2\]

• 90 ≤ Inclination angle < 150 (24):

(24)\[Nu=(0,825+\frac{0.387 \cdot (Ra\cdot sin(\alpha))^{\frac{1}{6}}}{(1+(\frac{0,492}{Pr})^{\frac{9}{16}})^{\frac{8}{27}}})^2\]

where \(\alpha\) is the plate inclination angle, [deg].

• 150 < Inclination angle < 180 (25):

(25)\[Nu=Nu(0)+\frac{Nu(150)-Nu(180)}{0,5}\cdot sin(\alpha)\]

where \(\alpha\) is the plate inclination angle, [deg].

• Inclination angle = 180 (26):

(26)\[Nu=0,27 \cdot Ra^{\frac{1}{4}}\]

Forced convection

• Laminar flow Re < 31836 (27):

(27)\[Nu=0,664 \cdot Ra^{\frac{1}{2}}\cdot Pr^{\frac{1}{3}}\]

• Turbulent flow Re > 31836 (28):

(28)\[Nu=0,037 \cdot Re^{\frac{4}{5}}\cdot Pr^{\frac{1}{3}}\]

1.  Properties

Property	Default value	Description	Units of measurement
Surface area	10	-	m\(^2\)
Convection type: Natural convection; Forced convection; Mixed convection	Natural convection	-	-
Flow around type: Straight pipes; Plane surface; Horizontal cylinder	Straight pipes	-	-
Characteristic dimension	10	-	m
Pipe length	10	Available, when Flow around type - Straight pipes	m
Wetted perimeter	30	Available, when Flow around type - Straight pipes	m
Cross-sectional area of medium flow	30	Available, when Flow around type - Straight pipes	m\(^2\)
Plate inclination angle	60	Available, when Flow around type - Plane surface	ᵒ
Plate width	20	Available, when Flow around type - Plane surface	m
Plate length	20	Available, when Flow around type - Plane surface	m

2.  Inputs and outputs

Parameters	Input/Output	Description	Units of measurement	Port type
Velocity	Input	-	m/s	Automation
Input	Input	-	-	Heat exchange
Output	Output	-	-	Heat exchange

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