A fundamental model based on thermal balances for core oil temperature and wax layer development with axial distance in a pipe transporting heated oil

In the papers by Needham, Johansson and Reeve [1] and Mason, Needham and Meyer [2], a new model was introduced to describe the formation and temporal evolution, under external cooling, of a wax layer precipitating onto the interior bounding surface of a circular pipe transporting hot oil, which contains dissolved paraffinic waxes. This model was first proposed to the authors in unpublished notes by Schulkes [3], and is based upon fundamental thermal energy balances across the circular cross section of the pipe, from the hot core oil, to the inner pipe surface, and across the pipe surface to the cooling bath. In [1] and [2], this model was formulated and analysed in detail in the simple situation where inlet and outlet effects are neglected, and it is assumed that the wax layer is uniform in the axial direction along the pipe. The key to this assumption being accurate is that the core oil temperature in the pipe remains reasonably independent of axial distance down the pipe, which is realised for local sections of pipe away from inlets and outlets. The conclusions from this model and analysis have explained a number of long standing paradoxes that have been recorded between the predictions of previous mechanical shear-induced mechanisms of wax layer formation and the results of detailed experiments. The thermal model has now been the subject of recent experimental work and verification (see, for example, the recent papers by Van Geest et al.: [4, 5] and [6]), and the thermal energy balance is now accepted as the fundamental mechanism associated with wax layer formation. A full discussion of this fundamental model, and its comparison with previous models is given in the introductions to [1] and [2], whilst recent verification of this model can be found in the investigations of Van Geest et al. cited above, and in the review article [7]. This, and the additional references therein, need not be repeated here, but we do emphasise that this paper should be read in close conjunction with [1] and [2].

The present paper addresses the next natural development of this model, which is to determine the axial development of the wax layer in a long (relative to pipe diameter) pipe, from the inlet to the far reaches of the pipe. The key to the axial development of the wax layer is the coupled modelling of the variation in core oil temperature, and its interactive effect on the local wax precipitation temperature, all of which ensues from the accumulative heat loss from the core oil, through the side walls to the coolant jacket, as the core oil flows down the pipe from the inlet. This is achieved by considering a local spatio-temporal thermal energy balance in the core oil and then coupling the local core oil temperature to a local wax precipitation temperature, via a local relation between soluble wax precipitation concentration and solvent temperature. We demonstrate that this relation is in full accord with recently reported experimental observations and data. This can then be coupled to the local thermal wax layer evolution model introduced and fully analysed in [1] and [2]. This leads to a closed spatio-temporal model which determines both the core oil temperature development, and the wax layer thickness evolution, both with axial distance from the pipe inlet and time. The model is then analysed, and typical situations are examined in detail for illustrative purposes. The key findings are as follows:The paper is structured as follows. Section 2 develops the thermal model to accommodate the dependence of the core oil temperature on axial distance down the pipe from the inlet, and its consequent effect on wax layer formation and thickness. Thereafter this extended thermal model is analysed in detail, and its implications are considered. In Sect. 3, an efficient numerical method is formulated, tested, and implemented in a number of specific typical cases, which fully illustrate the theory of Sect. 2. Conclusions are drawn in the final section.

