A composite slab is made of two rectangular solid slabs of equal cross-sectional

Question

A composite slab is made of two rectangular solid slabs of equal cross-sectional area (perpendicular to the plane of the figure) A, different thicknesses L_1 and L_2 and different thermal conductivities k_1 and k_2, as shown. The left side of the composite slab is held at temperature T_C, while the right side is held at T_H. Because the two slabs are joined "in series rate of heat transfer from the right to left must be equal through each slab. Use this fact to a symbolic equation for the temperature T_M at the surface where two slabs meet, in terms of only the given quantities. Simplify, as always. Derive an equation for the rate of heat transfer H in terms of only the given quantities (not in terms of T_M). Write this equation so that L_1, L_2, k_1 and k_2, all only appear in the denominator. What would the equation for H be if instead there were three slabs joined in series (the third slab having the same cross-sectional area A, but different thickness L_3 and thermal conductivity k_3 (You don't need to derive it from scratch. Just extend the result for two slabs, to allow for three.)

Explanation / Answer

rate of heat tranfer

H = dQ/dt = KA dT/dx

the temperature difference from tleft to the joint surface = Tc -T , where T is temperature at the join

H = k1A (Tc-T)/L1

sinilarly from the join to the right we have

H = k2A (T-Tm)L2

k1A (Tc-T)/L1 = k2A (T-Tm)L2

solving the baove we get

T = (k1L2Tc +k2L1Tm)/(k1L2 + k2L1)

b) we assume Tm > Tc , i.e heat flows from right to left

let us call L1/k1A    = R1 and L2/k2A    = R2

the we can write the heat tansfer equations as

H = (Tm -T) /R1   = (T-Tc)/R2

H = (Tm-Tc)/(R1+R2) , eleiminating T from both the equations.

suppose k is the conductivity of the composite slab then we can write

H = kA(Tm-Tc)/(L1+L2)

comparing the above two equations we have

(L1+L2)/kA = (R1+R2)

substituting for R1 and R2 and simplifyinf we get

k = (k1k2)(L1+L2) /(L1k1 +L2k2)    , conductivity of the composite slab

c) we define R = kA/(L1 + L2)

then the conductivity of the three slabs would be

     = 1/( R1 +R2 + R3)

      = 1/( k1A/L1 +k2A/L2 +k3A/L3)

      = (k1k2k3) (L1 + L2 +L3) /(L1k2k3 +L2k1k3 + L3 k1k2)

Related Questions

Navigate

• Browse (All)
• Browse A
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.