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464 T. Coumaressin & K. Palaniradja Fig. 2: Evaporator test section. One can obtain an analytical solution to the governing equations in the fully developed region ⎡ ⎛r⎞2⎤ u ( r ) = 2 U ⎢ ⎢ 1 − ⎜⎝ R ⎟⎠ ⎥ ⎥ ⎣⎦ (1) Where, R is the pipe radius. The length of the development region for laminar flow can be estimated using the following correlation (experimentally developed equation). Xfd,h=0.05ReD =20 (2) D For thermally fully – developed flow with a constant surface heat flux there is no longer any variation in the shape of the temperature profile in the axial x – direction. Again, one obtain an analytical solution to the governing equations in the fully – developed region as follows 2 pUR2q ⎡ 3 1 ⎛ r ⎞ 4 1 ⎛ r ⎞ 2 ⎤ T(r,x)=Ts(x)− kmL ⎢16+16⎜⎝R⎟⎠ −4⎜⎝R⎟⎠ ⎥ (3) ⎣⎦ Where p is the density, k is the thermal conductivity, m is the mass flow rate, and Ts is the surface temperature on the inside of the pipe wall. The length of the development region for laminar flow can be estimated using the following correlation Xfd,t=0.05ReD Pr=15 (4) D Based on an overall energy balance for the pile, the mean temperature distribution for a uniform surface heat flux in the developing and fully – developed region can be calculated from the following (which is valid for both laminar and turbulent flow): ⎛q⎞x Tm(x)=Tm,t+⎜mc ⎟L (5) ⎝p⎠ Where, Cp is the specific heat and the mean temperature is defined based on a mass – weighted average. Finally, the heat transfer coefficient for out pipe flow is defined as:PDF Image | Performance Analysis of a Refrigeration System Using Nano Fluid
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