In this section, we develop the thermal model introduced in [1] and [2] to include the possibility of axial variations in core oil temperature, and consequent on this, corresponding axial variations in the wax layer evolution. To avoid repetition, we will adopt the notation and model development used in Sect. 2 of [2], and concentrate on the additional model developments which we introduce to account for axial variations in the thermal energy balances. Throughout we will consider heated oil flowing with uniform cross-sectionally averaged velocity \(V_o\) through a long straight section of a horizontal pipe, with circular cross section, and internal radius R. The hot flowing cross-sectionally averaged core oil temperature is denoted by \(T_o\), and it has paraffinic waxes dissolved within at a saturation concentration which we denote by \(c_w\). In general, we may expect the core oil flow velocity and temperature to have some axially symmetric radial dependence. However to keep the model tractable, as a first approximation, it is not unreasonable to simplify the hot oil flow as a radially independent mean flow (particularly as the flow is generally turbulent) and replacing the effects of radial dependence on the heat transfer from the core oil to the wax layer at the wall simply by including an enhanced core oil thermal conductivity proportional to the core oil flow Nusselt number Nu (as represented in equation (23) which forms the principal thermal balance in the core oil flow) which enables the principal core oil heat transfer mechanisms to be taken into account. Of course, we acknowledge that improvements in the representation of the core oil flow details will naturally give improvements in the models performance, and as such can be considered as refinements to the key underlying model presented here. The cooling around the exterior of the pipe will, when sufficiently strong, lead to wax layer formation on the inner wall of the pipe, with crystalline wax deposit precipitating from the core oil. We will denote the radial wax layer thickness as h, and as discussed in [2], we may anticipate that \(h<<R\), which, following [2] requires that,which is generally satisfied, and we can then take full advantage of this in the model to reduce the problem to a planar geometry. Here, the parameters are as referred to in the introduction, with \(k_w\) also representing the crystalline wax thermal conductivity. We denote axial distance down the pipe by the coordinate z, and normal distance from the inner pipe wall towards the pipe axis by the coordinate x, whilst t represents time. The inlet to the pipe is taken to be at \(z=0\), with the pipe lying in \(z>0\). The pipe is surrounded by an aligned circular coolant jacket, and the coolant fluid within the coolant jacket is maintained at a constant cooling temperature \(T_c\). The details of this situation are illustrated in Fig. 1.

Throughout, the core oil mean velocity \(V_o\) can be taken as constant, independent of axial distance z and time t. However, the local saturation concentration \(c_w\) of paraffinic wax dissolved in the core oil will generally depend upon the core oil temperature \(T_o\), so thatand is an increasing function of \(T_o\), with specific details depending upon the physical characteristics of the oil and wax under consideration. The basis of this dependence is verified and supported in the experimental part of the paper of Singh et al. [8], where it is reported in the introduction and the section headed Physics of Wax Deposition that the solubility of paraffinic wax is temperature dependent, and is increasing with increasing temperature, and that the dependence is strong (an example of this dependence is well illustrated in [8, Fig. 4]). It can also be verified from the experimental graphs provided in [8] that the precipitation temperature for wax crystal deposition from the core oil, which we denote by \(T_h\), depends upon \(c_w\), and is increasing with increasing \(c_w\). This can be inferred from [8, Figs. 18 and 20], using the variable Time in both of these figures as a parameterisation of this dependence. Thus, we havewith again \(T_h\) increasing with \(c_w\), and the specific details of this relation depending upon the particular oil and wax under consideration. It now follows directly from relations (3) and (4) that for a given oil and wax, then \(T_h\) is a specified thermodynamic increasing function of \(T_o\), which we assume throughout to be a smooth function. Thus, we may writeWhen \(T_h^i\) is set to be the value of \(T_h\) at the pipe inlet, where the core oil temperature is prescribed at \(T_o=T_o^i\), we thus have the inlet relation,It is now convenient to introduce, as in [2], the dimensionless temperature variable,which vanishes when the core oil temperature is equal to the constant coolant temperature. In terms of \(u_o\), (5) becomeswhilst (6) may be written aswithwhich is the dimensionless core oil inlet temperature. We note that a fundamental requirement isand we will consider that the constant cooling temperature is chosen so thatwhich determinesIn dimensionless form, we now write the relation (8) asWe now review the function \(u_h:{\mathbb {R}}\rightarrow {\mathbb {R}}\). We will assume throughout that, at least,whilst the above discussion determines thatIn addition, via (10), we haveand, via (11),which, we recall, represents that the wax crystallisation temperature at core oil temperature \(u_o\) is always lower than the core oil temperature \(u_o\). We next observe, via (17) and (18), that there exists a unique value \(u_o = u_o^*\), and this value lies in the interval \((0,u_o^i)\), such that,withandand so,A sketch of a typical form for \(u_h(u_o)\) for \(u_o\in [0,u_o^i]\) is shown in Fig. 2.

