Simplified Regenerative Heat Transfer Design

Louis Herrington, P.E.
112 Wildoak Dr.
Daphne, AL 36526

This paper describes simple and understandable equations for the design of Regenerative Heat Transfer Systems.  Many bed designers rely on the complex time and temperature heat balance equations.  Debora, et al, used the energy balance equations to develop a time and temperature profile through the inert particulate bed that resulted from a flowing fluid of a different temperature.  Because of the complexities involved with the equations and the subsequent lack of understanding of the results, the Plant Engineer is likely to neglect energy savings from hot gas streams.  Other designers rely on Schumann’s Curves, but they also fail to provide understanding and offer no clue as to accuracy.  The most comprehensive treatment of Regenerative Heat Transfer is in Jacobs Text, Heat Transfer.  This paper describes a method that will not only replace Schumann’s Curves but will provide basic bed design criterion that is quite comprehensible.  With improved understanding of Regenerative Heat Transfer will come significant energy savings.  From this analysis new light is given to Schumann’s solution.  All discussion and derivations are for the case of hot fluid and cold bed starting form a bed of uniform temperature.  Equations for cold fluid and hot bed will be included without comment.

Double S-Curves

Figure 1: Transition Region

The most important concept for the design of Regenerative Beds is a simple heat balance.  Before proceeding with the heat balance it is important to reexamine the temperature Transition Region (TR).  A heat balance will not appear quite useful until it becomes clear how the two concepts are related.  After sufficient time, when hot gas passes through a colder bed or a cold gas through a hot bed, a region will exist between the hot and cold parts of the bed.  That transition for both bed and fluid will take the form of a double S-Curve.  Figure 1 illustrates and example of the two S-Curves of the TR.  The TR will travel through the bed in the direction of the gas flow.  Let the leading S-Curve be the hot fluid and the trailing S-Curve represent the cold bed.   The beginning and ending of both S-curves will come together.  It is easy to imagine this region to be long but the length of the region can be quite finite when starting with a uniform temperature bed.  All heat transfer to the bed occurs within the TR.  Vertical lines connecting the two S-Curves represent the temperature difference between bed and fluid at any point within the TR.  Area for heat transfer is provided by all particles within the TR.  Actual slope of the TR is controlled by the initial bed temperature configuration.

Transition Region Velocity

From a heat balance the velocity of the TR can be determined.  A second element of the bed design is to estimate when hot gas of a selected temperature will exit the bed.  In order to make that prediction the gas temperature profile is needed.  As can be seen in Figure 1 that profile is complex.  But there is a simple way to overcome that complexity.  A relatively simple equation can be used to predict the initial gas temperature profile.  A differential equation is derived and solved later that will predict the distance x along the bed for any gas temperature.  This equation is easily derived for a uniform temperature bed because as the gas first moves through the bed it is always in contact with the cold bed.

The leading edge of the initial hot gas profile will provide an accurate estimate of the extra bed length needed for the TR.  As an example to follow in the thought process, assign a quite reasonable TR velocity of 1 ft per hour for a selected gas flow.  Generally the bed designer calls for bed heating to end as a selected fluid temperature exits the bed.  It can be shown that the initial gas cooling temperature profile is established within seconds thus allowing the assumption that the profile exists at the start of heating.  If the initial hot gas temperature profile indicates the selected temperature is 0.5 ft ahead of the inlet at the start of heating then add that increment of distance to the bed.  If a 4 hour cycle is selected the required bed length becomes 4.5 ft.

Equations for initial uniform temperature bed design are listed below.

