Chapter 4
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 = ⊄ |
Step-by-Step
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)