We now continue the modelling by addressing the core oil flow. We consider that the pipe is long compared to the core oil flow length scale \(V_o t_s\), where \(t_s\) is the wax layer evolution time scale as given in [2, Eq. (12)] but now with \(T_h\) and \(T_o\) replaced by the constants \(T_h^i\) and \(T_o^i\), respectively. Both of these length scales are long compared to the pipe radius R. Thus, the core oil temperature can be considered as slowly varying down the pipe, and as a consequence, we can regard the core oil temperature \(u_o\) to be independent of the radial coordinate x (via interpreting \(u_o\) as an average over the core oil section), with its dependence being on axial distance down the pipe z and time t. Therefore, we may write \(u_o = u_o(z,t)\) with \((z,t)\in [0,\infty ) \times {\mathbb {R}}^+\). The fundamental thermal energy balance in the core oil flow then requireswith \(c_o,~\rho _o,~k_o\), and Nu being the core oil specific heat, density, thermal conductivity, and Nusselt number, respectively. On scaling time t with \(t_s\) and z with the natural flow length scale \(V_ot_s\), this equation takes on the dimensionless formwhere the dimensionless parameter \(\mu \) is given byOn using typical estimates as reported in [2], we can readily determine that, generally, \(\mu \) is large (\(t_s>> c_o \rho _o R^2/k_o Nu\)). It then follows immediately from equation (24) that temperature in the core oil flow will reach a steady state on the rapid dimensionless time scale \(t\sim O(\mu ^{-1})\), and therefore much more rapidly than the wax layer growth time scale \(t\sim O(1)\). Thus, (as detailed in Appendix A) when considering wax layer evolution, we can approximate the core oil temperature as being in steady state, and so we write \(u_o = {\overline{u}}_o(z)\), where now,subject toIn fact, the ODE in (26) only holds at those locations in \(z>0\) where \(h>0\), that is where a wax layer is present. This equation also reveals that, in the dimensionless variables, the axial length scale for the wax layer variation isIn the absence of a wax layer, at location z, then the thermal energy balance from the core oil across the pipe wall must have, in dimensionless form,where \(u_w\) represents the pipe wall temperature at the location z, and,with k being the dimensionless parameter and \(x_s\) the length scale introduced in [2] (specifically in equations (17) and (12) therein, with superscript i now being inserted on each term involving T). It follows directly from equation (29) that at each location \(z>0\) where \(h=0\), the pipe wall temperature is given bySpecifically, equation (26) should now be replaced by the modified form,where now,with the initial condition remaining as (27). It should be noted from above that, since \(u_o^i>1\), then the core oil temperature \({\overline{u}}_o(z)\), at any location \(z>0\), will be below \(u_o^i\). The two different forms in (33) are naturally connected by demanding continuity of \({\overline{u}}_o(z)\) at every \(z\ge 0\).

