Description and Analytical Modeling for a Solid Block Cross-flow High Temperature Heat Exchanger

In this report, the design specifics, and performance modeling results, for a simple, solid metal block, liquid–liquid cross-flow heat exchanger, intended for high temperature applications, are given. The design consists of a solid block of metal, with cylindrical channels providing the cross-flow passageways for two non-mixing liquids. In this design, all flow channels are separated by a certain minimum thickness of the host solid block material. This particular design is limited by the length of pores that can be machined from a solid block. In this study, a simple heat transfer model, appropriate for such an exchanger, was used to estimate what values of effectiveness might be obtainable while keeping the size of the exchanger as compact as possible. The effects of channel length and spacing, liquid specific heat and viscosity and block material conductivity on exchanger effectiveness are considered and results reported. The model predicts that for a cubic exchanger of side length 8.25 cm with 50 channels per side at a diameter of 3.0 mm each, for a particular high temperature situation using molten salts, with an inlet and outlet temperature difference of around 170 K, an effectiveness of 0.4 can be achieved with a total mass flow rate of 0.5 kg·\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\cdot$$\end{document}s-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{-1}$$\end{document} along with a Reynolds number of less than 2000.


Introduction
Most heat exchangers, commercially available, are generally of the shell and tube [1,2], shell and plate [3] or the tube in tube design [4,5]. Components in heat exchangers of this type in addition to being costly, introduce multiple modes for failure, including brazing or weld failure, vibration-induced fatigue, and inconsistent thermal expansion rates between differing materials thus introducing challenges for their high temperature application. Repair of these conventional heat exchangers can also be involved. This process might consist of removing a tube or shell and replacing it with another. Leakage can occur if brazed connections or welds fail. Vibrations that exist inside the heat exchanger can not only result in brazing or weld failures but can also produce noise, an undesirable situation in certain applications. In this report the description and modeling results are given for a heat exchanger that can potentially avoid many of the limiting issues mentioned above. The proposed design is for a simple, solid metal block cross flow, liquid to liquid, non-mixing heat exchanger. Compared with conventional heat exchangers, this design could be easier to manufacture, should have lower maintenance requirements, is durable and quiet. Its design makes it modular so that multiple units can be used for complex applications, and it can be easily and safely transported to remote locations for emergency and immediate needs, including medical, military, disaster, and conventional applications. Unlike with other exchangers which show promise for high temperature applications, including the plate-fin cross flow [6] the shuttle baffle [7], and the cross-flow tube bundle [8], the solid block exchanger allows for a higher degree of material isolation between the hot and cool side flow in addition to the qualities of durability, and coolant isolation inherit in a homogeneous solid block design.
Currently, most of the heat exchanger research work, that is reported in the literature and known to the authors, seems to be focused on designs other than the single block unit. A notable exception is recently reported results of a theoretical study for a porous ceramic, parallel flow, high temperature block exchanger [9]. However, apparently due to the demand for durable exchangers that can operate at high temperatures, there are solid block exchangers on the market, most notably the cylindrical solid graphite heat exchanger from GAB Neumann where a series of solid graphite blocks, of machinable size, are assembled in a stack to form the entire exchanger [10].
One of the chief design issues for the solid block exchanger is the limitation on the length of the fluid pores that can be machined from the solid block material. For the above mentioned graphite exchanger, this challenge was overcome by stacking several blocks of machinable size so as to create an exchanger with sufficient overall size so as to achieve a useful effectiveness value. A primary goal of this study was to determine what values of effectiveness can be obtained from a single solid block exchanger within the range of overall dimensions into which flow channels could be machined.
Therefore, any solid block cross-flow heat exchanger is likely to be relatively compact in size. Though this is often a desirable quality in an exchanger is does create flow and efficiency challenges. Often, these issues can be addressed somewhat by decreasing the flow channel radius and increasing the number of channels. These criteria are met with new designs for printed circuit board (PCB) heat exchangers [11] which show significant promise in the area of efficient compact heat transfer design. Compact solid block exchangers could serve as a durable alternative to the PCB exchanger.