TR velocity in consistent units:
Equation 1 v = Gcg/(bcs+rcg)
Or Equation 1A, v = GHg/(bHs+rHg)
Where: v TR velocity (Not superficial gas velocity)
cg Gas heat capacity
cs solid heat capacity
b=s(1-e) bed bulk density
r Gas density at Tgo
Hg Gas enthalpy, Hhot-Hcold
Hs Solid enthalpy, Hhot-Hcold
s Particle density
e Bed porosity
L Length of bed heated
Initial hot gas temperature profile:
Equation 2, Cold Bed x = -(Gcg)/[h’(1-e)]Ln[(Tg-Tso)/(Tgo-Tso)]
Hot Bed x = -(Gcg)/[h’(1-e)]Ln[(Tso-Tg)/(Tso-Tso)]
Where: (English units) x Distance along bed, ft
h’ Gas/bed film coefficient, btu/hr-ft3-‘F
Tgo Initial gas temperature, ‘F
Tg Gas temperature at any point, ‘F
Tso Initial bed temperature, ‘F
For later discussion h Gas /bed film coefficient, btu/hr-ft2-‘F
Ln Natural logarithm
And h is related to h’ by Equation 3.
Equation 3 h’=h(Ap/Vp)
Ap Particle area, ft2
Vp Particle volume, ft3
Therefore total bed length is given by Equation 4
Equation 4 L = vq + x(at selected exit Tg)
q Total heating time, hr
Calculation of the gas temperature profile verses x requires a value of h’ as does the Schumann Curves.  Equation 5 was recommended by Lof & Hawley and was used by Kern.  Although it has questionable precision, it provides a source for values of h’.  Since the major portion of the heated bed length can be independent of h’ the accuracy of the coefficient is of reduced significance.
Equation 5 h’ = 0.79(G/D)
Where: D Characteristic particle diameter, ft
Equation 5 is a dimensional equation and requires G in lb/hr-ft2 and D in ft.  The equivalent spherical diameter was used in development of the correlation.  Lof and Hawley developed Equation 5 from experimental studies with granite type gravel.  Temperatures were relatively low and were mostly with air at about 200’F and ambient beds.  After a review of other studies Lof and Hawley recommended a correction be made to the resulting prediction for higher temperatures.  The recommended correction was the ratio of the central range temperatures raised to the 0.3 power.
If the Designer wishes to construct a TR profile for the bed, additional equations are required.  The TR will not be established until the front face of the bed is heated to near the hot gas temperature, Tgo.
Bed face heat up time is expressed by the easily derived Equation 6:
Equation 6 Cold Bed t = -(cs?/h’)Ln[(Tgo-Ts)/(Tgo-Tso)]
Hot Bed t = -(cs?/h’)Ln[(Tso-Ts)/(Tso-Tgo)]

Important Considerations
Regenerative bed designs of good accuracy will be available through the application of Equations 1, through 5 if beds are uniform temperature.  There will be opportunities to improve the results through better measurements and correlations of Heat Transfer Film Coefficients.  Balakrishnan and Pei examined existing data and concluded that no general correlation was available.  For those who wish to review more recent efforts at predicting h a good start is the work of Thompson and Jacobs.  Lof & Hawley abandoned the plan they had developed and followed Furnes by using Schumann Curves to compare exit temperature profiles measured in experiments.  In light of Equation 2 the method does result in approximate values of h’.  One opportunity to compare methods was afforded because actual data from Run 10 of the study was included in Lof and Hawley’s report.  After examining the data it was apparent that the exit temperature profile was extended significantly due to heat losses to the surroundings.  In spite of huge heat losses the exit profile was used as an overlay on the Schumann Curves for selecting the parameter called Y.  When Equation 2 was applied to the data from Run 10 a higher value of h’ resulted.   That was not a surprise due to the effects of heat losses on Lof & Hawley’s profile.  Equation 5 resulted from Lof and Hawley’s work and improvements are needed in accuracy.

In spite of using the volumetric heat transfer coefficient, the simplified equation derived in this paper to predict bed face heating in theory requires particles of near equal size.  If not, the large particles would heat slower than the small particles and a range of temperatures would exist at the surface.  There are compensating circumstances.  A typical heating time for the face is somewhat long.  Thirty minutes could be a typical heating time and some radial heat transfer will occur helping to bring the temperatures together.  The simplifying assumption is Case 5 in McAdams with zero conduction parallel to flow and finite radially.