We now move on to the wax layer model. The modification to the constant core oil temperature model introduced and developed in [2] is straightforward, and particularly so since \(h<<R\) which guarantees that in the wax layer derivatives in the axial (z) direction can be neglected compared to derivatives in the radial (x) direction. At each location \(z\ge 0\), the wax model is, therefore, exactly as in [2] except that the appearance of the dimensionless wax precipitation temperature as unity in the boundary conditions on \(x=h\) is now replaced by the axially dependent wax precipitation temperature \(u_h({\overline{u}}_o(z))\). Thus, in the same dimensionless variables as introduced in [2], at each location \(z\ge 0\), where a wax layer develops, the wax layer thickness h(z, t) and temperature distribution u(x, z, t) are as in [2, Sect. 3, (19)–(24)], which in the present context giveswith \(\epsilon \) and k being the dimensionless parameters introduced in [2, Eq. (17)], with superscript i now being inserted on each term involving T) and D(u) being the prescribed diffusivity of the crystalline wax deposit, as also introduced in [2]. We recall from [2] that the parameter \(\epsilon \) measures the ratio of the time scale for heat conduction in the wax layer to the time scale for wax layer growth, whilst the parameter k gives a dimensionless measure of the heat energy cooling rate at the pipe wall. Throughout the rest of the paper, we will focus on the canonical case of constant diffusivity, which in dimensionless form amounts to setting \(D(u) \equiv 1\). This allows for attention to be given to the key contribution of axial dependency introduced and developed here. However, we note that more structured diffusivities can be included, with the theory developing in much the same way. For a given core oil distribution \({\overline{u}}_o(z)\) in \(z\ge 0\), the problem (34)–(39) provides a closed problem, at each \(z\ge 0\), for the wax layer evolution at that location z. We will refer to this problem as [IBVP(z)]. The theory developed in Sect. 3 of [2] for the problem [IBVP] now applies, with only very minor modification, to [IBVP(z)] at each \(z\ge 0\). Using this theory, we may first conclude that [IBVP(z)] has a solution at location \(z\ge 0\) if and only ifwhereHere, we recall that the dimensionless parameter k is a measure of the rate of heat energy loss through the pipe wall due to cooling (see [2], Sect. 2), whilst the left side of equation (40) is a reciprocal measure of the local transfer rate of heat energy into the wax layer from the core oil flow. Therefore, the inequality in condition (40) is a detailed requirement that the cooling rate at the pipe wall must be sufficiently strong compared to the local heat transfer from the core oil flow to enable the formation of a wax layer at that location. At each such location \(z\ge 0\), this solution has h(z, t) increasing with \(t>0\) and approaching a steady state value, so thatThe corresponding crystalline wax layer temperature field hasWe conclude, in particular, that a wax layer will develop (so \(h>0\)) at precisely those locations which have a core oil temperature \({\overline{u}}_o\) which satisfies the inequalityWe note from this inequality that a necessary condition for wax layer formation at locations where the associated core oil temperature is \({\overline{u}}_o\) requires \(u_h({\overline{u}}_o )> 0\). Moreover, throughout the interior of the wax layer, at this location \(0<u<u_h({\overline{u}}_o)\). We must now determine the range of values of \({\overline{u}}_o\), below \(u_o^i\), which satisfy inequality (44). To do this, we first examine G(X) for \(X\in [0,u_o^i]\). On using conditions (15)–(18), we have immediately that \(G \in C^1({\mathbb {R}})\) andwith \(u_o^* \in (0,u_o^i)\) being the only zero of G(X), which is negative at all lower arguments and positive at all higher arguments. From (41), we haveIt then follows thatwhilstThese conditions establish that there are two distinct generic cases (although other more unusual cases may occur), namely

(A) \(u_o^i u_h'(u_o^i)>1\) : In this case, we limit attention to the simplest scenario when G(X) is monotone increasing on \([0,u_o^i]\), with graph as given in Fig. 3.

Thus, for \(k\le 1\), inequality (44) fails to be satisfied at any core oil temperature in \([0,u_o^i]\), and consequently, a wax layer does not develop at any location \(z\ge 0\). In this case, it only remains to determine the core oil temperature. Since \(h=0\) at all locations \(z\ge 0\), then (27), (32) and (33) becomes simply,subject towhich has solution,Thus, in this case, wax layer formation is totally absent, and the core oil temperature simply drops off exponentially, under the cooling effect, with axial distance down the pipe from the inlet.

However, for \(k>1\), there exists a point \(u_o^s(k)\in (u_o^*,u_o^i)\) for which the inequality (44) is satisfied on \((u_o^s(k),u_o^i]\) but fails on \([0,u_o^s(k)]\). Here, \(u_o^s(k)\) is decreasing with \(k>1\), and satisfiesTherefore, a wax layer will develop at those locations where the core oil temperature lies in the interval \((u_o^s(k),u_o^i]\), but not otherwise. It then follows from (33) that in this case, we haveand a sketch of \(H({\overline{u}}_o)\) (recalling that \(\kappa>>1\)) on \([0,u_o^i]\) is given in Fig. 4.

This now completes the initial value problem (27) and (32) for the core oil temperature \({\overline{u}}_o(z)\) in \(z\ge 0\), which, on integration, is given implicitly bywherewhilstThus, wax layer formation does not occur at any location \(z\ge z_s(k)\), whilst a wax layer forms at all locations \(0\le z < z_s(k)\), with the wax layer terminal thickness given by, via (42),which is a smooth decreasing function of z, from \((1-k^{-1})\) at \(z=0\) to zero at \(z=z_s(k)\).

