Chapter 2 - Conduction 2012 - Web

# Chapter 2- - C HAPTER 2 – C ONDUCTION Consider a wall where the surface temperatures are such that T 1>T 2 q x = flow rate of heat in the x

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Unformatted text preview: C HAPTER 2 – C ONDUCTION Consider a wall where the surface temperatures are such that T 1 >T 2 q x = flow rate of heat in the x direction (vector!) [units of Watts] q x is dependent on THREE factors: 1. A - surface area perpendicular to the flow of heat (i.e. y x z ), 2. k – the capacity of material to transfer heat across its body (thermal conductivity) 3. L T – temperature gradient across the thickness of the wall, L T 1 T 2 q x L x These three factors are related together through FOURIER’S LAW : dx dT kA q x Q: why the negative sign? Or, in terms of heat flux q” (q” = q/A) dx dT k q x " Although we are only considering 1D heat flow in this case, Fourier’s Law may be expressed in 3 dimensions: dz dT k dy dT j dx dT i k T k q " where is the del operator NOTE: Fourier’s law is typically written individually for each dimension over which heat flow occurs. The driving force of conduction is the temperature gradient dx dT [ o C/m or K/m] Average gradient: L T T run rise dx dT 1 2 here dx dT , thus q x >0 Instantaneous gradient: may be some function of x If T(x)=Px 2 +Qx+R q x = -kA(dT/dx) = -kA(2Px+Q) T 2 T 1 L T 1 T 2 dT dx How can you determine if the temperature gradient is linear? Is there an internal source of heat? Is the material through which heat is flowing homogeneous? o Is k constant? Does the surface area perpendicular to the heat flow change with dx? o Increasing cross-sectional area = lower flux, more uniform temperature as a function of x r q wall T x T q H EAT T RANSFER P ROPERTIES Thermal Conductivity ( k ) : (W/m•K) High k values are good conductors Low k values are better insulators e.g. Material Temp (K) k (W/mK) Material Temp (K) k (W/mK) Brick 300 0.72 Air 300 0.026 Cork 300 0.039 Copper 300 401 Glass 300 1.4 Aluminum 300 237 More values available in Appendix A k is often given as a constant, but it does vary with T– get a value valid for the T range you are looking at. A more accurate relationship is: aT k k o 1 where k o and a are constants. a can be negative or positive depending on the material. For blends of materials A and B, use a weighted average to determine k : B A eff k a ak k 1 ( a = fraction of material A in the blend) For porous media consisting of a mixture of a stationary solid and a fluid, k depends on the geometry of the porous media (size distribution of pores, pore shape, etc.) Minimum k : heat conducts sequentially through a fluid region of length ɛ L and a solid region of length (1- ɛ ) L: ( fluid forms long, random fingers in solid) q Solid k s Pore (fluid) k f L L f s eff k k k 1 1 min , Maximum k : heat can conduct through either a continuous fluid region of cross-section ɛ w or a continuous solid region of cross- section (1- ɛ ) w: (interconnected solid) s f eff k k k...
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## This note was uploaded on 01/09/2012 for the course CHEM ENG 2A04 taught by Professor Toddhoare during the Spring '10 term at McMaster University.

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Chapter 2- - C HAPTER 2 – C ONDUCTION Consider a wall where the surface temperatures are such that T 1>T 2 q x = flow rate of heat in the x

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