Heat Diffusion Equation Apply this equation to a solid undergoing conduction heat transfer as shown where: E = mc p T= (  V)c p T =  (dxdydz)c p T x qxqx q x+dx = 0 Heat diffusion equation – mod 11/27/01 dy dx y Steady State Conduction The above equation states: Rate of change of total energy (dE/dt) = Rate of energy generated (dEg/dt) + Rate of Heat Transfer in ( q in ) - Rate of Heat Transfer out ( q out ) q in = q out =

Heat Diffusion Equation (cont’d, page 2) Generalized to three-dimensions Note: partial differential operator is used since T = T(x,y,z,t) Substituting in the Energy Balance equation, we obtain: dE/dt dE g /dt q in - q out =>

Heat Diffusion Equation (cont’d, page 3)

1-D, Steady State Conduction Example: T(x=0)=100°C=C 2 T(x=1 m)=20°C=C 1 +C 2, C 1 =-80°C T(x)=100-80x (°C) T x Constant

1-D Heat, Steady State Heat Transfer Thermal Resistance (Electrical Circuit Analogy) T1T1 T2T2 x L q (I) T 1 (V 1 ) T 2 (V 2 ) R (R) This is analogous to an electrical circuit I = (V 1 -V 2 )/R Constant

Thermal Resistance - Composite Wall Heat Transfer T2T2 T1T1 R 1 =L 1 /(k 1 A) R 2 =L 2 /(k 2 A) T T1T1 T2T2 L1L1 L2L2 k1k1 k2k2 T Use of the thermal resistance concept make the analysis of complex geometries relatively easy, as discussed in the example below. However, note that the thermal heat resistance concept can only be applied for steady state heat transfer with no heat generation. Example: Consider a composite wall made of two different materials

Composite Wall Heat Transfer (cont’d) R 1 =L 1 /(k 1 A) R 2 =L 2 /(k 2 A) T2T2 T1T1 T T1T1 T2T2 L1L1 L2L2 k1k1 k2k2 T Now consider the case where we have 2 different fluids on either sides of the wall at temperatures, T ,1 and T ,2, respectively. There is heat transfer by convection from the first fluid (on left) to material 1 and from material 2 to the second fluid (on right). Similar to conduction resistance, we can determine the convection resistance, where R conv = 1/hA T,1T,1 T,2T,2 T,1T,1 T,2T,2 R conv,1 = 1/(h 1 A) R conv,2 = 1/(h 2 A) where This approach can be extended to much more complex geometries (see YAC, 8-4 and 8-5)

“R” value of insulations Q: Why is the R value given by L/k ? A: From previous slide, we know that the thermal resistance to conduction is defined as the temperature difference across the insulation by the heat flux going through it, hence: Example: The typical space inside the residential frame wall is 3.5 in. Find the R-value if the wall cavity is filled with fiberglass batt. (k=0.046 W/m.K=0.027 Btu/h.ft.R) In the US, insulation materials are often specified in terms of their thermal resistance, denoted as R values, where R value = L/k (thickness/thermal conductivity) In the US, it has units of (hr ft 2 °F)/Btu (Note: 1 Btu=1055 J) R-11 for wall, R-19 to R-31 for ceiling.