A qualitative sketch of the wax layer thickness \(h_s(z)\) as it develops with axial distance z from the pipe inlet in the second of the cases delineated above, for \(k>1\) (there is no wax layer for the other case \(0<k\le 1)\), is shown in Fig. 5. In Fig. 5, as \(k\rightarrow 1^+\), the extent and maximum height of the wax layer shrink to zero.

(B) \(0<u_o^i u_h'(u_o^i)<1\) : In this case, the simplest scenario is when G(X) is decreasing on \((u_o^m,u_o^i]\) and increasing on \([0,u_o^m)\), and has a local maximum at \(X=u_o^m\) with value \(G_m>1\), as shown in Fig. 6.

Thus, in this case, for \(k\le 1/G_m (<1)\), inequality (44) fails to be satisfied at any core oil temperature in \([0,u_o^i]\), and consequently, a wax layer does not develop at any location \(z\ge 0\). In this case, it again only remains to determine the core oil temperature, and this is identical to the similar situation in case (A), with the core oil temperature being given by (51).

However, when \(1/G_m < k \le 1\), there exist two points \(u_o^1(k),u_o^2(k)\in (u_o^*,u_o^i)\) for which the inequality (44) is satisfied on \((u_o^1(k),u_o^2(k))\) but fails on \([0,u_o^1(k)] \cup [u_o^2(k),u_o^i]\). Here, \(u_o^1(k)\) is decreasing with \(k\in (1/G_m,1]\), and satisfieswhere \(u_o^l\) is the unique zero of the equation \(G(X)=1\) with \(X\in (u_o^*, u_o^m)\), whilst \(u_o^2(k)\) is increasing with \(k\in (1/G_m,1]\), and satisfiesTherefore, a wax layer will develop at those locations where the core oil temperature lies in the interval \((u_o^1(k),u_o^2(k))\), but not otherwise. It then follows from (33) that in this case, we haveand a sketch of \(H({\overline{u}}_o)\) (recalling that \(\kappa>>1\)) on \([0,u_o^i]\) is given in Fig. 7.

This now completes the initial value problem (27) and (32) for the core oil temperature \({\overline{u}}_o(z)\) in \(z\ge 0\), which, on integration, is given implicitly bywhereIn addition,whilst,Thus, wax layer formation does not occur at any location \(z\in [0,z_2^s(k)] \cup [z_1^s(k),\infty )\), whilst a wax layer forms at all locations \(z\in (z_2^s(k),z_1^s(k))\), with the wax layer terminal thickness given by, via (42),which is a smooth hump-like profile, being zero at the two end points. The maximum wax layer thickness, which occurs at the single local maximum, is given byTwo more key quantities are the distance from the inlet where the wax layer begins to form, and thereafter the down stream length over which the wax layer persists. These quantities are given, respectively, by \(z_2^s(k)\) and \((z_1^s(k) - z_2^s(k))\), which are determined directly from equations (62) and (63) above.

Finally, when \(k>1\), the situation is the same as in case (A), and we adopt the same notation, with the core oil temperature given as in (54)–(56), and the wax layer developing only when \(z\in [0,z_s(k))\), with the wax layer height as in (57). The only difference now is that, rather than being monotone decreasing, the wax layer has a single maximum height, which is again given by (67).

Qualitative sketches of the wax layer thickness \(h_s(z)\) as it develops with axial distance z from the pipe inlet in the latter two cases as delineated above (there is no wax layer for the other case \(0<k\le 1/G_m<1)\) are shown in Fig. 8. In Fig. 8a, as \(k\rightarrow 1^+\), the extent of the wax layer decreases to a finite value whilst retaining the form of a single hump, with the wax layer depth at the inlet reducing to zero. As k reduces further, so that now \((1/G_m< k<1)\), the single hump separates from the inlet and becomes located further down the pipe, as illustrated in Fig. 8b. Finally, as \(k\rightarrow (1/G_m)^+\), the extent of the hump and its maximum height both shrink to zero at a point which lies a finite axial distance from the inlet. This case is now complete.

