Theory

The outer surface is treated as a horizontal cylinder of diameter $D_{out}$ : the gas-side characteristic diameter $4V/A_{walls}$ (exact for a pipe, an equivalent length for a can-shaped muffler) plus twice the layer stack. The air properties are those of dry air at the film temperature $T_f = (T_s + T_a)/2$ and the ambient pressure: ideal-gas density, Sutherland viscosity, $c_p(T)$ of air, $k = \mu c_p/Pr$ with $Pr$ of the Material step, $\beta = 1/T_f$ .

Still air: natural convection from a horizontal cylinder, Churchill and Chu (Churchill and Chu, 1975), equation (1).
Vehicle moving: forced convection across a cylinder at the vehicle speed, Churchill and Bernstein (Churchill and Bernstein, 1977), equation (2).
Manual h: the value of Outer heat-transfer coefficient, used as it is.

The solver receives this convective coefficient; radiation is added by the boundary condition itself through the emissivity. Because $h$ depends on the outer surface temperature $T_s$ , it is evaluated first at a guessed $T_s$ and updated once from the solved heat flux after the start-up phase (Thermal condition of a wall).

Equations

Nu = \left[0.60 + \frac{0.387\,Ra^{1/6}}{\left(1 + (0.559/Pr)^{9/16}\right)^{8/27}}\right]^2, \qquad Ra = \frac{g\,\beta\,\lvert T_s - T_a\rvert\,D_{out}^3}{\nu\,\alpha}

(1)

Nu = 0.3 + \frac{0.62\,Re^{1/2}Pr^{1/3}}{\left(1 + (0.4/Pr)^{2/3}\right)^{1/4}}\left[1 + \left(\frac{Re}{282000}\right)^{5/8}\right]^{4/5}, \qquad Re = \frac{V D_{out}}{\nu}

(2)

h = \frac{Nu\,k}{D_{out}}, \qquad D_{out} = \frac{4V_{fluid}}{A_{walls}} + 2\sum_i t_i

(3)

where	meaning	unit
$Nu$	Nusselt number	-
$Ra$	Rayleigh number	-
$Re$	Reynolds number of the cross flow	-
$Pr$	Prandtl number (Material step)	-
$g$	Gravitational acceleration, 9.80665	m/s²
$\beta$	Expansion coefficient, $1/T_f$	1/K
$\nu, \alpha$	Kinematic viscosity and thermal diffusivity of air at $T_f$	m²/s
$k$	Conductivity of air at $T_f$	W/(m K)
$V$	Vehicle speed	m/s
$D_{out}$	Outer characteristic diameter	m
$V_{fluid}$	Fluid volume of the model	m³
$A_{walls}$	Area of the outer wall patches	m²
$t_i$	Thicknesses of shell and insulation	m

Default and rule

Default: Still airComputed by the program
Rule: Still air. The automatic $h$ needs the exported geometry (volume and wall area); until then it is pending.
Source: Correlations as cited. Representing the whole body by a single cylinder of equal volume-to-wall-area ratio is the program's simplification.

Valid range Still air | Vehicle moving | Manual h

One cylinder

The whole muffler is one horizontal cylinder of diameter $4V/A$ ; a flat-sided or vertical body has a different natural convection.

Cross flow

Churchill-Bernstein is for flow across a cylinder ( $Re\,Pr > 0.2$ ); flow along the muffler's axis, or under a vehicle floor, is not modelled.

Outer environment

Theory

Equations

Default and rule

Valid range Still air | Vehicle moving | Manual h

References 2

Theory

Equations

Default and rule

Valid range Still air | Vehicle moving | Manual h

Related 7

References 2