Question

Consider a double-pipe parallel-flow heat exchanger of length \(L\). The inner and outer diameters of the inner tube are \(D_{1}\) and \(D_{2}\), respectively, and the inner diameter of the outer tube is \(D_{3}\). Explain how you would determine the two heat transfer surface areas \(A_{j}\) and \(A_{o^{-}}\)When is it reasonable to assume \(A_{i} \approx A_{e} \approx A_{s}\) ?

Short answer

Based on the given information, briefly describe the conditions when it is reasonable to assume that Ai ≈ Ae ≈ As in a double-pipe parallel-flow heat exchanger, and calculate the inner (Ai) and annular (Ae) areas.

Step-by-step solution

Calculate the Inner Area (Ai)

To calculate the inner area (Ai), first find the inner radius of the inner tube (R1) by dividing the inner diameter (D1) by 2. Then, the inner area can be expressed as follows: $$ A_i = 2\pi R_1 L $$ where L represents the length of the heat exchanger.

Calculate the Annular Area (Ae)

To find the annular area (Ae), we first calculate the outer radius of the inner tube (R2) by dividing the outer diameter (D2) by 2 and the inner radius of the outer tube (R3) by dividing the inner diameter (D3) by 2. Then, the annular area can be found using the following formula: $$ A_e = 2\pi (R_3 - R_2) L $$

Calculate the Surface Area (As)

The surface area (As) of the inner tube can be calculated using the outer diameter (D2) and the length (L) as follows: $$ A_s = \pi D_2 L $$

Approximation Conditions

It is reasonable to assume that Ai ≈ Ae ≈ As when the difference in the diameters D1, D2, and D3 are relatively small and the overall heat transfer rate is primarily dependent on surface area and not significantly influenced by the thickness of the material. This usually occurs when the heat exchanger is operating at steady-state and the temperature gradients across the heat exchanger walls are relatively uniform.