Chapter 12, Problem 12.2P

Interpretation Introduction

Interpretation:

The temperature of the oil when it leaves the pipe and after it is mixed is to be determined. Also, the average heat transfer coefficient is to be calculated.

Concept Introduction:

The formula to calculate the Reynolds number for the flow of fluid in a pipe is:

$\mathrm{Re}=\frac{D\overline{V}\rho }{\mu }$ ..... (1)

Here, $D$ is the tube inner diameter, $\overline{V}$ is the velocity of the fluid flowing inside the pipe, $\rho$ is the density of the fluid, and $\mu$ is the viscosity of the fluid.

For laminar flow of fluid in a pipe, the correction factor is calculated as:

${\varphi }_v={\left(\frac{\mu }{{\mu }_W}\right)}^{0.14}$ ..... (2)

Here, $\mu$ is the viscosity of the fluid at its arithmetic mean temperature and ${\mu }_W$ is the viscosity of the fluid at the wall temperature.

Graetz number is calculated as:

$G_z=\frac{\dot{m}{C}_P}{kL}$ ..... (3)

Here, $\dot{m}$ is the mass flow rate of the fluid, $L$ is the length of the pipe, $C_P$ is the specific heat of the fluid, and $k$ is the thermal conductivity of the fluid.

The Nusselt number for the heat transfer to the fluid flowing inside the pipe is:

$Nu=\frac{h_{i}D}{k}$ ..... (4)

Here, $h_i$ is the heat transfer coefficient.

The relationship for the Nusselt number, Graetz number, and the correction coefficient is:

$Nu=2G_z^{1/3}{\varphi }_v$ ..... (5)

The log mean temperature difference for heating of a fluid flowing inside a pipe is written as:

$\Delta {\overline{T}}_L=\frac{\left(T_W-T_a\right)-\left(T_W-{\overline{T}}_b\right)}{\ln \left(\frac{T_W-T_a}{T_W-{\overline{T}}_b}\right)}$ ..... (6)

Here, $T_W$ is the wall or surface temperature of the pipe, T_a is the inlet fluid temperature, and ${\overline{T}}_b$ is the average outlet fluid temperature.

The heat transfer coefficient in terms of the log mean temperature difference is written as:

$h_i=\frac{\dot{m}{C}_P\left({\overline{T}}_b-T_a\right)}{\pi DL\Delta {\overline{T}}_L}$ ..... (7)