Though a cross-flow solid block heat exchanger is hardly a novel design, there does appear to be a lack of information in the literature on the thermal performance of such devices, especially for high temperature applications. The primary purpose of this work is to fill this void by developing a thermal model for such an exchanger which can then be used to study the performance of this device for a variety of high temperature situations.
The theoretical study discussed here is for a square solid block, cross-flow, nonmixing, liquid to liquid heat exchanger operating in high temperature situations; situations for which molten salts and liquid metals coolants are often utilized. A depiction of such an exchanger is shown in Fig. 1. The solid block material should be a metal alloy appropriate for high temperature applications. The flow pores are cylindrical with circular cross-section of common radius r.
Physical data for three different molten salts, as well as the alkali metals Na, K and NaK, were used to study high temperature heat transfer situations in the exchanger. Additionally, physical data for water and ethylene glycol were used to model a low temperature transfer process. Input temperatures, ranging from 278 K to 1248 K were considered with the limitation being that, temperatures at which physical data for the liquids had been previously tabulated and which were available to the authors, had to be used. Geometric influences upon performance were investigated such as the effect of varying channel spacing, number of flow channels and thus the overall exchanger size. What effect the solid block material has on exchanger performance was investigated by estimating effectiveness values over a range of thermal conductivities relevant for high temperature alloys. Finally, effectiveness values were estimated for a variety of coolant mass flow rates. Other quantities indicative of exchanger performance were also estimated including the pumping power, power density, pressure drop and Reynolds number.
After arriving at a temperature profile for the hot and cool exit side of the exchanger, it is assumed that the two unused faces of the exchanger (see Fig. 1) are adiabatically isolated and thus temperatures are known along the boundaries of a two-dimensional problem. Using Laplace's equation, an analytic expression is obtained which gives the temperature distribution in the block at steady state. Using a similar approach, the steady state temperature profile along all liquid flow channels is estimated.

Heat Transfer Model
In this section, the model used to estimate outlet temperatures for the exchanger, is outlined. Consider a cubic exchanger, as depicted in Fig. 1, with rows of N coolant flow channels crossed with N hot side channels. The total number of openings on Fig. 1 Semi-transparent view of proposed cubic solid block cross-flow heat exchanger system. All pores pass through the entire block. The entrance for the 50, that is (N 2 ∕2) , crossflow coolant passages are shown front to back in this semi-transparent view. Hot side passages run from top to bottom each entry and exit side are then N 2 ∕2 . In this design, N is restricted to be an even integer. Relevant distances on the exchanger face are denoted in Fig. 2 which shows a top view of the cube from Fig. 1. For convenience in this study, the buffer distance a, as denoted in Fig. 2, is set equal to z.
It was assumed that the exchanger was adiabatically isolated, the fluids at all times remained in the liquid phase, there was a constant and uniform fluid inlet temperature and flow rate and that there were no transverse temperature variations along the flow channels. Additionally, channel passages are assumed to be smooth and any friction introduced due to fouling is neglected. Figure 3 gives a depiction of the top left side of the cube in Fig. 1. for the first cool side flow channel and the left most row of hot side inlet channels. Our scheme can be describe by referring to this diagram.
The approximation was made that the thermal interaction between one layer of cool liquid channels and one layer of hot flow channels can be used to represent the condition for the remaining N∕2 − 1 bi-layers. Upon entry, a section of cool liquid in the channel, of length L as denoted in Fig. 2, interacts thermally with a corresponding section of length L of hot channel flow above it over some time period t that is set by the Fig. 2 Top view of the cubic block exchanger depicted in Fig. 1 with relevant distances denoted. N is the number of channels along a row and a is a buffer distance around the outside of the channel array  Fig. 1 flow rate of the two channels. For simplicity, the hot and cool mass flow rates were set to be equal in this study. Then, it was assumed the transfer rate between hot and cool channels, over this time period, could be approximated as an interaction between two channel segments that are isothermal over the time period t.
