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Chapter 11: Problem 15

Under what conditions can the overall heat transfer coefficient of a heat exchanger be determined from \(U=\) \(\left(1 / h_{i}+1 / h_{o}\right)^{-1} ?\)

Short Answer

Expert verified
Briefly describe the steps to determine the overall heat transfer coefficient under these conditions. Answer: The formula for the overall heat transfer coefficient can be used when the heat exchanger is operating under steady-state conditions, the internal and external heat transfer coefficients are constant along the heat exchanger, and the effect of the thermal conductivity of the material is negligible. To determine the overall heat transfer coefficient, follow these steps: 1. Ensure the heat exchanger is under steady-state conditions. 2. Determine the constant heat transfer coefficients at the inner and outer surfaces, \(h_i\) and \(h_o\), respectively. 3. Verify that the thermal conductivity of the heat exchanger material is negligible compared to the convective heat transfer coefficients. 4. Use the formula \(U=\left(1/h_{i} + 1/h_{o}\right)^{-1}\) to calculate the overall heat transfer coefficient.

Step by step solution

01

Steady-state condition

Ensure that the heat exchanger is operating under steady-state conditions. This means that the temperatures and flow rates of the fluids should be constant over time.
02

Heat transfer coefficients

Determine the heat transfer coefficients at the inner and outer surfaces, \(h_i\) and \(h_o\) respectively. These values should be constant along the entire length of the heat exchanger, and should account for both the convective and radiative heat transfer mechanisms.
03

Negligible thermal conductivity

Verify that the thermal conductivity of the heat exchanger material (e.g., pipe walls) is negligible compared to the convective heat transfer coefficients. This will ensure that the primary mode of heat transfer is via convection, and the formula for overall heat transfer coefficient can be used.
04

Calculate the overall heat transfer coefficient

Use the formula \(U=\left(1/h_{i} + 1/h_{o}\right)^{-1}\) to determine the overall heat transfer coefficient of the heat exchanger given the constant internal and external heat transfer coefficients.

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Most popular questions from this chapter

Consider a recuperative crossflow heat exchanger (both fluids unmixed) used in a gas turbine system that carries the exhaust gases at a flow rate of $7.5 \mathrm{~kg} / \mathrm{s}\( and a temperature of \)500^{\circ} \mathrm{C}$. The air initially at \(30^{\circ} \mathrm{C}\) and flowing at a rate of $15 \mathrm{~kg} / \mathrm{s}$ is to be heated in the recuperator. The convective heat transfer coefficients on the exhaust gas and air sides are $750 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\( and \)300 \mathrm{~W} / \mathrm{m}^{2}, \mathrm{~K}$, respectively. Due to long-term use of the gas turbine, the recuperative heat exchanger is subject to fouling on both gas and air sides that offers a resistance of \(0.0004\) \(\mathrm{m}^{2}\). $/ \mathrm{W}$ each. Take the properties of exhaust gas to be the same as that of air \(\left(c_{p}=1069 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\right)\). If the exit temperature of the exhaust gas is \(320^{\circ} \mathrm{C}\), determine \((a)\) if the air could be heated to a temperature of \(150^{\circ} \mathrm{C}\) and \((b)\) the area of the heat exchanger. (c) If the answer to part \((a)\) is no, then determine what should be the air mass flow rate in order to attain the desired exit temperature of \(150^{\circ} \mathrm{C}\) and \((d)\) plot the variation of the exit air temperature over a range of \(75^{\circ} \mathrm{C}\) to \(300^{\circ} \mathrm{C}\) with the air mass flow rate, assuming all the other conditions remain the same.

For a specified fluid pair, inlet temperatures, and mass flow rates, what kind of heat exchanger will have the highest effectiveness: double-pipe parallel- flow, double-pipe counterflow, crossflow, or multipass shell-and-tube heat exchanger?

Hot oil \(\left(c_{p}=2.1 \mathrm{~kJ} / \mathrm{kg} \cdot \mathrm{K}\right)\) at \(110^{\circ} \mathrm{C}\) and \(12 \mathrm{~kg} / \mathrm{s}\) is to be cooled in a heat exchanger by cold water \(\left(c_{p}=4.18\right.\) $\mathrm{kJ} / \mathrm{kg} \cdot \mathrm{K})\( entering at \)10^{\circ} \mathrm{C}$ and at a rate of \(2 \mathrm{~kg} / \mathrm{s}\). The lowest temperature that oil can be cooled in this heat exchanger is (a) \(10^{\circ} \mathrm{C}\) (b) \(24^{\circ} \mathrm{C}\) (c) \(47^{\circ} \mathrm{C}\) (d) \(61^{\circ} \mathrm{C}\) (e) \(77^{\circ} \mathrm{C}\)

The cardiovascular countercurrent heat exchanger has an overall heat transfer coefficient of \(100 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). Arterial blood enters at \(37^{\circ} \mathrm{C}\) and exits at \(27^{\circ} \mathrm{C}\). Venous blood enters at \(25^{\circ} \mathrm{C}\) and exits at $34^{\circ} \mathrm{C}$. Determine the mass flow rates of the arterial blood and venous blood in \(\mathrm{g} / \mathrm{s}\) if the specific heat of both arterial and venous blood is constant and equal to $3475 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\(, and the surface area of the heat transfer to occur is \)0.15 \mathrm{~cm}^{2}$.

Glycerin \(\left(c_{p}=2400 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\right)\) at \(20^{\circ} \mathrm{C}\) and \(0.5 \mathrm{~kg} / \mathrm{s}\) is to be heated by ethylene glycol $\left(c_{p}=2500 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\right)\( at \)60^{\circ} \mathrm{C}$ and the same mass flow rate in a thin-walled double-pipe parallelflow heat exchanger. If the overall heat transfer coefficient is \(380 \mathrm{~W} / \mathrm{m}^{2}\). \(\mathrm{K}\) and the heat transfer surface area is \(6.5 \mathrm{~m}^{2}\), determine \((a)\) the rate of heat transfer and \((b)\) the outlet temperatures of the glycerin and the glycol.
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