Volumetric Film Coefficients

If a true coefficient of heat transfer is to be measured and correlated it should be adjusted from volume to area.  Lof & Hawley, working toward a specific goal of characterizing gravel were not concerned about an area to volume ratio.  That freedom will certainly not be available when particle shapes change.  Present data shows that heat transfer resistance is predominately at the particle surface film.  It is useful to review how the area to volume ratio for particles can vary.  The simplest case is for spheres, with an area to volume ratio of 6/D.  Cylinders with length equal to diameter are also 6/D.  As cylinders lengthen, the constant begins to drop and approaches 4.  Constants for flat plates can increase without bounds as thickness decreases.  Hollow particles can impact the ratio in more than one way depending on the actual configuration.  The K/D ratio effects heat transfer that passes from flowing fluid into particles, so factors such as stacking of flat plates will have a large effect.  In some particles the K/D ratio for heat transfer will not likely be determined by other than experiment.  Previous and future experiments with known K/D values such as spherical particles will and have been quite rewarding.  Lof and Hawley presented experimental work by others.  Other work by Sanders and Ford with spherical particles found that h’ correlated with G/D with no exponent.  Since h’ is equal to hK/D the implication is that h is independent of D in the experimental regime.   Differences in heat transfer characteristics for other types of particles might be best attributed to variations in K/D based on experimentally measured values of h’.  A useful procedure for measuring h’ will be illustrated later.  Foust, et, al, show experimental results from crushed ore samples.  Surface area as a function of diameter is presented.  Examination of the data showed K increasing with diameter.  Correlations of data will require that such variables be considered.  Particle properties other than specific heat have little correlation with experimental measurements of h because small particles achieve thermal equilibrium with surface temperature within seconds.

Application to Bed Design, Example 1
Application of Equation 1 – 5 can be made to a bed design.  Kern presented an example from Lof and Hawley, in which a bed of 1 inch gravel was heated by 60 lb/hr-ft2 of hot air.  A bed length was to be determined that could absorb the heat from 200’F air that was available for 6 hours.  Other needed properties were porosity, 0.45, gravel density, 165 lb/ft3, and bed heat capacity of 0.25 btu/lb-‘F.  Air density was listed as 0.0806 lb/ft3 and the air heat capacity given as 0.0191 btu/ft3-’F.  In the Kern example, the solution was developed using the Schumann Curves that required the heat transfer coefficient h’ in units of volumetric heat transfer.  Equation 1 can be used to calculate the velocity of the TR.  An examination of the development of Equation 1 shows that the air density required in the solution is for air at 200’F (0.0600lb/ft3).

TR velocity:   v=60*0.237/(165*(1-0.45)*0.25+0.0600*.237)

Thus:                                      v= 0.627 ft/hr

Then for 6 hours of heating the TR will move 0.627*6 or 3.76 ft.  In the problem statement it was required that the hot gas emerge from the bed at 90’F after 6 hours.  From Equation 2, the distance x from the bed entrance can be calculated for the 90’F point on the initial gas temperature profile.  In order to make that application of Equation 2 requires the assumption that the leading end of the profile remains approximately intact.  That is, the bed heats from the inlet and reshapes the upper temperature regions of the gas profile into an S-Curve while the leading end of the profile moves along with the TR velocity.  While this will not be exactly true it offers a quite accurate estimate that is easily comprehended and well within design tolerance.  Note the heated length calculation is rigorous.  Saturated bed entry is stipulated for each cycle.

Equation 5 h’ =0.79(60/(1/12))^0.7 = 79 btu/hr-ft3-‘F
This value is substituted into:
Equation 2 x =- (60*0.237)/[79*(1-.45)]Ln[(90-50)/(200-50)]
x = 0.43 ft
Bed length via Equation 4 L = 6*0.627+0.43 = 4.2 ft verses 4.4 via Schumann.

There will be further comparison with Schumann’s Curves later.

