**Thermal Circuitry and Coffee Mugs**

**Which Coffee Cup is the Best?**

Thermal Circuit Analysis

1-D Radial Heat Transfer

Steady State

Not Considering Cup Design

Uniform Material

Solving for Heat Flux

Assumptions and Heat Transfer Calculations

Based on Ohm's Law

**Calculations**

V= iR

ΔT = qR

Heat Transfer

T = Change in temperature

Finding Total Thermal Resistance

Equivalent resistance in series

R = R + R + R

coffee

mug

air

eq

Conductive resistance = L/kA

Convective resistance = 1/hA

k = thermal conductivity, h = convection coefficient, L = length, A = area

q'' denotes the heat flux which is a measure of heat transfer per unit area

Ceramic (Earthenware) Mug

Rmug = L/kA

Glass Mug

Rmug = L/kA

Paper Cup

Which mug is the best choice?

(heat transfer rate per unit area)

**Use the principles of thermal circuitry to determine which mug is best to keep your coffee warm based on the rate of heat transfer.**

**Earthenware**

**Glass**

**Paper Cup**

total

R = Equivalent thermal resistance

q = Heat transfer rate

total

Other Considerations

Treat handle as a fin

Heat loss in second direction through open top and bottom neglected for simplicity

Analyzing heat loss through the walls of the vessel defined as a simple cylinder

L = Thickness of mug wall

k = Thermal Conductivity

k = 0.9 W/mK

Using the heat flux equation

q'' = 668 W/m

T = T - T

coffee

air

L = 1/16"

coffee

mug

air

L = 1/8"

h = 750 W/mK

h = 10 W/mK

coffee

air

T = 90 C

coffee

T = 20 C

air

2

L = Thickness of mug wall

k = Thermal Conductivity

k = 1.0 W/mK

Using the heat flux equation

q'' = 680 W/m

2

Rmug = L/kA

L = thickness of mug wall

L = 1/40"

k = Thermal Conductivity

k = 0.18 W/mK

Using the heat flux equation

q'' = 680 W/m

2

Conclusions are based on many assumptions

Earthenware will be the best choice

Coffee Joulies