The heat transfer between the two channels can be taken to be determined by three thermal resistances in series-two for the convective transfer at each channel wall and one for conduction in the solid material of conductivity k [12], so that the net thermal resistance R m is where h H and h C are the thermal transfer coefficients for convective transfer from the hot side liquid and the cool side liquid, respectively. A is the inner surface area of the channel segment of interest and S is a shape factor. A shape factor model described in Ref. [12] was used. It describes an isothermal cylindrical tube of diameter D at temperature T 1 immersed in a semi-infinite medium of conductivity k a distance z from a flat face at constant temperature T 2 . In terms of distances described here it reads This shape factor is valid for z > 3r so that z was restricted to z = 4r for all tests in this study. From Fig. 2, it is seen that L = z + r or since z = a = 4r , L = 5r . This then sets the side length, L t , of the cubic exchanger to be L t = (N − 1)5r + 9r . The exchanger volume is simply L t 3 . The minimum spacing between channels is then fixed at 3r.
Unlike other cross-flow systems where the hot and cool flows are separated by a thin barrier [13][14][15][16], due to the relatively large distance between pores in this exchanger, the approximation is made that h H A and h C A will both be much greater than Sk so that conductivity through the block will be the rate limiting step in the transfer process and therefore Consider a time interval t during exchanger flow given by where is the mass density of the coolant side liquid and ṁ c is the mass flow rate, in kg/s, of the coolant liquid. During this time period, energy conservation between the coolant liquid and the solid block, leads to the following equality of rates .
Here T H is the initial hot side temperature and T C the initial cool side temperature. c c is the specific heat for the cool side liquid. The left side of Eq. 5 gives the heat transfer rate between two constant temperatures: a cool channel with liquid initially at temperature T C and the hot side liquid, modeled via the shape factor as an isothermal surface, at the constant temperature T H , where obviously T H > T C . The right side of Eq. 5 gives the heat energy flow rate into the cool side liquid initially at temperature T C and then at the end of the time interval at the final temperature of Accordingly, the inverse relationship is used to describe the rate of heat energy exiting the hot side channel liquid: where ṁ h is the hot side flow rate, c h the hot side liquid specific heat and T H f the final hot side temperature at the end of the time interval t, where T H > T H f .
Moving from left to right in Fig. 3, Eq. 5 is used to estimate the change in the coolant temperature ΔT C n , where n denotes the channel number and n = 1, 2, 3, … , N . After passing by the first hot side inlet channel, On continuing this process down the coolant channel the change in coolant temperature after every hot channel encounter for n > 1 can be written as Therefore, the final coolant temperature upon exiting the first coolant channel, T C f 1 is given by To estimate the thermal properties for the entire exchanger the process is repeated at the next coolant channel in the by-layer. For example, from Fig. 1 on moving down to the second coolant flow passage from the top left side, the scheme outlined above is repeated. As with all coolant channels down this by-layer, fresh coolant enters at temperature T C . However, at the second coolant channel the hot side liquid temperature will have a decreased temperature due to its interaction with the first coolant channel. Therefore, after the first pass, the nth hot side flow temperature above the second coolant channel is lowered by the amount ΔT H n , for n = 1, 2, 3, … , N to a new initial value. Using Eq. 5 these values are for n = 1: and for n > 1, (10) Now, Eq. 8 is generalized to give ΔT C n l for the nth sector of the lth coolant row, for l = 1, 2, 3, … , N as where The final temperature upon exit from the lth cool side channel is Now, Eq. 11 is generalized to for n = 1 and for n > 1,

Here, T H n l
= T H for all n when l = 1 . The final hot side temperature upon exit from the lth hot side channel, T H f l , is given by Starting at the top by-layer, on the top, left side of Fig. 1, the above scheme can be used to estimate temperatures at every point where channels cross and at the exit portals. It is assumed here that the results of this by-layer will be equivalent to the temperature profile of the other N∕2 − 1 bi-layers. If at any time during this process T H n l ≤ T C n l the system has obtained equilibrium and there is no further heat transfer. In the next section, results from applying the above scheme to estimate outlet temperatures for a variety of high temperature liquids in the solid block exchanger are reported. In doing so, the overall effectiveness, and other relevant quantities for the exchanger, are estimated.

