Question

2. Two rods, with thermal conductivities k and k2, are used to transfer heat between two regions with temperatures TH and Tc (TH > Tc) as shown. Each rod has length L and cross sectional area A. The rods are first configured in parallel (left) and then in series (right). Which configuration has the largest heat flow rate? Write an expression for the heat flow of each configuration Tc Tc TH 2L

we are given the equation: delta Q/delta t =kA delta T /L with t=time and T=temperature in K

The one on the left would be (k1+k2)A delta T/L. I am stumped for the one on the right.

Answer 1

Using a formula, we have

$\Delta$Q / $\Delta$t = k A $\Delta$T / L

where, k = thermal conductivity of a material

L = length of each rod

A = cross sectional area

$\Delta$t = time taken

$\Delta$T = temperature difference = (T_H - T_C)

# An expression for the heat flow of left configuration which will be given as -

In parallel, we have

1 / k = 1 / k₁ + 1 / k₂

1 / k = (k₁ + k₂) / (k₁ . k₂)

k = [(k₁ . k₂) / (k₁ + k₂)]

Then, we using an above formula.

$\Delta$Q / $\Delta$t = k A $\Delta$T / L

$\Delta$Q / $\Delta$t = [(k₁ . k₂) / (k₁ + k₂)] A (T_H - T_C) / L

$\Delta$Q = [(k₁ . k₂) / (k₁ + k₂)] A (T_H - T_C) $\Delta$t / L

# An expression for the heat flow of right configuration which will be given as -

In series, we have

k = k₁ + k₂

Then, we using an above formula.

$\Delta$Q / $\Delta$t = k A $\Delta$T / L

$\Delta$Q / $\Delta$t = (k₁ + k₂) A (T_H - T_C) / 2L

$\Delta$Q = (k₁ + k₂) A (T_H - T_C) $\Delta$t / 2L