Objective: 1D (or higher if desired) steady-state heat transfer analysis for a heat source conducting through an insulating later to the air, where convection and radiation supply the boundary condition.
Accumulation = heat in – heat out + generation Þ
Nussault dimensionless number for heat flow with forced convection:
Radiation equation (Stefan-Boltzmann Equation):
Useful quantities:
Thermal conductivity k : kair = 0.0238 W/(mK), k of fabric = kc ~ 0.1 W/(mK)
Surface area of glove ~ 0.05 m2
a
= 5.67x10-8 W/(m2K4)
Figure: Model for heat transfer from a finger of the glove, considering the finger as a cylinder. The layers under consideration are shown. The heating element is embedded between two layers of glove fabric. There is thin layer of trapped air between the finger and the glove.
Temperature profile for layers: finger, heater, fabric and boundary layer (BL)
Dimensionless Numbers
Prandtl number (Pr) is a dimensionless number that applies to convection and controls the similarity between the velocity profile and the temperature profile.
where 
.
n
= kinematic viscosity = h
/r
(in units m2/s)
h
= viscosity (in units kg/m-s)
r
= density (in units kg/m3)
cp = specific heat (J/kg-K)
k = thermal conductivity (in units W/m-K)
*Pr for gases ranges from 0.7 - 1.0 and show almost no variation with temperature
Nusselt number (Nu) is a dimensionless number associated with conditions of forced convection
h = heat transfer coefficient (in units W/m2K)
D = distance
Reynolds number (Re) is a dimensionless number that also applies to conditions of forced convection and distinguishes between laminar and turbulent fluid flow
v = velocity of fluid (in units m/s)
In the following heat transfer calculation and using the suggestion of Prof. Powell, the hand will be modeled after a cylinder with the radius equal to the thickness of one finger. Even though the hand is not of this shape, it is adequate for our model, which also takes into account that most of the heat loss from the hand comes from the fingers. We will calculate heat loss from the hand, using the condition of forced convection and again using the condition of natural convection. Let's say that with natural convection, such as in an enclosed room at -10°C, your gloved hand (without a heating element) is comfortable. However, with forced convection, such as wind moving at 15 miles/hour as you're walking across the Massachusetts Avenue bridge in -10°C, your gloved hand is not comfortable. In this condition of forced convection, your hand is losing much more heat and a heating element is needed. Just how much heat should this heating element supply? Let's say that is equal to the difference in heat loss for the two situations: heat loss in forced convection - heat loss in natural convection. The following temperature diagram is used:
Heat loss from your hand through the glove fabric and through the boundary layer (BL) is modeled after a sum of resistances for these two layers:
Figure: Temperature profile from hand at body temperature (34°C) to ambient temperature (-10°C). Just a schematic for the purpose of showing the layers and values used in calculation, not an exact profile.
Forced Convection
According to the McAdams correlation, which correlates heat transfer coefficients for forced convection past a submerged, long cylindrical-shaped object (such as the finger) and is based on experiments with water, oils and air:
For 1 < Re < 5x103: Nu Pr -0.3 = 0.35 Re0.52
For 103 < Re < 5x104: Nu Pr -0.3 = 0.26 Re0.60
First Re must be calculated to determine which of the above correlations to use:
Natural Convection
The following equations calculate h for a horizonal pipe in the condition of natural convection of air:
For 5x10-6 < D3D T < 50: h = 1.22(D T/D)1/4
For 50 < D3D T < 5x104: h = 1.24(D T)1/3
For our situation: D3D T = (0.05)3(44) = 5.5x10-3
Therefore, the first equation will be used to calculate h in the condition of natural convection.
h = 1.22(D T/D)1/4 = 1.22(44/0.05)1/4 = 6.6 W/m2K
Now, we have the 2 values of h, for forced and for natural convection. These are evaluated into our equation for q (W/m2):
Therefore, according to the above calculations and assumptions, we want the heating element in our gloves to supply 5.2W - 2.5W = 2.7W.
The equations and the values for air used in the above calculations came from the 3.185 textbook, D.R. Poirier and G.H. Geiger, Transport Phenomena in Materials Processing, 1994.