sec3-1 - ChE 120B General Heat Transfer Equations (Set #3)...

Info iconThis preview shows pages 1–6. Sign up to view the full content.

View Full Document Right Arrow Icon
ChE 120B 3 - 1 General Heat Transfer Equations (Set #3) Fundamental Energy Postulate time rate of change of internal +kinetic energy = rate of heat transfer + surface work transfer (viscous & other deformations) + work done by body “shaft” work + Electromagnetic energy or other sources if we take an arbitrary volume of material let U — internal energy per unit mass 1 2 ρ v 2 — kinetic energy per unit volume internal + kinetic = 2 () 1 2 m Vt Uv d V ρρ ⎛⎞ + ⎜⎟ ⎝⎠ Total energy in volume
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
ChE 120B 3 - 2 Then the time rate of change of this quantity is: 2 () 1 2 m Vt D Uv d V Dt ρρ ⎛⎞ + ⎜⎟ ⎝⎠ for an observer moving with the volume element. Hence, it is called a material or total derivative heat transfer rate = m At dA −⋅ qn This is no more than a definition of q . From fluid mechanics, the force d F acting on the surface is d F =t n dA t n is the stress vector. Imagine the area moving in a direction λ by an amount Δ L in time Δ t. The work done on this surface is just the force times the distance: t n dA ⋅λΔ L and the rate of work is just n dA λ L t ⋅Δ Δ t
Background image of page 2
ChE 120B 3 - 3 In the limit as λ L t0 , t Δ Δ→ Δ then λ L v, t Δ = Δ The velocity vector and surface work on element dA = t n v and the total work done on volume is () m At n tv d A −⋅ similarly, body force work = ρ m Vt dV qv g is for gravity Let Φ = energy source/unit volume, and combining 2 () n 1 ρ 2 mm m m At D U v dV dA dA dV dV Dt ρρ ⎛⎞ += + + + Φ ⎜⎟ ⎝⎠ ∫∫ qn t v gv We can re-write the gravity term as D dV dV Dt φ ⋅= in which is the potential energy/mass and 2 1 ρ e+ ρ v+ ρ 2 m m D dV dA dA dV Dt =− + + Φ ( ) Tv dA ⋅⋅ n T is the stress tensor T= ρ + τ I or U + KE + PE = Q + W ± ±± ± ± which is just our old way of writing the conservation of energy. Limiting Cases Steady-state, v = 0 , 0 AV dA dV + Φ Now A m , V m are fixed in space
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
ChE 120B 3 - 4 Remember the divergence theorem from calculus (or physics)? AV dA dV ⋅= ∫∫ qn q so [An aside: () vv dV ⋅⋅= ∇ ⋅ ⋅ nT T ] v 0 dV =− ∇⋅ Φ ⎡⎤ ⎣⎦ q- For this to be true over an arbitrary volume, the integrand must be identically 0 0 −Φ= q or in various forms Rectangular: 0 y x z q q q xyz =++− Φ ∂∂∂ Cylindrical: 11 0 z r q q rq rr rz θ =+ + Φ ∂∂ Spherical: 2 2 1 0s i n sin sin r q rq q r r θθ + Φ ⎢⎥ In the general case, we have 2 1 U+ v + 2 V D dV Dt ρφ ⎛⎞ ⎜⎟ ⎝⎠ Tv V dV ∇⋅ −∇ −Φ q Reynold's transport theorem states 22 VV DD dV dV Dt Dt ρφρ φ = 2 1 2 V D Dt + ( ) v0 qd V − ∇⋅ ⋅ = T hence inside the integral is also zero to account for arbitrary volumes. ( ) 2 1 q v 2 D Dt =−∇ ⋅ +∇ ⋅ ⋅ +Φ T O r ( ) 2 1 U+ v q v v 2 D Dt ρρ ⋅ ⋅ + ⋅ +Φ Tg We can expand the substantial derivative 2 1 2 D tD t +⋅ = v f o r 0 = v we get U t ρ =−∇⋅ Φ q+ This is the transient energy balance in the absence of convection.
Background image of page 4
ChE 120B 3 - 5 Remember ˆ ˆˆ ij k x yz ∂∂∂
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 6
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 18

sec3-1 - ChE 120B General Heat Transfer Equations (Set #3)...

This preview shows document pages 1 - 6. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online