Question

Geothermal water \(\left(c_{p}=4250 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\right)\) at \(75^{\circ} \mathrm{C}\) is to be used to heat fresh water \(\left(c_{p}=4180 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\right)\) at \(17^{\circ} \mathrm{C}\) at a rate of \(1.2 \mathrm{~kg} / \mathrm{s}\) in a double-pipe counter-flow heat exchanger. The heat transfer surface area is \(25 \mathrm{~m}^{2}\), the overall heat transfer coefficient is \(480 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\), and the mass flow rate of geothermal water is larger than that of fresh water. If the effectiveness of the heat exchanger is desired to be \(0.823\), determine the mass flow rate of geothermal water and the outlet temperatures of both fluids.

Short answer

Based on the given information and step-by-step solution, complete the following short answer: To find the mass flow rate of geothermal water and the outlet temperatures of both fluids using the effectiveness-NTU method, we first determine the capacity rate ratio (C). We then calculate the NTU value using the given effectiveness and capacity rate ratio. Next, we find the mass flow rate of geothermal water by utilizing the NTU equation and the given information. Lastly, we determine the outlet temperatures of both fluids by relating the effectiveness to the temperature differences and applying energy balance equations.

Step-by-step solution

Determine the minimum specific heat rate and the capacity rate ratio (C)

First, we define the specific heat rate for both fluids: \(C_{1}=\dot{m_{1}} c_{p1}\) and \(C_{2}=\dot{m_{2}} c_{p2}.\) We don't know the mass flow rate of geothermal water \(\dot{m_{1}}\) yet, but we are given \(\dot{m_{2}} = 1.2\,\text{kg/s}\). Then we can find \(C_{2}\): \(C_{2}=\dot{m_{2}} c_{p2}=1.2 \times 4180 = 5028\,\mathrm{W/K}\). Now let's find the capacity rate ratio, which is defined as \(C=\frac{C_{min}}{C_{max}}\). Since the mass flow rate of geothermal water is larger than that of fresh water, we can claim \(C_{1} > C_{2}\), and thus \(C_{min}=C_{2}\) and \(C_{max}=C_{1}\). Therefore, the capacity rate ratio \(C\) is: \(C = \frac{C_{min}}{C_{max}} = \frac{C_2}{C_1} = \frac{5028}{C_{1}}\)

Calculate the NTU value using the given effectiveness and capacity rate ratio

We have the desired effectiveness, \(\epsilon = 0.823\), and we can use the effectiveness-NTU relation for counter-flow heat exchangers: $$ \epsilon = \frac{1 - \exp(-NTU(1-C))}{1 - C\cdot\exp(-NTU(1-C))} $$ We can solve this equation for NTU: $$ NTU = \frac{-\log(1 - \epsilon + C\cdot \epsilon)}{(1 - C)} $$ Plugging in the desired effectiveness \(\epsilon = 0.823\) and \(C = \frac{5028}{C_1}\), we can find the numerical value for NTU.

Calculate the mass flow rate of geothermal water

We know the definition of the NTU is the ratio of the heat transfer surface area (\(A\)) to the product of overall heat transfer coefficient (\(U\)) and the minimum specific heat rate (\(C_{min}\)): $$ NTU = \frac{A \cdot U}{C_{min}} $$ We can now use this equation to find the mass flow rate of the geothermal water: $$ \dot{m_1} = \frac{AU}{(NTU\,c_{p1})} $$ Plugging in the values \(A = 25\,\mathrm{m^2}\), \(U = 480\,\mathrm{W/m^2K}\), and \(c_{p1}= 4250\,\mathrm{J/(kg\cdot K)}\), we can find the numerical value for \(\dot{m_1}\).