Analysis
The heat transfer model presented in the previous section can be easily implemented using commercially available mathematical analysis software or a high level programming language. The model was used to estimate the exchanger outlet temperatures, for various liquids at different input temperatures, along a bi-layer of hot and cool flow channels which, as mentioned previously, were use to approximate the thermal state of the other N∕2 − 1 outlet channel bi-layers. Obviously, these outlet temperatures vary along a row of hot or cool outlet channels. Results indicate that the coolant output channels, along a row, get progressively cooler in the direction of the hot side flow and in a nearly linear fashion. Likewise, the hot side outlet channels get progressively warmer in approximately a linear fashion along the direction of coolant flow. In both cases, the temperature extremes occur at channels adjacent to the exchanger corners, the so called hot and cool corners of the cross-flow exchanger [16]. A plot giving the hot side exit temperatures for one of the tests, carried out and reported here, is shown in Fig. 4. As one might slightly detect from this Figure, the temperature change along an output row is mildly quadratic. However, for convenience, the curve taken from a linear fit to these output temperatures was used to describe the temperature over the outlet faces.
For purposes of reporting the channel exit temperatures in each test, the peak hot exit temperature is denoted as T fp while the minimum hot exit temperature is T fm . An analogous notation is used for the cool side extremes. Then, a slope, m s , was defined as and likewise for the cool temperature outlet side.
With the thermal and geometric characteristics of the proposed exchanger determined, the results were then used to compute a variety of quantities that are useful for the evaluation of the performance and general behavior of the device. These values include the exchanger effectiveness , the power density P d , the channel pressure drop ΔP and the pumping power P p . Other useful quantities are also computed in each case include the Reynolds number R e and the flow velocity v. Thermal  where v is the flow velocity which is obtained from the mass flow rate and the pore radius. is the liquid mass density and is the dynamic viscosity of the liquid. It is useful to compute a power density, P d , which is defined here as the rate of heat transfer in the exchanger divided by the volume of the exchanger. Consider the heat transferred to the cool liquid during one pass through the exchanger. In a particular layer, each of the N coolant channels will have a unique temperature upon exit. Therefore, the power density is defined as Here, ΔT n is the difference between the final and initial temperature for the nth channel row and L t 3 is the exchanger volume as discussed previously. The factor of N/2 accounts for transfer of all N/2 cool flow layers, each layer with N channels.
The Darcy-Weisbach-Fanning expression was used to compute the pressure drop, ΔP , for a turbulent fluid flowing through a pipe with a smooth inner surface [17].
where f is a friction coefficient, L t is the channel length and g the magnitude of the acceleration due to gravity. For f, the model of Blasius was utilized [17]: With the pressure drop computed, one can then compute the cool side pumping power, P p , defined as Here (N 2 ∕2)ṁ c gives the total flow rate for all N 2 ∕2 channels. An analogous expression was used to estimated the hot side pumping power.
An average change in volume due to an increase in temperature for the entire block exchanger can be computed in the following fashion. First, assume a linear thermal expansion relationship where the side length at room temperature, l o , is increased by the length Δl due to an increase in temperature ΔT as given by Δl = l o ΔT . Here, is the linear expansion coefficient values of which are tabulated for many high temperature metals. This leads to a volume expansion ΔV for the small cube being given by A more detailed finite element thermal expansion analysis for the proposed exchanger was not performed during this study and was left for future work.
