## Q. 4.5

Single-Tube Heat Exchanger

Consider a smooth curved pipe of unknown length (D = 12.5 mm) immersed in a boiling-water reservoir ($\rm T_R$ = 373K, $\rm p_R = p_{atm}$), where the tube water ($\rm \dot m$ = 0.0576 kg/s, $\rm T_{in}$ = 300K) should be heated to $\rm T_{out}$ = 360K . Determine the required tube length and total heat transfer rate. The property data for reference temperature $\rm T_{ref} = 0.5 (T_{in} + T_{out} )$ = 330K are: ρ = 984 kg / m³ , μ = 4.9 ×10$^{−4}$ N ⋅ s /m² , $\rm c_p$ = 4184 J /(kg ⋅ K), k = 0.65 W/(m·K); and Pr = 3.15.

$\rm\overline{Nu_D} =0.023\,Re_D^{0.8}Pr^{1/3}$                                    (4.26)

$\rm\ln\,{\frac{T_{{m}}({x})-T_{{w}}}{T_{{m,1}}-T_{{w}}}}={\frac{-\,\pi\,{D}\,{\overline{{h}}}}{{c_{p}}\,\dot{m}}}\,{x}$                                  (4.31a)

 Concepts Assumptions Sketch • $\rm T_w$ = constant Case • Steady 1-D flow without entrance and form-loss effects • Check $\rm Re_D$ > 4,000 • Constant properties • Average h from $\rm\overline{Nu_D}$ – correlation, Eq. (4.26) • Fully turbulent flow • Pipe length from Eq. (4.31a) • Boiling water assures $\rm T_{wall}$ = 373K = ⊄

## Verified Solution

• The total heat flow rate (Eq. (4.28)) is:

$\rm\dot Q_{1−2} ≡ \dot Q _L = \dot m c_p (T_{m,2} − T_{m,1})$                            (4.28)

$\rm\dot{{Q}}_{{total}}={\dot{m}}{c}_{{p}}[{T}_{{m}}({x}={L})-{T}_{{m}}({x}=0)]$                        (E.4.5.l)

= 0.0576· 4184(360 – 300)

$\rm\underline{\dot{{Q}}_{{total}}=14.46{~kJ/}{~s\,\hat = \,l4.46~kW}}$

• Equation (4.31a) can be rewritten as:

$\rm{L}={\frac{{\dot m}{c_{p}}}{{\bar h}\left(\pi\,{D}\right)}}\ln\left[{\frac{{T_{m}}\left({L}\right)-{T_{w}}}{{T_{m}}\left(0\right)-{T_{w}}}}\right]$                                    (E.4.5.2)

where $\rm\bar h$ is unknown,

• Equation (4.26) contains $\rm\bar h$ , i.e.,

$\rm {\overline{{{Nu_{D}}}}}\equiv{\frac{\bar{h}\;{D}}{{k}}}=0.023\,{Re}_{{D}}^{0\,8}\,{Pr}^{0.33}$                            (E.4.5.3)

where $\rm{{Re}}_{{D}}={\frac{4{{\dot m}}}{\pi{{~\mu~D}}}}:=12\times10^{3}$ , i.e., turbulent flow occurs.

Hence,

$\rm \overline{{{ h}}}={3}.47\,\frac{{k}{W}}{{ m}^{2}\cdot{ K}}$

Now, Eq. (E.4.5.2) can be evaluated with $\rm T_w$ = 373K to obtain the pipe length:

L = 3.05 m

Comments: • Figure 4.3a depicts the pipe-water heating process in detail. Note that Eq. (E.4.5.1) is a global energy balance.

• The pipe length is based on an average heat transfer coefficient (Eq. (E.4.5.3)) and hence would be incorrect if the pipe entrance effect would be significant.

• For a constant-wall-heat-flux ease, say, with $\rm q_w$ , D and L given, as well as $\rm T_m(x)$ and $\rm T_w (x)$ measured, where 0 ≤ x ≤ L , $\rm h_x$ = h = ⊄ can be readily obtained from Eq. (4.29) and then $\rm\dot Q_{ total} = q_w$ (π D L) .

$\rm\dot Q(x) = h_x (2r_0 π x) [T_w (x) − T_m(x)]$                                 (4.29)