Determine the outlet temperatures of both fluids

Using the relationships between the effectiveness and the temperatures, we can calculate the outlet temperatures for both fluids: $$ \Delta T_{2} = \epsilon \frac{C_{min}}{C_{2}} \Delta T_{1} $$ From the energy balance equation, $$ \Delta T_{1} = \frac{C_{max}}{C_{min}} \Delta T_{2} $$ By plugging in the calculated values of \(\epsilon\), \(C_{min}\), \(C_{max}\), and \(\Delta T_{1}\), we can find the value of \(\Delta T_{2}\), and thus the outlet temperature of the fresh water: \(T_{o2} = T_{i2} + \Delta T_{2}\). Now we find the outlet temperature of the geothermal water: \(T_{o1} = T_{i1} - \Delta T_{1}\). Following these steps, the mass flow rate of the geothermal water and the outlet temperatures of both fluids are determined.

Key concepts

Capacity Rate Ratio

When delving into the principles of heat exchangers, one essential concept is the Capacity Rate Ratio (CRR), a dimensionless value that helps in analyzing the performance of heat exchangers. It's defined as the ratio of the heat capacity rates of the two fluids involved in the heat exchange process. To be more precise, it's the ratio of the minimum heat capacity rate to the maximum heat capacity rate.

The importance of the CRR comes into play when calculating the effectiveness of the heat exchanger—how well the heat exchanger transfers heat relative to its maximum potential. A CRR closer to 1 implies a more balanced heat exchange between the fluids and can often lead to a higher effectiveness, while a CRR that deviates significantly from 1 can indicate a less efficient transfer. Understanding CRR not only enhances the comprehension of heat exchanger design but is also pivotal in thermal engineering and its applications.

NTU Method

The Number of Transfer Units (NTU) method is a powerful tool in the thermal analysis of heat exchangers. It is employed predominantly to estimate the heat exchanger's effectiveness based on the geometry and flow arrangement of the heat exchanger. NTU is a dimensionless quantity that encapsulates the size of the heat exchanger and the properties of the fluids.

A fundamental relationship connects the effectiveness of the heat exchanger, the NTU, and the capacity rate ratio. When designing or evaluating a heat exchanger, engineers aim for an optimal NTU that matches the desired effectiveness while considering economic and physical constraints. The NTU calculation, once mingled with the specific heat capacity of the fluids and the overall heat transfer coefficient, is central to determining the mass flow rate and, subsequently, the outlet temperatures of the fluids engaged in the heat transfer process.

Overall Heat Transfer Coefficient

A pivotal element in the analysis of heat exchangers is the Overall Heat Transfer Coefficient (U). This coefficient represents the ability of the heat exchanger to conduct heat through its materials from the hot fluid to the cold fluid. It encompasses attributes related to the heat exchanger's materials, geometry, and the nature of fluid flow.

The Overall Heat Transfer Coefficient is measured in watts per square meter Kelvin \(\text{W/m}^2\cdot\text{K}\) and is influenced by factors such as the type of materials used in the construction of the heat exchanger, the fouling of surfaces, and the flow regimes of the fluids involved. A higher U value signifies a more effective heat transfer capability, making it a crucial parameter in the sizing and evaluation of heat exchangers. This coefficient integrates with the area of the heat exchanger and the mean temperature difference between the fluids to calculate the heat exchange rate.

Thermal Energy Balance

At the heart of any heat exchanger analysis is the Thermal Energy Balance, capturing the conservation of energy within the system. It is premised on the first law of thermodynamics, which states that energy cannot be created or destroyed in an isolated system. In the context of a heat exchanger, this principle implies that the heat lost by the hot fluid must be equal to the heat gained by the cold fluid, accounting for any inefficiencies present.

The Thermal Energy Balance is critical for determining the outlet temperatures of the fluids after heat exchange. By applying the energy balance, we infer that the amount of heat removed from the hot fluid should equal the heat absorbed by the cold fluid when the system reaches a steady state. This balance guides the calculations to find out the temperature changes and, thus, allows the correct sizing of heat exchangers and prediction of their operational behavior.