Derivation of Equation 1
From a simple heat balance the velocity of the TR, can be determined.  To derive the expression for the velocity, consider a cold particulate bed of heat capacity cs, temperature Tso, and bulk density b.  A hot gas of heat capacity cg, mass rate per unit area G, and temperature Tgo flows into the bed.  At some point in the bed the hot gas has cooled to the temperature of the bed, Tso.  Cooling is the result of heat transferred to the bed.  After a time, in a perfectly insulated bed, the temperature of some portion of the bed is near the inlet gas temperature, Tgo.  The region between where the bed and fluid are hot and cold is the TR.  Now consider a heat balance across the inlet to the TR.  A simplification of the heat balance allows the TR to become stationary.  Then the heat enters the region from the left in the non-condensable hot gas and flows back to the left in a stream of particles.  Because the TR actually moves in the direction of the gas the concept of a non-moving TR requires a reduction in the fluid velocity relative to the region.  The reduction in velocity for the fluid is equal to the actual velocity of the TR.  The velocity of the solid bed is then determined from the heat balance by reducing the heat input from the fluid as if it had slowed by the velocity of the solid.  Heat in the fluid is Gcg(Tgo-Tso) less the reduced velocity component vrcg(Tgo-Tso), where the symbol r represents hot gas density.  Heat leaving the region back to the left in the solid particles is equal to vbcs(Tgo-Tso).  Equation A shows the heat balance.

Equation A                            Gcg(Tgo-Tso)-vrcg(Tgo-Tso)=vbcs(Tgo-Tso)

After canceling (Tgo-Tso) solving for velocity, v, reveals the velocity of the TR.

Equation 1                            v=Gcg/(bcs+rcg)

The presence of rcg in the denominator is somewhat mysterious because v can be derived as the heat, Gcg(Tgo-Tso) needed to saturate one foot of bed, bcs(Tgo-Tso).  If rcg is important to the equation it remains as an insignificant quantity in atmospheric gas problems.

As in all gas flow in beds, conditions should be selected to reduce the channeling of gases.  Pressure drop in the bed is also a necessary consideration.    Heat losses and heat capacity along the wall should be kept as low as possible for energy savings.  As temperatures are dropping along the insulated bed resulting from heat loss to the walls there will be only small impact on the velocity of the TR.  Any change will be expressed by Equation 1 or 1A where properties will be at the new temperature.

Derivation of Equation 2
To develop the initial gas cooling curve consider the small leading end of the hot gas as it flows through the cold bed.  Let the small portion of gas have the mass dw.  As it passes through the cold bed it cools by transferring heat to the cold bed.  The incremental heat lost from the gas is dwcg(dTg) where cg is the gas heat capacity, and dTg is the differential change is gas temperature.  The increment of heat transferred to the constant temperature particles in the increment of time dt is h?(Tg-Tso)dt.  In this expression ? is the area of particles in contact with the thin leading edge dw of hot gas.  Here the film coefficient is in units of ft2.  The volume of the end increment is dV=dw/?=dwRTg/(PM) when gas density is expressed in terms of the ideal gas law and where M is gas molecular weight and R and P are the gas constant and pressure.  For a unit cross-section of bed the particulate area ? exposed to the gas leading edge dw is equal to ApNdV.  Here Ap is the area of the average particle.  The parameter N is the number of particles per unit volume.  In more familiar terms N=(1-e)/Vp where e is the porosity and Vp is the volume of the average particle.  Combining these expressions plus Equation 3, clearing terms, and rearranging gives the differential expression for temperature change with time for the leading edge of gas.

Equation B                            dTg/(Tg(Tg-Tso))=(h’/cg)(R/MP)(1-e)dt

Because of slow bed heating by the hot gas this profile is also the initial gas temperature profile.  With the use of Equation B and numerical integration the value of Tg verses time, t, can be developed.  Also, note the dependence of the derived equation on the beds uniform temperature.  Equation B can be integrated using partial fractions.  The integrated expression for time and temperature is shown in Equation C.

Equation C                      (1/Tso)Ln[(Tgo/Tg)(Tg-Tso)/(Tgo-Tso)]= (-(h’/cg)(R/MP)(1-e))t

The reader can verify from this equation just how quickly the temperature profile for the gas is formed.  It required seven seconds to form in Example 2 to follow.

Subsequently, if use is made of the fact that dx=udt in Equation B, where u is superficial gas velocity, and Equations 7 and 8 are used to replace u,

Equation 7 u=G/r
Equation 8 r=PM/RT,  gas density
Where u        Superficial Gas velocity  
P        System pressure 
M       Molecular weight of gas 
R        Universal Gas Constant


Then integration and substitution of Equation 3, h’=h(Ap/Vp), results in a rearranged version of Equation 2

Equation 2(Rearranged)    h’/(cgG)(1-e)x=Ln[(Tg-Tso)/(Tgo-Tso)]

Figure 1. Developed from Lof & Hawley Data

Figure 1 illustrates the gas cooling profile for Example 1.  After a short time Equation 2 is no longer correct.  However, the leading part of the profile will remain closely intact and that is why it is useful for projection through the bed.  Note that the Integral of h’(1-e)(Tg-Tso)dx is equal to Gcg(Tgo-Tso).