Reported physical data for eight different liquids, six of which could serve as high temperature coolants for the solid block exchanger, were obtained from Ref. [18]. Liquid coolants considered include, Oak Ridge National Lab (ORNL) molten salt no. 30, ORNL molten salt no. 14, High Temperature Salt (HTS: NaNO 3 , KNO 3 , KNO 2 ), Na, K, NaK(56 % Na; 44 % K) water and ethylene glycol (EG). The tabulated data includes the specific heat, viscosity and mass density. These data were given in the cited reference for selected high temperatures at 1.0 atm. These temperatures determined our choices for the initial hot and cool liquid temperatures in the exchanger simulation tests.
The United States Department of Energy gives a listing of no fewer than twenty alloys that are useful for the types of high temperature applications being considered here [19]. In this list are Inconel 600 and stainless steel 321. Both of these alloys should be suitable for the temperatures encountered in this study. Since for most alloys the thermal conductivity falls with increasing temperature we use the value at the highest temperature listed in reference [20] that is, 25.0 W ⋅ m −1 ⋅ K −1 and 26.0 W ⋅ m −1 ⋅ K −1 for stainless steel and Inconel, respectively, both at 973 K. It was assumed that this value remained constant up to the highest temperature considered here, 1248.6 K. Therefore, a constant value for 26.0 W ⋅ m −1 ⋅ K −1 was used for most tests in this study. This approximation seems justified upon considering that the pure metals Cr and Ti have a thermal conductivity decrease of only around 0.4 % on going from 1400 K to 1600 K [20].
For each test, the temperature profile on exit, for the hot and cool side, were determined and the quantities discussed above, in Eqs. 19 through 25, were computed. Results are listed in Tables 1, 2 and 3. Specific heat, viscosity and density data for the liquids comes from Ref. [18] where it was listed for the input temperatures used here. T i is the inlet temperature and ṁ , the channel flow rate. In all test cases from Tables 1, 2  (26) in Eq. 18. ΔP h and ΔP c are the hot and cool channel pressures drops, respectively. v h and v c are the hot and cool side flow velocities, respectively. As mentioned above, k was set to 26.0 W ⋅ (m −1 ⋅ K −1 ) for all tests listed in Tables 1, 2 and 3. To investigate what effect the pore radius and the number of pores per layer has upon the overall performance of the exchanger, a promising test case from Table 1 was repeated, (test number 2), while varying radius at constant N and channels per layer, N, at constant radius. In all of these tests the restriction that z = 4r was retained. This restriction then connects the side length, and thus channel length of the exchanger, with N and r. The side length of the exchanger is therefore listed in each case. Results are listed in Tables 4 and 5.
Since, it is not uncommon for thermal conductivities for metals to range from 20.0 to more than 100.0 W ⋅ m −1 ⋅ K −1 , what effect varying k might have upon exchanger effectiveness was considered again for the liquids and inlet temperatures used in Test  Test no  Test no  Table 6.
Several comments are required to clarify the results reported in Table 4. First, the effectiveness increases with increasing channel radius but this is due to the fact that the cubic geometry of the model, and a constant value for N, causes the channel length and thus interior channel volume, to increase with increasing channel radius. The sole effect that varying channel radius has upon on effectiveness is difficult to determine using the model described here as Eq. 2 is only valid for z > 3r , so that as r increases or decreases, the smallest possible value for z varies in proportion to this change, thus keeping the ratio 4z/2r in Eq. 2 constant. As for the effect of inner channel spacing, z, on effectiveness, it is clear from Eq. 2 that, upon holding r fixed, as z → ∞ , R m → ∞ and thus effectiveness falls with increasing z.   Test no Test no    Test no  Test no The onset of complete turbulent flow is typically taken to be when R e > 2000 , even though well defined eddies have been reported to appear for cases when R e < 10 [21]. For the test cases considered here 604 ≤ R e ≤ 24, 834 so that turbulent conditions were assumed thus motivating the use of Eq. 23, a relation defined for turbulent flow, when computing channel pressure drops.
