What is the heat capacity rate? What can you say about the temperature changes of the hot and cold fluids in a heat exchanger if both fluids have the same capacity rate? What does a heat capacity of infinity for a fluid in a heat exchanger mean?

Short Answer

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What does it mean when a heat capacity of infinity is mentioned for a fluid in a heat exchanger? Answer: Heat capacity rate (C) refers to the rate at which a substance can absorb or release heat as its temperature changes, and it is calculated using the formula C = m * cp, where m is the mass flow rate (kg/s), and cp is the fluid's specific heat capacity (J/(kg·K)). In a heat exchanger with equal heat capacity rates for both the hot and cold fluids, the fluids will experience equal and opposite temperature changes, as their temperature differences will be equal. A heat capacity of infinity implies a hypothetical situation where a fluid has an unlimited capacity to absorb or release heat without experiencing any significant change in temperature, and it is used for approximate calculations when there is a vast difference in heat capacity rates between two fluids.

Step by step solution

01

Define Heat Capacity Rate

Heat capacity rate, denoted as C (also known as 'heat capacity flow rate'), is a property that defines the rate at which a substance can absorb or release heat as its temperature changes. It is a product of the mass flow rate (m) and the specific heat capacity (cp) of a fluid. The formula to find the heat capacity rate is: C = m * cp where C is the heat capacity rate, m is the mass flow rate (kg/s), and cp is the specific heat capacity of the fluid (J/(kg·K)). Heat capacity rate determines how fast a fluid's temperature changes with respect to the heat transfer rate (Q). Higher heat capacity rate indicates that the fluid can absorb or release larger amounts of heat with relatively smaller changes in temperature.
02

Temperature Changes in a Heat Exchanger with Equal Heat Capacity Rates

In a heat exchanger, heat is transferred between two fluids with different temperatures. One fluid is the hot fluid, which cools down while transferring its heat, and the other fluid is the cold fluid, which heats up by receiving the heat from the hot fluid. If both the hot and the cold fluids have the same heat capacity rate then they will experience equal and opposite temperature changes. This occurs because the heat transfer rate (Q) is constant throughout the exchanger, and since the heat capacity rates are the same, their temperature differences will be equal. This can be expressed mathematically as: ΔTh = -ΔTc where ΔTh is the temperature change of the hot fluid, and ΔTc is the temperature change of the cold fluid.
03

Heat Capacity of Infinity in a Heat Exchanger

A heat capacity of infinity for a fluid in a heat exchanger implies that the fluid has an unlimited capacity to absorb or release heat without experiencing any significant change in temperature. This is a hypothetical situation and is not practically possible. In reality, every fluid has a finite heat capacity. However, such scenarios can be used for approximate calculations when there is a vast difference in heat capacity rates between the two fluids. In this case, the fluid with a much greater heat capacity rate can be assumed to have infinite heat capacity. The temperature of this fluid will remain practically constant throughout the heat exchanger, while the other fluid will experience significant temperature changes.

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Most popular questions from this chapter

Air at \(18^{\circ} \mathrm{C}\left(c_{p}=1006 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\right)\) is to be heated to \(58^{\circ} \mathrm{C}\) by hot oil at \(80^{\circ} \mathrm{C}\left(c_{p}=2150 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\right)\) in a cross-flow heat exchanger with air mixed and oil unmixed. The product of heat transfer surface area and the overall heat transfer coefficient is \(750 \mathrm{~W} / \mathrm{K}\) and the mass flow rate of air is twice that of oil. Determine \((a)\) the effectiveness of the heat exchanger, \((b)\) the mass flow rate of air, and \((c)\) the rate of heat transfer.