Example 2
Example 2 is also taken from the work of Lof and Hawley.  Lof and Hawley made extensive measurements of heat transfer in a bed of length 3 ft, and 0.77 ft2 of area.  Heated air was allowed to flow through the bed using several particle sizes and flows.  Using exit temperature profiles and the curves of Schumann they reduced the data to obtain values of h’ in units of btu/hr-ft3-‘F, the coefficient of heat transfer between gas and bed.  In one case the actual temperature profile was given.  From that case referred to as Run 10, and Equations 1 to 5 it was possible to directly calculate a value for h’ in units of btu/ft3-‘F-hr.  The data from Run 10 is presented in Table 1.

Other Bed Parameters
G                                                204.5 lb/hr-ft2
cg                                               0.2379 btu/lb-‘F
s                                                 165   lb/ft3
cs                                               0.25    btu/lb-‘F
e                                                 0.454
Tgo                                            200     ‘F
Tso                                             66        ‘F

The volumetric film coefficient can be calculated by recognizing that the temperature point that emerged after 1 hour had moved along in the bed for all that time.  Knowing the TR velocity allows the starting point in the bed for that temperature to be calculated.  By substitution in Equations 1, 2 & 4,

v=2.16 ft/hr
Movement of 67.46(Tg@ 1hr.)=vt=2.16*(1hr)=2.16 ft
Location of point at start of heating=3.00-2.16=0.84 ft
h’=479.4 btu/hr-ft3-‘F

Precision of this result is not as high as this procedure has capability to obtain.  It is obvious that selecting an initial temperature of 67.46 leaves room for error when subtracted by 66’F.  Selecting a higher temperature would have introduced increasing heat losses as shown by the 4 hrs required to reach 196’F.  It can be shown that the hot gas had sufficient energy to heat the bed in 1.31 hrs.  A better way would be to use a shorter bed length so a portion of the hot gas profile would have emerged in a time measured in seconds.  Heat loss would not have had time to occur and the emerging temperature would have changed so slowly that a quality measurement of temperature could be taken.  It is always true that bed housing will have heat capacity and will withdraw heat from the stream.  Such losses should best be accounted separately and not lumped with a natural parameter.  Equation 5 calculated h’ as 419 for Run 10 and Lof & Hawley used Schumann’s Curves to estimate a value of 437.  The above value was 479.  All would give reasonable bed lengths.  Lower values of h’ give longer S-Curves and longer increments to add to bed length.

Think of Equation 1 as the plug flow part of the solution.  The term “plug flow part” is the recognition that Equation 1 represents that if the bed were to heat by plug flow then the extent of heating in the bed would be exactly as predicted by the product vq.  Because the hot end of the bed is represented by a symmetrical S-Curve, Equation 1 will represent the distance where 50%, or midpoint, of the temperature range is located.

Comparison to Schumann Solution

It came as a surprise to notice that Equation 4 without the x increment could be rearranged to give the parameters Y and Z.  It was later in this study that the relationship of the Equations to Schumann’s curves was first noticed.  Neglect the insignificant second part of the denominator in Equation 1.

L= vq ={Gcg/[scs(1-e)]}q

If both sides are multiplied through by h’ and divided through by Gcg a new equality is obtained.


These terms are exact duplicates of the terms Y and Z from Schumann’s Curves.  The relationship becomes clearer when the point where Y=Z is examined on the Curves.  In each case the parameter


Schumann adds more length to beds with exit temperature lower than that ratio and subtracts length from beds with larger ratios.

It is obvious that Schumann’s solution to the basic bed length prediction was quite accurate.  The method selected for presenting the information to the Scientific Community was confusing.  It has remained quite mysterious to some of us.  His choice for placement of the gas S-Curve might be a useful study.  Did he predict that the shape and length of the S-Curve would change as it traveled?  Why did the slopes change as Z increased?  Perhaps the variance was due to incomplete S-Curves at the start.  The conclusion from this analysis is that Schumann discovered the principles discussed in this Paper but the simplicity of the solution was obscured by his Curves.

