# Free Convection External Heat exchange due to natural convection for external surfaces

# Free Convection Internal Heat exchange due to natural convection for closed volumes

# 1.   Free convection (general)

## 1.1.   Parameters and variables

Parameter Description
$$g$$ earth acceleration (9.81 m/s 2)
$$T_s$$ surface temperature
$$T_\infty$$ fluid temperature (far from the surface)
$$p$$ fluid pressure
$$\mu$$ dynamic viscosity
$$\nu = \mu / \rho$$ kinematic viscosity
$$c_p$$ specific heat capacity at constant pressure
$$\beta=\frac{1}{\rho}\left(\frac{\partial\rho}{\partial T}\right)_{p}$$ isobaric thermal expansion coefficient
$$\lambda$$ thermal conductivity
$$\alpha$$ convection heat rate coefficient

## 1.2.   Dimensionless groups

Group Name Meaning
$$Pr = \frac{c_p \mu}{\lambda}$$ Prandtl number

ratio of momentum diffusivity

to thermal diffusivity

$$Gr =\frac{g s^3 \beta \Delta T}{\nu^2}$$ Grashof number

ratio of buoyancy forces

to viscous forces

$$Ra = Gr\cdot Pr$$ Rayleigh number
$$Nu = \frac{\alpha s}{\lambda}$$ Nusselt number

ratio of convective heat transfer

to conductive heat transfer

## 1.3.   Calculation algorithm

The following steps are taken to compute the convective heat exchange:

1. Compute the fluid properties
2. Determine the characteristic length and surface area for the particular geometry configuration
3. Compute the Grasshof number
4. Compute the Rayleigh number
5. Determine whether laminar or turbulent flow occurs
6. Use the appropriate Nusselt number correlation to compute the $$Nu=f\left(Ra,Pr\right)$$
7. Compute the convection coefficient
\begin{equation*} \alpha=\frac{Nu\cdot\lambda}{s} \end{equation*}
1. Compute the heat flow rate
\begin{equation*} \dot{Q}=\alpha\cdot A\cdot\Delta T \end{equation*}

# 2.   External Flow

The following geometry configurations have been implemented for external free convection:

## 2.1.   Vertical plane

Parameter Description
$$h$$ height
$$w$$ width
$$s = h$$ characteristic length
\begin{equation*} Nu=\left\{ 0.825+0.387\left[Ra\cdot f_{1}\left(Pr\right)\right]^{\frac{1}{6}}\right\} ^{2} \end{equation*}

where

\begin{equation*} f_{1}\left(Pr\right)=\left[1+\left(\frac{0.492}{Pr}\right)^{\frac{9}{16}}\right]^{-\frac{16}{9}} \end{equation*}

## 2.2.   Vertical cylinder

Parameter Description
$$h$$ height
$$d$$ diameter
$$s = h$$ characteristic length
\begin{equation*} Nu=Nu_{plate}+0.97\frac{h}{d} \end{equation*}

where $$Nu_{plate}$$ is the Nusselt number for a vertical plate with height $$h$$

## 2.3.   Inclined plane

Parameter Description
$$l$$ length (inclined)
$$w$$ width
$$\alpha$$ inclination angle ($$\alpha = 0$$ vertical, $$\alpha = 90^\circ$$ horizontal)
$$s = l$$ characteristic length

There are two cases:

### 2.3.1.   Top side of cold plane or bottom side of hot plane

The favorable pressure gradient stabilizes the boundary layer and pushes it towards the plate. The resulting Nusselt number can be obtained from the equation for vertical plane by substituting $$Ra_{\alpha}=Ra\cdot \cos (\alpha)$$ for $$Ra$$.

### 2.3.2.   Top side of hot plane or bottom side of cold plane

At low $$Ra$$ the same holds: substitute $$Ra_{\alpha}=Ra\cdot\cos(\alpha)$$ for $$Ra$$ in the equation for vertical plate. At $$Ra > Ra_{c}$$, the adverse pressure gradient tends to cause boundary layer separation. In this case

\begin{equation*} Nu=0.56\left(Ra_{c}\cdot \cos (\alpha)\right)^{\frac{1}{4}}+0.13\left(Ra^{\frac{1}{3}}-Ra_{c}^{\frac{1}{3}}\right) \end{equation*}

The critical Rayleigh number is a function of the angle $$\alpha$$ and is given by:

\begin{equation*} Ra_{c}=10^{\left(8.9-0.00178\cdot\alpha^{1.82}\right)} \end{equation*}

## 2.4.   Horizontal plane

Shape Parameter Description
Rectangle

$$l$$

$$w$$

$$s = [l\cdot w]/[2(l + w)]$$

length

width

characteristic length

Circle

$$d$$

$$s = d$$

diameter

characteristic length

Once again there are two cases:

### 2.4.1.   Top side of hot plane or bottom side of cold plane

The fluid flow is laminar for

\begin{equation*} Ra\cdot f_{1}\left(Pr\right)<7\cdot10^{4} \end{equation*}

and turbulent otherwise. The Nusselt number is found from:

\begin{equation*} Nu=\begin{cases} 0.766\cdot\left[Ra\cdot f_{1}\left(Pr\right)\right]^{\frac{1}{5}} & \textrm{if flow is laminar}\\ 0.15\cdot\left[Ra\cdot f_{1}\left(Pr\right)\right]^{\frac{1}{3}} & \textrm{if flow is turbulent} \end{cases} \end{equation*}

where

\begin{equation*} f_{1}\left(Pr\right)=\left[1+\left(\frac{0.322}{Pr}\right)^{\frac{11}{20}}\right]^{-\frac{20}{11}} \end{equation*}