Oil in an engine is being cooled by air in a cross-flow heat exchanger, where both fluids are unmixed. Oil \(\left(c_{p h}=\right.\) \(2047 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\) ) flowing with a flow rate of \(0.026 \mathrm{~kg} / \mathrm{s}\) enters the heat exchanger at \(75^{\circ} \mathrm{C}\), while air \(\left(c_{p c}=1007 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\right)\) enters at \(30^{\circ} \mathrm{C}\) with a flow rate of \(0.21 \mathrm{~kg} / \mathrm{s}\). The overall heat transfer coefficient of the heat exchanger is \(53 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) and the total surface area is \(1 \mathrm{~m}^{2}\). Determine \((a)\) the heat transfer effectiveness and \((b)\) the outlet temperature of the oil.

A single-pass cross-flow heat exchanger is used to cool jacket water \(\left(c_{p}=1.0 \mathrm{Btu} / \mathrm{lbm} \cdot{ }^{\circ} \mathrm{F}\right)\) of a diesel engine from \(190^{\circ} \mathrm{F}\) to \(140^{\circ} \mathrm{F}\), using air \(\left(c_{p}=0.245 \mathrm{Btu} / \mathrm{lbm} \cdot{ }^{\circ} \mathrm{F}\right)\) with inlet temperature of \(90^{\circ} \mathrm{F}\). Both air flow and water flow are unmixed. If the water and air mass flow rates are \(92,000 \mathrm{lbm} / \mathrm{h}\) and \(400,000 \mathrm{lbm} / \mathrm{h}\), respectively, determine the log mean temperature difference for this heat exchanger.

The cardiovascular counter-current heat exchanger mechanism is to warm venous blood from \(28^{\circ} \mathrm{C}\) to \(35^{\circ} \mathrm{C}\) at a mass flow rate of \(2 \mathrm{~g} / \mathrm{s}\). The artery inflow temperature is \(37^{\circ} \mathrm{C}\) at a mass flow rate of \(5 \mathrm{~g} / \mathrm{s}\). The average diameter of the vein is \(5 \mathrm{~cm}\) and the overall heat transfer coefficient is \(125 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). Determine the overall blood vessel length needed to warm the venous blood to \(35^{\circ} \mathrm{C}\) if the specific heat of both arterial and venous blood is constant and equal to \(3475 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\).

A counterflow double-pipe heat exchanger with \(A_{s}=\) \(9.0 \mathrm{~m}^{2}\) is used for cooling a liquid stream \(\left(c_{p}=3.15 \mathrm{~kJ} / \mathrm{kg} \cdot \mathrm{K}\right)\) at a rate of \(10.0 \mathrm{~kg} / \mathrm{s}\) with an inlet temperature of \(90^{\circ} \mathrm{C}\). The coolant \(\left(c_{p}=4.2 \mathrm{~kJ} / \mathrm{kg} \cdot \mathrm{K}\right)\) enters the heat exchanger at a rate of \(8.0 \mathrm{~kg} / \mathrm{s}\) with an inlet temperature of \(10^{\circ} \mathrm{C}\). The plant data gave the following equation for the overall heat transfer coefficient in W/m \({ }^{2} \cdot \mathrm{K}: U=600 /\left(1 / \dot{m}_{c}^{0.8}+2 / \dot{m}_{h}^{0.8}\right)\), where \(\dot{m}_{c}\) and \(\dot{m}_{h}\) are the cold-and hot-stream flow rates in kg/s, respectively. (a) Calculate the rate of heat transfer and the outlet stream temperatures for this unit. (b) The existing unit is to be replaced. A vendor is offering a very attractive discount on two identical heat exchangers that are presently stocked in its warehouse, each with \(A_{s}=5 \mathrm{~m}^{2}\). Because the tube diameters in the existing and new units are the same, the above heat transfer coefficient equation is expected to be valid for the new units as well. The vendor is proposing that the two new units could be operated in parallel, such that each unit would process exactly one-half the flow rate of each of the hot and cold streams in a counterflow manner; hence, they together would meet (or exceed) the present plant heat duty. Give your recommendation, with supporting calculations, on this replacement proposal.

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