ABSTRACT
Example 10.1: Laminar Flow in a Double-Pipe Heat Exchanger [1]
Two flows in steady state with the fully developed velocity profiles inside a double pipe are considered. The double pipe consists of an inner central tube and an outer annualar channel. Concurrent and countercurrent cases are studied. The thermal conduction in the fluids and viscose dissipation are neglected. The governing system includes energy equations for the inner and outer streams (indices 1 and 2) and for the separating wall, boundary and conjugate conditions:
u x r r r
1 0
∂ ∂
= ∂ ∂
+ ∂ ∂
= ∂θ θ θ θ θ( , ) , θ1 0 0 ∂
= r
x( , ) (10.1)
u x mc
R r rp
24 1 1∂
∂ = − +
∂ ∂
+ ∂θ λ θ θ ( )
( )∆ ∂
r2 (10.2)
θ θ θ θ θ2 20 2 20 20 0( , ) , ( , ) , ( , )r L r r x Ri= =
∂ ∂
= (10.3)
Pe r r r x x
1 0 0
∂ ∂
+ ∂ ∂
+ ∂ ∂
= ∂ ∂
θ θ θ θ ( , r
r L rs) ( , )= ∂
∂ =θ 0 (10.4)
θ θ θ λ θ1 11 1 1 1( , ) ( , ), ( , ) ( , )x x r x
w= ∂ ∂
= ∂ ∂
(10.5)
θ θ θ λ λ
21 1 1( , ) ( , ), ( , ) (x x r
x r
xw w w+ = + ∂
∂ + = ∂
∂ ∆ ∆ ∆ , )1+ ∆ (10.6)
The nondimensional variables are scaled as follows: linear sizes by internal radius of the inner duct Ri, velocities by mean axial velocity U, and temperatures by the inlet temperature of the inner fluid T0 1, , thermal conductivity λ2, and capacity ( )mcp 2 by corresponding values of the inner flow. The first and the second relations of boundary conditions in Equation (10.3) pertain to concurrent and countercurrent cases, respectively.
