Flow Distribution in U- and Z-Type Manifolds: Experimental and Numerical Investigation

Abstract

Heat exchangers are used to transfer energy from one medium to another and increase contact area for higher efficiency. In case of fluids, higher flow area would lead to high/low flow zones that severely affect the fluid dynamics and thermal performance. Channels are used to distribute the flow uniformly to fully utilize the whole heat exchanger area. The flow uniformity in a flow distribution manifold is dependent on several factors such as flow rate, inlet and exit locations, and the manifold and channel geometrical configuration. In the current work, flow distribution measurements were performed using a particle image velocimetry (PIV) technique in two types of rectangular manifolds. The experimental results are further verified against the results obtained from numerical modeling with similar trends. The flow distribution in U- and Z-type arrangements are evaluated and compared with ten channels incorporated in the design. It was found that the flow is more in the channels near the inlet for U-type design, while more near the outlet for the Z-type. An increase in the inlet flow rate enhances the flow distribution for the U-type while results in more maldistribution for the Z-type. For the U-type, the normalized velocity varies from 1.34 to 0.52 in a wide manifold, and between 2.82 and 0.18 for a narrow manifold. A U-type wider manifold is recommended for all conditions examined in this work since at lower flow rates, both have similar mirrored distribution, while at higher flow rates, U-type manifold has better flow distribution.

Appendix: Uncertainty Analysis

Measurement of the velocity with the use of PIV involves correlating multiple images to identify the change in the position of illuminated particles. In this method, the time is a fixed value based on the frame rate of the images taken by the imaging device. Hence, it is the distance measurement that has an uncertainty associated with it. Regarding the uncertainty associated with the PIV system, multiple values have been found in the literature. Sciacchitano et al. [49] discussed in detail the procedure to calculate the actual uncertainty in the PIV measurements by super-resolution principle, where the cross-correlated distance is compared with the residual between the particles in the paired images. This comparison was made for all the particles, and in this way, the uncertainties associated with the PIV measurements were evaluated, and 0.1 pixel was used as the accuracy of the PIV systems. For the current work, the velocity was about 1.6 pixels, which converts to about 6.25% uncertainty. It was also determined that the use of the larger image can result in large variations. It was also noticed in the current work that if the images of the complete channel were used, it resulted in an error of about an order of magnitude. For this reason, only a small portion of suitably illuminated particles was used for each channel that resulted in excellent results. Furthermore, the RMS error resulted in an accuracy of 0.01 pixels. In another study [47], the uncertainty of average velocities was found to be within 5% of the actual values. Hence, a value of 5% can be associated with the PIV system velocity measurements.

The uncertainty analysis is performed on data that are derived from multiple experimentally determined quantities. In this regard, the data reduction equation is of the form [50],

$$R = f\left( {X_{1} ,X_{2} ,X_{3} , \ldots } \right).$$

(3)

where R is a calculated variable as a function of parameters X₁, X₂, X₃, etc.

The uncertainty analysis of this equation is of the following form,

$$\left[ {\frac{{U_{R} }}{R}} \right]^{2} = \left[ {\frac{1}{R}\frac{\partial R}{{\partial X_{1} }}U_{{X_{1} }} } \right]^{2} + \left[ {\frac{1}{R}\frac{\partial R}{{\partial X_{2} }}U_{{X_{2} }} } \right]^{2} + \left[ {\frac{1}{R}\frac{\partial R}{{\partial X_{3} }}U_{{X_{3} }} } \right]^{2} + \cdots$$

(4)

where U represents the total uncertainty associated with a particular variable.

The uncertainty analysis equation provides information on how the uncertainties of the individually determined quantities are propagated to affect the uncertainty of the calculated variable. Concerning the uncertainty in the velocity itself (not its measurement), the following data reduction equation is used to find the relation between the independent measurements that affect the velocity [41].

$$u = \frac{{{{\dot{m}_{\text{in}} } \mathord{\left/ {\vphantom {{\dot{m}_{\text{in}} } {10}}} \right. \kern-0pt} {10}}}}{{\rho A_{\text{c}} }}.$$

(5)

This relation can be explained in more detail as follows. The flow is being measured by a rotameter; hence, we have uncertainty related to the flow measurement. This flow rate is divided into 10 channels, such that the average flow rate is 1/10th in each channel. This mass flow rate is converted into velocity by using the continuity equation, where the density is standard for water. The area, which is the product of width (b) and depth (h) of the channel, has its measurement uncertainty. Hence, the three factors affecting the uncertainty in velocity are the width and height of the channel in addition to the flowmeter.