The minimum required operating pressure, P, for one side of the exchanger, with N 2 ∕2 channels, can now be estimated using the computed pressure drops along with Bernoulli's equation as: P = (N 2 ∕2)[(1∕2) v 2 + ΔP] . As an example, consider ORNL #14 from test case 2 in Table 1. This approach leads to a minimum required pressure of P = 27, 601.2 Pa or about 0.27 atm.
The effect of thermal expansion will have to be dealt with upon development of any prototype heat exchanger of the sort described here. Though a complete finite element expansion analysis was not carried out in this work, one can get a quick order of magnitude estimate for situations such as those considered in Tables 1, 2 and 3 by using Eq. 26. Nickel and iron based alloys often have linear expansion coefficients on the order of = 12.0 × 10 −6 • C −1 [20]. Assuming the exchanger is initially at room temperature, from Table 1 it can be seen that a typical temperature increase would be about ΔT = 700 K. Using this value in Eq. 26, along with and an initial side length of 8.25 cm, one obtains ΔV ≈ 1.6 × 10 −7 m 3 , which corresponds to an increase in the side length, using l o ΔT , of approximately 0.7 mm. This results indicates the need to use similar metals for any connecting parts on such an exchanger.  A necessary component of any future thermal expansion analysis would be an analytic expression for the equilibrium temperature at all points in the solid body of the exchanger during use. In the following section, data used and results obtained in this section are used to estimate the steady state temperature at all non-adiabatic boundaries of the block. These boundary conditions are then used to obtain a continuous analytic expression for the steady state temperature throughout the solid body of the exchanger. Using a similar approach, a continuous relation for the fluid temperature within the flow channels is also determined.

Steady State Block Temperature
In this section, an analytic expression for the temperature profile in the exchanger block at equilibrium is derived. It is assumed that the linear exit temperatures along a layer, an example of which was shown in Fig. 4, can be used to describe the temperature profile over the entire hot outlet side. The cool exit analog of this is then used to describe the surface temperature on the cool outlet side. Letting the sides containing the input channels each have the uniform inlet temperatures of T H and T C the steady state temperature is then established on four sides of the block. The two unused sides of the exchanger are adiabatic, so that the interior temperature profile, can be considered as a two-dimensional problem, that is, T = T(x, y) . This situation is depicted below in Fig. 6 where w denotes the side length of the cube which was defined in terms of other relevant distance values in Fig. 2.
A solution to Laplace's equation, ∇ 2 T = 0 , is required for these boundary conditions. This presents a Dirichlet problem with four inhomogeneous boundary conditions which cannot be solved by separation of variables. However, as shown by Zill [22], a solution can be given by the superposition of the separable solutions for two similar problems with paired homogeneous boundary conditions. For purposes of visualization, experience has taught us that a superposition of four separable solutions for Laplace's equation in this region, each with three homogeneous boundary conditions and one nonhomogeneous condition relevant for our problem, is more useful. This process is shown pictorially in Fig. 7.
Our required solution is then For the four cases described by Fig. 7, separable solutions can be found for T 1 , T 2 , T 3 and T 4 . Using the superposition principle in each case, along with the Fourier series method for boundary conditions [23], one arrives at and where and where (27) T(x, y) = T 1 (x, y) + T 2 (x, y) + T 3 (x, y) + T 4 (x, y). The variable boundary conditions are of the form and For simplicity, let the slope m h and m c be (T fp − T fm )∕(w) and assume the extreme temperatures occur at the cube edge so that T H b is set to the minimum hot side exit temperature and T C b equals the maximum cool side exit temperature.
Using Eq. 34 in 30 and then Eq. 35 in 33 yields and Using Eq. 27, along with the results of Eqs. 28 through 37, one can then estimate the temperature distribution in the solid block at steady state. An example of this process was carried out for one of the test cases considered for which data is listed in Table 1 and the results are depicted graphically in Fig. 8.