It is important to note here, following (46), and the earlier regularity and monotonicity conditions on \(u_h(X)\), that the function G(X) has at most one turning point in \([0,u_o^i)\) when \(u''_h(X)\) is non-negative or non-positive on this interval. Therefore, weak convexity or weak concavity of \(u_h(X)\) on the interval \([0,u_o^i]\) guarantees that one or other of the scenarios detailed above will occur. The convexity must be broken for more exotic structures to emerge.

The theory is now complete, and the next section presents a numerical implementation of the theory to illustrate the key features in a number of relevant cases.

As an illustration of the theory developed in the last section, we numerically approximate the steady state of (34)–(37) for a number of representative parameter values. We focus attention on the key quantities, specifically the core oil temperature \({\overline{u}}_o(z)\) and the wax layer thickness \(h_s(z)\) as they develop with axial distance z down the pipe. To achieve this, we must first integrate the ODE (32), via (33), and initial condition (27), with condition (44) utilised directly to check if the wax layer exists at a given station z. Using the initial condition (27) at \(z=0\), we solve the ODE for \(u_o\) with step increases in z, using a first-order forward Euler method to approximate (32), after which \(h_s\) is subsequently updated at each step via (42). In all figures depicting \({\overline{u}}_o\) and \(h_s\), 1000 uniformly spread grid points were used. Moreover, the values in the z-domain in the following figures have been selected to illustrate where the wax layer has been computed to exist.

To aid comparison between the figures, we fix \(\mu =0.5\) and \(\kappa =10\) throughout. We note that increasing \(\mu \) primarily increases the decay rate of \({\overline{u}}_o\) which shortens the wax layer region, when it exists. Similarly varying \(\kappa \) primarily has the effect of varying the decay rate of \({\overline{u}}_o\) when the wax layer is present which can consequently lengthen or shorten the wax layer, when it exists.

To illustrate case (A) from the previous section it is convenient to fix \(u_o^i = 4\), and consider the situation when the functional form for \(u_h:{\mathbb {R}}\rightarrow {\mathbb {R}}\) has the simple, and in the absence of additional information, reasonable structureWith this choice, we use (41) to determine G(X), and this is graphed in Figure 9. As expected, we see that if \(k\le 1\) then no wax layer will be present, whereas for each \(k>1\), a wax layer will be present and exist on \([0,z_s(k)]\). These two scenarios are illustrated in Fig. 10a and b. We observe that \({\overline{u}}_o\) decays slightly less rapidly in Fig. 10b than in Fig. 10a, due to the presence of the wax layer. The difference is not large due to \({\overline{u}}_o\) being reasonably large compared to \(u_h({\overline{u}}_o)\).

To generate situations where the complimentary case (B) is present, it is convenient to consider suitable piecewise linear forms for \(u_h(X)\) which have been locally mollified to accommodate (15). Thus, for each \(\delta >0\), we introduce \(M_\delta :C({\mathbb {R}})\rightarrow C^1({\mathbb {R}})\) given byand take \(u_h:{\mathbb {R}}_+\rightarrow {\mathbb {R}}\) asfor \(\delta =0.1\) with \(f:{\mathbb {R}}\rightarrow {\mathbb {R}}\) given byrecalling that \(u_o^i>1\). Hereafter, we fix \(u_o^i=2\), and with these choices, a graph of G(X) is shown in Fig. 11 from which we conclude that, for \(k\le G_m^{-1}\approx (2.8)^{-1}\) no wax layer will form; for \(G_m^{-1}< k < 1\) a wax layer will form on \((z_1^s(k),z_2^s(k))\) with \(z_1^s(k)>0\) and for \(k \ge 1\) a wax layer forms on \([0,z_s(k))\). The numerical solution for the latter two cases is shown in Fig. 12a and b, respectively. Figure 12a and b highlights the effect of the presence of the wax layer in decreasing the decay rate of \({\overline{u}}_o\).

