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Chapter 11: Problem 111

A cross-flow heat exchanger with both fluids unmixed has an overall heat transfer coefficient of \(200 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\), and a heat transfer surface area of \(400 \mathrm{~m}^{2}\). The hot fluid has a heat capacity of \(40,000 \mathrm{~W} / \mathrm{K}\), while the cold fluid has a heat capacity of \(80,000 \mathrm{~W} / \mathrm{K}\). If the inlet temperatures of both hot and cold fluids are \(80^{\circ} \mathrm{C}\) and \(20^{\circ} \mathrm{C}\), respectively, determine the exit temperature of the cold fluid.

Short Answer

Expert verified
Based on the given information, calculate the exit temperature of the cold fluid in the heat exchanger. Solution: Step 1: Calculate the total heat exchanged in the heat exchanger: \(Q = U \cdot A \cdot \Delta T = 200 \frac{\mathrm{W}}{\mathrm{m}^2 \cdot \mathrm{K}} \times 400 \mathrm{~m}^{2} \times (80 \mathrm{~^\circ C} - 20 \mathrm{~^\circ C})\) Step 2: Use the heat exchange formula to find the exit temperature of the cold fluid: \(T_{cold, exit} = \frac{Q}{80000 \frac{\mathrm{W}}{\mathrm{K}}} + 20 \mathrm{~^\circ C}\) The exit temperature of the cold fluid is \(T_{cold, exit}\).

Step by step solution

01

Calculate the heat exchanged in the heat exchanger

First, we need to compute the total amount of heat exchanged by both fluids in the heat exchanger since the heat exchange is equal on both sides. Use the given overall heat transfer coefficient, surface area for heat transfer, and the temperature difference between the inlet temperatures of the hot and cold fluids: \(Q = U \cdot A \cdot \Delta T\) \(Q = 200 \frac{\mathrm{W}}{\mathrm{m}^2 \cdot \mathrm{K}} \times 400 \mathrm{~m}^{2} \times (80 \mathrm{~^\circ C} - 20 \mathrm{~^\circ C})\) \(Q = W\)
02

Use the heat exchange formula to find the exit temperature of the cold fluid

Now we can use the heat exchange formula, using the heat capacity rate of the cold fluid and the inlet temperature of the cold fluid, and the known total heat exchange (\(Q\)) value from Step 1: \(Q = C_{cold}(T_{cold, exit} - T_{cold, inlet})\) \(W = 80000 \frac{\mathrm{W}}{\mathrm{K}} \times (T_{cold, exit} - 20 \mathrm{~^\circ C})\) Now rearrange the equation to solve for \(T_{cold, exit}\): \(T_{cold, exit} = \frac{W}{80000 \frac{\mathrm{W}}{\mathrm{K}}} + 20 \mathrm{~^\circ C}\) \(T_{cold, exit} = T\) The exit temperature of the cold fluid is \(T\).

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Most popular questions from this chapter

There are two heat exchangers that can meet the heat transfer requirements of a facility. One is smaller and cheaper but requires a larger pump, while the other is larger and more expensive but has a smaller pressure drop and thus requires a smaller pump. Both heat exchangers have the same life expectancy and meet all other requirements. Explain which heat exchanger you would choose and under what conditions.

Can the temperature of the hot fluid drop below the inlet temperature of the cold fluid at any location in a heat exchanger? Explain.

A performance test is being conducted on a double pipe counter flow heat exchanger that carries engine oil and water at a flow rate of \(2.5 \mathrm{~kg} / \mathrm{s}\) and \(1.75 \mathrm{~kg} / \mathrm{s}\), respectively. Since the heat exchanger has been in service over a long period of time it is suspected that the fouling might have developed inside the heat exchanger that might have affected the overall heat transfer coefficient. The test to be carried out is such that, for a designed value of the overall heat transfer coefficient of \(450 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) and a surface area of \(7.5 \mathrm{~m}^{2}\), the oil must be heated from \(25^{\circ} \mathrm{C}\) to \(55^{\circ} \mathrm{C}\) by passing hot water at \(100^{\circ} \mathrm{C}\left(c_{p}=4206 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\right)\) at the flow rates mentioned above. Determine if the fouling has affected the overall heat transfer coefficient. If yes, then what is the magnitude of the fouling resistance?

A shell-and-tube heat exchanger is to be designed to cool down the petroleum- based organic vapor available at a flow rate of \(5 \mathrm{~kg} / \mathrm{s}\) and at a saturation temperature of \(75^{\circ} \mathrm{C}\). The cold water \(\left(c_{p}=4187 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\right)\) used for its condensation is supplied at a rate of \(25 \mathrm{~kg} / \mathrm{s}\) and a temperature of \(15^{\circ} \mathrm{C}\). The cold water flows through copper tubes with an outside diameter of \(20 \mathrm{~mm}\), a thickness of \(2 \mathrm{~mm}\), and a length of \(5 \mathrm{~m}\). The overall heat transfer coefficient is assumed to be \(550 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) and the latent heat of vaporization of the organic vapor may be taken to be \(580 \mathrm{~kJ} / \mathrm{kg}\). Assuming negligible thermal resistance due to pipe wall thickness, determine the number of tubes required.

Consider a condenser unit (shell and tube heat exchanger) of an HVAC facility where saturated refrigerant \(\mathrm{R} 134 \mathrm{a}\) at a saturation pressure of \(1318.6 \mathrm{kPa}\) and a rate of \(2.5 \mathrm{~kg} / \mathrm{s}\) flows through thin-walled copper tubes. The refrigerant enters the condenser as saturated vapor and it is desired to have a saturated liquid refrigerant at the exit. The cooling of refrigerant is carried out by cold water that enters the heat exchanger at \(10^{\circ} \mathrm{C}\) and exits at \(40^{\circ} \mathrm{C}\). Assuming initial overall heat transfer coefficient of the heat exchanger to be \(3500 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\), determine the surface area of the heat exchanger and the mass flow rate of cooling water for complete condensation of the refrigerant. In practice, over a long period of time, fouling occurs inside the heat exchanger that reduces its overall heat transfer coefficient and causes the mass flow rate of cooling water to increase. Increase in the mass flow rate of cooling water will require additional pumping power making the heat exchange process uneconomical. To prevent the condenser unit from under performance, assume that fouling has occurred inside the heat exchanger and has reduced its overall heat transfer coefficient by \(20 \%\). For the same inlet temperature and flow rate of refrigerant, determine the new flow rate of cooling water to ensure complete condensation of the refrigerant at the heat exchanger exit.
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