Forced convection in a pipe

Consider a system where fluid flowing in a pipe is being heated or cooled. The wall temperature of the pipe is fixed at a value of $T_{wall}$, which is different than the bulk temperature of the fluid, or $T_{bulk}$. For an incompressible fluid with a heat capacity that is independent of temperature, the internal energy balance is

$\frac{d}{dt}\underset{CV}{\iiint }\left(\mathrm{\rho }{c}_{v}T\right)dV=-\dot{m}{c}_p\left(T_2-T_1\right)+\dot{Q}+{\dot{E}}_{viscous}$

(1)

At steady-state, Equation (1) reduces to

$\dot{m}{c}_p\left(T_2-T_1\right)=\dot{Q}+{\dot{E}}_{viscous}$

(2)

If heat generation due to viscous flow is negligible, ${\dot{E}}_{viscous}\ll \dot{Q}$, then Equation (2) can be written as

$\dot{m}{c}_p\left(T_2-T_1\right)=\dot{Q}$

(3)

which states that the change in enthalpy of the flowing fluid is equal to the heat transferred at the system boundary of the control volume.

Radial Heat Transfer

An important thing to note is that heat transfer occurs in the radial direction due to the difference between the bulk temperature of the fluid and the wall temperature of the pipe. The local heat flux in the the radial direction is

${\dot{q}}_r(z)=h(T_{wall}-T_{bulk})$

(4)

where $h$ is the heat transfer coefficient. The total rate of heat transfer $\dot{Q}$ is

$\dot{Q}=\underset{CV}{\iint }\left(\dot{\mathbf{q}}\mathbf{\cdot }\hat{\mathbf{n}}\right)dA$

(5)

where $\hat{\mathbf{n}}$ is the unit normal defining the control surface. In this example, there is only heat transfer in the radial direction, hence $(\dot{\mathbf{q}}\cdot \hat{\mathbf{n}})dA={\dot{q}}_{r}Rdzd\theta$ and the total rate of heat transfer is

$\dot{Q}=\underset{CV}{\iint }(\dot{\mathbf{q}}\cdot \hat{\mathbf{n}})=\pi D\underset{0}{\overset{L}{\int }}{\dot{q}}_r\left(z\right)dz$

(6)

The above equation can also differentiated with respect to $z$ to find the differential rate of radial heat transfer in the axial direction.

$\frac{d\dot{Q}}{dz}=\pi D{\dot{q}}_r(z)$

(7)

Differential form of the internal energy balance

Equation (3) can be differentiated with respect to $z$ to account for the change in the bulk temperature along the axial or the $z$ direction:

$\dot{m}{C}_p\frac{d{T}_{bulk}}{dz}=\frac{d\dot{Q}}{dz}$

(8)

Combining Equations (7) and (8) yields

$\dot{m}{C}_p\frac{d{T}_{bulk}}{dz}=\pi D{\dot{q}}_r(z)$

(9)

Log-mean temperature difference

Equations (9) and (4) can be combined to yield

$\frac{d{T}_{bulk}}{dz}=\frac{h\pi D}{\dot{m}{C}_p}(T_{wall}-T_{bulk})$

(10)

If the wall temperature is constant along the pipe, the variables $T_{bulk}$ and $z$ in Equation (9) can be easily separated

$\underset{T_1}{\overset{T_2}{\int }}\frac{d{T}_{bulk}}{T_{wall}-T_{bulk}}=\frac{h\pi D}{\dot{m}{C}_p}\underset{0}{\overset{L}{\int }}dz$

(11)

Integrating and rearranging Equation (11), we get

$\dot{m}{C}_p(T_2-T_1)=-h\left(\pi DL\right)\left[\frac{(T_2-T_{wall})-(T_1-T_{wall})}{ln\frac{(T_2-T_{wall})}{(T_1-T_{wall})}}\right]$

(12)

We define the log-mean temperature difference or Log Mean Temperature Difference as

$\mathrm{\Delta }{T}_{LM}=\left[\frac{(T_2-T_{wall})-(T_1-T_{wall})}{ln\frac{(T_2-T_{wall})}{(T_1-T_{wall})}}\right]$

(13)

and Equation (18),

and writing Equation (12) in terms of $\mathrm{\Delta }{T}_{LM}$ we get

$\dot{m}{C}_p(T_2-T_1)=-h\left(\pi DL\right)\mathrm{\Delta }{T}_{LM}$

(14)

Design equation for forced convection in pipe flow

For fluid flow in which the wall temperature is known, the heat transfer equation is expressed as

$\dot{m}{C}_p(T_2-T_1)=-h_i{A}_i\mathrm{\Delta }{T}_{LM}$

(15)

where the subscript $i$ is meant to denote the inside of the pipe.

• $h_i$ refers to the heat transfer coefficient of the flowing fluid
• $A_i$ refers to the heat transfer area between the flowing fluid and the inner pipe wall. In this case $A_i=\pi D_{i}L$, where D_i is the inner pipe diameter.

See also

• Overall heat transfer coefficient
• Shell and tube heat exchanger