Finally, we consider \(u_h:{\mathbb {R}}\rightarrow {\mathbb {R}}\) given by (70) and \(\delta = 0.1\), but instead with \(f:{\mathbb {R}}\rightarrow {\mathbb {R}}\) given byfor \(N\in {\mathbb {N}}\) and \(u_o^i=2\). The function f(X) in (72) satisfies \(f(X)>\frac{1}{2} = \frac{1}{u_o^i}\) on N disjoint intervals. Hence, provided the alteration following mollification is sufficiently small then \(u_h(X)\) given by (70) and (72) also has N disjoint intervals in which \(u_h(X)>1/2\). The associated form of G(X) is illustrated for \(N=2\) in Fig. 13. These conditions give rise to N disjoint wax layer regions when \(k\approx 1\) and \(u_o^i=2\), which is depicted in Fig. 14b for \(N=2\). If k is increased so that there is only one interval in which \(G(X)>1/k\) then a single wax region will instead be present, as illustrated in Fig. 14a.

This completes our numerical illustrations.

In this paper, we have introduced, developed and analysed a principal extension to the thermal model describing the evolution of a wax layer on the cooled interior wall of a circular cylindrical pipe transporting heated oil. In particular, we have addressed key mechanisms responsible for the dependence of the core oil temperature with axial distance down the pipe from the inlet. The outcomes from this development are in line with, and provide a rational explanation for, the main features arising from experimental observations on wax layer evolution with axial distance from the pipe inlet. To see this, we will first estimate scales and parameters which emerge in the model by using the typical physical properties of hot crude oil and the precipitating paraffinic wax, together with the typical flow conditions of the hot crude oil as it emerges into and flows down a long, cooled pipeline, which typically has a radius of 0.5 m. We have in the model, typically,where use has been made of the estimates given in [1] and [2]. It then follows from (25) and (28) that the dimensional axial length scale for wax layer variation is determined from the model aswith the dimensionless model parameters having estimates,where \(Nu_c\) is the Nusselt number in the coolant flow exterior to the pipe and this typically varies between \(10^{-1}\) and \(10^2\). Now we refer to the numerical examples in Sect. 3, for the simple mollified forms of \(u_h(\cdot )\) used therein to illustrate each of the cases in possibilities (A) and (B). We observe from the respective figures giving the dimensionless detailed wax layer profiles, with increasing axial distance from the pipe inlet, that in all cases, the dimensionless axial span of the wax layer is of O(1) in z and the dimensionless depth of the wax layer is of O(1) in x. Thus, in the original dimensional form, the model is predicting in each case, that the overall axial length of the wax layer from the inlet scales with \(z_s \sim 2.2\times 10^4~\text {m} = 22~\text {km}\), whilst the overall thickness of the wax layer scales with \(x_s\sim 72~\text {mm}\), for the typical situation giving rise to the estimates in (73) and (74). We can make an order of magnitude comparison between these estimates from the model and the field measurements recorded from the Heimdal Brae De-Waxing Operation as reported, amongst other data, in a Statoil PPSA Seminar in 2013 by Fahre-Skau et al. [9]. For convenience, we refer to the figures in this report on p.13, p.20 and p.38 as graphs (1)–(3), respectively, each graph representing measurements of wax layer thickness, measured in millimetres, against axial distance down the transporting pipe from an inlet, measured in metres. We see immediately that in each graph, the axial extent of the wax layer varies from \(25\,\textrm{km}\,\,\textrm{to}\,\,100\) km, whilst the mean wax layer thickness varies between \(20\,\textrm{mm}\,\,\textrm{and}\,\,30\) mm. Given that no direct quantitative comparison should be expected, as the model estimates have not been computed for the exact materials and conditions prevailing in the field system on which the measurements were made (which are not available in the report), the order of magnitude agreement on the axial wax layer extent (being tens of kilometres) and the typical wax layer thickness (being tens of millimetres) between the model predictions and the field data is very encouraging. Advancing this comparison further, we see that each field data wax layer profile in graphs (1), (2) and (3) has profiles, which, when smoothed, shows qualitative similarity to the model profiles illustrated in Figs. 12 and 14. This gives reasonable preliminary confidence that the extended model may be used as a predictive tool to estimate the key features of wax layer development with axial distance from the pipe inlet.