### 2.4.2.   Top side of cold plane or bottom side of hot plane

\begin{equation*} Nu=0.6\left[Ra\cdot f_{1}(Pr)\right]^{\frac{1}{5}} \end{equation*}

on condition that

\begin{equation*} 10^{3}<Ra\cdot f_{1}\left(Pr\right)<10^{10} \end{equation*}

where

\begin{equation*} f_{1}\left(Pr\right)=\left[1+\left(\frac{0.492}{Pr}\right)^{\frac{9}{16}}\right]^{-\frac{16}{9}} \end{equation*}

## 2.5.   Horizontal cylinder

Parameter Description
$$d$$ diameter
$$l$$ length
$$s = d$$ characteristic length
\begin{equation*} Nu=\left\{ 0.60+0.387\left[Ra\cdot f_{1}\left(Pr\right)\right]^{\frac{1}{6}}\right\} ^{2} \end{equation*}

where

\begin{equation*} f_{1}\left(Pr\right)=\left[1+\left(\frac{0.559}{Pr}\right)^{\frac{9}{16}}\right]^{-\frac{16}{9}} \end{equation*}

## 2.6.   Sphere

Parameter Description
$$d$$ diameter
$$s = d$$ characteristic length
\begin{equation*} Nu=0.56\left[\left(\frac{Pr}{0.846+Pr}\right)Ra\right]^{\frac{1}{4}}+2 \end{equation*}

## 2.7.   Finned pipe

Parameter Description
$$d$$ core pipe diameter
$$h_f$$ fin height (above core pipe)
$$d_e = d + h_f$$ effective diameter
$$b$$ fin spacing
$$s = d_e$$ characteristic length
\begin{equation*} Nu=0.24\left(Ra\frac{b}{d}\right)^{\frac{1}{3}} \end{equation*}

Note: the accuracy of the correlation is $$\pm 25\%$$

# 3.   Internal Flow

The following geometry configurations have been implemented for internal free convection:

## 3.1.   Vertical planes

Parameter Description
$$h$$ height
$$w$$ width
$$d$$ distance between planes
$$s = d$$ characteristic length

If

\begin{equation*} \frac{h}{s}<80 \end{equation*}

for

\begin{equation*} 10^{4}<Ra<10^{7} \end{equation*}

then

\begin{equation*} Nu=0.42\cdot Pr^{0.012}\cdot Ra^{0.25}\left(\frac{h}{s}\right)^{-0.25} \end{equation*}

while for

\begin{equation*} 10^{7}<Ra<10^{9} \end{equation*}

the Nusselt number is derived from

\begin{equation*} Nu=0.049\cdot Ra^{0.33} \end{equation*}

In the case of

\begin{equation*} Ra>10^{9} \end{equation*}

the Nusselt correlation is unknown.

## 3.2.   Inclined planes

Parameter Description
$$l$$ length (inclined)
$$w$$ width
$$d$$ distance between planes
$$\alpha$$ inclination angle ($$\alpha = 0$$ vertical, $$\alpha = 90^\circ$$ horizontal)
$$s = d$$ characteristic length

There are two cases:

### 3.2.1.   Heat is transmitted upwards

\begin{equation*} Nu=C\cdot Ra^{0.33}\cdot Pr^{0.074} \end{equation*}

where $$C$$ is determined from $$\alpha$$ based on the following values:

$$\alpha$$ $$C$$
$$0^\circ$$ $$4.9\cdot10^{-2}$$
$$30^\circ$$ $$5.7\cdot10^{-2}$$
$$45^\circ$$ $$5.9\cdot10^{-2}$$
$$60^\circ$$ $$6.5\cdot10^{-2}$$
$$90^\circ$$ $$6.9\cdot10^{-2}$$

### 3.2.2.   Heat is transmitted downwards

If

\begin{equation*} 5\cdot10^{3}<Ra<10^{8} \end{equation*}

for

\begin{equation*} \alpha=45^{\circ} \end{equation*}

the Nusselt number is calculated from the formula

\begin{equation*} Nu=1+\frac{0.025\cdot Ra^{1.36}}{Ra+1.3\cdot10^{4}} \end{equation*}

In the other cases, the Nusselt correlation is unknown.

## 3.3.   Horizontal planes

Parameter Description
$$l$$ length
$$w$$ width
$$d$$ distance between planes
$$s = d$$ characteristic length

For

\begin{equation*} Ra>Ra_{c}\left(Ra_{c}=1708\right) \end{equation*}

if

\begin{equation*} 1708<Ra<2.2\cdot10^{4} \end{equation*}

the Nusselt number is determined by the correlation

\begin{equation*} Nu=0.208\cdot Ra^{0.25} \end{equation*}

while for

\begin{equation*} Ra<2.2\cdot10^{4} \end{equation*}

it can be obtained using the formula

\begin{equation*} Nu=0.092\cdot Ra^{0.33} \end{equation*}

For

\begin{equation*} Ra<Ra_{c}\left(Ra_{c}=1708\right) \end{equation*}

no convection occurs. Heat exchange is purely by conduction.

## 3.4.   Horizontal annuli

Parameter Description
$$l$$ length
$$r_{i}$$ inner radius
$$r_{o}$$ outer radius
$$s = r_{o}-r_{i}$$ characteristic length

For

\begin{equation*} Ra>7.1\cdot10^{3} \end{equation*}

if

\begin{equation*} \frac{r_{o}}{r_{i}}\leqq8 \end{equation*}

and if heat is transmitted outwards, the Nusselt correlation is:

\begin{equation*} Nu=0.2\cdot Ra^{0.25}\cdot\left(\frac{r_{o}}{r_{i}}\right)^{0.5} \end{equation*}

In the other cases, the Nusselt correlation is unknown.