Calculating the Bed S-Curve

Some designers might prefer rigor to simplicity in bed design.  For those, an equation has been developed for calculating the actual temperature profile for the bed.  While analyzing the results from numerical calculations a technique was discovered that allowed the bed profile to be constructed.  It required the addition of two exponential curves.  One curve was the bed profile that resulted from allowing the temperatures on the bed face to move into the bed at the TR velocity.  When area under that profile failed to contain all the heat supplied by the gas a second equation was discovered.  It represented heat carried into the bed at temperatures higher than the momentary face temperature.  After numerical construction of the beds temperature profile it became apparent that the two equations could be combined into one equation.  Equation 9 shows the bed temperature verses distance along the bed.  The bed profile is shown in Figure 2.

Equation 9 Cold Bed                   x=v{?-(scs/h’)Ln[(Ts-Tso)/(Tgo-Ts)]}

Hot Bed                      x=v{?-(scs/h’)Ln[(Tso-Ts)/(Ts-Tgo)]}

Figure 2. Developed from Lof &Hawley Data

Note that the initial slope of the S-Curve part is regulated by bed density, heat capacity, the heat transfer coefficient and initial bed temperature profile.  In verbal communications with Loren Hov, P.E., former Director of Energy Management of the former Stauffer Chemical Co., now part of Syngenta, who proof read the paper offered indisputable evidence that S-Curves flatten in slope each time reverse flow enters the bed until an equilibrium is established.  There is little doubt that the initial gas temperature profile dictates the initial S-Curve slope.  An existing S-Curve will increase the length of the cooling profile and add length to the S-Curve.  Design and operation of Regenerative beds must overcome that difficulty if the initial steep S-Curve is to remain.

Air and Bed Profile

The air temperature profile can be described in a logical way by moving the bed profile along an additional distance.  Figure 3 shows the double profile.  Correctly locating the air profile is done by trial.  A new bed profile is moved forward a selected distance by making ? equal to ?+??.  A numerical integral is computed using Equation 10 along the Double S-Curves where values of Tg-Ts are at equal x values.

Equation 10                                      Q=Integral [h’(1-e)(Tg-Ts)dx]=Gcg(Tgo-Tso)

Based on the outcome of trials the air profile is moved until Equation 10 is correct.  Values of Tg-Ts can be scaled from the graph.  If it is recognized that the points of interest on the two S-Curves fall where values of x are equal then a helpful expression can be derived.  Equation 11 resulted from setting common values of x equal and solving for the difference in temperature between bed and fluid.    Values for the temperature difference are then placed in the numerical integral table.  Next the numerical integral is developed for (Tg-Ts)dx.  Once the integral has been established and converted to total heat transferred then simply adjust ?? until the total heat is correct.  Equation 11 calculates values of (Tg-Ts) for selected values of Ts.

Equation 11 Tg-Ts=(Ts-Tso)[(e^a)-1]/(1+[(Ts-Tso)/(Tg-Ts)]e^a)
e is base of Natural Logarithm only in Eq. 11.

Figure 3. Air Bed Profile from Example 2


Equation 9 allows an accurate bed temperature profile to be drawn and Equation 10 allows the air profile to be placed correctly.  However, there is little if any improvement over the method applied to Example 1 unless a fluid exit temperature higher than 50% of range is selected.  Equations 1 to 5 will provide all the accuracy for bed design that will normally be needed.  The simplicity associated with the latter equations will allow quick designs to be developed for streams that were of no commercial value prior to the rise in energy cost.  Selection of equations will be the choice of the reader.  Equations in this Paper will also find useful application with Rotary Kilns.  The equations will not perform under a reverse flow arrangement where the new flow makes contact with and existing S-Curve.  Present designers who apply reverse flow in bed designs must operate with long S-Curves and short cycles.  Reverse flow greatly simplifies the materials of construction but sacrifices bed capacity.  That type Regenerator is described in the text “Heat Transfer” by Max Jacobs.


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