Steady State Fluid Temperature
As Laplace's equation was used in the preceding section to estimate the equilibrium temperature distribution in the solid block, this approach can also be used to estimate the steady state temperature profile within the channel liquid. As in the previous section, due to the symmetry of the exchanger, and the two unused sides, only the two-dimensional problem need be considered. It is assumed that the liquid temperature within the channels can be given by the solution to Laplace's equation with two inhomogenous boundary conditions, the inlet and outlet temperature. The other two boundaries are assumed to be adiabatic. This situation is depicted in Fig. 9 for hot liquid flow through the exchanger. An analogous situation is used to analyze the cool side flow channels. The resulting temperature profile, T(x, y), will only be valid for values of x and y which are within a flow channel.
Finding the separated solution for Laplace's equation for this situation, using the superposition principle, applying the adiabatic boundary conditions, and the condi-  Fig. 6. Distribution was generated for Test no. 1 from Table 1

Conclusions
In this paper, the results of a theoretical study for a high temperature solid block liquid to liquid non-mixing cross-flow heat exchanger are reported. A scheme is presented that approximates the energy transfer through the exchanger given knowledge of certain physical properties of the solid block and the coolants. The exchanger is A n = 2 w sinh(n ) (T H b + m h w) sin(n ) + m h w n (cos(n ) − 1) .

Fig. 10
Temperature map for estimated cool and hot side channel liquids for the case considered in Fig. 8 with 10 channels per layer envisioned to be machined from one solid block of high temperature metal alloy. Since the minimum distance between hot and cool flows are always maintained at three time the pore radius, it was assumed that the energy transfer process is rate limited by conduction through the solid body of the exchanger. A variety of high temperature liquid combinations were considered and exchanger effectiveness values, and other relevant quantities, were estimated. One case, in which two different ORNL molten salts were paired, shows particular promise yielding an effectiveness of 0.4 and a power density of 4.90 MW ⋅ m −3 . Though the effectiveness values and power densities estimated here may not be competitive with traditional low temperature heat exchangers on the market, this solid block design has the potential to be robust and durable for high temperature applications. For comparison, Fraas and Ozisik report power densities for four different molten salt/ liquid metal heat exchangers. These values range from 0.16 MW ⋅ m −3 to 266.0 MW ⋅ m −3 [21]. Additionally, results reported here indicate that the effectiveness improves dramatically upon increase in the solid block thermal conductivity. Therefore, the development of high temperature alloys, with improved thermal conductivity, could make the solid block design a more useful candidate for high and medium temperature applications.
Analytic expressions are derived, and visualized, for the temperature distribution in the block and flow liquids at steady state. In addition to giving the equilibrium temperature profile of the exchanger, these results could be used in the future to carry out a detailed thermal expansion analysis.
The ultimate validation of the model presented here should come from the development and testing of such an exchanger in the laboratory which is left for future work. Performance data for other similar systems, such as from Ref.
[10], was not available to the authors. However, comparison can be made with the results of Liu et al. [16] for an air to air cross-flow non-mixing exchanger with corrugated aluminum spacers used for heat and/or energy recovery. This exchanger is similar in design to the one proposed here only flow sides were separated by thin aluminum walls. They report, depending upon the application, effectiveness values, from both experimental and theoretical investigations, of 0.38 to 0.58. Values reported here range from 0.17 to 0.54. Additionally, Liu et al. give a steady state temperature distribution map for the exchanger that shows an interior temperature profile having the essential features reported here, in Fig. 8, for the solid block non-mixing cross-flow liquid-liquid exchanger.
It is important now that this study be followed by the development and testing of a metal alloy solid block cross-flow heat exchanger in the laboratory. Results given in this report should serve as an aid to researchers considering options for the overall design geometry, choice of block material and coolant liquids, for such an investigation.
Author Contributions DAB: Supervision, methodology, investigation, writing original draft, formal analysis. JBB: Conceptualization, validation, writing review and editing. SSH: Formal analysis, validation, writing review and editing.
Funding The authors' have no funding sources to acknowledge.