By performing uncertainty analysis Eq. (4) on the data reduction Eq. (5), we obtain the following expression

$$\left[ {\frac{{U_{u} }}{{\dot{m}_{u} }}} \right]^{2} = \left[ {\frac{1}{u}\frac{\partial u}{{\partial \dot{m}_{\text{in}} }}U_{{\dot{m}_{\text{in}} }} } \right]^{2} + \left[ {\frac{1}{u}\frac{\partial u}{{\partial A_{\text{c}} }}U_{{A_{\text{c}} }} } \right]^{2} .$$

(6)

where the two terms on the right side are evaluated as

$$\left[ {\frac{1}{u}\frac{\partial u}{{\partial \dot{m}_{\text{in}} }}U_{{\dot{m}_{\text{in}} }} } \right] = \left[ {\frac{1}{{\frac{{{{\dot{m}_{\text{in}} } \mathord{\left/ {\vphantom {{\dot{m}_{\text{in}} } {10}}} \right. \kern-0pt} {10}}}}{{\rho A_{\text{c}} }}}}\frac{{{1 \mathord{\left/ {\vphantom {1 {10}}} \right. \kern-0pt} {10}}}}{{\rho A_{\text{c}} }}U_{m} } \right] = \left[ {\frac{{U_{{\dot{m}_{\text{in}} }} }}{{\dot{m}_{\text{in}} }}} \right]$$

(7)

and

$$\left[ {\frac{1}{u}\frac{\partial u}{{\partial A_{\text{c}} }}U_{{A_{\text{c}} }} } \right] = \left[ {\frac{1}{{\frac{{{{\dot{m}_{\text{in}} } \mathord{\left/ {\vphantom {{\dot{m}_{\text{in}} } {10}}} \right. \kern-0pt} {10}}}}{{\rho A_{\text{c}} }}}}\frac{{{{ - \dot{m}_{\text{in}} } \mathord{\left/ {\vphantom {{ - \dot{m}_{\text{in}} } {10}}} \right. \kern-0pt} {10}}}}{{\rho A_{\text{c}}^{2} }}U_{m} } \right] = \left[ { - \frac{{U_{m} }}{{\dot{m}_{\text{in}} }}} \right]$$

(8)

Hence, Eq. (6) becomes

$$\left[ {\frac{{U_{u} }}{u}} \right]^{2} = \left[ {\frac{{U_{{\dot{m}_{\text{in}} }} }}{{\dot{m}_{\text{in}} }}} \right]^{2} + \left[ {\frac{{U_{{A_{\text{c}} }} }}{{A_{\text{c}} }}} \right]^{2} .$$

(9)

It is important to note that if there is a higher-order term in the data reduction equation, the corresponding uncertainty term of that variable has a higher contribution toward the total uncertainty. In the current cases, all the terms are of the first order, so a simple root-sum-square (RSS) of the uncertainties will provide the total uncertainty. Here, the area can be split into the width and height of the channel separately to account for their independent uncertainties as:

$$\left[ {\frac{{U_{u} }}{u}} \right]^{2} = \left[ {\frac{{U_{{\dot{m}_{\text{in}} }} }}{{\dot{m}_{\text{in}} }}} \right]^{2} + \left[ {\frac{{U_{b} }}{b}} \right]^{2} + \left[ {\frac{{U_{h} }}{h}} \right]^{2}$$

(10)

The measurements of the height and width of each channel were obtained using a digital Vernier caliber of resolution 0.01 mm. The maximum and minimum values, obtained, are presented in Table 4.

Table 4 The characteristics of the sample data for the calculation of channel area uncertainty

The standard deviation is found by summing the square of the difference of each point from the mean value, divided by one less than the total number of samples. The square root of this value is the standard deviation, as shown by Eq (11). The normalized value is obtained in percentage by dividing the standard deviation by the average value [50].

$$S_{x} = \sqrt {\frac{{\sum\nolimits_{i = 1}^{n} {\left( {x_{i} - \bar{x}} \right)^{2} } }}{n - 1}} .$$

(11)

The flow rate, being supplied by a continuous centrifugal pump, is measured with a rotameter which has a measurement uncertainty of 4%. By inserting the values of 4% for mass flow rate uncertainty of the rotameter, 5.33% for width, and 0.97% for height. The total uncertainty in the velocity of the channel is about 6.85%. This is the uncertainty associated with the actual velocity, not velocity measurements.

Combining the PIV measurement error (5%) with the calculated velocity uncertainty (6.85%.), we can obtain the total uncertainty in velocity measurement, for the current setup, using RSS to be about 8.48% which is presented as ± 4.24%. This value is used for all the experimental data presented in this work.